TITLE: Can someone explain how to solve/what is meant in the solution for these kinds of homogeneous differential equations?
QUESTION [0 upvotes]: Consider a differential equation with solution $y(x)$ of the form:
$$y'=F\left(\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}\right) $$
The interesting case is the one where $a_1x+b_1y+c_1 \neq K(a_2x+b_2y+c_2)$; as this one can be easily solved by the change of variables $z=a_1x+b_2y$.
This case I've read in books and online that is to be solved by "moving the origin of coordinates the intersection point of the two lines". I understand that each algebraic expression in the quotient is a line. However, I don't understand how to carry out this process nor how it can arrive to a solution. My books don't offer examples for this case and I haven't found any online, so I'd like to get either an example or a more comprehensive explanation of the process.

REPLY [2 votes]: Consider $$y'=F\left(\frac{a_1x+b_1y+c_1}{a_2x+b_2y+c_2}\right) $$
Define variables $$X=x+p, Y=y+q$$ so that the constant term is eliminated, i.e.$$a_1p+b_1q+c_1 =0$$ and $$a_2p+b_2q+c_2=0 $$This gives you a system of equations in two variables which can be solved for p,q. Then
we get $\frac{dY}{dX}=F(\frac{a_1X+b_1Y}{a_2X+b_2Y})………………(1)$ because $dx=dX, dy=dY.$
Now put $Y=vX$ where $v$ is a function of X,
and write $$\frac{dY}{dX}=v+x\frac{dv}{dX}$$ Now substitute into (1) and the resulting equation can usually be solved using separation of variables or other methods.


TITLE: Tuples of sums of two squares
QUESTION [12 upvotes]: This is in response to a question of evarist that was closed. Evarist's question was this: Do there exist infinitely many triples of consecutive sums of two squares? The topic actually has an interesting history: In 1903 an anonymous reader submitted the following question to the section Mathematical Questions in the British journal The Educational Times - Find all consecutive triples of sums of two squares. Elementary though non-explicit solutions were submitted by A. J. Champneys Cunningham and two other British academics (The reference is Mathematical Questions 14955. Educ. Times (2) 3 (1903), 41-43.) The Cunningham in question is the same one as the Cunningham project is named after. He was an artillery officer in the Indian Army who turned to elementary number theory after he retired.
Later J. E. Littlewood posed the following research problem which generalizes Evarist's first question: Given distinct positive integers h and k, do there exist infinitely many triples n,n+h,n+k that are simultaneously sums of two squares? This was solved in 1973 by C. Hooley, On the Intervals between Numbers that are sums of Two Squares, II. J. Number Theory 5 (1973), 215-217. Hooley's proof that there are infinitely many such triples is not long, but uses the theory of ternary quadratic forms.
The following problem is the reason that I have information relevant to Evarist's question, and together with another Littlewood problem about sums of two squares, the source of my interest in this topic:

For which choices of positive integers $h_1,\ldots,h_r$ are there infinitely many tuples $n + h_1,\ldots,n + h_r$ that are simultanously sums of two squares?

This ties up with Evarist's second question. Clearly a necessary condition is that $h_1,\ldots,h_r$ do not cover all four residue classes modulo $4$, since integers congruent to 3 modulo 4 are not sums of two squares.
EDIT: Greg Martin (see below) has observed that there are other necessary congruence conditions
than the one modulo 4. So the answer to the optimistic version of my question is NO, but the above highlighted question still stands. (The optimistic version was that the congruence condition modulo 4 would be sufficient).
(By the way, I have not been able to find Littlewood's statement of his research problem, and actually he may have considered the more general problem too. I could not find the n,n+h,n+k problem in the collection of research problems that he published, though his other and better known problem about sums of two squares is there).
I think the general case of the above problem is likely to be difficult. I had planned a project for a master's thesis to obtain more computational evidence, to establish special cases, and possibly to generalize the Cochrane-Dressler result discussed below, but no student with the requisite background was interested, so nothing came of it.
Actually, there is an unsolved problem directly connected to Evarist's first question too. In 1987
T. Cochrane and R. E. Dressler used the Selberg sieve to obtain un upper bound
$$
A(x) \ll \frac{x}{\log^{3/2}(x)}
$$
for the counting function $A(x)$ of the triples of consecutive sums of two squares. The reference is Consecutive triples of sums of two squares, Archiv der Math. 49 (1987), 301-304. One would expect a lower bound of the same order of growth, and actually an asymptotic estimate, but this is not known.
(The distribution of consecutive pairs of sums of two squares has been investigated, and more is known in this case).
Cochrane and Dressler give a very simple argument for Evarist's first question. The integers 8,9,10 are sums of two squares. Suppose n+1,n,n-1 are sums of two squares. Then $n^2 + 1$, $n^2$ and $n^2 - 1 = (n - 1)(n +1)$ are also sums of two squares, the last by the Brahmagupta-Fibonacci identity
$(x^2 + y^2)(u^2 + v^2) = (xu + yv)^2 + (xv - yu)^2$ that shows that products of sums of two squares are sums of two squares.

REPLY [2 votes]: As it stands, the problem is hopeless. The answer is yes for $h = (0,1,2)$ (by this I mean the vector $h_1 = 0$, $h_2 = 1$, $h_3 = 2$) by Cochrane and Dressler, and the same argument also works for $h = (0,1,2,5)$. Since $(0,1,2,3)$ is impossible, the "smallest" nontrivial problem is $(0,1,2,4)$. Is there a simple way of treating this one?
In general, the condition that the $h_i$ do not cover all residue classes modulo $4$ is much too weak. Consider e.g. $h = (0,2,4,6)$. If $n$ is odd, then $n$ or $n+2$ is $\equiv 3 \bmod 4$; if $n$ is even and $n+h$ is a sum of two squares, then so is $(n+h)/2$ because of
$$ a^2 + b^2 = \Big(\frac{a-b}2\Big)^2 + \Big(\frac{a+b}2\Big)^2. $$
Thus if the answer were positive for $h = (0,2,4,6)$, then it would also be positive for $h = (0,1,2,3)$, which it is not.


text:
\begin{document}

\title{Fundamental groupoid as a  terminal costack}

\author[I. Pirashvili]{Ilia  Pirashvili}
\address{
Department of Mathematics\\
University of Leicester\\
University Road\\
Leicester\\
LE1 7RH, UK} \email{ip96@leicester.ac.uk}
\maketitle

\begin{abstract} Let $X$ be a topological space. We denote by $\pi_0(X)$ the set of connected components of $X$ and by $\Pi_1(U)$ the fundamental groupoid. In this paper we prove that for good topological spaces the assignments $U\mapsto\pi_0(U)$ and $U\mapsto\Pi_1(U)$ are the terminal cosheaf and costack respectively.
\end{abstract}

\bigskip
\section{Introduction}

For quit some time now, stack theory, which is the 2-mathematical analogue of sheaf theory, has enjoyed a significant amount of popularity, especially in the last decade. But despite the fact that sheaves and stacks have been studied deeply for so long, the very closely related cosheaves and costacks have been virtually ignored. In this paper we aim to demonstrate their importance and usefulness. We will show that for good topological spaces the assignment $U\mapsto \pi_0(U)$ is the terminal cosheaf of sets, and the assignment $U\mapsto \Pi_1(U)$ is the terminal costack of groupoids.

Though the main result of this paper is conceptual rather then calculatory, Example. \ref{ex} demonstrates the advantage of 2-mathematics. Using 2-colimits rather then colimits, we are able to use Van Kampens theorem to calculate the fundamental groupoid of $S^1$ significantly easier.

This alone is a rather nice fact but it could have other important implications. This result generalises in two directions which will be in forthcomming papers. The first one is rather obvious, that the fundamental $n$-groupoid would most likely be the terminal co-$n$-stack with values in $n$-groupoids. The other is that if we define the fundamental groupoid to be the terminal costack, we don't need notions like paths and coverings. Indeed since the terminal costack (resp. terminal cosheaf) was the costackification (resp. cosheafification) of the constant functor with values in the one point category (resp. the one point set), Thm. \ref{vo} gives hope that we could define the fundamental groupoid for other more pathological spaces, and indeed for general 2-sites as well.

This paper is part of my PhD thesis at the University of Leicester under the supervision of Dr. Frank Neumann. He introduced me to the fundamental groupoid and insipred much of this paper, for which I would like to thank him.

\section{$\pi_0$ as a terminal cosheaf}

Let $X$ be a topological space. Denote by $\mathfrak{Off}(X)$ the poset of open subsets of $X$.

\begin{De}\label{cosh} Let $G$ be a (covariant) functor from the poset $\mathfrak{Off}(X)$ to a category $\bf A$. We say that $G$ is a cosheaf, if for any open covering  $U=\bigcup_iU_i$ the diagram
$$\xymatrix{\coprod_{i,j}G(U_i\cap U_j) \ar@<.5ex>[r]^{ \ \ \ \ \ b_0} \ar@<-.5ex>[r]_{ \ \ \ \ \ b_1} & \coprod_i G(U_i) \ar[r]^{ \ \ a} & G(U) \ar[r] & 1 }$$
is exact, in other words,
$G(U)$ is the coequiliser of $b_0,b_1$. That is to say, $G$ is a cosheaf if and only if for any object $S$ the functor $F$ defined by $F(U)= \hom_{\bf A}(G(U),S)$ is a sheaf of sets \cite{br}.
\end{De}

Let $G:\mathfrak{Off}(X)\rightarrow {\bf A}$ be a functor. We say that $\hat{G}:\mathfrak{Off}(X)\rightarrow {\bf A}$ is the associated cosheaf of $G$ if there is given a natural transformation  $\epsilon:\hat{G}\rightarrow G$, and for all cosheafs $G':\mathfrak{Off}(X)\rightarrow {\bf A}$ and all natural transformations $\varphi:G'\rightarrow G$, there is a unique natural transformation $\hat{\varphi}:G'\rightarrow \hat{G}$ such that the diagram
$$\xymatrix{G'\ar[r]^{\varphi}\ar[dr]_{\hat{\varphi}} & G \\
	           & \hat{G}\ar[u]_{\epsilon}}$$
commutes.

Observe the following theorem    \cite[Theorem  6.28] {jajr}.

\begin{Th}\label{vo} Assuming Vop\v{e}nka's  principle, every full subcategory of a locally presentable category $\bf K$, closed under colimits in $\bf K$, is coreflective.
\end{Th}

As a corollary of this theorem, we have immediately that, assuming Vop\v{e}nka's  principle, cosheafification exists whenever $G$ has values in sets (or many other categories, like groups etc). However, we do not make use of this fact. We are mainly interested in the terminal cosheaf, which is easy  to construct under the assumption that $X$ is locally connected. For a topological space $X$ we let $\pi_0(X)$ be the set of all connected components of $X$.

\begin{Le} Let $X$ be a locally connected topological space. Then $U\mapsto \pi_0(U)$ is a cosheaf.
\end{Le}

\begin{proof} To prove this, we will show that $U\mapsto \hom(\pi_0(U),S)$ is a sheaf for all $S$. But whenever $X$ is locally connected, $\hom(\pi_0(U),S)$ is the set of all locally constant functions on $U$ (LCF(U) for short) with values in $S$. Hence $U\mapsto LCF(U)$ is clearly a sheaf, which implies the lemma.
\end{proof}

\begin{Pro} Let $X$ be a locally connected topological space and consider the constant functor $p$, where $p(U)=1$, the one point set for any open set $U$. Then we have $\hat{p}(U)=\pi_0(U)$ where $\hat{p}$ is the cosheafification of $p$.
\end{Pro}

\begin{proof} The proof is done by the universality condition of the associated cosheaf. For this, we first need to construct a map $\alpha$ from $G$ to $\hat{p}$ for every cosheaf $G$ defined on $X$. Then we need to check that it is unique. To define $G(U)\rightarrow \hat{p}(U)$, consider the cover $\{U_i\}$ of $U$ by its connected components. Now it is clear that we have maps
$$\xymatrix{\coprod_{i,j}G(U_i\cap U_j) \ar@<.5ex>[r]^{ \ \ \ \ \ b_0} \ar@<-.5ex>[r]_{ \ \ \ \ \ b_1}\ar[d] & \coprod_i G(U_i) \ar[r]^{ \ \ a}\ar[d] & G(U) \ar[r] & 1 \\
	          \coprod_{i,j}\pi_0(U_i\cap U_j) \ar@<.5ex>[r]^{ \ \ \ \ \ b_0} \ar@<-.5ex>[r]_{ \ \ \ \ \ b_1} & \coprod_i \pi_0(U_i) \ar[r]^{ \ \ a} & \pi_0(U) \ar[r] & 1 }$$
since the intersection of connected components is either itself or empty, and $\pi_0$ of a connected space is 1, the one point set. It follows from the exactness of the sequences that there is a unique map $G(U)\rightarrow \pi_0(U)$. Since every map in the one point set is unique, the middle map is unique, which implies the uniqueness of $\alpha$. This finishes the proof.
\end{proof}

\begin{Co} The terminal object of the category of cosheafs over a locally connected topological space $X$ is given by $U\mapsto \pi_0(U)$.
\end{Co}

\section{$\Pi_1$ and Costacks}
Recall that a \emph{groupoid} is a category where all morphisms are isomorphisms. A groupoid $\sf G$ is \emph{simply connected} \cite[p. 36]{gz} provided for any two objects $a$ and $b$ of $\sf G$, there is exactly one morphism $a\to b$. By abuse of notations we let $\Grp$ be the category, as well as the $2$-category of small groupoids. Thus for any functor $\F:I\to \Grp$ we can talk not only of limits and colimits of $\F$, but also of 2-limits and 2-colimits (see Section \ref{2lim}) of $\F$.

\begin{De}\label{cost} Let $\G$ be a covariant functor from  the poset $\mathfrak{Off}(X)$ to $\Grp$. We say that $\G$ is a costack, if
$$\xymatrix{\coprod\limits_{i,j,k}\G(U_i\cap U_j\cap U_k) \ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r]&\coprod\limits_{i,j}\G(U_i\cap U_j) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \coprod\limits_i \G(U_i) \ar[r] & \G(U) \ar[r] & 0 }$$
is $2$-exact. That is to say, $\G(U)$ is the $2$-colimit of the diagram
$$\xymatrix{\coprod\limits_{i,j,k}\G(U_i\cap U_j\cap U_k) \ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r]&\coprod\limits_{i,j}\G(U_i\cap U_j) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \coprod\limits_i \G(U_i)  &  &  }.$$
Equivalently,
$\G$ is a costack iff  for any groupoid ${\sf G}$ the assignment $$U\mapsto \hom_{\Grp}(\G(U),{\sf G})$$ defines a stack \cite{br}.
\end{De}

Hence we can again talk about costackification, or associated costacks of functors, if they exist. Probably an analogue of Theorem \ref{vo} is still true for $2$-categories and hence under some set theoretical assumptions the costackification does exist, but we do not go in this direction and we restrict ourselves  only to the existence of 2-terminal objects. Recall that an object $t$ of a $2$-category $\bf A$ is  \emph{$2$-terminal}, provided for any object $x$ of $\bf A$ the category $\hom_{\bf A}(x,t)$ is a simply connected groupoid, which is the same to say that $\hom_{\bf A}(x,t)$ is equivalent to the groupoid $\bf 1$, with one object and one arrow.

For a topological space $X$, we let $\Pi_1(X)$ be the fundamental groupoid of $X$, see \cite{Brown} . Recall that the objects of $\Pi_1(X)$ are points of $X$, while morphisms are homotopy classes of paths. Then we have the following result.

\begin{Th}\label{Pi1} Let $X$ be a  topological space. Then the association $U\mapsto \Pi_1(U)$ defines simultaneously a cosheaf and costack with values in $\Grp$.
\end{Th}

The statement on cosheaves is basically the classical Van-Kampen theorem formulated in terms of fundamental groupoids, compare with \cite[Theorem II.7] {May}, where the author considers only coverings with connected open sets.  The last condition is unnecessary,  as it was demonstrated in \cite[p. 226, Statement 6.7.2]{Brown} where the author considers only coverings with two members. The statement about costacks is the main part of the paper and  the proof will be given in Section \ref{pr22}.

