text:
\begin{document}

\title{Jackson-type inequalities and widths of functional classes
  in the Musielak-Orlicz type spaces\thanks{This work was supported in part  by the Kyrgyz-Turkish Manas
University (Bishkek / Kyrgyz Republic),  project No.~KTM\"{U}-BAP-2019.FBE.02, the Ministry of Education and Science of Ukraine in the framework of the fundamental research No.~0118U003390  and the Volkswagen Foundation (VolkswagenStiftung), program ``From Modeling and Analysis to Approximation''.}}

\author{F.\,G.~Abdullayev, S.\,O.~Chaichenko, M.~Imashkyzy, A.\,L.~Shidlich}

\date{}

\maketitle

\begin{abstract}

In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$,   exact Jackson-type inequalities are obtained in terms of
best approximations of functions and  the averaged values of their generalized moduli of smoothness. The  values of
Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

\end{abstract}

\keywords{Kolmogorov width, Bernstein width, best approximation, module of smoothness, Jackson-type inequality, Musielak-Orlicz spaces.}

\subjclass{\, 41A17	\and 42A32}

\section{Introduction}

Let ${\bf M}=\{M_k(t)\}_{k\in {\mathbb Z}}$, $t\ge 0$, be a sequence of Orlicz functions. In other words, for every $k\in {\mathbb Z}$, the
function $M_k(t)$ is a nondecreasing convex function for which $M_k(0)=0$ and $M_k(t)\to \infty$ as $t\to \infty$.
The modular space (or  Musilak-Orlicz  type space) ${\mathcal S}_{\bf M}$  is the space of $2\pi$-periodic  complex-valued Lebe\-sgue summable functions $f$ ($f\in L$)   such that
the following quantity (which is also called the Luxemburg norm of $f$) is finite:
\begin{equation}\label{def_Lux_norm}
    \|{f}\|_{_{\scriptstyle  {\bf M}}}:=
    \|\{\widehat{f}(k)\}_{k\in {\mathbb Z}}\|_{_{\scriptstyle l_{\bf M}({\mathbb Z})}}:=
    \inf\bigg\{a>0:\  \sum\limits_{k\in\mathbb Z}  M_k(|{\widehat{f}(k)}|/{a})\le 1\bigg\},
\end{equation}
where $\widehat{f}(k):={[f]}\widehat{\ \ }(k)={(2\pi)^{-1}}\int_0^{2\pi}f(x) \mathrm{e}^{- \mathrm{i}kx}\mathrm{d}x$,
$k\in\mathbb Z$, are the Fourier coefficients of   the function
$f$.

 The spaces ${\mathcal S}_{\bf M}$ defined in this way are Banach spaces.  Functional  spaces of this type have been studied by mathematicians since the 1930s (see, for example, the monographs
  \cite{Lindenstrauss-1977}, \cite{Musielak-1983}, \cite{Rao_Ren_2002}). If all functions $M_k$ are identical (namely, $M_k(t)\equiv M(t)$,
 $k\in {\mathbb Z}$), the spaces ${\mathcal S}_{\bf M}$ coincide with the ordinary Orlicz type spaces ${\mathcal S}_{M}$
 \cite{Chaichenko_Shidlich_Abdullayev_2019}. If $M_k(t)=\mu_k t^{p_k}$,  $p_k\ge 1$, $\mu_k\ge 0 $, then ${\mathcal S}_{\bf M}$ coincide with the weighted spaces
${\mathcal S}_{_{\scriptstyle  \mathbf p,\,\mu}}$ with variable exponents \cite{Abdullayev_Chaichenko_Imash_kyzy_Shidlich_2020}.
If all $M_k(t)=t^p$, $p\ge 1$, then the spaces ${\mathcal S}_{\bf M}$ are the  known
spaces ${\mathcal S}^{p}$ (see, for example, \cite[Ch.~11]{Stepanets_M2005}),
which in the case $p=2$ coincide with ordinary Lebesgue spaces ${\mathcal S}^{2}=L_2$.

In the paper, we study the approximative properties of the spaces ${\mathcal S}_{\bf M}$. In particular,
 exact Jackson-type inequalities in ${\mathcal S}_{\bf M}$ are obtained in terms of
the best approximations of functions and the averaged values of their generalized moduli of smoothness. The  values of
Kolmogorov, Bernstein, linear, and projective widths in the spaces ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.

Jackson-type (or Jackson-Stechkin-type) inequalities are inequalities  that estimate  the values of the best approximations
of functions via the value  of their modulus of continuity (smoothness) at a certain point.
The first exact Jackson-type inequality for the best uniform approximations of $2\pi$-periodic continuous functions  by trigonometric polynomials was obtained by Korneichuk \cite{Korneichuk_1962} in 1962. A similar result for the best uniform approximations of continuous functions given on the real axis by entire functions of the exponential type
was obtained by Dzyadyk in \cite{Dzyadyk_1975}. In 1967,   Chernykh \cite{Chernykh_1967_1, Chernykh_1967_2} proved two unimprovable inequalities for  $2\pi$-periodic functions from the Lebesgue spaces $L_2$. In \cite{Chernykh_1967_2},  it was shown in particular that the averaged values of the moduli of smoothness can be more effective for characterizing the structural and approximative properties of the functions $f$ than the moduli themselves. In  \cite{Taikov_1976, Taikov_1979}  (see also \cite[Ch. 4]{Pinkus_1985}), Taikov originated systematic investigations of the problem of exact inequalities that estimate  the values of the best approximations of functions via the averaged values of their  moduli of smoothness. He first considered the
functional classes  of  $2\pi$-periodic  functions  defined by certain conditions on the averaged values of their moduli of
smoothness and found the exact values of the widths of such classes  in the spaces  $L_2$.
Later, similar topic was studied by  numerous mathematicians in various  functional  spaces  (see, for example,
\cite{Stepanets_Serdyuk_2002, Vakarchuk_2004, Israflov_Guven_2006, Akgun_Israflov_2011, Akgun_Kokilashvili_2011, Sharapudinov_2013, Babenko_Konareva_2019, Akgun_2019, Abdullayev_Serdyuk_Shidlich_2021}, etc.).
For more detailed information on the results obtained in this direction and the corresponding references, see also \cite{Shabozov_Yusupov_2011, Vakarchuk_2012, Vakarchuk_2016, Babenko_Konareva_2019}.

\section{Preliminaries }

\subsection{Orlicz  norm}

In addition  to the Luxemburg norm (\ref{def_Lux_norm}) of the space $
{\mathcal S}_{\bf M}$, consider  the
Orlicz norm that is defined as follows. Let
 ${\bf {M^*}}=\{{M^*_k}(v)\}_{k\in {\mathbb Z}}$, $v\ge 0$, be the sequence of  functions defined by the relations
 \[

    {M^*_k}(v):=\sup\{uv-M_k(u): ~u\ge 0\}, \quad k \in \mathbb{Z}.
 \]

Consider the set $\Lambda{=}\Lambda({\bf {M^*}})$ of
sequences of positive numbers $\lambda=\{\lambda_k\}_{k\in \mathbb{Z}}$
such that  $\sum_{k\in \mathbb{Z}}{M^*_k}(\lambda_k){\le} 1$. For any function  $f\in {\mathcal S}_{\bf M}$, define its Orlicz norm by the equality
\begin{equation} \label{def-Orlicz-norm}
    \|f\|^\ast_{_{\scriptstyle  {\bf M}}}:=\|\{\widehat{f}(k)\}_{k\in {\mathbb Z}}\|_{_{\scriptstyle l_{\bf M}^*({\mathbb Z})}}:= \sup \Big\{ \sum\limits_{k \in \mathbb{Z}}
    \lambda_k|\widehat{f}(k) |: \quad  \lambda\in \Lambda({\bf {M^*}})\Big\}.
\end{equation}

Further, we will mainly use the Orlicz norm for functions $f \in {\mathcal S}_{\bf M}$. However, taking into account the following Lemma \ref{Lemma_3}, some corollaries can also be formulated from the results obtained when considering the Luxemburg norm.

\begin{lemma}
      \label{Lemma_3}
      For any function $f \in {\mathcal S}_{\bf M}$, the following relation holds:
      \begin{equation} \label{estim-for-norms}
      \| f\|_{_{\scriptstyle  {\bf M}}} \le \| f\|^*_{_{\scriptstyle  {\bf M}}}\le 2 \,\| f\|_{_{\scriptstyle  {\bf M}}}.
      \end{equation}
\end{lemma}

Relation (\ref{estim-for-norms}) follows from the similarly relation for corresponding norms in the modular Orlicz sequence  spaces (see, for example \cite[Ch. 4]{Lindenstrauss-1977}).

\subsection{Generalized moduli of smoothness and their averaged values}

Let $\omega_\alpha(f,\delta)$ be the modulus of smoothness of a function  $f \in {\mathcal S}_{\bf M}$ of order
$\alpha>0$, i.e.,
\begin{equation}\label{usual_modulus}
    \omega_\alpha(f,t)_{_{\scriptstyle  {\bf M}}}^*:=
    \sup\limits_{|h|\le t}\|\Delta_h^\alpha f\|_{_{\scriptstyle  {\bf M}}}^*=
     \sup\limits_{|h|\le t} \Big\|\sum\limits_{j=0}^\infty (-1)^j {\alpha \choose j} f(\cdot-jh)
     \Big\|_{_{\scriptstyle  {\bf M}}}^*,
\end{equation}
where ${\alpha \choose j}=\frac {\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-j+1)}{j!}$ for $j \in \mathbb{N}$ and ${\alpha \choose j}=1$ for $j=0$. By the definition, for any ${k}\in {\mathbb Z}$, we have
\begin{equation}\label{modulus_difference_Fourier_Coeff}
|{[\Delta_h^\alpha f]}\widehat {\ \ }(k)|=|1-\mathrm{e}^{-\mathrm{i}kh}|^\alpha |\widehat{f}(k)|
=2^\frac \alpha 2 (1-\cos{kh})^\frac \alpha2 |\widehat{f}(k)|.
\end{equation}

Consider the set $\Phi$ of all continuous bounded nonnegative
pair functions $\varphi$ such that
$\varphi(0)=0$ and the Lebesgue measure of the set $\{t\in {\mathbb R}:\,\varphi(t)=0\}$ is equal to zero. For a fixed function $\varphi\in \Phi$, $h\in {\mathbb R}$ and for any $f \in {\mathcal S}_{\bf M}$, we denote by
$\{{[\Delta_h^\varphi f]}\widehat {\ \ }(k)\}_{k\in {\mathbb Z}}$ the sequence of numbers such that  for any $k\in {\mathbb Z}$
  \begin{equation}\label{modulus_generalize difference_Fourier_Coeff}
  {[\Delta_h^\varphi f]}\widehat {\ \ }(k)  =\varphi(kh)  \widehat{f}(k).
\end{equation}

If there exists a function $\Delta_h^\varphi f\in L$ whose Fourier coefficients coincide with the numbers
${[\Delta_h^\varphi f]}\widehat {\ \ }(k)$, $k\in {\mathbb Z}$, then, as above,
the expression  $\|\Delta_h^\varphi f\|^*_{_{\scriptstyle  {\bf M}}}$ denotes  the  Orlicz norm of the function
 $\Delta_h^\varphi f$.

 If such a function does not exist, then
 the  notation   $\|\Delta_h^\varphi f\|^*_{_{\scriptstyle  {\bf M}}}$ denotes the  norm  $\|\cdot\|_{_{\scriptstyle l_{\bf M}^*({\mathbb Z})}}$ of the sequence $\{{[\Delta_h^\varphi f]}\widehat {\ \ }(k)\}_{k\in {\mathbb Z}}$.

Similary to  \cite{Shapiro_1968}, \cite{Boman_Shapiro_1971}, \cite{Boman_1980},
define  the generalized  modulus of smoothness   of a function $f \in {\mathcal S}_{\bf M}$  by the equality
\begin{equation}\label{general_modulus}
    \omega_\varphi(f,t)_{_{\scriptstyle  {\bf M}}}^*=\sup\limits_{|h|\le t} \|\Delta_h^\varphi f\|_{_{\scriptstyle  {\bf M}}}^*.
 \end{equation}
It follows from  (\ref{modulus_difference_Fourier_Coeff}) that $\omega_\alpha(f,t)_{_{\scriptstyle  {\bf M}}}^*\!=\omega_\varphi(f,t)_{_{\scriptstyle  {\bf M}}}^*$  when
$\varphi(t)\!=\varphi_\alpha(t)=\!2^\frac \alpha 2 (1-\cos{kt})^\frac \alpha2$\!. In the general case, such modules were considered, in particular, in  \cite{Vasil'ev_2001}, \cite{Kozko_Rozhdestvenskii_2004},
\cite{Vakarchuk_2016}, \cite{Babenko_Konareva_2019}, etc.

Further, let  ${\mathcal M}(\tau )$, $\tau>0$, be the set of all functions $\mu  $, bounded
non-decreasing and non-constant on the segment   $[0, \tau]$. By $\Omega _\varphi(f, \tau, \mu  , u)_{_{\scriptstyle  {\bf M}}}^* $, $u>0$,   denote the average value of the generalized modulus of smoothness
$\omega _\varphi(f, t)_{_{\scriptstyle  {\bf M}}}^*$ of the function  $f$ with the weight $\mu   \in {\mathcal M}(\tau )$, that is,
 \begin{equation}\label{Mean_Value_Gen_Modulus}
  \Omega _\varphi(f, \tau, \mu  , u)_{_{\scriptstyle  {\bf M}}}^*  := \frac
   {1}{\mu    (\tau ) - \mu    (0)}\int
   _0^u\omega _\varphi(f, t)_{_{\scriptstyle  {\bf M}}}^*  {\mathrm d}\mu    \Big(\frac {\tau
   t}{u}\Big).
 \end{equation}

Note that for any $f\in {\mathcal S}_{\bf M},$ $\tau >0,$ $\mu
\in {\mathcal M}(\tau )$ and  $u>0$ the functionals $\Omega _\varphi(f, \tau, \mu  , u)_{_{\scriptstyle  {\bf M}}}^* $
do not exceed the value $\omega _\varphi(f,u)_{_{\scriptstyle  {\bf M}}}^*,$ and therefore in a number of questions
they can be more effective for characterizing the structural and approximative properties of the function $f$.

