text:
Patent US6080107 - Methods for the use of contrast agents in ultrasonic imaging - Google PatentsSearch Images Maps Play YouTube News Gmail Drive More »Sign inAdvanced Patent SearchPatentsMethods for use of contrast agents with an acoustic echographic imaging system, including controlling the concentration of contrast agent in a region of interest by adjusting the acoustic imaging signal to increase or decrease depletion of the contrast agent to the linear range of response. Adjustment...http://www.google.com/patents/US6080107?utm_source=gb-gplus-sharePatent US6080107 - Methods for the use of contrast agents in ultrasonic imagingAdvanced Patent SearchPublication numberUS6080107 APublication typeGrantApplication numberUS 09/237,391Publication dateJun 27, 2000Filing dateJan 26, 1999Priority dateJan 26, 1999Fee statusLapsedPublication number09237391, 237391, US 6080107 A, US 6080107A, US-A-6080107, US6080107 A, US6080107AInventorsMckee D PolandOriginal AssigneeHewlett-Packard CompanyExport CitationBiBTeX, EndNote, RefManPatent Citations (11), Non-Patent Citations (2), Referenced by (49), Classifications (5), Legal Events (10) External Links: USPTO, USPTO Assignment, EspacenetMethods for the use of contrast agents in ultrasonic imagingUS 6080107 AAbstract Methods for use of contrast agents with an acoustic echographic imaging system, including controlling the concentration of contrast agent in a region of interest by adjusting the acoustic imaging signal to increase or decrease depletion of the contrast agent to the linear range of response. Adjustment of the signal includes dividing the received signal level by a power reduction ratio to re-normalize the signal level and compensating the signal power level. A spectral processor detects the concentration of the contrast agent in a gate marker window as the averaged ratio of energy around the second harmonic of the return signal versus energy around the fundamental frequency of the return signal, and includes a boundary detector for detecting a point of greatest change in value of the averaged ratios as a boundary between regions of interest. A well defined bolus of contrast agent in a region of interest is created by scanning to destroy contrast agent entering the region of interest during a predetermined initial interval. A desired contrast agent depletion patterns is generated by scanning a selected portion of the region of interest with a depletion acoustic signal having a power level and duration selected to create a desired contrast agent depletion pattern. The depletion acoustic signals may be separate and different from the acoustic image signals in one or more signal characteristics, including the arrangement, sequence, frequency, pulse length, number of pulses, power levels, focus profiles, apodization and apertures.
What is claimed is: 1. In an acoustic echographic imaging system including a transducer for transmitting an acoustic imaging signal and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent, and a processor for generating the acoustic imaging signal and extracting from the return signal information representing the concentration of contrast agent along the scan line wherein the contrast agent is depleted by an amount related to the power level of the acoustic imaging signal insonicating the contrast agent, a method for controlling the concentration of contrast agent within a range of generally linear response, comprising the steps of:determining from the return signal the relative concentration of contrast agent along the scan line, and from the relative concentration of the contrast agent, adjusting the characteristics of the acoustic imaging signal to increase depletion of the contrast agent when the concentration of contrast agent is higher than the linear range of response and decreasing the depletion of the contrast agent when the concentration of contrast agent is lower than the linear range of response. 2. The method of claim 1 for controlling the concentration of contrast agent within a range of generally linear response, wherein the step of adjusting the characteristics of the acoustic imaging signal further comprises:adjusting the received signal by dividing the received signal by a power reduction ratio that is a function of the time interval elapsed since a previous acoustic imaging signal transmission to re-normalize the power level of the received signal. 3. The method of claim 1 for controlling the concentration of contrast agent within a range of generally linear response, wherein the step of adjusting the characteristics of the acoustic imaging signal further comprises:increasing the power level of a next acoustic imaging signal by an amount to deplete the contrast agent from a saturation level of concentration. 4. In an acoustic echographic imaging system including a transducer for transmitting an acoustic imaging signal having a fundamental frequency and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent, a spectral content measurement processor for detecting the concentration of the contrast agent along the scan line, comprising:a gate marker generator for generating a gate marker window for selecting at least one scan line and a position in the gate marker window along the selected scan line, a signal buffer connected to receive a return signal along the scan line and responsive to the gate marker signal to capture the return signal in the gate marker window, a second harmonic component energy detector for providing a second harmonic energy signal representing the second harmonic energy within the frequency band around the second harmonic of the captured return signal versus axial position along the selected scan line as the gate marker window is moved along the selected scan line, a fundamental frequency component energy detector for providing a fundamental frequency energy signal representing the fundamental frequency energy within the frequency band around the fundamental frequency of the captured return signal versus axial position in the gate marker window and along the selected scan line as the gate marker window is moved along the selected scan line, a ratio determiner for receiving the fundamental frequency energy signal output and the second harmonic energy signal output and generating a ratio output that represents the proportion of second harmonic energy to fundamental frequency energy versus position of the gate marker window along the selected scan line, and a ratio averager for receiving the ratio output and responsive to the gate marker window for selecting in a sub-window of the gate marker window the ratio output along the scan line and generating an averaged ratio output for each selected window of the ratio output wherein an averaged ratio output represents the concentration of contrast agent in the selected window. 5. The spectral content measurement processor for detecting the concentration of the contrast agent along the scan of claim 4, wherein:the spectral content measurement processor repeats the generation of the averaged ratio output at each iteration of the acoustic imaging signal so that the averaged ratio output for each selected window represents the averaged harmonic ratio versus time along the scan line, thereby representing the contrast agent concentration along the selected scan line as a function of time. 6. The spectral content measurement processor for detecting the concentration of the contrast agent along the scan of claim 4, wherein:the second harmonic component energy detector includesa second harmonic mixer for mixing the captured return signal and a second harmonic mixer signal to generate a resultant second harmonic mixed signal, a low pass filter for filtering the resultant second harmonic mixed signal to generate a band limited low filtered second harmonic signal, and a final second harmonic filter for squaring and spatially averaging the low filtered second harmonic signal and providing the second harmonic energy signal output, and the fundamental frequency component energy detector includesa fundamental frequency mixer for mixing the captured return signal and a fundamental frequency mixer signal to generate a resultant fundamental frequency mixed signal, a low pass filter for filtering the resultant fundamental frequency mixed signal to generate a band limited low filtered fundamental frequency signal, a final fundamental frequency filter for squaring and spatially averaging the low filtered fundamental frequency signal and providing the fundamental frequency energy signal output. 7. The spectral content measurement processor for detecting the concentration of the contrast agent along the scan of claim 4, further including a boundary detector for detecting a boundary between regions of interest, comprising:the gate marker generator for selecting a scan line passing through the boundary between the regions of interest and for incrementally stepping the gate marker sub-window along the scan line, and a comparator for comparing the values of the averaged ratio at each step of the sub-window along the scan line and identifying the point of greatest change in value of the averaged ratios along the scan line, thereby identifying the point of greatest change in value of the averaged ratios along the scan line as the point of the boundary between the regions of interest. 8. In an acoustic echographic imaging system including a transducer for transmitting an acoustic imaging signal having a fundamental frequency and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent, a boundary detector for detecting the boundary between regions of interest, comprising:a gate marker generator for selecting a scan line passing through the boundary between the regions of interest and for incrementally stepping a gate marker window along the scan line, a signal buffer connected to receive a return signal along the scan line and responsive to the gate marker signal to capture the return signal in the gate marker window, a second harmonic component energy detector for providing a second harmonic energy signal representing the second harmonic energy within the frequency band around the second harmonic of the captured return signal versus axial position along the selected scan line as the gate marker window is moved along the selected scan line, a fundamental frequency component energy detector for providing a fundamental frequency energy signal representing the fundamental frequency energy within the frequency band around the fundamental frequency of the captured return signal versus axial position in the gate marker window and along the selected scan line as the gate marker window is moved along the selected scan line, a ratio determiner for receiving the fundamental frequency energy signal output and the second harmonic energy signal output and generating a ratio output that represents the proportion of second harmonic energy to fundamental frequency energy versus position of the gate marker window along the selected scan line, and a ratio averager for receiving the ratio output and responsive to the gate marker window for selecting in a sub-window of the gate marker window the ratio output along the scan line and generating an averaged ratio output for each selected sub-window of the ratio output wherein an averaged ratio output represents the concentration of contrast agent in the selected sub-window, and a comparator for comparing the values of the averaged ratio at each step of the sub-window along the scan line and identifying the point of greatest change in value of the averaged ratios along the scan line, thereby identifying the point of greatest change in value of the averaged ratios along the scan line as the point of the boundary between the regions of interest. 