This already demonstrates the importance of costacks, but indeed more is true. Call a topological space $X$ \emph{good} if any open subset $U$ of $X$ possesses an open covering $U=\bigcup_iU_i$ such that all $U_i$ are simply connected, as well as all nonempty intersections $U_i\cap U_j$, $U_i\cap U_j\cap U_k$.

\begin{Th} Let $X$ be a good topological space. Consider the constant functor $P:U\mapsto {\bf 1}$, where ${\bf 1}$ is the one point groupoid (with only one object and one morphism). Then $P$ has an associated costack, which is the fundamental groupoid, i.e. $\hat{P}(U)=\Pi_1(U)$. Thus the costack  given by $U\mapsto \Pi_1(U)$ is a $2$-terminal object in the 2-category of all costacks on $X$.
\end{Th}

\begin{proof} By Theorem \ref{Pi1} $U\mapsto \Pi_1(U)$ is a costack. To prove that it is the costackification of $P$, we will use the universal property. It is clear that we have a morphism $\mathfrak{p}:\Pi_1\rightarrow P$ of 2-functors, so we only have to prove that given another 2-functor $Q:U\rightarrow {\Grp}$, there exists a morphism $\mathfrak{q}:\Pi_1\rightarrow Q$, where $Q$ is a costack. The fact that the compatibility conditions will be satisfied comes from the triviality of $P$.

Hence we only have to construct $\mathfrak{q}$. To define $\mathfrak{q}(U)$, cover $U$ by open subsets $\{U_i\}_{i\in I}$ such that all $U_i$ are simply connected  and the same holds for all their nonempty pairwise and triple intersections. We can do this by the assumption on $X$. Since $Q$ is a costack

$$\xymatrix{\coprod\limits_{i,j,k}Q(U_i\cap U_j\cap U_k) \ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r]&\coprod\limits_{i,j}Q(U_i\cap U_j) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \coprod\limits_i Q(U_i) \ar[r] & Q(U) \ar[r] & 1 }$$
is $2$-exact. Likewise for $\Pi_1$. Note that on the empty set $Q$ is empty, just like $\Pi_1$. For $U_i$-s and their non-empty intersections $U_i\cap U_j$ and $U_i\cap U_j\cap U_k$, $\Pi_1$ is the trivial groupoid, since they are simply connected. It follows that there is essentially a unique functor from $Q(U_i), Q(U_i\cap U_j)$ and $Q(U_i\cap U_j\cap U_k)$ to $\Pi_1(U_i), \Pi_1(U_i\cap U_j)$ and $\Pi_1(I_i\cap U_j\cap U_k)$ respectively.

Hence by $2$-exactness we get a map, unique up to natural transformation $\q(U):Q(U)\rightarrow \Pi_1(U)$ satisfying the compatibility condition. This finishes the proof.
\end{proof}

\section{Limits, colimits, $2$-limits and 2-colimts }\label{2lim}
First let us fix some notations. For functors and natural transformations
$$\xymatrix{{\sf A}\ar[r]^F&{\sf B}\ar@<.5ex>[r]^{ \ \ \ \ \ G_1} \ar@<-.5ex>[r]_{ \ \ \ \ \ G_2} & {\sf C}\ar[r]^T &{\sf D}}, \ \  \alpha:G_1 \Longrightarrow G_2. $$
one denotes by $\alpha \star F$ and  $T\star \alpha$ the
induced natural transformations $G_1F\Longrightarrow G_2F$, $TG_1\Longrightarrow TG_2$.

Let $I$ be a poset and let $\F:I^{op}\to \Grp$ be  a contravariant functor from the category $I$ to the category of groupoids. For an element $i\in I$ we let $\F_i$ be the value of $\F$ at $i$. For $i\leq j$ we let $\psi_{ij}:\F_j\to \F_i$ be the induced functor. Recall the construction of two groupoids $\lim\limits_{i} \F_i$ and $\tl\limits_i\F_i$, called respectively \emph{limit} and \emph{$2$-limit} of $\F$.

Objects of the groupoid $\lim\limits_i \F_i$ (or simply $\lim \F$) are  families $(x_i)$, where $x_i$ is an object of the category $\F_i$ such that for any $i\leq j$ one has $\psi_{ij}(x_j)=x_i$. A morphism $(x_i)\to(y_i)$ is  a family $(f_i)$, where $f_i:x_i\to y_i$ is a morphism of $\F_i$ such that for any $i\leq j$ one has $\psi_{ij}(f_j)=f_i$.

 Objects of the groupoid $\tl\limits_i\F_i$ (or simply $\tl \F$) are  collections $(x_i,\xi_{ij})$, where $x_i$ is an object of $\F_i$, while  $\xi_{ij}:\psi_{ij}(x_j)\to x_i$ for  $i\leq j$ is an isomorphism of the category $\F_j$ satisfying the 1-cocycle condition: For any $i\leq j\leq k$ one has
 $$\xi_{ik}=\xi_{ij} \circ \psi_{ij}(\xi_{kj}).$$
A morphism from $(x_i,\xi_{ij})$ to $(y_i,\eta_{ij})$ is  a collection $(f_i)$, where $f_i:x_i\to y_i$ is a morphism of $\F_i$ such that for any $i\leq j$ the following
$$\xymatrix{\psi_{ij}(x_j)\ar[r]^{\xi_{ij}}\ar[d]_{\psi_{ij}(f_j)}& x_i\ar[d]^{f_i}\\
\psi_{ij}(y_j)\ar[r]_{\eta_{ij}}& y_i}$$
 is a commutative diagram.  Let us state the following obvious Lemma.
 \begin{Le}\label{gamma} The functor $$\gamma:\lim \F\to \tl \F,$$ given by
 $$\gamma(x_i)=(x_i,Id_{x_i})$$
 is  full and faithful.
 \end{Le}

 Dually, for  a functor $\G:I\to \Grp$ one can define \emph{colimits} and \emph{$2$-colimits} of $\G$ as follows. $\colim\limits_i \G_i$ is a groupoid together with a collection of functors
$$\alpha_i:\G_i\to \colim\limits_i \G_i$$
 such that $\alpha_i=\alpha_j\psi_{ij}$ for any $i\leq j$. Additionally, one requires that for any groupoid ${\sf G}$ the canonical functor
$$c:\hom_{\Grp}(\colim\limits_i\G_i,{\sf G})\to \lim\limits_i\hom_{\Grp}(\G_i,{\sf G})$$
is an isomorphism of groupoids. Here the functor $c$ is given by $c(\chi)=(\chi\circ  \alpha_i)$,  where $\chi:\colim\limits_i\G_i\to \sf G$ is a functor. It is well-known that such $\colim\limits_i \G_i$ exists and it is unique up to isomorphism of groupoids \cite[p. 36]{gz}.

Quite similarly, $\tc\limits_i \G_i$ is a groupoid together with a family of functors
 $$\alpha_i:\G_i\to \tc\limits_i \G_i$$
 and natural transformations $\lambda_{ij}:\alpha_j\psi_{ij}\Longrightarrow \alpha_i$, $i\leq j$
 satisfying  the $1$-cocycle condition: For any $i\leq j\leq k$,
 $$\lambda_{ik}=\lambda_{ij}\circ (\lambda_{jk}\star\psi_{ij}).$$
Furthermore, one requires that for any groupoid ${\sf G}$, the canonical functor
$$\kappa:\hom_{\Grp}(\tc\limits_i\G_i,{\sf G})\to \tl\limits_i\hom_{\Grp}(\G_i,{\sf G})$$
is an equivalence of groupoids. Here the functor $\kappa$ is given by $\kappa(\chi)=(\chi\circ  \alpha_i, \chi_i\star \lambda_{ij})$. It is well-known that $\tc$ exists and is unique up to equivalence of groupoids, see, \cite[pp. 192-193] {br}.

The functors $c, \kappa$  and the comparison functor $\gamma$ in Lemma \ref{gamma} allow us to consider the composite functor:
$$\hom_{\Grp}(\colim\limits_i\G_i,{\sf G})\xto{c} \lim\limits_i\hom_{\Grp}(\G_i,{\sf G})\xto{\gamma} \tl\limits_i\hom_{\Grp}(\G_i,{\sf G}) \xto{\kappa^{-1}} \hom_{\Grp}(\tc\limits_i\G_i,{\sf G})$$
By the Yoneda lemma one obtains a  comparison functor
$$\delta:\tc\limits_i\G_i\to \colim\limits_i\G_i.$$
\begin{Le}\label{delta} The functor $\delta$ is an equivalence of categories iff for any groupoid ${\sf D}$, and any family of functors
 $$\beta_i:\G_i\to {\sf D}$$
 and natural transformations $\tau_{ij}:\beta_j\psi_{ij}\Longrightarrow \beta_i$, $i\leq j$
 satisfying  the $1$-cocycle condition [i.e. for any $i\leq j\leq k$,
 $$\tau_{ik}=\tau_{ij}\circ (\tau_{jk}\star\psi_{ij})],$$
 there exists another family of functors $\beta_i':\G_i\to {\sf D}$
 and natural transformations $\mu_i:\beta_i\Longrightarrow \beta'_i$ such that
 $\beta_i'=\beta'_j\circ \psi_{ij}$,  and $\tau_{ij}=\mu_i^{-1}\circ (\mu_j\star \psi_{ij})$ for all $i\leq j$.
\end{Le}
\begin{proof} By the Yoneda lemma for $2$-categories $\delta$ is an equivalence of categories iff for any groupoid ${\sf D}$ the induced functor
$$\hom_{\Grp}(\delta, {\sf D}):\hom_{\Grp}(\colim \G,{\sf D})\to \hom_{\Grp}(\tc \G,{\sf D})$$ is an equivalence of categories. Since
$$\hom_{\Grp}(\colim\limits_i  \G,{\sf D}) =\lim\limits_i \hom_{\Grp}(\G_i,{\sf D})$$
and
$$\hom_{\Grp}(\tc\limits_i  \G,{\sf D}) =\tl\limits_i \hom_{\Grp}(\G_i,{\sf D}),$$
we can use Lemma \ref{gamma} to conclude that the functor $\hom_{\Grp}(\delta, {\sf D})$ is always full and faithful. Hence it is an equivalence iff it is essentially surjective on objects. One easily sees that the last condition is exactly the assertion made in the statement and we are done.

\end{proof}
As we will see in the next section, the functor $\delta$ is often an equivalence of categories, which explains the fact that $U\mapsto \Pi_1(U)$ is simultaneously a cosheaf and a costack.

\section{When $\delta:\tc\to \colim$ is an equivalence of categories}
Recall that a poset $I$ is called \emph{filtered} provided for any elements $i,j\in I$ there is an element $k\in I$ such that $i\leq k$ and $j\leq k$.

\begin{Pro}\label{filtered} If $I$ is  a filtered poset, then for any functor $\G:I\to \Grp$ the canonical comparison functor $\delta:\tc\limits_i\G_i\to \colim\limits_i\G_i $ is an equivalence of categories.
\end{Pro}
\begin{proof}  In this case the construction of $\colim\limits_i\G_i $ given in \cite[p.36]{gz} has the following description.
Let  $X$ and $Y$ be objects of the  categories $\G_i$ and $\G_j$ respectively. We will say that $X$ and $Y$ are equivalent, if there exists $k\in I$, such that  $i\leq k,j\leq k$  and $\psi_{ik}(X)=\psi_{jk}(Y).$ The equivalence class of $X$ is denoted by $\{X\}$. The collection of such equivalence classes forms the class of all objects of the category $\colim\limits_i\G_i $.  If $\{X\}$ and $\{Y\}$ are two equivalence classes, assuming $X\in \G_i$ and $Y\in \G_j$, then a morphism $\{X\}\to \{Y\}$ is the equivalence class of morphisms $f:\psi_{ik}(X)\to \psi_{jk}(Y)$ of the category $\G_k$, where $i\leq k, j\leq k$. Here two morphisms $f:\psi_{ik}(X)\to \psi_{jk}(Y)$ and $g:\psi_{im}(X)\to \psi_{jm}(Y)$, $i\leq m, j\leq m$ are equivalent if there exists an element $n\in I$ such that $k\leq n, m\leq n$ and $\psi_{kn}(f)=\psi_{mn}(g).$

Recall that the construction of $\tc \G$ given in  \cite[pp. 192-193] {br} shows that the objects of $\tc \G$ are a disjoint union of the objects of $\G_i$, $i\in I$, while for $X\in \G_i$ and $Y\in \G_j$  one has
$$\hom_{\tc\limits_i \G_i}(X,Y)=\hom_{\colim\limits_i \G_i}(\{X\},\{Y,\}).$$
It follows that the functor $\delta$, which sends $X$ to $\{X\}$ is obviously full and faithful and surjective on objects. Hence $\delta$ is an equivalence of categories.
\end{proof}
\begin{Pro}\label{2} Let
$$\xymatrix{ {\bf A}\ar[r]^{i_1}\ar[d]_{i_2} & {\bf B} \\
	         {\bf C} & }$$
be a diagram of groupoids, where $i_1$ is injective on objects. Then the colimit and 2-colimit are equivalent.
\end{Pro}
Thanks to Lemma \ref{delta} the result follows  from the following deformation lemma.
\begin{Le} \label{deformation} Let $$\xymatrix{ {\bf A} \drtwocell\omit{\lambda}\ar[r]^{i_1}\ar[d]_{i_2} & {\bf B}\ar[d]^{j_1} &  \\ {\bf C}\ar[r]_{j_2} & {\bf D} &  }$$
be a $2$-diagram of groupoids, i.e. $\lambda:j_1i_1\Longrightarrow j_2i_2$ is a natural transformation.
If the functor $i_1$ is injective on objects, then there exists a functor $j_1':{\bf B}\to {\bf D}$ and a natural transformation $\kappa:j_1\Longrightarrow j_1'$ for which the diagram
$$\xymatrix{ {\bf A}\ar[r]^{i_1}\ar[d]_{i_2} & {\bf B}\ar[d]^{j_1'} \\ {\bf C} \ar[r]_{j_2}& {\bf D}}$$
commutes and $\lambda$ coincides with $\kappa\star i_1:j_1i_1\Longrightarrow j_1'i_1=j_2i_2.$
\end{Le}
\begin{proof} Define $j_1'$ on objects by
$$j_1'(b)=\begin{cases}j_1(b), & {\rm if} \ b\not\in Im(i_1)\\ j_2i_2(a),& {\rm if} \ b=i_1(a).\end{cases}$$
To define $j_1'$ on morphisms we proceed as follows.
Let $\beta:b_1\to b_2$ be a morphism in $\bf B$. To define $$j_1'(\beta):j_1'(b_1)\to j_1'(b_2)$$ we have to consider five different cases.

{\it Case 1}. $b_1=i_1(a_1)$ and $b_2\not\in Im(i_1)$. One defines $j_1'(\beta)$ to be the composite:
$$j_1'(b_1)=j_2i_2(a_1)\xto{\lambda^{-1}(a_1)}j_1i_1(a_1)=j_1(b_1)\xto{j_1(\beta)} j_1(b_2)=j_1'(b_2).$$

{\it Case 2}. $b_1=i_1(a_1)$, $b_2=  i_1(a_2)$, but $\beta \not \in Im(i_1)$. One defines $j_1'(\beta)$ to be the composite:
$$j_1'(b_1)=j_2i_2(a_1)\xto{\lambda^{-1}(a_1)}j_1i_1(a_1)=j_1(b_1)\xto{j_1(\beta)} j_1(b_2)=j_1i_1(a_2)\xto{\lambda(a_2)} j_2i_2(a_2)=j_1'(b_2).$$

{\it Case 3}. $\beta\in Im(j_1)$. We choose $\alpha:a_1\to a_2$ in $\bf A$ such that $\beta=i_1(\alpha)$. One defines $j_1'(\beta)$ by
$$j_1'(b_1)=j_2i_2(a_1)\xto{j_2i_2(\alpha)} j_2i_2(a_2)=j_1'(b_2).$$
To check that this is independent on the choice of $\alpha$, let us also consider also $\alpha':a_1\to a_2$ with the property $i_1(\alpha')=\beta$. Since $\lambda$ is a natural transformation, we have a commutative diagram
$$\xymatrix{j_1i_1(a)\ar[rr]^{\lambda(a)}\ar@<-.5ex>[d]_{_{j_1i_1(\alpha)}}\ar@<.5ex>[d]^{_{j_1i_1(\alpha')}} & &  j_2i_2(a)\ar@<-.5ex>[d]_{_{j_2i_2(\alpha)}}\ar@<.5ex>[d]^{_{j_2i_2(\alpha')}} \\
	          j_1i_1(a')\ar[rr]_{\lambda(a')} & & j_2i_2(a') }$$
Since the left vertical arrows are equal and horizontal ones are isomorphisms, it follows that the right vertical arrows are also equal. Hence $j_1'(\beta)$ is well-defined in this case.

