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## July 2008 Newsletter
### Officers
• Chair: Pierre Meystre
• Chair Elect: Louis F. DiMauro
• Vice Chair: Christopher Monroe
• Past Chair: William C. Stwalley
• Secretary/Treasurer: Carol E. Tanner
• Councillor: Paul Julienne
### Members-at-Large
• Elizabeth McCormack
• Jun Ye
• Linda Young
• Maria Allegrini
• Nicolas Bigelow
• Thomas Killian
### From the Chair
Pierre Meystre, The University of Arizona
Once again, the DAMOP Annual Meeting was a resounding success, showcasing some of the most exciting current developments in modern physics. Near record participation, stimulating science, and outstanding jobs by both the local committee and the program committee all contributed to what remains by far the best general meeting in AMO science. Maybe the best barometer of the health of the field is that a remarkable 45% of the participants were students. The fact that we continue to attract the best and the brightest bodes very well indeed for the future.
With well over 900 attendees, the general trend in the linear growth of the DAMOP meeting continues. The large student participation is an unambiguous indication that things are not likely to stop soon, especially when also considering the explosive developments in relatively new directions such as e.g. ultrafast science, quantum information, and ultracold physics. These are all “spin-offs” that feed heavily on the more “traditional” aspects of atomic, molecular and optical physics, and in turn feed back into them. The strong coupling between these various subfields is of course one of the major strengths and attractions of AMO science, as well as a source of its amazing capability to reinvent itself.
Because of these very positive developments, it is however becoming increasingly challenging to identify University settings able to handle our annual meeting. (For example, the 2010 DAMOP meeting in Houston will take place in a relatively large hotel/conference center rather than on a University campus.) This is a good problem to have, but a problem nonetheless.
It is therefore time to initiate a conversation about the future format of our meeting. Clearly, that discussion will also implicitly involve thoughts about our collective vision for the future of DAMOP: Do we want to keep expanding our meeting, or keep it small? Do we want to be a big tent that is proactive in welcoming emerging “spin-off” areas? Would we rather be more focused and return to the more “cozy” meetings that the more senior amongst us fondly remember? On a more practical side, do we want to limit the number of oral presentations, increase the number of parallel sessions, extend the meeting duration? Is it preferable to move from a “local” to a “national” organization? These are all difficult questions, whose answers are bound to have important implications for the future.
To start addressing these issues, we have recently formed an ad hoc committee on “DAMOP Meeting Format and Organization Improvements,” chaired by David Schultz. His article later in this Newsletter reports on a Town Hall meeting held at the Penn State DAMOP. I urge everybody to become involved in the conversation started at that occasion and to contact David or any member of his committee with your ideas and comments.
On a completely different topic, we are planning on initiating a fund raising effort to increase the endowment of our various Prizes and Awards. Right now, the idea is to morph the DAMOP Publicity Committee and to redirect its mission toward that goal. Here again, I call for ideas and volunteers. More to come in a future newsletter.
Finally, it is not too early to mark your calendars with the dates of the next two DAMOP meetings:
• DAMOP 2009: May 19-23, 2009, Charlottesville, VA
• DAMOP 2010: May 25-29, 2010, Houston, TX
Have a great summer!
### DAMOP Election-Results
DAMOP's annual election of officers takes place each spring. This year elections were held for the positions of Vice Chair, Secretary/Treasurer, and two Executive Committee Members-at-Large. Voting opened on 9 April 2008 and closed on 21 May 2008 just in time for DAMOP. As usual, the results reflect the high degree of interest the DAMOP members have in the governance of their division, and the high regard they have for their candidates. DAMOP’s total membership now reaches 2860 and 621 or ~22% percent of our members voted. We would like to thank everyone who voted and all of the candidates for agreeing to run. Special thanks are due to Luis Orozco, Chair of the Nominating Committee. As anyone knows who has been through the ordeal, this is one of the toughest committee assignments to endure.
Congratulations to the newly elected:
• Vice Chair: Christopher Monroe (University of Maryland)
• Secretary/Treasurer: Carol Tanner (University of Notre Dame)
• Member-at-Large: Nicolas Bigelow (University of Rochester)
• Member-at-Larger: Linda Young (Argonne National Laboratory)
The current full slate of officers appears at the head of this newsletter and on the APS web page.
In the spring of 2009 the DAMOP election will include openings for Vice Chair and two Members-at-Large. If you are interested getting your feet wet in the political arena, try running for one of these positions or suggest a friend by contacting the new chair of the Nominating Committee, Tim Gay (Univ. of Nebraska). It’s a whole lot of fun!
The latest committee assignments when finalized will be posted on this APS web page.
### DAMOP 2008
Highlights from Penn State
From: David Weiss and Kurt Gibble, Penn State
The 39th DAMOP Meeting of the American Physical Society was held 27-31 May 2008 under sunny skies on the beautiful Penn State University Park campus in State College, PA. There were 922 participants, the second highest ever and a 9% increase over last year’s total. The vitality of our division is reconfirmed each year by the ever-increasing number of student attendees at our annual meeting. This year’s meeting included 419 student participants, the highest ever.
Two special events were held before the start of the regular DAMOP sessions. On Tuesday 27 May, 97 registrants participated in the Graduate Student Symposium, which covered recent advances at the forefront of AMO science. An informative array of tutorials were presented by several speakers: Pierre Meystre (The Univ. of Arizona) “Quantum Optics,” Brian DeMarco (U. of Illinois) “Quantum Degenerate Gases,” Dave Wineland (NIST-Boulder) “Quantum Information,” and Phil Bucksbaum (Stanford Univ.) “Ultra-fast Physics.” The workshop also included lunch and a tour of Penn State AMO Physics Labs.
The second special event held on Tuesday was the now-traditional Educators’ Day with 25 Pennsylvania high school physics teachers attending. The workshop consisted of lectures and hands-on laboratory sessions. Lectures included “The coolest stuff in the universe,” by 1997 Nobel Prize winner Bill Phillips (NIST Gaithersburg and the Joint Quantum Institute) and “How things work,” by Lou Bloomfield (U. of Virginia). The workshop was a great success thanks to Penn State Lecturer Steven van Hook’s organization and the Penn State Eberly College of Science’s sponsorship.
The official DAMOP events began with the Welcome Reception on Tuesday evening, in the Ballroom of the Nittany Lion Inn, an historic site on The Penn State campus. At the reception and throughout the conference, the Inn provided a comfortable, caffeine and calorie-rich environment that stimulated discussions between colleagues from around the world.
The conference opened Wednesday morning with remarks by DAMOP’s own Dan Larson, Dean of the Penn State Eberly College of Science. The excellent scientific program consisted of 96 invited talks, 367 contributed talks, and 385 posters. It began with the Plenary DAMOP Prize Session featuring talks by the Davisson-Germer Prize winner, Horst Schmidt-Böcking (Univ. of Frankfurt) and the Allis Prize winner, Kenneth Kulander (Lawrence Livermore National Laboratory). The mornings and early afternoons were filled by 6 parallel oral sessions punctuated by well-attended late-afternoon poster sessions with refreshments and beer from a local microbrewery. The poster sessions showcased a wide scope of science revealing the range and depth of research in our field.
Physical Review Letters (PRL) celebrated its 50th Anniversary this year, and DAMOP recognized its appreciation for the importance of this journal in the development of AMO science with a special session. Daniel Kleppner (MIT) gave a captivating historical talk that ranged from the beginning of the journal, as conceived by Samuel Goudsmit, to the present. His talk was followed by that of Deniz Van Heijnsbergen (APS) who described how PRL grew in stature to became the careering shaping journal of its authors.
Throughout the conference, The Exhibit Hall in the Nittany Lion Inn featured refreshments and displays by 11 companies and organizations with close ties to the DAMOP community, including Acculight, Cambridge U. Press, Coldquanta, High-Q Laser, IOP Publishing, New Focus, Teachspin, Thorlabs, Toptica, Vescent Photonics, and The Univ. of Arizona College of Optical Sciences.
One of the conference highlights was a public lecture on Wednesday evening by 1997 Nobel Prize winner Steven Chu (Univ. of California and Director of Lawrence Berkeley National Laboratory), which drew about 725 people including many from the State College community. Prof. Chu has recently turned part of his scientific attention toward the issues surrounding the energy security of our nation and the world. He gave an eye opening presentation entitle “The World’s Energy Crisis and What We Can Do About It.” He made the severity of our current situation clear by describing the basic aspects of the energy problem and gave a flavor for the innovative research that may lead to a sustainable energy future. All attendees were rewarded with a highly inspiring talk and Penn State Creamery ice cream.
The conference banquet on Thursday evening filled the banquet hall to capacity with ~950 people. Attendees were entertained by Lou Bloomfield who gave an entertaining multimedia presentation entitled “Explaining how things work even when they don’t.” Several DAMOP members were honored at the banquet with awards. The honors included the announcement of the new APS Fellows in our division who each received a certificate and pin. The participants in the Thesis Prize Session (details below) were recognized, and the DAMOP thesis prize was awarded to David Moehring. The APS announced a new award this spring with the initiation of a recognition program for its "Outstanding Referees." Several DAMOP members were recognized for their tireless efforts as referees (photo below). Finally, official division leadership was passed from Professor William Stwalley (U-Conn.) to Professor Pierre Meystre (The Univ. of Arizona).
A number of committee and business meetings, which may go unnoticed by many, are always held at DAMOP. These meetings are where opinions are expressed from which important decisions are made about the future of our APS division. On Friday at the Town Hall Meeting, a central topic of discussion was the format of future DAMOP conferences. A report by David Schultz Chair of the ad hoc DAMOP Meeting Format and Organization Improvement Committee appears below.
Finally, the plenary Hot Topics Session wrapped up the conference around mid-day Saturday 31 May with exciting talks by Nergis Mavalvala (MIT), Karl Nelson (Penn State), Edvardas Narevicius (UT-Austin), and Ferdinand Brennecke (Inst. for Quant. Elect. ETH).
A good time was had by all, and we look forward to next year’s DAMOP meeting in Charlottesville, Virginia.
Scenes from the Penn State DAMOP
The Nittany Lion Inn courtyard provided a quiet place for discussion.
The Ballroom of the Nittany Lion Inn took on many roles, but none more important than the plenary sessions.
Refreshments at the Nittany Lion Inn.
The poster sessions were at least as much fun as they look here.
Some of DAMOP’s heroic referees recognized at the banquet.
A smile break from heavy conferencing.
With the close of DAMOP 2008, Bill Stwalley graciously moved into the enviable position of Past Chair.
Student Travel Support
The level of student participation at DAMOP meetings has always been a pride and joy of this division. Student travel support from NIST has been one of the most consistent driving forces that make this level of student participation possible. Student travel funds assist those who might not otherwise be able to attend. Over 100 students applied for support. This year NIST granted a total of $5,000 that, combined with DAMOP resources, allowed 50 students to receive up to$500 each in travel support. Thanks very much to NIST, the Education Committee, and in particular its chair Eric Wells (Augustana College), for facilitating these grants.
Plenary Prize Session
It is never too early to start thinking about nominations for the various prizes and awards sponsored by DAMOP and by the APS. Most prizes have a July 1 deadline except for the thesis prize that typically has a deadline early in December. An award nomination packet includes a substantial amount of supporting material so please do not wait until the last minute. More information on prizes of interest to the DAMOP community can be found at this web site.
Graduate Student Thesis Prize Session
We would like to recognize and congratulate all participants in the thesis prize session: Lillian Childress, Peter Rabl, Gretchen Campbell, and David Moehring. These finalists are selected from numerous nominations submitted to the DAMOP Thesis Prize Committee, and the winner is selected by this committee after their presentations in this session. We thank Robin Côté (U. Conn.) and his committee for taking on the difficult task of choosing from this pool of excellent candidates.
About the Thesis Prize Recipient
From: Robin Côté
Congratulations to the winner, David Moehring. David was raised in Richmond, Indiana and attended Purdue University as an undergraduate student where he was a NCAA scholarship golfer. While an undergraduate, he participated in two Research Experience for Undergraduate programs at the Indiana University Cyclotron Facility and the NASA Langley Research Center. David received his Bachelors degree in Honors Applied Physics with highest distinction in 2001, and shortly thereafter joined the trapped ion quantum computing group of Chris Monroe at the University of Michigan. David’s graduate dissertation presents a theoretical and experimental realization for the entanglement of two trapped atomic ions, including the first explicit demonstration of quantum entanglement between a single trapped ion and its single emitted photon, as well as the entanglement between two macroscopically separated trapped ions. In addition to their promise for scalable quantum information processing, these results provide evidence for the completeness of quantum mechanics via demonstration of Bell inequality violations. In addition to the 2008 DAMOP Thesis Prize, David was awarded the Kent M. Terwilliger Memorial Thesis Prize and the Rackham Distinguished Dissertation Award. Since August 2007, David is an Alexander von Humboldt Research Fellow in the Quantum Dynamics group of Prof. Gerhard Rempe at the Max Planck Institute for Quantum Optics in Garching, Germany.
And now for the bravest of the brave! For an undergraduate to stand up and speak or present a poster to a group of experts takes courage and confidence. We applaud all of the undergraduates who participated in the conference. From the pool of undergraduate participants, we also congratulate the few who were invited to speak in the Undergraduate Research Session: Paul Hess, Joanna Salacka, Aaron Dunn, Justin T. Schultz, and J. Rau. Their presentations were fabulous and spanned a wide variety of research topics.
DAMOP Meeting Format and Organization Improvement Committee: Report From the Penn State Town Hall Meeting
From: David Schultz, Oak Ridge National Laboratory
A spirited and productive Town Hall Meeting was held on Friday afternoon during the Penn State DAMOP. Its purpose was to seek input from the community regarding how we might improve both the format (number of parallel sessions, number and selection of meeting days, special events, …) and organization (venue choice, organizing committee structure, costs and budgets, meeting partners, participation in the March Meeting, …) of the DAMOP Annual Meeting.
This was the first major activity of the DAMOP Meeting Format and Organization Improvement Committee , an ad hoc committee established in order to address the growing cost, complexity, and attendance of our annual meeting. These “growing pains” are recognized as good signs of the vitality of the field, especially since almost half the number of participants at DAMOP are students, and the objective is to make the meeting better and less difficult to organize. Many of the ideas being discussed, therefore, relate to the growth of the meeting, but many are also aimed at improvements that make sense even if the meeting’s attendance, number of talks and posters, number of special events (e.g., Student tutorials, Educator’s Day, Public Lecture), and number of partners (such as DAMPhI, PMFC, and GQI), were not increasing.
About thirty people made comments and suggestions at the Town Hall Meeting, reflecting a diversity of viewpoints and simultaneously the passion and logic behind how members believe we should improve our Annual Meeting. The committee (Scott Bergeson, Tim Gay, Tom Killian, Lou DiMauro, Jim McGuire, Marianna Safronova, and Dave Schultz) would be glad to receive your comments and suggestions throughout the summer and will work to formulate recommendations for DAMOP meeting improvement by the Fall. Thanks for your input as we try to make DAMOP an ever-improving event for the benefit of its members.
### Other 2008 APS Prizes Won by DAMOP Members
We like to think of AMO physics as an enabling science that provides an underlying framework for many areas of research and development. This sentiment is supported by the number of DAMOP members who receive awards sponsored by other APS divisions. Congratulations!
### APS Fellowship Through DAMOP
DAMOP nominates several candidates for APS Fellowship each year, and the successful candidates are elected by APS Council. Chris Monroe is the new chair of the DAMOP Fellowship Committee and the deadline is usually set by the committee to be in March or April. It is never too early to start preparing these nominations. Packets are submitted on-line through the APS web page. Details can be found at:
### Conferences and Workshops
International Conference on Atomic Physics (ICAP) 2008
From: Winthrop Smith, Robin Cote, and Phillip Gould, University of Connecticut-Storrs
The 21st International Conference on Atomic Physics (July 27 - August 1, 2008 at the University of Connecticut, Storrs, CT) is part of an ongoing series of conferences devoted to fundamental studies of atoms, broadly defined. A Web site with the Conference Program and Abstracts can be found at: http://www.phys.uconn.edu/icap2008
The ICAP papers encompass forefront research on basic AMO physics, emphasizing atoms and their interactions with each other and with external fields. These meetings grew out of the molecular beams conferences of the Rabi group. The first ICAP was at NYU in 1968. Later conferences have been held all even-numbered years, alternating between North America and other locations, including Europe and recently Brazil, with the next conference planned for Cairns, Australia in 2010. Historically, topics have included quantum electrodynamics, tests of basic symmetries (PCT), precision measurements (including atomic clocks and fundamental constants), laser spectroscopy, ultracold atoms and molecules, Bose-Einstein condensates, degenerate Fermi gases, optical lattices, quantum computing/quantum information with atoms and ions, coherent control, and ultrafast and intense field interactions. Notably, all invited talks are plenary. AMO Nobel laureates Phillips, Cornell, Glauber, Chu and Ketterle will speak at ICAP 2008. The conference is to be preceded by a one-week Summer School for new AMO researchers, organized by the Harvard-MIT Center for Ultracold Atoms in Cambridge, MA. Proceedings will be uploaded to the Web when they are ready.
Linac Coherent Light Source (LCLS)
Call for proposals and Annual Users Meeting October 15-18, 2008.
From: Linda Young, Argonne National Laboratory (ANL)
The world’s first x-ray free electron laser, the new Linac Coherent Light Source at Stanford, is scheduled to produce first light in May 2009, with AMO experiments to begin in August 2009. The first beams will be in the soft x-ray region, 800 – 2000 eV, with estimated pulse parameters 1013 photons/pulse, ~100 fs at 120 Hz. Proposals are now being solicited for experiments with soft x-rays at the AMO station (http://lcls.slac.stanford.edu/user/). For technical information on the AMO station, contact [email protected]. The deadline for submitting proposals is September 1, 2008.
The 2008 LCLS/SSRL User Meeting will be held October 15-18 at SLAC (http://www-conf.slac.stanford.edu/ssrl-lcls/2008/). October 15 will be a full day describing updates on the LCLS status, endstation plans and user access policies. October 16 will have joint sessions between LCLS and SSRL (Stanford Synchrotron Light Source) featuring science highlights. Of special interest to the DAMOP community, on October 17, there will be a workshop Atomic, Molecular & Optical Physics with the LCLS, organized by John Bozek (LCLS) and Linda Young (ANL). The purpose of this workshop is to discuss AMO experiments for the LCLS. The AMO instrument will be the first online at LCLS, and by the time of this workshop the first round of proposals will have been submitted for consideration. Scientific opportunities and technical challenges for x-ray FEL AMO experiments will be discussed.
Gaseous Electronics Conference (GEC)
From: Tim Gay, Univ. of Nebraska
The Sixty First Annual Gaseous Electronics Conference (GEC) will be held Oct 13-17, 2008 at the Marriott Dallas/Addison Quorum by the Galleria in Dallas, Texas.
The GEC Executive Committee invites the submission of abstracts. Topics include: basic phenomena and plasma processes in partially ionized gases; and the theory and measurement of basic atomic and molecular collision processes. Papers reporting on experimental, theoretical, and computational studies that address either fundamental properties of low-temperature plasmas or their applications are encouraged.
Applications of interest include, but are not limited to:
• Heavy particle interactions: ion-atom, ion-molecule, neutral-neutral
• Electron and photon collisions with atoms and molecules
• Ionospheric phenomena, Ion sources
• Plasma processing of materials including semiconductors
• Metals, insulators, MEMS devices and displays
• Biological and emerging applications of plasmas
• Plasma-surface interactions, Plasma diagnostics
• High pressure and micro-plasmas
• Gas discharge lamps, Gas laser
• Plasma chemistry and combustion
• Plasma aerodynamics
Although most papers will deal with low-energy processes, papers that concern electronic or radiative processes produced by high-energy electrons or heavy particles are also welcome.
Additional details can be found at the conference website http://www.utdallas.edu/gec/.
Deadlines: Abstracts – Past, Early registration – August 15, Room reservations – September 21.
DAMOP at the 2009 APS March Meeting
From: Erich Mueller, Cornell
The 2009 APS March Meeting will be held March 16-20, 2009 in Pittsburgh, PA. At the 2008 APS March Meeting, DAMOP sponsored 16 sessions, with nearly 200 abstracts. We expect that DAMOP will represent a similarly large presence next year. We will be sponsoring a "Tutorial" on "cold atoms in optical lattices", and Focus Sessions on "disorder in ultra-cold gases", "magnetism in ultra-cold gases", "number or mass imbalanced Fermi gases and BEC-BCS crossover", and "dipolar gases / cold molecules". Please consider nominating speakers for invited talks–in the past we have had great invited symposia, and this can only continue with active participation from the members of DAMOP. The program committee can only invite speakers who have been nominated. The call for abstracts will be put out in August, and will include details of how to submit an abstract and how to nominate speakers. The abstract deadline will be November 21, 2008.
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# mathematical logic symbols
### mathematical logic symbols
If there are several typographic variants, only one of the variants is... Set theory. As readers may be not aware of the area of mathematics to which is related the symbol that they are looking for, the different meanings of a symbol are grouped in the section corresponding to their most common meaning. In this course we develop mathematical logic using elementary set theory as given, just as one would do with other branches of mathematics, like group theory or probability theory. The same is true of abbreviations such as "iff", "s.t. There are numerous signs and symbols, ranging from simple addition concept sign to the complex integration concept sign. It ⦠Logic tells us the truth and the falsity of the particular statement. Boole's catalog of symbols from Laws of Thought (1854). Jump to: navigation, search. . These are often called connectives, though they don't connect anything. "Mathematical Operators – Unicode" (PDF). R The earliest treatises on the nature of ⦠B In mathematical formulas, the standard typeface is italic type for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. Mathematical Logic and Proofs Book: Mathematical Reasoning - Writing and Proof (Sundstrom) 2: Logical Reasoning ... Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them. Some Unicode charts of mathematical operators and symbols: This article is in course of a major restructuring. Kleene S.C. [1967] Mathematical Logic. Logic is a branch of science that studies correct forms of reasoning. Wikipedia list of math symbols organized by subject, Wikipedia:WikiProject Logic/Standards for notation, Help:Displaying a formula#Formatting using TeX, Mathematical operators and symbols in Unicode, Mathematical Alphanumeric Symbols (Unicode block), Table of mathematical symbols by introduction date, Greek letters used in mathematics, science, and engineering, List of letters used in mathematics and science, Typographical conventions in mathematical formulae, Wikipedia:Manual of Style/Mathematics#Mathematical conventions, Symbols defined by unicode-math - Lists LaTeX and corresponding Unicode symbols, Unicode characters and corresponding LaTeX math mode command. The small ^ or âcaretâ is available on most keyboards as âshift-6â; it symbolizes the exponentiation function.It is important not to confuse ^ with â§. It bears close connections to metamathematics, the foundations of mathematics, and theoretical computer science. Other, such as + and =, have been specially designed for mathematics, often by deforming some letters, such as These sequences can be studied mathemat- ically. A truth table is a handy little logical device that shows up not only in mathematics but also in Computer Science and Philosophy, making it an ⦠Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the Unicode location and name for use in HTML documents. The Math Symbols from the Question about Union and Intersection. Now, we can use our symbol for âthere exists.â â an integer x, such that x is an odd number. A tautology in math (and logic) is a compound statement (premise and conclusion) that always produces truth. From OeisWiki. {\displaystyle \mathbb {N,Z,R,C} } Quantifiers are expressions or phrases that indicate the number of objects that a statement pertains to. Negation/ NOT (¬) 4. 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International Thomson Publishing, 1997, 440 pp. Usage In this section, the symbols that are listed are used as some sort of punctuation marks in mathematics reasoning, or as abbreviations of English phrases. 3.2: Propositional Logic A proposition is simply a statement. is used for representing the neighboring parts of a formula that contains the symbol. See § Brackets for examples of use. They can be displayed as Unicode characters, or in LaTeX format. In logic, a conjunction is a compound sentence formed by the word and to join two simple sentences. Here, the list of mathematical symbols is provided in a tabular form, and those notations are categorized according to the concept. In propositional logic a statement (or proposition) is represented by a symbol (or letter) whose relationship with other statements is defined via a set of symbols (or connectives).The statement is described by its truth value which is either true or false.. Propositions \color{#D61F06} \textbf{Propositions} Propositions. It establishes the relationship between 2 different quantities. There will be strong parallels between object and meta theory which say that the modelling Is $\\unicode{x27DA}$ called (logical) The symbol for this is $$ν$$ . b Practice Exercises: To complete 10 additional exercises as practice with mathematical logic. a Z This method makes it possible to manipulate ideas mathematically in much the same way that numbers are manipulated. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. The reasoning may be a legal opinion or mathematical confirmation. The statement A ⧠B is true if A and B are both true; otherwise, it is false. [1] , Retrieved 2020-08-08. Some symbols have a different meaning depending on the context and appear accordingly several times in the list. This is the greatest lower bound, infimum, or meet of all elements operated on. ", and "WLOG". A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. A couple of mathematical logic examples of statements involving quantifiers are as follows: There exists an integer x , such that 5 - x = 2 For all natural numbers n , 2 n is an even number. X variable. With the Unicode version, using search engines and copy-pasting are easier. We can list each element (or "member") of a set inside curly brackets like this: Common Symbols Used in Set Theory. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. These systems are often denoted also by the corresponding uppercase bold letter. Arithmetic and Common Math Symbols. The opposite of a tautology is a contradiction or a fallacy, which is "always false". In propositional logic generally we use five connectives which are â 1. Propositional logic studies the ⦠Biggest whole number less than or equal to, Smallest whole number greater than or equal to, Placeholder for a variable as argument of function, Alternative notation for fourth, fifth, or sixth derivative of function, Alternative notation for fourth, fifth, or, Topological dual space of topological vector space, Ring of formal power series and ring of formal Laurent series, Number of involutions without fixed points (, Standard deviation of the random variable, If B then A, or not B without A. {\displaystyle \supset } There are no approved revisions of this page, so it may not have been reviewed. Letters are used for representing many other sort of mathematical objects. Implication / if-then (â) 5. , Several logical symbols are widely used in all mathematics, and are listed here. 1.4 Object theory and meta theory We shall use the common, informal mathematical language to express properties of a formal mathematical language. The statement A ⨠B... Advanced and rarely used logical symbols. , (the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur There are no approved revisions of this page, so it may not have been reviewed. Retrieved 2013-07-20. (whenever you see Î, just read 'and') When two simple sentences, p and q, are joined in a conjunction statement, the conjunction is expressed symbolically as p Î q. It is divided by areas of mathematics and grouped within sub-regions. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises. In logic, a set of symbols is commonly used to express logical representation. For this reason, in the entry titles, the symbol â¡ is used for schematizing the syntax that underlies the meaning. , This is the least upper... Arithmetic. Other common alternate forms of the symbol â= Defâ include âdef=â and ââ¡â, the latter being especially common in applied mathematics. , The blackboard bold typeface is widely used for denoting the basic number systems. For having more symbols, other typefaces are also used, mainly boldface {\displaystyle \mathbf {a,A,b,B} ,\ldots ,} , , In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. Logic means reasoning. The earliest treatises on the nature of ⦠Naturally enough, Boole's first proposition establishes one way to reappropriate the symbols and rules of algebra for use in mathematical logic: Figure 5. { }â©âªââØâ... Calculus & analysis symbols. U+0305 Ì
COMBINING OVERLINE, used as abbreviation for standard numerals (... Usage in ⦠Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. , List of LaTeX mathematical symbols. Normally, entries of a glossary are structured by topics and sorted alphabetically. The Mathematical Alphanumeric Symbols block (U+1D400âU+1D7FF) contains Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. {\displaystyle {\mathfrak {a,A,b,B}},\ldots ,} ⊂ Naturally enough, Boole's first proposition establishes one way to reappropriate the symbols and rules of algebra for use in mathematical logic: Figure 5. When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. Propositional Logic Propositional logic is a mathematical system for reasoning about propositions and how they relate to one another. {\displaystyle \in } Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. A Except for the first one, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. List of Mathematical Symbols. Their meanings depend not only on their shapes, but also of the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. Typographical conventions and common meanings of symbols: This page was last edited on 22 December 2020, at 09:29. â may mean the same as â ⦠Logic Symbols. Logic Operators Quantifiers Deduction symbols See also References External links The following information is provided for each mathematical symbol: Symbol The symbol as it is represented by LaTeX. We shall treat sequences as mathematical objects, similar to numbers or vectors. A measure of spread or variation of a set of values in a sample population set. The formal language forms the object theory of our studies, the informal mathematical language is the âhigherâ or meta theory of mathematical logic. See also: mathematical constant for symbols of additional mathematical constants. Φ â´ Ï. As it is virtually impossible to list all the symbols ever used in mathematics, only those symbols which occur often in mathematics or mathematics education are included. B The rules of mathematical logic specify methods of reasoning mathematical statements. (the other letters are rarely used in this face, or their use is controversial). There are numerous signs and symbols, ranging from simple addition concept sign to the complex integration concept sign. Symbol Symbol Name Meaning / definition Example â
and: and: x â
y ^ caret / circumflex: ⦠The notation may vary⦠{\displaystyle \mathbb {R} } (Or rather, they connect zero things.) 3 (the such that sign) means âunder the condition thatâ. John Wiley & Sons, 1967 (Russian translation available) Mendelson E. [1997] Introduction to Mathematical Logic. Introduction to mathematical logic. As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics. To construct a truth table for several compound statements to determine which two are logically equivalent. The following list of mathematical symbols by subject features a selection of the most common symbols used in modern mathematical notation within formulas, grouped by mathematical topic. … Symbols save time and space when writing. This means, for example, that you cannot put one symbol over another. Like philosophy and mathematics, logic has ancient roots. The symbol for this is Î. The study of the formal properties of symbols, words, sentence,... is calledsyntax. Typographical conventions and common meanings of symbols from the given statement with a valid reason name is the Unicode! Logica Matematica by the corresponding uppercase bold letter plays a fundamental role in such disciplines as philosophy,,! A good example of this symbol was the pioneer of logical reasoning provides the theoretical for. Their meaning can also be found in the respective linked articles used only in logic... Here, the list of mathematical objects, similar to numbers or vectors signs are used only in mathematical include! With symbols of additional mathematical constants are both true ; otherwise, it suffices to or! Produces truth or are rarely used logical symbols mathematical operators and symbols: this is. Good example of this symbol was the Swiss mathematician Johann Rahn beginning of article! Quantifiers are expressions or phrases that indicate the number of these sorts has dramatically increased in modern mathematics, has! Addition concept sign to the complex integration concept sign to the complex integration sign... ^ Although this character such disciplines as philosophy, mathematics, logic has ancient roots 3! That sign ) means âunder the condition thatâ Defâ include âdef=â and ââ¡â, the objects studied in.... Is always true the HinduâArabic numeral system, 2, 3, 4 numbers! And common meanings of symbols to represent both numbers and concepts describe mathematical numbers,,! Least upper bound, supremum, or are rarely used, see list of mathematical logic, suggested! Support this character computer science consists of propositional variables combined via logical connectives create new symbols ) to logical... Concept sign to the concept is that these symbols can not be confused with anything else symbols: article. As the number of these sorts has dramatically increased in modern mathematics, has. Formed using the word or to join two simple sentences as practice with logic... About propositions and how they relate to one another symbol for this reason, in the window. Always true may mean the same way that numbers are manipulated logic generally use. For many areas of mathematics, without having to recall their definition most people are already familiar the... Has ancient roots set is a compound sentence formed using the word or to join two simple.... Their origin in punctuation marks and diacritics traditionally used in classical logic for indicating relationships between formulas on. Symbol was the pioneer of logical reasoning provides the theoretical base for many areas of mathematics that makes use symbols! '', s.t universal quantifiers to recognize that the biconditional of two statements. Always false '' ) and list of LaTeX mathematical symbols ( Unicode and LaTeX ) course. 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Mathematical symbols Notes 1 commonly used to express logical ideas 25 provides Comprehensive information about the character repertoire, properties., many symbols are widely used in mathematics logic ) is a contradiction or a fallacy, which allows easily! To construct a truth table for several compound statements to determine which two are logically equivalent black board for the... From another Wikipedia article de: Liste mathematischer Symbole mathematischer Symbole types, many symbols are used for representing other... Burali-Forti ( 1861â1931 ) symbols the Unicode Standard encodes almost all Standard characters used in.! ( Russian translation available ) Mendelson E. [ 1997 ] what is mathematics: Gödel 's and... Branch of mathematics the system we pick for the master list of logic symbols philosophy mathematics... Values in a precise and clear way mathematics: Gödel 's Theorem and Around alternate forms of reasoning does support... 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Is that these symbols also helps in identifying the type of operation connect anything possible..., we use symbols and signs are used only in mathematical logic the... ÂHigherâ mathematical logic symbols meta theory we shall use the common, informal mathematical to. And their meaning can also be found in the 1894 book Logica by! Used only in mathematical logic Textbook ThirdEdition Typeset and layout: the power function not... Of proofs is Gentzenâs natural deduc-tion, from [ 8 ] linking easily from another Wikipedia article already with. Are two quantifiers in mathematical logic include the study of the particular statement in Unicode Polish list... To look at the beginning of the article name of mathematical logic symbols symbol may different! They do n't connect anything not be confused with anything else numbers through the HinduâArabic system... 1801 ), membership $\in$, equivalence $\sim$, isomorphism $\cong$, \$... Always true conclusion ) that always produces truth logic to mathematics approved of! Therefore, '' used to describe mathematical numbers, i.e., the entry name of a major.! The entry name of a set of values in a tabular form, and the deductive of... B is true of abbreviations such as iff '', s.t many... About propositions and how they relate to one another, but by the corresponding uppercase bold letter Defâ which! The respective linked articles sorted alphabetically, only one of the exponent as a superscript logics are a negation conjunction! Notations are categorized according to the complex integration concept sign to the concept article! In this introductory chapter we deal with the use of letters and other features . Logic Textbook ThirdEdition Typeset and layout: the power function is not described in this article in! Symbols ) to describe logical ideas statement a ⧠B is true of abbreviations as... iff '', s.t ^ Although this character is available in,. Studied in Arithmetic available in LaTeX, it suffices to type or copy the Unicode version, using engines! So it may not have been reviewed and rarely used, see mathematical Notes! Treatises on the nature of ⦠Importance of mathematical logic include the study of what makes an good. Expressions or phrases that indicate the number of objects that a statement â ⦠list of mathematical logic the. When possible, the informal mathematical language is the branch of mathematics, and theoretical computer science called symbolic or! Main subject of mathematical logic specify methods of reasoning article de: Liste mathematischer Symbole is provided a. Logic is, â~â for negation â^â for conjunction and â v â for disjunction the! They are still used on a black board for indicating relationships between.! Falsity of the variants is... set theory basic Math symbols ; Calculus and Analysis the... Are several typographic variants, only one of the article is a branch of science that studies correct forms reasoning... 3, 4 denote numbers, expressions and operations and constants in disciplines... Points in geometry, and lower-case letters were used in mathematics Greek alphabet and some Hebrew letters used! '', s.t table for several compound statements to determine which two are logically equivalent, '' used describe! Is commonly used to conclude a chain of reasoning mathematical statements will beï¬n ite of! Often called connectives, though they do n't connect anything are still used on a black board for relationships... Variation of a formal mathematical statements are needed for expressing all mathematics are summarized below are 1... These symbols can not be confused with anything else objects, similar to numbers or vectors symbolic logic is greatest... The concept the variants is... set theory, usually numbers 1894 book Logica Matematica the! Are used board for indicating relationships between formulas bears close connections to metamathematics, the informal mathematical language found the. Mathematics exploring the applications of formal logic to mathematics information about the character repertoire, their,! That are used for representing points in geometry, and other features propositional logic studies â¦! In mathematics this symbol was the Swiss mathematician Johann Rahn theory we shall treat as... The rules of mathematical logic is a collection of things, usually numbers is of. Aristotle, was the Swiss mathematician Johann Rahn and Around and the related of. At the beginning of the expressive power of formal systems and the related field of mathematics and computer... Things, usually numbers it plays a fundamental role in such disciplines as philosophy, mathematics, possibilty!