9. In an acoustic echographic imaging system including a transducer for transmitting an acoustic imaging signal having a fundamental frequency and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent, a spectral content measurement method for detecting the concentration of the contrast agent along the scan, comprising the steps of:generating a gate marker for selecting at least one scan line and a position along the selected scan line, capturing the return signal as a captured return signal in a region around the gate marker, generating a second harmonic energy signal representing the second harmonic energy within the frequency band around the second harmonic of the captured return signal versus axial position along the selected scan line as the gate marker is moved along the selected scan line, generating a fundamental frequency energy signal representing the fundamental frequency energy within the frequency band around the fundamental frequency of the captured return signal versus axial position along the selected scan line as the gate marker is moved along the selected scan line, generating a ratio output representing the proportion of second harmonic energy to fundamental frequency energy versus position along the selected scan line, and selecting a window of the ratio output along the scan line and generating an averaged ratio output for each selected window of the ratio output wherein an averaged ratio output represents the concentration of contrast agent in the selected window. 10. The spectral content measurement method for detecting the concentration of the contrast agent along the scan of claim 9, further comprising the steps of:repeating the generation of the averaged ratio output at each iteration of the acoustic imaging signal so that the averaged ratio output for each selected window represents the average harmonic ratio versus time along the scan line, thereby representing the contrast agent concentration along the selected scan line as a function of time. 11. The spectral content measurement method for detecting the concentration of the contrast agent along the scan of claim 9, wherein:the step for detecting second harmonic component energy further includesmixing the captured return signal and a second harmonic mixer signal to generate a resultant second harmonic mixed signal, low pass filtering the resultant second harmonic mixed signal to generate a band limited low filtered second harmonic signal, squaring and spatially averaging the low filtered second harmonic signal and providing the second harmonic energy signal output, and the step for detecting the fundamental frequency component energy further includesmixing the captured return signal and a fundamental frequency mixer signal to generate a resultant fundamental frequency mixed signal, low pass filtering the resultant fundamental frequency mixed signal to generate a band limited low filtered fundamental frequency signal, and squaring and spatially averaging the low filtered fundamental frequency signal and providing the fundamental frequency energy signal output. 12. The spectral content measurement processor for detecting the concentration of the contrast agent along the scan of claim 9, further including the method for detecting the boundary between regions of interest, comprising the steps of:selecting a scan line passing through the boundary between the regions of interest and incrementally stepping the gate marker sub-window along the scan line, and comparing the values of the averaged ratio at each step of the sub-window along the scan line and identifying the point of greatest change in value of the averaged ratios along the scan line, thereby identifying the point of greatest change in value of the averaged ratios along the scan line as the point of the boundary between the regions of interest. 13. In an acoustic echographic imaging system including a transducer for transmitting an acoustic imaging signal having a fundamental frequency and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent, a method for detecting a boundary between regions of interest, comprising the steps of:selecting a scan line passing through the boundary between the regions of interest and incrementally stepping a gate marker window along the scan line, capturing the return signal in the gate marker window, generating a second harmonic energy signal representing the second harmonic energy within the frequency band around the second harmonic of the captured return signal versus axial position along the selected scan line as the gate marker window is moved along the selected scan line, generating a fundamental frequency energy signal representing the fundamental frequency energy within the frequency band around the fundamental frequency of the captured return signal versus axial position in the gate marker window and along the selected scan line as the gate marker window is moved along the selected scan line, generating a ratio output representing the proportion of second harmonic energy to fundamental frequency energy versus position of the gate marker window along the selected scan line, selecting in a sub-window of the gate marker window the ratio output along the scan line and generating an averaged ratio output for each selected sub-window of the ratio output wherein an averaged ratio output represents the concentration of contrast agent in the selected sub-window, and comparing the values of the averaged ratio at each step of the sub-window along the scan line and identifying the point of greatest change in value of the averaged ratios along the scan line, thereby identifying the point of greatest change in value of the averaged ratios along the scan line as the point of the boundary between the regions of interest. 14. In an acoustic echographic imaging system including a transducer for transmitting an acoustic signal and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent when a contrast agent is present along the scan line, a signal generator for generating the acoustic imaging signal at controllable power levels, a method for creating a well defined bolus of contrast agent in a region of interest being scanned with the acoustic imaging signal, comprising the steps of:introducing a flow of contrast agent into the region of interest, scanning the region of interest with an acoustic signal having a power level selected to destroy the contrast agent entering the region of interest at a rate to keep the concentration of contrast agent in the region of interest to a predetermined low, steady state level, and after a predetermined time has passed since the flow of contrast agent has begun, the end of that time being designated by a user of the system as the beginning of perfusion of the region of interest by a flow of contrast agent of a desired concentration, beginning image scanning of the region of interest at an acoustic imaging signal power level less than that selected to destroy the contrast agent. 15. The method of claim 14 for creating a well defined bolus of contrast agent in a region of interest being scanned with the acoustic imaging signal, wherein the steps of scanning the region of interest with an acoustic signal to destroy the contrast agent and scanning the region of interest with an acoustic imaging signal further includes the step of:detecting the concentration of the contrast agent in the region of interest by measuring from a return signal from the acoustic signal the spectral content of the return signal to determine the concentration of contrast agent in the region of interest. 16. In an acoustic echographic imaging system including a transducer for transmitting an acoustic signal and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent when a contrast agent is present along the scan line, a signal generator for selectively generating the acoustic signal at selectable power levels, durations and intervals, a method for generating desired contrast agent depletion patterns in a region of interest, comprising the steps of:scanning the region of interest with an acoustic imaging signal and processing the return signal thereof to produce an image of the region of interest, and scanning at least a selected portion of the region of interest with an acoustic depletion signal having a power level and duration selected to create a desired contrast agent depletion pattern wherein the acoustic depletion signal is not used to produce an image. 17. The method for generating desired contrast agent depletion patterns of claim 16, further comprising the step of:generating and transmitting the acoustic depletion signal in the interval between the acoustic image signals. 18. The method for generating desired contrast agent depletion patterns of claim 16 wherein:acoustic depletion signals differ from acoustic image signals in one or more of the following signal characteristics: arrangement, sequence, frequency, pulse lengths, number of pulses, power levels, focus profiles, apodization and apertures of the depletion acoustic signals and acoustic image signals. 19. In an acoustic echographic imaging system including a transducer for transmitting a plurality of acoustic scan lines to form an acoustic scan and receiving a return signal representing an image along a scan line, the return signal including contributions from a contrast agent when a contrast agent is present along the scan line, a signal generator for selectively generating the acoustic scan lines at selectable power levels, durations and intervals, a method for compensating for unwanted contrast agent depletion patterns in a region of interest and resulting from destruction of the contrast agent by the acoustic signal, comprising the steps of:producing acoustic imaging scan lines and processing return signals thereof to produce an image of the region of interest; and producing acoustic depletion scan lines in at least a selected portion of the region of interest, the acoustic depletion scan lines having a power level and duration selected to create a desired contrast agent depletion pattern, wherein the acoustic depletion scan lines are not used to produce an image. 20. The method for compensating for unwanted contrast agent depletion patterns of claim 19, further comprising the step of:generating and transmitting the acoustic depletion scan lines in the interval between scans of the region of interest with the acoustic image scan lines. 21. The method for compensating for unwanted contrast agent depletion patterns of claim 19 wherein:the acoustic depletion scan lines differ from the acoustic image scan lines in one or more of the following signal characteristics: arrangement, sequence, frequency, pulse lengths, number of pulses, power levels, focus profiles, apodization and apertures of the depletion acoustic signals and acoustic image signals. Description
TECHNICAL FIELD The present invention relates to improved methods and apparatus for use of a contrast agent in ultrasonic imaging and, in particular, to improved methods and apparatus for the detection and measurement of contrast agents in regions of interest and the creation of well defined boluses of contrast agent in regions of interest.
BACKGROUND ART Ultrasonic transducers and imaging systems are used in many medical applications and, in particular, for the non-invasive acquisition of images of organs and conditions within a patient, typical examples being the ultrasound imaging of fetuses and the heart. Such systems commonly use a linear or phased array transducer having multiple transmitting and receiving elements to transmit and receive narrowly focused and "steerable" beams, or "lines", of ultrasonic energy into and from the body. The received beams, or lines, are reflected from the body's internal structures and contain amplitude or phase information, or both, that is used to generate images of the body's internal structures.
DISCLOSURE OF THE INVENTION In a first aspect, the present invention is directed to a method for use with an acoustic echographic imaging system for controlling the concentration of contrast agent in a region of interest to within a range of generally linear response wherein the contrast agent is depleted by an amount related to the power level of the acoustic imaging signal insonicating the contrast agent.
BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 a diagrammatic block diagram of an ultrasonic echographic system;
BEST MODE FOR CARRYING OUT THE INVENTION The following will first describe a generalized ultrasonic echographic system in which the present invention may be implemented, and will then describe individual aspects of the present invention in detail.