\subsection{Definition of $\psi$-integrals, $\psi$-derivatives   and functional classes}

Let  $\psi=\{\psi(k)\}_{k\in {\mathbb Z}}$ be an arbitrary sequence of complex numbers.
If for a given function $f\in L$ with the Fourier series
 $\sum
 _{k\in {\mathbb Z}}\widehat {f}(k){\mathrm e}^{{\mathrm i}k
 x}
 $
 the series
 $
 \sum
 _{k\in {\mathbb Z}}\psi(k)\widehat {f}(k){\mathrm e}^{{\mathrm i}k
 x}
 $
 is the Fourier series of  a certain function  $F\in L$, then   $F$ is called  (see, for example, \cite[Ch.~11]{Stepanets_M2005})
 $\psi$-integral of the function  $f$ and is denoted as $F={\cal J}^{\psi }(f, \cdot)$. In turn,
 the function $f$ is called the $\psi$-derivative of the function $F$ and is denoted as  $f=F^{\psi}$.
 The Fourier coefficients of functions $f$ and $f^{\psi }$ are related by the equalities
\begin{equation} \label{Fourier_Coeff_der}
 \widehat f(k)=\psi (k)\widehat f^{\psi }(k), \quad k \in {\mathbb Z}.
  \end{equation}
The set of  $\psi$-integrals of
functions  $f$ of $L$ is denoted as  $L^{\psi }$. If
 ${\mathfrak {N}}\subset L$, then  $L^{\psi }{\mathfrak {N}}$ denotes the set of $\psi$-integrals of
 functions $f\in {\mathfrak {N}}$. In particular,    $L^{\psi }{\mathcal S}_{\bf M}$ is the set
 of $\psi$-integrals of    functions $f\in {\mathcal S}_{\bf M}$.

For arbitrary fixed $\varphi\in \Phi$, $\tau>0$ and $\mu  \in  M(\tau)$, define the following functional classes:
  \begin{equation} \label{L^psi(varphi,n)}
  L^{\psi }(\varphi,\tau ,\mu   , n)_{_{\scriptstyle  {\bf M}}}^*:=
  \Big\{f\in L^{\psi }{\mathcal S}_{\bf M}:\quad
  \Omega_\varphi\Big(f^{\psi }, \tau, \mu   ,\frac{\tau }n\Big)_{_{\scriptstyle  {\bf M}}}^* \le 1, \quad n \in
  {\mathbb{N}}\Big\},
  \end{equation}
   \begin{equation} \label{L^psi(varphi,Phi)}
   L^{\psi }(\varphi, \tau, \mu   , \Omega )_{_{\scriptstyle  {\bf M}}}^*  :=
    \Big\{f\in L^{\psi }{\mathcal S}_{\bf M}:\
    \Omega _\varphi(f^{\psi }, \tau , \mu   ,  u)_{_{\scriptstyle  {\bf M}}}^* \le \Omega  (u),\ 0\le u\le \tau \Big\},
  \end{equation}
where $\Omega  (u)$ is a fixed continuous monotonically increasing function of the variable   $u\ge 0$ such that $\Omega  (0)=0$.

Note that  in the Lebesgue spaces $L_2$,  for  $\psi (k)=k^{-r}$, $r\in \mathbb{N}$,
$\varphi(t)=2^\frac \alpha 2 (1-\cos{kt})^\frac \alpha2$, and the weight function
$\mu  (t)=t$, Taikov     \cite{Taikov_1976, Taikov_1979} (see also \cite[Ch. 4]{Pinkus_1985}) first considered the
functional classes  similar to (\ref{L^psi(varphi,n)}) and (\ref{L^psi(varphi,Phi)}).  He found
the exact values of the widths of such classes  in the spaces  $L_2$   in the case when the majorants $\Omega$
of the averaged values of  the moduli of smoothness satisfied some constraints. The problem of finding the exact values
of the widths
in different  spaces of functional classes of this kind
was also studied in   \cite{Yussef_1988, Shalaev_1991, Esmaganbetov_1999, Serdyuk_2003, Vakarchuk_2004, Vakarchuk_2016, Abdullayev_Serdyuk_Shidlich_2021}, etc.

\subsection{Best approximations and widths of functional classes}

Let   ${\mathscr T}_{2n+1}$, $n=0,1,\ldots$, be the set of all trigonometric polynomials
${T}_{n}(x) = \sum_{|k|\le n}  c_{k}\mathrm{e}^{\mathrm{i}kx}$ of the order  $n$, where $c_{ k}$
are arbitrary complex numbers.

For any function $f\in {\mathcal S}_{\bf M}$ denote by $E_n (f)_{_{\scriptstyle  {\bf M}}}^*$ its best approximation by the trigonometric polynomials   ${T}_{n-1}\in {\mathscr T}_{2n-1}$ in the space  ${\mathcal S}_{\bf M}$ with respect to the norm $\|\cdot\|_{_{\scriptstyle  {\bf M}}}^*$, i.e.,
 \begin{equation}\label{Best_Approx_Deff}
    E_n (f)_{_{\scriptstyle  {\bf M}}}^* :=
    \inf\limits_{{T}_{n-1}\in {\mathscr T}_{2n-1} }\|f-{T}_{n-1}\|_{_{\scriptstyle  {\bf M}}}^*
    .
 \end{equation}

From  relation (\ref{def-Orlicz-norm}),  it follows \cite[Lemma 2]{Abdullayev_Chaichenko_Shidlich_2021} that for any   $f \in {\mathcal S}_{\bf M}$ and all $n=0,1,\ldots$,
           \begin{equation} \label{Best_app_all}
           E_n (f)_{_{\scriptstyle  {\bf M}}}^* =\|f-{S}_{n-1}({f})\|_{_{\scriptstyle  {\bf M}}}^*
             = \sup \Big\{
           \sum _{|k |\ge  n}  \lambda_k|\widehat{f}(k) |: \quad  \lambda\in \Lambda \Big\},
           \end{equation}
where $S_{n-1}(f)=S_{n-1}(f,\cdot)= \sum _{|k|\le n-1}\widehat{f}(k) {\mathrm{e}^{\mathrm{i}k\cdot}}$ is the partial Fourier sum
of the order  $n-1$ of the function $f$.

Further, let  $K$ be a convex centrally symmetric subset of ${\mathcal S}_{\bf M}$ and
 let $B_{_{\scriptstyle  {\bf M}}}^*$ be a unit ball of the space  ${\mathcal S}_{\bf M}$ with respect to the norm $\|\cdot\|_{_{\scriptstyle  {\bf M}}}^*$. Let also  $F_N$ be an arbitrary $N$-dimensional subspace of the space ${\mathcal S}_{\bf M}$, $N\in {\mathbb N}$, and
$\mathscr{L}({\mathcal S}_{\bf M}, F_N)$ be a set of linear operators from ${\mathcal S}_{\bf M}$ to $F_N$.
 By    $\mathscr {P}({\mathcal S}_{\bf M}, F_N)$ denote the subset of projection operators of the set ${\mathscr{L}}({\mathcal S}_{\bf M}, F_N)$,  that is, the set of the operators  $A$ of linear projection onto the set
  $F_N$ such that $Af = f$ when $f\in F_N$.  The quantities
 \[
 b_N(K, {\mathcal S}_{\bf M})=\sup\limits _{F_{N+1}}\sup\{\varepsilon>0: \varepsilon B_{_{\scriptstyle  {\bf M}}}^*\cap F_{N+1}
 \subset K\},
 \]
 \[
 d_N(K, {\mathcal S}_{\bf M})=\inf\limits _{F_N}\sup \limits _{f\in K}
 \inf \limits _{u\in F_N}\|f - u \|_{_{\scriptstyle  {\bf M}}}^* ,
 \]
 \[
 \lambda _N(K,{\mathcal S}_{\bf M})=
 \inf \limits _{F_N}\inf \limits_{A\in {\mathscr {L}}({\mathcal S}_{\bf M}, F_N)}\sup \limits _{f\in K}
 \|f - Af\|_{_{\scriptstyle  {\bf M}}}^* ,
 \]
 \[
 \pi _N(K, {\mathcal S}_{\bf M})=\inf \limits_{F_N}\inf \limits _{A\in {\mathscr {P}}({\mathcal S}_{\bf M},F_N)}
 \sup \limits _{f\in K}\|f - Af\|_{_{\scriptstyle  {\bf M}}}^* ,
 \]
 are called Bernstein, Kolmogorov, linear, and projection $N$-widths of the set $K$ in the space ${\mathcal S}_{\bf M}$, respectively.

\section{Main results}

\subsection{Jackson-type inequalities}

In this subsection,  Jackson-type inequalities are obtained  in terms of the   best approximations and the  averaged values of generalized moduli of smoothness in the spaces ${\mathcal S}_{\bf M}$. To state these results,  denote by $\Psi$ the set of arbitrary
sequences $\psi=\{\psi (k)\}_{k\in {\mathbb Z}}$  of complex numbers such that  $|\psi (k)|=|\psi (-k)| \ge |\psi (k+1)|$ for $k\in {\mathbb N}$. Here and below, we also assume that the sequence ${\bf {M^*}}=\{{M^*_k}(v)\}_{k\in {\mathbb Z}}$  satisfies the condition \begin{equation}\label{M_condition}
M_k^*(v)>M_k^*(1)=1, \quad v>1,\ k\in {\mathbb Z}.
 \end{equation}

 \begin{theorem}
       \label{Th.3.1}
       Assume that $f\in L^{\psi }{\mathcal S}_{\bf M}$, condition $(\ref{M_condition})$ holds,    $\varphi\in \Phi$, $\tau>0$,  $\mu  \in  {\mathcal M}(\tau )$
       and $\psi\in \Psi$.         Then for any   $n\in {{\mathbb N}}$ the following inequality is true:
 \begin{equation}\label{Jackson_Type_Ineq}
    E_n(f)_{_{\scriptstyle  {\bf M}}}^* \le
     \frac {\mu   (\tau ) - \mu   (0)}
    {I_{n,\varphi}(\tau ,\mu   )}  |\psi  (n)|\,
    \Omega _\varphi\Big(f^{\psi}, \tau,\mu   , \frac{\tau }n\Big)_{_{\scriptstyle  {\bf M}}}^*,
 \end{equation}
where
 \begin{equation}\label{I_n,varphi,p}
      I_{n,\varphi}(\tau ,\mu   ):=
      \mathop{\inf\limits _{k\ge n}}\limits_{k \in {\mathbb N}} \int\limits _0^{\tau }\varphi \Big(\frac {k t}n\Big)
      {\mathrm d} \mu     (t).
 \end{equation}
If, in addition, the function $\varphi$ is non-decreasing on the interval
$[0,\tau]$  and the condition
 \begin{equation}\label{I_n,varphi,p_Equiv_n}
      I_{n,\varphi }(\tau ,\mu   )=\int\limits _0^{\tau }\varphi (t) {\mathrm d} \mu     (t),
 \end{equation}
holds, then  inequality $(\ref{Jackson_Type_Ineq})$ can not be improved and therefore,
 \begin{equation}
       \label{Jackson_Type_Exact}
       \mathop {\sup\limits _{f\in L^{\psi}{\mathcal S}_{\bf M}}}\limits _{f\not ={\rm const }}
       \frac {E_n(f)_{_{\scriptstyle  {\bf M}}}^* }
       {\Omega_\varphi (f^{\psi }, \tau, \mu  , \frac{\tau }{n} )_{_{\scriptstyle  {\bf M}}}^* }=
        \frac {\mu  (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi (t) {\mathrm d} \mu     (t)} |\psi  (n)|.
 \end{equation}

\end{theorem}

\begin{proof} Let $f\in L^{\psi }{\mathcal S}_{\bf M}$. By virtue of  (\ref{Fourier_Coeff_der}) and (\ref{Best_app_all}), we have
 \[
 E_n(f)_{_{\scriptstyle  {\bf M}}}^* =\sup \Big\{
           \sum _{|k |\ge  n}  \lambda_k|\widehat{f}(k) |: \  \lambda\in \Lambda \Big\} \le
           \sup \Big\{
           \sum _{|k |\ge  n}  \lambda_k \Big|\frac {\psi (n)
 }{\psi (k)}\Big||\widehat{f}(k) |: \   \lambda\in \Lambda \Big\}
 \]
 \begin{equation}
       \label{A6.91}
       =|\psi  (n)|  \sup \Big\{
           \sum _{|k |\ge  n}  \lambda_k \Big|\frac {\widehat f(k)}{\psi (k)}\Big|: \quad  \lambda\in \Lambda \Big\} =
           |\psi  (n)|  E_n (f^{\psi })_{_{\scriptstyle  {\bf M}}}^* .
   \end{equation}
As shown in  \cite[Proof of Theorem 1]{Abdullayev_Chaichenko_Shidlich_2021}, for any   $g \in {\mathcal S}_{\bf M}$,   $\tau >0$, $\varphi\in \Phi$,  $\mu   \in {\mathcal M}(\tau )$ and   $n\in {\mathbb N}$
 \begin{equation}\label{Jackson_type_OLD}
       E_n (g)_{_{\scriptstyle  {\bf M}}}^* \le
       \frac 1{I_{n,\varphi}(\tau ,\mu   )}
       \int \limits _0^{\tau }
       \omega_\varphi \Big(g, \frac t{n}\Big)_{_{\scriptstyle  {\bf M}}}^* {\mathrm d} \mu    (t).
   \end{equation}
Let us set  $g =f^{\psi }$ in (\ref{Jackson_type_OLD}). Then
   \begin{equation}\label{A6.93}
        E_n (f^{\psi})_{_{\scriptstyle  {\bf M}}}^* \le
        \frac {\mu   (\tau ) - \mu   (0)}{I_{n,\varphi}(\tau ,\mu   )}
        \frac {\int  _0^{\tau }\omega _\varphi (f^{\psi }, \frac t{n})_{_{\scriptstyle  {\bf M}}}^* {\mathrm d} \mu    (t)}{\mu   (\tau ) - \mu   (0)} \le
        \frac {\mu   (\tau ) - \mu   (0)}{I_{n,\varphi}(\tau ,\mu   )}
        \Omega _\varphi \Big(f^{\psi }, \tau, \mu   , \frac{\tau }n\Big)_{_{\scriptstyle  {\bf M}}}^* .
   \end{equation}
Combining  inequalities (\ref{A6.91}) and (\ref{A6.93}), we obtain   (\ref{Jackson_Type_Ineq}).