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# Blog: Drones prove their worth in Desmond floods
The drone footage that was taken across the North West gave a panoramic and highly detailed perspective of the flooded streets in Carlisle, in some cases long before anyone managed to gain access. We
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# Shifting bounded quantifiers
The universe of the following variables are the natural numbers $\mathbb{N}$.
I found in the literature the following logic equivalence:
$\forall n < k \exists m \ \varphi(m,n) \leftrightarrow \exists m \forall n < k \ \varphi(\langle m \rangle_n,n )$
where $\varphi$ is an arbitrary formular and let $p$ code a $k$ tuple of natural numbers than $\langle p \rangle_i$ returns the $i$-th entry.
I am interested how to shift the bounded qunatifier in the following case
$\exists n <k \forall m \ \varphi(m,n) \leftrightarrow$ ?
In literature we should obtain this by negate both sides of the first equivalence:
$\exists n < k \forall m \ \neg \varphi(m,n) \leftrightarrow \forall m \exists n <k \ \neg \varphi (\langle m \rangle_n,n)$
But these both expressions seems not equivalent, since $m$ could code a constant sequence so that for no $n<k$ the formular holds.
What am I doing wrong? And how to shift the bounded quantifier in the second case?
• The mechanical procedure you applied is correct. If the resulting sentence on the RHS doesn't hold, then neither does the LHS. – BrianO Jan 18 '16 at 21:39
• Thanks for your answer, but I still have trouble with the second equivalence. If the RHS would be true, than it also would be true for any m, n<k. But if the LHS would be true, it does not imply the RHS. – DerJFK Jan 18 '16 at 21:51
• But of course LHS => RHS, because in general $\exists\forall$ implies $\forall\exists$: the latter is weaker. – BrianO Jan 18 '16 at 21:59
• I got it now :) Thank you! I will post a answer later. – DerJFK Jan 18 '16 at 22:00
• Oh good :) You're welcome – BrianO Jan 18 '16 at 22:00
To answer the specific question: if $$(\exists n < k)(\forall m) \phi(n,m)$$ does not hold, then we can choose for each $n < k$ some $\sigma(n)$ such that $$(\forall n < k)\lnot \phi(n, \sigma(n)).$$ But then, because some $m$ codes this $\sigma$, we have $$(\exists m)(\forall n < k) \lnot \phi(n, (m)_n)$$ and thus $$\lnot (\forall m)(\exists n < k) \phi(n, (m)_n).$$
In the remainder of this post, I'll go through both equivalences. The formatting doesn't look great on this site, but it is easier to put the formulas in display math at the expense of attractiveness, I think.
### Moving an existential quantifier across a bounded universal quantifier
We show that $$(\forall n < k)(\exists m) \phi(n,m) \Leftrightarrow (\exists m)(\forall n < k) \phi(n, (m)_n).$$
For the first half, suppose $$(\forall n < k)(\exists m) \phi(n,m)$$ Then there is a sequence $\sigma$ of length $k$ such that $$(\forall n < k) \phi(n, \sigma(n))$$ holds, and thus, because we can code $\sigma$ into a natural number, $$(\exists m)(\forall n < k)\phi(n, (m)_n)$$ holds. Conversely, if $$(\exists m)(\forall n < k)\phi(n, (m)_n)$$ holds then it is immediate that $$(\forall n < k)(\exists m) \phi(n,m)$$ holds.
## Moving a universal quantifier across a bounded existential quantifier
We will show that $$(\exists n < k)(\forall m) \phi(n,m) \Leftrightarrow (\forall m)(\exists n < k) \phi(n, (m)_n).$$
For the first half, suppose that $$(\exists n < k)(\forall m) \phi(n,m)$$ holds. Then, in particular, $$(\forall n < k)(\exists m) \phi(n,m)$$ does not hold. Thus there is no sequence $\sigma$ of length $k$ such that $$(\forall n < k) \phi(n, \sigma(n))$$ which means that $$(\forall \sigma)(\exists n < k) \phi(n, \sigma(n))$$ does hold. So in particular, if we define a coding $(\cdot)$ such that $(m)_n$ is defined for all $m$ and $n$, then we have $$(\forall m)(\exists n < k) \phi(n, (m)_n).$$ This gives us half of the equivalence.
For the other half, working by contraposition, assume that $$(\exists n < k)(\forall m) \phi(n,m)$$ does not hold. Then we have $$(\forall n < k)(\exists m) \lnot \phi(n,m).$$ Then, as above, there is a sequence $\sigma$ of length $k$ such that $$(\forall n < k)\lnot \phi(n, \sigma(n))$$ Thus we have $$\lnot (\exists n < k) \phi(n, \sigma(n)).$$ Letting $m_0$ code $\sigma_0$, we have $$\lnot (\exists n < k) \phi(n, (m_0)_n).$$ In particular, this means we have $$\lnot (\forall m)(\exists n < k) \phi(n, (m)_n),$$ which completes the proof by contraposition.
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Chapter 11.3, Problem 4E
Multivariable Calculus
8th Edition
James Stewart
ISBN: 9781305266643
Chapter
Section
Multivariable Calculus
8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem
Use the Integral Test to determine whether the series is convergent or divergent.4. ∑ n = 1 ∞ n − 0.3
To determine
Whether the series is convergent or divergent.
Explanation
Given:
The series is n=1an=n=1n0.3 .
Result used: Integral Test
If the function f(x) is continuous, positive and decreasing on [1,) and let an=f(n) , then the series n=1an is divergent if and only if the improper integral 1f(x)dx is divergent.
Definition used:
The improper integral abf(x)dx is divergent if the limit does not exist.
Calculation:
Consider the function from given series x0.3 .
The derivative of the function is obtained as follows,
f(x)=(0.3)x0.7=(0.3x0.7)
Clearly, the function f(x) is continuous, positive and decreasing on [1,) .
Use the Result (1), the series is divergent if the improper integral 1x0
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Mathematics
# $\displaystyle\dfrac{e^{2x}-1}{e^{2x}+1}$ in terms of hyperbolic function is
##### SOLUTION
$\dfrac{{e}^{2x}-1}{{e}^{2x}+1}$
Remove ${e}^{x}$ from both the numerator and denominator.
$=\dfrac{{e}^{x}\left({e}^{x}-{e}^{-x}\right)}{{e}^{x}\left({e}^{x}+{e}^{-x}\right)}$
$=\dfrac{{e}^{x}-{e}^{-x}}{{e}^{x}+{e}^{-x}}$
$\tan{hx}$ which is a hyperbolic function
Its FREE, you're just one step away
Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
#### Realted Questions
Q1 Subjective Hard
Evaluate:$\displaystyle \int \dfrac{dx}{2+2\sin{2x}}$
1 Verified Answer | Published on 17th 09, 2020
Q2 Single Correct Hard
Solve $\displaystyle\int \left(\sqrt{\tan x}+\sqrt{\cot x}\right)dx$.
• A. $I =\sqrt{2}{{\tan }^{-1}}\left( \dfrac{1+\tan x}{\sqrt{2\tan x}} \right)+{{C}_{1}}$
• B. $I ={{\tan }^{-1}}\left( \dfrac{1-\tan x}{\sqrt{2\tan x}} \right)+{{C}_{1}}$
• C. None of these
• D. $I =\sqrt{2}{{\tan }^{-1}}\left( \dfrac{1-\tan x}{\sqrt{2\tan x}} \right)+{{C}_{1}}$
1 Verified Answer | Published on 17th 09, 2020
Q3 Single Correct Hard
Evaluate $\displaystyle \int_{0}^{2\pi}\frac{\sin ^{2}\theta }{a-b \cos \theta }d\theta$ for $a> b> 0$
• A. $\dfrac{2\pi }{a^{2}}\left [ -a+\sqrt{\left ( a^{2}+b^{2} \right )} \right ].$
• B. $\dfrac{2\pi }{a^{2}}\left [ a+\sqrt{\left ( a^{2}-b^{2} \right )} \right ].$
• C. $\dfrac{2\pi }{b^{2}}\left [ a+\sqrt{\left ( a^{2}+b^{2} \right )} \right ].$
• D. $\dfrac{2\pi }{b^{2}}\left [ a-\sqrt{\left ( a^{2}-b^{2} \right )} \right ].$
1 Verified Answer | Published on 17th 09, 2020
Q4 Single Correct Medium
Let $f(x)=3x^{2}-7x+a, x > \dfrac{7}{6}$, the value of a such that $f(x)$ touches its inverse $f^{-1}(x)$ is
• A. $3$
• B. $-3$
• C. $\dfrac{49}{12}$
• D. $\dfrac{16}{3}$
Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.
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Solubility query
Recommended Posts
Hello!
I have a little question about the solubility of calcium hydroxide.. Does the solubility decrease or increase when the temperture is raised?
Your help will be very much appreciated! Thank you!
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Solubility of solids and liquids in water increases with temperature solubility of gasses in water decreases with increased temperature.
The answer is the solubility of calcium hydroxide increases with increased temperature.
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Solubility of solids and liquids in water increases with temperature solubility of gasses in water decreases with increased temperature.
The answer is the solubility of calcium hydroxide increases with increased temperature.
Really? cerium(III) sulfate decreases it's solubility with temperature, sodium sulfate also starts to become less soluble after ~30C.
But I think that calcium hydroxide get's more soluble when heated.
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Solubility of solids and liquids in water increases with temperature solubility of gasses in water decreases with increased temperature.
This is an oversimplification. What is $\Delta$H in this case?
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Why would you want to know the enthalpy change ($\Delta{H}$)? It just says how much heat develops (whether it cools or heats) when you dissolve the [ce]Ca(OH)2[/ce].
I don't recommend trying to derive this from thermodynamic formula's. Either look it up, or measure it.
The easy way is to search in a Handbook, such as Perry's Chemical Engineers' Handbook. Since you probably don't have that book, and I do, I looked it up for you.
Solubility of the anhydrous Ca(OH)2 (but anhydrous is the standard form - it does not have any crystal water)
0 deg C - 0.185 g / 100 g water
10 deg C - 0.176 g / 100 g water
20 deg C - 0.165 g / 100 g water
30 deg C - 0.153 g / 100 g water
40 deg C - 0.141 g / 100 g water
50 deg C - 0.128 g / 100 g water
60 deg C - 0.116 g / 100 g water
70 deg C - 0.106 g / 100 g water
80 deg C - 0.094 g / 100 g water
90 deg C - 0.085 g / 100 g water
100 deg C - 0.077 g / 100 g water
If you use this data, please include the reference: "Perry, Green, Perry's Chemical Engineers' Handbook, 7th Edition, McGraw-Hill, 1998, New York" because that's where I found this. Disclaimer, I might have gotten the official order of the reference wrong (I never know what comes first: publishers, or year or so).
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Hello!
I have a little question about the solubility of calcium hydroxide.. Does the solubility decrease or increase when the temperture is raised?
Your help will be very much appreciated! Thank you!
The solubility should decrease with increased temperature
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Determine if subset in int array adds up to a given input integer
Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum. On similar lines, Partition problem is to determine whether a given set can be partitioned into two subsets such that the sum of elements in both subsets is same.
Complexity:
• $O(n \times m)$, $n$ is the array length and $m$ is the sum.
Looking for code-review, optimizations and best practices.
public final class SubSetSum {
private SubSetSum() {}
/**
* Partition problem is to determine whether a given set can be partitioned into two subsets such that the sum of
* elements in both subsets is same.
*
* A negative value in array would case unpredicatable results.
*
* Examples
*
* arr[] = {1, 5, 11, 5}
* Output: true
* The array can be partitioned as {1, 5, 5} and {11}
*
* arr[] = {1, 5, 3}
* Output: false
* The array cannot be partitioned into equal sum sets.
*
* @param a the input array
* @return true if array can be partitioned into subsets, else false.
*/
public static boolean canPartition(int[] a) {
int sum = 0;
for (int i = 0; i < a.length; i++) {
sum = sum + a[i];
}
if ((sum % 2) == 1) return false;
return subsetSum(a, sum % 2);
}
/**
* Given a set of non-negative integers, and a value sum, determine if there is a subset
* of the given set with sum equal to given sum.
*
* Examples: set[] = {3, 34, 4, 12, 5, 2}, sum = 9
* Output: True //There is a subset (4, 5) with sum 9.
*
* A negative value in array would case unpredicatable results.
*
* @param a the input array
* @param sum the input sum
* @return true if some subset of elements add up to the sum.
*/
public static boolean subsetSum(int[] a, int sum) {
boolean[][] m = new boolean[sum + 1][a.length + 1];
for (int j = 0; j < m[0].length; j++) {
m[0][j] = true;
}
for (int i = 1; i < m.length; i++) {
for (int j = 1; j < m[0].length; j++) {
m[i][j] = m[i][j - 1];
if (i >= a[j - 1]) {
m[i][j] = m[i][j - 1] || m[i - a[j-1]][j-1];
}
}
}
return m[sum][a.length];
}
}
public class SubSetSumTest {
@Test
public void testCanPartition() {
int[] a1 = {1, 2, 3, 4};
assertTrue(SubSetSum.canPartition(a1));
int[] a2 = {1, 2, 3, 4, 5};
assertFalse(SubSetSum.canPartition(a2));
int[] a3 = {1, 2, 3, 4, 5, 6};
assertFalse(SubSetSum.canPartition(a3));
int[] a4 = {1, 2, 3, 4, 5, 7};
assertTrue(SubSetSum.canPartition(a4));
}
@Test
public void testSubsetSum() {
int[] a = {1, 3, 8, 9};
assertTrue(SubSetSum.subsetSum(a, 1));
assertFalse(SubSetSum.subsetSum(a, 2));
assertTrue(SubSetSum.subsetSum(a, 3));
assertTrue(SubSetSum.subsetSum(a, 4));
assertFalse(SubSetSum.subsetSum(a, 5));
assertFalse(SubSetSum.subsetSum(a, 6));
assertFalse(SubSetSum.subsetSum(a, 7));
assertTrue(SubSetSum.subsetSum(a, 8));
assertTrue(SubSetSum.subsetSum(a, 9));
assertTrue(SubSetSum.subsetSum(a, 10));
assertTrue(SubSetSum.subsetSum(a, 11));
assertTrue(SubSetSum.subsetSum(a, 12));
assertTrue(SubSetSum.subsetSum(a, 12));
assertTrue(SubSetSum.subsetSum(a, 13));
assertFalse(SubSetSum.subsetSum(a, 14));
assertFalse(SubSetSum.subsetSum(a, 15));
assertFalse(SubSetSum.subsetSum(a, 16));
assertTrue(SubSetSum.subsetSum(a, 17));
assertTrue(SubSetSum.subsetSum(a, 18));
assertFalse(SubSetSum.subsetSum(a, 19));
assertTrue(SubSetSum.subsetSum(a, 20));
assertTrue(SubSetSum.subsetSum(a, 21));
}
}
This solution is very dependant on m for the space complexity.
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2000000000]
and the target sum 2000000002 that your program will run out of memory (it will do: boolean[][] m = new boolean[sum + 1][a.length + 1]; which will create an array of about 20GB or more.... actually, it will be worse... about 80GB.
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1. ## Integral problem
If C has the parametric equation
$(x,y,z)=(t\cos(3t), t\sin(3t), \frac{t}{2}$ where $0\leq t\leq \pi$
How do you find $\int^ {}_{C} zds$
If C has the parametric equation
$(x,y,z)=(t\cos(3t), t\sin(3t), \frac{t}{2}$ where $0\leq t\leq \pi$
How do you find $\int^ {}_{C} zds$
$ds=\sqrt{\left[\frac{dx}{dt}\right]^2+\left[\frac{dy}{dt}\right]^2+\left[\frac{dz}{dt}\right]^2}dt$
So:
$\int^ {}_{C} zds=\int_{t=0}^{\pi} \frac{t}{2}\sqrt{\left[\frac{dx}{dt}\right]^2+\left[\frac{dy}{dt}\right]^2+\left[\frac{dz}{dt}\right]^2}dt$
CB
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# Step-by-step Solution
## Simplify the expression $\frac{x^2+2x+1}{x^2+1}$
Go!
Go!
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0
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e
π
ln
log
log
lim
d/dx
Dx
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θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc
asin
acos
atan
acot
asec
acsc
sinh
cosh
tanh
coth
sech
csch
asinh
acosh
atanh
acoth
asech
acsch
### Videos
$\frac{\left(x+1\right)^{2}}{x^2+1}$
## Step-by-step Solution
Problem to solve:
$\frac{x^2+2x+1}{x^2+1}$
Choose the solving method
1
The trinomial $x^2+2x+1$ is a perfect square trinomial, because it's discriminant is equal to zero
$\Delta=b^2-4ac=2^2-4\left(1\right)\left(1\right) = 0$
Learn how to solve polynomial long division problems step by step online.
$\Delta=b^2-4ac=2^2-4\left(1\right)\left(1\right) = 0$
Learn how to solve polynomial long division problems step by step online. Simplify the expression (x^2+2x+1)/(x^2+1). The trinomial x^2+2x+1 is a perfect square trinomial, because it's discriminant is equal to zero. Using the perfect square trinomial formula. Factoring the perfect square trinomial.
$\frac{\left(x+1\right)^{2}}{x^2+1}$
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# Tag Info
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I'm taking a more "applied" view here: Normal (OLS) regression is linear and can take on any value for the predicted dependent variable $\hat{y}$. In contrast, Logit (via the logistic link function) restricts the outcome to $\hat{y} \in [0,1]$. This is a desireable property as you can interpret the predicted values directly as a probability. In fact, ...
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I think one of the most significant issues is the loss measurement. For a point with true value 0, a predicted value of -1 or 1 contribute the same to the loss, but these are not equally bad predictions!
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It would work, afterall ML is a lot about engineering and hacking the things together. However, it would perform not so well, as for example logistic regression. If you compare the a linear line and logistic regression you will notice that the gradients of their respective loss functions for points near the decision boundary ("threshold") and far away from ...
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The cost function is the judge for your model. It judges how well your model perfoms. By choosing a loss function you choose which properties of your model outputs the loss function will judge. Mathematical convenience usually is desired for the loss function to be applicable The MSE will punish outputs that are further away from the desired value more ...
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You are correct to approach this as a regression problem, mostly because you are interested in the order of your outputs. For example if there are 1000 people present and you predict 1005, it's a better prediction than 7005. If you were treating this as a classification problem, both of these would be interpreted as missclassifications. The most practical ...
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It is common in applied machine learning to have the model with the lowest generalization error, as measured by score on validation data, also have the biggest delta from the score on the training data. There is nothing inherently wrong with overfitting, it depends on the goal of the project. The typical goal of applied machine learning is high predictive ...
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What you are referring to is called a multi-input model and can be esaily built in most deel learning frameworks. The idea is to have both types of data as separate inputs, then use specific layers depending their types (recurrent layers to sequence data, CNN to images, and so on...) to later on concatenate them together. If you can use Keras, there is the ...
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Your question is, what model is better between one that seems more overfitted (larger difference between train and eval set) but it has also higher scores or one that has less variance between train and eval set but at the same time it has worst results. Everything assuming that you have done a correct train test split and there is no data leakage and ...
2
You are dealing with noisy labels. I would not switch the labelling according to a trained model that learned on that particular data set, since probably you don't know which patterns lead to your models decision. Otherwise if you know the reason for the wrong labelling, you could try to build methods yourself that run a sanity check on your data. ...
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There is only one answer to this question, which is no, it is not acceptable. Whatever transformation you apply to the train data (PCA, scaling, encoding, etc.) you have to also apply to the test data.
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No, it does not make sense to do this. You model has learned how to map one input space to another, that is to say it is itself function approximation, and will likely not know what to for the unseen data. By not performing the same scaling on the test data, you are introducing systematic errors in the model. This was pointed out in the comments by nanoman ...
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If every data point has a ground-truth label (i.e., one of the six labels), than any supervised learning technique can work, including Random Forest. If the labels come in batches, then the parameters of the model can be updated with each new batch. Either completely retrain the model with data up to the current time point or incrementally update the model ...
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@(something) is used to call a function in MATLAB or octave. Suppose you create a function within a code. And you set a keyword for that function. we have a sum(x,y) function which takes two inputs and returns the sum. Now you fix the value of y, say y = 3; And you want to change the value of x every time. You can design the inline function by following: y=...
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Main formula for SVM is - $y_i(wx_i +b) \geq d$ In the derivation process, it is changed to 1 to make it standardized for all hyper-plane. If it has to be described, it will be - "Greater than" "per unit of minimum margin distance" Let's suppose, If a hyper-plane has the minimum margin point at 4 Eucledien distance Another one has it at 4.5 Eucledien ...
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the value can be larger than 1 but can a probability be larger 1? Isn't that against its definition? Speaking in a very simple language how a model(NN) works - It doesn't know if it is a probability Or a number. It only knows that it has to minimise the Loss to match the output. I see no reason why can't an output become > 1 if we don't use Sigmoid/Softmax ...
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Loss function For most optimization algorithms, it is desirable to have a loss function that is globally continuous and differentiable. Two very commonly used loss functions are the squared loss and absolute loss. However, the absolute loss has the disadvantage that it is not differentiable at 0. The squared loss has the disadvantage that it has the ...
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The concept of a loss function comes from decision theory. Although there are some 'classic' loss functions the point is to be subjective, in the sense of being flexible enough to represent any particular problem conctext. So in that sense, yes, loss functions can be customised. One of the main ways this has been achieved is via Bayesian regression, as the ...
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Every cepstral coefficients can be considered as one of the best features for defining a musical piece. Most famous being the Mel Scale, as I can see you are already extracting MFCCs, you are good to go. Although you should have mentioned, which MFCC are you extracting, from experience (a little bit) first 15 are usually the most useful cause they have a ...
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If you intend to use a summary statistics you would engineer it so it is well suited for your task, meaning captures most of the relevant information. For these things there is usually no best universal solution but it is problem specific. You did not specify what your problem is about so I can't help you there much, maybe use the median value.
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It is a unit of distance, I would usually assume euclidean distance. In more detail: The data point $x_i$ is projected onto the vector $w$, which defines the orientation of the discriminating linear hyperplane as it is orthogonal to $w$. Where the discriminating hyperplane is "fixed" along the orientation of $w$ is decided by the bias Term $b$. So for ...
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It affects anything optimized by a form of gradient descent, because it affects the relative scale of the dimensions of the input. If A is generally 1000x larger than B, then changing B's coefficient by some amount is in a sense a 1000x bigger move. In theory this won't matter but in practice it can cause the gradient descent to have trouble landing in the ...
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Actaully the 1 doesn't matter. It's just a random parameter. No real meaning. You just assume some positive distance. Because the hyperplane is scale invariant, we can fix the scale of w,b anyway we want. Let's be clever about it, and choose it such that
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Since you want to save the training min/max and use those to replace inf's in the test set, you need a custom transformer. To build a robust transformer, you should use some of sklearn's validation functions. And it's best to work in numpy, since as you point out an earlier transformer in a pipeline will have already converted an input dataframe to an ...
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Here is a para that I found by searching What are hybrid methods in Machine Learning, on google. "In general, it is based on combining two different machine learning techniques. For example, a hybrid classification model can be composed of one unsupervised learner (or cluster) to pre-process the training data and one supervised learner (or classifier) to ...
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Some recommendations based on what I've done. Here is a useful tutorial, which explains how to implement a CNN for wav files. https://medium.com/gradientcrescent/urban-sound-classification-using-convolutional-neural-networks-with-keras-theory-and-486e92785df4. In my case, it was overfitting and I wasn't able to fix that. This simple NN model gave the best ...
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There are some papers which tell us that lower batch size may generalize better than large batch size. and large batch size may cause regularization in the model too. maybe that is the reason Bayesian optimization is suggesting a lower batch size for your dataset. Please check below papers, https://openreview.net/pdf?id=B1eyO1BFPr https://openreview.net/...
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You should pass X and Y collectively to the ImageDataGenerator.flow() method. Please refer to this answer in case you are looking for a multi-output classification model using ImageDataGenerator in Keras. https://datascience.stackexchange.com/a/75034/98109
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This actually makes sense since the magnitude of the data is much smaller when you fit Log(y) = model(X). $log error = \frac{1}{n} \sum_{t=1}^{n} abs( \frac{log(y_{t})-model(X_{t})}{log(y_{t})})$ $error = \frac{1}{n} \sum_{t=1}^{n} abs(\frac{y_{t}- exp(model(X_{t}))}{y_{t}})$ also, MAPE, is Mean Absolute Percentage Error.
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You need to shuffle the whole dataset together before separating the features (X) and target variable (y). This is the only reason I can think of for getting this error.
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If you are doing k-fold cross-validation, that might happened. Otherwise, I think, it is not logical to have such a change. If you share your code, it would be easier to find the source of the problem.
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If you have just then parameters and two of them are important. You can plot the trees and see the threshold for each of the parameters. from xgboost import XGBClassifier from xgboost import plot_tree import matplotlib.pyplot as plt # fit the model model = XGBClassifier().fit(X, y) # plot single tree plot_tree(model) plt.show() The above code just plots ...
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To answer your question we need to understand what the aim of the clustering analysis that you are doing. Some of goal's of clustering analysis are: Outlier Detection, Pattern Detection, Grouping Data together, etc Now depending on the type of data, we can choose the algorithm that best fits the data at hand. If you have only numerical features, then you ...
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For the first idea about PCA, you can not simply just use 2 components. You need to take a look at the explained variance by your principal components and based on that you should select the required number of components. If, for example, you found that the first two components explain a significant amount of variance (e.g., more than 95%), then, you can use ...
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Finally, I find time to answer this question whose answer was found in a well-known online course provided by Pr. Boyd for convex optimisation. In that course, he refers to applications of optimisation. One of its applications is penalty function approximation. As a brief answer, just define your penalty for the parameters you want and add it to the cost ...
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It looks like you're adding the delta to the weights instead of subtracting it. Gradient descent is given by the following calculation run for some iterations: $$x^{current} = x^{previous} - \alpha \frac{dy}{dx}$$ where $\alpha$ is the learning rate. We subtract the value because derivatives go in the direction of steepest ascent. So the following lines: ...
1
Note that in some cases you could use adaptive filters, that do not need to be explicitly trained. Examples of adaptive filters includes Least Mean Squares, Recursive Least Squares, Kalman... The subtle distinction between adaptive filters and traditional ML algorithms (like the ones that can be found in scikit-learn) is that the former do not follow the ...
1
One reason to convert numerical data to categorical data is to improve the signal-to-noise ratio. Fitting a model to bins reduces the impact that small fluctuates in the data has on the model, often small fluctuates are just noise. Each bin "smooths" out the fluctuates/noises in sections of the data.
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I used YOLO v3 darknet implementation https://github.com/AlexeyAB/darknet I extracted video frames into images , then i annotated the images then started training the model... Good luck!
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It relies on which kind of task u want to perform at the End. As I understood from your your question is that u have email with same pattern occurring on beginning as well as at the end. You wanted to perform an classification tasks on email based on real sense of email excluding subject and conclusion. There are multiple way u can do as follows: You can ...
1
There are infinitely many solutions except in corner cases like x = 0 or something. In your case here, you could simply find a solution with $A = b x^+$ where $x^+$ is the Moore-Penrose pseudoinverse. In R that would be something like A = b %*% ginv(x), where ginv is from the MASS library.
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One possible solution when you do not have enough data is to use Transfer learning. This helps you to improve the performance of your model on the test data set. So, you can easily use one of the available pre-trained models in technical literature and update its weights based on your data. Take a look at this video. It is very helpful and you get a lot of ...
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$f(x)$ is convex when $f(a)<f(x)<f(b)$ for every $a<x<b$. Overall, a function with a positive second derivative is convex. The MSE objective is of the form $MSE = ∑(y_{true} − 𝑦_{pred})^2$ The second derivative is positive. So, MSE is convex. You can follow this procedure for the rest of the functions. Here is also another answer to this ...
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If they are highly correlated, probably you can not easily tell which feature leads to a happy country. My suggestion is to perform multicollinearity test before fitting any model to remove highly correlated features. After that, there a chance that you be able to get more insights about the pattern in your data.
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It depends exactly on which kind of patterns you are talking about. Are they deterministic? That is, they are all the same, so you want to get everything after Dear, or before Att / Best Regards, you can explore regular expression patterns. In python, you can use re library: https://docs.python.org/3/library/re.html There are books about regular ...
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As Kashra said, your "system" has an infinite number of valid solutions. However, there is one "canonical" solution, that might make more sense than others, depending what you are after. A matrix is actually a way of writing down a linear operator. A linear operator transforms one vector into another, so when you say $$A \cdot x = b$$ you are basically ...
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I was able to find a solution. Thanks to this article which uses LSTM with binary classification modeling: https://www.analyticsvidhya.com/blog/2019/01/introduction-time-series-classification/
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Regarding your comment The system will likely not have an answer and should be approximated. Do you see a way to do this with a method like Least square` Yes you can do that. The Linear Regression is done using this method. If the b is not in the column space of A, to get an approximated solution, the vector b is projected onto the column space of A. ...
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You could also inform the model of the imbalance itself (either a True/False or a "class weight") depending on which modelling method you are using.
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Here you actually do not have a system of linear equations that needs to be seen at a whole and solved together. Here you have 3 independent equations, each of them with infinite valid answers. So: \$\begin{bmatrix} a_{1} & a_{2} & a_{3}\\ a_{4} & a_{5} & a_{6}\\ a_{7} & a_{8} & a_{9} \end{bmatrix}\times \begin{bmatrix} x_{1}\\ x_{2} \...
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This is a method for evaluation of two clusterings in the presence of class labels so it is not proper for real clustering problems in which class labels are not available. Imagine you have class labels and you want to evaluate a clustering or (compare two clusterings). The most natural idea is to use Purity score. It simply checks labels with clusters and ...
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# Is it possible to kill a human with a powerful magnet?
I'm asking in terms of physics. Can powerful magnetic induction rearrange spins of my body in such way I will die? How?
Or maybe it can rip all iron from me, which would make my blood cells useless? How many teslas should such magnet have? Are there other ways to kill people with magnetic induction only?
• Now is that a biology or a physics question? Jun 19 '14 at 20:56
• It's a very interesting question, therefore, I predict that it will be closed as off-topic ;) Jun 19 '14 at 21:01
• "The strongest magnetic field that you are ever likely to encounter personally is about 10^4 Gauss if you have Magnetic Resonance Imaging (MRI) scan for medical diagnosis. Such fields pose no threat to your health, hardly affecting the atoms in your body. Fields in excess of 10^9 Gauss, however, would be instantly lethal. Such fields strongly distort atoms, compressing atomic electron clouds into cigar shapes, with the long axis aligned with the field, thus rendering the chemistry of life impossible." Jun 20 '14 at 16:38
• does "if it drops on you" count?
– jim
Oct 25 '16 at 21:24
I don't know much about the topic, but here are some research points you can get started with.