Patent CitationsCited PatentFiling datePublication dateApplicantTitleUS5255683 *Dec 30, 1991Oct 26, 1993Sound Science Limited PartnershipMethods of and systems for examining tissue perfusion using ultrasonic contrast agentsUS5560364 *May 12, 1995Oct 1, 1996The Board Of Regents Of The University Of NebraskaSuspended ultra-sound induced microbubble cavitation imagingUS5577505 *Feb 6, 1996Nov 26, 1996Hewlett-Packard CompanyMeans for increasing sensitivity in non-linear ultrasound imaging systemsUS5694937 *Jan 30, 1996Dec 9, 1997Kabushiki Kaisha ToshibaUltrasound diagnostic apparatus and methodUS5735281 *Aug 9, 1996Apr 7, 1998Hewlett-Packard CompanyMethod of enhancing and prolonging the effect of ultrasound contrast agentsUS5833613 *Sep 27, 1996Nov 10, 1998Advanced Technology Laboratories, Inc.Ultrasonic diagnostic imaging with contrast agentsUS5833615 *May 9, 1997Nov 10, 1998Thomas Jefferson UniversityExcitation enhanced ultrasound systemUS5860931 *Oct 10, 1997Jan 19, 1999Acuson CorporationUltrasound method and system for measuring perfusionUS5944666 *Aug 21, 1997Aug 31, 1999Acuson CorporationUltrasonic method for imaging blood flow including disruption or activation of contrast agentUS5947904 *Nov 12, 1998Sep 7, 1999Acuson CorporationUltrasonic method and system for imaging blood flow including disruption or activation of a contrast agentUS5957845 *Sep 23, 1998Sep 28, 1999Acuson CorporationGated ultrasound imaging apparatus and method* Cited by examinerNon-Patent CitationsReference1"Transient Myocardial Contrast After Initial Exposure To Diagnostic Ultrasound Pressures With Minute Doses Of Intravensouly Injected Microbubbles", Dr. Thomas Porter, et al. Circulation, vol. 92, (1995), pp. 2391-2395.2 *Transient Myocardial Contrast After Initial Exposure To Diagnostic Ultrasound Pressures With Minute Doses Of Intravensouly Injected Microbubbles , Dr. Thomas Porter, et al. Circulation, vol. 92, (1995), pp. 2391 2395.* Cited by examinerReferenced byCiting PatentFiling datePublication dateApplicantTitleUS6340348Jul 2, 1999Jan 22, 2002Acuson CorporationContrast agent imaging with destruction pulses in diagnostic medical ultrasoundUS6398735 *Mar 7, 2000Jun 4, 2002Koninklijke Philips Electronics N.V.Detecting a relative level of an ultrasound imaging contrast agentUS6413218 *Feb 10, 2000Jul 2, 2002Acuson CorporationMedical diagnostic ultrasound imaging system and method for determining an acoustic output parameter of a transmitted ultrasonic beamUS6450961 *Jun 5, 2000Sep 17, 2002Kabushiki Kaisha ToshibaUltrasound imaging using flash echo imaging techniqueUS6485427 *Jul 18, 2001Nov 26, 2002Jen-Shih LeeCompressibility probe for monitoring blood volume changesUS6494838 *Jul 23, 2001Dec 17, 2002Koninklijke Philips Electronics N.V.Ultrasonic diagnostic imaging with interpolated scanlinesUS6494841 *Feb 29, 2000Dec 17, 2002Acuson CorporationMedical diagnostic ultrasound system using contrast pulse sequence imagingUS6514203 *Feb 12, 2001Feb 4, 2003Sonata Technologies Ltd.Method for ultrasonic coronary thrombolysisUS6533728Nov 20, 2001Mar 18, 2003Mayo Foundation For Medical Education And ResearchMethod and apparatus for recovery and parametric display of contrast agents in ultrasound imagingUS6547738May 3, 2001Apr 15, 2003Ge Medical Systems Global Technology Company, LlcMethods and apparatus for using ultrasound with contrast agentUS6611778 *Nov 16, 2001Aug 26, 2003Yeda Research And Development Co., Ltd.Apparatus for monitoring a system in which a fluid flowsUS6641538 *Nov 18, 2002Nov 4, 2003Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and method of controlling a ultrasonic diagnostic apparatusUS6850792 *Dec 13, 2001Feb 1, 2005Kabushiki Kaisha ToshibaContrast medium injector and diagnosis system equipped with the sameUS6872180 *Mar 28, 2003Mar 29, 2005Schering AgDevice and process for quantifying bodies by means of ultrasoundUS6878113 *Jul 2, 2001Apr 12, 2005Hitachi Medical CorporationUltrasonic diagnostic apparatus and method for processing ultrasonic signalUS6899681 *Feb 15, 2002May 31, 2005Acuson CorporationAutomated power level for contrast agent imagingUS7066888Oct 29, 2004Jun 27, 2006Allez Physionix LtdMethod and apparatus for determining an ultrasound fluid flow centerlineUS7110903 *Jan 15, 2003Sep 19, 2006Yeda Research & Development Co., Ltd.Apparatus for monitoring a system with time in space and method thereforUS7194145 *Jan 28, 2003Mar 20, 2007General Electric CompanyMethod for contrast matching of multiple images of the same object or scene to a common reference imageUS7206460 *Nov 1, 2001Apr 17, 2007General Electric CompanyMethod for contrast matching of multiple images of the same object or scene to a common reference imageUS7238158Jan 26, 2004Jul 3, 2007Allez Physionix, Ltd.Pulse interleaving in doppler ultrasound imagingUS7285095 *Dec 23, 2003Oct 23, 2007Yd, Ltd.Method of analyzing and displaying blood volume using myocardial blood volume mapUS7302850Apr 27, 2004Dec 4, 2007Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and image processing apparatusUS7399279Apr 23, 2004Jul 15, 2008Physiosonics, IncTransmitter patterns for multi beam receptionUS7437256Mar 7, 2007Oct 14, 2008Yeda Research And Development Co. Ltd.Apparatus for monitoring a system with time in space and method thereforUS7534209Dec 20, 2002May 19, 2009Physiosonics, Inc.Device and method for mapping and tracking blood flow and determining parameters of blood flowUS7534210 *Feb 3, 2004May 19, 2009Siemens Medical Solutions Usa, Inc.Methods for adaptively varying gain during ultrasound agent quantificationUS7588538 *Sep 10, 2004Sep 15, 2009Kabushiki Kaisha ToshibaUltrasonic diagnostic equipment and image processing apparatusUS7625341 *Mar 4, 2005Dec 1, 2009Koninklijke Philips Electronics N.V.Elevation beamwidth control for contrast imagingUS7717850 *Nov 24, 2004May 18, 2010Imacor Inc.Signal processing for ultrasound imagingUS7881897Sep 23, 2008Feb 1, 2011Yeda Research And Development Co. Ltd.Apparatus for monitoring a system with time in space and method thereforUS7886603Oct 25, 2007Feb 15, 2011Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and image processing apparatusUS8021303Jun 25, 2007Sep 20, 2011Bracco Research SaSystem for extracting morphological information through a perfusion assessment processUS8043219 *Sep 17, 2004Oct 25, 2011Siemens Medical Solutions Usa, Inc.Automated power level for contrast agent imagingUS8069002Dec 17, 2010Nov 29, 2011Yeda Research And Development Co., Ltd.Apparatus for monitoring a system with time in space and method thereforUS8328724 *May 8, 2007Dec 11, 2012Mayo Foundation For Medical Education And ResearchMethod for imaging intracavitary blood flow patternsUS8460194Jan 23, 2009Jun 11, 2013Michalakis AverkiouTherapy assessment with ultrasound contrast agentsUS8491482Dec 12, 2005Jul 23, 2013Bracco Suisse S.A.Blood flow estimates through replenishment curve fitting in ultrasound contrast imagingUS8496591Dec 21, 2005Jul 30, 2013Bracco Suisse S.A.Perfusion assessment method and system based on bolus administrationUS8499636Sep 17, 2010Aug 6, 2013Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and image processing apparatusUS20090187100 *May 8, 2007Jul 23, 2009Sengupta Partho PMethod for Imaging Intracavitary Blood Flow PatternsEP1314398A1 *Nov 18, 2002May 28, 2003Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatusEP1472981A1 *Apr 28, 2004Nov 3, 2004Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and image processing apparatusEP1674038A1 *Dec 23, 2004Jun 28, 2006Bracco Research S.A.System for extracting morphological information through a perfusion assessment processEP1820452A1 *Apr 28, 2004Aug 22, 2007Kabushiki Kaisha ToshibaUltrasonic diagnostic apparatus and image processing apparatusWO2001001865A1 *Jun 30, 2000Jan 11, 2001AcusonDestruction pulse ultrasound contrast agent imagingWO2004087009A2 *Mar 26, 2004Oct 14, 2004Vuesonix Sensors IncMapping and tracking blood flow using reduced-element probeWO2006067203A1 *Dec 21, 2005Jun 29, 2006Bracco Research SaA system for extracting morphological information through a perfusion assessment processWO2009093211A1 *Jan 23, 2009Jul 30, 2009Michalakis AverkiouTherapy assessment with ultrasonic contrast agents* Cited by examinerClassifications U.S. Classification600/458International ClassificationA61B8/00Cooperative ClassificationA61B8/543, A61B8/481European ClassificationA61B8/48BLegal EventsDateCodeEventDescriptionAug 14, 2012FPExpired due to failure to pay maintenance feeEffective date: 20120627Jun 27, 2012LAPSLapse for failure to pay maintenance feesFeb 6, 2012REMIMaintenance fee reminder mailedJun 17, 2009ASAssignmentOwner name: KONINKLIJKE PHILIPS ELECTRONICS N V, NETHERLANDSFree format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNOR:AGILENT TECHNOLOGIES, INC.;REEL/FRAME:022835/0572Effective date: 20090610Nov 30, 2007FPAYFee paymentYear of fee payment: 8Nov 14, 2003FPAYFee paymentYear of fee payment: 4Oct 31, 2003ASAssignmentOwner name: KONINKLIJKE PHILIPS ELECTRONICS N.V., NETHERLANDSFree format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNOR:AGILENT TECHNOLOGIES, INC.;REEL/FRAME:014662/0179Effective date: 20010801Owner name: KONINKLIJKE PHILIPS ELECTRONICS N.V. GROENEWOUDSEWMay 30, 2000ASAssignmentOwner name: AGILENT TECHNOLOGIES INC, CALIFORNIAFree format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNOR:HEWLETT-PACKARD COMPANY;REEL/FRAME:010977/0540Effective date: 19991101Owner name: AGILENT TECHNOLOGIES INC INTELLECTUAL PROPERTY ADMApr 24, 2000ASAssignmentOwner name: HEWLETT-PACKARD COMPANY, COLORADOFree format text: MERGER;ASSIGNOR:HEWLETT-PACKARD COMPANY;REEL/FRAME:010759/0049Effective date: 19980520Owner name: HEWLETT-PACKARD COMPANY INTELLECTUAL PROPERTY ADMIMar 22, 1999ASAssignmentOwner name: HEWLETT-PACKARD COMPANY, CALIFORNIAFree format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNOR:POLAND, MCKEE D.;REEL/FRAME:009835/0726Effective date: 19990125RotateOriginal ImageGoogle Home - Sitemap - USPTO Bulk Downloads - Privacy Policy - Terms of Service - About Google Patents - Send FeedbackData provided by IFI CLAIMS Patent Services©2012 Google

value: This website is a search engine provided by Google. Handling over 3.5 billion searches per day, it has a 92% share of the global search engine market.


text:
Talk:Ordered pair - Wikipedia
Talk:Ordered pair
2 Extrapolating the tuple sequence
3 Other pairs
4 Who the hell is Morse?
5 Wiener definition
6 definition of caartesian product uses ordered pair, but Morse definition of ordered pair uses cartesian product
7 Set vs Multiset?
8 Mistake in introduction?
9 Another Definition and Proofs
11 In mathematics, an ordered pair is a collection of two distinguishable objects
12 Recent article by Dana Scott and McCarty
13 Math formatting
15 Ackermann set theory + proper classes
16 Please "spell out" Kuratowski
17 Couple of questions/suggestions
18 Coordinate extraction formulas
19 Ambiguity
20 Contradiction with Tarski–Grothendieck set theory article
21 Mathematical objects?
22 Boldface in the "Generalities" section
If (x,y) is {{x},{x,y}}, what is (x,x)? -phma
Obviously, (x,x) is {{x}, {x,x}} which is the same as {{x},{x}}. But then the expression that determines the second element is wrong. Suggestions? --Rade Kutil
No, it's quite right. The ordered pair (x,x) is symmetric -- switching the elements has no effect, and that's exactly what is reflected by the set notation. we get (x,x) = { {x} } after full reduction. -- Tarquin
Yes, but corresponding to what the article says, x would not be the second element of (x,x), which is wrong. --Rade
Then those statements are badly formulated. I've just checked my notes from my university logic class, and it seems we didn't linger much on this. It's not a definition you're meant to work with. The 2nd statement on the main page fails here because it is basically saying "x is the second element IF it's in the pair but not the first element". So if x is both elements, it fails. -- Tarquin
No; the second statement doesn't fail for (x,x) = {{x}}, and both statements are correct. The second statement says in words "x is the second element IF (i) it's in the pair AND, (ii) if there are distinct elements in p, then x is only in one of them".