Now assume that the function $\varphi$ is non-decreasing on  the interval
$[0,\tau]$ and
condition (\ref{I_n,varphi,p_Equiv_n}) holds.
Then by virtue of (\ref{Jackson_Type_Ineq}), we have
 \begin{equation}\label{A6.94}
      \mathop {\sup\limits _{f\in L^{\psi}{\mathcal S}_{\bf M}}}\limits _{f\not ={\rm const }}
       \frac {E_n(f)_{_{\scriptstyle  {\bf M}}}^* }
       {\Omega_\varphi (f^{\psi }, \tau, \mu  , \frac{\tau }n )_{_{\scriptstyle  {\bf M}}}^* }\le
             \frac {\mu  (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(u) {\mathrm d} \mu     (u)}|\psi  (n)|
       .
   \end{equation}
 To prove the unimprovability of  inequality (\ref{A6.94}),   consider  the function
 \[
  f_n(x)=\gamma  +\varepsilon_{-n}\delta   {\mathrm e}^{-{\mathrm i}nx} +
  \varepsilon_{n} \delta    {\mathrm e}^{{\mathrm i}nx},
  \]
where  $\gamma$ and  $\delta $ are arbitrary complex numbers, and   $\varepsilon_k$, $k\in \{-n,n\}$,
are integers such that  $|\varepsilon_{n}|+|\varepsilon_{-n}|=1$
. Taking into account relations (\ref{modulus_generalize difference_Fourier_Coeff}), (\ref{Fourier_Coeff_der}) and (\ref{M_condition}), we have
 \begin{equation}\label{A6.95}
 \|\Delta_h^\varphi f_n^\psi\|_{_{\scriptstyle  {\bf M}}}^*=|\delta |\frac {\varphi (n h)}{|\psi(n)| }\sup \Big\{
           |\varepsilon_{-n}|\lambda_{-n}+
           |\varepsilon_{n}|\lambda_{n}: \quad  \lambda\in \Lambda \Big\}
 =  |\delta |\frac {\varphi (nh)}{|\psi(n)| }.
 \end{equation}
Since the  function $\varphi(nh)$ is non-decreasing on the interval  $[0, \frac {\tau}n]$,
then for $0\le t\le \tau$,
 \begin{equation}\label{A6.96}
\omega _\varphi (f_n^\psi, t) =    |\delta  |
 \frac {\varphi
 (nt)}{|\psi  (n)|}.
 \end{equation}
Taking into account (\ref{Mean_Value_Gen_Modulus}),  (\ref{A6.96}) and the equality
 $
    E_n
  (f_n)_{_{\scriptstyle  {\bf M}}}^*  =|\delta  |
  ,
 $
we see that
  \[
  \mathop {\sup\limits _{f\in L^{\psi }{\mathcal S}_{\bf M}}}\limits _{f\not ={\rm const}}
  \frac {E_n  (f)_{_{\scriptstyle  {\bf M}}}^* }
  {\Omega _\varphi  (f^{\psi }, \tau , \mu   , \frac {\tau }n)_{_{\scriptstyle  {\bf M}}}^* }
  \ge
  \frac {E_n  (f_n)_{_{\scriptstyle  {\bf M}}}^* }
  {\Omega _\varphi  (f_n^{\psi }, \tau, \mu  , \frac {\tau }n)_{_{\scriptstyle  {\bf M}}}^* }
  \]
  \begin{equation}\label{A6.97}
  = \frac {|\delta  | (\mu   (\tau )-\mu   (0))  |\psi  (n)|}
   {\int _0^{\tau /n}|\delta  | \varphi (nt){\mathrm d} \mu    (nt) }=
         \frac {\mu  (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(u) {\mathrm d} \mu     (u)}  |\psi  (n)|.
 \end{equation}
Relations (\ref{A6.94}) and (\ref{A6.97}) yield (\ref{Jackson_Type_Exact}).

\end{proof}

\subsection{Widths of the classes $L^{\psi }(\varphi, \mu, \tau, n)_{_{\scriptstyle  {\bf M}}}^*  $}

In this subsection,  the values of Kolmogorov, Bernstein, linear, and projection widths are found for the
 classes $L^{\psi}(\varphi, \mu  , \tau, n)_{_{\scriptstyle  {\bf M}}}^*  $ in the case when the sequences
$\psi (k)$ satisfy some natural restrictions.

\begin{theorem} \label{Th.3.2}
       Assume that    $\psi \in \Psi$, $\tau>0$,  condition $(\ref{M_condition})$ holds, the function
       $\varphi\in \Phi$ is non-decreasing on the interval $[0,\tau]$ and  $\mu   \in {\mathcal M}(\tau )$.
       Then for any  $n\in {\mathbb N}$ and  $N\in \{2n-1,  2n\}$  the following inequalities are true:
\begin{equation}\label{A6.105}
    \frac{\mu   (\tau ) - \mu   (0)}
   {\int_0^{\tau }\varphi (t) {\mathrm d} \mu     (t)} |\psi (n)|
   \le P_{N} \Big(L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}} \Big)
    \le  \frac {\mu   (\tau ) - \mu  (0)}
   {I_{n,\varphi}(\tau ,\mu   )}    |\psi (n)|,
  \end{equation}
where the quantity $I_{n,\varphi}(\tau ,\mu   )$ is defined by $(\ref{I_n,varphi,p})$, and $P_N$
is any of the widths $b_N$, $d_N$, $\lambda _N$ or $ \pi _N$. If, in addition, condition $(\ref{I_n,varphi,p_Equiv_n})$ holds, then
    \begin{equation}\label{A6.106}
          P_{N} \Big(L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big )=
           \frac{\mu   (\tau ) - \mu   (0)}
          {\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)} |\psi (n)|.
      \end{equation}
\end{theorem}

\begin{proof}  The proof of Theorems \ref{Th.3.2} and  \ref{Th.3.3}  basically repeats the proof  of corresponding
theorems in the spaces   ${\mathcal S}^p$ (see \cite{Serdyuk_2003, Abdullayev_Serdyuk_Shidlich_2021}) and   is adapted  in accordance with the properties of the spaces ${\mathcal S}_{\bf M}$. Based on Theorem \ref{Th.3.1}, taking into account the definition of the set $\Psi$, for an arbitrary function
$f\in L^{\psi }(\varphi, \tau , \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  $, we have
     \begin{equation}\label{A6.107}
            E_n(f)_{_{\scriptstyle  {\bf M}}}^*  \le
                 \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )}
                 \Omega _\varphi\Big(f^{\psi}, \tau , \mu  , \frac {\tau}n\Big)_{_{\scriptstyle  {\bf M}}}^*  |\psi (n)|
      \le   \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )}
                |\psi (n)|.
      \end{equation}
Then, taking into account the definition of the projection width $\pi _{N}$, and  relations
(\ref{Best_app_all}) and  (\ref{A6.107}), we conclude that
     \begin{equation}\label{A6.109}
           \pi _{2n-1}\Big(L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)
           \le   \sup\limits _{f\in L^{\psi }(\varphi, \tau,\, \mu  ,\, n)_{_{\scriptstyle  {\bf M}}}^*  }
          E_n(f)_{_{\scriptstyle  {\bf M}}}^*
 \le
           \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )} |\psi (n)|.
      \end{equation}
Since the widths  $b_N$, $d_N$, $\lambda _N$ and  $\pi _N$ do not increase with increasing $N$  and
     \begin{equation}\label{A6.110}
         b_N(K, X)\le   d_N(K, X) \le \lambda_N(K, X)\le \pi _N(K, X)
      \end{equation}
(see, for example, \cite[Ch.~4]{Tikhomirov_M1976}), then by virtue of (\ref{A6.109}), we get  the upper estimate in
(\ref{A6.105}).

To obtain the necessary lower estimate, it suffices to show that
     \begin{equation}\label{A6.111}
              b_{2n}\Big(L^{\psi }(\varphi, \mu  , \tau, n)_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)\ge
               \frac{\mu   (\tau ) - \mu   (0)}   {\int_0^{\tau }\varphi(u) {\mathrm d} \mu     (u)} |\psi (n)|=:R_n.
      \end{equation}

In the $(2n+1)$-dimensional space ${\mathscr T}_{2n+1}$ of trigonometric polynomials of order $n$, consider
the ball $B_{2n+1}$, whose radius is equal to the number $R_n$ defined in  (\ref{A6.111}), that is,
  $$
  B_{2n+1}=\Big\{t_n\in {\mathscr T}_{2n+1}
  : \|t_n\|_{_{\scriptstyle  {\bf M}}}^*  \le R_n\Big\}.
  $$
and prove the embedding  $B_{2n+1}\subset L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  $.

For an arbitrary polynomial $T_n\in B_{2n+1}$, due to (\ref{general_modulus}) and the parity of the function
$\varphi$, we have
 $$
 \omega_\varphi(T_n^{\psi }, t)_{_{\scriptstyle  {\bf M}}}^*
 =\sup\limits_{0\le v\le t}\sup\Big\{
 \sum _{|k|\le n} \lambda_k\varphi(kv) |\widehat T_n^{\psi }(k)|:\lambda\in \Lambda\Big\} .
 $$
Then, taking into account   relation (\ref{Fourier_Coeff_der}) and the nondecreasing of the function $\varphi$  on $[0,a]$,
for $\tau \in (0,a]$  we get
  \[
  (\mu   (\tau ) -\mu   (0))  \Omega _\varphi \Big (T_n^{\psi }, \tau,\mu  , \frac {\tau }n\Big )_{_{\scriptstyle  {\bf M}}}^*
   =\int\limits _0^{\tau }\omega _\varphi  \Big(T_n^{\psi }, \frac t{n}\Big)_{_{\scriptstyle  {\bf M}}}^*  {\mathrm d} \mu    (t)
   \]
     \[
    = \int\limits _0^{\tau } \sup\limits_{0\le v\le \frac t{n}}\sup\Big\{
 \sum _{|k|\le n} \lambda_k\varphi(kv) |\widehat T_n^{\psi }(k)|:\lambda\in \Lambda\Big\}  {\mathrm d} \mu    (t)
 \]
 \[
 = \int\limits _0^{\tau } \sup\limits_{0\le v\le \frac t{n}}\sup\Big\{
 \sum _{|k|\le n} \lambda_k\varphi(kv) \Big| \frac{\widehat T_n(k)}{\psi(k)}\Big|:\lambda\in \Lambda\Big\}  {\mathrm d} \mu    (t)
 \]
 \[
 \le \frac 1{|\psi(n)|}\int\limits _0^{\tau } \varphi(t) \sup\Big\{
 \sum _{|k|\le n} \lambda_k  |  \widehat T_n(k) |:\lambda\in \Lambda\Big\}  {\mathrm d} \mu    (t)
 = \frac {\|T_n\|_{_{\scriptstyle  {\bf M}}}^*    }
    {|\psi(n)| }
    \int\limits _0^{\tau } \varphi(t) {\mathrm d} \mu    (t).
   \]
Therefore, given the inclusion $T_n\in B_{2n+1}$ it follows that
$\Omega _\varphi   (T_n^{\psi }, \tau,\mu  , \frac {\tau }n  )_{_{\scriptstyle  {\bf M}}}^* \!\le \!1$. Thus,
$T_n\in L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^* $, and the embedding
$B_{2n+1}\subset L^{\psi }(\varphi,\mu   , \tau, n)_{_{\scriptstyle  {\bf M}}}^* $ is true.  By
the definition of  Bernstein width, the inequality  (\ref{A6.111}) holds. Thus, relation  (\ref{A6.105}) is proved.
It is easy to see that, under the condition (\ref{I_n,varphi,p_Equiv_n}), the upper and lower bounds for
the quantities $P_N (L^{\psi}(\varphi, \tau, \mu  , n)_{_{\scriptstyle  {\bf M}}}^*  ,
{\mathcal S}_{_{\scriptstyle  {\bf M}}} )$ coincide and, therefore, equalities (\ref{A6.106}) hold.

\end{proof}

\subsection{Widths of the classes   $L^{\psi }(\varphi, \mu  , \tau, \Omega )_{_{\scriptstyle  {\bf M}}}^*  $}

Let us find  the widths of the classes $L^{\psi }(\varphi, \mu  , \tau,
\Omega )_{_{\scriptstyle  {\bf M}}}^*  $   that are defined by a majorant $\Omega$ of the averaged values of  generalized moduli of smoothness.

\begin{theorem}
       \label{Th.3.3}
       Let   $\psi \in \Psi$,  condition $(\ref{M_condition})$ holds, the function
       $\varphi\in \Phi$ be non-decreasing on a certain interval $[0,a]$, $a>0$, and
       $\varphi(a)=\sup\{\varphi(t):\, t\in {\mathbb R}\}$.
       Let also  $\tau\in (0,a]$,  the function  $\mu   \in {\mathcal M}(\tau )$
       and  $\Omega  (u)$ be a fixed continuous monotonically
       increasing function of the variable   $u \ge  0$  such that $\Omega  (0)=0$ and for all $\xi>0$ and $0<u\le a $,  the condition
     \begin{equation}\label{A6.113}
     \Omega  \Big(\frac u{\xi }\Big) \int\limits _0^{\xi \tau}\varphi_{*} (t) {\mathrm d} \mu
     \Big(\frac t{\xi }\Big)
     \le \Omega  (u)\ \int \limits _0^{\tau }\varphi(t)   {\mathrm d} \mu    (t)  ,
      \end{equation}
      is satisfied, where
     \begin{equation}\label{A6.114}
      \varphi_{*}(t):=\left \{\begin{matrix} \varphi(t),\quad \hfill & 0\le t\le a, \\
      \varphi(a),\quad \hfill &  t\ge a.\end{matrix}\right.
      \end{equation}
 Then for any  $n\in {\mathbb N}$ and  $N\in \{2n-1,  2n\}$  the following inequalities are true:
\[
             \frac{\mu   (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)}
            |\psi (n)|\,\,\Omega  \Big(\frac {\tau }n\Big)\le
            P_N\Big(L^{\psi }(\varphi, \tau, \mu  , \Omega)_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)
            \]
                    \begin{equation}\label{A6.115}
                    \le
    \frac {\mu   (\tau ) - \mu  (0)} {I_{n,\varphi}(\tau ,\mu   )}   |\psi (n)|
            \,\,\Omega  \Big(\frac {\tau }n\Big),
      \end{equation}
where the quantity $I_{n,\varphi}(\tau ,\mu   )$ is defined by $(\ref{I_n,varphi,p})$, and $P_N$
is any of the widths $b_N$, $d_N$, $\lambda _N$ or $ \pi _N$. If, in addition,  condition $(\ref{I_n,varphi,p_Equiv_n})$ holds, then
     \begin{equation}\label{A6.116}
      P_{N}(L^{\psi } \Big(\varphi, \tau,  \mu  , \Omega  )_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)=
      \frac{\mu   (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)}
            |\psi (n)|\,\,\Omega  \Big(\frac {\tau }n\Big).
      \end{equation}