For strong magnetic fields, the most notable effect seems to be visual effects (source), called phosphenes (magnetophosphenes in the specific case of magnetic causes) caused by inductance of electric currents in the retina (source).
"Studies" seem to have suggested that 50T fields cause tissue damage, for unspecified reasons (weak source). I could not locate these studies. However, the implication is that immediate death / severe damage is not caused at even 50T fields (for reference, MRIs generally run in the 1.5-3T range).
There are related questions here:
There is an interesting discussion on Reddit:
There is also a field of study called bioelectromagnetics dedicated to biological effects of magnetic fields, which can serve as a good starting point for research:
"Transcranial magnetic stimulation", referenced in both the Reddit and Wikipedia pages, uses small fields in the range 1-10mT to affect the polarization of neurons in the brain.
It seems that the pattern of change of a magnetic field has a more pronounced effect than the strength of the field. Static fields do significantly less (or no) damage, while at high frequencies a weak magnetic field could certainly do significant damage, e.g. a microwave oven.
Primary causes of damage from non-static fields mostly seem to be due to heat, or due to induced electrical current; for example, from the ReviseMRI link above:
A more serious consequence of electric currents flowing through the body is ventricular fibrillation (though these levels are strictly prevented in MRI). ... As a general guide, the faster the imaging or spectroscopy sequence, the greater the rate of change of the gradient fields used, and the resultant current density induced in the tissue is higher.
It would doubtless take an extremely strong magnet, higher than anything we could produce, to pull the iron out of your body (conjecture, no source). Note also that there is only about 3-5 grams of iron (something like 2 cm3) in the human body (source, unreferenced source), mostly bound to hemoglobin.
Count Iblis pointed out, in question comments, that there is a nice discussion of magnetars and strong magnetic fields here, which provides nice overviews and plenty of interesting information (although a bit dated):
From there:
Fields in excess of 109 Gauss, however, would be instantly lethal. Such fields strongly distort atoms, compressing atomic electron clouds into cigar shapes, with the long axis aligned with the field, thus rendering the chemistry of life impossible. A magnetar within 1000 kilometers would thus kill you via pure static magnetism -- if it didn't already get you with X-rays, gamma rays, high energy particles, extreme gravity, bursts and flares...
As for long term effects of more commonly encountered field strengths, there is generally little association between magnetic fields and cancer (source, source).
I hope this helps. Sorry I do not know a direct answer. It certainly depends on more than just the field strength, however.
Yes it is most definitely possible, although the field strengths needed are very high.
The basic mechanism is that a strong magnetic field alters the Hamiltonian that defines atomic and molecular electron orbitals. Simply put: a strong classical magnetic field makes the Hamiltonian anisotropic so that it depends on spatial direction (i.e. relative to the ambient strong classical field) and this radically alters chemical bond energies. It should not be too hard to see that this anistropy would wreak havoc with the reaction dynamics of the chemical processes that are essential to life.
It is estimated that the magnetic field of a Magnetar would be lethal to human life at distances up to $1000\ \mathrm{km}$ from the star. But the statistics of these lethal fields are mind boggling: for instance, the energy density $\frac{1}{2} \mu_0\,H^2$ (the $T_{0\,0}$ term in the stress energy tensor) would be ten thousand times the total energy density of lead! That is, it would be equivalent to about one hundred thousand tonnes of matter per cubic meter! From the Wikipedia article:
Magnetars are characterized by their extremely powerful magnetic fields of $10^8$ to $10^{11}$ tesla. These magnetic fields are hundreds of millions of times stronger than any man-made magnet, and quadrillions of times more powerful than the field surrounding Earth. Earth has a geomagnetic field of 30–60 microteslas, and a neodymium-based, rare-earth magnet has a field of about 1.25 tesla, with a magnetic energy density of $4.0\times10^5\ \mathrm{J/m^3}$. A magnetar's $10^{10}$ tesla field, by contrast, has an energy density of $4.0\times10^{25}\ \mathrm{J/m^3}$, with an $E/c^2$ mass density ${>}10^4$ times that of lead. The magnetic field of a magnetar would be lethal even at a distance of $1000\ \mathrm{km}$ due to the strong magnetic field distorting the electron clouds of the subject's constituent atoms, rendering the chemistry of life impossible....
At a more Earthly level: high magnetic fields of hundreds of millitesla (i.e. a few tenths of a tesla) can be lethal to people with certain kinds of prostheses. These days prostheses wherever possible are made of non ferromagnetic material but in the past there have been deaths of people e.g. imaged by NMR machines with early, ferromagnetic pacemakers, or with ferromagnetic clips in the brain to shore up vascular aneurysms there.
I'll answer more in a clinical perspective. I don't know about extreme situations when the spins of your organism's atoms are rearranged in a lethal way, but as far as MRI magnets go, the first concern when using MRI equipment is the possible induction of electric currents inside the human body.
These concerns are more prevalent for investigation MRI magnets, which are more powerful than the ones typically used at an hospital (they go up to 3 Tesla, while the ones used in academic research can go up to 9 Tesla. Record-breaking magnets reach 11 Tesla).
The human body starts to suffer from current induction when the exterior magnetic field reaches around 7 to 8 Tesla. Symptoms include increase in body temperature, diminishing brain functions and even hallucinations (never witnessed this one, but I've heard it's possible. Still, take this last symptom with a grain of salt). All this considering patients don't have metallic implants, obviously, or exposure can be lethal. The "Specific Absorption Ratio" is commonly used to try to measure these changes in the human body (expressed in Watt/Kilogram).
Conclusion: Before the human body dies of "iron loss", the effects of induction can be lethal for weaker magnetic fields. MF's around 8T can have a noticeable effect in the human body.
Sources: My knowledge comes from my work and my studies, and I can't reveal them here (I'm sorry). However I cited this article for one of my projects, and I believe it sums up my point nicely:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2705217/
• From a quick scan, I don't think that article says what you're saying. Current induction is only important for time varying magnetic fields; static fields cannot induce current. The paper seems to say that static magnetic fields do begin to affect biology in the tesla range, and speculates on the cause being slight changes to the electron orbitals (the comments about the lowered cat retina activity) but that generally, although the effect seems real (statistically significant), mechanisms for harm are poorly understood. Is that a reasonable summary? Jun 22 '16 at 0:59
• Yes, it is a good analysis, particularly at the "effect is statistically significant but mechanisms are poorly understood" level. A proof of that fact are the common discussions about changing MRI Safety guidelines related to the maximum SAR allowed for each part of the organism. However, current induction can be a problem because while the MF's are static, the patients tend to move inside the MRI scanner, particularly those with claustrophobia. This is mainly the cause for current induction. Jun 22 '16 at 9:55
• While most studies into this matter are mainly theoretical (I'll post some of sources here) it is general consensus that current induction due to rapid movements can be hazardous. But 3 Tesla MF's are generally not considered strong enough for this effect. A teacher of mine said they could be dangerous, but I never witnessed such symptoms in a clinical environment. Sources: aifmrm.files.wordpress.com/2013/03/crozier-2005.pdf elettra2000.it/pdf/reports/pubblicazioni2011/… Jun 22 '16 at 10:02
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# On the cauchy problem for the fractional Camassa-Holm equation
Duruk Mutlubaş, Nilay (2019) On the cauchy problem for the fractional Camassa-Holm equation. Monatshefte für Mathematik . ISSN 0026-9255 (Print) 1436-5081 (Online) Published Online First http://dx.doi.org/10.1007/s00605-019-01278-6
In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $H^{s}({\Bbb R})$, $s>{\frac{5}{2}}$.
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# What is a “variable”?
The idea of “variable” is widely used in maths, but in most maths education never really explained. And it’s quite tricky. If you want to explore it more, look at:
https://matheducators.stackexchange.com/questions/10673/what-is-a-variable
The way I’ve explained it to year 8 and 9 students is that a variable is what’s inside a container covering a number (or other mathematical entity) chosen from a particular set, but you may not know which one. For example, a real variable is a container with some real number or other in it. I have coloured paper beakers placed on a desk (upside down, so no-one can see what’s inside) to display this idea, with the convention that two beakers the same colour are the same container appearing twice in the mathematical statement.
Since we want to be able to do maths with pencil and paper, without colours, we use letters instead of coloured beakers: “x” means “the number [or whatever] under the cover marked x”.
Sometimes (as in the picture above) you have enough info to find out what is inside the containers. Sometimes (as e.g. if you had only the first row in the picture above) you have info about what is in the containers, but not enough to pin it down exactly.
What we call a “constant” in maths is often a variable, only one which has the same value (same number or thing inside) over a range of statements. We may even know what the value is, but just find it shorter to refer to it as “the number in the lime-green container” (it may be easier to write N, when we’re doing physical chemistry, for Avogadro’s number, rather than 6.02214076×1023).
In s = ½gt2, usually “g” is just short for “9.81 in SI units”. We know what is in the container. If we consider the same equation across a range of planets, then g is a “variable” variable. But for calculations on any one planet, it is a “constant” variable.
If we consider F = ma for a range of forces F but the same mass m, then m is a “constant” (though we may not know exactly what it is). If we consider the same equation for the same F but a range of masses m, then F is the “constant” and m and a are the variables (you might even say, the variable “variables”).
We also have dummy variables, i.e. variables which do all their varying backstage from the statement they appear in.
In $\sum_{r=1}^{10} r \;$ , “r” is a container for which we are told: put 1 in it, take a running total, then put 2 in it, recalculate the running total, put 3 in it, recalculate… put 10 in it, recalculate. Once you’ve finished the calculation, you’re done with using that “container”, and it didn’t matter whether it was yellow, turquoise, orange, whatever. The contents of the container vary all right, but only within the sum calculation.
In an equation like $\sum_1^n r = 55 \;$ , r has already done all its varying within the sum calculation, so it’s backstage in the context of the actual equation, which is an equation giving you info about what’s in the container called “n”.
Likewise the dummy variable x in $\int_1^{10} f(x) \; dx$.
Notice you can change the name of the dummy variable and it makes no difference to the sum or integral or whatever. If you add the ages of the students in a class, and do it by adding one after another in alphabetical order of first name, it makes no difference whether you call the counter “first name”, “given name”, “prénom”, “Vorname”, or whatever.
$\sum_{i=1}^{10} i \;$ , $\sum_{j=1}^{10} j \;$ , $\sum_{r=1}^{10} r \;$ , whatever, are all the same, all equal to 55. $\int_0^1 x \; dx \;$ , $\int_0^1 t \; dt \;$ , etc., are all the same, all equal to ½.
A random variable in statistics is really a function, not a variable in the sense above.
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If a compound has a number of individual dipoles, then. Question: Which Of The Following Substances Would You Expect To Have A Nonzero Dipole Moment? Lv 4. 1. Close. If the individual bond dipole moments cancel one another, there is no net dipole moment. Dipole moment PCl3 and PCl5 melting point difference Chemistry pp3 doubt How do I know which bonds are polar or not? How many of the following molecules possess dipole moments? Ask Ethan: If Mass Curves Spacetime, How Does It Un-Curve Again? Bond dipole moment. A) CH4 B) CCl4 C) CO2 D) SO3 E) none of these . Which of the following types of molecules always has a dipole moment? ... of these there is a linear structure and in the other there is a trigonal structure which both result in no permanent dipoles. It is an ionic solid compound exhibiting as a cation or anion in a solid-state. 1 Approved Answer . Whilst the dipole moment of pyrrole faces away from the nitrogen as a result of greater resonance effects. The polarity of a molecule is directly proportional to the difference in their electronegativity. It was first discovered by a famous English chemist Humphry Davy in the year 1808. Hence, the bonds formed in this molecule are Polar Bonds. Want to see the step-by-step answer? Hence, {eq}PCl_3 {/eq} has a net dipole moment and would therefore have larger dipole-dipole forces. Because due to the symmetrical structure, the polarity in such a molecule gets canceled by each other.eval(ez_write_tag([[250,250],'techiescientist_com-large-leaderboard-2','ezslot_7',107,'0','0'])); Phosphorus pentachloride is nonpolar in nature because of its geometrical structure. Click to Chat. PCl5 has zero dipole moment. In the case of Cl_2, the 2 atoms are identical, so no polarization of the bond is possible, and the dipole moment is zero. Which out of pcl5 and pcl3 has higher dipole moment and why 2 See answers Brainly User Brainly User As far as boiling point is concerned,PCl3 does have a lower boining point than PCl5 because of the greater polarity as PCl3 has a trigonal pyramidal structure with a net dipole moment while PCl5 is non polar. Nonpolar. Which of the following molecules has a dipole moment? It is one of the most important phosphorus chlorides, others being PCl3 and POCl3. both compounds are sp3d2 hybridised but in pcl3br2 the 2 br occupy the equitor × Enroll For Free Now & Improve Your Performance. PCl5 has five chlorine atoms bonded onto the central phosphorous atom. Since the molecule Phosphorus Pentachloride (PCl5) is symmetrically arranged around the central (P) atom, there is no charge distributed unevenly. The equation for dipole moment is … BH3, CH4, PCl5, H2O, HF, H2. It is prominent that all bond angles of 90 degrees and 120 degrees formed in trigonal bipyramidal geometry are not identical. Polar molecules must contain polar bonds due to a difference in electronegativity between the bonded atoms. 2. These atoms also have the same electronegativity value and therefore forms a zero dipole moment. Due to which the polarity of P-CL bonds gets canceled by each other. If you don't know how to do Lewis Structures...then it is a bit difficult to explain. 4 6 D P C l 2 F 3 , C H 2 C l 2 - The dipole moment of C H 2 C l 2 is μ = 1. Similarly, in the P-Cl bond, the Chlorine atom attracts the bonded electron nearer towards it. 1 year ago. Therefore, it is possible that a molecule can be nonpolar even if there are polar bonds present in it. As a result, the chlorine atom gains partial negative charge and the Phosphorus atom gets a partial positive charge. Trigonal pyramid molecules. 0 0. ran d. vor 1 Jahrzehnt. However, as $\ce{PCl3}$ has a dipole moment, it has dipole-dipole intermolecular attractions, implying that $\ce{PCl3}$ has a higher melting point than $\ce{PCl5}$, which only has London dispersion forces. Answer = PoCl3 ( PHOSPHORUS OXYCHLORIDE ) is Polar What is polar and non-polar? for example Cl is more electronegative than P, thus the bonds in PCl5 are polar, but the molecule overall is nonpolar due to the arrangement of Cl atoms in a trigonal bipyramidal structure where they share electrons equally. Examples of such molecules are CO2, O2, PCl5, etc. Lewis Structure of Phosphorous Pentachloride. Sketch given below; F 3 SSF has asymmetrical structure so there is net dipole moment in it making it polar. Thus that leaves us with HCCl3. you can check out the reason for the polarity of H2O. B. PCl5 has five chlorine atoms bonded onto the central phosphorous atom. We know that Cl is more electronegative than P, therefore the bonds formed in PCl5 are polar, but the molecule as overall is nonpolar. share | improve this answer | follow | answered Mar 21 '17 at 21:49. rashi rashi. It readily gets decomposed in water to form phosphoric acid. This problem has been solved! But, on the other hand, BrF5 does have a dipole moment due to the asymmetric structure as shown earlier in … The net dipole is the measurable, which is called the dipole moment. The dipole moment of a molecule is therefore the vector sum of the dipole moments of the individual bonds in the molecule. Dipole moment is a vector quantity. If a compound has a number of individual dipoles, then. Chemistry. Chemical Bonding and Molecular Structure. Additionally,the 2P-Cl bonds perpendicular to plane opposite each other causing nullification of the dipoles. One electron is shared by each chlorine atom. It is polar overall. This is because of the symmetrical geometrical structure as in the case of PCl5. Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes! The electron pull balances out electronegativity, in both the vertical and horizontal axis of the Phosphorous Pentachloride molecule. Want to see this answer and more? Part I. BH3, CH4, PCl5, H2O, HF, H2. Phosphorus trichloride is a chemical compound of phosphorus and chlorine, having the chemical formula PCl 3.It is a toxic and volatile liquid which reacts violently with water to release HCl gas. PCl5 has zero dipole moment. Lattice energy is the largest factor in determining the stability of an ionic solid. But the dipole moment of NH 3 is 1.46D which is higher than dipole moment of NF 3 which is 0.24D. Lv 4. 2 0. imai. So guys if you have any questions regarding the polarity of PCl5, you can ask them in the comment section. This cancellation is responsible for nonpolar nature as there is no dipole moment and lone pair of electrons. CS2; SeS2; CCl2F2; PCl5; ClNO; check_circle Expert Answer. See the answer. D) SO3. The energy of these 3d orbitals is very close to the energy of 3s and 3p orbitals. Source(s): https://shorte.im/a76m7. How many of the following molecules possess dipole moments? Such is the case for CO 2, a linear molecule (part (a) in Figure 2.2.8). What is the dipole moment of Phosphorus Pentachloride (PCl5)? Why? It is polar overall. 2 Answers. [1] SF6 [2] PCl5 [3] BF3 [4] SF4 [5] CCl4? Thus, PCl5 becomes unstable due to weak axial bonds. Many students may have a query about whether PCl5 is polar or not. 1 Approved Answer . Required fields are marked *. 1. Which of the following types of molecules always has a dipole moment? Hence if can be broken down into a vertical and horizontal component. It is a crystalline solid compound with an irritating odor and is highly toxic when a hydrolysis reaction takes place. What are the points to determine the polarity of pcl5? As a result, the net dipole moment of PCl5 comes out to be zero. Because PCl5 has a trigonal bipyramidal structure, there is no net dipole moment since the molecule is symmetric around the central phosphorus atom. Question = Is PCl5 ( PHOSPHORUS PENTACHLORIDE ) polar or nonpolar ? SF4: Has four S-F bonds and since F is the most electronegative and S is moderately electro negative.S-F bonds will be polar. (27) Which of the following has a dipole moment? See Answer. Pcl3 Dipole Moment. molecular-structure electronegativity. Polar "In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. Which of the following molecules has a dipole moment? Lv 7. As far as boiling point is concerned, PCl3 does have a lower boining point than PCl5 because of the greater polarity as PCl3 has a trigonal pyramidal structure with a net dipole moment while PCl5 … PCl5 has symmetric charge distribution of Chlorine atoms around the central atom Phosphorous. Phosphorus pentachloride is nonpolar in nature as it is symmetrical in shape. These p orbitals are singularly occupied and together they form all the 5 P–Cl sigma bonds in PCl5. So we have 7+7+7+1 = 22 valence electrons. The molecular geometry of Phosphorous Pentachloride (PCl5 ) is trigonal bipyramidal. And the remaining two P-Cl bonds are at two ends of an axis passing through the plane of three P-Cl bonds. How would I go about finding which one has a higher dipole moment? This heat can be sufficient enough to ignite any combustible material such as plastic. B) 2. Therefore, it is very important to understand that although PCl5 has polar bonds within it, it is a nonpolar in nature due to its symmetrical shape. CCl4- 0D) (Original post by Nator) I see your point, is there any way you know if the dipole moment is equal to another of opposite direction? The five orbitals i.e. The lone pair in PCl3 is the main cause of this. The molecule of PCl5 has chlorine and phosphorus atoms having an electronegativity difference of 0.97D that determines the polarity in the P-Cl bond. BH3, CH4, PCl5, H2O, HF, H2 A) 1 B) 2 C) 3 D) 4 E) 5. Relevance. Bond Parameters. It also acts as a catalyst for cyclization and condensation reactions and hence, is very useful. At higher concentrations, the second equilibrium becomes more prevalent: Phosphorus Pentachloride is used as a chlorinating agent. It has its great use as an intermediate compound in the manufacturing of pesticides and water treatment. As a result, the hybridization including 3s, 3p, and 3d or 3d, 4s, and 4p are all feasible. A) CO2 B) SeO3 C) XeF4 D) SF4 E) BeCl2. Question: Which Of These Molecules Have A Dipole Moment? Hence, the PCl5 molecule has a sp3d hybridization and a trigonal bipyramidal geometry shape. Whenever a molecule is not "symmetrical" it has a dipole moment. Check out a sample Q&A here. Nonpolar Molecules: These are the molecules that have zero dipole moment. 1. Therefore due to its asymmetrical structure, the dipole moment of one direction is not canceled by the dipole moment of other directions. and BF3 is trigonal planar. Favourite answer. The chemical compound PCl5 is also known as Phosphorus Pentachloride. Expert Answer . Choose the statement that best describes the PbCl4 molecule in the gas phase. This results in bond angles closer to 90°. The molecule PCl5 is observed not to have a dipole moment. 1 year ago. It exhibits a trigonal bipyramidal geometrical shape. The dipole moment of a polar molecule is always equaled to non zero and nonpolar molecules always have zero dipole moment. In molecules containing more than one polar bond, the molecular dipole moment is just the vector combination of what can be regarded as individual "bond dipole moments".Mathematically, dipole moments are vectors; they possess both a magnitude and a direction.The dipole moment of a molecule is therefore the vector sum of the dipole moments of the … Its commercial samples can sometimes appear to be greenish-yellow in color when contaminated with hydrogen chloride. The term dipole moment can be defined either with respect to chemical bonds or with respect to molecules. From just the IB chemistry bonding chapter information, you would normally deduce that $\ce{PCl3}$ and $\ce{PCl5}$ are both covalent molecules. The more acute bonds would then result in a higher dipole moment. In a trig bypyrid, there is a trigonal pyramid on the x-axis and a linear model on the y-axis (google may have a good picture of it). Which of the following substances would you expect to have a nonzero dipole moment? The electronegativity of Phosphorus is 2.19 and that of Chlorine is 3.16. It is a completely nonpolar compound forming polar ionic bonds. So technically PCl3 should have higher melting point cause of permanent dipole but in reality PCl3 is a colorless liquid at r.t while PCl5 is a white solid. The chemical compound PCl5 is pungent in smell. Das Dipolmoment. It is one of the common chlorinating reagents. Select the molecule among the following that has a dipole moment. How many of the molecules have no dipole moment? COMMENTARY: Why men are willing to donate sperm — and are using Facebook to make the arrangements. Polar "In chemistry, polarity is a separation of electric charge leading to a molecule or its chemical groups having an electric dipole or multipole moment. [1] SF6 [2] PCl5 [3] BF3 [4] SF4 [5] CCl4? Polar Molecules: These are the molecules that have a net dipole moment equals to non zero. The molecular structure of PCl5 doesn’t carry any unbalanced localized charge or rather any dipoles in this chemical compound. The molecule PCl5 is observed not to have a dipole moment. C is trigonal planar i believe, without drawing the lewis structure. 0rohfxodu 0rgholqj 0lqqhdsrolv &rppxqlw\\ dqg 7hfkqlfdo &roohjh y 2emhfwlyhv 7r frqvwuxfw sk\\vlfdo dqg frpsxwhu dlghg prghov ri prohfxohv dqg phdvxuh prohfxodu sdudphwhuv lqfoxglqj erqg ohqjwk erqg dqjoh dqg glsroh prphqw Best Answer 100% (1 rating) Previous question Next question Get more … If the individual bond dipole moments cancel one another, there is … The electrons in a covalent bond connecting two different atoms are not equally shared by the atoms due to the electronegativity difference between the two elements. zero) dipole moment. C. The polarity of the P-Cl bonds cancel out due to the geometry of the molecule. If we see the lewis structure of this molecule! The 3P-Cl bonds of bond angle 120 fgrom each other cancels each other dipole moment. The characteristics that determine whether or not a molecule is dipolar are: -the electron distribution around the central atom is not symmetrical and/or -atoms coming off the central atom are not … SF4: Has four S-F bonds and since F is the most electronegative and S is moderately electro negative.S-F bonds will be polar. Phosphorus pentachloride is the chemical compound with the formula PCl5. Its dipoles all cancel. IUPAC name of PCl5 is phosphorus pentachloride. Join Now. As charge distribution is equal and there is no net dipole moment therefore, PCl5 molecule is nonpolar. A) CO2 B) SeO3 C) XeF4 D) SF4 E) BeCl2. This is because the polarity of such bonds gets canceled with each other due to the symmetrical geometrical structure of the molecule.eval(ez_write_tag([[468,60],'techiescientist_com-medrectangle-4','ezslot_3',104,'0','0'])); . At a temperature of 80 °C, the vapor pressure of PCl5 is 1.11 kPa. Whilst the dipole moment of pyrrole faces away from the nitrogen as a result of greater resonance effects. Plus, Cl is more electronegative than P in PCl5. Therefore, the polarity of a molecule is also directly proportional to its dipole moment. Bond Dipole Moment. In every other case except H_2S, the polarization of charge associated with each bond is exactly cancelled by the other bonds, resulting in no net dipole moment. PCl5 is a colorless crystal in appearance at room temperature. × Thank you for registering. So the dipole moment of furan faces towards the oxygen, a result of greater inductive effects than resonance effects. This extraction procedure can be accomplished using the garage chemist’s procedure only. Question = Is PoCl3 polar or nonpolar ? Exposure to a high concentration of PCl5 can lead to serious health problems. Each C–O bond in CO 2 is polar, yet experiments show that the CO 2 molecule has no dipole moment. PCl5 finds use as a chlorinating reagent. Whenever a molecule is not "symmetrical" it has a dipole moment. Bobby. Which of the following molecules have net molecular dipole moments? It is unstable due to weak axial bonds. These two atoms have unequal electronegativity and therefore have a non zero dipole moment. 5 9 D and 1,4-dichloro-2,3-dimethyl benzene (C 8 H 8 C l 2 ) Answered By . Why PCl5 is nonpolar even with an odd number of vectors around the central atom? However, I don't know how to look at these three factors collectively to determine the net dipole moment. D) SF4. C is trigonal planar i believe, without drawing the lewis structure. 5 points for each argument. BrF5, on the other hand, does have a dipole moment due to the asymmetric structure. This occurs due to an atoms' electronegativity - where one atom has the ability to attract electrons towards it (In other words, electrons wants to spend other time around it) giving it a negative charge and the other a positive charge. The valence electrons of Phosphorus are 5 electrons and that of Chlorine is 7. The covalent bonds can be the polar or nonpolar that depends on different factors like electronegativity, geometrical shape, and dipole moment. There are five chlorine atoms attached to the phosphorus and all bonds are covalent in nature. Dipole moment: It is the measure of the polarity of a molecule. Hemanth A answered on January 22, 2016. These molecules are overall a nonpolar in nature even if it contains a polar bond within it. A CO 2 molecule contains two polar bonds but the net dipole moment is zero. Bonnie. 3) Assign the point group of the molecules below. Thus, PCl5 exists as an ionic solid. This PCl5 compound can be easily obtained by chlorinating Phosphorus Trichloride with an elemental chlorine. P-CL bond ensures some nonzero dipole moment but due to symmetrical geometrical structure, the polarity of the P-Cl bond gets canceled by other P-Cl bonds. It is one of the common chlorinating reagents. As, both the N-H bond are in same direction it adds to the bond moment of the lone pair, while N-F bond are in opposite direction so they partly cuts the bond moment of lone pair. Electronegativity of an atom is its strength to attract the bonded electron pairs towards it. A. PCl_5 And SiCl_4 B. POCl_3, SO_2Cl_2, And SO_3 C. POCl_3 And SO_2Cl_2 D. PCl_5, POCl_3, SOCl_2, SO_3, And SiCl_4 E. PCl_5, SF_6, SO_3, And SiCl_4. The three bonds lie in a single plane in such a way that three P-Cl bonds make 120 degrees angle with each other and lie at corners of an equilateral triangle. So, Is PCl5 Polar or Nonpolar? [A] 4 [B] 2 [C] 3 [D] 1 [E] They are all polar . In this article, I will answer this question and will cover the surrounding topics too. since the electronegativity of chlorine is higher than hydrogen, hence the dipole moment will be higher in the case of PCl3 than PH3. The polarity of the P-Cl polar bonds cancel out each other due to the trigonal bipyramidal geometry of the molecule. Answer Save. Why does PCl5 exist as a cation and anion in the solid-state ( i.e. Lv 4. Increased lattice energy and better packing efficiency! 5 years ago. If the dipole moments were equal in all directions then the molecule wouldn't be polar (e.g. Even though the molecule carries no dipole charge, a large amount of electrons still leads the molecule to form strong temporary forces allowing it, a melting point of 161˚C as an outcome. Such is the case for CO 2, a linear molecule (part (a) in Figure 2.2.8). (a) 18. The elements present in the third period comprise of d orbitals along with s and p orbitals in the periodic table. Also determine the polarity and whether or not it has a dipole moment. There are no lone pairs of electrons on the central atom. In the solid PCl5 is ionic PCl4+ PCl6- In the gas and liquid phases molecular PCl5 is present which does not have a permanent dipole moment. How many of the following molecules possess dipole moments? Here, in PCl5 the bonds are symmetric but due to the electronegativity of chlorine being higher, the molecule has polar bonds canceling their vector sum. The term dipole moment can be defined either with respect to chemical bonds or with respect to molecules. Select the molecule among the following that has a dipole moment. It is because _____ (a) the molecule has symmetrical linear geometry (b) the molecule is non-linear (c) the electronegativity difference between the two atoms is too large (d) the electronegativity difference between the two atoms is too small. Polar if these two atoms covalently bonded forms a zero dipole moment and PCl3 have dipole! Structures... then it is a common chlorinating agent by a famous English chemist Humphry Davy in manufacturing! The largest factor in determining the stability of an atom is its.... C ) XeF4 D ) SF4 E ) BeCl2 in color when contaminated with hydrogen chloride PCl5... Of bond angle 120 fgrom each other causing nullification of the 3-dimensional arrangement (.... Out to you as soon as possible each C–O bond in CO 2 molecule has a net dipole?... Are 5 electrons and that of chlorine atoms bonded onto the central atom phosphorous geometries included this... Pairs on the phosphorus that all bond angles of 90 degrees and 120 degrees in! Phosphorus OXYCHLORIDE ) is polar and non-polar be the polar or not D ] 1 [ E ] are. Found to be polar because of the following molecules possess dipole moments and are Facebook... To another in PCl5 atoms is said to be greenish-yellow in color when contaminated with hydrogen chloride compound has dipole... And the remaining two P-Cl bonds cancel out each other causing nullification of molecules! Including 3s, 3p, and dipole pcl5 dipole moment about the chemical compound due a... Between different atoms will be polar this situation is also a result of greater resonance effects °F... An atom is its polarity in all directions then the molecule has no dipole moment equals zero. Greater the dipole moment of NH 3 is μ = 1 atoms are arranged that... { \rm XeF_2 } { \rm PCl_5 } this problem has been solved know how to do Structures. 3P orbitals nonpolar substance with polar bonds cancel out their charges, the 2P-Cl bonds perpendicular the! Counsellors will contact you within 1 working day are perpendicular to the lone pairs on the atom! °C or 332.2 °F and p orbitals in the P-Cl bond, the chlorine atom gains partial charge! The charge and the remaining two P-Cl bonds of an axis passing through plane... Of pyrrole faces away from the directions of dipole moments increase with increasing dipole moment of polar. Cl atoms in PCl5 covalent bonds can be broken down into a vertical horizontal. Character and decrease with covalent bond character } has a number of.. Always has a dipole moment is equal and there is a trigonal bipyramidal geometry are not identical [ ]. Of our academic counsellors will contact you within 1 working day moment therefore, PCl5 you. Hf, H2 bonded forms a nonpolar bond if the atoms ' electronegativities. D and 1,4-dichloro-2,3-dimethyl benzene ( C 8 H 8 C l 2 ) by. According to VSEPR theory ) l 2 ) answered by other directions 2 is polar non-polar! Or not it has a dipole moment is equal and there is net dipole equals. 5Sp3D hybrid orbitals ( according to VSEPR theory ) moments of C C l )... The term dipole moment geometry of phosphorous Pentachloride molecule partial negative charge and the distance how... The lewis structure down into a vertical and horizontal axis of the molecule are polar or non-polar localized or... Situation is also a result of greater inductive effects than resonance effects to have dipole! Has pcl5 dipole moment solved pcl3br2 the 2 Br 3 dipole moment is - H! Concentration of PCl5 is observed not to have a Nonzero dipole moment POCl3 and PCl3 along with and... Net molecular dipole moments were equal in all directions then the molecule would be! ( C 8 H 8 C l 3 and NH 3, without drawing the structure... C C l 2 ) answered by an equatorial and axial bond first by. Due to its Cl atoms in PCl5 electric dipole moment then it measured! For dipole moment C ) CO2 B ) SeO3 C ) CO2 B ) C. Ionic solid case of PCl5 does not PCl5 can lead to serious health problems yellowish and contaminated hydrogen... As the product of the other there is a net dipole moment in it plane opposite each other when with. In contact with water to provide step-by-step solutions in as fast as 30 minutes are! One has a number of vectors around the central atom so 3 ; so 3 ; PCl 4 ; 3-Part! Is directly proportional to the lone pairs of electrons molecules has a number of vectors around central... The centers of positive and negative charge solutions in as fast as 30!... Bond, the hybridization including 3s, 3p, and dipole moment, colorless, and orbitals. Bond in NF 3 which is asymmetric in nature tend to be.... Xef_2 } { \rm XeF_2 } { \rm CH_3NH_2 } { \rm CH_3NH_2 } { \rm CH_3NH_2 {! Localized charge or rather any dipoles in this molecule are polar or nonpolar that depends on different like. The trigonal bipyramidal geometry shape axial bonds SF4 E ) BeCl2 consists 5... Degrees and 120 degrees formed in this molecule are polar bonds partial positive charge according to theory., yet experiments show that the CO 2 molecule has a dipole moment lone... Behind this determination 3-Part II this browser for the polarity of the other there is no net dipole moment furan... Becomes more prevalent: phosphorus Pentachloride ) is trigonal bipyramidal, in the P-Cl bonds gets by! Nature ie ; trigonal bipyramidal geometry shape chlorine is 7 the phosphorous Pentachloride molecule 80 °C the... To measure a chemical bond ’ s procedure only central atoms ) CH4 B ) C! Fast as 30 minutes molecules has a dipole moment bonds perpendicular to plane opposite each other dipole of! By chlorinating phosphorus Trichloride with an elemental chlorine equal to the asymmetric structure the formula.. Check out the reason for the next time I comment you as soon as possible square structure. Greenish-Yellow in color when contaminated with hydrogen chloride linear, trigonal planar,,! 3P-Cl bonds of bond angle 120 fgrom each other causing nullification of the following molecules possess dipole moments of bond! Because the shape and determine whether They are polar or nonpolar that on! 3 which is asymmetric in nature answer = POCl3 ( phosphorus OXYCHLORIDE ) is polar and non-polar in. Pcl_5 } this problem has been solved helps in the pharmaceutical industry, it a! Contains two polar bonds but the net dipole moment due larger atomic radii of other... Its strength to attract the bonded electron pair slightly more towards it PbCl4 molecule in the bond... Greenish-Yellow in color when contaminated with hydrogen chloride, whereas the trigonal bipyramidal molecular geometry of the share! So the dipole moment of other directions experts are waiting 24/7 to provide step-by-step solutions in as as. At two ends of an equatorial and axial bond of metals are perpendicular to plane opposite other. With this electronic configuration polar What is polar and non-polar as there no... The two bonds axial to the difference between the centers of positive negative. Bonds will be polar because of the most electronegative and s is moderately electro negative.S-F bonds will be because. A dipole moment can be nonpolar even with an odd number of vectors chloride ion one! Go about finding which one has a dipole moment PCl 5 has trigonal bipyramidal geometry of... Charges, the PCl5 molecule is the case for CO 2 molecule contains polar! Repulsion and hence, the whole molecule has a dipole moment in it of charge a Study.com member to this. Also responsible for its boiling point which is asymmetric in nature even if are! Atom contains a total of 5 chlorine atoms bonded onto the central atom is higher than dipole moment be! Penicillin and cephalosporin heat can be nonpolar even if it contains a total of 5 chlorine atoms bonded onto central. Is 0.24D PCl5, you can check out the reason for the polarity of the molecule with... Contains a polar bond within it are overall a nonpolar molecule, yet experiments show that the 2.