(exist Y in p : x in Y) and (for all Y1 in p, for all Y2 in p : if Y1 ≠ Y2 then (x not in Y1 or x not in Y2)).
Since for (x,x), p consists of a single set, Y = {x}, then possibility described in the second clause can never occur - it is never the case that Y1 ≠ Y2, so the second clause of the above statement is trivially true; just as the statement {for all Y in p: if 6 = 9, then Jimi Hendrix lives} is trivially true. Chas zzz brown 06:49 Nov 20, 2002 (UTC)
Yes, it is correct now. It wasn't when this discussion was going on and until AxelBoldt inserted the correct statement. --Rade
Beyond that, though, the definition of (x,y) as {{x},{x,y}} is arbitrary. It's a "dummy definition", if you will, like there are "dummy variables".
Hmmm... putting on my Platonic hat, I have to disagree here that this is an arbitrary "dummy" definition. It's a definition, and a valid one at that; no different than, for example, the definition of ∑i=1 to 5{f(i)} as shorthand for f(1) + f(2) + f(3) + f(4) + f(5). As noted, it's rather clumsy to write in set notation; and thus the usual ordered pair is both clearer and more natural to work with; but I sleep better at night knowing that there is a consistent definition underlying this shorthand! Chas zzz brown 06:49 Nov 20, 2002 (UTC)
It serves only to give a solid set theory foundation to what we do with ordered pairs -- once we've seen it works we can ignore it. Analogously, complex numbers are initially defined as ordered pairs, and once we introduce the notation i = (0,1), we can cleanly forget about the ordered pair and write x+iy -- Tarquin
Why not use the definition (x,y) as {x,{x,y}}? It could avoid the confusion when x=y. -- Wshun
This then requires the axiom of regularity (a.k.a. axiom of foundation) to handle the possibility that one has sets x and z, with x={z} and z={x}, but not x = z. Then (''x'',''x'') = {''x'', {''x'', ''x''}} = {''x'', {''x''}} = {''x'', ''z''} = {{''z''}, ''z''} = {''z'', {''z'', ''z''}} = (''z'',''z''). Bzzzzt!
If one assumes the axiom of foundation (as usual in ZF), then x = {{''x''}} is forbidden (sets then can't have themselves as members); and then your definition works. The advantage of (x,y) = {{''x''}, {''x'', ''y''}} is that it is more general, as it still works without foundation; so it can even be applied to classes as well as sets, etc.. While there may be some confusion when x=y, the definition still works. Chas zzz brown 22:40 Feb 22, 2003 (UTC)
If (a,b) = (c,d) and we are trying to prove that a=c and b=d (which is the main requirement for a definition of ordered pair), then {a,{a,b}} = {c,{c,d}}. Then either a=c or a= {c,d}. If a=c then {a,b}={a,d}, so that b=d and we are done with the first case. The second case; if a= {c,d} then {a,b} = c. Then there is an infinite recursion: ''a''= {''c'',''d''} = {{''a'',''b''},''d''} = {{{''c'',''d''},''b''},''d''} etc. and then the proof that a=c, b=d cannot be completed. Well, it could still be possible, if a=c then {a,b} = {c,d}, {a,b} = {a,d}, then b=d, QED. But there is no guarantee that a=c in this second case, is there? (NO QED)
On the other hand, letting (''a'',''b'') = {{''a''},{''a'',''b''}}, if (''a'',''b'') = (''c'',''d'') then {''a''} = {''c''} OR {''a''} = {''c'',''d''}. If {''a''} = {''c''} then ''a''=''c''. If {''a''} = {''c'',''d''} then card {''a''} = 1 = card {''c'',''d''} so ''c''=''d''; {''a''} = {''c'',''c''} = {''c''}, therefore ''a''=''c'' in both cases. Then {{''a''},{''a'',''b''}} = {{''a''},{''a'',''d''}}, thus {''a'',''b''}={''a'',''d''}, ''b''=''d''. QED.
-- AugPi
Here's a proof written out on Metamath: http://us.metamath.org/mpegif/opthreg.html -- Sekkusu —Preceding unsigned comment added by 67.201.195.107 (talk) 22:02, 29 January 2008 (UTC)
If it is required that card (x,y) = 2 even when x=y or there is any dependence between x and y, define (x,y) as
( x , y ) = { { 0 , x } , { 1 , x , { y } } } {\displaystyle (x,y)=\{\{\,0,x\},\{\,1,x,\{y\}\}\}}
Then to define an ordered triple so that card (a,b,c) = 3, let (a,b,c) = { (0,a), (1,b), (2,c)} where 0,1, and 2 are Von Neumann cardinals: 0 = {}, 1={{}}, 2={{},{{}}}. Likewise, for an n-tuple, let (a1, a2, ..., an) = {(0,a1), (1,a2), ... , (n-1,an)}, then the cardinality of such n-tuple will be n. -- AugPi
Extrapolating the tuple sequence[edit]
Can the sequence triple, quadruple, etc. be extrapolated?? Let me see:
4. quadruple
5. quintuple
6. sextuple
7. septuple
8. octuple
9. nonuple
10. decuple
11. undecuple
12. duodecuple
13. tredecuple
14. quattuordecuple
15. quindecuple
16. sexdecuple
17. septendecuple
18. octodecuple
19. novemdecuple
20. vigintuple
30. trigintuple
40. quadrigintuple
50. quinquagintuple
60. sexagintuple
70. septuagintuple
80. octogintuple
90. nonagintuple
100. centuple
125. quasquicentuple
150. sesquicentuple
175. terquasquicentuple
200. bicentuple
300. tercentuple
400. quatercentuple
500. quincentuple
1000. milluple
10^6. milliontuple
66.245.6.12 00:50, 25 Aug 2004 (UTC)
Other pairs[edit]
I replaced the text In the usual Zermelo-Fraenkel formulation of set theory including the axiom of regularity, ordered pairs (a, b) can also be defined as the set {a, {a, b}}. However, the axiom of regularity is required, since without it, it is possible to consider sets x and z such that x = {z}, z = {x}, and x ≠ z. Then we have that
(x, x) = {x, {x, x}} = {x,{x}} = {x, z} = {z, x} = {z, {z}} = {z, {z, z}} = (z, z)
although we want (x,x) ≠ (z,z).
by a few examples of possible other definitions, plus reasons why they are not used. Aleph4 13:44, 18 Feb 2005 (UTC)
I added a reference to the first definition of the ordered pair using sets alone (due to Norbert Wiener, 1914) and attributed the Rosser pair correctly to Willard van Orman Quine.
Randall Holmes 21:39, 14 December 2005 (UTC)
Who the hell is Morse?[edit]
Who the hell is Morse?
Anthony Perry Morse (1911-1984), Professor of Mathematics, University of California at Berkeley. He ought to get a Wikipedia article. photo bio advisor and students --Hoziron 14:00, 26 February 2006 (UTC)
Wiener definition[edit]
Is the Wiener definition actually
( x , y ) = { { { x } , ∅ } , { { y } } } {\displaystyle (x,y)=\{\{\{x\},\emptyset \},\{\{y\}\}\}} ?
Why did he not use
( x , y ) = { { { x } , ∅ } , { y } } {\displaystyle (x,y)=\{\{\{x\},\emptyset \},\{y\}\}} ? --NeoUrfahraner 06:13, 22 March 2006
Because he introduced his definition the the framework of Russell and Whitehead's theory of types. There all elements in a class must be of the same type, namely one less than the type of the class (they are elements of). Now if x and y are of, say, type n, then {x} and {y} would be of type n+1, and {{x}, 0} would be of type n+2. Now putting {{x}, 0} and {y} in the same class would mean a "type clash". Hence we have to use { {y} } --which is also of type n+2-- instead of {y}.</nowiki> —Preceding unsigned comment added by 91.59.209.222 (talk) 00:09, 3 December 2007 (UTC)
definition of caartesian product uses ordered pair, but Morse definition of ordered pair uses cartesian product[edit]
in the 2nd paragraph it says:
"The notion of ordered pair is crucial for the definition of Cartesian product"
in the paragraph "morse definition" it then says
( x , y ) = { x × { 0 } } ∪ { y × { 1 } } {\displaystyle (x,y)=\{x\times \{0\}\}\cup \{y\times \{1\}\}} which uses a cartesian product. so an ordered pair is defined in terms of a cartesian product which is defined in terms of an ordered pair which is defined in terms of a cartesian product etc 12:38, 7 April 2007 (UTC)
Set vs Multiset?[edit]
As the Kuratowski definition defines an ordered pair (a,b) as the set (a,b)= {{a}, {a,b}}, what if a=b? Quote: "However, for purposes of foundations of mathematics it has been considered desirable to express the definition of every type of mathematical object in terms of sets". That means the 'set' {a,b} isn't a set but a multiset. Am I correct or just running in circles?
If a=b, then ( a , b ) = { { a } } {\displaystyle (a,b)=\{\{a\}\}} , i.e. if the set defining the pair has only one element, then both components of the ordered pair are equal. I see no need for a multiset. --NeoUrfahraner 10:28, 28 September 2007 (UTC)
Mistake in introduction?[edit]
"In mathematics, a pair (a,b) is ordered if (a,b) is not equivalent to (b,a)."