\end{theorem}

\begin{proof}
Based on  inequality (\ref{Jackson_Type_Ineq}), for an arbitrary function
$f\in L^{\psi }(\varphi, \tau, \mu  , \Omega)_{_{\scriptstyle  {\bf M}}}^* $
   \begin{equation}\label{A6.117}
            E_n(f)_{_{\scriptstyle  {\bf M}}}^*  \le
                 \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )}
                |\psi (n)| \Omega _\varphi\Big(\frac {\tau}n\Big),
      \end{equation}
whence, taking into account the definition of the width $\pi _{N}$ and relation (\ref{Best_app_all}), we obtain
   \begin{equation}\label{A6.118}
          \pi _{2n-1}\Big(L^{\psi }(\varphi, \mu  , \tau, \Omega  )_{_{\scriptstyle  {\bf M}}}^*  ,
          {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)=\sup\limits _{f\in L^{\psi }(\varphi, \tau,\mu   , \Omega)_{_{\scriptstyle  {\bf M}}}^*  }\!\!
           E_n(f)_{_{\scriptstyle  {\bf M}}}^*
             \le  \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )}
                |\psi (n)| \Omega _\varphi\Big(\frac {\tau}n\Big).
      \end{equation}
To obtain the necessary lower estimate, let us show that
    \begin{equation}\label{A6.119}
    b_{2n}\Big(L^{\psi }(\varphi, \mu   , \tau, \Omega  )_{_{\scriptstyle  {\bf M}}}^*  ,{\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)\ge
     \frac {\mu   (\tau ) - \mu  (0)}{I_{n,\varphi}(\tau ,\mu   )}
    |\psi (n)| \Omega _\varphi\Big(\frac {\tau}n\Big)=: R_n^*.
      \end{equation}
For this purpose, in the $(2n+1)$-dimensional space ${\mathscr T}_{2n+1}$ of trigonometric polynomials of order $n$, consider
ball $B_{2n+1}$, whose radius is equal to the number $R_n$ defined in (\ref{A6.119}), that is,
   \[
   B_{2n+1}^*=\Big\{T_n\in {\mathscr T}_{2n+1}
  : \|T_n\|_{_{\scriptstyle  {\bf M}}}^*  \le R_n^*\Big\}
  \]
and prove the validity of the  embedding  $B_{2n+1}^*\subset L^{\psi }(\varphi, \mu   , \tau, \Omega  )_{_{\scriptstyle  {\bf M}}}^* $.

Assume that $T_n\in B_{2n+1}^*$. Taking into account the non-decrease of the function
$\varphi$ on $[0,a]$  and relations  (\ref{Fourier_Coeff_der}) and (\ref{A6.114}), we have
  \[
  (\mu   (\tau ) -\mu   (0))\cdot \Omega _\varphi  (T_n^{\psi }, \tau,\mu  , u )_{_{\scriptstyle  {\bf M}}}^*
   =\int\limits _0^{u}\omega _\varphi  (T_n^{\psi }, t)_{_{\scriptstyle  {\bf M}}}^*
   {\mathrm d} \mu    \Big(\frac {\tau t}{u}\Big)
   \]

 \[
 = \int\limits _0^{\tau } \sup\limits_{0\le v\le t}\sup\Big\{
 \sum _{|k|\le n} \lambda_k\varphi(kv) \Big| \frac{\widehat T_n(k)}{\psi(k)}\Big|:\lambda\in \Lambda\Big\}  {\mathrm d} \mu    \Big(\frac {\tau t}{u}\Big)
 \]
 \[
 \le  \frac{1}{|\psi(n)|}\int\limits _0^{\tau } \varphi_*(n t)\sup\Big\{
 \sum _{|k|\le n} \lambda_k   | \widehat T_n(k)|:\lambda\in \Lambda\Big\}  {\mathrm d} \mu    \Big(\frac {\tau t}{u}\Big)
 \]
 \[
 =
  \frac {\|T_n\|_{_{\scriptstyle  {\bf M}}}^*   }
    {|\psi(n)|}
    \int\limits _0^{u} \varphi_*(n t) {\mathrm d} \mu    \Big(\frac {\tau t}{u}\Big)=
    \frac {\|T_n\|_{_{\scriptstyle  {\bf M}}}^*   }
    {|\psi(n)|}
    \int\limits _0^{nu} \varphi_*(t) {\mathrm d} \mu    \Big(\frac {\tau t}{nu}\Big).
   \]

From the inclusion of  $T_n\in B_{2n+1}^*$ and  relation (\ref{A6.113}) with $\xi =\frac {nu}{\tau} $, it follows  that
   \[
           \Omega _\varphi  (T_n^{\psi }, \tau,\mu  , u )_{_{\scriptstyle  {\bf M}}}^* \le
             \frac {\int _0^{nu} \varphi_*(t) {\mathrm d} \mu    (\frac {\tau t}{n u})}
           {\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)}
           \Omega  \Big(\frac {\tau}n\Big)\le \Omega  (u).
    \]
Therefore, indeed  $B_{2n+1}^*\subset L^{\psi }(\varphi, \tau,\mu   , \Omega  )_{_{\scriptstyle  {\bf M}}}^*  $
and by the definition of Bernstein width, relation (\ref{A6.119}) is true. Combining relations
(\ref{A6.110}), (\ref{A6.117}) and (\ref{A6.119}), and taking into account
monotonic non-increase of each of the widths $b_N,$  $d_N,$ $\lambda _N$ and $\pi _N$ on $N$, we get
(\ref{A6.115}).  Under the additional condition (\ref{I_n,varphi,p_Equiv_n}), the upper and lower estimates of the quantities
$ P_N\Big(L^{\psi }(\varphi,  \tau, \mu  , \Omega  )_{_{\scriptstyle  {\bf M}}}^*  , {\mathcal S}_{_{\scriptstyle  {\bf M}}}\Big)$ coinside in relation (\ref{A6.115}) and hence, equalities (\ref{A6.116}) are true.

\end{proof}

\subsection{Some corollaries and remarks}

As mentioned above, the functional classes  similar to the classes  of the kind (\ref{L^psi(varphi,n)}) and (\ref{L^psi(varphi,Phi)}) were studied by many  authors. To compare our results with the results of these studies, let us give some notations.

Consider the case where all functions $M_k(t)=t^p\Big(p^{-1/p}q^{-1/q}\Big)^p$, $p>1$, $1/p+1/q=1$.
  In this case, all functions $M^*_k(v)=v^q$, the set $\Lambda({\bf {M^*}})$ is a set of all sequences of
  positive numbers $\lambda=\{\lambda_k\}_{k\in \mathbb{Z}}$ such that  $\|\lambda\|_{_{l_q({\mathbb Z})}}\le 1$.
  Then the spaces ${\mathcal S}_{\bf M}$ coincide with the  mentioned above spaces ${\mathcal S}^{p}$   of functions $f\in L$  with the finite norm
 \begin{equation}\label{norm_Sp}
    \|f\|_{_{\scriptstyle {p}}}:=
    \|f\|_{_{\scriptstyle {\mathcal S}^p}}=\|\{\widehat f({k})\}_{{k}\in\mathbb  Z}
    \|_{l_p({\mathbb Z})}=\Big(\sum_{{ k}\in\mathbb  Z}|\widehat f({k})|^p\Big)^{1/p}, \quad 1\le p<\infty.
 \end{equation}
  As shown in \cite{Abdullayev_Chaichenko_Shidlich_2021},  for any $f\in {\mathcal S}^{p}$ and $p>1$,
    $\|f\|^\ast_{_{\scriptstyle  {\bf M}}}=\|f\|_{_{\scriptstyle {p}}}$.

In the case $p=1$, the similar equality for norms $ \| f\|^\ast_{_{\scriptstyle  {\bf M}}}=
\|f\|_{_{\scriptstyle {1}}}$
 obviously can be obtained if we consider all $M_k(u)=u$, $k\in {\mathbb Z}$, and the set
$\Lambda$ is a set of all sequences of  positive numbers $\lambda=\{\lambda_k\}_{k\in \mathbb{Z}}$ such that  $\|\lambda\|_{_{l_\infty({\mathbb Z})}}=\sup_{k\in \mathbb{Z}}\lambda_k \le 1$.

In particular, if all functions $M_k(t)=\frac {t^2}4$, then the space ${\mathcal S}_{\bf M}$ coincides with the
Lebesgue space $L_2$ of functions $f\in L$ with finite norm
 \[
 \|f\|_{_{\scriptstyle  L_2}}= \bigg(\frac 1{2\pi}\int\limits_{0}^{2\pi}|f(t)|^2
 {\mathrm d}t\bigg)^{1/2}.
 \]

In the spaces ${\mathcal S}^{p}$, we denote the generalized modulus of smoothness of a function $f$  by  $ \omega_\alpha(f,t)_{_{\scriptstyle  p}}$, and  the  best approximation of $f$ by the trigonometric polynomials    ${T}_{n-1}\in {\mathscr T}_{2n-1}$ is denoted by   $E_n (f)_{_{\scriptstyle  p}}$. By $\Omega _\varphi(f, \tau, \mu  , u)_{_{\scriptstyle  p}}$, $u>0$, we  denote the average value of the generalized modulus of smoothness
$\omega _\varphi(f, t)_{_{\scriptstyle  p}}$ of the function  $f$ with the weight $\mu   \in {\mathcal M}(\tau )$. Theorems \ref{Th.3.1},
\ref{Th.3.2} and \ref{Th.3.3} yields the following corollaries:

\begin{corollary}
       \label{Cor.3.1}
       Assume that $f\in {\mathcal S}^p$, $1\le p<\infty$,   $\varphi\in \Phi$, $\tau>0$,  $\mu  \in  {\mathcal M}(\tau )$
       and $\psi\in \Psi$.     Then for any   $n\in {{\mathbb N}}$ the following inequality is true:
 \begin{equation}\label{Jackson_Type_Ineq_Sp}
    E_n(f)_{_{\scriptstyle {p}}} \le
     \frac {\mu   (\tau ) - \mu   (0)}
    {I_{n,\varphi }(\tau ,\mu   )}  |\psi  (n)|\,
    \Omega _\varphi\Big(f^{\psi}, \tau,\mu   , \frac{\tau }n\Big)_{_{\scriptstyle {p}}},
 \end{equation}
where   the quantity $I_{n,\varphi}(\tau ,\mu   )$ is defined by $(\ref{I_n,varphi,p})$.
If, in addition, the function $\varphi$ is non-decreasing on
$[0,\tau]$  and condition $(\ref{I_n,varphi,p_Equiv_n})$ holds, then  inequality $(\ref{Jackson_Type_Ineq_Sp})$ can not be improved and therefore,
 \begin{equation}
       \label{Jackson_Type_Exact_1}
       \mathop {\sup\limits _{f\in L^{\psi}{\mathcal S}^p}}\limits _{f\not ={\rm const }}
       \frac {E_n(f)_{_{\scriptstyle {p}}} }
       {\Omega_\varphi (f^{\psi }, \tau, \mu  , \frac{\tau }{n} )_{_{\scriptstyle {p}}} }=
        \frac {\mu  (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi (t) {\mathrm d} \mu     (t)} |\psi  (n)|.
 \end{equation}

\end{corollary}

\begin{corollary} \label{Cor.3.2}
       Assume that  $1\le p<\infty$,  $\psi \in \Psi$, $\tau>0$,   the function
       $\varphi\in \Phi$ is non-decreasing on the interval $[0,\tau]$ and  $\mu   \in {\mathcal M}(\tau )$.
       Then for any  $n\in {\mathbb N}$ and  $N\in \{2n-1,  2n\}$  the following inequalities are true:
\begin{equation}\label{A6.1051}
    \frac{\mu   (\tau ) - \mu   (0)}
   {\int_0^{\tau }\varphi (t) {\mathrm d} \mu     (t)} |\psi (n)|
   \le P_{N} \Big(L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle {p}}}  , {\mathcal S}^p \Big)
    \le  \frac {\mu   (\tau ) - \mu  (0)}
   {I_{n,\varphi}(\tau ,\mu   )}    |\psi (n)|,
  \end{equation}
where the quantity $I_{n,\varphi}(\tau ,\mu   )$ is defined by $(\ref{I_n,varphi,p})$, and $P_N$
is any of the widths $b_N$, $d_N$, $\lambda _N$ or $ \pi _N$. If, in addition, condition $(\ref{I_n,varphi,p_Equiv_n})$ holds, then
    \begin{equation}\label{A6.1061}
          P_{N} \Big(L^{\psi }(\varphi, \tau, \mu  , n)_{_{\scriptstyle {p}}}  , {\mathcal S}^p\Big )=
           \frac{\mu   (\tau ) - \mu   (0)}
          {\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)} |\psi (n)|.
      \end{equation}
\end{corollary}

\begin{corollary}
       \label{Cor.3.3}
       Let  $1\le p<\infty$, $\psi \in \Psi$,   the function
       $\varphi\in \Phi$ be non-decreasing on a certain interval $[0,a]$, $a>0$, and
       $\varphi(a)=\sup\{\varphi(t):\, t\in {\mathbb R}\}$.
       Let also  $\tau\in (0,a]$,  the function  $\mu   \in {\mathcal M}(\tau )$
       and for all $\xi>0$ and $0<u\le a $, $\Omega  (u)$ be a fixed continuous monotonically
       increasing function of the variable   $u \ge  0$  such that $\Omega  (0)=0$ and   condition $(\ref{A6.113})$.
      is satisfied.  Then for any  $n\in {\mathbb N}$ and  $N\in \{2n-1,  2n\}$  the following inequalities are true:
\[
             \frac{\mu   (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)}
            |\psi (n)|\,\,\Omega  \Big(\frac {\tau }n\Big)\le
            P_N\Big(L^{\psi }(\varphi, \tau, \mu  , \Omega)_{_{\scriptstyle {p}}}  , {\mathcal S}^p\Big)
            \]
                   \begin{equation}\label{A6.1151}
                   \le
    \frac {\mu   (\tau ) - \mu  (0)} {I_{n,\varphi}(\tau ,\mu   )}   |\psi (n)|
            \,\,\Omega  \Big(\frac {\tau }n\Big),
      \end{equation}
where the quantity $I_{n,\varphi}(\tau ,\mu   )$ is defined by $(\ref{I_n,varphi,p})$, and $P_N$
is any of the widths $b_N$, $d_N$, $\lambda _N$ or $ \pi _N$. If, in addition,  condition $(\ref{I_n,varphi,p_Equiv_n})$ holds, then
     \begin{equation}\label{A6.1161}
      P_{N}(L^{\psi } \Big(\varphi, \tau,  \mu  , \Omega  )_{_{\scriptstyle {p}}}  , {\mathcal S}^p\Big)=
      \frac{\mu   (\tau ) - \mu   (0)}{\int_0^{\tau }\varphi(t) {\mathrm d} \mu     (t)}
            |\psi (n)|\,\,\Omega  \Big(\frac {\tau }n\Big).
      \end{equation}