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## Conditional Equational Specifications of Data Types with Partial Operations for Inductive Theorem Proving
• We propose a specification language for the formalization of data types with par-tial or non-terminating operations as part of a rewrite-based logical frameworkfor inductive theorem proving. The language requires constructors for designat-ing data items and admits positive/negative conditional equations as axioms inspecifications. The (total algebra) semantics for such specifications is based onso-called data models. We present admissibility conditions that guarantee theunique existence of a distinguished data model with properties similar to thoseof the initial model of a usual equational specification. Since admissibility of aspecification requires confluence of the induced rewrite relation, we provide aneffectively testable confluence criterion which does not presuppose termination.
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Author: Ulrich Kühler, Claus-Peter Wirth urn:nbn:de:hbz:386-kluedo-3672 SEKI Report (96,11) Preprint English 1999 1999 Technische Universität Kaiserslautern 2000/04/03 Fachbereich Informatik 0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011
$Rev: 13581$
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Session 19 - Solar & Planetary Systems.
Display session, Monday, January 13
Metropolitan Ballroom,
## [19.05] Quasi-stationary States of Dust Flows Under Pointing-Robertson Drag: New Analytical Solutions
N. Gor'kavyi (Crimean Obs.), L. Ozernoy (CSI/GMU and GSFC/NASA), J. Mather (GSFC/NASA), T. Taidakova (Crimean Obs.)
The effect of solar/stellar radiation on dust particles trajectories (the P-R drag) has been studied by a number of authors and applied to interplanetary dust dynamics in numerical computations. Meanwhile some important features of dust flows can be studied analytically by implementing the continuity equation written in particle's orbital elements as coordinates (Gor'kavyi, Ozernoy, amp; Mather 1997). By employing this approach and integrating the continuity equation, we are able to find two integrals of motion when the P-R drag dominates the dissipative forces in the dust flow. In this case, the integrals of motion are C_1=ae^-4/5(1-e^2) and C_2=ne^1/5\sqrt1-e^2. The integral C_1 that describes the trajectory of the dust flow in the space of particle's orbital elements, coincides with that for the motion of individual particles (Wyatt amp; Whipple 1950), and the integral C_2 (which is a new result) allows to determine the density of the flow along its trajectory. Taken together, C_1 and C_2 imply conservation of the particle's flux in the flow under the P-R drag. These integrals of motion enable us to explore basic characteristics of dust flows from any sources in the Solar system (such as asteroids, comets, Kuiper belt, etc.) or in another planetary system. In particular, we have reproduced the clasical solution n\propto 1/r that represents approximately the overall distribution of dust in the Solar system. We have also investigated the factors that could be responsible for deviations of the power law index in n\propto r^-\alpha from \alpha=1: non-uniform distribution of dust sources around the observer, eccentricity of particle orbits, and the change of particle's sizes due to evaporation. Comparison with the measured dust distribution in the Solar system is done.
References:
Gor'kavyi, N., Ozernoy, L. amp; Mather, J. 1997, ApJ 474 No.1 (in press)
Wyatt, S.P. amp; Whipple, F.L. 1950, ApJ 111, 134
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2005AJ....129...86H - Astron. J., 129, 86-103 (2005/January-0)
Chandra-SDSS normal and star-forming galaxies. I. X-ray source properties of galaxies detected by the Chandra X-ray observatory in SDSS DR2.
HORNSCHEMEIER A.E., HECKMAN T.M., PTAK A.F., TREMONTI C.A. and COLBERT E.J.M.
Abstract (from CDS):
We have cross-correlated X-ray catalogs derived from archival Chandra X-Ray Observatory ACIS observations with a Sloan Digital Sky Survey Data Release 2 (DR2) galaxy catalog to form a sample of 42 serendipitously X-ray-detected galaxies over the redshift interval 0.03<z<0.25. This pilot study will help fill in the redshift gap'' between local X-ray-studied samples of normal galaxies and those in the deepest X-ray surveys. Our chief purpose is to compare optical spectroscopic diagnostics of activity (both star formation and accretion) with X-ray properties of galaxies. Our work supports a normalization value of the X-ray-star formation rate correlation consistent with the lower values published in the literature. The difference is in the allocation of X-ray emission to high-mass X-ray binaries relative to other components, such as hot gas, low-mass X-ray binaries, and/or active galactic nuclei (AGNs). We are able to quantify a few pitfalls in the use of lower resolution, lower signal-to-noise ratio optical spectroscopy to identify X-ray sources (as has necessarily been employed for many X-ray surveys). Notably, we find a few AGNs that likely would have been misidentified as non-AGN sources in higher redshift studies. However, we do not find any X-ray-hard, highly X-ray-luminous galaxies lacking optical spectroscopic diagnostics of AGN activity. Such sources are members of the X-ray-bright, optically normal galaxy'' (XBONG) class of AGNs.
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# Subsequences and Accumulation Values
A sequence that does not converge may still have converging subsequences.
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## Dominos
• 105 Likes
• World of Warcraft
• Supports: 6.2.0
• Updated 07/25/2015
• Created 06/15/2008
• 6,435 Favorites
• Project Site
• Release Type: Release
Support development! **
#### Dominos is an action bar addon intended to do the following:
• Reuse as much standard blizzard action button code as possible.
• Contain a relatively minimal feature set
• Be easy to use
• Be stable
#### It includes the following features:
• 10 action bars, pet, class, menu, and bag bars. Each one has customizable settings for padding, spacing, columns, scale, and opacity.
• Customizable paging. You can switch pages on: action bar pages, modifier keys, forms, and targeting.
• Customizable show states. You can tell your bar under which macro options to show.
• Fading bars. You can set your bars to fade out to a certain opacity when not moused over.
• The ability to customize showing empty buttons or not
• Keybound support
• Sticky frames
• The ability to move buttons in combat
• A movable casting bar (optional)
• A movable xp/reputation bar (optional)
• Configurable right click targeting
• Configurable self cast key settings
#### Here's how you use it:
• To see the current list of slash commands, type /dom ? or /dominos ?
• To open up the options menu, either go into interface options, or type /dom
• To move bars around, either go into the options menu and press the "Enter Config Mode" button, or type /dom lock
• To bind keys, enter binding mode via /kb or /keybound, or press the "Enter Binding Mode" button in the options menu.
#### You can use the following to add extra functionality:
• tullaRange - Colors action buttons red when out of range
• OmniCC - Adds cooldown count text
• Masque - Allows you to change the look of the action, pet, and class buttons
6.2.7b
* Fixed a typo I somehow missed.
6.2.7
Workaround to try to ensure the encounter bar works in combat.
Fix out of date roll and xp bars.
6.2.6
* Added a fix for cooldown pulses showing up on bars that are completely transparent.
6.2.5
* Add flyout support to Dominos_ActionSets
6.2.4
* Yet another encounter bar bugfix (stupid typos)
6.2.3
• Encounter bar bugfix
• Fix missing print version command
6.2.2
• Encounter bar workaround.
6.2.1
• Added Dominos_ActionSets. This is a module attaches action button placements to your Dominos profile. So when you switch profiles, your action buttons can also switch. Dominos_ActionSets is disabled by default, so you need to enable from the addons menu.
6.2.0
• Updated for WoW 6.2.0
• Fixed an issue where Dominos would apply a default skin to buttons regardless of if Masque was running
• Refactored the layout code for menu buttons.
6.1.9
• Fix padding issues with frame layouts
6.1.8
• Moved Masque support into its own module.
• Fixed a bug causing flyout buttons to not properly update direction
6.1.7
6.1.6
• Resolved errors when opening the menus of the encounter and roll bars.
6.1.5
• The main bag button should be positioned properly again
• Resolved an issue causing the vehicle exit button to not work after being used once
• And yes, more bugfixes
6.1.4
• Resolved vehicle button display issues (hopefully!)
6.1.3
• Added ability to cycle through overlapped in config mode frames via modifier key + mousewheel (thanks Gorwanaws)
• Bugfixes
6.1.2
• Added support for cancelling flights.
• Lots of code reorganization (make sure you delete everything and restart WoW after installing)
6.1.1
• Added missing dungeon finder button.
6.1.0
• Updated for WoW 6.1
• Dominos will now properly skin buttons when Masque is disabled for a particular group (on /reloadui)
6.0.13
• Bugfix for an error that would occur when an actionbar had less buttons than normal.
6.0.12
• Updated Italian localization (thanks to Kuzunkhaa)
6.0.11
• Added Italian localization (thanks to Kuzunkhaa)
6.0.10
• Reorganized folder layout. Make sure that you delete all Dominos folders before installing the addon, and restart the game.
• Adjusted the configuration mode look and feel.
• Added the ability to nudge bars: In configuration mode, hover over a bars and press an arrow key
6.0.9
• Updated Shadow Dance paging state
• Fixed UNKNOWN_STATE display for some stance bars.
6.0.8
• Added new option, /dom configstatus : Shows if Dominos_Config is enabled or not.
6.0.7
• Added support for Stance of the Spirited Crane and Gladiator Stance
6.0.6b
• Implemented a workaround to hide cooldown spirals on transparent action bars.
6.0.6
• Implemented a fix to try and hide cooldown spirals on action bars that have zero opacity.
6.0.5
• Removed Berserker stance settings, since it no longer exists.
• Reimplemented defaults for stance paging for Warriors
6.0.4
• Updated Ace libraries.
6.0.3
• Resolved an issue causing the XP bar to not automatically hide when in pet battles/vehicles
6.0.2
• Fixed issue causing the Dominos_Config not to load when only enabled on some characters.
6.0.1
6.0.0 (Warlords of Draenor)
• Updated the TOC for 6.0.0
• Fixed some issues causing the menu bar to not show up properly during pet battles/vehicles.
5.4.12
• Fixed an issue when tracking a reputation of a friendly faction, and already at max level (hopefully)
5.4.11
• Resolved an issue causing the Dominos override/vehicle bar to not display any icons for certain encounters (ex, Noodle Time, Naz'Jar Battlemaiden). Thanks to mlangen & Wetxius for insight and testing.
5.4.10
• Paging bugfix
5.4.9
• Added a fix for the faction reputation error.
• Setting opacity with the mousewheel now properly uses a bar's base opacity (thanks ckaotik).
• Implemented hopeful fixes to resolve override bar issues.
5.4.8
• Fixed combat lockdown errors.
5.4.7
• Reworked code used to hide Blizzard frames and reuse the MultiActionBarButtons, hopefully finally preventing the addon blocked errors.
5.4.6
• Fixed a regression that was causing the help button to not display again.
5.4.5
• "Resolved" addon blocked error in a way that hopefully allows you to switch talents
5.4.4
• Resolved taint errors related to the Blizzard store button.
5.4.3
• Added back the help button
5.4.2
• Fixed a regression in keybinding code
• Fixed slider display issues
5.4.1
• Added a new option to disable Blizzard artwork on the extra action button (available from its right click menu).
• Fixed issues preventing the options menu from displaying properly/
• Fixed issues with friendship reputation display
5.4.0
• Updated for WoW 5.4.0
• Adjusted override bar controller to use the new [shapeshift] macro conditional. This should hopefully resolve issues with fights that use the temp shapeshift bar.
• Adjusted reputation bar to handle friendship reputation (thanks to b-morgan)
• Code cleanup
5.3.1
• Simplified surrogate binding code and limited it to only custom buttons (ex, DominosActionButtonXX)
• Lowered surrogate button binding priority.
5.3.0
• Updated the TOC for 5.3.0. You can still use Dominos 5.3.0 with WoW 5.2, but you'll need to check the Enable Out of Date AddOns checkbox in the addons menu on the character select screen.
• The Encounter Bar (PlayerPowerBarAlt) should now show up when using the Blizzard Override/Vehicle UI.
• The class/stance bar has been rewritten. It now reuses the standard stance buttons and bindings and should no longer produce an error if you happen to level up and gain a new stance/form while in combat.
• The extra bar has been rewritten to now display and use the standard bindings for the bar.
• Adjusted hide blizzard code to hopefully reduce tainting issues.
• Implemented full "Cast on Key Down" support.
5.2.0:
• TOC bump for WoW 5.2.0
• Added Dominos_Encounter: A new module for moving around the PlayerPowerBarAlt that shows up in some encounters. Thanks to Goranaws for the original version.
• Adjusted layout code to hopefully prevent issues when changing screen resolution (ported from LibWindow)
5.1.1
• Rewrote override page calculations to take advantage of some new functions available to me in the restricted environment
5.1.0
• TOC bump for WoW 5.1.0
5.0.29
• Fixed flyout button positions (thanks StephenClouse)
• Added a hack to fix errors when hiding the achievement micro button
• Simplified menu bar positioning code when the override ui
5.0.28
• Fixed an error when attempting to disable a menu button that no longer exists
5.0.27
• Fixed an issue that would cause the menu bar to be hidden when the world map was shown.
5.0.26
5.0.25
• Replaced custom minimap button code with LibDBIcon-1.0
5.0.24
• Resolved issues with empty action buttons sometimes being shown.
5.0.23
• Fixed some tainting issues
• Fixed an issue causing Dominos to think that the vehicle bar was shown upon loading
• Added a bit more debug information to the /dom statedump command.
5.0.22
• Fixed some typos
• Added new slash command /dom statedump: Please tell me what you get from this when reporting vehicle state bugs
5.0.21
• Fixed an error that was caused by attempting to read information about what the main action bar's page in combat.
5.0.20
• Fixed an error when switching profiles
• Added a new hack: If you enter a vehicle or something where you should get a new action bar, but are not, tap any modifier key. Your bar should hopefully switch :)
5.0.19
• Fixed a bug where your bar would get stuck on vehicle/override actions after exiting a vehicle
5.0.18
• Fixed show states being broken.
5.0.17
• Reworked how the override bar works a bit again. Should hopefully handle the state of a vehicle without a vehicle ui.
5.0.16
• Resolved issues with the possess bar not showing actions
5.0.15
• Resolved issue with buttons not working after the override ui is hidden.
5.0.14
• Added new global option: Use Blizzard Override UI. When enabled, shows the Blizzard Override UI interface.
• Added new advanced bar options: Show with Override UI, Show with Pet Battle UI. These control what bars will show up when the override ui/pet battle ui are shown.
5.0.13
• Fixed a bug that was causing the pet action bar background to show up
• Fixed a bug that caused errors when switching profiles
5.0.12
• Fixed missing pet bar issue
• Simplified layout code for menu bar.
5.0.11
• Reworked override ui code to better handle cases where action bars should change, but the UI does not.
• Renamed Possess Bar option to Override Bar.
5.0.10
• Reworked the roll bar.
5.0.9b
• Accidentally reverted the paging conditional for Shadow Dance; this has now been resolved.
5.0.9
• Fixed up Tree of Life bar switching.
• Added the roll bar back (I have not tested it, though)
• Renamed the Shadow Dance slider to Shadow Dance/Vanish to reflect that it controls paging for both abilities.
• Minor UI cleanup.
5.0.8
• Added a dedicated bar for the vehicle exit button.
• Fixed a bug causing click through settings to not work on the menu bar
5.0.7
• So it turns out that Show Lua Errors is disabled by default on the beta :P
• Entering/exiting a vehicle should now work properly in combat
• Shadow dance does exists, its vanish that I can't detect separately anymore :P
• Warlock metamorphosis should work properly again.
5.0.6
• Fixed an error when entering a vehicle
5.0.5
• Adjusted states for warrior stances so that they work again
5.0.4
• Fixed a bug that would occur when switching profiles
5.0.3
• Re-enabled the advanced layout options for the menu bar (including disabling buttons)
• Adjusted vehicle bar button placements based upon vehicle bar size
5.0.2
• Made Dominos hide itself when the pet battle interface is shown.
• Made Dominos hide itself when the vehicle interface is shown. Dropped the Dominos vehicle bar. NEEDS TESTING WITH A VARIETY OF VEHICLES.
• Updated the possess bar to use the new [possessbar] macro conditional.
• Updated the extra action bar to use the new [extractionbar] macro conditional.
5.0.1
• Fixed divide by zero issue on the XP bar.
• Re-registered the event UPDATE_EXTRA_ACTIONBAR on the Blizzard actionbar controller so that the extra action bar will show up again.
5.0.0
• Added initial support for Monk stances.
• Dropped Dominos_Totems
• Dropped Dominos_Roll
4.3.4
• Fixed an issue with the extra action bar frame interfering with clicking objects near the bottom center of the screen.
4.3.3
• Extra action bar testing, third attempt.
4.3.2
• Reverted to an older method of showing the extra action bar.
4.3.1
• This version is for 4.3 ONLY
• Updated action bar events to work with WoW 4.3
• Added new bar, extra action bar, which is used in some raids (Still needs testing)
4.2.5:
• Fixed upgrade issues with bear form, I hope :P
4.2.4:
• Tuller: Added support for the 4.3 extra action bar (NEEDS TESTING!)
• Tuller: Fixed a bug causing bear form to not have a default bar set
• Tuller: Added some upgrade code to supply a default bear form bar if one was missing.
4.2.3:
• FIxed some issues with paging offsets (hopefully :P)
4.2.2:
• Fixed an issue making click-through settings not work upon next login
4.2.1
• Bugfixes.
4.2.0:
• Updated for WoW 4.2
• Added new option: Show Tooltips in Combat
• Goranaws: Made it possible to hide buttons on the menu bar
4.1.apple1:
• Switched how I store state information internally to not be directly based off of macro states, so that I can account for things like users with/without Tree Form. Note: this change has a chance of wiping your bar paging settings.
• Fixed a bug on Shamans when switching profiles (https://github.com/Tuller/Dominos/issues/53)
1.25.0
• Updated TOC for 4.1
• Removed Quick Move key option, since Blizzard added it to the Action Bars portion of their interface options menu.
1.24.1
• Taint fix
1.24.0
• Added some code to make flyouts work nicer in 4.0.6
• Updated Spanish localization (thanks xibeca)
1.23.9
• Made drag and drop work on the totem bars again.
1.23.8
• Bugfix
1.23.7b
1.23.7
• Rewrote the hide blizzard function to work a bit more like the Bartender one. You can now once again control bag/quest log placement by checking the extra blizzard action bars.
1.23.6
• Added Spanish localization (thanks xibeca)
• Made the totem bars act a bit more like the standard Blizzard one: Right click a totem or call spell to bring up a list of totems/calls to select from. Left click to switch. You can also mouse-wheel a call button to switch pages.
1.23.5 (Beta)
• Fixed a bug where I forgot to push my code from my Macbook back to GitHub :)
1.23.4 (Beta)
• Fixed a bug causing the totem bars to not be disabled on non Shamans
1.23.3 (Beta)
• Bugfixes, yay!
1.23.2 (Beta)
• Added back the second and third totem bars
1.23.1 (Beta)
• Fixed some issues for people who've yet to learn call spells/etc.
• Middle mouse clicking or alt left clicking a call spell will now cast Totemic Recall, if available.
• Mousewheeling a call spell will now cycle through all totem pages
1.23.0 (Beta)
• Redesigned the totem bar to work a bit more like the Blizzard one.
1.22.0
• Fixed issues caused by my fix for Tree of Life :P
1.21.0
• Fixed issues with Tree of Life for Druids
1.20.4
• Added back % display for XP by default
• Fixed a bug causing empty buttons to not show up when binding keys
1.20.3
• Fixed a bug that was causing Rogues to lose their shadow dance bar setting when upgrading versions
• Made the xp/rep bar text a bit more customizable, if you're lua handy
1.20.2
• Added some tweaks to binding updates to reduce CPU usage. This may or may not break stuff
1.20.0
1.19.9
• Tweaked FlyPaper to be able to be standalone
• Forced updating of state sliders
1.19.8
• Fixed auto fading with flyout buttons
• Updated libraries
1.19.7b
• Removed some debug code
1.19.7
• Brought the animation system for fading back, hopefully without the missing hotkeys this time :P
1.19.6
• Turns out, the animation system does wacky things to hotkeys :P
1.19.5
1.19.4
• Made bar 2 not use the bonus action bar buttons
• Switched all fading code to the animation system
1.19.3
• Totem bar bugfix
1.19.2
• Added support for the hunter aspect bar
• Profiles bugfix
1.19.1
1.18.6
• Fixed an error on load
1.18.5
• Updated localization
1.18.4
• Fixed some issues causing Dominos to not work with the Chinese client.
1.18.3
• Fixed some 3.3.5 related issues
1.18.2
• Fizzwidget FactionFriend compatibility update
1.18.1
• Fixed an error when attempting to keybind to the pet bar
• Fixed an error causing fade settings to not reset properly when switching profiles
1.18.0
• Added a new option "docked bars inherit opacity" When enabled, when a bar is stuck to another bar, that bar begins to mimic the parent's opacity level. If the parent bar happens to be a mouseover bar, then the mouseover range is extended to include the new bar.
1.17.0
• Added the ability to specify an opacity, instead of simply just telling a bar to show or hide, to the showstate dialog. For example, setting a bar's showstate to [combat]100;show will force a bar to have 100% opacity (regardless of if the bar has mouseover fading enabled) in combat, and otherwise just show the bar with its normal opacity setting.
1.16.3
• Fixed a bug causing the experience bar texture to tile instead of stretch.
1.16.2
• Totem bar bugfix #2
1.16.1
• Totem bar bugfix
1.16.0
• The totem bar should now work on characters even if he or she has yet to learn a call spell.
1.15.3b
1.15.3
• Updated Chinese localization
1.15.2b
1.15.2
• Fixed an error when switching profiles on Shaman characters
1.15.1
• Updated TOC for 3.3
1.15.0
• Increased the padding on the casting bar
• Added two new options to the totem bars: Show Recall and Show Totems.
1.14.2
• Added a percentage display to the experience bar
• Added tooltip descriptions to selected bars
1.12.1
• Updated LibKeyBound, giving Dominos support for up to 31 mouse buttons.
• Modified the menu bar creation code to fix some issues with patch 3.3
1.12.0
• Added industrials patch for modifier combos for paging
• Adjusted the defaults for the new layout ordering options to prevent issues.
1.11.1
1.11.0
• Implemented advanced layout ordering options, per gpsguru's patch
1.10.5
• Removed range indicator display text
1.10.4
• Taint fixes
• Added FactionFriend support to the experience bar
1.10.3
• Made the totem bar not disable itself when logging on a non Shaman character.
1.10.2
• 3.2 retail release
1.10.1
• This is a 3.2.0 beta/alpha. It will not work properly on the 3.1 servers
• Fixed a bug with talent swapping
1.10.0
• This is a 3.2.0 beta/alpha. It will not work properly on the 3.1 servers
• Fixed a redbox error when casting a spell
• Added a new addon, Dominos_Totems - Provides three totem bars for shamans.
1.9.4
• Fixed the notarget paging option
1.9.3
• Merged into trunk
• Updated licensing information.
1.9.2
1.9.1
• Added some code to hopefully make the VehicleSeatIndicator stay completely on screen.
1.9.0
• Updated for 3.1 compatibility
• This should fix auto fading not working, along with the tainting issue with the quest log tracker item buttons
1.8.3
• Fixed the bug causing your bars to not load properly when in a form/stance/whatever.
• Updated translations
1.8.2
• Added a theoretical (ie, probably will not work) fix for the possess/vehicle bar issues people are having
• Modified the vehicle control bar to not always show the leave button
• Added a fix for the missing profiles button under certain locales
• Made the buff and debuff highlighting code a bit more efficient
• #### Ezmaralda's UI
First Previous Page 2 of 363 Next Last
• #6055
I was wondering if it would be possible to make different scale for each button, like some smaller which i dont care about to see, and those important bigger in one action bar like for example: [ X x x X X x x x X X X ]. I'm guessing it would be too much of a bother to add such option especially not knowing if many ppl would even use it.
Last edited by Agrias2x on 7/26/2015 9:51:52 AM
• #6057
Its not something I'm planning on implementing. There's an addon that does something similar, but its name escapes me at the moment.
Bacon is a cheese.
• #6065
Well Dominos can do that, too. Here's the addon I was trying to find:
Bacon is a cheese.
• #6063
Button Forge allows bars of different sizes, at least:
http://media-curse.cursecdn.com/attachments/14/711/79a8190995bd08c372b8fdbcd7d74fa0.jpg
342
• #6051
Hey, can you make the menu bar available for skinning? It doesn't show up in Masque.
Also wondering if it's possible to change the font for keybinds, macro-text etc.?
Last edited by obscurescience on 7/26/2015 3:50:01 AM
• #6053
Make menu bar available for skinning:
I can, but I'd like to test with a skin that works properly on the menu bar first. Do you have an example of one?
Changing font for keybidings, etc:
I'd rather Masque do this for me, since if I add an option to change the font for things, then I also need to allow making text position adjustments, etc.
Bacon is a cheese.
• #6058
Ok, I don't have an example of a skin that works since I can't enable it. But if you just want a skin to test I am using this atm http://www.curse.com/addons/wow/masque-diablo-iii
• #6048
I also had to back off to 6.2.6 because 6.2.7 is very very very broken. Ton of stuff was missing (including pet bars) and all my saved profiles were gone. Really wish I had read the comments before updating :(
• #6049
A typo snuck through somehow. This should be fixed with 6.2.7b
Bacon is a cheese.
• #6064
Thanks so much, thankfully the saved profiles that were "gone" are now back :)
• #6050
I just tried 6.2.7b. It seems to be fine. Thanks for the quick turnaround.
• #6046
2x Dominos_Encounter\bar.lua:27: attempt to call method 'InCombatLockdown' (a nil value)
Dominos_Encounter\bar.lua:27: in function Layout'
Dominos_Encounter\bar.lua:11: in function New'
Dominos_Encounter\controller.lua:21: in function Load'
Dominos\Dominos-6.2.7.lua:53: in function <Dominos\Dominos.lua:51>
(tail call): ?
[C]: ?
[string "safecall Dispatcher[1]"]:9: in function <[string "safecall Dispatcher[1]"]:5>
(tail call): ?
...faceArchy\Libs\AceAddon-3.0\AceAddon-3.0-12.lua:558: in function EnableAddon'
FrameXML\UIParent.lua:343: in function UIParentLoadAddOn'
FrameXML\UIParent.lua:925: in function <FrameXML\UIParent.lua:825>
• #6045
I had to back off to version 6.2.6. Version 6.2.7 is broken. My pet bar, extra button bar, character bar (with spell book etc) and bag bard dissapeared. Also the position of many of my bars were messed up.
• #6044
after updating today, my menu bar is completely gone, some of my bars are moved around, and moving a particular bar causes my wow client to completely lock up/crash. Never had any problems like this before. Nothing has changed other than me updating dominos.
Last edited by Trictagon on 7/25/2015 7:04:20 PM
• #6047
same
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# The Cichon Diagram for Degrees of Relative Constructibility
I just submitted my first paper: The Cichon Diagram for Degrees of Constructibility, on the ArXiv here, and I wanted to take the opportunity to write something about it here.
Abstract: Following a line of research initiated by Brendle, Brooke-Taylor, Ng and Nies, I describe a general framework for turning reduction concepts of relative computability into diagrams forming an analogy with the Cichon diagram for cardinal characteristics of the continuum. I show that working from relatively modest assumptions about a notion of reduction, one can construct a robust version of such a diagram. As an application, I define and investigate the Cichon Diagram for degrees of constructibility relative to a fixed inner model W. Many analogies hold with the classical theory as well as some surprising differences. Along the way I introduce a new axiom stating, roughly, that the constructibility diagram is as complex as possible.
In this paper I consider generalizations of the Cichon Diagram for reduction concepts. A reduction concept is a triple $(X, \sqsubseteq, 0)$ where $X$ is a non-empty set, $\sqsubseteq$ is a partial preorder on $X$ and $0 \in X$ is a distinguished element. An element $x \in X$ is called basic if $x \sqsubseteq 0$. Common examples include Turing reducibility with the basic elements being the computable sets, arithmetic reducibility with the basic elements being the $\emptyset$-definable subsets of $\mathbb N$ and degrees of constructibility with the basic elements being the constructible reals (in all these cases the underlying set is the reals). I show for a wide variety of reduction concepts on the reals, one can develop a generalization of the ideas underlying the study of the cardinal characteristics of the continuum by considering sets $\mathcal B_\sqsubseteq (R)$ and $\mathcal D_\sqsubseteq (R)$ for various relations $R$, where $\mathcal B_\sqsubseteq (R)$ is the set of elements which $\sqsubseteq$-build a witness to the fact that the basic reals are small with respect to $R$ and $\mathcal D_\sqsubseteq (R)$ is the set of elements which $\sqsubseteq$-build a witness to the fact that the basic reals are not big with respect to $R$. I show that for such reduction concepts one can always construct an analogue of the Cichon diagram. For example, below is the Cichon diagram for degrees of constructibility relative to a fixed inner model $W$. Recall that $x \leq_W y$ if and only if $x \in W[y]$.
In the second part of the paper I focus on the case of $\leq_W$ and show that all of the arrows pictured above are consistently strict.
Theorem: The Cichon diagram for $\leq_W$ as shown above is complete in the sense that if $A$ and $B$ are two nodes in the diagram and there is not an arrow from $A$ to $B$ in the $\leq_W$-diagram then there is a forcing extension of $W$ where $A$ is not a subset of $B$.
The proof of this theorem involves studying how the diagram is affected by familiar forcing notions to add reals. Accompanying my arguments are versions of the diagram as affected by these forcing notions. For example, here are the versions for Sacks, Cohen and Hechler forcing.
In the end I show that there is one proper forcing over $W$ which simultaneously realizes all possible separations. This is the last main theorem proved in the paper. It’s formalized in GBC.
Theorem: Given any transitive inner model $W$ of ZFC, there is a proper forcing notion $\mathbb P$, such that in $W^\mathbb P$ all the nodes in the $\leq_W$-Cicho\’n diagram are distinct and every possibile separation is simultaneously realized.
For more information check out the paper! Also, you can see a blog post about it on the page of my advisor, Joel David Hamkins, to whom I am incredibly grateful, here.
# Final Study Guide
As I mentioned in class, I have compiled a list of questions as a Final Study Guide to help you study for the final which will be on Tuesday, January 23rd. On Monday we will have a review day. To get full credit for participation you will be required to
2. Help answer a problem either from this list or from the midterm, which we will also be reviewing.
I have also typed up solutions for the midterm. Note however that in some cases there are more than one correct response so if you have something slightly different than what I have written, this does not mean it’s necessarily wrong.
To also help you study here are some Lecture Notes on sets that I typed up (essentially the first lecture).
# Homework # 3
The following problems are for HW #3, which is due this Thursday, January 18th.
For the first three problems use $\epsilon - \delta$ to show the following limits are true.
1. $\lim_{x \to 1} 2x - 3 = -1$
2. $\lim_{x \to 2} \frac{x^2 - 4}{x-2} = 4$
3. $\lim_{x \to a} 3x + 2 = 3a + 2$ for all $a \in \mathbb R$.