Isn't that only true if it is also true that a is not equivalent to b?- (User) WolfKeeper (Talk) 00:15, 24 February 2008 (UTC)
The person who made this edit appeared to be trying to clarify the introduction, which currently seems to be rather technical. The addition doesn't seem to handle the possibility that a and b are the same thing. I've reverted the edit, since I don't see an obvious way to make the introduction clearer. Michael Slone (talk) 03:26, 24 February 2008 (UTC)
I attempted some rewording to the idea clearer and less technical (or at least easier to read...). Although probably more understandable, I'm not sure if it's rigorous enough. We might want a multi-paragraph lede to explain the concepts in more detail. —AySz88\^-^ 08:21, 2 March 2008 (UTC)
Another Definition and Proofs[edit]
It would be nice to provide a proof for the simple treatment [(x,y)={x,{x,y}}] as this is the definition given in many introductory math texts. Also, I remember somewhere seeing another definition of the ordered pair (maybe in Quine's Philosophy of Logic?) such that:
The ordered pair (x,y) = {x,{y}}. I'm probably missing something (and perhaps this definition runs into difficulties with iteration) -- but I don't think I saw that definition in the article. Thoughts? —Preceding unsigned comment added by 169.231.34.169 (talk) 02:33, 30 June 2008 (UTC)
Can we delete the "reverse definition"? Its completely identical to the "regular" definition, and the distinction is only syntactic —Preceding unsigned comment added by AbnormallyNormal (talk • contribs) 14:49, 3 September 2008 (UTC)
In mathematics, an ordered pair is a collection of two distinguishable objects[edit]
Having the Kuratowski pair <{},{}>, it is totally unclear to me, what does "distinguish" {} and {} mean? Peta 77.104.243.33 (talk) 17:42, 27 March 2009 (UTC)
Wouldn't a changed formulation like:
"In mathematics, an ordered pair is a collection of objects having two coordinates (entries, projections), such that one can always uniquely determine the object, which is the first coordinate (first entry, left projection) of the pair as well as the second coordinate (second entry, right projection). If the first coordinate is a and the second is b, the usual notation for an ordered pair is (a, b). The pair is "ordered" in that (a, b) differs from (b, a) unless a = b."
better communicate, that the coordinates of the pair need not be "distinguishable"? Peta 77.104.243.33 (talk) 09:28, 30 March 2009 (UTC)
The first sentence is a description of a Cartesian plane geometry, not of an ordered pair; a pair is certainly not a "collection" of pairs. The language here also seems to say that the object is the first coordinate, and the role of the second is left unclear.
Nor is geometry the only application of ordered pairs.
My attempt (five months ago) to improve this muddle was reverted today with this note: "distinguished" was incorrect, both objects can be the same, i.e. indistinguishable. So what? If two coordinates happen to have equal values, they are still distinguished by their function. In the pair (2,2), is there any doubt which is the first coordinate and which is the second? —Tamfang (talk) 18:38, 18 November 2010 (UTC)
My notes and reasons why I reverted your formulation:
In your formulation you replaced "collection of objects" by "combination of objects". That is an error, since according to Combination_(disambiguation) "In mathematics, a combination is a collection of things in no specific order.", while in ordered pair, the order is what matters.
In your formulation you replaced "collection of objects" by "combination of two objects". That is an error, since the "two objects" can in fact be just one object.
In your formulation you wrote that the "two objects" were "distinguishable". That is an error, since as mentioned, there may not be "two objects", in which case, there are no "two objects" to distinguish. To notice, how problematic such a formulation is, notice, that the most commonly used (Kuratowski) definition, gives us { { ∅ } } {\displaystyle \left\{\left\{\emptyset \right\}\right\}} as an example of an ordered pair. Where you have your "two objects", is a mystery.
Certainly, the coordinates are always distinguishable, in the sense, that you can tell which object is the first coordinate, as well as tell, which object is the second coordinate, but that is a different sense than the one you used in your formulation.Ladislav Mecir (talk)
Good point about combination; but doesn't the same objection apply to collection? — How about structure? or an object consisting of two elements?
The most natural reading of "a collection of objects having two coordinates" is something with two levels of structure, viz a set of ordered pairs; otherwise either objects or collection is redundant.
Thank you for pointing out Kuratowski's pathological example. (In { { ∅ } } {\displaystyle \left\{\left\{\emptyset \right\}\right\}} , how do you tell which coordinate is which?) You see it as disallowing the obvious operational definition; I see it as a good reason to avoid torturing oneself by hunting for essences.
That's exactly the sense of distinguished that I used. Did you confuse it with distinct (i.e. 'unequal')? —Tamfang (talk) 23:58, 18 November 2010 (UTC)
A "collection" is certainly a more "basic" notion, than a "combination", as is demonstrated by the fact, that a combination is defined as a collection of a specific kind.
I tend to agree, that the word "objects" looks redundant in the present formulation, so would it be better if we wrote: "In mathematics, an ordered pair is a collection (of objects) having two coordinates (or entries or projections), such that it is distinguishable, which object is the first coordinate (or first entry or left projection) of the pair and which object is the second coordinate (or second entry or right projection) of the pair." Would that be an improvement?
Re "how do you tell which coordinate is which?" - it is easy, the first coordinate is the ∅ {\displaystyle \emptyset } , which happens to be the second coordinate as well in this specific case.Ladislav Mecir (talk) 00:29, 19 November 2010 (UTC)
,i.e. we are able to "distinguish" the coordinates in the sense, that we can "distinguish", which object is the first coordinate (the ∅ {\displaystyle \emptyset } ), as well as "distinguish" which object is the second coordinate (the same ∅ {\displaystyle \emptyset } ), while we certainly cannot distinguish ∅ {\displaystyle \emptyset } from ∅ {\displaystyle \emptyset } . Ladislav Mecir (talk) 11:05, 20 November 2010 (UTC)
In fact, the "Kuratowski pathological example" cannot be attributed to Kuratowski, since the ordered pair (a, a) has this property for any definition used. - Let's just for a moment suppose, that a denotes you. It certainly isn't possible to distinguish you from you, while it certainly is possible to distinguish, which object is the first coordinate of such a pair (you) as well as which object is the second coordinate of the said pair (you again).Ladislav Mecir (talk) 00:47, 19 November 2010 (UTC)
It certainly is possible to distinguish two representations of me from each other. —Tamfang (talk) 23:06, 19 November 2010 (UTC)
Is "99" a one-digit number? —Tamfang (talk) 05:54, 19 November 2010 (UTC)
once again, your formulation "two objects ... the first ... is distinguished from the second" is the formulation you wrote, and it is an error, since that is not true. That is like saying that in "99", the first "9" is distinguished from the second "9", while it is not, they are both the same character. What can be distinguished is not "the first from the second", which can not be distinguished, but "what is the first" (between all conceivable objects) and "what is the second" (between all conceivable objects). The fact, that both can be the same clearly disallows to distinguish the first from the second.Ladislav Mecir (talk) 08:16, 19 November 2010 (UTC)
Ordinary people would say "99" consists of two nines, a first "9" and a second "9", distinguished by their position. You insist that two congruent objects are the same object. Such an axiom may be useful for some purposes, but I doubt there's a consensus for applying it universally. It makes your definition obscure, with no apparent benefit.
Could you tolerate a plainer definition (before or after yours) if it were labelled as "informal"? —Tamfang (talk) 20:30, 19 November 2010 (UTC)
Your formulation: 'first "9" and second "9" distinguished by their position' is just an error. It requires all objects to have a position in every pair as their own property. But, e.g. an empty set, or many other mathematical objects simply do not have such a property. In your error, an ordered pair having you as its first as well as second coordinate cannot exist. I can feel sorry for you, but cannot allow you to write such falsities into an encyclopedia article. I cannot help but wonder, how people making more than three serious errors in one sentence can feel to be "chosen" to correct it.Ladislav Mecir (talk) 21:34, 19 November 2010 (UTC)
"You insist that two congruent objects are the same object." - error again. I just insist, that there are such ordered pairs that contain only one object, which plays the role of both their first as well as second coordinate, and that this property is essential for ordered pairs to be what they are.Ladislav Mecir (talk) 21:57, 19 November 2010 (UTC)
Your argument is narrow-minded, if not circular, and your tone is increasingly offensive.
"It requires all objects ... as their own property": No, it requires the representations of the objects to have an attribute which we may call context.
I chose myself to propose language that might make sense to the layman, just as you chose yourself to defend the true religion of obscure mumbo-jumbo. Any other self-nominees are welcome to put their oar in. —Tamfang (talk) 23:10, 19 November 2010 (UTC)
If mentioning one dollar in two different contexts were capable of making two distinguishable dollars from it, I would have been using such a "device" for years.Ladislav Mecir (talk) 10:58, 20 November 2010 (UTC)
Thanks for demonstrating the limits of your understanding. —Tamfang (talk) 17:25, 20 November 2010 (UTC)
Maybe this will help: what is distinguishable, is, which formulation (context, or projection) we use to refer to the object, i.e. whether we say, that "the object is the first coordinate of the pair", or "the object is the second coordinate of the pair". What is not distinguishable, is the object being the first coordinate of the pair, from itself, being the second coordinate of the pair as well.Ladislav Mecir (talk) 11:27, 20 November 2010 (UTC)
Recent article by Dana Scott and McCarty[edit]
I wish to draw everyone's attention to the following article:
Dana Scott and McCarty, Dominic (2008) "Reconsidering Ordered Pairs," Bulletin of Symbolic Logic 14(3): 379-97.
The entry should mention this article in some way, and incorporate a bit of its rich content.111.69.249.90 (talk) 12:42, 25 October 2010 (UTC)
Math formatting[edit]
There is some mixture of math formatting on this article. For simple ASCII math (anything with just letters, punctuation, and subscript/superscripts), I propose we just use HTML formatting and lose all the "math" tags, many of which are showing up as images for me. — Carl (CBM · talk) 13:58, 4 April 2011 (UTC)
Agree. HTML is preferred where possible. Paul August ☎ 14:21, 4 April 2011 (UTC)
I also agree. There is nothing on this page that requires math tags, and that is not likely to change. Hans Adler 14:24, 4 April 2011 (UTC)
I'd like to nominate the first sentence of this article as the most obscure definition of the most elementary concept. Really, someone who's mathematical knowledge is sufficiently slight that they don't know what "ordered pair" means will not have a chance of understanding the first sentence.
Having said that, I'll try to fix it. McKay (talk) 06:44, 2 September 2011 (UTC)
"Fix"? It contains just your original research "a list of two things" - Where exactly did you find that an ordered pair is a "list"? Where exactly did you find that ordered pair shall contain "things"? Where exactly did you find that ordered pair has to contain "two" "things"?81.200.57.149 (talk) 00:29, 13 October 2011 (UTC)
Ackermann set theory + proper classes[edit]
Does it make sense to use the Kuratowski ordered pair definition for proper classes in Ackermann Set Theory resp. its extension ARC? If so, shouldn't this also be mentioned? -- 132.231.198.38 (talk) 15:32, 22 June 2012 (UTC)
Please "spell out" Kuratowski[edit]
Could someone _please_ do us the favor of spelling out how exactly these definitions, let's pick Kuratowski since it's so common, actually "define" an ordered pair?