{\it Case 4}. $b_1\not\in Im(i_1)$ and $b_2=i_1(a_2)$. One defines $j_1'(\beta)$ to be the composite:
$$j_1'(b_1)=j_1(b_1)\xto{j_1(\beta)} j_1(b_2)=j_1i_i(a_2)\xto{\lambda(a_2)} j_2i_2(a_2)=j_2'(b_2)$$

{\it Case 5}. $b_1\not\in Im(i_1)$  and $b_2\not\in Im(i_1)$. One defines $j_1'(\beta)$ as
$$j_1'(b_1)=j_1(b_1)\xto{j_1(\beta)} j_1(b_2)=j_1'(b_2)$$
Checking case by case shows that $j_1'$ is really a functor, with $j_1'i_1=j_2i_2$. Define $\kappa$  by
$$\kappa(b)=\begin{cases} Id_{j_1(b)}, & {\rm if} \ b\not\in Im(i_1)\\ \lambda(a),& {\rm if} \ b=i_1(a). \end{cases} $$
One easily sees that $j_1'$ and $\kappa$ satisfy the assertions of the Lemma.

\end{proof}

It follows that the classical  Van Kampen theorem  \cite[p. 226]{Brown} can be restated as follows.

\begin{Th} \label{vk} (Van Kampen) Let $U$ and $V$ be open subsets of $X$. Assume $X= U \cup V$. Then the diagram is simultaneously a pushout and $2$-pushout of groupoids:
$$\xymatrix{\Pi_1(W) \ar[d]\ar[r] & \Pi_1(U)\ar[d]\\ \Pi_1(V)\ar[r] & \Pi_1(X),}$$
where  $W=U \cap V$.
\end{Th}

The advantage of  $2$-pushouts is that the fundamental groupoids involved can be changed by equivalent ones. Here is an example to demonstrate the usefulness of $2$-pushouts.

\begin{Ex} \label{ex}  Let us cover $S^1$ by $U$ and $V$, where both are open subsets of $S^1$, with a single point removed (of course different). Since $\Pi_1(U)$ and $\Pi_1(V)$ are equivalent to the groupoid ${\bf 1}$ with one object and one morphism, and $\Pi_1(W)$ is equivalent to the groupoid
${\bf 1} \coprod {\bf 1}$, we see that $\Pi_1(S^1)$ is equivalent to the $2$-pushout (and hence pushout) of the diagram
$$\xymatrix{{\bf 1} \coprod {\bf 1} \ar[r] \ar[d] & {\bf 2} \\ {\bf 1} &}$$
where $\bf 2$ is the simply connected groupoid with two objects and the horizontal arrow is bijective on objects. Clearly the pushout of this diagram is the groupoid with one object and the infinite cyclic group as the group of automorphisms. Compare this computation with the computation in \cite[pp. 233-234] {Brown} which uses the more complicated groupoid $\Pi_1XA$, where $A$ is an additional set.
\end{Ex}

\section{Proof of Theorem \ref{Pi1}}\label{pr22}
Let $X$ be a topological space and $\F:\mathfrak{Off}(X)^{op}\rightarrow \mathsf{Grp}$ be a functor.

\begin{De}
Let $n\geq 2$ be an integer. We will say that $\F$ satisfies the property ${\mathsf sh}(n)$ (resp. ${\mathsf st(n)})$ if for any open subset $U$ of $X$ and any open cover $U_1,\cdots, U_n$ of $U$, the natural functor
$$\xymatrix{\F(U)\ar[r] & \lim [\prod_i \F(U_i)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \prod_{i,j}\F(U_i\cap U_j)]}$$
is an isomorphism of categories, (resp.
$$\xymatrix{\F(U)\ar[r] & \tl [\prod_i \F(U_i)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \prod_{i,j}\F(U_i\cap U_j)\ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r] & \prod\limits_{i,j,k}\F(U_i\cap U_j\cap U_k)]}$$
is an equivalence of categories). That is to say, $\F$ satisfies the sheaf (resp. stack) condition for all coverings having $n$-members.
\end{De}

The aim of this section is to prove Theorem \ref{Pi1}, which will be done with formal arguments. In Thm. \ref{vk} we have shown that $U\rightarrow \hom_{\Grp}(\Pi_1(U),{\sf G})$ satisfies the $sh(2)$ as well as the $st(2)$ property for any groupoid $G$. In order to show that it is a sheaf (resp. a stack), we have to show that these conditions hold for general coverings, see Definition \ref{cosh} and Definition \ref{cost}. To do so, we will first show that $sh(2)$ (resp. $st(2)$) implies $sh(n)$ (resp. $st(n)$) for any $n\in\mathbb{N}$ and then use \ref{filtered} to prove the general case.

\begin{Le} \label{finite} If $\F$ satisfies the condition ${\mathsf sh}(2)$, (resp. ${\mathsf st}(2)$) then $\F$ satisfies the condition ${\mathsf sh(n)}$ (resp. ${\mathsf st}(n)$) for any $n\geq 2$.
\end{Le}

\begin{proof} We consider only the case $n=3$, since the only difference between this and the general case is notation. By definition, the objects of
$$\xymatrix{\tl[\prod_i \F(U_i)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \prod_{i,j}\F(U_i\times_UU_j)\ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r] & \prod\limits_{i,j,k}\F(U_i\times_UU_j\times_UU_k)]}$$
are $$((g_1,g_2,g_3),\alpha_{12}, \alpha_{13},\alpha_{23})$$
where $g_i$ is an object of $\F(U_i)$, $i=1,2,3$ and
$\alpha_{12}:g_1|_{U_{12}}\rightarrow g_2|_{U_{12}}$, $\alpha_{13}:g_1|_{U_{13}}\rightarrow g_3|_{U_{13}}$  and $\alpha_{23}:g_2|_{U_{23}}\rightarrow g_3|_{U_{23}}$ are morphisms respectively in $\F(U_{12})$, $\F(U_{13})$  and $\F(U_{23})$. Here $U_{ij}=U_i\cap U_j$. One also requires that these data satisfy
the 1-cocylce condition. The morphisms are $(h_1:g_1\rightarrow g'_1, h_2:g_2\rightarrow g'_2, h_3:g_3\rightarrow g'_3)$, such that restricted on the intersections, the obvious diagrams commute.

We set $V=U_1\cup U_2$. Since $\F$ satisfies the condition $st(2)$, we can assume that the objects of $F(U)=F(V \cup U_3)$ are
the same as of the category
$$\xymatrix{\tl[\F(V)\times\F(U_3) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \F(V\cap U_3)]}$$
So we can assume that they are of the form
$$\{((g_v,g_3),\gamma:g_v|_{V\cap U_3}\rightarrow g_3|_{V\cap U_3})\},$$
where  $g_v$ is an object of $\F(V)$. But since $V=U_1\cup U_2$ and $\F$ satisfies the condition $st(2)$ we may assume that the objects of $\F(V)$ have the form
$$g_v=\{((g_1,g_2),\delta:g_1|_{U_{12}}\rightarrow g_2|_{U_{12}})\}.$$
Thus the objects  of $\F(V\cup U_3)$ are
$$\{((g_v,g_3),\gamma:g_v|_{V\cap U_3}\rightarrow g_3|_{V\cap U_3})\}.$$
Hence the objects of $\F(U)=\F(V\cup U_3)$, can be seen as
$$\{((g_1,g_2,g_3),\delta:g_1|_{U_{12}}\rightarrow g_2|_{U_{12}}, \gamma_1:g_1|_{U_{13}}\rightarrow g_3|_{U_{13}}, \gamma_2:g_2|_{U_{23}}\rightarrow g_3|_{U_{23}})\},$$
such that
$$\xymatrix{ g_1|_{U_{123}}\ar[r]\ar[d]_{\delta|_{U_{123}}} & g_3|_{U_{123}}\ar[d]^{id} \\
	           g_2|_{U_{123}}\ar[r] & g_3|_{U_{123}} }$$commutes.
	           Since the last condition  is exactly  the 1-cocycle condition, we can see that the categories $\F(U)$ and $\xymatrix{\tl[\prod_i \F(U_i)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \prod_{i,j}\F(U_i\times_UU_j)\ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r] & \prod\limits_{i,j,k}\F(U_i\times_UU_j\times_UU_k)]}$ have essentially the same objects. Observe that the morphims of $\F(U)=\F(V\cup U_3)$ are $\{(\varphi_V:g_V\rightarrow g'_V, \varphi_{U_3}:g_3\rightarrow g_3')\}$, such that the obvious diagram commutes. Then morphisms of $\F(V)$ itself are $\{(\lambda_{U_1}:g_1\rightarrow g_1',\lambda_{U_2}:g_2\rightarrow g_2')\},$ such that again the obvious diagram commutes. Hence putting them together, we get $\{(\varphi_V:g_{U_1\cup U_2}\rightarrow g'_{U_1\cup U_2}, \varphi_{U_3}:g_3\rightarrow g_3')\}$ i.e. $\{(\varphi_{U_1}:g_1\rightarrow g'_1, \varphi_{U_2}:g_2\rightarrow g_2'),\varphi_{U_3}:g_3\rightarrow g_3')\}$ such that on the intersections, the restrictions agree. This shows the equivalence of $\F(U)$ and $\xymatrix{\tl[\prod_i \F(U_i)\ar@<.5ex>[r]\ar@<-.5ex>[r] & \prod_{i,j}\F(U_i\times_UU_j)\ar@<.7ex>[r]\ar[r]\ar@<-.7ex>[r] &\prod\limits_{i,j,k}\F(U_i\times_UU_j\times_UU_k)]}$. Hence $st(3)$ holds.

The fact that ${\mathsf sh}(2)$ implies ${\mathsf sh}(3)$ is a special case of the above proof, by simply taking $\alpha_i$'s  to be identities.
\end{proof}

To prove the general case, we basically only have to prove the following lemma:

\begin{Le} \label{general} Let $\G:\mathfrak{Off}(X)\to {\Grp}$ be  a functor which commutes with filtered colimits. Then $\G$ is a cosheaf (resp. costack) if and only if for all groupoids ${\sf G}$, the functor $\F$ defined by $\F(U)=\hom_{Grp}(\G(U),{\sf G})$ satisfies the ${sh}(n)$ (resp.${st}(n)$) condition for all $n$.
\end{Le}

\begin{proof} The 'only if' part is obvious. Assume $\F$ satisfies the condition ${\mathsf sh}(n)$ for all $n$.  We have to prove that $\G$ is a cosheaf. Let $U$ be an open set and $\{U_i\}_{i\in I}$ be a cover of $U$. Denote by $f(I)$ the set of finite subsets. We know that $f(I)$ is a filtered system for all $I$. For a fixed $\lambda\in f(I)$, denote by $U_\lambda$ the union of all $U_i$, with $i\in\lambda$. By assumption, $\G$ satisfies the cosheaf condition for any finite covering. Hence we have
$$\xymatrix{\G(U)=\G(\colim\limits_{\lambda}U_{\lambda})=\colim\limits_{\lambda}\G(U_{\lambda})=
\colim\limits_{\lambda}(\colim[\coprod\limits_{i,j\in \lambda}\G(U_{ij}) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \coprod\limits_{i\in\lambda}\G(U_i)])= }$$
$$\xymatrix{
\colim([\colim\limits_{\lambda}\coprod\limits_{i,j\in \lambda}\G(U_{ij}) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \colim\limits_{\lambda}\coprod\limits_{i\in\lambda}\G(U_i)])=\colim[\coprod\limits_{i,j\in I}\G(U_{ij}) \ar@<.5ex>[r]\ar@<-.5ex>[r] & \coprod\limits_{i\in I}\G(U_i)] }$$
as desired. This shows that $G$ is a cosheaf.

If $\F$ satisfies the condition $st(n)$, then by Proposition \ref{filtered} we have
$$\G(U)=\colim\limits_{\lambda}\G(U_{\lambda})\cong \tc\limits_{\lambda}\G(U_{\lambda})$$
Thus we can repeat the above computation replacing $\colim$ with $\tc$.
\end{proof}

\begin{proof} [Proof of Theorem \ref{Pi1}] By Theorem   \ref{vk} we know that the cosheaf  condition for the functor $U\mapsto \Pi_1(U)$ holds for coverings having only two members. By Proposition \ref{2}, for the same coverings the costack condition holds as well. Take  a groupoid $\sf G$ and consider the functor $F$ defined by
$$\F(U)=\hom_{\Grp}(\Pi_1(U),{\sf G})$$
Then $F$ satisfies both $sh(2)$ and $st(2)$ conditions. Hence for $F$  the conditions $sh(n)$ and $st(n)$ hold as well, thanks to  Lemma \ref{finite}. Since $\Pi_1$ commutes with filtered colimits, we can use Lemma \ref{filtered} to show that $U\mapsto \Pi_1(U)$ is both a cosheaf and a costack.
\end{proof}

config: arxiv


TITLE: About a claim in limits
QUESTION [3 upvotes]: Let $a_n\geq 0$.
If $\displaystyle \lim_{n\to +\infty} a_n=L$ then
$\displaystyle\lim_{n\to +\infty} \sqrt[k]{a_n}=\sqrt[k]{L}$, $(k\in N)$
I am trying to show this using that:
$a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+...+b^{k-1})$.