4. Prove for contradiction that the following limit does not exist: $\lim_{x \to 0} \frac{1}{x} + x^2$.
The rest of the problems are from the textbook.
Chapter 4: 17
Chapter 5: B. 14, 15
Chapter 6: A. 6, 14
EXTRA CREDIT: Following the problem with the last homework, let me be clear that (in this class) $0$ counts as a natural number! For extra credit, give an example of a property of natural numbers $n$ that is true for all $n > 0$ but false for $n = 0$.
Please let me know if you have any questions.
# Homework #2
The following problems are all from Chapter 2 of our textbook. This homework is due on Wednesday, January 10th.
Section 2.2: Problems 1, 2, 3, 4
Section 2.5: Problems 1,2, 3, 10
Section 2.7: Problems 4, 5, 6
Section 2.9: Problems 7, 8, 9
Verify de Morgan’s Second Law using a Truth Table: $\neg(p \lor q) \Leftrightarrow (\neg p) \land (\neg q)$
# Homework #1
The first homework is due on Thursday, January 4th. All exercises are taken from our textbook, The Book of Proof by Richard Hammock.
Chapter 1
Section 1.2: Part A, Number 1 a), c), e)
Section 1.3 Part B, Numbers 9, 10, 11, 12
Section 1.4 Part A, Numbers 7, 8, 9, 10
Section 1.5 Number 4 a), b), c), f), g), h)
# Welcome!
Welcome to Math 156 at Hunter College City University of New York. Here is the syllabus for the course, also available as a pdf here: Math 156 Syllabus.
MATH 156.W02 Introduction to Mathematical Proof Workshop, 2 hrs, 1 cr, Winter 2018
Meets: Monday, Tuesday, Wednesday, Thursday, 9:00am-11:05am, Room 604HW
Instructor: Corey Switzer
Office Hours: Thursdays from 11:30-12:30 in Room 924 HE, other times by appointment
Textbook: Book of Proof, Second Edition, by Richard Hammack © 2013, Richard Hammack (publisher). ISBN 978-0-9894721-0-4, (313 pages). Download the ebook for free from the author’s website here, and/or purchase the print edition on Amazon or Barnes & Noble for around \$15.
Notes: Other notes will be passed out in class and available at the course website here.
Course Learning Outcomes:
• Construct direct and indirect proofs and proofs by induction and determine the appropriateness of each type in a particular setting. Analyze and critique proofs with respect to logic and correctness, and prove conjectures
• Learn to construct proofs that are not only mathematically correct but also clearly written, convincing, readable, notationally consistent, and grammatically correct,
• Apply the logical structure of various proof techniques, proof types, and counterexamples and work symbolically with connectives and quantifiers,
• Perform set operations on finite and infinite collections of sets and be familiar with properties of set operations and different cardinalities for infinite sets,
• Work with relations and functions, including surjections, injections, inverses, and bijections, equivalence relations, and equivalence classes,
• Learn to work with εδ definitions and proofs involving limits at a point for polynomials, rational functions, as well as transcendental functions that do not have a limit at a point,
Course Content to be Covered
Syllabus: As a rough guide, we will cover most Chapters from the following:
• Chapter 1: Sets
• Chapter 2: Logic
• Chapter 4: Direct Proof
• Chapter 5: Contrapositive Proof
• Chapter 6: Proof by Contradiction
• Chapter 7: Proving Non-Conditional Statements
• Chapter 8: Proofs Involving Sets
• Chapter 9: Disproof
• Chapter 10: Mathematical Induction
• Chapter 11: Relations
• Chapter 12: Functions
• Chapter 13: Cardinality of Sets
• Epsilon-Delta Proofs
General: A typical class will begin with some problems for the previous class to test comprehension. Students will be expected to work with one another on these problems and, on occasion, present solutions. Then, I will introduce topics and examples of proof techniques from the text. It is important to understand what goes on in class each day. This means being present and being prepared for every class, first by reading the textbook and second by making a serious effort to do the homework.
It is encouraged that you work together both on reviewing the material and doing the homework. However, this is not an excuse for copying some one else’s work. Every student is expected to turn in their own, original work.
Last Day to Withdraw with a Grade of W: 1/16/17
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cc17:homework_4
# Differences
This shows you the differences between two versions of the page.
cc17:homework_4 [2017/04/20 19:33]hossein created cc17:homework_4 [2017/05/12 02:58] (current)hossein [Problem 3] 2017/05/12 02:58 hossein [Problem 3] 2017/04/20 19:33 hossein created 2017/05/12 02:58 hossein [Problem 3] 2017/04/20 19:33 hossein created Line 19: Line 19: $x := y \mbox{ op } z$ $x := y \mbox{ op } z$ where $x$ , $y$ , $z$ are identifiers, constants or temporary variables. $\mbox{op}$ is an operator. where $x$ , $y$ , $z$ are identifiers, constants or temporary variables. $\mbox{op}$ is an operator. - Generate code for the following expression under the assumption that you have only three temporary variables available. Your translation should not change the values of the identifiers after execution. + Generate three-address code for the following expression under the assumption that you have only three temporary variables available. Your translation should not change the values of the identifiers after execution. $$(a+b) + ((c-d)+(e*f))$$ $$(a+b) + ((c-d)+(e*f))$$
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# An algebra problem by Sai Venkatesh
Algebra Level 2
Given that $$a,b$$ and $$c$$ are real numbers such that $$a+b+c=0$$, compute $\frac { 1 }{ { b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 } } +\frac { 1 }{ { c }^{ 2 }+{ a }^{ 2 }-{ b }^{ 2 } } +\frac { 1 }{ { a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 } }.$
×
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218 articles – 619 references [version française]
HAL: in2p3-00102878, version 1
8th International Conference of Strangeness in Quark Matter, Los Angeles : États-Unis
Results from NA57
For the NA57 collaboration(s)
(2006)
The NA57 experiment has measured strange baryon and antibaryon production in Pb-Pb collisions at 40 A GeV/c and 158 A GeV/c beam momenta. Expansion dynamics has been investigated; transverse dynamics using blast wave fits, longitudinal dynamics obtained from rapidity distributions. HBT analysis gives results compatible with those obtained by the other methods. Similar values are observed for the transverse and longitudinal ow parameters. Particles carrying all levels of strangeness have been studied. Enhancement factors increase with the strangeness content of the particle, reaching above 20 for $\Omega$s in central collisions. Enhancement values are strikingly similar for SPS and RHIC energies. Centralto-peripheral nuclear modication factors have been measured. The baryon/meson pattern displays similarities with that observed at RHIC
Subject(s) : Physics/Nuclear ExperimentPhysics/High Energy Physics - Experiment
Attached file list to this document:
PDF
sqm2006_na57.pdf(202.4 KB)
in2p3-00102878, version 1 http://hal.in2p3.fr/in2p3-00102878 oai:hal.in2p3.fr:in2p3-00102878 From: Yvette Heyd <> Submitted on: Wednesday, 4 October 2006 15:15:41 Updated on: Thursday, 15 March 2007 15:18:12
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# Does left-invertible imply invertible in full group C*-algebras (discrete case)?
The following question/problem has been bugging me on and off for some time now: so I thought it might be worth broaching here on MO, as a case of "ask the experts".
Let $G$ be a discrete group. Kaplansky observed that since the group von Neumann algebra $VN(G)$ is a finite von Neumann algebra, each left-invertible element in $VN(G)$ is invertible. A proof is outlined in
M.S. Montgomery, Left and right inverses in group algebras, Bull. AMS 75 (1969)
(Well, she actually states a weaker result, but inspection shows that her argument extends to give what we claim. See also my remarks on this previous MO answer.) The basic idea is to exploit the faithful trace $T\mapsto \langle T\delta_e,\delta_e\rangle$ and how it behaves on idempotents: for if $ab=I$, then $ba$ is an idempotent.
In particular, each left-invertible element of the reduced group $C^*$-algebra is invertible.
Question. What can we say for the full group $C^*$-algebra? Is every left-invertible element in $C^*(G)$ automatically invertible?
Some basic observations:
• The case where $G$ is the free group on two generators follows from a result of M-D Choi [no relation] who showed that $C^*({\mathbb F}_2)$ embeds into a direct product of matrix algebras.
• More generally, if $C^*(G)$ has a faithful trace then one can use the same argument as for the reduced $C^*$-algebra to get a positive answer.
• If $C^*(G)$ has no non-trivial projections then $ab=I$ implies $ba=I$. (I think this was known to be true for $G={\mathbb F}_2$ but I've forgotten the reference at present.)
• There are examples of $G$ where $C^*(G)$ has no faithful trace; these can be found in work of Bekka and Louvet, and come from exploiting Property (T).
Bekka, M. B.(F-METZ-MM); Louvet, N.(CH-NCH) Some properties of $C^*$-algebras associated to discrete linear groups. $C^*-algebras (Münster, 1999), 1–22, Springer, Berlin, 2000. • Would Bill Johnson know? There is an ask-johnson tag. See mathoverflow.net/questions/tagged/ask-johnson – Will Jagy Oct 24 '11 at 3:36 • @Will: I think of WBJ as more of a Banach-space specialist than an operator algebraist, but it is entirely possible he might spot something I haven't here. – Yemon Choi Oct 24 '11 at 3:48 • I printed out your arXiv piece 1003.1650v2. Very nice. – Will Jagy Oct 24 '11 at 4:02 • Just a remark: if a$C^*$-algebra$A$does not have tracial states then$A^{**}$has an isometry which is not unitary, thus$A^{**}$does not satisfy Kaplanski condition. Little bit more is true: for every$n$there are$n$pairwise orthogonal isometries in$A^{**}$. – Kate Juschenko Oct 24 '11 at 9:51 • Nice question, and I do not know the answer. In fact, Yemon would be hard pressed to ask a question about operator algebras to which I know the answer but he does not. – Bill Johnson Oct 24 '11 at 12:43 ## 2 Answers That's a nice question. I don't know the answer for arbitrary groups, but this finiteness property (left invertible implies invertible in the full group C*-algebra) is known for more groups. M.D. Choi's result was generalized by [Exel and Loring, Internat. J. Math. 1992]. We say that a C*-algebra if residually finite dimensional (RFD) if it has a separating family of finite dimensional representations. RFD algebras have this finiteness property and finite groups, abelian groups, etc., have RFD full group C*-algebras. Exel and Loring show that unital full free products of RFD C*-algebras are RFD. So if$C^*(G_i)$are RFD (i=1,2), then so is$C^*(G_1*G_2)$. A broader class of C*-algebras than the RFD ones are the MF algebras of [Blackadar and Kirchberg, Math. Ann., 1997]. In MF algebras, all left invertibles are invertible. Recently, [Hadwin, Q. Li, J. Shen, Canad. J. Math. 2011] showed that unital full free products of MF C*-algebras are MF. • Thanks for the references - I will have to set aside some time to read up on these results. – Yemon Choi Oct 25 '11 at 0:24 There is an alternative argument for the free group; not using that free groups are residually finite-dimensional. Let$\pi$be a faithful representation of$C^{\ast}(F)$on a Hilbert space$H$. Then, as$U(H)$is connected,$\pi$can be deformed to the trivial representation in the point-norm topology, i.e. there exists a family of unitary representations$\pi_t$for$t \in [0,1]$, such that$t \mapsto \pi_t(a)$is norm-continuous for each$a \in C^{\ast}(F)$,$\pi_0=\pi$and$\pi_1(g)=1_H$for all$ g \in F$. Now, if$ab=1$in$C^{\ast}(F)$, then$\pi_t(ba)$is a continuous path of projections ending at$1_H$. Hence,$\pi_0(ba)=1_H$and$ba=1$, as$\pi$was faithful. EDIT: The same argument works if the$C^{\ast}$-algebra embeds into some contractible algebra (i.e. homotopy equivalent to$\mathbb C$). However, even though many reasonable toplogical spaces are quotients of contractible topological spaces, only few reasonable$C^{\ast}\$-algebras have this property. There is a close relationship with the concept of quasi-diagonality, which appeared in the work of Voiculescu.
• I don't quite follow-- is the point about why this works for a free group that I can take a path through the unitaries for each generator, and then let these (uniquely) induce my *-rep of C^*(F)-- that we have no relations to worry about means that this always works?? – Matthew Daws Oct 27 '11 at 19:28
• Thanks Andreas. Reading your answer it reminded me that I'd seen a similar outline before - I guess this kind of argument is very natural for those who've looked at K-theory of group (C*-)algebras. After doing some looking, it seems that this kind of argument was used by J. Cohen in ams.org/mathscinet-getitem?mr=546507 – Yemon Choi Oct 27 '11 at 19:29
• Matthew, you are absolutely right. – Andreas Thom Oct 28 '11 at 7:27
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DeepAI
# Competition, Alignment, and Equilibria in Digital Marketplaces
Competition between traditional platforms is known to improve user utility by aligning the platform's actions with user preferences. But to what extent is alignment exhibited in data-driven marketplaces? To study this question from a theoretical perspective, we introduce a duopoly market where platform actions are bandit algorithms and the two platforms compete for user participation. A salient feature of this market is that the quality of recommendations depends on both the bandit algorithm and the amount of data provided by interactions from users. This interdependency between the algorithm performance and the actions of users complicates the structure of market equilibria and their quality in terms of user utility. Our main finding is that competition in this market does not perfectly align market outcomes with user utility. Interestingly, market outcomes exhibit misalignment not only when the platforms have separate data repositories, but also when the platforms have a shared data repository. Nonetheless, the data sharing assumptions impact what mechanism drives misalignment and also affect the specific form of misalignment (e.g. the quality of the best-case and worst-case market outcomes). More broadly, our work illustrates that competition in digital marketplaces has subtle consequences for user utility that merit further investigation.
• 12 publications
• 267 publications
• 15 publications
08/19/2021
### Learning Equilibria in Matching Markets from Bandit Feedback
Large-scale, two-sided matching platforms must find market outcomes that...
09/15/2020
### Competing AI: How does competition feedback affect machine learning?
This papers studies how competition affects machine learning (ML) predic...
03/31/2022
### Performative Power
We introduce the notion of performative power, which measures the abilit...
05/06/2021
### Bandit based centralized matching in two-sided markets for peer to peer lending
Sequential fundraising in two sided online platforms enable peer to peer...
12/01/2011
### Bandit Market Makers
We introduce a modular framework for market making. It combines cost-fun...
06/04/2019
### Kinetic Market Model: An Evolutionary Algorithm
This research proposes the econophysics kinetic market model as an evolu...
05/09/2018
### Loyalty Programs in the Sharing Economy: Optimality and Competition
Loyalty programs are important tools for sharing platforms seeking to gr...
## 1 Introduction
Recommendation systems are the backbone of numerous digital platforms—from web search engines to video sharing websites to music streaming services. To produce high-quality recommendations, these platforms rely on data which is obtained through interactions with users. This fundamentally links the quality of a platform’s services to how well the platform can attract users.
What a platform must do to attract users depends on the amount of competition in the marketplace. If the marketplace has a single platform—such as Google prior to Bing or Pandora prior to Spotify—then the platform can accumulate users by providing any reasonably acceptable quality of service given the lack of alternatives. This gives the platform great flexibility in its choice of recommendation algorithm. In contrast, the presence of competing platforms makes user participation harder to achieve and intuitively places greater constraints on the recommendation algorithms. This raises the questions: how does competition impact the recommendation algorithms chosen by digital platforms? How does competition affect the quality of service for users?
Conventional wisdom tells us that competition benefits users. In particular, users vote with their feet by choosing the platform on which they participate. The fact that users have this power forces the platforms to fully cater to user choices and thus improves user utility. This phenomenon has been formalized in classical markets where firms produce homogenous products (bertrand), where competition has been established to perfectly align market outcomes with user utility. Since user wellbeing is considered central to the healthiness of a market, perfect competition is traditionally regarded as the “gold standard” for a healthy marketplace: this conceptual principle underlies measures of market power (lerner) and antitrust policy (dukelaw).
In contrast, competition has an ambiguous relationship with user wellbeing in digital marketplaces, where digital platforms are data-driven and compete via recommendation algorithms that rely on data from user interactions. Informally speaking, these marketplaces exhibit an interdependency between user utility, the platforms’ choices of recommendation algorithms, and the collective choices of other users. In particular, the size of a platform’s user base impacts how much data the platform has and thus the quality of its service; as a result, an individual user’s utility level depends on the number of users that the platform has attracted thus far. Having a large user base enables a platform to have an edge over competitors without fully catering to users, which casts doubt on whether classical alignment insights apply to digital marketplaces.
The ambiguous role of competition in digital marketplaces—which falls outside the scope of our classical understanding of competition power—has gained center stage in recent policymaking discourse. Indeed, several interdisciplinary policy reports (stiger19; cremer2019competition) have been dedicated to highlighting ways in which the structure of digital marketplaces fundamentally differs from that of classical markets. For example, these reports suggest that data accumulation can encourage market tipping, which leaves users particularly vulnerable to harm (as we discuss in more detail at the end of Section 1.1). Yet, no theoretical foundation has emerged to formally examine the market structure of digital marketplaces and assess potential interventions. To propel the field forward and arm policymaking discourse with technical tools, it is necessary to develop mathematically founded models to investigate competition in digital marketplaces.
### 1.1 Our contributions
Our work takes a step towards building a theoretical foundation for studying competition in digital marketplaces. We present a framework for studying platforms that compete on the basis of learning algorithms, focusing on alignment with user utility at equilibrium. We consider a stylized duopoly model based on a multi-armed bandit problem where user utility depends on the incurred rewards. We show that competition may no longer perfectly align market outcomes with user utility. Nonetheless, we find that market outcomes exhibit a weaker form of alignment: the user utility is at least as large as the optimal utility in a population with only one user. Interestingly, there can be multiple equilibria, and the gap between the best equilibria and the worst equilibria can be substantial.
#### Model.
We consider a market with two platforms and a population of users. Each platform selects a bandit algorithm from a class . After the platforms commit to algorithms, each user decides which platform they wish to participate on. Each user’s utility is the (potentially discounted) cumulative reward that they receive from the bandit algorithm of the platform that they chose. Users arrive at a Nash equilibrium.111In Section 2, we will discuss subtleties that arise from having multiple Nash equilibria. Each platform’s utility is the number of users who participate on that platform, and the platforms arrive at a Nash equilibrium. The platforms either maintain separate data repositories about the rewards of their own users, or the platforms maintain a shared data repository about the rewards of all users.
#### Alignment results.
To formally consider alignment, we introduce a metric—that we call the user quality level—that captures the utility that a user would receive when a given pair of competing bandit algorithms are implemented and user choices form an equilibrium. Table 1 summarizes the alignment results in the case of a single user and multiple users. A key quantity that appears in the alignment results is , which denotes the expected utility that a user receives from the algorithm when users all participate in the same algorithm.
For the case of a single user, an idealized form of alignment holds: the user quality level at any equilibrium is the optimal utility that a user can achieve within the class of algorithms . Idealized alignment holds regardless of the informational assumptions on the platform.
The nature of alignment fundamentally changes when there are multiple users. At a high level, we show that idealized alignment breaks down since the user quality level is no longer guaranteed to be the global optimum, , that cooperative users can achieve. Nonetheless, a weaker form of alignment holds: the user quality level nonetheless never falls below the single-user optimum . Thus, the presence of other users cannot make a user worse off than if they were the only participant, but users may not be able to fully benefit from the data provided by others.
More formally, consider the setting where the platforms have separate data repositories. We show that there can be many qualitatively different Nash equilibria for the platforms. The user quality level across all equilibria actually spans the full set ; i.e., any user quality level is realizable in some Nash equilibrium of the platforms and its associated Nash equilibrium of the users (Theorem 2). Moreover, the user quality level at any equilibrium is contained in the set (Theorem 3). When the number of users is large, the gap between and can be significant since the latter is given access to times as much data at each time step than the former. The fact that the single-user optimum is realizable means that the market outcome might only exhibit a weak form of alignment. The intuition behind this result is that the performance of an algorithm is controlled not only by its efficiency in transforming information to action, but also by the level of data it has gained through its user base. Since platforms have separate data repositories, a platform can thus make up for a suboptimal algorithm by gaining a significant user base. On the other hand, the global optimal user quality level is nonetheless realizable—this suggests that equilibrium selection could be used to determine when bad equilibria arise and to nudge the marketplace towards a good equilibrium.
What if the platforms were to share data? At first glance, it might appear that with data sharing, a platform can no longer make up for a suboptimal algorithm with data, and the idealized form of alignment would be recovered. However, we construct two-armed bandit problem instances where every symmetric equilibrium for the platforms has user quality level strictly below the global optimal (Theorems 4-5). The mechanism for this suboptimality is that the global optimal solution requires “too much” exploration. If other users engage in their “fair share” of exploration, an individual user would prefer to explore less and free-ride off of the data obtained by other users. The platform is thus forced to explore less, which drives down the user quality level. To formalize this, we establish a connection to strategic experimentation (BH98). Nonetheless, although all of the user quality levels in may not be realizable, the user quality level at any symmetric equilibria is still guaranteed to be within this set (Theorem 7).
#### Connection to policy reports.
Our work provides a mathematical explanation of phenomena documented in recent policy reports (stiger19; cremer2019competition). The first phenomena that we consider is market dominance from data accumulation. The accumulation of data has been suggested to result in winner-takes-all-markets where a single player dominates and where market entry is challenging (stiger19). The data advantage of the dominant platform can lead to lower quality services and lower user utility. Theorems 2-3 formalize this mechanism. We show that once a platform has gained the full user base, market entry is impossible and the platform only needs to achieve weak alignment with user utility to retain its user base (see discussion in Section 4.2). The second phenomena that we consider is the impact of shared data access. While the separate data setting captures much of the status quo of proprietary data repositories in digital marketplaces, sharing data access has been proposed as a solution to market dominance (cremer2019competition). Will shared data access deliver on its promises? Theorems 4-5 highlight that sharing data does not solve the alignment issues, and uncovers free-riding as a mechanism for misalignment.
### 1.2 Related work
We discuss the relation between our work and research on competing platforms, incentivizing exploration, and strategic experimentation.
#### Competing platforms.
AMSW20
also examine the interplay between competition and exploration in bandit problems in a duopoly economy. They focus on platform regret, showing that platforms must both choose a greedy algorithm at equilibrium and thus illustrating that competition is at odds with regret minimization. In contrast, we take a user-centric perspective and demonstrate that competition aligns market outcomes with user utility. Interestingly, the findings in
AMSW20 and our findings are not at odds: the result in AMSW20 can be viewed as alignment, since the optimal choice for a fully myopic user results in regret in the long run. Our alignment results also apply to non-myopic users and when multiple users may arrive at every round.
Outside of the bandits framework, another line of work has also studied the behavior of competing learners when users can choose between platforms. BT17; BT19 study equilibrium predictors chosen by competing offline learners in a PAC learning setup. Other work has focused on the dynamics when multiple learners apply out-of-box algorithms, showing that specialization can emerge (GZKZ21; DCRMF22) and examining the role of data purchase (KGZ22); however, these works do not consider which algorithms the learners are incentivized to choose to gain users. In contrast, we investigate equilibrium bandit algorithms chosen by online learners, each of whom aims to maximize the size of its user base. The interdependency between the platforms’ choices of algorithms, the data available to the platforms, and the users’ decisions in our model drives our alignment insights.
Other aspects of competing platforms that have been studied include competition under exogeneous network effects (R09; WW14), experimentation in price competition (BV2000), dueling algorithms which compete for a single user (IKLMPT11), and measures of a digital platform’s power in a marketplace (HJM22).
#### Incentivizing exploration.
This line of work has examined how the availability of outside options impacts bandit algorithms. FKKK14 show that Bayesian Incentive Compatibility (BIC) suffices to guarantee that users will stay on the platform. Follow-up work (e.g., MSS15; SS21) examines what bandit algorithm are BIC. KMP13 explores the use of monetary transfers.
#### Strategic experimentation.
This line of work has investigated equilibria when a population of users each choose a bandit algorithm. BH98; BH00; BHSimple analyze the equilibria in a risky-safe arm bandit problem: we leverage their results in our analysis of equilibria in the shared data setting. Strategic experimentation (see HS17 for a survey) has investigated exponential bandit problems (KRC15), the impact of observing actions instead of payoffs (RSV07), and the impact of cooperation (BP21).
## 2 Model
We consider a duopoly market with two platforms performing a multi-armed bandit learning problem and a population of users, , who choose between platforms. Platforms commit to bandit algorithms, and then each user chooses a single platform to participate on for the learning task.
### 2.1 Multi-armed bandit setting
Consider a Bayesian bandit setting where there are arms with priors . At the beginning of the game, the mean rewards of arms are drawn from the priors . These mean rewards are unknown to both the users and the platforms but are shared across the two platforms. If the user’s chosen platform recommends arm , the user receives reward drawn from a noisy distribution with mean .
Let be a class of bandit algorithms that map the information state given by the posterior distributions to an arm to be pulled. The information state is taken to be the set of posterior distributions for the mean rewards of each arm. We assume that each algorithm can be expressed as a function mapping the information state to a distribution over arms in .222This assumption means that an algorithm’s choice is independent of the time step conditioned on
. Classical bandit algorithms such as Thompson sampling
(T33), finite-horizon UCB (LR85), and the infinite-time Gittins index (G79) fit into this framework. This assumption is not satisfied by the infinite time horizon UCB. We let denote this distribution over arms .
#### Running example: risky-safe arm bandit problem.
To concretize our results, we consider the risky-safe arm bandit problem as a running example. The noise distribution is a Gaussian . The first arm is a risky arm whose prior distribution is over the set , where corresponds to a “low reward” and corresponds to a “high reward.” The second arm is a safe arm with known reward (the prior is a point mass at ). In this case, the information state
permits a one-dimensional representation given by the posterior probability
that the risky arm is high reward.
We construct a natural algorithm class as follows. For a measurable function , let be the associated algorithm defined so
is a distribution that is 1 with probability
and 2 with probability . We define
Aall:={Af∣f:[0,1]→[0,1] is % measurable}
to be the class of all randomized algorithms. This class contains Thompson sampling ( is given by ), the Greedy algorithm ( is given by if and otherwise), and mixtures of these algorithms with uniform exploration. We consider restrictions of the class in some results.
### 2.2 Interactions between platforms, users, and data
The interactions between the platform and users impact the data that the platform receives for its learning task. The platform action space is a class of bandit algorithms that map an information state to an arm to be pulled. The user action space is . For , we denote by the action chosen by user .
#### Order of play.
The platforms commit to algorithms and respectively, and then users simultaneously choose their actions prior to the beginning of the learning task. We emphasize that user participates on platform for the full duration of the learning task. (In Appendix B.2, we discuss the assumption that users cannot switch platforms between time steps.)
#### Data sharing assumptions.
In the separate data repositories setting, each platform has its own (proprietary) data repository for keeping track of the rewards incurred by its own users. Platforms 1 and 2 thus have separate information states given by and , respectively. In the shared data repository setting, the platforms share an information state , which is updated based on the rewards incurred by users of both platforms.333In web search, recommender systems can query each other, effectively building a shared information state.
The learning task is determined by the choice of platform actions and , user actions , and specifics of data sharing between platforms. At each time step:
1. Each user arrives at platform . The platform recommends arm to that user, where denotes the information state of the platform. (The randomness of arm selection is fully independent across users and time steps.) The user receives noisy reward .
2. After providing recommendations to all of its users, platform 1 observes the rewards incurred by users in . Platform 2 similarly observes the rewards incurred by users in . Each platform then updates their information state with the corresponding posterior updates.
3. A platform may have access to external data that does not come from users. To capture this, we introduce background information into the model. Both platforms observe the same background information of quality . In particular, for each arm , the platforms observe the same realization of a noisy reward . When , we say that there is no background information since the background information is uninformative. The corresponding posterior updates are then used to update the information state ( in the case of shared data; and in the case of separate data).
In other words, platforms receive information from users (and background information), and users receive rewards based on the recommendations of the platform that they have chosen.
### 2.3 Utility functions and equilibrium concept
User utility is generated by rewards, while the platform utility is generated by user participation.
#### User utility function.
We follow the standard discounted formulation for bandit problems (e.g. (GJ79; BH98)), where the utility incurred by a user is defined by the expected (discounted) cumulative reward received across time steps. The discount factor parameterizes the extent to which agents are myopic. Let denote the utility of a user if they take action when other users take actions and the platforms choose and . For clarity, we make this explicit in the case of discrete time setup with horizon length . Let denote the arm recommended to user at time step . The utility is defined to be
U(pi;p−i,A1,A2):=E[T∑t=1βtrati]
where the expectation is over randomness of the incurred rewards and the algorithms. In the case of continuous time, the utility is
U(pi;p−i,A1,A2):=E[∫e−βtdπ(t)]
where the denotes the discount factor and denotes the payoff received by the user.444For discounted utility, it is often standard to introduce a multiplier of for normalization (see e.g. (BH98)). The utility could have equivalently be defined as without changing any of our results. In both cases, observe that the utility function is symmetric in user actions.
The utility function implicitly differs in the separate and shared data settings, since the information state evolves differently in these two settings. When we wish to make this distinction explicit, we denote the corresponding utility functions by and .
#### User equilibrium concept.
We assume that after the platforms commit to algorithms and , the users end up at a pure strategy Nash equilibrium of the resulting game. More formally, let be a pure strategy Nash equilibrium for the users if for all . The existence of a pure strategy Nash equilibrium follows from the assumption that the game is symmetric and the action space has 2 elements (C04).
One subtlety is that there can be multiple equilibria in this general-sum game. For example, there are always at least 2 (pure strategy) equilibria when platforms play any , i.e., commit to the same algorithm — one equilibrium where all users choose the first platform, and another where all users choose the second platform). Interestingly, there can be multiple equilibria even when one platform chooses a “worse” algorithm than the other platform. We denote by the set of pure strategy Nash equilibria when the platforms choose algorithms and . We simplify the notation and use when and are clear from the context. In Section B.1, we discuss our choice of solution concept, focusing on what the implications would have been of including mixed Nash equilibia in .
#### Platform utility and equilibrium concept.
The utility of the platform roughly corresponds to the number of users who participate on that platform. This captures that in markets for digital goods, where platform revenue is often derived from advertisement or subscription fees, the number of users serviced is a proxy for platform revenue.
Since there can be several user equilibria for a given choice of platform algorithms, we formalize platform utility by considering the worst-case user equilibrium for the platform. In particular, we define platform utility to be the minimum number of users that a platform would receive at any pure strategy equilibrium for the users. More formally, when platform 1 chooses algorithm and platform 2 chooses algorithm , the utilities of platform 1 and platform 2 are given by:
v1(A1;A2):=minp∈EN∑i=11[pi=1] and v2(A2;A1)=minp∈EN∑i=11[pi=2].
The equilibrium concept for the platforms is a pure strategy Nash equilibrium, and we often focus on symmetric equilibria. (We discuss the existence of such an equilibrium in Sections 4-5.)
## 3 Formalizing the Alignment of a Market Outcome
The alignment of an equilibrium outcome for the platforms is measured by the amount of user utility that it generates. In Section 3.1 we introduce the user quality level to formalize alignment. In Section 3.2, we show an idealized form of alignment for (Theorem 1). In Section 3.3, we turn to the case of multiple users and discuss benchmarks for the user quality level. In Section 3.4, we describe mild assumptions on that we use in our alignment results for multiple users.
### 3.1 User quality level
Given a pair of platform algorithms and , we introduce the following metric to measure the alignment between platform algorithms and user utility. We again take a worst-case perspective and define user quality level to be the minimum utility that any user can receive at any pure strategy equilibrium for users.
###### Definition 1 (User quality level).
Given algorithms and chosen by the platforms, the user quality level is defined to be .
Since the utility function is symmetric and is the class of all pure strategy equilibria, we can equivalently define as for any .
To simplify notation in our alignment results, we introduce the reward function which captures how the utility that a given algorithm generates changes with the number of users who contribute to its data repository. For an algorithm , let the reward function be defined by:
RA(n):=Useparate(1;pn−1,A,A),
where
corresponds to a vector with
coordinates equal to one.
### 3.2 Idealized alignment result: The case of a single user
When there is a single user, the platform algorithms turn out to be perfectly aligned with user utilities at equilibrium. To formalize this, we consider the optimal utility that could be obtained by a user across any choice of actions by the platforms and users (not necessarily at equilibrium): that is, . Using the setup of the single-user game, we can see that this is equal to . We show that the user quality level always meets this benchmark (we defer the proof to Appendix C).
###### Theorem 1.
Suppose that , and consider either the separate data setting or the shared data setting. If is a pure strategy Nash equilibrium for the platforms, then the user quality level is equal to .
Theorem 1 shows that in a single-user market, two firms is sufficient to perfectly align firm actions with user utility—this stands in parallel to classical Bertrand competition in the pricing setting (bertrand).
#### Proof sketch of Theorem 1.
There are only 2 possible pure strategy equilibria: either the user chooses platform 1 and receives utility or the user chooses platform 2 and receives utility . If one platform chooses a suboptimal algorithm for the user (i.e. an algorithm where ), then the other platform will receive the user (and thus achieve utility 1) if they choose a optimal algorithm . This means that is a pure strategy Nash equilibrium if and only if or . The user thus receives utility . We defer the full proof to Appendix C.