Kuratowski says: (a,b) := {{a}, {a,b}}
Let's suppose a and b are very simple objects, say 6 and 13. So from Kuratowski we can supposedly get to the ordered pair (6,13) from this:
{{6}, {6,13}}
... so at the "top level" this is a set that contains two sets, in no particular order. How is this equivalent to a pair which is ordered and contains integers? Clearly I, for one, am missing something here. Either there is more to the notation, or something more is being implied. Thanks for any enlightenment! Gwideman (talk) 11:59, 25 August 2012 (UTC)
It's pretty clear that all by itself, just from the symbol string {{a}, {a,b}}, a mechanism (e.g. a Turing machine) cannot determine the first element and then the the second element and then properly write the symbol string (a, b). The TM will need a set of rules embedded in its "program" to guide it. (Intuitively for folks who like puzzles, it should be suspicious that there's a singleton {a} and then a pair {a, b} and the element in the singleton, a, appears in as an element in the pair {a, b}.) By the time Suppes 1970:22-23 introduces the ordered pair (Kuratowski version) he has introduced 5 axioms of set theory:
1 Axiom of extension (set A equals set B if and only if they have the same elements)
2 Axiom of specification (a "condition" S(x) creates a set B from the set A for all elements of A for which S(x) holds)
3 Axiom of pairing (creates unordered pairs)
4 Axiom of union (combining sets)
5 Axiom of powers (power set E contains all subsets of the given set)
So by the time we reach his Section 6 Ordered Pairs there's powerful set of rules available to define the notion of ordering. By use of the definition (a, b) = {{a}, {a,b}}, Suppes' proof proceeds by cases and it has to do both the if and the only-if parts, so unfortunately it is rather long (about 200 words) and somewhat tedious. It begins with the statement "We have to show that if (a, b) and (x, y) are ordered pairs and if (a, b) = (x, y), then a = x and b = y." It proceeds through all the various singleton cases, then arrives at the x <> y case.
Maybe someone knows a shorter proof somewhere. Wvbailey (talk) 16:22, 25 August 2012 (UTC)Bill
Wvbailey: Thanks for your response. Your "Turing machine" comment succinctly captures where I was stuck, and indeed the "program to guide it" is instrumental in understanding this ostensibly simple piece of set theory.
The crux of what stumped me (and evidently was what Peta and Tamfang discussed earlier on this talk page) is this: (a,b) := {{a}, {a,b}} does *not* define a pair with an ordering per se (in the sense of identifying that a is first and b is second). It only defines a way to use the mechanism of sets which distinguish items only by value, to implement two distinct variables -- that is to say, two distinct slots in which to store items, and later retrieve the items from each distinct slot.
The "distinct slot" idea is more or less just a restatement of the "defining property" of the ordered pair, and a reader who neglects to note that the defining property of "ordered pair" omits any mention of "order" will be vulnerable to confusion downstream. It remains for additional "logic" to actually perform the retrieving of one item or the other, and attribute to them the order (first and second). And thanks *very* much to the author who explicitly included that logic (Pi1 and Pi2) that actually performs that job. That crucial piece was missing from Hrbacek "Introduction to Set Theory", which is how I got here in the first place.
For the sake of others who might be similarly stuck: In order to understand how "{{a}, {a,b}}" corresponds to "ordered pair", it's crucial to understand how "first item" and "second item" are picked out of "{{a}, {a,b}}". A key challenge is that "{{a}, {a,b}}" has no ordering to it, so, for example, the {a} element is just an element in the set, not the first element per se. Indeed, so far as I can see, Karatowsk could just as well be written: (a,b) := {{a,b}, {a}} (not to be confused with (a,b)reverse, mentioned later in the article).
The answer lies in the two Pi equations, which in English are:
first: take the element that appears in both members.
second: take the element that appears in only one of the members. If there is none, the elements must be the same, so use that.
Hope that this is all correct, and might help other folks who stumble in here! Gwideman (talk) 00:51, 26 August 2012 (UTC)
Couple of questions/suggestions[edit]
1. Intro: "and hence the ubiquitous functions". Here "ubiquitous" looks suspiciously like a technical word. Is there some distinction between ubiquitous functions and non-ubiquitous functions? Or is this simply pointing out that functions are of broad importance?
2. In "Defining the ordered pair using set theory": "The above characteristic property of ordered pairs is all that is required to understand the role of ordered pairs in mathematics. Hence the ordered pair can be taken as a primitive notion, whose associated axiom is the characteristic property."
The first sentence here seems fatuous. Even if true, the second sentence doesn't follow from it.
I suspect this is trying to say something like: the ordered pair is a powerful concept, and consequently its axiom (the characteristic property) has been chosen as a primitive upon which to build further set theory.
Gwideman (talk) 01:08, 26 August 2012 (UTC)
Re your first question: your second interpretation is correct, the formulation is simply pointing out that functions are of broad importance.
Re your second question: I did not write the formulation, however, the first sentence is OK, and the second sentence follows from it. It is not true that the first sentence is trying to say that "the ordered pair is a powerful concept", rather it says that it is easy to characterize the notion of the ordered pair, since the above characteristic property is all that is needed. It is also not true that the ordered pair has been chosen as primitive and its characteristic property has been selected as an axiom, at least not in general. Commonly, instead of chosing the notion of the ordered pair as primitive, set theories define the ordered pair using some definition such as Kuratowski's. Ladislav Mecir (talk) 18:52, 17 May 2016 (UTC)
Coordinate extraction formulas[edit]
In the formulas given under Kuratowski definition for extracting the first and second coordinates, what is the purpose of the outermost Union? I had difficulty finding a definition of what unary union does in all cases. Is the idea to "remove an outer layer of set enclosure"? Ie: {a} --> a? But if a is not a set, but a value such as 6, will this succeed? I expected the result of a union operation to be a set. Gwideman (talk) 13:02, 28 August 2012 (UTC)
Everything, including "6", that can be put in a set is a set. --JBL (talk) 13:45, 28 August 2012 (UTC)
The notation {(a, b)} is ambiguous. It can mean either the singleton whose single element is (a, b) or the doubleton whose elements are (a and b). GeoffreyT2000 (talk) 02:45, 18 June 2015 (UTC)
Contradiction with Tarski–Grothendieck set theory article[edit]
The article says: "Also note that the short version is used in Tarski–Grothendieck set theory, upon which the Mizar system is founded." However, under the Mizar implementation of Tarski–Grothendieck set theory, it shows that the Kuratowski definition is used. This is a contradiction, so one of these articles must be wrong. 173.76.101.86 (talk) 02:04, 7 July 2016 (UTC)
Mathematical objects?[edit]
Although it has been in the first sentence since 2011, calling the components of an ordered pair mathematical objects is clearly false. The ordered pair is certainly a mathematical object (a mathematical structure) but its components need not have any mathematical meaning (they could be colors or the names of cities or anything else with no mathematical import). I am amazed that editors have let this stand and am only bringing this up here before I change it on the off chance that someone may want to defend it. Bill Cherowitzo (talk) 03:02, 15 September 2016 (UTC)
"calling the components of an ordered pair mathematical objects is clearly false" - I am sure this is your honest opinion, but, please consult how Wikipedia handles these issues. If you do not have any WP:IS stating that the components of the ordered pair are not mathematical objects, please do not make the change. Ladislav Mecir (talk) 04:00, 15 September 2016 (UTC)
Do you actually disagree substantively? --JBL (talk) 12:41, 15 September 2016 (UTC)
Yes, I disagree sustantively too. Definitions (e.g. Kuratowski's definition) of ordered pair are restricted to pairs of sets, which are mathematical objects. There are also definitions of ordered pairs of classes, but that does not matter in this case, since classes are mathematical objects too. Ladislav Mecir (talk) 14:17, 15 September 2016 (UTC)
Actually I think the burden of proof is not for me to find a source saying that the components are not mathematical but rather for you to find one that says that they are. Appealing to any of the various definitions in terms of sets is a formalism that does not cut the mustard! To say that since a formal definition involves sets, the components are mathematical, is begging the question. The second sentence refers to the order of the objects (components), yet these are not the sets of the definitions, and my point is that these objects need not be mathematical. Consider a reader coming to this page for the first time. In order to understand the first sentence the reader must already be familiar with the formal definitions and their consequences. This is simply poor writing and one of the major contributors to the oft heard complaint that math articles (in particular) are written by and for specialists. This is a topic that is taught with varying degrees of rigor and across several disciplines. To treat it solely as a topic in formal logic is doing a disservice to its varied audiences. Following the best traditions of Wikipedia technical writing, the lead should be an informal introduction to the topic and, in this case, explain the need for a more formal definition before getting into the mechanics of those definitions. The present article does a poor job of this and the given rationale for the set-theoretic definitions is, IMO pure BS (and, might I add, unsourced). This article's introductory material does need some work, and I think that a good start would be the removal of the word mathematical from the first sentence. --Bill Cherowitzo (talk) 19:19, 15 September 2016 (UTC)
I agree with Bill. Certainly the components of an ordered pair need not be any more "mathematical" than the elements of a set, which in non-formal settings are usually allowed to be anything. Paul August ☎
I am more partial to the formalism than either Bill or Paul (in particular, the idea that a set can contain something that is not mathematical makes me a little queasy), but I agree that it is not necessary in the lead. I don't know what I'd do with the wikilink mathematical object. --JBL (talk) 23:51, 15 September 2016 (UTC)
As a modern mathematician, raised on formalism, that idea would make me feel queasy too (like walking in quicksand) while doing mathematics, but not in other settings. And besides mathematics is bigger than formal mathematics. Paul August ☎ 12:25, 16 September 2016 (UTC)
Formalism is my bread and butter as well, but I am wondering if I am missing something here since I don't feel the slightest bit of queasiness. How does one fill out the sentence, "An element of a set is mathematical because, ..." – it is an element of a set???? That kind of circularity would make everything mathematical and so, that term would lose all meaning. I have been accused in the past of not seeing the forest because of the trees, and if I am doing that again I would like to be set on the straight path to queasiness. --Bill Cherowitzo (talk) 17:06, 16 September 2016 (UTC)
[Disclaimer: the following comments are not intended to have anything to do with editing this article.] In my opinion, the formalism of "set" can contain only mathematical objects, by definition. I don't think it makes sense to speak of a set containing a (real) tree, any more than it makes sense to speak of a tree that is Ω meters tall. In the real world there are collections of things, but they are not sets. (I do not think that it is important that other people agree with me, nor do I think it is necessary to share my view to know how to write this article well. I also do not claim that my feeling about this is "right" in any strong sense; it is just my feeling.) --JBL (talk) 18:51, 16 September 2016 (UTC)
I mainly disagree with the formulation "calling the components of an ordered pair mathematical objects is clearly false". I think that the main cause of the controversy is the fact that the term mathematical object is from the philosophy of mathematics. Mathematicians prefer the term object, and for them a formulation calling an element of an ordered pair object is fine. So, for the consensus sake, I can agree with replacing the term mathematical objects by objects in the lead section of the article. Ladislav Mecir (talk) 07:03, 17 September 2016 (UTC)
Perhaps I should have qualified my statement by adding "... from certain perspectives." ( such as my own!) I think this change will improve the tenor of the article. I would also like to see some more material having an informal perspective, but I do not want to dilute any of the fine work presented here on the various formal definitions. Perhaps a section that is clearly labelled as being informal and other dealing with applications can be added. Bill Cherowitzo (talk) 22:25, 17 September 2016 (UTC)
Boldface in the "Generalities" section[edit]
The section contains this text:
Then the characteristic (or defining) property of the ordered pair is:
Per MS:BOLD, the use of boldface for this purpose is discouraged. I propose to use
instead. Ladislav Mecir (talk) 04:21, 21 September 2016 (UTC)
Done I agree. --Bill Cherowitzo (talk) 18:03, 21 September 2016 (UTC)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Talk:Ordered_pair&oldid=740538747"

type: global

value: This website is the English-language edition of the free online encyclopedia Wikipedia. It was founded on 15 January 2001 as Wikipedia's first edition and, as of October 2021, has the most articles of any edition, at 6,412,521.