\end{corollary}

Let us note that in ${\mathcal S}^p$, the statements similar to Corollaries \ref{Cor.3.1}-\ref{Cor.3.3} were also obtained in \cite{Abdullayev_Serdyuk_Shidlich_2021} and \cite{Serdyuk_2003}. In \cite{Abdullayev_Serdyuk_Shidlich_2021},
instead of the  average values
$\Omega _\varphi(f, \tau, \mu  , u)_{_{\scriptstyle  p}}$ of the generalized moduli of smoothness
$\omega _\varphi(f, t)_{_{\scriptstyle  p}}$, the authors considered the quantities
 \begin{equation}\label{Mean_Value_Gen_Modulus_Sp}
  \Omega _\varphi(f, \tau, \mu ,s , u )_{_{\scriptstyle  p}}  =
   \Big(\frac{1}{\mu    (\tau ) - \mu    (0)}\int
   _0^u\omega _\varphi^s(f, t)_{_{\scriptstyle  p}}  {\mathrm d}\mu    \Big(\frac {\tau
   t}{u}\Big)\Big)^{1/s}
 \end{equation}
 in the case where $s=p$. In  \cite{Serdyuk_2003}, the quantities $\Omega _\varphi(f, \tau, \mu,p  , u )_{_{\scriptstyle  p}} $
  were considered when the function $\varphi(t)=\varphi_\alpha(t)=2^\frac \alpha 2 (1-\cos{kt})^\frac \alpha2$.  In the case $s=p=1$ and $\psi\in \Psi$, the results of  Corollaries \ref{Cor.3.1}-\ref{Cor.3.3} coincide with the corresponding results of \cite{Abdullayev_Serdyuk_Shidlich_2021}.

 In the case  $p=2$,  $\psi (k) = ({\mathrm i}k)^{-r}$, $r=0, 1,\ldots$ and  $\mu  _1(t) = 1 - \cos t$,  equalities of the kind as  (\ref{Jackson_Type_Ineq_Sp})
 with the quantities $\Omega _{\varphi_\alpha}(f, \tau, \mu  _1,2  , u)_{_{\scriptstyle  2}}$  follow from  the result of Chernykh  \cite{Chernykh_1967_2} when $\alpha=1$, and from results of   Yussef \cite{Yussef_1988}
 when $\alpha=k\in {{\mathbb N}}$ and $n\in {\mathbb N}$. For the weight function $\mu  _2(t)=t$ and
 $\psi (k) = ({\mathrm i}k)^{-r}$, $r=0, 1,\ldots$,    equalities of the kind as  (\ref{Jackson_Type_Ineq_Sp})
 with the quantities $\Omega _{\varphi_\alpha}(f, \tau, \mu  _2,2  , u)_{_{\scriptstyle  2}}$ were obtained by
 Taikov  \cite{Taikov_1976, Taikov_1979}  ($k=1$ or $r\ge 1/2$ and $k \in {{\mathbb N}}$).

The widths of the classes
   \[
   L^{\psi }(\varphi, \tau, \mu,s   , \Omega )_{_{\scriptstyle  p}}=  \Big\{f\in L^{\psi }{\mathcal S}^p:\
    \Omega _\varphi(f^{\psi }, \tau , \mu ,s  ,  u)_{_{\scriptstyle  p}}  \le \Omega  (u),\ 0\le u\le \tau \Big\},
  \]
when $p=s=2,$ $\mu (t)=\mu  _2(t)=t$, $\varphi(t)=\varphi_\alpha(t)=2^\frac \alpha 2 (1-\cos{kt})^\frac \alpha2$,  $\psi (k)=({\mathrm i}k)^{-r}$, for $r\ge 0$ and $\alpha=1$ or $r\ge 1/2$ and $\alpha\in {{\mathbb N}},$  were obtained in  \cite{Taikov_1976, Taikov_1979} (see also \cite[Ch. 4]{Pinkus_1985}), where the existence
 of functions $\Omega$ satisfying condition similar to $(\ref{A6.113})$ was also proved.

\footnotesize


TITLE: How to prove that $q+1$ divides $q^k - 1$ for $k \in 2 \mathbb N$
QUESTION [0 upvotes]: I have to prove that if $k \in 2 \mathbb N $ every $q+1$ divides $q^k-1$ and I really have trouble understanding how to do this.

REPLY [1 votes]: $((q+1)-1)^{2m}-1=$
$\displaystyle{\sum_{k=0}^{2m}}\binom{2m}{k}(q+1)^{2m-k}(-1)^k-1=$
$\displaystyle{\sum_{k=0}^{2m-1}}\binom{2m}{k}(q+1)^{2m-k}(-1)^k.$
All terms contain a factor $(q+1)$.


TITLE: How to find the height of sand in an hourglass?
QUESTION [9 upvotes]: EDIT: I have made an important correction to the "previous question" link below... it was accidentally pointing to an unrelated question before.
I would also like to emphasize that I welcome any completely different approach to the question (solving for height as a function of "P") -- not just a correction to/extension of my potentially flawed approach... the simpler the better, of course.
Also, I'm realizing that it's unclear whether I'm asking for a general solution for any shape that conforms to the described properties, or whether I'm asking for a specific, perhaps more elegant, solution specifically for the lemniscate of Bernoulli... or thirdly, for help completing my specific attempt at a solution.  I am interested in all those things, so, honestly I don't really know what to do about that other than to break the question up into multiple separate questions.  Hopefully it's OK as is.

Let's say we have an hourglass/"container-type" shape that when viewed in 2 dimensions is both horizontally and vertically symmetrical around its center... with its 3-dimensional "counterpart" being radially symmetric around its vertical axis.  I have chosen the lemniscate of Bernoulli:

We imagine the "top" half is "full" of "sand" -- or some imaginary substance that has a perfectly level surface.
Anyway, given that some percentage of the sand, P, has fallen to the bottom, how can we calculate the height of that sand as a function of P?

So that is the basic question.  It's been years since the calculus days (assuming the answer can't be found an easier way), and I have spend days getting extremely confused by this.

My attempt at an answer and the reasoning for it:
Apparently it is true that it makes no difference if we use a 2 or 3-dimensional world in which to make this calculation (I have verified this with a previous question).
Further, since the shape is symmetrical, we can literally, cut this problem in half by plotting half of the lemniscate in the following way (in fact, we probably only need a quarter of it):

Setting a = 1:

Because we have cut the problem in half, for our purposes now, the the total amount of sand is equal to one quarter the area of the original lemniscate -- the area under the curve of the above function on the interval (0,sqrt(2)):

This area = 0.5 because the area of a lemniscate is:
Area squared = 2 * a squared. (and a=1 as established already).
So let's say some amount of sand has fallen:

Now, we should be able to find the height, s, of that percent, P, of the sand in the bottom half... by solving for the upper limit of integration of:

Where f(x) is the "y =" function shown earlier.
However I'm stuck here because I can't find the integral of that function, presumably because it's not possible?  I tried using integral-calculator.com and it says it's not possible.
To summarize, my general approach (for any shape) is:

Establish a "f(x)" function representing half of the shape in its horizontal orientation
Calculate the "total area of sand"
For a given percentage of sand that has fallen, solve for the height of sand by using a limit of integration.

More importantly, though, I'm sure there is a better way to do this whole problem... maybe an ultra-simple geometric way exists for the lemniscate due to its unique properties?

REPLY [2 votes]: @Intelligenti pauca gave the solution and
$$\int \big[-(1+t^2)+\sqrt{4 t^2+1}\big]\,dt=-t-\frac{1}{3}t^3+\frac{1}{2}t \sqrt{4 t^2+1} +\frac{1}{4} \sinh ^{-1}(2 t)$$ Computing the definite integrals we have
$$\left(3 \sinh ^{-1}\left(2 \sqrt{2}\right)-2 \sqrt{2}\right)\,P=$$
$$-4 h^3+12 \sqrt{2} h^2+6 \left(\sqrt{4 h \left(h-2 \sqrt{2}\right)+9}-6\right) h+$$ $$3
   \left(-2 \sqrt{8 h \left(h-2 \sqrt{2}\right)+18}+6 \sqrt{2}+\sinh ^{-1}\left(2
   \sqrt{2}\right)\right)-3 \sinh ^{-1}\left(2 \left(\sqrt{2}-h\right)\right)$$ which cannot be inversed. However, you could have a good approximation of it using an empirical model such as
$$h=\frac {\sum_{k=1}^m a_k \,P^k}{1+\sum_{k=1}^n b_k \,P^k}$$ For example, with $m=n=3$, $R^2>0.999999$, all paremeters being highly significant. The maximum absolute error is $0.003328$ and the mean error is $0.000020$.
Just make the regression.


TITLE: What is the length of one side of the square
QUESTION [2 upvotes]: $a)$ In the figure, $ABCD$ is a square. $E$ and $F$ are the midmpoint of the sides $[AD]$ and $[CD]$ respectively. The are of the shaded region is $6cm^2$. What is the length of one side of the square.

My attemp is:$S_{\Delta ABG}=\frac{AB\cdot GG_1}{2},$ where $GG_1\perp AB$
$\Rightarrow$ $S_{\Delta ABG}=\frac{AB\cdot BC}{6},$ becouse i soppose that $GG_1=\frac{BC}{3},$ but I didint now it is correct. Now we have: $AB=6$
Please help me. Thaky very much.

REPLY [2 votes]: Consider the image:

$$A = \dfrac{3x^2}{2} = 6$$
So $x = 2$
Therefore side length $S = 3x = 6$


TITLE: Exercises of polynomial and turing reductions
QUESTION [3 upvotes]: I'm following a graduated course in theoretical computer sicence. A good part the theory we see in this course has to do with polynomial and Turing reductions of NP problem (to prove NP-completeness), mostly including gadgets. However the course has very few exercises to practice ourselves and the teacher is reluctant to give us some more. Is there any resources, web or books, that could fill this gap for me? I'm talking about problems and possibly their solutions and some explanation about the process of finding the reduction.
Thanks.

REPLY [7 votes]: A good start is Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson. Chapter 3.2 discusses techniques and 3.3 has exercises. You can find more in the books about complexity everyone should read.

REPLY [4 votes]: Two suggestions:

On a historical note, look at Karp's paper "Reducibility Among Combinatorial Problems", which gives 21 NP-complete problems. This paper showed that many well-known problems were NP-complete for the first time, and gave clever reductions between them.
Introduction to the Theory of Computation by Michael Sipser has a nice collection of reduction exercises between NP-complete problems in the time complexity chapter.


TITLE: Product of non-zero eigenvalues (pseudo-determinant) of a singular matrix
QUESTION [1 upvotes]: Consider a real symmetric matrix $P$ that is partitioned as
$$
P = \begin{bmatrix}
A & AB \\
B^\top A & C+B^\top A B
\end{bmatrix},
$$
where P is singular, while $A$ is positive definite, and $C$ is a non-zero matrix.
I am wondering if the following relation is true:
\begin{equation}
Det(P) = \det(A) Det(C),  \quad (1)
\end{equation}
where $Det$ denotes the pseudo determinant, i.e. $Det(P)$ is the product of all the non-zero eigenvalues of $P$.
The above relation is straightforward when only the determinant is concerned, in which case we can apply the following formula:
$$
\det(P) = \det(A) \det[(C+B^\top A B) - (B^\top A) A^{-1} (AB) ] = \det(A) \det(C).
$$
However, I do not know if $(1)$ holds.

REPLY [1 votes]: This doesn't hold. E.g. when $A=B=1$ and $C=0$, we have
$$
P=\pmatrix{1&1\\ 1&1}
$$
and hence
$$
Det(P)=2\ne 1=\det(A)Det(C).\tag{2}
$$
This counterexample can be easily extended to the case where $C$ is nonzero or even entrywise nonzero. First, by appending $1$s to $A,B$ and $C$, i.e. when $A=B=I_2$ and $C=\operatorname{diag}(0,1)$, the new $P$ is (by a simultaneous permutation of rows and columns) similar to $\pmatrix{1&1\\ 1&1}\oplus\pmatrix{1&1\\ 1&2}$. Therefore $Det(P)=2$ and the inequation $(2)$ still holds. Next, to make $C$ entrywise nonzero, we can simply replace $C$ by $QCQ^T$ for some appropriate real orthogonal matrix $Q$. In particular, the inequation $(2)$ is still true when $A=B=I_2$ and
$$
C=\left[\frac{1}{\sqrt{2}}\pmatrix{1&1\\ 1&-1}\right]\pmatrix{0&0\\ 0&1}\left[\frac{1}{\sqrt{2}}\pmatrix{1&1\\ 1&-1}\right]=\frac12\pmatrix{1&-1\\ -1&1}.
$$


TITLE: Adjust the Fibonacci function for different initial conditions
QUESTION [5 upvotes]: Someone on stackoverflow asked a question about tracking score in their game with a Fibonacci like sequence, just starting at different values $(F_1 = 50,~ F_2 = 100)$.  I know there's a formula to calculate Fibonacci numbers (based on Wikipedia) but I'm not familiar with most of the math that formula is based on.
Is there a way to adjust the formula to be used with these new initial conditions?