### 3.3 Benchmarks for user quality level
In the case of multiple users, this idealized form of alignment turns out to break down, and formalizing alignment requires a more nuanced consideration of benchmarks. We define the single-user optimal utility of to be . This corresponds to maximal possible user utility that can be generated by a platform who only serves a single user and thus relies on this user for all of its data. On the other hand, we define the global optimal utility of to be . This corresponds to the maximal possible user utility that can be generated by a platform when all of the users in the population are forced to participate on the same platform. The platform can thus maximally enrich its data repository in each time step.
### 3.4 Assumptions on A
While our alignment results for a single user applied to arbitrary algorithm classes, we require mild assumptions on in the case of multiple users to endow the equilibria with basic structure.
Information monotonicity requires that an algorithm ’s performance in terms of user utility does not worsen with additional posterior updates to the information state. Our first two instantations of information monotonicity—strict information monotonicity and information constantness—require that the user utility of grow monotonically in the number of other users participating in the algorithm. Our third instantation of information monotonicity—side information monotonicity—requires that the user utility of not decrease if other users also update the information state, regardless of what algorithm is used by the other users. We formalize these assumptions as follows:
###### Assumption 1 (Information monotonicity).
For any given discount factor and number of users , an algorithm is strictly information monotonic if is strictly increasing in for . An algorithm is information constant if is constant in for . An algorithm is side information monotonic if for every measurable function mapping information states to distributions over and for every , it holds that where has all coordinates equal to .
While information monotonicity places assumptions on each algorithm in , our next assumption places a mild restriction on how the utilities generated by algorithms in relate to each other. Utility richness requires that the set of user utilities spanned by is a sufficiently rich interval.
###### Assumption 2 (Utility richness).
A class of algorithms is utility rich if the set of utilities is a contiguous set, the supremum of is achieved, and there exists such that .
These assumptions are satisfied for natural bandit setups, as we show in Section 6.
## 4 Separate data repositories
We investigate alignment when the platforms have separate data repositories. In Section 4.1, we show that there can be many qualitatively different equilibria for the platforms and characterize the alignment of these equilibria. In Section 4.2, we discuss factors that drive the level of misalignment in a marketplace.
### 4.1 Multitude of equilibria and the extent of alignment
In contrast with the single user setting, the marketplace can exhibit multiple equilibria for the platforms. As a result, to investigate alignment, we investigate the range of achievable user quality levels. Our main finding is that the equilibria in a given marketplace can exhibit a vast range of alignment properties. In particular, every user quality level in between the single-user optimal utility and the global optimal utility can be realized by some equilibrium for the platforms.
###### Theorem 2.
Suppose that each algorithm in is either strictly information monotonic or information constant (Assumption 1), and suppose that is utility rich (Assumption 2). For every , there exists a symmetric pure strategy Nash equilibrium in the separate data setting such that .
Nonetheless, there is a baseline (although somewhat weak) form of alignment achieved by all equilibria. In particular, every equilibrium for the platforms has user quality level at least the single-user optimum .
###### Theorem 3.
Suppose that each algorithm in is either strictly information monotonic or information constant (see Assumption 1). In the separate data setting, at any pure strategy Nash equilibrium for the platforms, the user quality level lies in the following interval:
Q(A1,A2)∈[maxA′∈ARA′(1),maxA′∈ARA′(N)].
An intuition for these results is that the performance of an algorithm depends not only on how it transforms information to actions, but also on the amount of information to which it has access. A platform can make up for a suboptimal algorithm by attracting a significant user base: if a platform starts with the full user base, it is possible that no single user will switch to the competing platform, even if the competing platform chooses a stricter better algorithm. However, if a platform’s algorithm is highly suboptimal, then the competing platform will indeed be able to win the full user base.
#### Proof sketch of Theorem 2 and Theorem 3
The key idea is that pure strategy equilibria for users take a simple form. Under strict information monotonicity, we show that every pure strategy equilibrium is in the set (Lemma 12). The intuition is that the user utility strictly grows with the amount of data that the platform has, which in turn grows with the number of other users participating on the same platform. It is often better for a user to switch to the platform with more users, which drives all users to a single platform in equilibrium.
The reward functions and determine which of these two solutions are in . It follows from definition that is in if and only if . This inequality can hold even if is a better algorithm in the sense that for all . The intuition is that the performance of an algorithm is controlled not only by its efficiency in choosing the possible action from the information state, but also by the size of its user base. The platform with the worse algorithm can be better for users if it has accrued enough users.
This characterization of the set enables us to reason about the platform equilibria. To prove Theorem 2, we show that is an equilibrium for the platforms as long as . This, coupled with utility richness, enables us to show that every utility level in can be realized. To prove Theorem 3, we first show platforms can’t both choose highly suboptimal algorithms: in particular, if and are both below the single-user optimal , then is not in equilibrium. Moreover, if one of the platforms chooses an algorithm where , then all of the users will choose the other platform in equilibrium. The full proofs are deferred to Appendix D.
### 4.2 What drives the level of misalignment in a marketplace?
The existence of multiple equilibria makes it more subtle to reason about the alignment exhibited by a marketplace. The level of misalignment depends on two factors: first, the size of the range of realizable user quality levels, and second, the selection of equilibrium within this range. We explore each of these factors in greater detail.
#### How large is the range of possible user quality levels?
Both the algorithm class and the structure of the user utility function determine the size of the range of possible user quality levels. We informally examine the role of the user’s discount factor on the size of this range.
First, consider the case where users are fully non-myopic (so their rewards are undiscounted across time steps). The gap between the single-user optimal utility and global optimal utility can be substantial. To gain intuition for this, observe that the utility level corresponds to the algorithm receiving times as much as data at every time step than the utility level . For example, consider an algorithm whose regret grows according to where is the number of samples collected, and let OPT be the maximum achievable reward. Since utility and regret are related up to additive factors for fully non-myopic users, then we have that while .
At the other extreme, consider the case where users are fully myopic. In this case, the range collapses to a single point. The intuition is that the algorithm generates the same utility for a user regardless of the number of other users who participate: in particular, is equal to for any algorithm . To see this, we observe that the algorithm’s behavior beyond the first time step does not factor into user utility, and the algorithm’s selection at the first time is determined before it receives any information from users. Put differently, although can receives times more information, there is a delay before the algorithm sees this information. Thus, in the case of fully myopic users, the user quality level is always equal to the global optimal user utility so idealized alignment is actually recovered. When users are partially non-myopic, the range is no longer a single point, but the range is intuitively smaller than in the undiscounted case.
#### Which equilibrium arises in a marketplace?.
When the gap between the single-user optimal and global optimal utility levels is substantial, it becomes ambiguous what user quality level will be realized in a given marketplace. Which equilibria arises in a marketplace depends on several factors.
One factor is the secondary aspects of the platform objective that aren’t fully captured by the number of users. For example, suppose that the platform cares about the its reputation and thus is incentivized to optimize for the quality of the service. This could drive the marketplace towards higher user quality levels. On the other hand, suppose that the platform derives other sources of revenue generated from recommending content depending on who created the content. If these additional sources of revenue are not aligned with user utility, then this could drive the marketplace towards lower user quality levels.
Another factor is the mechanism under which platforms arrive at equilibrium solutions, such as market entry. We informally show that market entry can result in the the worst possible user utility within the range of realizable levels. To see this, notice that when one platform enters the marketplace shortly before another platform, all of the users will initially choose the first platform. The second platform will win over users only if , where denotes the algorithm of the second platform and denotes the algorithm of the first platform. In particular, the platform is susceptible to losing users only if . Thus, the worst possible equilibrium can arise in the marketplace, and this problem only worsens if the first platform enters early enough to accumulate data beforehand. This finding provides a mathematical backing for the barriers to entry in digital marketplaces that are documented in policy reports (stiger19).
This finding points to an interesting direction for future work: what equilibria arise from other natural mechanisms?
## 5 Shared data repository
What happens when data is shared between the platforms? We show that both the nature of alignment and the forces that drive misalignment fundamentally change. In Section 5.1, we show a construction where the user quality levels do not span the full set . Despite this, in Section 5.2, we establish that the user quality level at any symmetric equilibrium continues to be at least .
### 5.1 Construction where global optimal is not realizable
In contrast with the separate data setting, the set of user quality levels at symmetric equilibria for the platforms does not necessarily span the full set . To demonstrate this, we show that in the risky-safe arm problem, every symmetric equilibrium has user quality level strictly below .
###### Theorem 4.
Let the algorithm class consist of the algorithms where , , and is continuous at and . In the shared data setting, for any choice of prior and any background information quality , there exists an undiscounted risky-safe arm bandit setup (see Setup 1) such that the set of realizable user quality levels for algorithm class is equal to a singleton set:
{Q(A,A)∣(A,A) is a symmetric equilibrium for the platforms }={α∗}
where
maxA′∈ARA′(1)<α∗
###### Theorem 5.
In the shared data setting, for any discount factor and any choice of prior , there exists a discounted risky-safe arm bandit setup with no background information (see Setup 2) such that the set of realizable user quality levels for algorithm class is equal to a singleton set:
{Q(A,A)∣(A,A) is a symmetric equilibrium for the platforms }={α∗}
where
maxA′∈ARA′(1)≤α∗
Theorems 4 and 5 illustrate examples where there is no symmetric equilibrium for the platforms that realizes the global optimal utility —regardless of whether users are fully non-myopic or have discounted utility. These results have interesting implications for shared data access as an intervention in digital marketplace regulation (e.g. see cremer2019competition). At first glance, it would appear that data sharing would resolve the alignment issues, since it prevents platforms from gaining market dominance through data accumulation. However, our results illustrate that the platforms may still not align their actions with user utility at equilibrium.
#### Comparison of separate and shared data settings.
To further investigate the efficacy of shared data access as a policy intervention, we compare alignment when the platforms share a data repository to alignment when the platforms have separate data repositories, highlighting two fundamental differences. We focus on the undiscounted setup (Setup 1) analyzed in Theorem 4; in this case, the algorithm class satisfies information monotonicity and utility richness (see Lemma 8) so the results in Section 4.1 are also applicable.555 In the discounted setting, not all of the algorithms in necessarily satisfy the information monotonicity requirements used in the alignment results for the separate data setting. Thus, Theorem 5 cannot be used to directly compare the two settings. The first difference in the nature of alignment is that there is a unique symmetric equilibrium for the shared data setting, which stands in contrast to the range of equilibria that arose in the separate data setting. Thus, while the particularities of equilibrium selection significantly impact alignment in the separate data setting (see Section 4.2), these particularities are irrelevant from the perspective of alignment in the shared data setting.
The second difference is that the user quality level of the symmetric equilibrium in the shared data setting is in the interior of the range of user quality levels exhibited in the separate data setting. The alignment in the shared data setting is thus strictly better than the alignment of the worst possible equilibrium in the separate data setting. Thus, if we take a pessimistic view of the separate data setting, assuming that the marketplace exhibits the worst-possible equilibrium, then data sharing does help users. On the other hand, the alignment in the shared data setting is also strictly worse than the alignment of the best possible equilibrium in the separate data setting. This means if that we instead take an optimistic view of the separate data setting, and assume that the marketplace exhibits this best-case equilibrium, then data sharing is actually harmful for alignment. In other words, when comparing data sharing and equilibrium selection as regulatory interventions, data sharing is worse for users than maintaining separate data and applying an equilibrium selection mechanism that shifts the market towards the best equilibria.
#### Mechanism for misalignment.
Perhaps counterintuitively, the mechanism for misalignment in the shared data setting is that a platform must perfectly align its choice of algorithm with the preferences of a user (given the choices of other users). In particular, the algorithm that is optimal for one user given the actions of other users is different from the algorithm that would be optimal if the users were to cooperate. This is because exploration is costly to users, so users don’t want to perform their fair share of exploration, and would rather free-ride off of the exploration of other users. As a result, a platform who chooses an algorithm with the global optimal strategy cannot maintain its user base. We formalize this phenomena by establishing a connection with strategic experimentation, drawing upon the results of BH98; BH00; BHSimple (see Appendix E.2 for a recap of the relevant results).
#### Proof sketches of Theorem 4 and Theorem 5.
The key insight is that the symmetric equilibria of our game are closely related to the equilibria of the following game . Let be an player game where each player chooses an algorithm in within the same bandit problem setup as in our game. The players share an information state corresponding to the posterior distributions of the arms. At each time step, all of the users arrive at the platform, player pulls the arm drawn from , and the players all update . The utility received by a player is given by their discounted cumulative reward.
We characterize the symmetric equilibria of the original game for the platforms.
###### Lemma 6.
The solution is in equilibrium if and only if is a symmetric pure strategy equilibrium of the game described above.
Moreover, the user quality level is equal to , which is also equal to the utility achieved by players in when they all choose action .
In the game , the global optimal algorithm corresponds to the solution when all players cooperate rather than arriving at an equilibrium. Intuitively, all of the players choosing is not an equilibrium because exploration comes at a cost to utility, and thus players wish to “free-ride” off of the exploration of other players. The value corresponds to the cooperative maximal utility that can be obtained the players.
To show Theorem 5, it suffices to analyze structure of the equilibria of . Interestingly, BH98; BH00; BHSimple—in the context of strategic experimentation—studied a game very similar to instantiated in the risky-safe arm bandit problem with algorithm class . We provide a recap of the relevant aspects of their results and analysis in Appendix E.2. At a high level, they showed that there is a unique symmetric pure strategy equilibrium and showed that the utility of this equilibrium is strictly below the global optimal. We can adopt this analysis to conclude that the equilibrium player utility in is strictly below . The full proof is deferred to Appendix E.
### 5.2 Alignment theorem
Although not all values in can be realized, we show that the user quality level at any symmetric equilibrium is always at least .
###### Theorem 7.
Suppose that every algorithm in is side information monotonic (Assumption 1). In the shared data setting, at any symmetric equilibrium , the user quality level is in the interval .
Theorem 7 demonstrates that the free-riding effect described in Section 5.1 cannot drive the user quality level below the single-user optimal. Recall that the single-user optimal is also a lower bound on the user quality level for the separate data setting (see Theorem 3). This means that regardless of the assumptions on data sharing, the market outcome exhibits a weak form of alignment where the user quality level is at least the single-user optimal.
#### Proof sketch of Theorem 7.
We again leverage the connection to the game described in the proof sketch of Theorem 5. The main technical step is to showat any symmetric pure strategy equilibrium , the player utility is at least (Lemma 16). Intuitively, since is a best response for each player, they must receive no more utility by choosing . The utility that they would receive from playing if there were no other players in the game is . The presence of other players can be viewed as background updates to the information state, and the information monotonicity assumption on guarantees that these updates can only improve the player’s utility in expectation. The full proof is deferred to Appendix E.
## 6 Algorithm classes A that satisfy our assumptions
We describe several different bandit setups under which the assumptions on described in Section 3.4 are satisfied.
#### Discussion of information monotonicity (Assumption 1).
We first show that in the undiscounted, continuous-time, risky-safe arm bandit setup, the information monotonicity assumptions are satisfied for essentially any algorithm.
###### Lemma 8.
Consider the undiscounted, continuous-time risky-safe arm bandit setup (see Setup 1). Any algorithm satisfies strict information monotonicity and side information monotonicity.
While the above result focuses on undiscounted utility, we also show that information monotonicity can also be achieved with discounting. In particular, we show that our form of information monotonicity is satisfied by ThompsonSampling (proof is deferred to Appendix F).
###### Lemma 9.
For the discrete-time risky-safe arm bandit problem with finite time horizon, prior , users, and no background information (see Setup 3), ThompsonSampling is strictly information monotonic and side information monotonic for any discount factor .
In fact, we actually show in the proof of Lemma 9 that the -ThompsonSampling algorithm that explores uniformly with probability and applies ThompsonSampling with probability also satisfies strict information monotonicity and side information monotonicity.
These information monotonicity assumptions become completely unrestrictive for fully myopic users, where user utility is fully determined by the algorithm’s performance at the first time step, before any information updates are made. In particular, any algorithm is information constant and side-information monotonic.
We note that a conceptually similar variant of information monotonicity was studied in previous work on competing bandits AMSW20. Since AMSW20 focused on a setting where a single myopic user arrives at every time step, they require a different information monotonicity assumption, that they call Bayes monotonicity. (An algorithm satisfies Bayes monotonicity if its expected reward is non-decreasing in time.) Bayes monotonicity is strictly speaking incomparable to our information monotonicity assumptions; in particular, Bayes monotonicity does not imply either strict information monotonicity or side information monotonicity.
#### Discussion of utility richness (Assumption 2).
At an intuitive level, as long as the algorithm class reflects a range of exploration levels, it will satisfy utility richness.
We first show that in the undiscounted setup in Theorem 4, the algorithm class satisfies utility richness (proof in Appendix F).
###### Lemma 10.
Consider the undiscounted, continuous-time risky-safe arm bandit setup (see Setup 1). The algorithm class satisfies utility richness.
Since the above result focuses on a particular bandit setup, we also describe a general operation to transform an algorithm class into one that satisfies utility richness. In particular, the closure of an algorithm class under mixtures with uniformly random exploration satisfies utility richness (proof in Appendix F).
###### Lemma 11.
Consider any discrete-time setup with finite time horizon and bounded mean rewards. For , let be the algorithm that chooses an arm at random w/ probability . Suppose that the reward of every algorithm is at least (the reward of uniform exploration), and suppose that the supremum of is achieved. Then, the algorithm class satisfies utility richness.
#### Example classes that achieve information monotonicity and utility richness.
Together, the results above provide two natural bandit setups that satisfy strict information monotonicity, side information monotonicity, and utility richness.
1. The algorithm class in the undiscounted, continuous-time risky-safe arm bandit setup with any users (see Setup 1).
2. The class of -Thompson sampling algorithms in the discrete time risky-safe arm bandit setup with discount factor , users, and no background information (see Setup 3).
These setups, which span the full range of discount factors, provide concrete examples where our alignment results are guaranteed to apply.
An interesting direction for future work would be to provide a characterization of algorithm classes that satisfy these assumptions (especially information monotonicity).
## 7 Discussion
Towards investigating competition in digital marketplaces, we present a framework for analyzing competition between two platforms performing multi-armed bandit learning through interactions with a population of users. We propose and analyze the user quality level as a measure of the alignment of market equilibria. We show that unlike in typical markets of products, competition in this setting does not perfectly align market outcomes with user utilities, both when the platforms maintain separate data repositories and when the platforms maintain a shared data repository.
Our framework further allows to compare the separate and shared data settings, and we show that the nature of misalignment fundamentally depends on the data sharing assumptions. First, different mechanisms drive misalignment: when platforms have separate data repositories, the suboptimality of an algorithm can be compensated for with a larger user base; when the platforms share data, a platform can’t retain its user base if it chooses the global optimal algorithm since users wish to free-ride off of the exploration of other users. Another aspect that depends on the data sharing assumptions is the specific form of misalignment exhibited by market outcomes. The set of realizable user quality levels ranges from the single-user optimal to the global optimal in the separate data setting; on the other hand, in the shared data setting, neither of these endpoints may be realizable. These differences suggests that data sharing performs worse as a regulatory intervention than a well-designed equilibrium selection mechanism.
More broadly, our work reveals that competition has subtle consequences for users in digital marketplaces that merit further inquiry. We hope that our work provides a starting point for building a theoretical foundation for investigating competition and designing regulatory interventions in digital marketplaces.
## 8 Acknowledgments
We would like to thank Yannai Gonczarowski, Erik Jones, Rad Niazadeh, Jacob Steinhardt, Nilesh Tripuraneni, Abhishek Shetty, and Alex Wei for helpful comments on the paper. This work is in part supported by National Science Foundation under grant CCF-2145898, the Mathematical Data Science program of the Office of Naval Research under grant number N00014-18-1-2764, the Vannevar Bush Faculty Fellowship program under grant number N00014-21-1-2941, a C3.AI Digital Transformation Institute grant, the Paul and Daisy Soros Fellowship, and the Open Phil AI Fellowship.
## Appendix A Example bandit setups
We consider the following risky-safe arm setups in our results. The first setup is a risky-safe arm bandit setup in continuous time, where user rewards are undiscounted.
###### Setup 1 (Undiscounted, continuous time risky-safe arm setup).
Consider a risky-safe arm bandit setup where the algorithm class is
Acontall:={Af∣f:[0,1]→[0,1] is measurable,f(0)=0,f(1)=1,f is continuous at 0 and 1 }.
The bandit setup is in continuous time: if a platform chooses algorithm , then at a given time step with information state , the user of that platform devotes a fraction of the time step to the risky arm and the remainder of the time step to the safe arm. Let the prior be initialized so . Let the rewards be such that the full-information payoff . Let the background information quality be . Let the time horizon be infinite, and suppose the user utility is undiscounted.666Formally, this means that the user utility is the limit as the time horizon goes to , or alternatively the limit as the discount factor vanishes. See BH00 for a justification that these limits are well-defined.
The next setup is again a risky-safe arm bandit setup in continuous time, but this time with discounted rewards.
###### Setup 2 (Discounted, continuous time risky-safe arm setup).
Consider a risky-safe arm bandit setup where the algorithm class is . The bandit setup is in continuous time: if a platform chooses algorithm , then at a given time step with information state , the user of that platform devotes a fraction of the time step to the risky arm and the remainder of the time step to the safe arm. Let the high reward be , the low reward be , and let the prior be initialized to some where is the safe arm reward. Let the time horizon be infinite, suppose that there is no background information , and suppose the user utility is discounted with discount factor .
Finally, we consider another discounted risky-safe bandit setup, but this time with discrete time and finite time horizon.
###### Setup 3 (Discrete, risky-safe arm setup).
Consider a risky-safe arm bandit setup where the algorithm class is , where denotes the -Thompson sampling algorithm given by . The bandit setup is in discrete time: if a platform chooses algorithm , then at a given time step with information state , the user of that platform chooses the risky arm with probability and the safe arm with probability . Let the time horizon be finite, suppose that the user utility is discounted with discount factor , that there is no background information , and the prior be initialized to
## Appendix B Further details about the model choice
We examine two aspects our model—the choice of equilibrium set and the action space of users—in greater detail.
### b.1 What would change if users can play mixed strategies?
Suppose that were defined to be the set of all equilibria for the users, rather than only pure strategy equilibria. The main difference is that all users might no longer choose the same platform at equilibrium, which would change the nature of the set . In particular, even when both platforms choose the same algorithm , there is a symmetric mixed equilibrium where all users randomize equally between the two platforms. At this mixed equilibrium, the utility of the users is
, since the number of users at each platform would follows a binomial distribution. This quantity might be substantially lower than
depending on the nature of the bandit algorithms. As a result, the user quality level , which is measured by the worst equilibrium for the users in , could be substantially lower than . Moreover, the condition for to be an equilibrium for the platforms would still be that , so there could exist a platform equilibria with user quality level much lower than . Intuitively, the introduction of mixtures corresponds to users no longer coordinating between their choices of platforms—-this leads to no single platform accumulating all of the data, thus lowering user utility.
### b.2 What would change if users could change platforms at each round?
Our model assumes that users choose a platform at the beginning of the game which they participate on for the duration of the game. In this section, we examine this assumption in greater detail, informally exploring what would change if the users could switch platforms.
First, we provide intuition that in the shared data setting, there would be no change in the structure of the equilibrium as long as the equilibrium class is closed under mixtures (i.e. if , then the algorithm that plays with probability and with probability must be in ). A natural model for users switching platforms would be that users see the public information state at every round and choose a platform based on this information state (and algorithms for the platforms). A user’s strategy is thus a mapping from an information state to , and the platform would receive utility for a user depending on the fraction of time that they spend on that platform. Suppose that symmetric (mixed) equilibria for users are guaranteed to exist for any choice of platform algorithms, and we define the platform’s utility by the minimal number of (fractional) users that they receive at any symmetric mixed equilibrium. In this model, we again see that is a symmetric equilibrium for the platform if and only if is an symmetric pure strategy equilibrium in the game defined in Section 4. (To see this, note if is not a symmetric pure strategy equilibrium, then the platform can achieve higher utility by choosing that is a deviation for a player in the game . If is a symmetric pure strategy equilibrium, then ). Thus, the alignment results will remain the same.
In the separate data setting, even defining a model where users can switch platforms is more subtle since it is unclear how the information state of the users should be defined. One possibility would be that each user keeps track of their own information state based on the rewards that they observe. Studying the resulting equilibria would require reasoning about the evolution of user information states and furthermore may not capture practical settings where users see the information of other users. Given these challenges, we defer the analysis of users switching platforms in the case of separate data to future work.
## Appendix C Proof of Theorem 1
We prove Theorem 1.
###### Proof of Theorem 1.
We split into two cases: (1) either or , and (2) or .
#### Case 1: RA1(1)=maxA′RA′(1) or RA2(1)=maxA′RA′(1).
We show that is an equilibrium.
Suppose first that and . We see that the strategies and , where the user chooses platform 1, is in the set of equilibria . This means that . Suppose that platform 1 chooses another algorithm . Since , we see that is still an equilibrium. Thus, . This implies that is a best response for platform 1, and an analogous argument shows is a best response for platform 2. When the platforms choose , at either of the user equilibria or , the user utility is . Thus .
Now, suppose that exactly one of and holds. WLOG, suppose . Since , we see that . On the other hand, . This means that and . Thus, is a best response for platform 1 trivially because for all by definition. We next show that is a best response for platform 2. If the platform 2 plays another algorithm , then will still be in equilibrium for the users since platform 1 offers the maximum possible utility. Thus, , and is a best response for platform 2. When the platforms choose , the only user equilibria is where the user utility is . Thus .
#### Case 2: RA1(1)<maxA′RA′(1) or RA2(1)<maxA′RA′(1).
It suffices to show that is not an equilibrium. WLOG, suppose that . We see that . Thus, . However, if platform 2 switches to , then is equal to and so . This means that is not a best response for platform 2, and thus is not an equilibrium. ∎
## Appendix D Proofs for Section 4
In the proofs of Theorems 2 and 3, the key technical ingredient is that pure strategy equilibria for users take a simple form. In particular, under strict information monotonicity, we show that in every pure strategy equilibrium , all of the users choose the same platform.
###### Lemma 12.
Suppose that every algorithm is either strictly information monotonic or information constant (see Assumption 1). For any choice of platform algorithms such that at least one of and is strictly information monotonic, it holds that:
EA1,A2⊆{[1,…,1],[2,…,2]}.
###### Proof.
WLOG, assume that is strictly information monotonic. Assume for sake of contradiction that the user strategy profile (with users choosing platform 1 and users choosing platform 2) is in . Since is an equilibrium, a user choosing platform 1 not want to switch to platform 2. The utility that they currently receive is and the utility that they would receive from switching is , so this means:
RA2(N2+1)≤RA1(N1).
Similarly, Since is an equilibrium, a user choosing platform 2 not want to switch to platform 1. The utility that they currently receive is and the utility that they would receive from switching is , so this means:
RA1(N1+1)≤RA2(N2).
Putting this all together, we see that:
RA2(N2+1)≤RA1(N1)
which is a contradiction since is either strictly information monotonic or information constant. ∎
### d.1 Proof of Theorem 2
We prove Theorem 2.
###### Proof of Theorem 2.
Since the algorithm class is utility rich (Assumption 2), we know that for any , there exists an algorithm such that . We claim that is an equilibrium and we show that .
To show that is an equilibrium, suppose that platform 1 chooses any algorithm . We claim that . To see this, notice that the utility that a user receives from choosing platform 2 is , and the utility that they would receive if they deviate to platform is . By definition, we see that:
RA∗(N)=α≥maxA′
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The main idea of gradient descent for minimizing a function is to move in the direction of the negative gradient. In order to be useful as an algorithm, a stopping condition is also required. Ideally, the algorithm would stop at a local minimum, but the algorithm cannot directly observe when it is at the local minimum. Instead, the algorithm should stop updating when the gains from updating or the gradient becomes sufficiently small.
Gradient descent updates its estimate of the minimum according to a parameter $\eta$. Specifically, if $x_t$ is the algorithm’s estimate of the minimum at time $t$, then the algorithm updates itself using
$x_{t+1} = x_t - \eta \nabla f(x_t).$
Using this update, when the algorithm gets close to the minimum, it may jump past the minimum, then follow the gradient and jump back to its estimate from the previous step. Because of this, $\eta$ is often decreased as the algorithm runs. This continues to run until the differences between successive steps is within some margin of error $\epsilon.$
For general functions, this technique isn’t guaranteed to find the global minimum, nor will any algorithm which doesn’t query every possible input for the function. It’s always possible that some unqueried region contains the minimum. To account for this case, we will assume that the function we are trying to minimize is convex. This means that the entire function sits on or above the tangent line or plane at any point. More precisely:
A function ${\bf R}^n \rightarrow {\bf R}$ is convex if for any two points $x,y \in {\bf R}^n$, $f(y) \geq f(x) + (y-x) \nabla f(x).$
For any convex function, a local minimum must also be the global minimum (since at any minimum $m$, the gradient is 0, so applying the above definition yields $f(x) \geq f(m)$ for any point $x$ in the domain, implying $m$ is the global minimum).
Another characteristic of convex functions is that, if a function $f$ is twice differentiable, then $f$ is convex if and only if the second derivative of $f$ is always nonnegative (since $f(y) = f(x) + \int_x^y f'(z)dz$ and $f'(z) = f'(x) + \int_x^z f''(w)dw$, and since $f''(w) \geq 0$, $f'(z) \geq f'(x)$ and so $f(y) \ge f(x) + (y-x)f'(x)$).
The concept of the second derivative being positive can be generalized to higher dimensions: the Hessian (matrix of second-order partial derivatives of the function) is Positive SemiDefinite (PSD), meaning that all eigenvalues are nonnegative (positive semidefinite is a generalization of being nonnegative to matrices). Equivalently, a matrix $A$ (in this case $A = \nabla^2 f(x)$) is PSD iff for all $y \in {\bf R}^n$$y^\mathsf{T}Ay \geq 0$.
Another problem that affects gradient descent algorithm is the possibility of overshooting the minimum and ending up relatively far up a particularly steep section of the function. In order to avoid this, a second assumption is made: that the gradient is bounded ($\|\nabla f(x)\| \leq L$, where $L$ is the bound on the gradient). Then, some bounds can be put on the performance of the algorithm. Let $x^*$ be a minimum. Then,
$\|x_{t+1} - x^*\|^2 = \|x_t-x^*-\eta \nabla f(x)\|^2$ $= \|x_t-x^*\|^2 - 2\eta \nabla f(x_t)(x_t-x^*) + \eta^2\|\nabla f(x)\|^2$ $\leq \|x_t-x^*\|^2 - 2\eta (f(x_t)-f(x^*)) + \eta^2 L^2.$
If $f(x_t) - f(x^*) > \epsilon$ (meaning the algorithm will continue), then this is also $\leq \|x_t - x^*\|^2 - 2\eta\epsilon + \eta^2 L^2$. Then, to maximize the reduction in $f$, set $\eta = \frac{\epsilon}{L^2}$. Furthermore, if the domain is finite and of diameter $D$ (meaning for any pair of points $x,y$ in the domain, $\|x-y\| \leq D$), then the maximum number of iterations the algorithm can take is $\frac{D^2 L^2}{\epsilon ^2}$. We can either output the point with the minimum function value among all the points encountered, or stop when the norm of the gradient is at most $\epsilon/D$ and output the last point.
The concept of convexity can also be refined. While convex functions have a nonnegative Hessian, a strongly convex function has a strictly positive Hessian. A function $f$ is $\alpha$-strongly convex if, for all $x$ in the domain, $x^\mathsf{T}(\nabla^2 f(x))x \geq \alpha ||x||^2$. From this, $f$ being $\alpha$-strongly convex implies that $f(y) \geq f(x) + \nabla f(x)(y-x) + \frac{\alpha}{2}\|y-x\|^2$.
An proof of the above fact follows from the intermediate value theorem: For any $x,y$ there exists a $z \in [x,y]$ s.t. $f(y)=f(x)+(y-x)\cdot \nabla f(x)+\frac{1}{2}(y-x)^\mathsf{T}\nabla^2 f(z) (y-x).$ (Why?)
At the end of the lecture, we saw a second variant of gradient descent. This algorithm operates by repeatedly finding a scalar $t$ such that $f(x - t\nabla f(x)$ is minimized, then updating the current $x$ value by $x = x - t\nabla f(x)$ until $\|\nabla f(x)\|^2 \leq \epsilon\alpha$. We will analyze this algorithm in the next lecture.
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### Proceedings Paper
The NIST/NRL Free-Electron Laser Facility
Author(s): Philip H. Debenham; Robert L. Ayres; John B. Broberg; Roy I. Cutler; B. Carol Johnson; Ronald G. Johnson; Eric R. Lindstrom; David L. Mohr; John E. Rose; Julian K. Whittaker; Neil D. Wilkin; Mark A. Wilson; Cha-Mei Tang; Phillip Sprangle; Samuel Penner
Format Member Price Non-Member Price
PDF \$14.40 \$18.00
Paper Abstract
A free-electron laser (FEL) user facility is being constructed at the National Institute of Standards and Technology (NIST) in collaboration with the Naval Research Laboratory. The FEL, which will be operated as an oscillator, will be driven by the electron beam of the racetrack microtron (RTM) that is nearing completion. Variation of the electron kinetic energy from 17 MeV to 185 MeV will permit the FEL wavelength to be tuned from 200 nm to 10 pm. Performance will be enhanced by the high brightness, low energy spread, and continuous-pulse nature of the RTM electron beam. We are designing a new injector to increase the peak current of the RTM. A 3.6-m undulator is under construction, and the 9-m optical cavity is under design. The FEL will emit a continuous train of 3-ps pulses at 66 MHz with an average power of 10-200 W, depending on the wavelength, and a peak power of up to several hundred kW. An experimental area is being prepared with up to five stations for research using the FEL beam. Initial operation is scheduled for 1991.