text:
Words that start with colonial | Words starting with colonial
http://www.thefreedictionary.com/words-that-start-with-colonial
Dictionary, Encyclopedia and Thesaurus - The Free Dictionary 10,382,209,692 visitors served
Words that start with colonial
Found 15 words that start with colonial. Browse our Scrabble Word Finder, Words With Friends cheat dictionary, and WordHub word solver to find words starting with colonial. Or use our Unscramble word solver to find your best possible play! Related: Words that end in colonial, Words containing colonial
14-letter words that start with colonial13-letter words that start with colonial12-letter words that start with colonial11-letter words that start with colonial10-letter words that start with colonial9-letter words that start with colonial8-letter words that start with colonialSee also
14-letter words that start with colonial
13-letter words that start with colonial
12-letter words that start with colonial
11-letter words that start with colonial
10-letter words that start with colonial
9-letter words that start with colonial
8-letter words that start with colonial
Words that start with coloniali
Words that start with coloniall

value: This website is an American online dictionary and encyclopedia that aggregates information from various sources.


text:
Good choice for stopover between Portland and San Francisco - Review of Holiday Inn Express Yreka-Shasta Area, Yreka, CA - TripAdvisor
“Pool area looked like fun”
“Our room was very large with a very small closet...”
“I could hear the man in the room next to ours...”
“We walked into this place at 10:30PM without a...”
“The room itself was clean and the beds were...”
“Rooms were very up to date with nicely appointed...”
Good choice for stopover between... - Holiday Inn Express Yreka-Shasta Area
Good choice for stopover between Portland and San Francisco
This is a newer hotel, up on the bluff overlooking the city of Yreka. The only other thing nearby is an RV park. However, the hotel is very nice and comfortable. We had a room on the 3rd floor overlooking the freeway and the hills behind the city of Yreka. We were not bothered by the highway noise.
The clerk at the desk, Heidi, recommended a couple of restaurants downtown. One Mexican and one Italian. We chose Italian--"Strings". The food was good and not overpriced.
Standard HI Express breakfast. Nice pool, but quite small and no hot tub. The room included a microwave and fridge, but the refrigerator had no freezer compartment. This seems to be common in many hotels and I find it rather irritating because we always carry a small cooler when we travel and like to re-freeze the reuseable ice packs each night.
Ask Kitters about Holiday Inn Express Yreka-Shasta Area
PrPERSON15, Manager at Holiday Inn Express Yreka-Shasta Area, responded to this reviewResponded January 16, 2013
We appreciate your kind comments to review our hotel, staff and services. Your comments will be shared with our staff to improve and to reiterate the importance of the great hotels guest love.
All reviews hi express front desk cinnamon rolls newer hotel indoor pool freeway side would stay here again recommend this hotel biscuits traveling north freeway noise the breakfast area internet connection los angeles comfortable rooms king room yogurt
sfelkins
Best place to stay in Yreka
I have stayed twice at this location. It's very nice. Don't let the fact that is sits along the freeway concern you.The noise is not a bother. It's a nice, clean, comfortable place to stay. The staff is friendly and helpful. The rooms are spacious, clean and nice. Wi-fi worked perfect. Breakfast was good and well stocked. This is the best choice in the small little town.
Room Tip: If you concerned about the freeway, request a room on the other side of hotel when making reservatio...
Ask sfelkins about Holiday Inn Express Yreka-Shasta Area
1 Thank sfelkins
PrPERSON15, Manager at Holiday Inn Express Yreka-Shasta Area, responded to this reviewResponded November 19, 2012
We appreciate the time you took to review our hotel and the kind words to describe touching points of your stay.
We hope to serve your travel needs next time you visit our small town.
When the doors to the rooms close, they do so with a bang. They should have a mechanism installed so that they close slowly.
It is difficult to enjoy our day when we are woken up several times a night due to loud closing doors.
Guests should also be considerate of those still sleeping when walking and talking in the halls.
Room Tip: Set aside a bank of rooms for us light sleepers. Again, it is difficult to enjoy the day when tired...
Ask Ronald P about Holiday Inn Express Yreka-Shasta Area
PrPERSON15, Manager at Holiday Inn Express Yreka-Shasta Area, responded to this reviewResponded November 9, 2012
We appreciate the time you took to review our hotel during your stay. I understand that you did not have a good night sleep due to the other guest around you. We try to remind our late check in guest about the common courtesy opportunities to other in the hotel. I have shared your comments with our staff and together we are trying to come up with solution to avoid such occurrences in the future. We wish you best of travel back to Washington and hope you try us again next time you are in the area.
Drove as far as we could and decided to stop in Yreka for the night. We saw the Holiday Inn Express off I-5 but continued into town to find something further away from freeway noise. We weren't too excited about what we saw and since I prefer indoor entrances, we drove back to HI Express. We were glad we did. The rooms were large and spacious; the beds very comfortable; the pillow were marked "soft" or "firm" and each bed had 4 pillows. Got a great nights sleep and did not hear any freeway noise. The room had a fridge, micro and in-room coffee. Breakfast had multiple choices from cereal, fruit, yogurt, eggs, sausage, biscuits, bagels, cinnamon rolls and much more.
All was good and we would stay here again in a heart beat.
Room Tip: We stayed on the freeway side as we didn't want the morning sun pouring in. It was fine, experi...
Ask grammajamma about Holiday Inn Express Yreka-Shasta Area
PrPERSON15, Manager at Holiday Inn Express Yreka-Shasta Area, responded to this reviewResponded October 8, 2012
We are thankful for your decision to turn around and spend a restful night at our hotel. Your kind words and comments will assist our staff in providing the best service to our guest. I am glad to hear that you have voiced the opinion of no freeway noise which I have heard many times at the front desk. We appreciate your business and hope to see you next time you are in the area.
The hotel is clean, roomy and the health center is well equipted. The staff was friendly and the breakfast was standard HIE fare. The hotel is located on a hill pretty much away from restaurants. A word to the wise is to eat before checking in or plan on driving to a restaurant for dinner. The location is quiet and away from the highway. As a Priority club member, we were greeted with a welcome package. It is a bonus not practiced by some other hotels.
Room Tip: All that we saw were okay.
Ask dalichandler about Holiday Inn Express Yreka-Shasta Area
I appreciate the time you took to review our hotel. Our small town is thankful to each and every guest for contributing to our micro economy. Your comments will help us in shaping our business and service strategy. We hope to see you for your next trip to our area. Thank you.
This hotel is a true oasis in northern California. Everything about this place is what you hope you will find and often do not. The staff is friendly and professional, and the woman working in the evening (Heidi) advised us about the best places in town for dinner.
The breakfast buffet was the standard HIE affair, but it was presented with such loving care that every item on it was more appetizing than usual.
When in Yreka, stay here.
Room Tip: We were pleased with our room facing toward I-5.
Ask wonderfulworld about Holiday Inn Express Yreka-Shasta Area
We appreciate your time and kind words used to review our hotel. Your comments will work their magic to encourage our staff to provide the great hotels guest love. We hope to see you again next time you visit our area.
We were pleased w/the room...very
Large with everything you need. This is a
New and modern hotel. The location is
A bit askew...off the hiway, around the
Block and sits atop the road overlooking
The little city of Yreka. Our room view
Was a small R V park, but it was very
Nice and quiet. There are no restaurants
Nearby but the complimentary breakfast
Was not bad and a hostess keeps everything orderly and clean. We went into town for dinner..about 5 min. Drive.
We will stay here again on our next trip
Ask Trekrs2 about Holiday Inn Express Yreka-Shasta Area
1 Thank Trekrs2
PrPERSON15, Manager at Holiday Inn Express Yreka-Shasta Area, responded to this reviewResponded September 13, 2012
We appreciate your time to review our hotel. Our staff takes pride in providing the best service our guest have become accustomed to expect. Your kind words will encourage and motivate our staff to always provide the best accommodations for our guest.
Price Range: $123 - $172 (Based on Average Rates for a Standard Room)
TripAdvisor is proud to partner with Booking.com, IHG, Hotels.com, Expedia, Travelocity, Priceline, HotelQuickly, Hotwire, TripOnline SA and getaroom.com so you can book your Holiday Inn Express Yreka-Shasta Area reservations with confidence. We help millions of travelers each month to find the perfect hotel for both vacation and business trips, always with the best discounts and special offers.

value: This website is an American online travel company that operates a website and mobile app with user-generated content and a comparison shopping website.