REPLY [2 votes]: If $L=0$ the result is true (you can check directly by defining limit). If $L \neq 0$, how $a_n \geq 0$ then $L > 0$. Then  using that $a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+...+b^{k-1})$ with $a=\sqrt[k]{a_n}$ and $b=\sqrt[k]{L}$, we obtain
$$ \sqrt[k]{a_n}-\sqrt[k]{L} = \dfrac{a_n-L}{ (\sqrt[k]{a_n})^{k-1}+(\sqrt[k]{a_n})^{k-2}L+...+L^{k-1} }.$$
Then
$$|\sqrt[k]{a_n}-\sqrt[k]{L}| = \dfrac{|a_n-L|}{ |(\sqrt[k]{a_n})^{k-1}+(\sqrt[k]{a_n})^{k-2}L+...+L^{k-1} |} \leq \dfrac{|a_n-L|}{ |L|^{k-1}}$$
How $a_n$ is convergent, given $\varepsilon$ choose $n_0 \in \mathbb{N}$ such that for $n \geq n_0$ implies $|a_n-L| < |L|^{k-1}\varepsilon$. So from the above equation results
$$|\sqrt[k]{a_n}-\sqrt[k]{L}|  \leq \dfrac{|L|^{k-1}\varepsilon}{ |L|^{k-1}} = \varepsilon.$$


text:
\begin{document}

\title{
An upper bound for higher order eigenvalues of symmetric graphs
}
\author{Shinichiro Kobayashi}
\address{Mathematical Institute, Tohoku University, Sendai 980-8578, Japan}
\email{shin-ichiro.kobayashi.p3@dc.tohoku.ac.jp}
\subjclass[2010]{Primary 35P15, 05C50; Secondary 58C40}
\keywords{graph Laplacian; higher-order eigenvalue; regular graph; symmetric graph.}
\date{June 13, 2020.}
\maketitle

\begin{abstract}
In this paper, we derive an upper bound for higher order eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower order eigenvalues.
\end{abstract}

\section{Introduction}
Let $G$ be a connected, finite, simple and undirected graph of $N$ vertices.
Let $\Delta$ be the normalized Laplace operator assiciated with $G$. The operator $-\Delta$ is identified with a non-negative definite real symmetric matrix of size $N$.
Denote by $\lambda_0\leq\lambda_1\leq \dots\leq\lambda_{N-1}$ all eigenvalues of $\Delta$ counted with multiplicity.
For any connected graph, we have $\lambda_0=0$ and its multiplicity is $1$. All the eigenvalues lie in the interval $[0,2]$.
We consider the following question:
Are there other contraints on the spectrum $\{\lambda_i\}_{i=0}^{N-1}$?
In particular, is $\lambda_{k+1}$ controlled by past eigenvalues, $\lambda_1,\dots,\lambda_{k}$?
This question is a discrete analogue of the so-called Payne-P\'{o}lya-Weinberger's inequality.
For the Dirichlet eigenvalues $0<\lambda_1<\lambda_2\leq \lambda_3\leq \cdots \uparrow \infty$ of the Laplacian on a bounded domain in the Euclid plane, Payne-P\'{o}lya-Weinberger \cite{MR73046, MR84696} proved that
\[
\lambda_{k+1}-\lambda_{k}\leq \frac2k\sum_{i=1}^k\lambda_i.
\]
This result is extended to arbitrary dimension by Thompson \cite{MR257592}.
Later, Hile and Protter \cite{MR578204} and Yang \cite{Yang} proved sharper inequalities.
In particular, Yang \cite{Yang} proved that
\begin{equation}\label{Yangineq}
\sum_{i=1}^k(\lambda_{k+1}-\lambda_i)^2\leq \frac4n\sum_{i=1}^k(\lambda_{k+1}-\lambda_i)\lambda_i.
\end{equation}
Chung and Oden proposed to study of the discrete analogue of their results.
For the Dirichlet eigenvalues $\{\lambda_i\}_{i\geq 1}$ of the normalized Laplacian on a connected finite subgraph in the integer lattice of rank $n$, Hua, Lin and Su \cite{HLS} proved that
\[
\sum_{i=1}^k(\lambda_{k+1}-\lambda_i)^2(1-\lambda_i)\leq \frac4n\sum_{i=1}^k(\lambda_{k+1}-\lambda_i)\lambda_i.
\]

How about the case of the Laplacian without boundary conditions?
Unlike the case of the Dirichlet boundary condition, $0$ is always an eigenvalue.
For the eigenvalues $\{\lambda_i\}_{i\geq 0}$ with $\lambda_0:=0$ of the Laplacian on a compact Riemannian homogeneous manifold, Cheng and Yang \cite{MR2115463} proved that
\begin{equation}\label{thm: CY}
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)^2\leq \sum_{i=1}^k(\lambda_{k+1}-\lambda_i)(4\lambda_i+\lambda_1).
\end{equation}
In this paper, we consider a discrete analogue of \eqref{thm: CY}.
More precisely, for a finite symmetric graph, we prove a discrete analogue of \eqref{thm: CY}.
\begin{thm}\label{main thm}
Let $G$ be an symmetric finite graph with $N$ vertices.
Denote by $0=\lambda_0<\lambda_1\leq \lambda_2\leq\cdots\leq\lambda_{N-1}$ the all eigenvalues of the normalized Laplace operator. Then, for any non-zero eigenvalue $\lambda$ of $\Delta$, we have
\[
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)^2(1-\lambda_i)\leq
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)(2(2-\lambda)\lambda_i+\lambda).
\]
\end{thm}
By using Chebyshev's sum inequality, we obtain an upper bound of $\lambda_{k+1}$ in terms of $\lambda_1,\dots,\lambda_k$.
\begin{thm}\label{main thm 2}
In the same setting as Theorem \ref{main thm}, we have
\[
\lambda_{k+1}\leq \frac{(k+1)\lambda_1 + \sum_{i=1}^k((5-2\lambda_1)\lambda_i-\lambda_i^2)}{\sum_{i=0}^k(1-\lambda_i)}.
\]
\end{thm}
Let $\mu_1:=\lambda_1$ and $m$ be the multiplicity of $\mu_1$. If $G$ is not a complete graph, then we can consider $\mu_2:=\lambda_{m+1}$, i.e., the second smallest positive eigenvalue.
We have a upper bound for the ratio $\mu_2/\mu_1$ in terms of the multiplicity of $\mu_1$.
\begin{cor}\label{main cor}
In the same setting as Theorem \ref{main thm}, let $m$ be the multiplicity of $\mu_1$ and put $\mu_2:=\lambda_{m+1}$. Then, we have
\[
\frac{\mu_2}{\mu_1}\leq 3m+1.
\]
\end{cor}

\begin{ack*}
The author would like to thank Professor Takashi Shioya for helpful comments.
\end{ack*}

\section{Preliminaries}
In this section, unless otherwise stated, we assume that all graphs are connected, finite, simple and undirected.
We recall some basic facts on the theory of eigenvalues of a regular graph.
Let $G=(V,E)$ be a $d$-regular graph, $d\geq 1$, and put $N:=\#V$.
If two vertices $x,y\in V$ are adjacent, then we denote this situation by $x\sim y$. Note that since $G$ is undirected, $x\sim y$ if and only if $y\sim x$.
The \emph{normalized Laplace operator} $\Delta$ acting on the space $C(V)$ of functions on $V$ is defined by
\[
\Delta u(x):=\frac{1}{d}\sum_{y\sim x}\left(u(y)-u(x)\right),\, u\in C(V), x\in V.
\]
The normalized Laplace operator is identified with the real-symmetric matrix $D^{-1}A-I$, where $D$ is the scalar matrix with diagonal entries $d$, $A$ is the adjacency matrix of $G$ and $I$ is the identity matrix.
A complex number $\lambda$ is called an \emph{eigenvalue} of $\Delta$ if there exists $u\in C(V)\setminus \{0\}$ such that $\Delta u+\lambda u=0$ holds. In this case, the function $u$ is called an \emph{eigenfunction} with eigenvalue $\lambda$. For an eigenvalue $\lambda$ of $\Delta$, we denote by $W_{\lambda}$ the space of all functions $u\in C(V)$
satisfying $\Delta u+\lambda u=0$ and we call the dimension of $W_{\lambda}$ \emph{multiplicity} of $\lambda$.
Let us denote the eigenvalues of $\Delta$ by $\lambda_0\leq\lambda_1\leq \lambda_2\leq \dots\leq \lambda_{N-1}$, counted with multiplicity.
We define a inner product $\langle\cdot,\cdot\rangle$ on $C(V)$ by
\[
\langle u,v\rangle:=\sum_{x\in V}u(x)v(x)d.
\]
We denote by $\|\cdot\|$ the norm induced by the inner product $\langle \cdot,\cdot\rangle$.
We list up some elementary facts on eigenvalues and eigenfunctions without proofs.
\begin{itemize}
\item $0$ is an eigenvalue of multiplicity $1$ and constant functions are eigenfunctions with eigenvalue $0$.
\item All eigenvalues lie in the interval $[0,2]\subset \mathbb{R}$.
\item There exists a orthonormal basis $\{u_{i}\}_{i=0}^{N-1}$ of $C(V)$ such that each function $u_{i}$ is an eigenfunction with eigenvalue $\lambda_i.$
\end{itemize}
By the min-max formula, each eigenvalue $\lambda_k$ has a variational characterization:
\[
\lambda_k = \inf\left\{\frac{\sum_{x\sim y}(u(y)-u(x))^2}{2d\sum_{V}u^2}\setrel u\neq 0, \langle u,u_i\rangle=0,\, i=0,\dots, k-1\right\},
\]
where the symbol $\sum_{x\sim y}$ means the summation over all unordered pairs $(x,y)$ such that $x\sim y$.
In particular, we have
\begin{equation}\label{spectral gap}
\lambda_1 = \inf\left\{\frac{\sum_{x\sim y}(u(y)-u(x))^2}{2d\sum_{V}u^2}\setrel u\neq 0, \sum_Vu=0 \right\}.
\end{equation}
We shall derive a general upper bound for $\lambda_1$.
\begin{lem}\label{lem: upperbound}
For any regular graph $G$ but a complete graph, we have
\[
\lambda_1\leq 1.
\]
\end{lem}
\begin{proof}
Since $G$ is not complete, there exist two vertices $x_0,y_0\in V$ such that $x_0\not \sim y_0$. We define a function $u\in C(V)$ by
\[
u(x):=
\begin{cases}
1 & \text{if}\ x=x_0, \\
-1 & \text{if}\ x=y_0, \\
0 & \text{otherwise}.
\end{cases}
\]
Clearly, the function $u$ satisfies $\sum_Vu=0$. From \eqref{spectral gap}, we have
\[
\lambda_1\leq \frac{\sum_{x\sim y}(u(y)-u(x))^2}{2d\sum_{V}u^2}=1.
\]
\end{proof}
\begin{remark}
If $G$ is the complete graph of degree $d$, then $\lambda_1=1+1/d$.
\end{remark}
Let $\Gamma\colon C(V)\times C(V)\to C(V)$ be the \emph{carr\'{e} du champ operator} associated to $\Delta$, i.e., for $u,v\in C(V)$,
\[
\Gamma(u,v):=\frac12\left(\Delta(uv)-(\Delta u)v-u\Delta v\right).
\]
For two vertices $x,y\in V$ with $x\sim y$, we define the \emph{difference operator} $\nabla_{xy}\colon C(V)\to C(V)$
by
\[
\nabla_{xy}u:=u(y)-u(x),\, u\in C(V).
\]
By a simple calculation, we have
\[
\Gamma(u,v)(x)=\frac{1}{2d}\sum_{y\sim x}(\nabla_{xy}u)(\nabla_{xy}v),\, x\in V.
\]
The carr\'{e} du champ $\Gamma(u,v)$ is an analogy of $\langle \nabla u,\nabla v\rangle$ in the context of Riemannian geometry, where $\nabla$ is the gradient operator.
We list up some identities for $\Gamma$.
\begin{lem}\label{lem: calculus}
Let $u,v, v_1,v_2\in C(V)$.
\begin{enumerate}
\item $\langle u, \Delta v\rangle = -\sum_{V}\Gamma(u,v)d.$
\item For any $x\in V$, we have
\begin{align*}
\Gamma(u,v_1v_2)(x) = \Gamma(u, v_1)&v_2(x) + \Gamma(u, v_2)v_1(x) \\
&+\frac{1}{2d}\sum_{y\sim x}(\nabla_{xy}u)(\nabla_{xy} v_1)(\nabla_{xy}v_2).
\end{align*}
In particular,
\[
\sum_{V}\Gamma(u, v_1v_2) =\sum_{V} (\Gamma(u, v_1)v_2 + \Gamma(u, v_2)v_1).
\]
\end{enumerate}
\end{lem}
Making use of the min-max formula and appropriate trial functions, we have the following lemma.
\begin{lem}\label{CYineq}
Let $k\geq 1$ be an integer.
For any function $h\in C(V)$, we have
\[
\frac12\sum_{i=0}^{k}(\lambda_{k+1}-\lambda_i)^2\Phi_i(h)\leq \sum_{i=0}^{k}(\lambda_{k+1}-\lambda_i)\|2\Gamma(h,u_i) + u_i\Delta h\|^2,
\]
where $\Phi_i(h)=\sum_{x\sim y}u_i(x)u_i(y)(\nabla_{xy}h)^2$.
\end{lem}
\begin{proof}
Let $h\in C(V)$. For $i=0,\dots,k$, define $\varphi_i\in C(V)$ as the orthogonal projection of $hu_i$ to the subspace spanned by $\{u_{k+1},\dots,u_{N-1}\}$, i.e.,
\[
\varphi_i:=hu_i-\sum_{j=0}^ka_{ij}u_j, \, a_{ij}:=\langle hu_i,u_j\rangle.
\]
Clearly the function $\varphi_i$ is perpendicular to $u_0,\dots,u_{k}$. The min-max formula yields
\begin{equation}\label{minmax}
\lambda_{k+1}\|\varphi_i\|^2 \leq \frac12\sum_{x\sim y}(\nabla_{xy}\varphi_i)^2 = \sum_{V}\Gamma(\varphi_i,\varphi_i)d.
\end{equation}
From (1) in Lemma \ref{lem: calculus} and the fact that $\langle\varphi_i,u_j\rangle=0$ for $j=0,\dots,k$, we have
\begin{align*}
\sum_V\Gamma(\varphi_i,\varphi_i)d&= -\langle \varphi_i,\Delta \varphi_i\rangle \\
&=-\langle\varphi_i, 2\Gamma(h,u_i)+u_i\Delta h-\lambda_iu_ih+\sum_{j=0}^ka_{ij}\lambda_ju_j\rangle \\
&=-\langle\varphi_i, 2\Gamma(h,u_i)+u_i\Delta h-\lambda_iu_ih\rangle\\
&= -\langle\varphi_i,2\Gamma(h,u_i)+u_i\Delta h\rangle + \lambda_i \|\varphi_i\|^2.
\end{align*}
From \eqref{minmax}, we obtain
\begin{equation}\label{first}
(\lambda_{k+1}-\lambda_i)\|\varphi_i\|^2 \leq -\langle\varphi_i,2\Gamma(h,u_i)+u_i\Delta h\rangle.
\end{equation}
Let $A_i$ be the right hand side of (\ref{first}).
We estimate $A_i$ in two ways.
First, we claim that
\begin{equation}\label{second}
A_i = \frac12\sum_{x\sim y}u_i(x)u_i(y)(\nabla_{xy}h)^2 + \sum_{j=0}^{k} (\lambda_i-\lambda_j)a_{ij}^2.
\end{equation}
To see \eqref{second}, we use Lemma \ref{lem: calculus}. By the definition of $\varphi_i$,
\begin{align*}
A_i = \sum_{j=0}^{k}a_{ij}\langle u_j, u_i\Delta h +2\Gamma(h, u_i)\rangle - d\sum_{V}(hu_i^2\Delta h +2hu_i\Gamma(h,u_i)).
\end{align*}
The first term is equal to $\sum_{j=0}^k(\lambda_i-\lambda_j)a_{ij}^2$. Indeed, by the definition of $\Gamma(h,u_i)$ and Lemma \ref{lem: calculus}, we have
\begin{align}\label{bij}
\nonumber \langle u_j, u_i\Delta h +2\Gamma(h, u_i)\rangle &= \langle u_j, \Delta(hu_i) + \lambda_ihu_i \rangle\\
&= \nonumber \lambda_ia_{ij} - \langle \lambda_ju_j,hu_i\rangle \\
&= (\lambda_i-\lambda_j)a_{ij}.
\end{align}
The second term is equal to $\sum_{x\sim y}u_i(x)u_i(y)(\nabla_{xy}h)^2/2$. Indeed,
\begin{align*}
-\langle hu_i^2, \Delta h\rangle &= \sum_V\Gamma(hu_i^2, h)d \\
&= \sum_{V}(h\Gamma(u_i^2,h) + u_i^2 \Gamma(h,h))d \\
&= \frac{1}{2}\sum_{x\sim y}\left((\nabla_{xy}u_i)^2h(x)(\nabla_{xy}h)+u_i(x)^2(\nabla_{xy}h)^2\right) \\
&+ \sum_{V}2hu_i\Gamma(h,u_i)d \\
&= \frac{1}{2}\sum_{x\sim y}u_i(x)u_i(y)(\nabla_{xy}h)^2+\sum_{V}2hu_i\Gamma(h,u_i)d.
\end{align*}