REPLY [2 votes]: The recursion relation $F(n+2) = F(n+1)+F(n)$ suggests the generating function $x^2f(x) = xf(x)+f(x) \rightarrow f(x) = \frac 1 {x^2-x-1} = 1/[(x-\phi)(x+\phi^{-1})]$, where $\phi = \frac {1+\sqrt 5}2$. This fraction can be decomposed by partial fractions into $\frac A {x-\phi}+\frac B {x+\phi^{-1}}$. This is then two geometric sequences $A(\sum (\phi x)^n+B(\sum (-\phi^{_1})^n)$ or  $\sum (A\phi^n+B(-\phi^{-1})^n)x^n$.
I've left $A$ and $B$ as variables rather than solving for them, as the above doesn't take into account the initial terms, and the values of $A$ and be $B$ will depend on those terms; when we plug those in, we have two unknowns ($A$ and $B$), and two constraints (the two initial terms). In fact, given any $a_{i},a_{j}$, we can solve for $A$ and $B$ with $A\phi^{i}+B(-\phi^{-1})^{i}=a_{i}$ and $A\phi^{j}+B(-\phi^{-1})^{j}=a_{j}$.
In the particular case you gave, we have that the standard Fibonacci sequence goes $0,1,1,2,3,5,8...$ (although conventions vary whether it starts with $0$ or $1$). Multiply these numbers by $50$ yields $0,50,50,100,150,250,400...$. Your numbers $50,100$ correspond to $a_2,a_3$ of that sequence (with zero indexing), and so if $b_n$ denotes the $n$-th number in your sequence (with $b_1 = 50$, $b_2=100$), and $F_n$ denotes the $n$-th number of the Fibonacci sequence ($F_0=0$, $F_1=1$, $F_2=1$, ...), then $b_n = 50F_{n+1}$.


text:
\begin{document}

\title{\OURS: An Adaptive Second Order Optimizer for Machine Learning}

\author{
Zhewei Yao\thanks{Equal contribution.}$^{,1}$, Amir Gholami$^{*,1}$, Sheng Shen$^{1}$, Mustafa Mustafa$^{2}$, Kurt Keutzer$^{1}$, Michael W. Mahoney$^{1}$\\
$^{1}$University of California, Berkeley, $^{2}$NERSC, Lawrence Berkeley National Laboratory\\
{\tt\small \{zheweiy, amirgh, sheng.s, keutzer, mahoneymw\}@berkeley.edu}, \tt\small mmustafa@lbl.gov
}

\maketitle
\pagestyle{plain}

\input{_s0_abstract.tex}
\input{_s1_intro.tex}
\input{_s2_related_work.tex}
\input{_s3_method.tex}
\input{_s4_results.tex}
\input{_s5_discussion.tex}
\input{_s6_conclusion.tex}

\section*{Acknowledgments}
This work was supported by a gracious fund from Amazon Machine Learning Research Award (MLRA).
The UC Berkeley team also acknowledges gracious support from Intel corporation, Intel VLAB team, Google Cloud, Google TFTC team, and Nvidia.
Amir Gholami was supported through funding from Samsung SAIT.
Michael Mahoney would like to acknowledge the DARPA, NSF, and ONR via its BRC on RandNLA for providing partial support of this work.  Our conclusions do not necessarily reflect the position or the policy of our sponsors, and no official endorsement should be inferred.

{

\fontsize{9.5pt}{10.5pt} \selectfont

\bibliography{ref.bib}

}
\clearpage
\onecolumn
\input _s7_appendix.tex

\end{document}

config: arxiv


TITLE: Prove or disprove existence of a sequence converging weakly to $0$ in an infinite dim Hilbert space
QUESTION [7 upvotes]: This is a problem on an old analysis qual, the prompt is:
"Prove or give a counter example: if $H$ is an infinite dimensional Hilbert space and $0$ is the zero vector in $H$, then there exists a sequence $\{x_n\}$ in $H$ so that $||x_n|| \ge 1$ and $\{x_n\}$ converges weakly to the zero vector $0$ in $H$."
I know that the unit ball is not necessarily weakly compact in an infinite dimensional space if it is not reflexive. But this is specifying the existence of a single sequence, which doesn't say anything about every sequence having a convergent subsequence etc.
Since it is a Hilbert space I know it is equivalent to $(x_n,y) \rightarrow 0$ for all $y \in H$ for such a space. I was tempted to assume a countable orthonormal  use Parseval's Identity to show $||x_n||^2$ could be made less than 1, but this would seem to require $(x_n,e_k)$ to converge uniformly (ie independently of $k$ where the $e_k$'s are the orthonormal set).
Anyway, I am stuck. Any suggestions? Thanks!

REPLY [4 votes]: Okay, based on David Mitra's comment we can construct an orthonormal sequence. For ease of showing the required result let's call this $\{x_n\}$. Then Bessel's inequality gives for all $y \in H$ that:
$\sum_{n=1} ^\infty |(x_n,y)|^2 \le ||y||^2$. Assume $||y||^2$ is not infinite, which seems reasonable then this is a convergent sequence which implies the terms much approach zero, hence $\lim_{n \rightarrow \infty} (x_n,y) = 0$, thus by definition we have a sequence that converges to the 0 vector (and since it is orthonormal $||x_n|| = 1$ so it satisfies $||x_n|| \ge 1$.
I wonder why the question specified the norm being greater than or equal to 1. Just to throw to make it more confusing?


TITLE: When is a matrix function the Jacobian matrix of another mapping
QUESTION [5 upvotes]: Suppose $J(x)$ is a continuous matrix function $\mathbb{R}^D \to \mathbb{R}^D \times \mathbb{R}^D$. Do there always exist a mapping $f: \mathbb{R}^D \to \mathbb{R}^D$ so that $J = \nabla f$. If not, are there well-known conditions such that this mapping exists?

REPLY [2 votes]: Let $J_{1}, \ldots, J_{D}$ denote the columns of $J$. Then each
$J_{i}:\mathbb{R}^{D}\rightarrow\mathbb{R}^{D}$ and so you are trying to find
functions $f_{i}:\mathbb{R}^{D}\rightarrow\mathbb{R}$ such that $J_{i}=\nabla
f_{i}$ for every $i=1, \ldots, D$. To construct counter-examples is easy. If
$J$ is $C^{1}$ and not just continuous, since the domain is $\mathbb{R}^{D}$,
then a necessary and sufficient condition for each $J_{i}$ to be the gradient
of a function is that $J_{i}$ is irrotational, that is, $\frac{\partial
J_{i,j}}{\partial x_{k}}=\frac{\partial J_{i,k}}{\partial x_{j}}$ for all $j$,
$k$, where $J_{i}=(J_{i,1},\ldots,J_{i,D})$. In $\mathbb{R}^{2}$ take
$J_{1}(x,y)=(y,2x)$ and anything you want for $J_{2}$. Then $\frac{\partial
}{\partial y}(y)=1\neq\frac{\partial}{\partial x}(2x)=2$, and so $J_{1}$ is
not irrotational.
If $J$ is just continuous, then a necessary and sufficient condition for each
$J_{i}$ to be the gradient of a function is that $\int_{\gamma}J_{i}=0$ for
every closed curve $\gamma$. This is not so easy to use because you have to check every closed curve, but if you find one for which the integral is nonzero, then you immediately know that $J_{i}$ cannot be the gradient of a function.
You can find all this stuff in Fleming "Functions of several variables". Look
for exact differential forms.


TITLE: Prove that sets being equipotent is an equivalence relation
QUESTION [0 upvotes]: Let $X$ and $Y$ two subsets of $E$. We know that $X$ and $Y$ are equipotent, $X\sim Y$, if there exists a bijective function so that $f:X\rightarrow Y$. Proof that $\sim$ defines an equivalence relation in $\mathcal{P}(E)$.
I don't know how to do that, please help me!

REPLY [1 votes]: To see reflexivity, observe that the identity map on $X$ is a bijection. To see symmetry, observe that bijections have inverses, so if $X\sim Y$, then there is a bijection $f:X\to Y$, so there is a bijection $f^{-1}:Y\to X$. To see transitivity, observe that the composition of two bijections is a bijection, so if $X\sim Y$ and $Y\sim Z$, then there are bijections $f:X\to Y$ and $g:Y\to Z$, so you have a bijection $g\circ f:X\to Z$ which implies $X\sim Z$.


TITLE: Proving this convex hull lemma
QUESTION [0 upvotes]: I am trying to brush up on some computational geometry, and I am having trouble trying to prove the following:
given a planar point set P in general position, let k denote the number of hulls generated by the repeated hull process. There exists a point q in the plane such that every closed half plane whose bounding line l passing through q contains at least k points of P.
I got was looking at the questions from this PDF: https://www.cs.umd.edu/class/spring2012/cmsc754/Handouts/cmsc754-spring12-handouts.pdf
Problem 1c.
Any hints would be greatly appreciated.

REPLY [1 votes]: Hint 1 : intuition (?)

 When you're at point $q$, whatever direction you look you have $k$ points "in front" of you. The repeated hull process guarantees that you have $k$ "layers" of points in your point set $P$, so intuitively to have at least $k$ points in every direction, you want to put $q$ in the "innermost layer" of $P$.

Hint 2 :
a result that can be useful (formal proof omitted)

 If $q$ belongs to the convex hull of $P$, then any closed half-plane whose bounding line passes through $q$ contains at least one vertex of that convex hull.

Hint 3 :  basically the solution

 Take a point $q$ in the $k$-th convex hull ($H_k$ with the pdf's notation). Then $q$ belongs to every $H_i$, $1\le i\le k$.


TITLE: A High School Calculus inequality
QUESTION [0 upvotes]: The problem, encountered in a high school math textbook in the exercises on the MVT,goes as following:

Let $f(x)=e^x-ex, x\geq 0$ and $f(\ln{2})<2$, prove that $y=(1-e)x+1$ is tangent to the graph of $f, C_f,$ and find $b, c \in [0,+\infty)$ such that $$f(b)+f(c-2)\leq (1-e)(b+c-2)+2$$
holds.

I have started by finding where $y$ is tangent to $f(x)$, so it must be $y'=f'(x)\Rightarrow 1-e=e^x-e\Rightarrow x=0$. So it must be tangent at point $A(0, 1)$. My first thought was breaking down the inequality to obtain $f(b)+f(c-2)\leq y(b)+y(c-2)$. Could I perhaps use the Mean Value Theorem on a new function $g(x)=f(x)-y(x)$ in the interval $(0,1)$? Any help would be appreaciated!

REPLY [1 votes]: What a weird, contrived and unnecessary problem. Anyway, if this statement does not have an error, then you it must be the case that $b = c - 2 = 0.$ That's because, your function $f(x)$ is convex on the entire real line, meaning that it's graph lies above any tangent line i.e.,:
$$f(x)\geq y(x)$$
for all $x$ and the inequality is strict if $x\neq 0.$ Therefore, the only way you can have
$$f(b)+f(c-2)\leq y(b)+y(c-2)$$
is if $b = c - 2 = 0. $


TITLE: N white and black balls and N boxes Probability
QUESTION [1 upvotes]: Given $N$ boxes , $N$ white balls and $N$ black balls
chooses one box randomly and one ball from it randomly
Find the probability of getting white ball.

REPLY [1 votes]: It depends on how the balls are distributed among the boxes. We will assume each box has at least one ball. Then we could put a white into $N-1$ of the boxes, and all $N$ black and the remaining white into the last box. The probability of white is then $\frac{N-1}{N}+\frac{1}{N(N+1)}$. This is close to $1$ if $N$ is large. Similarly, by an analogous distribution, we can make the probability of black close to $1$.
If on the other hand we put $2$ balls in each box by a process that does not pay attention to the colours of the balls, then by symmetry the answer is $1/2$.


TITLE: An inequality for positive definite matrices
QUESTION [0 upvotes]: If $A$ and $B$ are poitive definite matrices, then
$n({\rm det} A. {\rm det} B)^{1/n} ≤ {\rm tr}AB$
I was wondering what are the sufficient and necessary conditions for the equality to hold ?

REPLY [2 votes]: Hint. Positive definite matrices always admit (unique) positive definite matrix square roots. Also, for square matrices, the trace of $XY$ is equal to the trace of $YX$. It follows that the inequality in question can be rewritten as
$$
\sqrt[n]{\det(B^{1/2}AB^{1/2})}\le\frac{\operatorname{tr}(B^{1/2}AB^{1/2})}n.
$$
If you express both sides in terms of the eigenvalues of $B^{1/2}AB^{1/2}$, the answer will become clear.


TITLE: Forces of tension in a pulley rope system
QUESTION [0 upvotes]: I have found similar questions to this topic on PSE but none of them aswered the questions I have.
So when we consider the pulley-rope system below, where the are two weights ($m_2>m_1$) and a pulley which is considered as homogeneous cylinder $m_3$. The rope is considerd massless. My question is first, are $T_1$ and $T_2$ the same magnitude? If not, would they have the same magnitude if the pulley was massless? And second, Where did we get $T_1'$ and $T_2'$ (and what's their magnitude)? (Someone said It is because of Newtons third law but I dont get why the opposite force to $T_1$ isn't $T_2$ but $T_1'$?)
(I am confused because my textbook says the magnitude of $T_1$ and $T_2$ will be the same even if $m_2>m_1$ (when the pulley is considered massless) and it also doesn't say anything about the existence of forces $T_1'$ and $T_2'$.)
Thanks.

REPLY [1 votes]: The rope is considered massless and inextensible and so the tension will be the same throughout the rope except when some other external force acts on the rope eg friction.
A rope with such properties has the following functions:
1 transmit a force from one position to another
and/or
2 change the direction of the transmitted force.
With the pulley not being massless and there being friction between the rope and the pulley, remembering that the function of the rope is to transmit forces, the labelled forces in your diagram can be defined as follows.
Force $\vec T_2$ can be thought of as the force on mass $m_2$ due to the pulley and
force $\vec T’_2$ can be thought of as the force on the pulley due to mass $m_2$.
So in a sense they are a Newton third law pair being equal in magnitude and opposite in direction.
The same is true of forces $\vec T_1$ and $\vec T'_1$
If the pulley has mass and hence a moment of inertia and there is friction between the rope and the pulley then to accelerate the pulley a torque must be applied and this is provided by forces $\vec T'_2$ and $\vec T'_1$.
If the pulley is massless and therefore has zero moment of inertia then there can be no torque applied on it otherwise it would suffer an infinite angular acceleration.
So in this case the magnitudes of forces $\vec T'_2$ and $\vec T'_1$ would be equal and there directions are opposite.