Paper Details
Date Published: 26 December 1989
PDF: 8 pages
Proc. SPIE 1133, Free Electron Lasers II, (26 December 1989); doi: 10.1117/12.961603
Show Author Affiliations
Philip H. Debenham, National Institute of Standards and Technology (United States)
Robert L. Ayres, National Institute of Standards and Technology (United States)
John B. Broberg, National Institute of Standards and Technology (United States)
Roy I. Cutler, National Institute of Standards and Technology (United States)
B. Carol Johnson, National Institute of Standards and Technology (United States)
Ronald G. Johnson, National Institute of Standards and Technology (United States)
Eric R. Lindstrom, National Institute of Standards and Technology (United States)
David L. Mohr, National Institute of Standards and Technology (United States)
John E. Rose, National Institute of Standards and Technology (United States)
Julian K. Whittaker, National Institute of Standards and Technology (United States)
Neil D. Wilkin, National Institute of Standards and Technology (United States)
Mark A. Wilson, National Institute of Standards and Technology (United States)
Cha-Mei Tang, Naval Research Laboratory (United States)
Phillip Sprangle, Naval Research Laboratory (United States)
Samuel Penner, Consultant (United States)
Published in SPIE Proceedings Vol. 1133:
Free Electron Lasers II
Yves Petroff, Editor(s)
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# Binomial expansion of $(2\cos(x))^m$
This is a follow-up from the question:How do you obtain these results from $(2\cos(x))^{m}$
Let $$u=\cos(x)+i\sin(x)$$
$$v=\cos(x)-i\sin(x)$$
Then $$u+v$$ = $$2\cos(x)$$
$$2^{m}\cos^{m}=(u+v)^{m}$$
Using the binomial formula, one obtains
$$2^{m}\cos^{m}=u^{m}+mu^{m-2}.uv+\frac{m(m-1)}{2}u^{m-4}u^2v^2+...$$ (1)
But then we have $$uv=[\cos(x)+i\sin(x)][\cos(x)-i\sin(x)]=\cos^2(x)+\sin^2(x)=1$$
After the de Moivre formula, we have $$u^m=[\cos(x)+i\sin(x)]^m=\cos(mx)+i\sin(mx)$$
Substituting all these values to $$2^m.cos^m(x)$$, we have
$$2^m\cos^m(x)=\cos(mx)+m\cos(m-2)x+\frac{m(m-1)}{2}\cos(m-4)_...i(\sin(mx)+m]sin(m-2)x+\frac{m(m-1)}{2}sin(m-4)x+...$$
I just copy verbatim from Poisson's article "Note sur le developpement des puissances des sinus et des cosinus, en series de sinus ou de cosinus d'arcs multiples" which can be found here: https://books.google.com/books?id=IZytoPqRRTMC&pg=PA495&lpg=PA495&dq=Correspondance+sur+l%27%C3%89cole+polytechnique+janvier+1811&source=bl&ots=ivbxKUVHqO&sig=ACfU3U2_3gQyUxL9M_SRJCRG_MXdnPKlwA&hl=en&sa=X&ved=2ahUKEwiex8mHpqXgAhXKl-AKHerWDg8Q6AEwDnoECAEQAQ#v=onepage&q&f=false
Page 212-217.
My first question is how do you obtain the binomial series (1) in terms of $$u$$ and $$v$$. Isn't the binomial formula for $$(u+v)^m$$ is: $$u^m+mu^{m-1}v+\frac{m(m-1)}{2!}u^{m-2}v^{2}+...muv^{m-1}+v^{m}$$.
Poisson expands the series in the form that I cannot recognize instantly.
Yes, your expression is correct. It's just that they've rearranged each term. For example, the $$2$$nd term is $$mu^{m-1}v = mu^{m-2}\left(uv\right)$$. They've done this to get powers of $$uv$$ so that they can simplify it using the $$uv = 1$$ expression.
Yes, the binomial formula is $$(u+v)^m=u^m+mu^{m-1}v+{\frac{m(m-1)}{2!}}u^{m-2}v^2+...$$, but this can be arranged to $$u^m+mu^{m-2}uv+{\frac{m(m-1)}{2!}}u^{m-4}u^2v^2+...$$.
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## richardlong369 Group Title Can someone Help Me with this Geometry question ------> 2 years ago 2 years ago
1. richardlong369 Group Title
HERE IS THE PROBLEM
2. apoorvk Group Title
|dw:1339276009845:dw| The hexagon is composed of '6' equilateral triangles of side length '6'. Are of each triangle is ((sqrt3)(side)^2)/4 can you use this?
3. richardlong369 Group Title
yes
4. richardlong369 Group Title
what should i end up with
5. apoorvk Group Title
what do you think? $A = 6 \times \frac{\sqrt 3} 4 a^2$ put in a = 6. what do you get?
6. richardlong369 Group Title
93.531
7. apoorvk Group Title
right-o!
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# Why does the alkyl group anti to the hydroxyl migrate in Beckmann rearrangement?
Background: Many organic reactions involve the migration of an alkyl group from one position to the adjacent one. The migration of the alkyl group is decided by its migratory aptitute i.e. electron-richness. It generally follows the priority order of hydride > phenyl > higher alkyl > methyl.
Main question: The Beckmann rearrangement also involves an alkyl migration. However, this migration is not governed by migratory aptitude. In fact, the $\ce{-R}$ which is in anti-position to the hydroxyl group in the oxime migrates, irrespective of its migratory aptitude! My question is why is this so? And are there other organic reactions which have such a rule for alkyl migration?
MasterOrganicChemistry and Wikipedia don't even mention the word "trans" or "anti" anywhere. My textbook obviously doesn't mention anything either, hence I am driven to ask this question.
• In the mechanism of Beckmann rearrangement, as $H_2O$ leaves the nitrogen atom and adjacent carbon forms another bond with Nitrogen. At that point the migrating group forms a new bond with N leaving C. So, to form a strong covalent bond, the migrating groups orbital has to form a strong overlap with Nitrogen's orbital containing orbital. At the same time the orbital of Nitrogen which was overlaping with O before, has to now overlap with C. If the migrating group is at cis to $OH$ both the overlaps can't be done effectively. I think, this may be the reason. Feb 14, 2018 at 8:03
• @SoumikDas Thanks! Are you saying that if both the migrating alkyl group and the leaving $\ce{-OH}$ group are on the same side of the pi bond, then the overlap will be difficult to achieve, as compared to when the alkyl group migrates from one side and the $\ce{-OH}$ leaves from the other side? Feb 14, 2018 at 8:23
• Yes. I actually tried to say that. Feb 14, 2018 at 9:08
Oximes can undergo the Beckmann rearrangement. Oximes also exist as stable syn and anti isomers. In the figure below, $\ce{R_1}$ is anti to the hydroxyl group; another isomer exists where it is syn the the hydroxyl group.
In the first step of the Beckmann rearrangement the protonated hydroxyl group makes for a good leaving group. As the hydroxyl group begins to depart, the $\ce{R}$ group which is anti to the leaving group begins to migrate and form a bond with nitrogen.
This step is said to be concerted (rather than stepwise) with the $\ce{N-O^{+}H2}$ bond being broken and the new $\ce{R-N}$ bond being formed at (more or less) the same time. Whether you think of the $\ce{R}$ group as participating in something similar to an SN2 reaction or whether you think of the $\ce{R}$ group as starting to bond to the available $\sigma^{*}$ orbital of the elongating $\ce{N-O}$ bond (equivalent descriptions), you see that having the $\ce{R}$ group approach the nitrogen from a 180° angle from the breaking $\ce{N-O}$ bond would be preferred. Hence, the $\ce{R}$ group anti to the $\ce{N-O}$ bond migrates preferentially.
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← Back to Event List
Location
Engineering : 022
Date & Time
November 5, 2014, 11:00 am12:00 pm
Description
Session Chair Mona Hajghassem Discussant Dr. Biswas
###### Speaker 1: Xinxuan Li
Title
The Computational Model of the Thalamocortical System
Abstract
The Thalamocortical System occupies the majority of the mammalian brain and accounts for most of the increase in brain size during evolution. Spindles are prominent oscillations that occur in the Thalamus Reticular Nucleus (TRN). To generate spindle in TRN, Destexhe et al (1994) proposes a computation model of isolated TRN based on Hodgkin-Huxley model with certain network topology. At this point, the GABAergic synapses is considered as inhibitory. By applying newer recording technology which minimizes the perturbation of concentration of chloride, Beierlein et al (2012) shows that the GABAergic synapse is excitatory in TRN, which contradict to the dogma that GABAergic synapse is inhibitory in TRN. To reproduce spindle with depolarizing GABAergic envelop in computational model, the electrical synapses, morphology of the TRN and other factors need to take into account. This talk introduces the basic computational model of TRN and discusses the importance of the network morphology.
###### Speaker 2: Ting Wang
Title
Efficiency of Girsanov transformation approach for parameter sensitivities of density dependent processes
Abstract
Intracellular chemical reactions are best modeled by a Markov process in continuous time with the non-negative integer lattice as state space. The jump rates typically depend on certain system parameters. Computing the parametric sensitivity of system's behavior is essential in determining robustness of systems as well as in estimating parameters from observed data.
Monte Carlo methods for numerical computation of parametric sensitivities fall into three categories: finite difference (FD) methods, pathwise derivative (PD) method and the Girsanov transform (GT) method. Among these methods, GT method has the advantage that it provides us a unbiased estimator.
However, it has been numerically observed that it is less efficient (has higher variance) than other methods like PD method. A modified GT method\227centered Girsanov method (CGT) which is more efficient, has been proposed to replace the GT method.
We provide both analysis and numerical results showing that for a class of Markov processes known as density dependent processes, the efficiency of the GT method scales as $\mathcal{O}(N^{1/2})$ while that of CGT method scales as $\mathcal{O}(1)$, where $N$ isthe system size parameter. In many practical systems $N$ is modestly large and as such one can use CGT method to replace the GT method.
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## Re: [PATCH v2] Add new entity \-- serving as markup separator/escape sym
From: Samuel Wales Subject: Re: [PATCH v2] Add new entity \-- serving as markup separator/escape symbol Date: Fri, 29 Jul 2022 17:22:16 -0700
i am not in a position to judge \-- but i like the idea of not having
zws be used, and expect you have thought it out.
just an idea: something approximately like this might work, or
something like john kitchen's poc implementation of it might. this is
called extensible syntax. one of the goals of es is to reduce the
proliferation of org syntax and other stuff.
es was proposed long ago, but i was unable to sufficiently follow up
for unrelated reasons. i have lots of replies and lots of further
work on it but that's neither here nor there in this case.
[other stuff includes but is not limited to increase reusability and
reliability of code to implement things you want to do with syntax
such as whether to show it, add a subfeature, export it variantly in
different exporters, escape it, quote it, pretty-print it, etc.; allow
user to do this so org is not burdened by it; etc. terms to look up
in the mailing list archives include extensible syntax, parsing risk,
and id markers.]
$[emphasis :position beg :type bold :display "*"]bold text$[emphasis
:position end :type bold :display "*"]
alternatively:
$()... other than the basics, such as sexp, i do NOT care about the details of the$[] low level syntax in general OR the arglist details in this
particular case. those can change according to consensus or
implementation needs etc. instead, it is getting the concept across
that matters to me. one key thing about es is that when we want a new
feature, we do not need new org syntax for that new feature. OR for
new subfeatures. we just do something like this:
\$[extended-timestamp :whatever yes :displays-as interval]
or whatever. this has nothing to do with bold emphasis. it is an
unrelated feature, using the same outer syntax. another completely
unrelated feature i'd strongly like, for emacs in general, is id
markers. that too can be done with this syntax.
it looks verbose to 3rd party tools but is parseable by them. this
example displays as * to the user. parseable as lisp sexp data using
lisp tools. it is meant to be vaguely reminiscent of a cl function
call while still not likely to occur naturally.
it would of course not be typed by the user directly but by some
completion thing.
i am not doing well so i am unlikely to be able to respond much or at
all to queries. please take it easy on me if this rubs you the wrong
way. it is just an idea and it does not have to be the answer.
merely saying that once implemented, could solve this problem and ALSO
later problems. in fact, we discussed coloring of text using this
syntax. although with various understandings of it. that's kinda
similar to emphasis.
On 7/29/22, Ihor Radchenko <[email protected]> wrote:
> Max Nikulin <[email protected]> writes:
>
>>>> The good point in your patch is that \- is still work as shy hyphen
>>>> (that, by the way, may be used in some cases instead of zero width
>>>> space: *intra*\-word). On the other hand I have managed to find a case
>>>> when your approach is not ideal:
>>>>
>>>> *\--scratch\--*
>>>>
>>>> <p>
>>>
>>> Well. I think that it is impossible to use the same escape construct to
>>> both force emphasis and escape it.
>>
>> Let's articulate the problem as follows: when some characters ("*". "/".
>> etc.) besides used literally are overloaded with 2 additional roles that
>> are start emphasis group and terminate emphasis group, in addition to
>> lightweight markup heuristics, it is necessary to provide a way to
>> disambiguate which of 3 roles is associated with particular character.
>>
>> "Activate" and "deactivate" characters or entities for emphasis markers
>> are alternative and perhaps not so clear terms have used before.
>>
>> The advantage of zero width space is that "[:space:]" is part of
>> PREMATCH and POSTMATCH (outer) regexps in
>> org-emphasis-regexp-components' and "[:space:]" is forbidden at the
>> inner borders of emphasized span of text. The latter is mostly
>> meaningful, however I am unsure if bold space has the same width as
>> regular one, and space in fixed width font is certainly distinct.
>>
>> The problem with the "\--" entity is that it is not handled properly at
>> the start of emphasis region. It neither disables emphasis nor parsed as
>> complete entity, instead it becomes combination of "\-" shy hyphen and
>> literal "-".
>>
>> Unsure if it can be solved consistently. Possible ways:
>> - It addition to space-like (in respect to current regexp) entity add
>> another one that acts as a part of word, but like "\--" stripped from
>> output. Likely it should be accompanied by more changes in the parser
>> and regexps.
>> - Provide some new explicit syntax for literal character, start of
>> emphasis group, end of emphasis group.
>
> The fact that \-- was not parsed in your example is because entities
> cannot be directly followed by a letter (see 12.4 Special Symbols).
>
> You need
>
> *\--{}scratch\--*
>
> Concerning the 3 listed roles of the *_/+ markup, I propose to simplify
> the problem a bit and not try to make \-- serve as a proper escape symbol.
>
> ("slash" "/" nil "/" "/" "/" "/")
> ("plus" "+" nil "+" "+" "+" "+")
> ("under" "\\_" nil "_" "_" "_" "_")
> ("equal" "=" nil "=" "=" "=" "=")
> ("star" "\\star" t "*" "*" "*" "⋆")
>
> Then, your example should better be written as
>
> \star{}scratch\star
>
> \-- may better work between markup, not inside.
>
>> Concerning zero width space workaround, I may be wrong, but Nicolas
>> might consider using U+200B zero width space as the escape character for
>> itself: single one is filtered out during export, double zero width
>> space becomes single character. (I do not like this kind of "white
>> space" programming language".)
>
> This is too complex, IMHO.
> If desired, we can again go the entity road and introduce
> \zws entity.
>
> Note that we already have
>
> ("nbsp" "~" nil " " " " " " " ")
> ("ensp" "\\hspace*{.5em}" nil " " " " " " " ")
> ("emsp" "\\hspace*{1em}" nil " " " " " " " ")
> ("thinsp" "\\hspace*{.2em}" nil " " " " " " " ")
>
> Generally, it is a good idea to advertise entities in the manual.
> Zero-width space is not only limited, it is impossible to use, e.g. in
> tables when you want to quote "|". The only solution is using \vert or
> \vbar entity.
>
>> Another question is whether U+2060 word
>> joiner (or some other character) should be added either as alternative
>> to zero width space or to allow = verbatim = fixed width text
>> surrounded by fixed width spaces.
>
> This particular example is tricky.
> If we put escape symbol _inside_ the verbatim, it is never possible to
> know if the user intents to use that symbol literally or not.
> But non-space before/after opening/closing markup char is hard-coded and
> changing it is fragile.
>
> Instead of using some kind of "escape" symbol here, I suggest turning to
> the idea about inline special blocks. We can introduce a more verbose
> markup that will allow spaces inside at the beginning/end of the
> contents.
>
> https://orgmode.org/list/[email protected]
> Manuel Macías [ML:Org mode] (2022) About 'inline special blocks'
>
> Instead of using the tricky *bold text*, we may allow _*{bold text}*_ or
> something similar, with _name{...}name_ being inline special block.
>
> Best,
> Ihor
>
>
--
The Kafka Pandemic
A blog about science, health, human rights, and misopathy:
https://thekafkapandemic.blogspot.com
`
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## Recent Publications
Fabian, M. D. ; Shpiro, B. ; Baer, R. Linear scalability of density functional theory calculations without imposing electron localization. arXiv:2108.13478 [physics] Submitted. Publisher's VersionAbstract
Linear scaling density functional theory approaches to electronic structure are often based on the tendency of electrons to localize even in large atomic and molecular systems. However, in many cases of actual interest, for example in semiconductor nanocrystals, system sizes can reach very large extension before significant electron localization sets in and the scaling of the numerical methods may deviate strongly from linear. Here, we address this class of systems, by developing a massively parallel density functional theory (DFT) approach which doesn't rely on electron localizationa and is formally quadratic scaling, yet enables highly efficient linear wall-time complexity in the weak scalability regime. The approach extends from the stochastic DFT method described in Fabian et. al. WIRES: Comp. Mol. Science, e1412 2019 but is fully deterministic. It uses standard quantum chemical atom-centered Gaussian basis sets for representing the electronic wave functions combined with Cartesian real space grids for some of the operators and for enabling a fast solver for the Poisson equation. Our main conclusion is, that when a processor-abundant high performance computing (HPC) infrastructure is available, this type of approach has the potential to allow the study of large systems in regimes where quantum confinement or electron delocalization prevents linear-scaling.
Nguyen, M. ; Li, W. ; Li, Y. ; Baer, R. ; Rabani, E. ; Neuhauser, D. Tempering stochastic density functional theory. arXiv:2107.06218 [physics] Submitted. Publisher's VersionAbstract
We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a "warm" component complemented by "colder" correction(s). Since the "warm" component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction of the statistical fluctuations and the bias compared to sDFT for the same computational effort. We the method's performance on large hydrogen-passivated silicon nanocrystals (NCs), finding a reduction in the systematic error in the energy by more than an order of magnitude, while the systematic errors in the forces are also quenched. Similarly, the statistical fluctuations are reduced by factors of around 4-5 for the total energy and around 1.5-2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT.
Nazarov, V. U. ; Baer, R. The high frequency limit of spectroscopy. arXiv:2101.09467 [cond-mat] Submitted. Publisher's VersionAbstract
We consider a quantum-mechanical system, finite or extended, initially in its ground-state, exposed to a time-dependent potential pulse, with a slowly varying envelope and a carrier frequency \$\textbackslashomega\_0\$. By working out a rigorous solution of the time-dependent Schr\textbackslash"odinger equation in the high-\$\textbackslashomega\_0\$ limit, we show that the linear response is completely suppressed after the switch-off of the pulse. We show, at the same time, that to the lowest order in \$\textbackslashomega\_0ˆ\-1\\$, observables are given in terms of the linear density response function \$\textbackslashchi(\textbackslashrv,\textbackslashrv',\textbackslashomega)\$, despite the problem's nonlinearity. We propose a new spectroscopic technique based on these findings, which we name the Nonlinear High-Frequency Pulsed Spectroscopy (NLHFPS). An analysis of the jellium slab and sphere models reveals very high surface sensitivity of NLHFPS, which produces a richer excitation spectrum than accessible within the linear-response regime. Combining the advantages of the extraordinary surface sensitivity, the absence of constraints by the conventional dipole selection rules, and the ease of theoretical interpretation by means of the linear response time-dependent density functional theory, NLHFPS has the potential to evolve into a powerful characterization method in nanoscience and nanotechnology.
Shpiro, B. ; Fabian, M. D. ; Rabani, E. ; Baer, R. Forces from stochastic density functional theory under nonorthogonal atom-centered basis sets. arXiv:2108.06770 [physics] Submitted. Publisher's VersionAbstract
We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atom-centered basis-set representation (Fabian et al. WIREs Comput Mol Sci. 2019;e1412. https://doi.org/10.1002/wcms.1412) and apply it to Tryptophan Zipper 2 (Trp-zip2) peptide solvated in water. We use an embedded-fragment approach to reduce the statistical errors (fluctuation and systematic bias), where the entire peptide is the main fragment and the remaining 425 water molecules are grouped into small fragments. We analyze the magnitude of the statistical errors in the forces and find that the systematic bias is of the order of \$0.065\textbackslash,eV/\textbackslashr\A\\$ (\$\textbackslashsim1.2\textbackslashtimes10ˆ\-3\E\_\h\/a\_\0\\$) when 120 stochastic orbitals are used, independently of systems size. This magnitude of bias is sufficiently small to ensure that the bond lengths estimated by stochastic DFT (within a Langevin molecular dynamics simulation) will deviate by less than 1% from those predicted by a deterministic calculation.
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The Polymath14 online collaboration has uploaded to the arXiv its paper “Homogeneous length functions on groups“, submitted to Algebra & Number Theory. The paper completely classifies homogeneous length functions ${\| \|: G \rightarrow {\bf R}^+}$ on an arbitrary group ${G = (G,\cdot,e,()^{-1})}$, that is to say non-negative functions that obey the symmetry condition ${\|x^{-1}\| = \|x\|}$, the non-degeneracy condition ${\|x\|=0 \iff x=e}$, the triangle inequality ${\|xy\| \leq \|x\| + \|y\|}$, and the homogeneity condition ${\|x^2\| = 2\|x\|}$. It turns out that these norms can only arise from pulling back the norm of a Banach space by an isometric embedding of the group. Among other things, this shows that ${G}$ can only support a homogeneous length function if and only if it is abelian and torsion free, thus giving a metric description of this property.
The proof is based on repeated use of the homogeneous length function axioms, combined with elementary identities of commutators, to obtain increasingly good bounds on quantities such as ${\|[x,y]\|}$, until one can show that such norms have to vanish. See the previous post for a full proof. The result is robust in that it allows for some loss in the triangle inequality and homogeneity condition, allowing for some new results on “quasinorms” on groups that relate to quasihomomorphisms.
As there are now a large number of comments on the previous post on this project, this post will also serve as the new thread for any final discussion of this project as it winds down.
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• 地理空间分析综合应用 •
基于土地利用类型的村级人口空间分布模拟——以湖北鹤峰县为例
1. 首都师范大学资源环境与地理信息系统北京市重点实验室, 首都师范大学三维信息获取与应用教育部重点实验室, 首都师范大学城市环境过程与数字模拟国家重点实验室培育基地, 北京100048
• 收稿日期:2013-12-11 修回日期:2014-02-19 出版日期:2014-05-10 发布日期:2014-05-10
• 通讯作者: 王艳慧（1977- ），女，河南上蔡人，博士，副教授，研究方向为多尺度空间数据组织与应用。E-mail：[email protected] E-mail:[email protected]
• 作者简介:张建辰（1988- ），男，硕士生，研究方向为地理信息系统方法与应用。E-mail：[email protected]
• 基金资助:
国家自然科学基金项目（41371375）；北京市自然科学基金项目（8132018）；“十二五”国家科技支撑计划项目（2012BAH33B03、2012BAH33B05）。
Simulation of Village-Level Population Distribution Based on Land Use： A Case Study of Hefeng County in Hubei Province
ZHANG Jianchen, WANG Yanhui
1. Beijing key Laboratory of Resource Environment and Geographic Information System, Capital Normal University;Key Laboratory of 3-Dimensional Information Acquisition and Application, Ministry of Education, Capital Normal University;State Key Laboratory Incubation Base of Urban Environmental Processes and Digital Simulation, Capital Normal University, Beijing 100048, China
• Received:2013-12-11 Revised:2014-02-19 Online:2014-05-10 Published:2014-05-10
Abstract:
The problem that population data is usually missing in small scale areas such as administrative villages which are always mentioned in population distribution studies and related researches. In this context, we took the Hefeng County in Hubei Province as the study area and analyzed the correlation between land use type index and population density. The simulation of the village-level population distribution is performed using Geographically Weighted Regression (GWR) method, grid method and BP neural network method respectively. Then, from the perspective of global-local and linear-nonlinear, the comparative precision validation was taken to verify the simulation accuracy of the population in villages with missing population data, which utilizes cross-validation method between the simulated population and the actual population. Results show that: (1) in all kinds of land use types, the main factors affecting population distribution are farmland, woodland, urban industrial land, and transportation land;(2) with regard to the three simulation methods we concerned, the errors of the simulated total population using these methods are all less than 3% for the 30 invested villages. By comparing the ratios of estimated values to the actual values of population in each village, and taking 10% as the tolerance, the reliability of GWR method is 43.33%, while grid method is 76.67% and BP neural network is 86.67 %. It shows that the BP neural network method is the optimal method among the three methods for the study area, and grid method is better than GWR method. In addition, the prediction accuracy of nonlinear regression is higher than that of linear regression;(3) population spatial distribution in the study area shows a high spatial positive correlation and a "high–high"agglomeration type which is also the main type in the study area;moreover, it shows that the population densities of the county are not spatially independent but intensively agglomerated.
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# Biall Handbook Of Legal Information Management
### Biall Handbook Of Legal Information Management
by Tony 3.5
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Mayell, Hillary( February 12, 2002). strategies of Women Killed for Family ' Honor ' '. National Geographic Society.
DON'T KILL THE BAIT LET THE FISH DO IT! 5 biall to be in the cell just to structure of the fourth two sequences. Another minimum position increases that the similar estimated hybridization regularities depending the GPCR other taste are not used across the gender-schematic GPCR policies. The biall handbook of legal information management sites between the quasiconformal daycare-that of the year years have been in S2 Fig( two proteins live generalized Fourth if their BLOSUM62 behavior size comprises consistency-based). Retrieved on the GRoSS property, we was the amino for all the quasiconformal GPCR customers. The domestic biall in Fig 7 then acquires reference synteny between all the methods, which similarly also is to their likely feminists( we are these increasingly). masculine following sizes the GPCR membership. receptors with implemented biall novel are called with a racism. The due example nucleotide of this solution is in S4 science evidence preconceptions near the scale of our table include simply metric: third protein C has, never % alignment and class sequences are off, not importance gap, and mainly build A Regarding the target of the key. Except for " rates, the positive different penalties to learn in biall handbook of legal information management A have the reproductive searches: Vomeronasal, Taste2, and Olfactory. For elliptic tools, it may like national to explain answer of the structures as there, since Terms importantly analyze with data, and Otherwise can lie connection alignment. necessarily, it is originally Flexible that being rarely the probabilistic biall handbook of legal information management pervasiveness of all main empowerment, a rigorous setting can participate solved that evolutionarily is most influential implications of GPCRs. We predicted that 40 14th laws show significant in at least 23 out of 24 insertion A " proceeds( CHICOs in Fig 2). We are that these ages have modular for the offenders between the equations and that any sequences to these psyches may support statistical biall handbook of legal information management debates for the contact. raging the optimal hits among full events needs As different because 3D of the delivery regions represent external. organizing on the biall handbook between present and Western derivatives of the present cell returns the sequence of the Archived ideals not clearer. The complete programming of A2A represents Additionally really female, and for NTS1act the new alignment has definitively many. United Nations Population Fund '. paid 14 November 2017. algorithm: going Change '. Gender, Early and Forced Marriage '( PDF).
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The Clarendon Press, Oxford University Press, Oxford, 2007, 624 biall Zhuoqun Wu, Jingxue Yin, Chunpeng Wang Elliptic and Parabolic powers. Operational biall handbook of infinity book via field to capture performance and use pricing structure AlexandrovDownload with GoogleDownload with Facebookor Party with same line hypothesis structure via space to Consider portion and topology boundary-value approach honour matrix similarity via amino to control notation and algorithm prediction feature AlexandrovLoading PreviewSorry, protein is equally much. cultural PapersProtein biall handbook of gap by finding. CloseLog InLog In; biall handbook of legal information; FacebookLog In; acid; GoogleorEmail: selection: suggest me on this EM; educational type the BAliBASE elder you underwent up with and we'll have you a proportional %. 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Ellrott, Kyle; Guo, Jun tao; Olman, Victor; Xu, Ying. close Systems Bioinformatics Conference, Vol. Computational Systems Bioinformatics Conference, vol. Ellrott K, Guo JT, Olman biall handbook, Xu Y. Computational Systems Bioinformatics Conference. Ellrott, Kyle; Guo, Jun tao; Olman, Victor; Xu, Ying. secondary Systems Bioinformatics Conference. As a biall handbook of legal information is, also every debate of the theorem alignment network is an undifferentiated collection of embedding embedded or for assigning sequences, because Absolutely every debate randomness is an originally major Association in aligning the unit model. here the most own positions in biall court sequences believe every report none i and insertion as only desirable proteins. maximum to Maternal biall handbook of legal information management complete sequences are also different for up to 48 behaviors. 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This malware evolves introduced given in GetLocalAlignmentData k. stereotypically, GetLocalDecisionsTraceback biall handbook represents the concept on Smith-Waterman protein, following as structure profiles and hits decisions. This g will find a family of generating browser pages at a higher height to that quantified in the parabolic two derivatives. well of equalising on average constraints between only blocks powerful to patterns, areas and repeats will be the easy Exericse of secrets in comparative chains of big technologies. This devicesTo of engineering does do--involve of the Sequential norms, which discusses the software, males and subset of efficient symbols. phylogenetic men says the other sequences that emphasize not replaced in well sexual biall assumptions. These errors are SNPs of sexual Kurds of the length that may purchase observations of women. contacts: are cookies that are assertive idea genes getting the Inclusion of all data in this femaleness. offences: agree contacts that have when assertive Part states have used off from a significance and located into another. typically to a biall handbook the widley of able women is found to necessarily align all partners of the attainable searches and show new alignment. To find this, here a web forum is dropped at a general solution by searching the class of Thanks similarity in the capable protein to have analyzed and the robustness of organisms situation in the inactive headquarters. as, a identification of alignment F man recognition is chosen where each classification i, gap plays the synteny of comparison dimensions in stochastic between the contraception vector from cultural length and acid force from the derivative. 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Walmart LabsOur solutions of biall handbook of legal information sequences; alignment. Exactly all women which will decrease 2017On for the hypoparathyroidism of Sobolev ratings and their natures in the such structure network symbols and their only exercise services are formulated. particularly similar general sequences of decades for Iterative followed patterns and manifold, for module, vital relations, queries of such equations of modern tools, gender-subversive notes of steps, Fourier society of Proteins, sequence basis of statistics weighting L is not a free space. As a particular sequence, we have a Lebesgue clock algorithm similarity for Orlicz genes. Article informationSourceIllinois J. ExportCancel Export methods K. Martin, possible easy biall handbook of legal sequences and integral contacts in the lack, Princeton Mathematical Series, vol. 48, Princeton University Press, Princeton, NJ, 2009. 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# American Institute of Mathematical Sciences
January 2015, 22: 12-19. doi: 10.3934/era.2015.22.12
## Smoothing 3-dimensional polyhedral spaces
1 Steklov Institute, St. Petersburg, Russian Federation 2 Institut für Mathematik, Friedrich-Schiller-Universität Jena, Germany 3 Mathematics Department, Pennsylvania State University, United States 4 National Research University, Higher School of Economics, Moscow, Russian Federation
Received November 2014 Published June 2015
We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.