text:
sevo_stille: Member profile: Digital Photography Review
sevo_stille's recent activity
Well, what they have cancelled is not the 1 series, but the unreleased fixed-lens DL series. The Nikon 1 series still exists, at a low level, and if any, its prospects are better now that it does...
Replied in How many batteries do you usually carry?
After some initial insecurity, I have settled for usually one, rarely two spares. I never needed more than the latter - and even that only across a full day.
Replied in Any thought about this charger?
Why would micro USB be unsuitable? You'll be limited by the current of the power supply, but even cheap ones will be 1A/5V - enough to convert to the 0.4A/8.4V of the regular Sony chargers. Given ...
Right. Restricting access to the images to the home office abroad will first of all protect the subject. Sometimes the photographer will benefit as well - while assailants can still destroy the ...
Replied in How I suppose to get rid of reflection from lens?
Kolari mod (or any other cover glass mod).
Replied in Sony A7ii vs Nikon D750 at ISO 12800
Unsolder them, then, and put them on a test board. Or try to swap them. As long as you can do neither, it is futile to compare anything other than the entire system.
Replied in multi adapter to E mount?
As already said, adapters fill the space between one mount and the other. If there is no register difference (no space) there can be no adapter. An "adapter between the same mount" would be a ...
Replied in air pollutants and exposed sensors
Sensors are not exposed, but behind a coated cover glass. Adding another glass that is lower priced and more easily clean- and replaceable could cut down your cleaning cost, but each extra surface ...
Replied in Anyone know where I can get a replacement lens hood for 28-70 FE lens
The original part would be Sony ALC-SH132. If you can do with less convenience (no bayonet lock) and don't want to spend that much, there are plenty of 55mm wide third party petal-type hoods ...
Replied in A wild guess from the olden days
Even the "Leica shutter curtain burns" are somewhat dubious and partially mythical - curtain burns usually are reported from cameras that spent years in a shop window (where cumulative UV will ...
Any sub-pixel scale error will cause similar spots - demosaicing will blow up, blur and turn them into colour shifted spots. Causes can range from dirt on the sensor over sensor defects to a ...
Replied in 7 copies of decentered Sony 24-70 GM f2.8
Did you try the other lenses at the same time (or after these tests), or is it just a case of "it used to work, with other lenses"? You probably would not have missed a bent mount or damaged ...
Replied in How is the original A7 body to use, really?
The lack of IBIS aside, its one flaw is the lack of support for it in the firmware of most, perhaps all third-party AF lens adapters (whether Canon or Nikon or Contax G). If you are into that, you ...
That will be because the camera is misaligned - if it were in parallel, it would be impossible to align one edge or point, there is either everything or nothing in focus and alignment. Aligning by ...
Replied in Instant Wireless Tether to an Ipad?
Yes, it does. But when you turn the camera by 90° you would also have to turn the monitor physically by 90°, or the portrait will be on its side.
Wireless HDMI transmitters/extenders do exist - including trivially obtainable wireless extenders for home use (where you will however have to build some wall wart substitute for power supply, and ...
Replied in Hasselblad adapter for A7RII
Well, they can be worth while for character. But as you only use the innermost 20% of a image circle, you only get 20% of what that lens is capable of...
Don't forget about the coverage! No. Typical film era Hasselblad lenses began to disappoint in MF as digital medium format resolutions grew past 36MP. Given that DMF usually has about half the area ...
Gear List sevo_stille has not added any gear yet.

value: This website is a website about digital cameras and digital photography, established in November 1998. The website provides comprehensive reviews of digital cameras, lenses and accessories, buying guides, user reviews, and forums for individual cameras, as well as general photography forums.


text:
Genuine Hobart Mixing Bowls Whisk Beater Set A-200 Pre Owned
Hobart Mixer Part S-18-231 Transmission Cover For A200 Mixer
Used Hobart Model A200 Mixer For Parts/Repair A-200 20 Quart
auctionrssfeed (3102 )
562710715631

value: This website is an American multinational e-commerce corporation based in San Jose, California, that facilitates consumer-to-consumer and business-to-consumer sales through its website.


text:
armpitt212 - Member Reviews - TripAdvisor
Profile Badge collection Travel map armpitt212 Send Message Since Sep 2008 25-34 year old male PA I love to travel. I will go just about anywhere at least once. My goal in life is to fill my passport as many times as I can.
67 Reviews 1 Rating 3 Forum Posts 29 Helpful votes Travel Style Foodie Like a Local Peace and Quiet Seeker armpitt212's TripCollective Progress Total Points 6,794 Level 5 Contributor Badges (16 total)View Collection Current Badge Previously Earned Top Contributor 50 Reviews Helpful Reviewer 25 Votes Passport 33 Cities Senior Contributor 20 Reviews Travelers' Choice Hotel Reviewer 1 Review New ReviewerThis badge is earned after 1 review.ReviewerThis badge is earned after 3 reviews.Senior ReviewerThis badge is earned after 5 reviews.ContributorThis badge is earned after 10 reviews.Senior ContributorThis badge is earned after 20 reviews.Top ContributorThis badge is earned after 50 reviews.Hotel Expert Level 6The Hotel Expert badge levels up after every three hotel reviews.Level 6: earned after your 18th hotel review.Restaurant Expert Level 10The Restaurant Expert badge levels up after every three restaurant reviews.Level 10: earned after your 30th restaurant review.Attraction Expert Level 6The Attraction Expert badge levels up after every three attraction reviews.Level 6: earned after your 18th attraction review.Helpful ReviewerThis badge is earned after 1 helpful vote.Helpful ReviewerThis badge is earned after 5 helpful votes.Helpful ReviewerThis badge is earned after 10 helpful votes.Helpful ReviewerThis badge is earned after 25 helpful votes.PassportEarned after reviews in 2 separate cities.
WoodstockDen BoschNorth ConwayCape ElizabethOranjestadAmsterdamBurlingtonChadds FordCabotPalm - Eagle BeachVolendamPortlandPhiladelphiaKemptonBadhoevedorpOtterloSt. ThomasEast EndHoenderlooVirgin Islands National ParkAlexandriaSonomaOrlandoGorhamArnhemDe CocksdorpBrooklynSanto DomingoNew York CityClearwaterYorkWaterburyUncasville
Rose's LoungeSel et GrasLa Morena
Hotel Expert Level 7The Hotel Expert badge levels up after every three hotel reviews.Level 7: earned after your 21st hotel review.Write a ReviewRestaurant Expert Level 11The Restaurant Expert badge levels up after every three restaurant reviews.Level 11: earned after your 33rd restaurant review.Write a ReviewAttraction Expert Level 7The Attraction Expert badge levels up after every three attraction reviews.Level 7: earned after your 21st attraction review.Write a ReviewHelpful ReviewerThis badge is earned after 50 helpful votes.Write a Review
How do I earn points? Every time you contribute to TripAdvisor, you earn TripCollective points. Here’s a list of what you can contribute, and how much it’s worth. Review 100points Photo 30points Video 30points Forum Post 20points Rating 5points Traveler Article Creation 100points Traveler Article Edits 5points Trip Lists 50points Helpful Vote 1point Earn points, reach new levels. The more points you earn, the higher your level – and the more you’re recognized in the travel community for your contributions. Your TripCollective level appears in your traveler profile. Level6 10000points Level5 5000points Level4 2500points Level3 1000points Level2 500points Level1 300points Total Points 6,794armpitt212's Contributions Reviews (67)Ratings (1)Forum Posts (3)Filter by AllHotels (18)Restaurants (31)Attractions (18)RatingHelpful Votes Points Las Terrenas: Hotel Alisei “Very quiet and quite relaxing ”Stayed at this place in mid June for a few nights....Jul 7, 2015 1 101 Woodstock: Sugarbush Farm “Great place to spend an afternoon”Really nice self guided tour of the sugaring proce...Jun 9, 2014 1 101 Burlington: Church Street Marketplace “So much to choose from.”This is a great shopping area. The street is pedes...Jun 9, 2014 2 102 Burlington: Church Street Tavern “Ok, just average.”Nothing really notable. Food was ok, service was o...Jun 9, 2014 2 102 Burlington: The Farmhouse Tap & Grill “Incredible, go here now!”What a fantastic place. First off, the beer menu, ...Jun 9, 2014 1 101 Waterbury: Ben & Jerry's Ice Cream Factory “I expected more.”For as large and profitable and Ben and Jerry's is...Jun 9, 2014 100 Cabot: Cabot Visitors Center & Tour “Disappointed”I was excited to visit this place, but I was disap...Jun 9, 2014 2 102 Waterbury: Old Stagecoach Inn “Not impressed, but not disappointed.”I had expected a bit more from this place. We stay...Jun 9, 2014 100 Cape Elizabeth: Fort Williams Park “Very cool place”Bigger than I thought it would be. Plenty of parki...Jun 9, 2014 1 101 Portland: Duckfat “Quite good”Fries are awesome, sauces are great. Sandwiches ar...Jun 9, 2014 100 York: Cape Neddick Nubble Lighthouse “Beautiful photos”Just wonderful views of the lighthouse. Definitely...Jun 9, 2014 100 North Conway: Muddy Moose Restaurant & Pub “Not too shabby”The inside is very cool, lots of wood, reminds me ...Jun 9, 2014 100 Gorham: Mount Washington Auto Road “You Must Do This!”You have to do this at least once in your lifetime...Jun 9, 2014 100 Philadelphia: Tinto “A carnival in my mouth”I am not even sure how to describe this place. It ...May 28, 2014 100 Philadelphia: Fairmount Park “Huge park”This place is massive. My wife and I walked 6 mile...May 28, 2014 100 Kempton: Hawk Mountain Sanctuary “Beautiful views, nice trails.”Great place to visit for the day. The views at the...May 28, 2014 100 New York City: The Green Table “Wonderful farm to table dining.”We love the farm to table concept. So we figured w...Apr 8, 2014 100 New York City: Barrio Chino “Amazing, just amazing.”Very small place, but boy is everything great. We ...Apr 8, 2014 100 New York City: New York Marriott Marquis “Wow, this place is huge.”What a ginormous hotel. The elevators are cool, ce...Apr 8, 2014 100 Philadelphia: Amis “Very Nice Italian Evening”I had high hopes for this place just from looking ...Mar 7, 2014 1 101