Second, we claim that
\begin{equation}\label{third}
(\lambda_{k+1}-\lambda_i)A_i\leq \|u_i\Delta h + 2\Gamma(u_i,h)\|^2 - \sum_{j=0}^{k}(\lambda_i-\lambda_j)^2a_{ij}^2.
\end{equation}
From the definition of $A_i$, we have
\[
A_i=-\langle \varphi_i, 2\Gamma(h,u_i)+u_i\Delta h -\sum_{j=0}^k(\lambda_i-\lambda_j)a_{ij}u_j\rangle.
\]
Applying the Cauchy-Schwartz inequality to the definition of $A_i$ and taking account into \eqref{first} and \eqref{bij}, we have
\[
(\lambda_{k+1}-\lambda_i)A_i^2 \leq A_i(\|2\Gamma(h,u_i)+u_i\Delta h\|^2-\sum_{j=0}^k(\lambda_i-\lambda_j)^2a_{ij}^2).
\]
From \eqref{second} and \eqref{third}, we obtain
\begin{align*}
&\frac12\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)^2\sum_{x\sim y}u_i(x)u_i(y)(\nabla_{xy}h)^2 + \sum_{i,j=0}^k(\lambda_{k+1}-\lambda_i)^2(\lambda_i-\lambda_j)a_{ij}^2\\
&\leq \sum_{i=0}^k(\lambda_{k+1}-\lambda_i)\|2\Gamma(h,u_i)+u_i\Delta h\|^2-\sum_{i,j=0}^k(\lambda_{k+1}-\lambda_i)(\lambda_i-\lambda_j)^2a_{ij}^2.
\end{align*}
Since $\sum_{i,j=0}^k(\lambda_{k+1}-\lambda_i)^2(\lambda_i-\lambda_j)a_{ij}^2=-\sum_{i,j=0}^k(\lambda_{k+1}-\lambda_i)(\lambda_i-\lambda_j)^2a_{ij}^2$, we complete the proof.
\end{proof}

\section{proof of main theorem}
In this section, we give a proof of Theorem 1.1. In order to complete the proof, we use some symmetries of eigenfunctions on a symmetric graph.
\subsection{Symmetries of eigenfunctions on a symmetric graph}
We derive some properties of  eigenfunctions on a symmetric graph.
In particular, Lemma \ref{lem: const2} is peculiar to symmetric graphs.
A graph $G=(V,E)$ is said to be \emph{symmetric} if for any two edges $(x,y), (x',y')\in E$, there exists an automorphism $\gamma$ of $G$ such that $x'=\gamma x$ and $y'=\gamma y$ hold.
We denote by $\aut (G)$ the group of automorphisms of $G$.
Note that symmetric graphs are vertex-transitive, i.e., $\aut (G)$ acts transitively on $V$, and thus regular.
We say that a vector subspace $W$ of $C(V)$ is \emph{invariant} if for any $u\in W$ and $\gamma\in \aut (G)$, $\gamma u\in W$, where $\gamma u$ is defined by $\gamma u(x):=u(\gamma x), x\in V$.
\begin{lem}\label{lem: const1}
Let $G=(V,E)$ be a vertex-transitive graph.
Let $W$ be an invariant vector subspace of $C(V)$ of dimension $m$ and let $\{ u_{\alpha} \}_{\alpha=1}^{m}$
be an orthonormal basis of $W$. Then, the function $\lvert u_1\rvert^2+\dots+\lvert u_m\rvert^2$ is constant
and its value is $m/d\#V$.
\end{lem}

\begin{proof}
Put $f(x):=\lvert u_1(x)\rvert^2+\dots+\lvert u_m(x)\rvert^2$. By the invariance of $W$, the family $\{\gamma u_{\alpha}\}_{\alpha=1}^m$ is also an orthonormal basis of $W$ for any $\gamma\in \aut (G)$.
For fixed $x\in V$, it is easy to see that the sum $\lvert u_1(x)\rvert^2+\dots+\lvert u_m(x)\rvert^2$ is independent of the choice of an orthonormal basis $\{u_{\alpha}\}$. Thus,
\[
f(\gamma x)=\sum_{\alpha=1}^m\lvert \gamma u_{\alpha}(x)\rvert^2 = \sum_{\alpha=1}^m\lvert u_{\alpha}(x)\rvert^2=f(x).
\]
The transitivity of the action of $\aut(G)$ yields that $f$ is constant.
Let $C$ be the value of $\lvert u_1(x)\rvert^2+\dots+\lvert u_m(x)\rvert^2$.
By multiplying $d$ and summing over $x\in V$, we have
\[
Cd\#V=\sum_{\alpha=1}^m\sum_{x\in V}\lvert u_{\alpha}(x)\rvert^2d=m.
\]
\end{proof}

\begin{lem}\label{lem: const2}
Let $G$ be a symmetric graph.
Let $\lambda$ be an eigenvalue of $\Delta$ and let $\{u_{\alpha}\}_{\alpha=1}^m$ be an orthonormal basis of $W_{\lambda}$. Then, the function $g(x,y):=\sum_{\alpha=1}^m\lvert\nabla_{xy}u_{\alpha}\rvert^2, x\sim y$, is constant
and its value is $m\lambda/\#E$.
\end{lem}

\begin{proof}
Since $W_{\lambda}$ is an invariant vector subspace of $C(V)$, the family $\{\gamma u_{\alpha}\}_{\alpha=1}^m$ is also an orthonormal basis of $W_{\lambda}$ for any $\gamma\in \aut(G)$.
Since the sum $\sum_{\alpha=1}^m\lvert\nabla_{xy}u_{\alpha}\rvert^2$ is independent of the choice of an orthonormal basis $\{u_{\alpha}\}$, we have
\[
g(\gamma x,\gamma y)=\sum_{\alpha=1}^m\lvert\nabla_{xy}(\gamma u_{\alpha})\rvert^2=\sum_{\alpha=1}^m\lvert\nabla_{xy}u_{\alpha}\rvert^2 = g(x,y).
\]
The symmetry of $G$ yields that $g$ is constant. Let $C'$ be the value of $g$. By summing over $x\sim y$, we have
\[
2C'\#E=\sum_{\alpha=1}^m\sum_{x\sim y}\lvert \nabla_{xy}u_{\alpha}\rvert^2 = 2\lambda m.
\]
\end{proof}

\begin{cor}\label{cor: const3}
Let $G$ be a symmetric graph.
Let $\lambda$ and $\{u_{\alpha}\}$ be as in Lemma \ref{lem: const2}.
Then, the function $f_3(x,y)=\sum_{\alpha=1}^mu_{\alpha}(x)\nabla_{xy}u_{\alpha}$ is constant and its value is
$-\lambda m/2\#E$.
\end{cor}

\begin{proof}
The constancy of $f_3$ immediately follows from Lemma \ref{lem: const1} and Lemma \ref{lem: const2}.
Let $C_3$ be the value of $f_3$. By summing over $x\sim y$, we have
\[
2C_3\#E=\sum_{\alpha=1}^m\sum_{x\sim y}u_{\alpha}(x)\nabla_{xy}u_{\alpha}.
\]
By interchanging $x$ and $y$,
\[
\sum_{x\sim y}u_{\alpha}(x)\nabla_{xy}u_{\alpha} = -\sum_{x\sim y} u_{\alpha}(y)\nabla_{xy}u_{\alpha}.
\]
Thus, we obtain
\[
2C_3\#E = \frac12 \sum_{\alpha=1}^m\sum_{x\sim y}(u_{\alpha}(x)-u_{\alpha}(y))\nabla_{xy}u_{\alpha} =-\lambda m.
\]
\end{proof}
\subsection{Proof of main theorem}
We prove Theorem \ref{main thm}, Theorem \ref{main thm 2} and Corollary \ref{main cor}.
First, we prove Theorem \ref{main thm}.
\begin{proof}[Proof of Theorem 1.1]
Let $\{u_{\alpha}\}_{\alpha}$ be an orthonormal basis of $E_{\mu}$. Then, we have
\begin{align}\label{term: lhs}
\nonumber \sum_{\alpha=1}^m\sum_{x\sim y}u_i(x)u_i(y)\lvert\nabla_{xy}u_{\alpha}\rvert^2 &=
\frac{\lambda m}{\#E}\sum_{x\sim y}u_i(x)u_i(y) \\
\nonumber&=\frac{\lambda m}{\#E}\sum_{x\in V}u_{i}(x)d\cdot \frac1d\sum_{y\sim x}u_i(y) \\
\nonumber&=\frac{\lambda m}{\#E}(1-\lambda_i)\sum_{x\in V}u_i(x)^2d \\
&=\frac{\lambda m}{\#E}(1-\lambda_i).
\end{align}
Next, we evaluate $\sum_{\alpha}\|2\Gamma (u_{\alpha},  u_i) + u_i\Delta u_{\alpha}\|^2$.
By Jensen's inequality, we have
\[
4\Gamma(u_i,u_{\alpha})(x)^2 = \left(\frac{1}{d}\sum_{y\sim x}(\nabla_{xy}u_i)(\nabla_{xy}u_{\alpha})\right)^2
\leq \frac{1}{d}\sum_{y\sim x}(\nabla_{xy}u_i)^2(\nabla_{xy}u_{\alpha})^2,
\]
which yields
\begin{equation}\label{term: rhs1}
4\sum_{\alpha=1}^m\sum_{x\in V}\Gamma(u_i,u_{\alpha})(x)^2d \leq \frac{2\lambda\lambda_i m}{\#E}.
\end{equation}
By Lemma \ref{lem: const1}, we have
\begin{equation}\label{term: rhs2}
\sum_{\alpha=1}^m\sum_{x\in V}(u_i(x)\Delta u_{\alpha}(x))^2d=\frac{\lambda^2 m}{2\#E}.
\end{equation}
By Lemma \ref{cor: const3},
\begin{align}\label{term: rhs3}
\nonumber -4\lambda\sum_{\alpha=1}^m\sum_{x\in V}u_i(x)u_{\alpha}(x)\Gamma(u_{\alpha},u_i)(x)d &=
\frac{\lambda^2 m}{\#E}\sum_{x\in V}u_i(x)\sum_{y\sim x}\nabla_{xy}u_i \\
&= -\frac{\lambda^2\lambda_im}{\#E}.
\end{align}
By letting $h=u_{\alpha}$ in Lemma \ref{CYineq}, summing over $\alpha=1,\dots,m$ and taking account into \eqref{term: lhs}, \eqref{term: rhs1}, \eqref{term: rhs2} and \eqref{term: rhs3}, we obtain
\[
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)^2(1-\lambda_i)\leq
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)(2(2-\lambda)\lambda_i+\lambda).
\]
\end{proof}
In order to prove Theorem \ref{main thm 2}, we need some lemmas.
\begin{lem}[Chebyshev's sum inequality]\label{Chebyineq}
Let $N\geq 1$ be an integer and $\{a_i\}_{i=1}^{N}, \{b_i\}_{i=1}^{N}$ two sequences of real numbers.
If both of $\{a_i\}_{i=1}^{N}, \{b_i\}_{i=1}^{N}$ are non-increasing, then
\[
\frac1N\sum_{i=1}^Na_ib_i \geq \left(\frac1N\sum_{i=1}^Na_i\right)\left(\frac1N\sum_{i=1}^Nb_i\right).
\]
\end{lem}

\begin{lem}\label{lem: admatrix}
For any $0\leq k\leq N-1$,
\[
\sum_{i=0}^k(1-\lambda_i)\geq 0
\]
and the equality holds if and only if $k=N-1$.
\end{lem}

\begin{proof}
Let $A$ be the adjacency matrix of $G$ and $\nu_0\geq \nu_1\geq \dots\geq \nu_{N-1}$ be all eigenvalues of $A$.
Since any diagonal entry of $A$ is $0$, $\sum_{i=0}^{N-1}\nu_i$ is also $0$ and $\sum_{i=0}^k\nu_i\geq 0$ for any $k$,
with the equality holds if and only if $k=N-1$.
By the relation between $\Delta$ and $A$, we have
\[
\sum_{i=0}^k(1-\lambda_i) = \frac1d\sum_{i=0}^k\nu_i \geq 0
\]
and equality holds if and only if $k=N-1$.
\end{proof}
Next, we prove Theorem \ref{main thm 2} and Corollary \ref{main cor}.
\begin{proof}[Proof of Theorem 1.2]
By letting $\lambda=\lambda_1$ in Theorem \ref{main thm}, we have
\[
\sum_{i=0}^k(\lambda_{k+1}-\lambda_i)(\lambda_i^2-(\lambda_{k+1}-2\lambda_1+5)\lambda_i +\lambda_{k+1}-\lambda)\leq 0.
\]
Clearly, $\lambda_{k+1}-\lambda_i$ is non-increasing in $i$.
Put $f(x):=x^2-(\lambda_{k+1}-2\lambda_1+5)x$. Then, the function $f$ is non-increasing in the interval $(-\infty,(\lambda_{k+1}-2\lambda_1+5)/2]$. From Lemma \ref{lem: upperbound}, $(\lambda_{k+1}-2\lambda_1+5)/2\geq 2$.
Since $0\leq \lambda_i \leq 2$, $\lambda_i^2-(\lambda_{k+1}-2\lambda_1+5)\lambda_i +\lambda_{k+1}-\lambda$ is non-increasing in $i$. We may use Lemma \ref{Chebyineq} and thus
\[
\left(\lambda_{k+1}-\sum_{i=0}^k\frac{\lambda_i}{k+1}\right)\left(\sum_{i=0}^k\frac{(1-\lambda_i)\lambda_{k+1}+\lambda_i^2 -(5-2\lambda_1)\lambda_i}{k+1}-\lambda_1\right)\leq 0.
\]
If $k\geq m(\lambda_1)$, then $\lambda_{k+1}-\sum_{i=0}^k\lambda_i/(k+1)$ is strictly positive. In this case, we have
\[
\frac{1}{k+1}\sum_{i=0}^k((1-\lambda_i)\lambda_{k+1}+\lambda_i^2 -(5-2\lambda_1)\lambda_i-\lambda_1)\leq 0.
\]
By Lemma \ref{lem: admatrix}, we obtain
\[
\lambda_{k+1}\leq \frac{(k+1)\lambda_1+\sum_{i=1}^k((5-2\lambda_1)\lambda_i-\lambda_i^2)}{\sum_{i=0}^k(1-\lambda_i)}.
\]
This inequality also holds for $k<m(\lambda_1)$.
\end{proof}
\begin{proof}[Proof of Corollary 1.3]
If $k=m(\lambda_1)$, then $\lambda_{k+1}=\mu_2$ and $\lambda_1=\cdots=\lambda_{k-1}=\mu_1$. By Theorem \ref{main thm 2},  we have
\[
\frac{\mu_2}{\mu_1}\leq \frac{6m+1-3m\mu_1}{m+1-m\mu_1}.
\]
Let $g(x):=(6m+1-3mx)/(m+1-mx)$. The function $g$ is increasing. By Lemma \ref{lem: upperbound},
\[
\frac{\mu_2}{\mu_1}\leq g(1)=3m+1.
\]
\end{proof}

\section{On the non-triviality of Corollary 1.3}
In this section, we consider symmetric graphs, other than complete graphs.
Let $\mu_1$ and $\mu_2$ be the first and the second smallest positive eigenvalue, respectively.
If $(3m+1)\mu_1$ is not less than $2$, then the inequality in Corollary \ref{main cor} is trivial since $\mu_2\leq 2$ always holds.
In this section, we see that there exist infinitely many graphs such that $(3m+1)\mu_1$ is strictly less than $2$.