To include the rope in a more formal way you would need to consider the force on the top of the rope due to the pulley and its Newton third law pair the force on the pulley due to the top of rope.
Also you would need to consider the force on the bottom of the rope due to the mass and its Newton third law pair the force on the mass due to the bottom of rope.
In this case the magnitude of the force on the top of the rope due to the pulley would equal the magnitude of the force on the bottom of the rope due to the mass and these forces would be in opposite directions ensuring a net zero force on the massless rope.
This would introduce extra (unnecessary?) forces to deal with.


TITLE: Extending a function continuously from a subset to the whole set
QUESTION [0 upvotes]: We are given two sets $E$ and $F$ such that $F \subset E \subset \mathbb{R}$. We are given a continuous function $f$ defined on $F$. Can we always extend it to a continuous function on E (not necessarily unique). I am not getting any ideas as to how to do this . If anyone has any ideas it would be great. Thanks.
Edit 1: As suggested below I am interested in bounded functions.

REPLY [4 votes]: No. $x \mapsto \sin(1/x)$ defined on $F=(0,1)$ cannot be extended continuously to $E=[0,1]$.


text:
\section{Optimal Power Flow}
\label{sec:opf}

We first set up the OPF framework, which enables us to compute the optimal set point for a collection of controllable DERs.
While we consider both singe- and three-phase networks, for the sake of brevity, we provide the derivation of OPF for single-phase settings only, directing the reader to three-phase OPF formulations in Section~\ref{sec:results_3ph}.
Adopting the full nonlinear \emph{DistFlow} model~\cite{baran_optimal_1989} for single-phase networks, we model a network $\G := (\N,\E)$ with buses or nodes $\N$ and edges or branches $\E$. The branch flow equations are
\begin{IEEEeqnarray}{Rl}
P_{mn} =& r_{mn} \ell_{mn} + p_{n}^\text{c} - p_{n}^\text{g} + u_n^p + \sum_{(n,k) \in \E, k \neq m} P_{nk} , \IEEEyesnumber\label{eqn:BFeqs1}\\
Q_{mn} =& \chi_{mn} \ell_{mn} + q_{n}^\text{c} - q_{n}^\text{g} + u_n^q + \sum_{(n,k) \in \E, k \neq m} Q_{nk}  , \IEEEyesnumber \label{eqn:BFeqs2} \\
y_{m} - y_{n} =& 2 \left( r_{mn} P_{mn} + \chi_{mn} Q_{mn} \right) \nonumber \\
& - \left( r_{mn}^2 + \chi_{mn}^2 \right) \ell_{mn} \,, \IEEEyesnumber \label{eqn:BFeqs3} \\
\ell_{mn} =& \frac{P_{mn}^2 + Q_{mn}^2}{y_m} \IEEEyesnumber\label{eqn:CurEq} \ , \ \forall (m,n) \in \E \,,
\end{IEEEeqnarray}
where $P_{mn},Q_{mn}$ denote the real and reactive power flowing out of node~$m$ towards node~$n$, $p_{n}^\text{c}$ and $q_{n}^\text{c}$ are the uncontrollable real and reactive power consumption at node~$n$, $p_{n}^\text{g}$ and $q_{n}^\text{g}$ are the uncontrollable real and reactive power generation at node~$n$, $y_n := V_n^2$ is the squared voltage magnitude, and $\ell_{mn} := I_{mn}^2$ is the squared current magnitude on branch~$(m,n)$.
We assume we are given a subset $\C \subset \N$ of buses that are equipped with a controllable source or load.
$u_n^p$ and $u_n^q$, for $n \in \mathcal C$, are the controllable parts of the real and reactive nodal power.
$r_{mn}$ and $\chi_{mn}$ are the resistance and reactance of branch~$(m,n)$. Here, we assume the nodal power consumption to be constant and not sensitive to voltage or current fluctuations.
This assumption is justified as loads are increasingly behaving as constant power due to many devices controlling their power intake or injection.
Extension to include voltage sensitivity is possible but omitted here. For a formulation, see~\cite{farivar_inverter_2011}.

\subsection{Optimal Power Flow Objectives and Constraints}
We consider different controller architectures that can deliver real and reactive power $u_n^p,u_n^q$.
Whether these are available depends on the controllable parts of the inverter-interfaced system, which may include both controllable generation, storage or load.

We consider the following optimal power flow formulation:
\begin{IEEEeqnarray}{Rl}
\min_{\pmb{u}} \hspace{8pt} & f_o (\pmb{x},\pmb{u})

\IEEEyesnumber \label{eqn:OPF} \\
\text{s.t.} \hspace{8pt} & \eqref{eqn:BFeqs1}-\eqref{eqn:BFeqs3}, \nonumber \\
& \ell_{mn} \ge \frac{P_{mn}^2 + Q_{mn}^2}{y_m} \,, \forall (m,n) \in \E \,, \label{eqn:ConvRel} \\
& \text{capacity constraints on } u^p_i , u^q_i \,, \nonumber \\
& \underline{y} \leq y_{n} \leq \overline{y} \,, \forall n \in \N \,, \label{eqn:VoltConst}
\\
\pmb{x} =& \left(y_n, \ell_{mn} , P_{mn}, Q_{mn} \right),\, \pmb{u} = \left( u^p_i , u^q_i \right) \,, \nonumber \\
&\forall n \in \N \ , \ \forall (m,n) \in \E \ , \ \forall i \in \C \,. \nonumber
\end{IEEEeqnarray}

In our case studies in Section~\ref{sec:results}, we optimize various objectives relevant in distribution grid operations.
\begin{align}
 f_{o} :=& \ \alpha \displaystyle \sum_{(m,n) \in \E} r_{mn}\ell_{mn} + \beta \sum_{n \in \mathcal N} \left( y_n - y_{\text{ref}}  \right)^2 \nonumber \\
 &+ \gamma   \left( P_{01}^2 + Q_{01}^2  \right) \,.
   \label{eqn:ObjFunOp}
\end{align}
Firstly, to ensure the solution lies on the feasibility boundary of the inequality constraint~\eqref{eqn:ConvRel}, the convex relaxation requires an objective strictly increasing in the squared current magnitude~$\ell_{mn}$ and no upper bounds on the loads, thereby satisfying the original equality constraint~\eqref{eqn:CurEq}, as explained in~\cite{farivar_branch_2013}.
We also use this objective to minimize real power losses.
Secondly, we minimize voltage deviations from a reference and voltage variability due to the intermittent nature of generation and consumption.
This objective aims to actively keep the voltage in a safe and desirable operable regime.
Lastly, we aim to make a network \emph{self-sufficient} by minimizing power delivered from the transmission grid, injected at the top of a radial network, thereby maximizing the use of distributed energy resources to provide loads nearby.
$\alpha, \beta,\gamma$ denote hyperparameters, and $y_{\text{ref}}$ the reference voltage throughout the network.
In this paper, we are primarily concerned with attaining certain qualitative behaviors with OPF. If one has an economic consideration, the hyper parameters can be scaled to reflect the economic cost of various objectives.
The nonlinear equality constraint \eqref{eqn:CurEq} is relaxed to \eqref{eqn:ConvRel}, enabling a convex second order cone program~\cite{farivar_branch_2013}.
Equation~\eqref{eqn:VoltConst} formulates the responsibility of American utilities to maintain service voltage within $\pm 5\%$ of 120$V$ as specified by ANSI Standard C84.1.

Two constraints are included to account for inverter capacity and voltage goals.
Each controller $i \in \C$ is ultimately limited by a local capacity on total apparent power capacity $\bar{s}_i$, which depends on the other functions an inverter executes.
Most inverters are able to provide real and reactive power in a coupled fashion through a \emph{four-quadrant} configuration, which yields a disk constraint,
\begin{equation}
(u_{i}^p)^2(t) + (u_{i}^q)^2(t) \le \bar{s}_i^2(t) \,.
\label{eqn:InvCap1}
\end{equation}

In some cases, capacity may be assigned in a decoupled fashion for real and/or reactive power, in which an inner approximation of the disk constraint leads to
\begin{equation}
\underline{p}_i(t) \le u_{i}^p(t) \le \overline{p}_i(t) \ , \ \underline{q}_i(t) \le u_{i}^q(t) \le \overline{q}_i(t) \,,
\label{eqn:InvCap}
\end{equation}
with $\underline{p}_i,\overline{p}_i$ denoting the lower and upper capacity constraints on nodal power injection.
Lastly, we consider inverters that also interface a system with a local photovoltaic (PV) installation, an electric vehicle (EV) or battery charging system, smart loads or a combination thereof.
While incorporating the temporal capacity of batteries and its intertemporal dependencies as a constraint is out-of-scope for this work, we can indirectly model time-varying capacity when historical data of capacity is available or can be computed.
Due to the nature of these subsystems having some hierarchy of priority, it can happen that the capacity available for the OPF problem is dependent on the scheduling and activity of these other systems.
In this paper, we consider a case where inverters can only deliver or consume reactive power based on the remaining capacity \emph{after} injecting surplus energy from the PV installation into the grid.
In this case, the reactive power capacity $\bar{q}_i(t)$ at time $t$ of an inverter is limited by the total apparent power capacity $\bar{s}$ (constant) minus the real power generation $p_i^{\text{g}}(t)$.
As a result, the demand of reactive power does not interfere with real power generation, which is formulated as
\begin{equation}
\left| u^q_i(t) \right|  \leq \bar{q}_i (t) = \sqrt{\bar{s}_i^2 - (p_i^{\text{g}}(t))^2} \,.
\label{eqn:InvCap2}
\end{equation}
We will use this constraint in case 1, where we assume that each inverter has some overcapacity with respect to the maximum real power output, e.g. $\bar{s}_i = 1.05 \bar{p}_i$.

\subsection{A Principled Partitioning of Variables}

To formalize the reconstruction, we first reformulate the state variables per bus as
\begin{equation}
x_i :=
\left[
\begin{matrix}
V_i \\
\delta_i \\
p_i \\
q_i \\
\end{matrix}
\right]
\in \R^{4} \,.
\label{eqn:state}
\end{equation}
Based on the bus type, we can partition the state~$x$ into controllable inputs~$u$, uncontrollable inputs or disturbances~$d$ and endogenous variables~$x^{\text{end}}$, as suggested in~\cite{hauswirth_online_2017} and summarized in Table~\ref{tab:bus_types} for basic bus types. Note that an actual bus may be a combination of basic bus types (such as combining load and generation, or a controllable load), and $p_i,q_i$ denote the net nodal power.

\begin{table}[h]
\centering
\caption{Partitioning of variables for all bus types, as proposed in~\cite{hauswirth_online_2017}.}
\begin{tabular}{lccc}
\toprule
 				& \multicolumn{2}{c}{Exogenous} 			& Endogenous 		\\ \cmidrule(r){2-3}
				& Controlled - $u$	& Uncontrolled - $d$ 	& 	$x^{\text{end}} := x_{\backslash \{ u,d \} }$			\\ \midrule
PQ generation		& $p_n,q_n$		&					& $V_n,\delta_n$	\\
PQ load			& 				& $p_n,q_n$			& $V_n,\delta_n$	\\
PV generation		& $p_n,V_n$		& 					& $q_n, \delta_n$	\\
slack bus			& $V_0$			& $\delta_0$			& $p_0,q_0$		\\ \bottomrule
\end{tabular}
\label{tab:bus_types}
\end{table}

An important reason to distinguish between endogenous and exogenous variables, is that these have a different effect on the dynamics when chosen as controller variables. Endogenous variables implicate closed-loop feedback control, while exogenous variables lead to an open-loop feedforward control scheme.
In this paper we focus our attention on the latter, using exogenous disturbances $d_i$ as our controller variables.
This choice circumvents the potential for instability inherent to feedback policies, such as those known to exist for volt-var control schemes~\cite{farivar_equilibrium_2013}.

In addition to exogenous variables directly impacting the optimal power flow problem, as listed in Table~\ref{tab:bus_types}, we consider other exogenous variables $d^{\text{other}}$ that may help explain our data, such as time and weather variables that are readily available.
In a separate effort, we compare open-loop and closed-loop configurations~\cite{karagiannopoulos_data-driven_2019}.

\subsection{Building A Data Set Of Optimal Set Points}
We treat the $i^{\text{th}}$ agent's optimal action in the centralized problem as a random variable $u_i^*$, which we define as an element in the set of minimizers of problem~\eqref{eqn:OPF}-\eqref{eqn:VoltConst}, and which depends on the global state variables $x = (x_1,\hdots,x_{|\N|})$.
In the OPF context, the goal of the machine learning procedure is to find a model for each DER $i \in \C$ that approximates the optimal power injection $u^*_i$ based \emph{solely} on local measurements~$d_i$ available at node $i$.
We construct a \emph{central training set} consisting of power flow scenarios that are representative of future behavior on the grid. This data set can contain historical measurements taken from advanced metering infrastructure (AMI) in the grid, augmented with proxy data for variables that were not measured or for scenarios that did not occur in the past, but are anticipated to occur in the future. For example, a load not metered by AMI, may be approximated with an average load profile.
Examples of anticipated future behaviors not captured in historical measurements are connections of new electric vehicles or solar installations.
If known a priori, these may be simulated and added to the training set.
Together, the central training set contains $T$ power flow scenarios augmented with the optimal control set points $\{x(t), u^*(t)\}_{t=1}^T$, which were computed in a centralized OPF problem as defined in Section~\ref{sec:opf}.


text:
\section{Boundary Integral Equation}
\label{sec:BIE}

An elastic body $\Omega$ subject to external loading without body forces is considered. Its behaviour in terms of displacements $\utens$ is described by the partial differential equation
\begin{align}
\label{eq:PDE}
  \op{L}\utens\ofpt{x} = -\left( \lambda + 2 \mu \right)
  \div \grad \utens \ofpt{x} + \mu \curl (\curl \utens \ofpt{x}) =
  0 & & \pt{x} \in \Omega
\end{align}
where $\op{L}$ denotes the Lam\'{e}-Navier operator with the Lam\'{e} constants $\lambda$ and $\mu$ \cite{kupradze1979}. The closed boundary of the domain is denoted by $\Gamma$ and the surface normal $\vek{n}$ points outside the considered domain.
In the following sections,
\revdel{almost everything}{}
\revadd{equation~\protect\eqref{eq:PDE} is modified, so that it}{}
is described by means of boundary data. Therefore the boundary trace
\begin{align}
  \label{eq:trace}
  \Tr \utens \ofpt{x} &= \lim_{\vek{x}\rightarrow\vek{y}} \utens \ofpt{x} =
  \utens \ofpt{y} & & \pt{x} \in \Omega,\, \pt{y} \in \Gamma
\end{align}
and the conormal derivative
\begin{align}
  \label{eq:conormal-derivative}
  \dTr \utens \ofpt{x} & = \lambda \div \utens \ofpt{y} \vek{n}\ofpt{y} +
  2\mu \grad \utens \ofpt{y} \cdot \vek{n}\ofpt{y} + \mu \vek{n}\ofpt{y}
  \times (\curl \utens \ofpt{y}) & & \pt{x} \in \Omega,\, \pt{y} \in
  \Gamma
\end{align}
are introduced \cite{steinbach2008}.
The trace operator \eqref{eq:trace} maps displacements~$\utens\ofpt{x}$ in the domain to boundary displacements~$\utens\ofpt{y}$. With the material law, the conormal derivative \eqref{eq:conormal-derivative} maps $\utens\ofpt{x}$ to surface traction $\ttens\ofpt{y}$.