Let $C_{N}$, $N\geq 3$, be the cycle graph with $N$ vertices.
Cycle graphs are symmetric.
The spectra of cycle graphs are well-known.
\begin{lem}
The smallest positive eigenvalue of the normalized Laplace operator associated with $C_{N}$ is
$1-\cos (2\pi/N)$ and its multiplicity is $2$.
\end{lem}
Since $1-\cos (2\pi/N)$ is decreasing in $N$ and tends to $0$ as $N\to\infty$,
there exists a number $N_0$ such that $(3m+1)\mu_1=7(1-\cos (2\pi/N))$ is strictly less than $2$ for any $N\geq N_0$. In fact, we can take $N_0=9$.
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\end{document}


text:
\begin{document}
\maketitle

\begin{abstract}
Steven Romans Logarithmic Binomial Formula analogue has been found
and is presented here also for the case of  fibonomial
coefficients - which recently have been given a combinatorial
interpretation by the present author.
\end{abstract}
Mathematics Subject Classification :  11B37,  05A40, 81S99

\textit{to appear in} volume 9 (1),\textit{Advanced Studies in
Contemporary Mathematics}

\section{Introduction}
The aim of this note is to find out  - as in \cite{14,15} of
Steven Roman - the form of "Fib-corresponding" Logarithmic
Fib-Binomial Formula. In \cite{14,15} Steven Roman introduced The
Logarithmic Binomial Formula :
$$\lambda_{n}^{(t)}(x+a)=\sum_{k\geq 0}\left[ {n \atop k}\right]\lambda^{(t)}_{n-k}(a)x^{k};
\;\;\;t=0,1;\;\;\;|x|<a;\;\;\;n \in {\bf Z}$$
where
$$[n]=\left\{{n\;\;\;\;n\neq 0 \atop 1\;\;\;\;n=0}\right.$$
 and the  Roman factorial is given by
 \begin{equation}\label{1.1}
 [n]!=\left\{{n!\;\;\;\;\;\;\;\;\;\;n\geq 0 \atop
 \frac{(-1)^{n+1}}{(-n-1)!}\;\;\;n<0}\right.
 \end{equation}
while hybrid binomial coefficients \cite{10}  (Roman coefficients
\cite{11}) read:
\begin{equation}\label{1.2}
\left[{n \atop k}\right]=\frac{[n]!}{[k]![n-k]!}
\end{equation}
 One may show that  (Propositions 3.2 , 4.1, 4,2,
4.3  in \cite{11} )
\begin{equation}\label{1.3}
\left[{n \atop k}\right]=\left\{ \begin{array}{lll}
\binom{n}{k}&&n,k\geq 0\\
\\
(-1)^{k}\binom{-n-1+k}{k}&&k\geq 0 \geq n\\
\\
(-1)^{k+n}\binom{-k-1}{n-k}&&0>n\geq k\\
\\
(-1)^{(n+k)}\left[\Delta^{n}\frac{1}{x-k}\right]_{k=0}&&k>n\geq 0
\end{array}\right.
\end{equation}
\textrm{}\\
$$\left[{0 \atop k}\right]=\left[ {0 \atop -k}\right]=\frac{(-1)^{k+1}}{k}$$
\textrm{}\\
\begin{equation}\label{1.4}
\left[{n \atop k}\right]=\left[ {n \atop
n-k}\right],\;\;\;\;\;\;\; \left[{n \atop j}\right]\cdot \left[ {j
\atop k}\right]=\left[{n \atop k}\right]\cdot \left[ {n-j \atop
j-k}\right]
\end{equation}
\textrm{}\\
\begin{equation}\label{1.5}
\left[{n \atop k}\right]=\left[ {n-1 \atop k-1}\right]+\left[{n-1
\atop k}\right]
\end{equation}
\textrm{}\\
 As seen from the above  the hybrid binomial
coefficients (Roman coefficients) are the intrinsic natural
extension  of binomial coefficients .

 The Logarithmic Binomial Formula  extends the notion of binomiality of polynomials as  used
in the Generalized Umbral Calculus  (see Chapter 6 in \cite{16}
for functional formulation and see \cite{8} for  abundant
references on Finite Operator Calculus of Rota formulation).

The great invention of Steven Roman - among others - relies on the
fact that the real i.e. $R$-linear span $L$  of  the basis
functions (harmonic logarithms-see: Proposition 4.1 in \cite{14} )
$$L=span\left| \left\{ \lambda_{n}^{(t)}\right\}_{n\in {\bf Z},\;t=0,1}\right|$$
allows the Fundamental Theorem of Calculus to hold on  $L$ i.e.
$D^{-1}D = DD^{-1} = id_{L}$. Here $D^{-1}$  depending on whether
$t=0$ or $t=1$ acts as follows:
$$D^{-1}=\int_{0}^{x}\;\;\;on\;\;\;\lambda_{n}^{(0)}\;\;\;n\neq -1,
\;\;n\in {\bf Z}\;\;\;and\;\; gives\;\;0\;\;for\;\;n=-1$$
\textrm{}\\
$$and\;\;\;D^{-1}=\int_{1}^{x}\;\;\;\;on\;\;\;\lambda_{n}^{(1)};\;\;\;n\in {\bf Z}$$

\section{Fibonomial Coefficients}
In \cite{5}  Fibonomial coefficients \cite{12,2,3,4} have been
given a combinatorial interpretation as counting the number of
finite "birth-selfsimilar" subposets of an infinite poset. We
shall  use here the following notation: {\em Fibonomial
coefficients} are defined as
$\binom{n}{k}_{F}=\frac{F_{n}!}{F_{k}!F_{n-k}!}$ or  - usefully
for our purpose here $\binom{n}{k}_{F}\equiv
\frac{n^{\underline{k}}_{F}}{k_{F}!}$ where we make an analogy
driven \cite{8} identifications: ($n>0$), $n_{F}\equiv F_{n}\neq
0$,$n_{F}!\equiv n_{F}(n-1)_{F}(n-2)_{F}(n-3)_{F}\ldots
2_{F}1_{F};\;\;0_{F}!=1$;\\
$n_{F}^{\underline{k}}=n_{F}(n-1)_{F}\ldots (n-k+1)_{F}$. This is
the appropriate specification of notation from \cite{8} for the
purpose Fibonomial Finite Operator Calculus case investigation
(see Example 2.1 in \cite{9}).

Let us now introduce an infinite poset $P$ (for further details
see: \cite{5}) via its finite part subposet $P_{m}$ Hasse diagram
to be continued ad infinitum in an obvious way as seen from the
figure below.  It looks like the Fibonacci tree with a specific
"cobweb": see Figure 1. One sees that the  $P_{m}$  is the
subposet of $P$ consisting of points up to $m$-th level points
$$\bigcup_{s=1}^{m}\Phi_{s}\;\; ;\;\Phi_{s} \;\;is\;\; the\;
\;set\;\;of\;\;elements\;\;of\;\;the\;\;s-th\;\;level$$

How many $P_{m}$`s  \underline{rooted at the $k$-th} level might
be found ?

 We answer this question in the following sequence of
observations right after Figure 1.

\begin{center}

\includegraphics[width=75mm]{Fig1.eps}

\vspace{2mm}

\noindent {\small Fig.~1. The construction of the  Fibonacci
"cobweb"  poset} \end{center}
 \begin{obs}
 The number of maximal chains starting from the root  (level  $F_{1}$ )
  to reach any point at the $n$-th level  labeled by  $F_{n}$  is equal
  to $n_{F}!$
  \end{obs}
 \begin{obs}
 The number of maximal chains starting from the level labeled by  $F_{k}$ to
reach any point at the $n$-th level labeled by  $F_{n}$  is equal
to $n_{F}^{\underline{m}},\;\;(n = k+m )$.
\end{obs}
\begin{obs}
Let $n = k+m$. The number of subposets $P_{m}$ rooted at the level
labeled by $F_{k}$ and ending at the $n$-th level labeled by
$F_{n}$ is equal to
$$ \binom{n}{m}_{F}=\binom{n}{k}_{F}=\frac{n^{\underline{k}}_{F}}{k_{F}!}.$$
\end{obs}

\section{The Logarithmic Fibonomial Case}
We shall now adopt the $\ast_{\psi}$  product  formalism \cite{9}
(see also Appendix  in  \cite{6}  and \cite{7}) to the Fibonomial
case with
$\exp_{F}\left\{x\right\}=\sum_{k=0}^{\infty}\frac{x^{k}}{k_{F}!}$
defining the $F$-exponential series.

\subsection{$\ast_{F}$ product}
Let   $n >0$ and  let $\partial_{F}$ be a linear operator acting
on formal series  and defined accordingly by
$\partial_{F}x^{n}=n_{F}x^{n-1};\;\;n\geq
0,\;\;\partial_{F}x^{0}=0$ .

 We shall call the $F$-multiplication the new  $\ast_{F}$
 product of functions or formal series  specified below.
 \begin{nt}\label{not1}
 $x\ast_{F}x^{n}=\hat{x}_{F}(x^{n})=\frac{(n+1)}{(n+1)_{F}}x^{n+1};\;\;n\geq
 0$ hence $x\ast_{F}1=x$ and $x\ast_{F}\alpha 1=\alpha 1\ast_{F}x=
 x\ast_{F}\alpha =\alpha \ast_{F}x=\alpha x;\;\;\;\forall x,\alpha \in {\bf
 R},\;\;f(x)\ast_{F}x^{n}=f(\hat{x}_{F})x^{n}$.

 For $k\neq n$ $x^{n}\ast_{F}x^{k}\neq x^{k}\ast_{F}x^{n}$ as well
 as $x^{n}\ast_{F}x^{k}\neq x^{n+k}$ - in general.
 \end{nt}
 \begin{defn}
  With  Notation \ref{not1} adopted   define  the  $\ast_{F}$
powers of  $x$  according to
$$x^{n^{\ast}_{F}}\equiv x\ast_{F}x^{(n-1)^{\ast}_{F}}=
\hat{x}\left(
x^{(n-1)^{\ast}_{F}}\right)=x\ast_{F}x\ast_{F}\ldots\ast_{F}x=\frac{n!}{n_{F}!}x^{n};\;\;\;n\geq
0.$$
\end{defn}
Note that
$x^{n^{\ast}_{F}}\ast_{F}x^{k^{\ast}_{F}}=\frac{n!}{n_{F}!}x^{(n+k)^{\ast}_{F}}
\neq
x^{k^{\ast}_{F}}\ast_{F}x^{n^{\ast}_{F}}=\frac{k!}{k_{F}!}x^{(n+k)^{\ast}_{F}}$
for $k\neq n$ and $x^{0^{\ast}_{F}}=1$.

This noncommutative $F$-product $\ast_{F}$ is devised so as to
ensure the observations below.
\begin{obs}\label{obs3.1} \textrm{}\\
\renewcommand{\labelenumi}{(\alph{enumi})}
\begin{enumerate}
\item $\partial_{F}x^{n^{\ast}_{F}}=nx^{(n-1)^{\ast}_{F}};\;\;
n\geq0$;
 \item $\exp_{F}[\alpha x]\equiv \exp\left\{
\alpha\hat{x}_{F}\right\}{\bf 1};$
 \item $\exp[\alpha
x]\ast_{F}\left\{\exp_{F}\left\{\beta\hat{x}_{F}\right\}1\right\}=
\exp_{F}\left\{[\alpha+\beta]\hat{x}_{F}\right\}{\bf 1}$;
 \item $\partial_{F}(x^{k}\ast_{F}x^{n^{\ast}_{F}})=
(Dx^{k})\ast_{F}x^{n^{\ast}_{F}}+x^{k}\ast_{F}(\partial_{F}x^{n^{\ast}_{F}})$;
\item Leibniz rule\;\;\;\;\;
$\partial_{F}(f\ast_{F}g)=(Df)\ast_{F}g+f\ast_{F}(\partial_{F}g)$;
$f,g$- formal series; \item
$f(\hat{x}_{F})g(\hat{x}_{F})1=f(x)\ast_{F}\tilde{g};\;\;
\tilde{g}(x)=g(\hat{x}_{F}){\bf 1}$.
\end{enumerate}
\end{obs}
\subsection{$F$-Integration}
 Let :  $\partial_{0}x^{n}=x^{n-1}$. The linear operator $\partial_{0}$ is
 identical with  divided difference operator. Let $\hat{Q}f(x)=f(qx)$.
 Recall  that to the Jackson $\partial_{q}$  derivative \cite{8}  there corresponds
 the $q$-integration  which is a right inverse operation to "$q$-difference-ization".
 Namely \cite{8}
\begin{equation}\label{3.2}
F(z):=\left(
\int_{q}\varphi\right)(z):=(1-q)z\sum_{k=0}^{\infty}\varphi\left(
q^{k}z\right)q^{k}
\end{equation}
\begin{equation}\label{3.3}
F(z)\equiv\left(\int_{q}\varphi\right)(z)=(1-q)z\left(
\sum_{k=0}^{\infty}q^{k}\hat{Q}^{k}\varphi\right)(z)=\left((1-q)\hat{z}\frac{1}{1-q\hat{Q}}\varphi\right)(z)
\end{equation}
where $(\hat{z}\varphi)(z)=z\varphi(z)$.

 Of course
\begin{equation}\label{3.4}
\partial_{q}\circ \int_{q}=id
\end{equation}
  as
\begin{equation}\label{3.5}
\frac{1-q\hat{Q}}{1-q}\partial_{0}\left(
(1-q)\hat{z}\frac{1}{1-q\hat{Q}}\right)=id
\end{equation}
Naturally (\ref{3.5}) might serve to define a right inverse to
Jackson's "$q$-difference-ization"
$\left(\partial_{q}\varphi\right)(x)=\frac{1-q\hat{Q}}{1-q}\partial_{0}\varphi(x)$
and consequently the "$q$-integration " as represented by
(\ref{3.2}) and (\ref{3.3}. As it is well known  the definite
$q$-integral is an numerical approximation of  the definite
integral obtained in the $q\longrightarrow 1$ limit.

Finally we introduce the analogous representation  for
$\partial_{F}$ difference-ization
\begin{equation}\label{3.6}
\partial_{F}=\hat{n}_{F}\partial_{0};\;\;\;\hat{n}_{F}x^{n-1}=n_{F}x^{n-1};\;\;\;n\geq
1.
\end{equation}
Then
\begin{equation}\label{3.7}
\int_{F}x^{n}=\left(
\hat{x}\frac{1}{\hat{n}_{F}}\right)x^{n}=\frac{1}{(n+1)_{F}}x^{n+1};\;\;\;n\geq
0
\end{equation}
 and of course
\begin{equation}\label{3.8}
\partial_{F}\circ \int_{F}=id.
\end{equation}
  Naturally  ($\int_{F}\equiv \int d_{F}$)
  $$\partial_{F}\int_{a}^{x}f(t)d_{F}t=f(x).$$
The formula of   "per partes" $F$-integration  is easily
obtainable from  Observation (\ref{obs3.1}) and it  reads:
\begin{equation}\label{3.9}
\int_{a}^{x}(f\ast_{F}\partial_{F}g)(t)d_{F}t=\left[
(f\ast_{F}g)(t)\right]^{x}_{a}-\int_{a}^{x}(Df\ast_{F}g)(t)d_{F}t.
\end{equation}

Now in order to have  $\partial_{F}^{-1}$- an  $F$-analogue  of
$D^{-1}$ as in \cite{13,14}( thus causing the fundamental Theorem
of Calculus to hold for $\partial_{F}$-difference-ization and
$\int_{F}$- integration on some linear space $L_{F}$ being the
linear span of "$F$-{\em harmonic logarithms}" )  - we shall
proceed exactly as Steven Roman  in \cite{14,15}.

\section{The Logarithmic Fib-Binomial Formula}
As in  \cite{14,15} of  Roman  -  we have also  {\em The
Logarithmic Fib-Binomial Formula} (see: Propositions
\ref{prop4.1}, \ref{prop4.2} below):
$$\phi_{n}^{(t)}(x+_{F}a)\equiv \left[ \exp\left\{ a\partial_{F}\right\} \phi_{n}^{(t)}\right](x)
=\sum_{k \geq 0}\left[{n \atop
k}\right]_{F}\phi_{n-k}^{(t)}(a)x^{k}\;\;\;t=0,1;\;|x|<a;\;n\in
{\bf Z}$$
 where ( more on "$+_{F}$ " see \cite{8,9})
 $$[n_{F}]=\left\{\begin{array}{lll}
 n_{F}&&n\neq 0\\
 1&&n=0\end{array}\right.$$
 and the  Roman Fib-factorial is given by
\begin{equation}\label{4.1}
[n_{F}]!=\left\{\begin{array}{lll} n_{F}!&&n\geq 0\\
\frac{(-1)^{n+1}}{(-n-1)_{F}!}&&n<0\end{array}\right.
\end{equation}
while {\em Fib-hybrid binomial coefficients} or Roman
Fib-coefficients (see: \cite{10,11}) read:
\begin{equation}\label{4.2}
\left[ {n \atop k}\right]_{F}=\frac{[n]!}{[k]![n-k]!}
\end{equation}
One observes   (as in  Propositions 3.2 , 4.1, 4,2, 4.3  in
\cite{11} ) that:
\begin{equation}\label{4.3}
\left[ {n \atop k}\right]_{F}=\left\{\begin{array}{lll}
\binom{n}{k}_{F}&&n,k\geq 0\\
\\
(-1)^{k}\binom{-n-1+k}{k}_{F}&&k\geq 0>n\\
\\
(-1)^{k+n}\binom{-k-1}{n-k}_{F}&&0>n\geq k\\
\\
(-1)^{(n+k)}\left[\Delta_{F}^{n}\frac{1}{x-k}\right]_{k=0}&&k>n\geq0
\end{array}\right.
\end{equation}
\textrm{}\\
$$\left[{0 \atop k}\right]_{F}=\left[ {0 \atop -k}\right]_{F}=\frac{(-1)^{k+1}}{k_{F}}$$
\textrm{}\\
\begin{equation}\label{4.4}
\left[{n \atop k}\right]_{F}=\left[ {n \atop
n-k}\right]_{F},\;\;\;\;\;\;\; \left[{n \atop j}\right]_{F}\cdot
\left[ {j \atop k}\right]_{F}=\left[{n \atop k}\right]_{F}\cdot
\left[ {n-j \atop j-k}\right]_{F}
\end{equation}
\textrm{}\\
\begin{equation}\label{4.5}
\left[{n \atop k}\right]_{F}=\left[ {n-1 \atop
k-1}\right]_{F}+\left[{n-1 \atop k}\right]_{F}
\end{equation}
\textrm{}\\
where (see: pp.333-334 in  \cite{8} )
$$\Delta_{F}=\exp_{F}\left\{ \partial_{F}\right\}-id.$$

Fib-Roman coefficients (as  seen  from the above) are then also
natural "relative" of binomial coefficients among the family of
$\psi$ - binomial ones  \cite{8}  (consult also Example 2.1 in
\cite{9}).

 The Logarithmic Fib-Binomial Formula  extends the
notion of binomiality of \underline{polynomials} as  used in the
Generalized Umbral Calculus (see Chapter 6 in \cite{16} for
functional formulation and see \cite{8} for  abundant  references
on Finite Operator Calculus of Rota formulation)- to sequences of
\underline{functions} - (compare with \cite{1}).

Here the importance of the great invention of Steven Roman - among
others - relies on the fact that the $R$-linear span $L_{F}$ of
now basis {\em Fib-harmonic logarithms} functions
$$\left\{ \phi_{n}^{(t)}\right\}_{n\in {\bf Z},\;t=0,1},\;\;\;\;\;L_{F}
=span\left| \left\{ \phi_{n}^{(t)}\right\}_{n\in {\bf
Z},\;t=0,1}\right|$$ allows the Fundamental Theorem of Calculus to
hold also on $L_{F}$, i.e.\\
$\partial_{F}^{-1}\partial_{F}=id_{L_{F}}$ for $\partial_{F}$-
difference-ization and  $\int_{F}$ - integration acting on \\a
linear space $L_{F}$ being the linear span of "{\em Fib-harmonic
logarithms}". Here anti-difference-ization operator
$\partial_{F}^{-1}$ - depending on whether $t = 0$  or  $t = 1$\\
- acts as follows on  {\em Fib-harmonic logarithm} functions :
$$\partial_{F}^{-1}=\int_{0}^{x}d_{F}\;\;\;on\;\;\;\phi_{n}^{(0)}\;\;\;n\neq -1,
\;\;n\in {\bf Z}\;\;\;and\;\; gives\;\;0\;\;for\;\;n=-1$$
$$and\;\;\;\partial_{F}^{-1}=\int_{1}^{x}d_{F}\;\;\;\;on\;\;\;\phi_{n}^{(1)};\;\;\;n\in {\bf Z}$$
 Let us define these {\em Fib-harmonic logarithms}
 $$\left\{ \phi_{n}^{(t)}\right\}_{n\in {\bf Z},\;t=0,1}$$
-(see  Proposition 2.2 in \cite{14}) - as solutions of {\em
Fib-harmonic} $t$-binomiality conditions. Thus  {\em Fib-harmonic
logarithm \underline{functions}} are unique solutions of {\em
Fib-harmonic} $t$-binomiality conditions; $t = 0,1$  (\ref{4.6}) -
(compare with  \cite{1} and relaxation Lemma 2.12 therein):
\begin{equation}\label{4.6}
\begin{array}{l}
1)\;\;
\phi_{0}^{(0)}(x)=1,\quad\quad\;\;\;2)\;\;\phi_{n}^{(0)}(0)=0,\;\;n\ni
{\bf Z}\backslash\{ 0\},\\
\\3)\;\;\partial_{F}\phi_{n}^{(0)}=[n_{F}]\phi_{n-1}^{(0)},\;\;\;n\in
{\bf Z}\\ \\ \\
1)\;\;\phi_{0}^{(1)}(x)=\ln x,\quad
\quad2)\;\;\phi_{n}^{(1)}(x)\;has\; no\;constant\;term,\;\;n\in
{\bf Z},\\
\\3)\;\;\partial_{F}\phi_{n}^{(1)}=[n_{F}]\phi_{n-1}^{(1)},\;\;\;n\in
{\bf Z}
\end{array}
\end{equation}
The {\em Fib-harmonic} $t$-binomiality conditions;  $t = 0,1$
(\ref{4.6}) yield \cite{14}  what follows:
\begin{prop}\label{prop4.1}
\begin{multline*}
\phi_{n}^{(0)}(x)=\left\{\begin{array}{lll}
x^{n}&&n\geq 0\\
0&&n<0\end{array}\right.,\;\;\;
\phi_{n}^{(1)}(x)=\left\{\begin{array}{lll} x^{n}(\ln
x-f_{n})&&n\geq 0\\
x^{n}&&n<0\end{array}\right. , \\
\;\;f_{0}=0,\;\;f_{n}=1+\frac{1}{2_{F}}+\frac{1}{3_{F}}+\ldots+\frac{1}{n_{F}},\;\;n\in
{\bf N}
\end{multline*}
\end{prop}
We shall call $f_{n},\;n\in {\bf N}$ the {\em
\underline{Fib-harmonic numbers}} ($f_{0}=0$), (see: \cite{13}).
\begin{prop}\label{prop4.2}
The linear  anti-difference-ization unique operator\\
$\partial_{F}^{-1}:L_{F}\longrightarrow L_{F}$;
$\partial_{F}^{-1}\partial_{F}=id_{L_{F}}$ is given by
\textrm{}\\
$$\partial_{F}^{-1}\phi_{n}^{(0)}=\left\{\begin{array}{lll}
\frac{1}{[n+1]_{F}}\phi_{n+1}^{(0)}&&n\neq -1\\
\\
0&&n=-1\end{array}\right.,\;\;\;\partial_{F}^{-1}\phi_{n}^{(1)}=\frac{1}{[n+1]_{F}}\phi_{n}^{(1)},\;\;n\in
{\bf Z}.$$
\textrm{}\\
\end{prop}
{\bf REMARK.} Instead of  Roman Fib-coefficients and  Roman
Fib-factorial  one may - (replace $F$ by $\psi$)- start to
consider Roman  $\psi$ -coefficients, $\psi$-{\em harmonic
logarithms} etc. However these seemingly might lack any
"reasonable" combinatorial interpretation.

 As the generally useful reading - also for this purpose one recommends here:\\
$[$LR$]$ Loeb D.,  Rota G-C. {\em "Recent Contributions to the
calculus of Finite Differences: a Survay"}  Lecture Notes in Pure
and Appl. Math. vol. \underline{132}(1991), pp. 239-276,  ArXiv:
math.co/ 9502210 V1 9 Feb 1995, see also:\\
http://arxiv.org/list/math.CO/9502


text:
{\bf Problem.} Evaluate $\left\lceil3\left(6-\frac12\right)\right\rceil$.
{\bf Level.} Level 3
{\bf Type.} Algebra
{\bf Solution.} Firstly, $3\left(6-\frac12\right)=18-1-\frac12=17-\frac12$.  Because $0\le\frac12<1$, we have $\left\lceil17-\frac12\right\rceil=\boxed{17}$.

set_name: MATH


TITLE: Four-indexed infinite matrix
QUESTION [0 upvotes]: I've found on an ancient book a structure of the kind described below and I do not even know if it is a commonly known structure nowadays. For me it doesn't match neither a direct sum nor a direct product.
Can someone tell me something about this? (available references will be appreciate)
The complex vector space of functions defined by the following base $\{\alpha_{ij}\}= \{ \eta_i \tau_j\}$ where $\{\eta_i\}$ and $\{\tau_j\}$ are respectively ortonormal bases of trigonometric polynomials. An element $\overrightarrow{v}$ of this space will be of the form: $\overrightarrow{v} = \sum_{i,j=1}^\infty \alpha_{ij}\eta_i\tau_j$. A linear operator (matrix) over this space will be an infinite dimensional matrix with four indexes which I'm not sure if has anything to do with tensors.

REPLY [1 votes]: If you require $\sum_{i,j} |\alpha_{ij}|^2 < \infty$, the vector space is the Hilbert space tensor product $\mathcal H = \mathcal H_\eta \otimes \mathcal H_\tau$ of the Hilbert space $\mathcal H_\eta$ generated by the
$\eta_i$ and $\mathcal H_\tau$ generated by the $\tau_j$.


text:
\chapter*{Introduction}

	Let $S$ be a Noetherian separated scheme of finite Krull dimension, and let
	$\stablehomotopy$ denote the motivic stable homotopy category of Morel
	and Voevodsky.  In order to get a motivic version of the Postnikov tower, Voevodsky \cite{MR1977582} constructs
	a filtered family of triangulated subcategories of $\stablehomotopy$:
		\begin{equation}
				\label{equation.introd.slicefiltration}
			\cdots \subseteq \stablehomotopyeffqplusone \subseteq
			\stablehomotopyeffq \subseteq \stablehomotopyeffqminusone \subseteq \cdots
		\end{equation}
	The work of Neeman \cite{MR1308405}, \cite{MR1812507}, shows that the inclusion:
		$$\xymatrix{i_{q}:\stablehomotopyeffq \ar@{^{(}->}[r]& \stablehomotopy}
		$$
	has a right adjoint $r_{q}$, and that the following functors are exact:
		$$\xymatrix{f_{q},s_{q}:\stablehomotopy \ar[r]& \stablehomotopy}
		$$
	where $f_{q}=i_{q}r_{q}$, and for every $T$-spectrum $X$, $s_{q}(X)$ fits in
	the following distinguished triangle:
		$$\xymatrix{f_{q+1}X\ar[r]& f_{q}X \ar[r]& s_{q}X \ar[r]& \Sigma _{T}^{1,0}f_{q+1}X}
		$$
	We say that $f_{q}(X)$ is the ($q-1$)-connective cover of $X$ and that $s_{q}(X)$ is the
	$q$-slice of $X$.

	Let $\motivicsymmTspectra$ be Jardine's model category of symmetric $T$-spectra
	\cite{MR1787949} and $\symmstablehomotopy$ its associated homotopy category.
	The symmetrization functor induces an equivalence of categories between $\stablehomotopy$ and $\symmstablehomotopy$
	(see \cite[theorem 4.31]{MR1787949}), we will denote
	by $f_{q}^{\Sigma}$, $s_{q}^{\Sigma}$ the functors on $\symmstablehomotopy$
	that correspond to $f_{q}$, $s_{q}$.

	We consider
	a cofibrant ring spectrum $A$ with unit in
	$\motivicsymmTspectra$, such that the natural map $f_{0}^{\Sigma}(A)\rightarrow A$
	is an isomorphism in $\symmstablehomotopy$.  Let
	$Y, Y', Z, Z'$ be arbitrary symmetric
	$T$-spectra, and $p, p', q, q' \in \mathbb Z$.
	Our main results (see theorems \ref{thm.3.5.colocal===>covers-slices-inherit-module-structures},
	\ref{thm.3.5.change-of-coefficients}, \ref{thm.3.5.q-slices=====HZ-modules}, \ref{thm.3.5.q-slices=====big-motives},
	\ref{thm.3.5.external-pairings}, \ref{thm.3.5.qslices-modules-over-zeroslice},
	and \ref{thm.3.5.pairings-Atiyah-Hirzebruch-ss}) are the following:
		\begin{enumerate}
			\item	For every $A$-module $M$ in $\motivicsymmTspectra$, its $q$-slice
						$s_{q}^{\Sigma}(M)$
						is again an $A$-module in $\motivicsymmTspectra$.
						(see lemma
						\ref{lemma.3.6.connecting-functors-with-model-structures}(\ref{lemma.3.6.connecting-functors-with-model-structures.a})
						and theorem \ref{thm.3.5.colocal===>covers-slices-inherit-module-structures}).  This means
						that the $q$-slice of every $A$-module inherits an $A$-module structure
						which is defined not just up to homotopy,
						but in a very strict sense.
			\item	If the unit map $u:\symmspherespectrum \rightarrow A$
						becomes an isomorphism after applying the zero slice functor $s_{0}^{\Sigma}$
						in $\symmstablehomotopy$,
						then $s_{q}^{\Sigma}(Y)$ has a natural structure of
						$A$-module in $\motivicsymmTspectra$
						(see lemma
						\ref{lemma.3.6.connecting-functors-with-model-structures}(\ref{lemma.3.6.connecting-functors-with-model-structures.d})
						and theorem \ref{thm.3.5.change-of-coefficients}), i.e. the $q$-slice of every symmetric $T$-spectrum
						is enriched with an $A$-module structure which is defined not just up to homotopy, but in
						a very strict sense.
			\item	As a consequence, if the base scheme $S$ is a perfect field,
						we are able to prove a conjecture of M. Levine (see \cite[corollary 11.1.3]{MR2365658}),
						which says that for every symmetric $T$-spectrum $X$,
						its $q$-slice $s_{q}^{\Sigma}(X)$ is naturally equipped with a module structure in $\motivicsymmTspectra$
						over the motivic Eilenberg-MacLane spectrum $H\mathbb Z$ (see theorem \ref{thm.3.5.q-slices=====HZ-modules}).
						Restricting the field even further to the case of characteristic zero, we get that
						all the slices $s_{q}^{\Sigma}(X)$ are big motives in the sense of Voevodsky
						(see theorem \ref{thm.3.5.q-slices=====big-motives}).
			\item	The smash product of symmetric $T$-spectra
						induces the following natural external pairing in $\symmstablehomotopy$
						(i.e. up to homotopy):
							$$\xymatrix{s_{p}^{\Sigma}(Y)\wedge s_{q}^{\Sigma}(Z)\ar[r]^-{\cup ^{s}_{p,q}}& s_{p+q}^{\Sigma}(Y\wedge Z)}
							$$
			\item	As a consequence, if $B$ is a ring in $\symmstablehomotopy$ (i.e. up to homotopy)
						and $N$ is a $B$-module
						in $\symmstablehomotopy$ (i.e. also up to homotopy) then:
							\begin{enumerate}
								\item	The zero slice of $B$, $s_{0}^{\Sigma}(B)$ is a ring spectrum in $\symmstablehomotopy$
											(i.e. up to homotopy).
								\item	The $q$-slice of $N$,
											$s_{q}^{\Sigma}(N)$ is a module in $\symmstablehomotopy$ (i.e up to homotopy)
											over $s_{0}^{\Sigma}(B)$.
								\item	The direct sum of all the slices of $B$, $s^{\Sigma}(B)=\oplus _{n\in \mathbb Z}s_{n}^{\Sigma}(B)$
											is a graded ring spectrum in $\symmstablehomotopy$
											(i.e. up to homotopy).
								\item	The direct sum of all the slices of $N$, $s^{\Sigma}(N)=\oplus _{n\in \mathbb Z}s_{n}^{\Sigma}(N)$
											is a graded module in $\symmstablehomotopy$ (i.e. up to homotopy)
											over $s^{\Sigma}(B)$.
							\end{enumerate}
			\item	The smash product of symmetric $T$-spectra induces natural external pairings in the motivic
						Atiyah-Hirzebruch spectral sequence generated by the slice filtration
						(see definition \ref{def.3.5.Atiyah-Hirzebruch-spectral-sequence}):
							$$\xymatrix{E_{r}^{p,q}(Y;Z)\otimes E_{r}^{p',q'}(Y';Z') \ar[r]& E_{r}^{p+p',q+q'}(Y\wedge Y';Z\wedge Z')}
							$$
		\end{enumerate}