The boundary is split into a
\revmod{Neumann part $\Gamma_N$ and a Dirichlet part $\Gamma_D$, such that }{}
${\Gamma = \Gamma_N \cup \Gamma_D}$ and ${\Gamma_N \cap \Gamma_D = \emptyset}$. This leads to the following homogeneous boundary value problem (BVP):
Find a displacement field $\utens\ofpt{x}$ so that
\begin{equation}
\label{eq:BVP}
  \begin{aligned}
    \op{L} \utens\ofpt{x} & = 0
    & & \forall \pt{x} \in \Omega\\
    \dTr \utens\ofpt{x} & = \ttens \ofpt{y}  = {g}_N\ofpt{y}
    & & \forall \pt{y} \in \Gamma_N\\
    \Tr \utens \ofpt{x} &= \utens \ofpt{y} = g_D\ofpt{y}
    & & \forall \pt{y} \in \Gamma_D.
  \end{aligned}
\end{equation}
Here, ${g}_N$ is the prescribed Neumann data in terms of surface tractions and ${g}_D$ represents the prescribed Dirichlet data in terms of displacements.

The variational solution of the BVP~\eqref{eq:BVP} is obtained by means of a boundary integral equation~\cite{hsiao2008}. For linear elasticity, the starting point is
\begin{align}
  \label{eq:somiglianas_identity}
  \Tr \utens\ofpt{x} = \Tr \int_\Gamma \fund{U}(\pt{x},\pt{y}) \ttens \ofpt{y}
  \dgamma{y} - \Tr \int_\Gamma \fund{T}(\pt{x},\pt{y}) \utens \ofpt{y}
  \dgamma{y} & & \pt{y} \in \Gamma.
\end{align}
This equation is known as \emph{Somigliana's identity}, on which the trace operator \eqref{eq:trace} is applied. Hence, all points $\pt{x}$ are shifted to the boundary. In elasticity, $\fund{U}(\pt{x},\pt{y})$ is Kelvin's fundamental solution for displacements and $\fund{T}(\pt{x},\pt{y}) = \dTr_y\fund{U}(\pt{x},\pt{y})$ that for tractions \cite{beer2008}. The subscript $y$ denotes the variable on which the conormal derivative $\dTr$ is applied. The fundamental solutions do not in fact depend directly on the variables $\pt{x}$ and $\pt{y}$ but on their distance $r=|\pt{x}-\pt{y}|$ and are therefore translationally invariant. Moreover, their values behave like $\order(r^{-s})$ with $s\geq 1 \in\N$ in three dimensions and tend to infinity if $r\rightarrow 0$. Hence, the integrals in equation~\eqref{eq:somiglianas_identity} become singular. For the purpose of readability, the boundary integral equation
\begin{align}
  \label{eq:BIE}
  \operate{(\op{C} + \op{K}) \utens}\ofpt{x} & = \operate{\op{V} \ttens} \ofpt{x}
  && \forall \pt{x} \in \Gamma
\end{align}
is reformulated in terms of boundary integral operators. Here,
\begin{align}
  \label{eq:SL}
  \operate{\op{V} t }\ofpt{x} &= \int_\Gamma \fund{U}(\pt{x},\pt{y}) \ttens \ofpt{y}
  \dgamma{y} & & \forall \pt{x},\pt{y} \in \Gamma
\end{align}
denotes the weakly singular ($s=1$) single layer operator and
\begin{align}
  \label{eq:DL}
  \operate{\op{K} \utens}\ofpt{x} &= \int_{\Gamma} \fund{T}(\pt{x},\pt{y})
  \utens \ofpt{y} \dgamma{y} & & \forall \pt{x},\pt{y} \in \Gamma
  \setminus B_\eps\ofpt{x}
\end{align}
the strongly singular ($s=2$) double layer operator. The latter integral only exists as a \emph{Cauchy principal value}, where the radius $r_{\eps}$ of a sphere $B_\eps$ around $\pt{x}\in\Gamma$ is treated in a limiting process $r_{\eps}\rightarrow 0$. The remainder of that process is an integral free term which is
\begin{align}
  \operate{\op{C} \utens} \ofpt{x} &= c \utens \ofpt{x} && \forall \pt{x} \in \Gamma
\end{align}
with $c=1/2$ on smooth surfaces.

\revdel{For non-smooth boundaries, the free term is calculated according to Mantic[39].}{This has been moved to section 3.3. }


TITLE: Calculating an expression using recursive formulas
QUESTION [0 upvotes]: How can I calculate this:
($\dots$(((2 * 3) * 4) * 5) * $\dots$) * 2017 ;
if x * y = $\frac{x + y}{1 + xy}$, (notice x*y is not multiplication). This should be calculated using recursive formulas only.

REPLY [0 votes]: Note:
$$x*y*z=x*z*y$$
$$2*4=\frac23; 2*4*6*8=\frac45; \cdots 2*4*\cdots *2014*2016=\frac{1008}{1009}.$$
$$3*5=\frac12; 3*5*7*9=\frac23; \cdots 3*5*\cdots *2015*2017=\frac{504}{505}
$$
$$\frac{1008}{1009}*\frac{504}{505}=...$$


TITLE: Determine the regions in which each of the following series is convergent?
QUESTION [0 upvotes]: $(i) \sum_{n=1}^ \infty \frac{z^n}{3^n + 1} $
$(ii) \sum_{n=1}^ \infty \frac{1}{(n + 1)z^n} $
Comment:I have attempted it and used the ratio test on both.
$(i)$ I have got to: limit at n tends of infinity of modulus of z*$(3^n+1))/(3^{n+1}+1)$ which gives $\mod(z/3)$ when n tends to infinity.So, the series converges for mod(z)less than 3. Is that correct?
$(ii)$ I have got to:limit at n tends of infinity of modulus of $((n+1)/((n+2)*z))$ which gives mod(1/z) when n tends to infinity. Not sure what to do next.

REPLY [1 votes]: 1) $lim_{n\rightarrow \infty}|\frac{z^n}{3^{n+1}+1}\frac{3^n+1}{z^n}| = lim_{n\rightarrow \infty}|z|\frac{1 +\frac{1}{3^n}}{3+\frac{1}{3^n}} = \frac{|z|}{3} \Rightarrow |z|<3$.
2) Let's make a substitution, let $\frac{1}{z} = a$
$lim_{n\rightarrow \infty}|\frac{a^{n+1}}{(n+1)+1}\frac{n+1}{a^n}| = lim_{n\rightarrow \infty}|a|\frac{n+1}{n+2} = |a| \Rightarrow |a| < 1$.  Then, $|a| = |\frac{1}{z}| < 1 \Leftrightarrow |z|>1$


TITLE: The smallest positive eigenvalue and the length of the shortest geodesic
QUESTION [3 upvotes]: I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ be a compact (connected) Riemann surface of genus $g\geq 2$.
Since the complex upper half plane $\mathfrak{h}$ is the universal cover of $X$, we have that $X$ inherits the structure of a Riemannian manifold from  $\mathfrak{h}$. The length of the shortest geodesic with respect to the smooth volume form on $X$ induced by the hyperbolic metric, denoted by $\ell_X$, is well-defined in this case. Let $\lambda_X$ be the smallest positive eigenvalue of the Laplace operator on $L^2(X)$.
Question. What's the relation between $\ell_X$ and $\lambda_X$? Is there some kind of correspondance?
Now, let's suppose that $b_1,\ldots,b_n$ are points in $X$. Then $X$ is the compactification of a quotient $G\backslash \mathfrak{h} = X\backslash \{b_1,\ldots,b_n\}$ by adding the ``cusps'' $b_1,\ldots,b_n$. (Note that $G \backslash \mathfrak{h}$ inherits the structure of Riemannian manifold from $\mathfrak{h}$.) In this case there is no shortest geodesic on $X$ (due to the existence of cusps). Let $\lambda_G$ be the smallest positive eigenvalue of the Laplace operator on $L^2(G\backslash \mathfrak{h}$.
Question. Does $\lambda_X $ equal $\lambda_G$?

REPLY [9 votes]: You can have a sequence surfaces $X_i$ with arbitrarily short geodesics but $\lambda_{X_i}$ bounded away from zero. The geodesics are non-separating, so when they shrink to length zero, the Cheeger constant stays bounded away from zero, and therefore so does $\lambda$. You may see this explicitly using Fenchel-Nielsen coordinates.
On the other hand, there exist surfaces $X_i$ with $\lambda_{X_i}\to 0$ but the shortest geodesic is bounded away from zero. Just take a surface whose shortest geodesic is $\geq N$, and take a large cyclic cover to make $\lambda_X$ arbitrarily small.
As for filling in cusps, $\lambda_X \neq \lambda_G$ (if I understand your statement correctly). With further geometric information, you can find some relation between them. Robert Brooks has investigated this.
There is some relation between the lengths of geodesics and the eigenvalues of the Laplacian. In principle, Selberg's trace formula can determine one from the other. If there is a very short geodesic, then one obtains many eigenvalues near $\frac14$, because the spectrum is approaching the continuous spectrum of a cusped surface.


TITLE: Quasi-isomorphism and homotopical equivalence
QUESTION [4 upvotes]: I am currently studying some homological algebra, I have a couple of questions concerning the notion of quasi-isomorphism and homotopical equivalence.
For two complexes on an abelian category $X^\bullet$ and $Y^\bullet$ to be homotopically equivalent means that there are morphisms $u:X^\bullet\rightarrow Y^\bullet$ and $v:Y^\bullet\rightarrow X^\bullet$ such that $v\circ u \sim 1_{X^\bullet}$ and $u\circ v \sim 1_{Y^\bullet}$ (where by "$\sim$" I mean the homotopical equivalence for morphisms).
In this case one can prove that $u$ and $v$ are quasi-isomorphism. I read that the converse is false in the sense that if there exists a quasi-isomorphism $u:X^\bullet \rightarrow Y^\bullet$ then $X^\bullet$ may not be homotopical equivalent to $Y^\bullet$. A counterexample is this: https://math.stackexchange.com/a/93284/142179. But what happens if we have two quasi-isomorphisms $u:X^\bullet \rightarrow Y^\bullet$ and $v:Y^\bullet \rightarrow X^\bullet$? Is it true then that $X^\bullet \sim Y^\bullet$?
I have also a softer question: what is a topological analogous of a quasi-isomorphism? I mean, is there a "property" that two continuous functions $u:X\rightarrow Y$ and $v:Y\rightarrow X$ have to satisfy if $v\circ u \sim \text{id}_X$ and $u\circ v \sim \text{id}_Y$?

REPLY [7 votes]: No, they needn't be homotopy equivalent. Let $$X = \dots \rightarrow \Bbb Z/4\Bbb Z \xrightarrow{2\times} \Bbb Z/4\Bbb Z\xrightarrow{2\times} \Bbb Z/4\Bbb Z \rightarrow \dots$$ and $Y$ be the zero complex. Then $X$ and $Y$ have trivial homology and the zero maps between them are quasi-isomorphisms; but the identity map on $X$ is not null-homotopic, so that $X \not\sim Y$.


TITLE: A possible Grothendieck Topology/ Coverage of $\mathbf{Top}$?
QUESTION [2 upvotes]: For any topological space $X$ we have a canonical functor from its category (poset) of open subsets $\mathcal{O}(X)$ to category of topological spaces $\mathbf{Top}$ defined in the obvious way. This induces a functor between presheaf categories $$L_X : \mathbf{Set}^{\mathbf{Top}^{\mathbf{op}}}\to \mathbf{Set}^{\mathcal{O}(X)^{\mathbf{op}}}$$ and we can define a presheaf $F\in \mathbf{Set}^{\mathbf{Top}^{\mathbf{op}}}$ to be a sheaf if $L_XF$ is a sheaf for all $X.$
In fact, the same strategy works for any category $\mathcal{C}$ that associate with a family of functors $\mathcal{O}(X)\to\mathcal{C}$ for all $X\in\mathbf{Top}.$
I have read that the minimum structure require for the notion of sheaves is a coverage.
Now, my question is, what is the underneath coverage or Grothendieck topology over $\mathbf{Top}$ induced by functors $L_X,$ if any?
If there is no such coverage/ Grothendieck topology, what are the other occasions that we can define sheaves without a coverage and a Grothendieck topology?

REPLY [3 votes]: This is no trouble: there is a Grothendieck topology on $\mathrm{Top}$ in which the coverings are jointly surjective families of open inclusions, and your notion of sheaf is precisely the same as the notion of sheaf for this covering. There is an issue in talking about the category of presheaves on $\mathrm{Top}$, since the latter is a large category; you can handle this either by restricting the size of your spaces or by allowing presheaves with values in large sets, or, if you really want to consider presheaves on arbitrary spaces valued in small sets, by giving up a few of the nice properties of legitimate Grothendieck toposes.