	To prove the results mentioned above, we need to carry out a very detailed analysis of the
	multiplicative properties (with respect to the smash product of spectra) of the filtration
	(\ref{equation.introd.slicefiltration}) considered above.  It turns out that the natural framework to do this, is provided by
	Jardine's category of motivic symmetric $T$-spectra \cite{MR1787949}.
	Our approach consists basically of three steps:
		\begin{enumerate}
			\item First, we lift Voevodsky's slice filtration to the usual category of $T$-spectra equipped
						with the Morel-Voevodsky motivic stable model structure (see section \ref{section.3.2.modstrslicefilt}).
			\item	Then, using the Quillen equivalence given by the symmetrization and forgetful functors
						\cite{MR1787949}, we are able to
						promote the previous lifting to the category of symmetric $T$-spectra (see section \ref{section.3.3.symmmodstrslicefilt}).
			\item	Finally, we describe the multiplicative properties of the slice filtration using
						the symmetric monoidal structure given by the smash product of symmetric $T$-spectra
						(see sections \ref{section.3.4.multiplicativeproperties-slicefiltration}
						and \ref{section.3.5furthermultiplicativeproperties}).
		\end{enumerate}

	We use Hirschhorn's approach to localization of model categories for
	the construction of the lifting of the slice filtration to the model category setting.
	In order to apply Hirschhorn's techniques, it is necessary to show that the Morel-Voevodsky motivic
	stable model structure is cellular; for this we rely on Hovey's general approach to spectra
	\cite{MR1860878} and on an unpublished result of Hirschhorn (see theorem \ref{thm2.2.injmod=>cellular}).
	For the description of the multiplicative properties of the slice filtration in the model category
	setting, we use Hovey's results on symmetric monoidal model categories \cite{MR1650134}*{chapter 4}.

	We now give an outline of this thesis.  In chapter \ref{chapter1}, we just recall some standard results
	about Quillen model categories.  The reader who is familiar with the terminology of model categories
	may skip this chapter.

	In chapter \ref{chapter-mot-hom-theory}, we review the definitions of the Morel-Voevodsky stable model structure
	for simplicial presheaves and Jardine's stable model structure for symmetric $T$-spectra.  We also show that
	these two model structures are cellular, therefore it is possible to apply Hirschhorn's technology to construct
	Bousfield localizations.
	In section \ref{section.2.8Modules-Algebras}
	we recall the construction of the model structures for the categories
	of $A$-modules and $A$-algebras,
	where $A$ denotes a cofibrant ring spectrum
	with unit in Jardine's motivic symmetric stable model category.
	We verify that the category of $A$-modules equipped with this model structure also
	satisfies Hirschhorn's cellularity condition.
	The reader who is familiar with these model structures may either skip this chapter
	or simply look at
	sections \ref{section.cellularity-injective},
	\ref{section-cellularity-motivic-stable}, \ref{section-cellularity-motivic-symmetric-stable} and
	\ref{section.2.8Modules-Algebras} where we prove that the cellularity condition holds.

	Finally in chapter \ref{chap-slice-filtration}, we carry out the program sketched above.
	In section \ref{section-slice-filtration}, we review Voevodsky's construction for the slice
	filtration in the setting of simplicial presheaves.  In section \ref{section.3.2.modstrslicefilt},
	we apply Hirschhorn's localizations techniques to the Morel-Voevodsky stable model structure
	in order to construct
	three families of model structures, namely
	$\qconnectedTspectra$, $\weightqTspectra$ and $\qsliceTspectra$ ($q\in \mathbb Z$).  The first
	family, $\qconnectedTspectra$ is constructed by a right Bousfield localization
	with respect to the Morel-Voevodsky stable model structure
	(see theorem \ref{thm.3.2.connective-model-structure}), and it provides a lifting
	of Voevodsky's slice filtration to the model category level
	(see theorem \ref{thm.3.2.liftingslicefilt-modelcats}).  Moreover, this family has the property
	that the cofibrant replacement functor $C_{q}$ provides an alternative description
	for the functor $f_{q}$ (($q-1$)-connective cover) defined above (see theorem \ref{thm.3.2.Rq-models-f<q}).
	In order to get a lifting for the slice functors $s_{q}$ to the model category level,
	we need to introduce the model structures $\weightqTspectra$ and $\qsliceTspectra$.
	The model category $\weightqTspectra$ is defined as a left Bousfield localization with respect to the Morel-Voevodsky
	stable model structure
	(see theorem \ref{thm.3.2.Lqmodelstructures}); its main property is that its fibrant replacement functor $W_{q}$ gives
	an alternative description for the cone of the natural map $f_{q}X\rightarrow X$
	(see theorems \ref{thm.3.1.motivictower} and \ref{thm.3.2.Lq-models-s<q}).  On the other hand, the model structure
	$\qsliceTspectra$ is constructed using right Bousfield localization with respect to the model category
	$\weightqplusoneTspectra$ (see theorem \ref{thm.3.2.Sq-modelcats}), and it gives the desired lifting
	for the slice functor $s_{q}$ to the
	model category level (see theorem \ref{thm.3.2.Sq-models-sq}).

	In section \ref{section.3.3.symmmodstrslicefilt}, we promote the model structures defined above
	(section \ref{section.3.2.modstrslicefilt}) to the setting of symmetric $T$-spectra.
	In this case, Hirschhorn's localization technology applied to Jardine's stable model structure
	for symmetric $T$-spectra allows us to introduce three families of model structures
	which we denote by $\qconnectedsymmTspectra$,
	$\weightqsymmTspectra$ and $\qslicesymmTspectra$; where the underlying category
	is given by symmetric $T$-spectra
	(see theorems \ref{thm.3.3.symmetricconnective-model-structure},
	\ref{thm.3.3.Lqsymmetricmodelstructures} and \ref{thm.3.3.symmetricSq-modelcats}).  Using the
	Quillen equivalence \cite{MR1787949}  given by the symmetrization and the forgetful functors,
	we are then able to show
	that these new families of model structures are also Quillen equivalent to the ones introduced
	in section \ref{section.3.2.modstrslicefilt}
	(see theorems \ref{thm.3.3.symmetrization-qconnected-Quillen-equiv}, \ref{thm.3.3.symmetrization-Quillen-equivalence-weight<q}
	and \ref{thm.3.3.symmetrization-qslice-Quillen-equiv}).
	Therefore, these model structures give liftings for the functors $f_{q}$ and $s_{q}$
	to the model category level
	(see corollary \ref{cor.3.3.symmetric-fq,s<q,sq}, and theorems \ref{thm.3.3.symmRq-models-symmf<q}(\ref{thm.3.3.symmRq-models-symmf<q.c}),
	\ref{thm.3.3.symmSq-models-symm-sq}(\ref{thm.3.3.symmSq-models-symm-sq.c})),
	with the great technical advantage that the underlying categories
	are now symmetric monoidal.  Hence, we have a natural framework for the study of the multiplicative properties
	of Voevodsky's slice filtration.

	In section \ref{section.3.4.multiplicativeproperties-slicefiltration}, we show that
	the smash product of symmetric $T$-spectra
		$$\xymatrix{\pconnectedsymmTspectra \times \qconnectedsymmTspectra \ar[r]^-{-\wedge -}& \pplusqconnectedsymmTspectra}
		$$

		$$\xymatrix{\pslicesymmTspectra \times \qslicesymmTspectra \ar[r]^-{-\wedge -}& \pplusqslicesymmTspectra}
		$$
	is in both cases a Quillen bifunctor in the sense of Hovey
	(see theorems \ref{thm.3.4.smash-Quillenbifunctor-RpxRq----->Rqplusp}
	and \ref{thm.3.4.smash-Quillenbifunctor-SpxSq----->Sqplusp}).

	In section \ref{section.3.5furthermultiplicativeproperties}, we will promote
	(using the free $A$-module functor $A\wedge -$) the model structures
	constructed in section \ref{section.3.3.symmmodstrslicefilt} to the category of $A$-modules,
	where $A$ is a cofibrant ring spectrum with unit in $\motivicsymmTspectra$.  We will denote these
	new model structures by $\qconnectedAmod$, $\weightqAmod$ and $\qsliceAmod$.  These
	new model structures will be used as an analogue of the slice filtration for
	the motivic stable homotopy category of $A$-modules, as well as a tool to describe
	the behavior of the slice functors $s_{q}^{\Sigma}$ when they are restricted to the category
	of $A$-modules.
	We will see that if one imposes some natural additonal conditions on the ring spectrum $A$, then
	the free $A$-module functor $A\wedge -$ induces a strict compatibility
	between the slice filtration in the categories of symmetric $T$-spectra and $A$-modules
	(see theorems \ref{thm.3.5.inheritingmodulestructures1}, \ref{thm.3.5.inheritmodelstructures2},
	\ref{thm.3.5.inheritmodelstructures-slices} and \ref{thm.3.5.sliceinvarianceofcoefficients}).

	In section \ref{section.3.5.applications}, we will rely in all the previous results
	to show that
	if we have a cofibrant ring spectrum $A$ with unit in $\motivicsymmTspectra$
	which also satisfies some additional hypothesis, then for every $q\in \mathbb Z$
	and for every $A$-module $M$ in $\motivicsymmTspectra$
	(see theorem \ref{thm.3.5.colocal===>covers-slices-inherit-module-structures}):
		\begin{enumerate}
			\item $f_{q}^{\Sigma}(M)$ is again an $A$-module in $\motivicsymmTspectra$ (not just up to homotopy,
						but in a very strict sense).
			\item $s_{q}^{\Sigma}(M)$ is again an $A$-module in $\motivicsymmTspectra$ (not just up to homotopy,
						but in a very strict sense).
		\end{enumerate}
	Furthermore, if the unit map $u:\symmspherespectrum \rightarrow A$ fulfills some mild conditions,
	then the free $A$-module functor $A\wedge -$ induces for every symmetric
	$T$-spectrum $X$ (see theorem \ref{thm.3.5.change-of-coefficients}),
	a natural structure of $A$-module in $\motivicsymmTspectra$ (i.e. not just up to homotopy, but in a very strict sense)
	on its $q$-slice $s_{q}^{\Sigma}(X)$.

	Finally, we will be able to prove a conjecture of M. Levine (see \cite[corollary 11.1.3]{MR2365658}),
	which says that if the base scheme $S$ is a perfect field, then for every $q\in \mathbb Z$ and for every symmetric
	$T$-spectrum $X$, its $q$-slice $s_{q}^{\Sigma}(X)$ is naturally equipped with a module structure in $\motivicsymmTspectra$
	over the motivic Eilenberg-MacLane spectrum $H\mathbb Z$ (see theorem \ref{thm.3.5.q-slices=====HZ-modules}).
	If we restrict the field even further, considering a field of characteristic zero, then as a consequence we will prove that
	all the slices $s_{q}^{\Sigma}(X)$ are big motives in the sense of Voevodsky
	(see theorem \ref{thm.3.5.q-slices=====big-motives}).

	We will also show that for every $p,q\in \mathbb Z$, the smash product of symmetric $T$-spectra induces
	up to homotopy natural pairings
	(see theorem \ref{thm.3.5.external-pairings}):
		$$\xymatrix{f_{p}^{\Sigma}(X)\wedge f_{q}^{\Sigma}(Y)\ar[r]^-{\cup ^{c}_{p,q}}& f_{p+q}^{\Sigma}(X\wedge Y)}
		$$

		$$\xymatrix{s_{p}^{\Sigma}(X)\wedge s_{q}^{\Sigma}(Y)\ar[r]^-{\cup ^{s}_{p,q}}& s_{p+q}^{\Sigma}(X\wedge Y)}
		$$
	As a consequence, if $A$ is a ring spectrum in $\symmstablehomotopy$ (i.e. up to homotopy)
	and $M$ is an $A$-module in $\symmstablehomotopy$, then
	(see theorem \ref{thm.3.5.qslices-modules-over-zeroslice}):
		\begin{enumerate}
			\item  The ($-1$)-connective cover of $A$, $f_{0}^{\Sigma}(A)$ is a ring spectrum (up to homotopy)
							in $\symmstablehomotopy$.
			\item	 For every $q\in \mathbb Z$, the ($q-1$)-connective cover of $M$, $f_{q}^{\Sigma}(M)$ is a module
							(up to homotopy) over $f_{0}^{\Sigma}(A)$.
			\item	 The direct sum of all the connective covers of $A$, $f^{\Sigma}(A)=\oplus _{n\in \mathbb Z}f_{n}^{\Sigma}(A)$
						 is a graded ring (up to homotopy) in $\symmstablehomotopy$.
			\item	 The direct sum of all the connective covers of $M$, $f^{\Sigma}(M)=\oplus _{n\in \mathbb Z}f_{n}^{\Sigma}(M)$
						 is a graded module (up to homotopy) over $f^{\Sigma}(A)$.
			\item  The zero slice of $A$, $s_{0}^{\Sigma}(A)$ is a ring spectrum (up to homotopy) in $\symmstablehomotopy$.
			\item	 For every $q\in \mathbb Z$, the $q$-slice of $M$, $s_{q}^{\Sigma}(M)$ is a module
							(up to homotopy) over $s_{0}^{\Sigma}(A)$.
			\item  The direct sum of all the slices of $A$, $s^{\Sigma}(A)=\oplus _{n\in \mathbb Z}s_{n}^{\Sigma}(A)$
					  	is a graded ring (up to homotopy) in $\symmstablehomotopy$.
			\item	The direct sum of all the slices of $M$, $s^{\Sigma}(M)=\oplus _{n\in \mathbb Z}s_{n}^{\Sigma}(M)$
						is a graded module (up to homotopy) over $s^{\Sigma}(A)$.
		\end{enumerate}
	We will also see that the smash product of symmetric $T$-spectra induces
	(via the external pairings $\cup ^{c}$ and $\cup ^{s}$)
	natural external pairings in the motivic Atiyah-Hirzebruch spectral sequence
	(see definition \ref{def.3.5.Atiyah-Hirzebruch-spectral-sequence} and
	theorem \ref{thm.3.5.pairings-Atiyah-Hirzebruch-ss}):
		$$\xymatrix@R=0.5pt{E_{r}^{p,q}(Y;X)\otimes E_{r}^{p',q'}(Y';X') \ar[r]& E_{r}^{p+p',q+q'}(Y\wedge Y';X\wedge X')\\
									     (\alpha, \beta) \ar@{|->}[r]& \alpha \smile \beta }
		$$

\section*{Acknowledgements}

	I would like to thank my advisor Marc Levine for his generosity in sharing
	his infinite knowledge, and for his unconditional support.  I would also like to
	thank the Mathematics Department at the University of Essen, especially
	professors H\'{e}l\`{e}ne Esnault and Eckart Viehweg
	for their hospitality and their financial support during the year in which
	this thesis was prepared.

	I would like to thank the Mathematics Department
	at Northeastern University for the financial support
	during my graduate studies.  Finally, I would like to thank
	my friends Shih-Wei Yang and Shouxin Dai for all their help and encouragement.