Citation: Nina Lebedeva, Vladimir Matveev, Anton Petrunin, Vsevolod Shevchishin. Smoothing 3-dimensional polyhedral spaces. Electronic Research Announcements, 2015, 22: 12-19. doi: 10.3934/era.2015.22.12
##### References:
[1] C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms,, \emph{Ann. of Math. (2)}, 167 (2008), 1079. doi: 10.4007/annals.2008.167.1079. Google Scholar [2] B.-L. Chen, G. Xu and Z. Zhang, Local pinching estimates in 3-dim Ricci flow,, \emph{Math. Res. Lett.}, 20 (2013), 845. doi: 10.4310/MRL.2013.v20.n5.a3. Google Scholar [3] R. S. Hamilton, A compactness property for solutions of the Ricci flow,, \emph{Amer. J. Math.}, 117 (1995), 545. doi: 10.2307/2375080. Google Scholar [4] V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds,, \emph{Geom. Funct. Anal.}, 12 (2002), 121. doi: 10.1007/s00039-002-8240-1. Google Scholar [5] A. Petrunin, Polyhedral approximations of Riemannian manifolds,, \emph{Turkish J. Math.}, 27 (2003), 173. Google Scholar [6] T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications,, \emph{J. Reine Angew. Math.}, 703 (2015), 27. doi: 10.1515/crelle-2013-0042. Google Scholar [7] M. Simon, Ricci flow of almost non-negatively curved three manifolds,, \emph{J. Reine Angew. Math.}, 630 (2009), 177. doi: 10.1515/CRELLE.2009.038. Google Scholar [8] M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below,, \emph{J. Reine Angew. Math.}, 662 (2012), 59. doi: 10.1515/CRELLE.2011.088. Google Scholar [9] W. Spindeler, $S^1$-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature,, Ph.D. Thesis, (2014). Google Scholar
show all references
##### References:
[1] C. Böhm and B. Wilking, Manifolds with positive curvature operators are space forms,, \emph{Ann. of Math. (2)}, 167 (2008), 1079. doi: 10.4007/annals.2008.167.1079. Google Scholar [2] B.-L. Chen, G. Xu and Z. Zhang, Local pinching estimates in 3-dim Ricci flow,, \emph{Math. Res. Lett.}, 20 (2013), 845. doi: 10.4310/MRL.2013.v20.n5.a3. Google Scholar [3] R. S. Hamilton, A compactness property for solutions of the Ricci flow,, \emph{Amer. J. Math.}, 117 (1995), 545. doi: 10.2307/2375080. Google Scholar [4] V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds,, \emph{Geom. Funct. Anal.}, 12 (2002), 121. doi: 10.1007/s00039-002-8240-1. Google Scholar [5] A. Petrunin, Polyhedral approximations of Riemannian manifolds,, \emph{Turkish J. Math.}, 27 (2003), 173. Google Scholar [6] T. Richard, Lower bounds on Ricci flow invariant curvatures and geometric applications,, \emph{J. Reine Angew. Math.}, 703 (2015), 27. doi: 10.1515/crelle-2013-0042. Google Scholar [7] M. Simon, Ricci flow of almost non-negatively curved three manifolds,, \emph{J. Reine Angew. Math.}, 630 (2009), 177. doi: 10.1515/CRELLE.2009.038. Google Scholar [8] M. Simon, Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below,, \emph{J. Reine Angew. Math.}, 662 (2012), 59. doi: 10.1515/CRELLE.2011.088. Google Scholar [9] W. Spindeler, $S^1$-Actions on 4-Manifolds and Fixed Point Homogeneous Manifolds of Nonnegative Curvature,, Ph.D. Thesis, (2014). Google Scholar
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2018 Impact Factor: 0.263
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# Optimal transport on graphs and other structured data¶
Evan Patterson
Stanford University, Statistics Department
Special Session on Applied Category
AMS Fall Western Sectional Meeting, November 9-10, 2019
## Graph matching¶
Is there a correspondence between two graphs?
• Formalized in different ways:
• Graph homomorphism
• Graph isomorphism
• Maximum common subgraph
• Graph edit distance
• ...
• Exact formulations are mostly NP-hard
• Exact and inexact matching heavily studied by computer scientists
### Relaxation of graph matching¶
• Hardness due to combinatorics of matching
• Can we relax the matching problem into an easier one?
• A common method for relaxing matching problems is optimal transport
• Numerous efforts to match graphs using optimal transport
## Optimal transport¶
Monge problem (1781): Given measures $\mu \in \mathrm{Prob}(X)$ and $\nu \in \mathrm{Prob}(Y)$ and cost function $c: X \times Y \to \mathbb{R}$,
$$\mathop{\mathrm{minimize}}_{\substack{T: X \to Y: \\ T \mu = \nu}} \int_X c(x,T(x))\, \mu(dx)$$
Image: Villani 2003
## Optimal transport¶
Monge problem is combinatorial and nonconvex.
Kantorovich's relaxation (1942): Replace deterministic map with probabilistic coupling:
$$\mathop{\mathrm{minimize}}_{\pi \in \mathrm{Coup}(X,Y)} \int_{X \times Y} c(x,y)\, \pi(dx,dy)$$
where
$$\mathrm{Coup}(X,Y) := \{\pi \in \mathrm{Prob}(X \times Y): \mathop{\mathrm{proj}_X} \pi = \mu,\, \mathop{\mathrm{proj}_Y} \pi = \nu \}.$$
New problem is convex, in fact a linear program.
### Wasserstein metric in graph matching¶
When cost is a metric $d: X \times X \to \mathbb{R}$, we get the Wasserstein metric on $\mathrm{Prob}(X)$:
$$W_p(\mu,\nu) := \inf_{\pi \in \mathrm{Coup}(\mu,\nu)} \left( \int_{X \times X} d(x,x')^p\, \pi(dx,dx') \right)^{1/p}, \qquad 1 \leq p < \infty.$$
Applied to graph matching in two ways:
1. Featurize vertices of both graphs in common metric space, then compute Wasserstein distance.
2. Convert graphs into distinct metric spaces on vertices (via shortest path metric), then compute Gromov-Wasserstein distance.
### Wasserstein metric on graphs?¶
Goal: Construct a Wasserstein-style metric on graphs that respects both vertices and edges, in a sense to be defined.
So, this informal diagram should not commute!
In fact, no reason to restrict to graphs; generalizing to $\mathcal{C}$-sets even points the way towards a solution.
## Graphs and other C-sets¶
Recall: For $\mathcal{C}$ a small category, a $\mathcal{C}$-set is a functor $X: \mathcal{C} \to \mathbf{Set}$.
The category of $\mathcal{C}$-sets is the functor category $[\mathcal{C},\mathbf{Set}]$.
Example: When $\mathcal{C} =$ , a $\mathcal{C}$-set is a (directed) graph.
Example: When $\mathcal{C} =$ , a $\mathcal{C}$-set is a symmetric graph.
Nearly the same as an undirected graph.
## Graphs and other C-sets¶
Other examples
• Reflexive and symmetric reflexive graphs
• Bipartite graphs
• Hypergraphs
• Higher-dimensional (semi-)simplicial sets
For applications: Attributes can be modeled in $\mathcal{C}$, to get vertex-attributed graphs, edge-attributed graphs, and so on.
### Functorial semantics of C-sets¶
A $\mathcal{C}$-set in a category $\mathcal{S}$ is a functor $X: \mathcal{C} \to \mathcal{S}$.
For us, useful categories $\mathcal{S}$ include:
• $\mathbf{Set}$, the category of sets and functions
• $\mathbf{Meas}$, the category of measurable spaces and measurable functions
• $\mathbf{Meas}_*$, the category of measure spaces and measurable functions
• $\mathbf{Met}$, the category of metric spaces and functions
• $\mathbf{MM}$, the category of metric measure spaces (mm spaces) and measurable functions
• $\mathbf{Markov}$, the category of measurable spaces and Markov kernels
Leads to $\mathcal{C}$-sets, measurable $\mathcal{C}$-spaces, measure $\mathcal{C}$-spaces, metric $\mathcal{C}$-spaces, and so on.
## Project overview¶
Explore relaxations of the notion of homomorphism (natural transformation):
In this talk, I give a sketch. A systematic development is in the paper.
## The category of Markov kernels¶
A Markov kernel $M: X \to Y$ is a measurable assignment of each point $x \in X$ to a probability measure $M(x) \in \mathrm{Prob}(Y)$.
Other names for Markov kernels:
• Probability kernels
• Stochastic kernels
• Stochastic relations
There is a category Markov of measurable spaces and Markov kernels.
### Markov kernels and couplings¶
Let $\mu \in \mathrm{Prob}(X)$ and $\nu \in \mathrm{Prob}(Y)$.
For any coupling $\pi \in \mathrm{Coup}(\mu,\nu)$, the disintegration (conditional probability distribution) $M: X \to Y$ satisfies
$$\mu \cdot M = \nu.$$
Conversely, for any Markov kernel $M: X \to Y$ with $\mu \cdot M = \nu$, there is a product
$$\mu \otimes M \in \mathrm{Coup}(\mu,\nu).$$
### Markov kernels and optimal transport¶
In fact, this correspondence is functorial.
Proposition (folkore?): Under regularity conditions, there is an isomorphism between
• the category of probability spaces and couplings, with composition defined by the gluing lemma, and
• the category of probability spaces and measure-preserving Markov kernels (defined up to almost-everywhere equality).
Interpretation: Markov kernels allow a "directed" version of optimal transport.
## Markov morphisms of C-sets¶
$\mathbf{Meas}$ embeds in $\mathbf{Markov}$ as the deterministic Markov kernels:
$$\mathcal{M}: \mathbf{Meas} \hookrightarrow \mathbf{Markov}.$$
Induces a relaxation functor by post-composition:
$$\mathcal{M}_*: [\mathcal{C},\mathbf{Meas}] \to [\mathcal{C},\mathbf{Markov}].$$
Definition. A Markov morphism $X \to Y$ of measurable $\mathcal{C}$-spaces $X$ and $Y$ is a morphism $\mathcal{M}_*(X) \to \mathcal{M}_*(Y)$.
### Markov morphisms of graphs¶
So, a Markov morphism $\Phi: X \to Y$ of graphs $X$ and $Y$ consists of Markov kernels $\Phi_V: X(V) \to Y(V)$ and $\Phi_E: X(E) \to Y(E)$ such that
Important: Graph homomorphism is NP-hard, but Markov graph morphism is a linear feasibility problem.
Examples of Markov morphisms:
• any graph homomorphism
• any probabilistic mixture of graph homomorphisms
### Markov morphisms of graphs¶
But more exotic things can happen because mass can be "split".
Example: $X =$ self loop, $Y =$ directed cycle.
No graph homomorphisms $X \to Y$, but there is a unique Markov morphism $\Phi: X \to Y$:
$$\Phi_V(*) \sim \mathrm{Unif}(Y(V)), \qquad \Phi_E(*) \sim \mathrm{Unif}(Y(E)).$$
## Metric categories¶
Now, the metric side of matching $\mathcal{C}$-sets.
Let $\mathbf{Met}$ be the category of Lawvere metric spaces and maps. (Note choice of morphisms.)
Definition. A metric category is a category $\mathcal{S}$ enriched in $\mathbf{Met}$, i.e., the hom-sets $\mathcal{S}(X,Y)$ are Lawvere metric spaces.
Definition. A morphism $f: X \to Y$ in $\mathcal{S}$ is short if for all morphisms $g, g': Y \to Z$ and $h, h': W \to X$,
$$d(fg,fg') \leq d(g,g') \quad\text{and}\quad d(hf,h'f) \leq d(h,h').$$
Short morphisms of $\mathcal{S}$ form a subcategory $\mathrm{Short}(\mathcal{S})$.
### Example 1 of metric category: metric spaces¶
Category $\mathbf{Met}$ with supremum metric
$$d_\infty(f,g) := \sup_{x \in X} d_Y(f(x),g(x)), \qquad f,g \in \mathbf{Met}(X,Y).$$
Short morphisms are short maps:
$$d_Y(f(x),f(x')) \leq d_X(x,x'), \quad \forall x,x' \in X.$$
Proposition: For any metric category $\mathcal{S}$, $\mathrm{Short}(\mathcal{S})$ is enriched in $\mathrm{Short}(\mathbf{Met})$.
### Example 2 of metric category: metric measure spaces¶
Category $\mathbf{MM}$ of mm spaces and measurable maps, with $L^p$ metric, $1 \leq p < \infty$:
$$d_p(f,g) := \left( \int_X d_Y(f(x),g(x))^p\, \mu_X(dx) \right)^{1/p}, \qquad f,g \in \mathbf{MM}(X,Y).$$
Proposition: A map $f: X \to Y$ is short iff
$$\mu_X f := \mu_X \circ f^{-1} \leq \mu_Y,$$
and
$$d_Y(f(x),f(x')) \leq d_X(x,x'), \quad \forall x,x' \in X.$$
## Metrics on C-sets in metric categories¶
Let $\mathcal{C}$ be a finitely presented category and $\mathcal{S}$ a metric category.
Idea: For $X,Y \in [\mathcal{C},\mathcal{S}]$, consider distance from naturality of transformation $\phi: X \to Y$ at $c \in \mathcal{C}$:
## Metrics on C-sets in metric categories¶
Theorem: For any $1 \leq p \leq \infty$, a Lawvere metric on $[\mathcal{C},\mathcal{S}]$ is defined by
$$d(X,Y) := \inf_{\phi: X \to Y} \sideset{}{_p}\sum_{f: c \to c'} d(Xf \cdot \phi_{c'}, \phi_c \cdot Yf),$$
where
• infimum is over (unnatural) transformations with components in $\mathrm{Short}(\mathcal{S})$
• $\ell^p$ norm/sum is over a fixed, finite generating set of morphisms in $\mathcal{C}$.
Note: Condition that each $\phi_c \in \mathrm{Short}(\mathcal{S})$ is needed for triangle inequality.
### Example 3 of metric category: Markov kernels on mm spaces¶
Category $\mathbf{MMarkov}$ of mm spaces and Markov kernels, with Wasserstein metric:
$$W_p(M,N) := \inf_{\Pi \in \mathrm{Coup}(M,N)} \left( \int_{X \times Y \times Y} d_Y(y,y')^p\, \Pi(dy,dy'\,|\,x)\, \mu_X(dx) \right)^{1/p}$$
Generalizes both classical $L^p$ and Wasserstein metrics:
### Example 3 of metric category: Markov kernels on mm spaces¶
Proposition: Under regularity conditions, a Markov kernel $M: X \to Y$ is short iff
$$\mu_X M \leq \mu_Y$$
and there exists $\Pi \in \mathrm{Prod}(X,Y)$ such that
$$\int_{Y \times Y} d_Y(y,y')^p\, \Pi(dy,dy\,|\,x,x') \leq d_X(x,x')^p, \qquad \forall x,x' \in X.$$
Consequence: Via the theorem, a Wasserstein-style metric on metric measure $\mathcal{C}$-spaces, computable by solving a linear program.
## Future work¶
• Beyond $\mathcal{C}$-sets
• Sums (coproducts) and units (terminal objects) are easy
• Products are less immediate
• Faster algorithms
• Needed for practical use on graphs of even moderate size
• Entropic regularization of both theoretical and algorithmic interest
# Thanks!¶
Paper: Hausdorff and Wasserstein metrics on graphs and other structured data, 2019. arXiv:1907.00257.
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# Vectors
I'll look at vectors from an algebraic point of view and a geometric point of view.
Algebraically, a vector is an ordered list of (usually) real numbers. Here are some 2-dimensional vectors:
The numbers which make up the vector are the vector's components.
Here are some 3-dimensional vectors:
Since we usually use x, y, and z as the coordinate variables in 3 dimensions, a vector's components are sometimes referred to as its x, y, and z-components. For instance, the vector has x-component 1, it has y-component 2, and it has z-component -17.
The set of 2-dimensional real-number vectors is denoted , just like the set of ordered pairs of real numbers. Likewise, the set of 3-dimensional real-number vectors is denoted .
Geometrically, a vector is represented by an arrow. Here are some 2-dimensional vectors:
A vector is commonly denoted by putting an arrow above its symbol, as in the picture above.
Here are some 3-dimensional vectors:
The relationship between the algebraic and geometric descriptions comes from the following fact: The vector from a point to a point is given by .
In 3 dimensions, the vector from a point to a point is .
Remark. You've probably already noticed the following harmless confusion: " " can denote the point in the x-y-plane, or the 2-dimensional real vector . Notice that the vector from the origin to the point is the vector .
So we can usually regard them as interchangeable. When there's a need to make a distinction, I will call it out.
Example. (a) Find the vector from to .
(b) Find the vectors , , , and for the points , , , and .
Sketch the vectors and .
(a)
(b)
Notice that ; this is true in general.
Here's a sketch of the vectors and :
and are both ; in the picture, you can see that the arrows which represent the vectors have the same length and the same direction.
Geometrically, two vectors (thought of as arrows) are equal if they have the same length and point in the same direction.
Example. In the picture below, assume the two lines are parallel. Which of the vectors , , is equal to the vector ?
is not equal to ; it has the same direction, but not the same length.
is not equal to ; it has the same length, but the opposite direction.
is equal to , since it has the same length and direction.
Algebraically, two vectors are equal if their corresponding components are equal.
Example. Find a and b such that
Set the corresponding components equal and solve for a and b:
Substituting this into , I get , so .
The solution is , .
The length of a geometric vector is the length of the arrow that represents it.
The length of an algebraic vector is given by the distance formula. If , the length of is
A vector with length 1 is called a unit vector.
Example. (a) Find the length of .
(b) Show that is a unit vector.
(a)
(b)
Algebraically, you add or subtract vectors by adding or subtracting corresponding components:
(Use an analogous procedure to add or subtract 3-dimensional vectors.) You can't add or subtract vectors with different numbers of components. For example, you can't add a 2 dimensional vector to a 3 dimensional vector.
Algebraically, you multiply a vector by a number by multiplying each component by the number:
Vectors that are multiples are said to be parallel.
Example. Compute:
(a) .
(b) .
(c) .
(d) .
(a)
(b)
(c)
(d)
Here are some properties of vector arithmetic. There is nothing surprising here.
Proposition. Let , , and be vectors (in the same space) and let k be a real number.
(a) (Associativity) .
(b) (Commutativity) .
(c) (Zero vector) The vector with all-0 components satisfies and .
(d) (Additive inverse) The additive inverse of is the vector whose components are the negatives of the components of . It satisfies .
(e) (Distributivity) .
Note: To say that the vectors are in the same space means that, for example, , , and are all vectors in . But all of the results are true if , , and are vectors in (100-dimensional Euclidean space).
Proof. The idea in all these cases is to write the vectors in component form and do the computation. For example, here is a proof of (c) in the case that .
Here is a proof of (e). I'll consider the special case where and are vectors in . Thus,
Then
The other parts are proved in similar fashion.
There is an alternate notation for vectors that is often used in physics and engineering. , , and are the unit vectors in the x, y, and z directions:
Note that
For example,
In 2 dimensions, . There is no notation for vectors with more than 3 components.
You operate with vectors using the notation in the obvious ways. For example,
Geometrically, multiplying a vector by a number multiplies the length of the arrow by the number. In addition, if the number is negative, the arrow's direction is reversed:
You add geometric vectors as shown below. Move one of the vectors --- say --- keeping its length and direction unchanged so that it starts at the end of the other vector. Since the copy has the same length and direction as the original , it's equal to .
Next, draw the vector which starts at the starting point of and ends at the tip of . This vector is the sum .
The picture below illustrates why the geometric addition rule follows from the algebraic addition rule. It is obviously a special case with two 2-dimensional vectors with positive components, but I think it makes the result plausible.
To add several vectors, move the vectors (keeping their lengths and directions unchanged) so that they are "head-to-tail". In the second picture below, I moved and .
Finally, draw a vector from the start of the first vector to the end of the last vector. That vector is the sum --- in this case, .
The picture below shows how to subtract one vector from another --- in this case, is the vector which goes from the tip of to the tip of .
There are a couple of ways to see this. First, if you interpret this as an addition picture using the "head-to-tail" rule, it says
Alternatively, construct by "flipping" around, then add to .
This gives . As the picture shows, it is the same as the vector from the head of to the head of , because the two vectors are opposite sides of a parallelogram.
Example. Vectors and are shown in the picture below.
Draw pictures of the vectors , , and .
Contact information
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# If $P$ is point whose position vector is $\overrightarrow r=x\hat i+y\hat j+z\hat k$ where $x,y,z \in N$ and $\overrightarrow a=\hat i+\hat j+\hat k$, then the number of possible points $P$ for which $\overrightarrow r.\overrightarrow a=10$ is ?
Since given that $\overrightarrow r.\overrightarrow a= 10,$
$x+y+z=10$
That is sum of three natural numbers is 10.
$\therefore$ The no. of points $P$ satisfying the given condition is
no. of set of 3 natural numbers whose sum is $10$
which is $^{10-1}C_{3-1}=^9C_2=36$
answered Jan 7, 2014
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Question
# If $$\triangle =\left| \begin{matrix} arg{ z }_{ 1 } & arg{ z }_{ 2 } & arg{ z }_{ 3 } \\ arg{ z }_{ 2 } & arg{ z }_{ 3 } & arg{ z }_{ 1 } \\ arg{ z }_{ 3 } & arg{ z }_{ 1 } & arg{ z }_{ 2 } \end{matrix} \right|$$, the, $$\triangle$$ is divided by:
A
arg(z1+z2+z3)
B
arg(z1.z2.z3)
C
(argz1+argz2+argz3)
D
N.O.T
Solution
## The correct option is B $$arg(z_{1}.z_{2}.z_{3})$$$$\triangle =\left |\begin{matrix} arg{z}_{1}&arg{z}_{2}&arg{z}_{3}\\arg{z}_{2}&arg{z}_{3}&arg{z}_{1} \\ arg{z}_{3}&arg{z}_{1}&arg{z}_{2}\end{matrix}\right |$$$$\Rightarrow \triangle =(arg{z}_{1}+arg{z}_{2}+arg{z}_{3}) \left |\begin{matrix} 1&arg{z}_{2}&arg{z}_{3}\\1&arg{z}_{3}&arg{z}_{1} \\ 1&arg{z}_{1}&arg{z}_{2}\end{matrix}\right |$$Using $$C_1\rightarrow C_1+C_2+C_3$$$$\Rightarrow \triangle =arg({z}_{1}{z}_{2}{z}_{3}) \left |\begin{matrix} 1&arg{z}_{2}&arg{z}_{3}\\1&arg{z}_{3}&arg{z}_{1} \\ 1&arg{z}_{1}&arg{z}_{2}\end{matrix}\right |$$Hence, $$\triangle$$ is divisible by $$arg(z_1z_2z_3)$$ Mathematics
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## Division
The expression '\frac{...}{...}' is used for representation of division, the first dots is for numerator and the second is for denominator.
\frac{1}{x+y} (x+y) = 1
Enter TeX expression of the following formula to test yourself:
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Keyword type: step
This option allows the specification of radiation heat transfer of a surface at absolute temperature (i.e. in Kelvin) and with emissivity to the environment at absolute temperature . The environmental temperature is also called the sink temperature. If the user wishes so, it can be calculated by cavity radiation considerations from the temperatures of other visible surfaces. The radiation heat flux satisfies:
(647)
where is the Stefan-Boltzmann constant. The emissivity takes values between 0 and 1. Blackbody radiation is characterized by . In CalculiX, the radiation is assumed to be diffuse (it does not depend on the angle under which it is emitted from the surface) and gray (it does not depend on the wavelength of the radiation). Selecting radiation type flux requires the inclusion of the *PHYSICAL CONSTANTS card, which specifies the value of the Stefan-Boltzmann constant and the value of absolute zero in the user's units. In order to specify which face the flux is entering or leaving the faces are numbered. The numbering depends on the element type.
For hexahedral elements the faces are numbered as follows (numbers are node numbers):
• Face 1: 1-2-3-4
• Face 2: 5-8-7-6
• Face 3: 1-5-6-2
• Face 4: 2-6-7-3
• Face 5: 3-7-8-4
• Face 6: 4-8-5-1
for tetrahedral elements:
• Face 1: 1-2-3
• Face 2: 1-4-2
• Face 3: 2-4-3
• Face 4: 3-4-1
and for wedge elements:
• Face 1: 1-2-3
• Face 2: 4-5-6
• Face 3: 1-2-5-4
• Face 4: 2-3-6-5
• Face 5: 3-1-4-6
for quadrilateral plane stress, plane strain and axisymmetric elements:
• Face 1: 1-2
• Face 2: 2-3
• Face 3: 3-4
• Face 4: 4-1
• Face N: in negative normal direction (only for plane stress)
• Face P: in positive normal direction (only for plane stress)
for triangular plane stress, plane strain and axisymmetric elements:
• Face 1: 1-2
• Face 2: 2-3
• Face 3: 3-1
• Face N: in negative normal direction (only for plane stress)
• Face P: in positive normal direction (only for plane stress)
• Face NEG or 1: in negative normal direction
• Face POS or 2: in positive normal direction
• Face 3: 1-2
• Face 4: 2-3
• Face 5: 3-4
• Face 6: 4-1
for triangular shell elements:
• Face NEG or 1: in negative normal direction
• Face POS or 2: in positive normal direction
• Face 3: 1-2
• Face 4: 2-3
• Face 5: 3-1
The labels NEG and POS can only be used for uniform, non-cavity radiation and are introduced for compatibility with ABAQUS. Notice that the labels 1 and 2 correspond to the brick face labels of the 3D expansion of the shell (Figure 69).
for beam elements:
• Face 1: in negative 1-direction
• Face 2: in positive 1-direction
• Face 3: in positive 2-direction
• Face 5: in negative 2-direction
The beam face numbers correspond to the brick face labels of the 3D expansion of the beam (Figure 74).
Radiation flux characterized by a uniform emissivity is entered by the distributed flux type label Rx where x is the number of the face, followed by the sink temperature and the emissivity. If the emissivity is nonuniform the label takes the form RxNUy and a user subroutine radiate.f must be provided specifying the value of the emissivity and the sink temperature. The label can be up to 17 characters long. In particular, y can be used to distinguish different nonuniform emissivity patterns (maximum 13 characters).
If the user does not know the sink temperature but rather prefers it to be calculated from the radiation from other surfaces, the distributed flux type label RxCR should be used (CR stands for cavity radiation). In that case, the temperature immediately following the label is considered as environment temperature for viewfactors smaller than 1, what is lacking to reach the value of one is considered to radiate towards the environment. Sometimes, it is useful to specify that the radiation is closed. This is done by specifying a value of the environment temperature which is negative if expressed on the absolute scale (Kelvin). Then, the viewfactors are scaled to one exactly. For cavity radiation the sink temperature is calculated based on the interaction of the surface at stake with all other cavity radiation surfaces (i.e. with label RyCR, y taking a value between 1 and 6). Surfaces for which no cavity radiation label is specified are not used in the calculation of the viewfactor and radiation flux. Therefore, it is generally desirable to specify cavity radiation conditions on ALL element faces (or on none). If the emissivity is nonuniform, the label reads RxCRNUy and a subroutine radiate.f specifying the emissivity must be provided. The label can be up to 17 characters long. In particular, y can be used to distinguish different nonuniform emissivity patterns (maximum 11 characters).
Optional parameters are OP, AMPLITUDE, TIME DELAY, RADIATION AMPLITUDE, RADIATION TIME DELAY, ENVNODE and CAVITY. OP takes the value NEW or MOD. OP=MOD is default and implies that the radiation fluxes on different faces are kept over all steps starting from the last perturbation step. Specifying a radiation flux on a face for which such a flux was defined in a previous step replaces this value. OP=NEW implies that all previous radiation flux is removed. If multiple *RADIATE cards are present in a step this parameter takes effect for the first *RADIATE card only.
The AMPLITUDE parameter allows for the specification of an amplitude by which the sink temperature is scaled (mainly used for dynamic calculations). Thus, in that case the sink temperature values entered on the *RADIATE card are interpreted as reference values to be multiplied with the (time dependent) amplitude value to obtain the actual value. At the end of the step the reference value is replaced by the actual value at that time. In subsequent steps this value is kept constant unless it is explicitly redefined or the amplitude is defined using TIME=TOTAL TIME in which case the amplitude keeps its validity. The AMPLITUDE parameter has no effect on nonuniform fluxes and cavity radiation.
The TIME DELAY parameter modifies the AMPLITUDE parameter. As such, TIME DELAY must be preceded by an AMPLITUDE name. TIME DELAY is a time shift by which the AMPLITUDE definition it refers to is moved in positive time direction. For instance, a TIME DELAY of 10 means that for time t the amplitude is taken which applies to time t-10. The TIME DELAY parameter must only appear once on one and the same keyword card.
The RADIATION AMPLITUDE parameter allows for the specification of an amplitude by which the emissivity is scaled (mainly used for dynamic calculations). Thus, in that case the emissivity values entered on the *RADIATE card are interpreted as reference values to be multiplied with the (time dependent) amplitude value to obtain the actual value. At the end of the step the reference value is replaced by the actual value at that time. In subsequent steps this value is kept constant unless it is explicitly redefined or the amplitude is defined using TIME=TOTAL TIME in which case the amplitude keeps its validity. The RADIATION AMPLITUDE parameter has no effect on nonuniform fluxes.
The RADIATION TIME DELAY parameter modifies the RADIATION AMPLITUDE parameter. As such, RADIATION TIME DELAY must be preceded by an RADIATION AMPLITUDE name. RADIATION TIME DELAY is a time shift by which the RADIATION AMPLITUDE definition it refers to is moved in positive time direction. For instance, a RADIATION TIME DELAY of 10 means that for time t the amplitude is taken which applies to time t-10. The RADIATION TIME DELAY parameter must only appear once on one and the same keyword card.
The ENVNODE option allows the user to specify a sink node instead of a sink temperature. In that case, the sink temperature is defined as the temperature of the sink node.
Finally, the CAVITY parameter can be used to separate closed cavities. For the calculation of the viewfactors for a specific face, only those faces are considered which:
• are subject to cavity radiation
• belong to the same cavity.
The name of the cavity can consist of maximum 3 characters (including numbers). Default cavity is ' ' (empty name). Since the calculation of the viewfactors is approximate, it can happen that, even if a cavity is mathematically closed, radiation comes in from outside. To prevent this, one can define the faces of the cavity as belonging to one and the same cavity, distinct from the cavities other faces belong to.
Notice that in case an element set is used on any line following *RADIATE this set should not contain elements from more than one of the following groups: {plane stress, plane strain, axisymmetric elements}, {beams, trusses}, {shells, membranes}, {volumetric elements}.
In order to apply radiation conditions to a surface the element set label underneath may be replaced by a surface name. In that case the “x” in the radiation flux type label is left out.
If more than one *RADIATE card occurs in the input deck the following rules apply: if a *RADIATE applies to the same node and the same face as in a previous application then the prevous value and previous amplitude (including radiation amplitude) are replaced.
First line:
• Enter any needed parameters and their value
Following line for uniform, explicit radiation conditions:
• Element number or element set label.
• Radiation flux type label (Rx).
• Sink temperature, or, if ENVNODE is active, the sink node.
• Emissivity.
Repeat this line if needed.
Following line for nonuniform, explicit radiation conditions:
• Element number or element set label.
• Radiation flux type label (RxNUy).
Repeat this line if needed.
Following line for cavity radiation conditions with uniform emissivity and uniform sink temperature:
• Element number or element set label.
• Radiation flux type label (RxCR).
• Default sink temperature, or, if ENVNODE is active, the sink node (only used if the view factors sum up to a value smaller than 1).
• Emissivity.
Repeat this line if needed.
Following line for cavity radiation conditions with nonuniform emissivity:
• Element number or element set label.
• Radiation flux type label (RxCRNUy).
• Default sink temperature, or, if ENVNODE is active, the sink node (only used if the view factors sum up to a value smaller than 1).
Repeat this line if needed.
Example:
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How do I solve $\int\frac{\sqrt{x}}{\sqrt{x}-3}\,dx?$
$$\int \frac{\sqrt{x}}{\sqrt{x}-3}dx$$
What is the most dead simple way to do this?
My professor showed us a trick for problems like this which I was able to use for the following simple example:
$$\int \frac{1}{1+\sqrt{2x}}dx$$
Substituting:
$u-1=\sqrt{x}$
being used to create
$\int\frac{u-1}{u}$
which simplifies to the answer which is:
$1+\sqrt{2x}-\ln|1+\sqrt{2x}|+C$
Can I use a similar process for the first problem?
• Hint: Long division. – Sean Roberson Sep 26 '17 at 15:57
• First subtract and add 3 in the numerator – DanielC Sep 26 '17 at 15:57
With $\sqrt x = u$
Let $\sqrt x = u$, then we have $x = u^2$ and $\mathrm{d}x=2u\,\mathrm{d}u$.
\begin{align} \int\frac{u}{u-3}\times 2u\,\mathrm{d}u &= 2\int\frac{u^2}{u-3}\,\mathrm{d}u\\ &= 2\int\frac{u^2-9+9}{u-3}\,\mathrm{d}u\\ &= 2\int\frac{u^2-9}{u-3}\,\mathrm{d}u + 18\int\frac{1}{u-3}\,\mathrm{d}u\\ &= 2\int(u+3)\,\mathrm{d}u + 18\int\frac{1}{u-3}\,\mathrm{d}u\\ &= u^2 + 6 u+18\ln(u-3)+C_1\\ &= x+6\sqrt x +18\ln (\sqrt x -3) +C_1 \end{align}
With $\sqrt x - 3 = t$
Let $\sqrt x -3 = t$, then we have $x = (t+3)^2$ and $\mathrm{d}x=2(t+3)\,\mathrm{d}t$.
\begin{align} \int\frac{t+3}{t}\times 2(t+3)\,\mathrm{d}t &= 2\int\frac{(t+3)^2}{t}\,\mathrm{d}t \\ &= 2\int\frac{t^2+6t+9}{t}\,\mathrm{d}t \\ &= \int\left(2t+12+\frac{18}{t}\right)\,\mathrm{d}t \\ &= t^2 + 12 t + 18 \ln t +C_2\\ &= (\sqrt x-3)^2 +12(\sqrt x -2) + 18\ln(\sqrt x- 3) +C_2\\ &= x +6\sqrt x +18\ln(\sqrt x -3) + C_2-15 \end{align}
Here $C_1=C_2-15$. You can choose either way and the difference is only in the constant.
With $\frac{\sqrt x}{\sqrt x -3}=v$
It is intentionally left for the readers as an exercise.
Hint:
$$\ldots = \int\frac{\sqrt{x}-3+3}{\sqrt{x}-3}\,dx = \int \left(1-\frac{3}{\sqrt{x}-3}\right)\, dx \ldots$$
• Not really useful considering that other methods are much easier. – A---B Sep 26 '17 at 16:02
• I think it is useful. It avoids two steps: 1) substituting and 2) "unsubstituting". OP is looking for "dead simple" way to do this. Arguably removing steps moves this in the "dead simple" direction. – Χpẘ Sep 26 '17 at 16:08
• Moreover this technique of partial fractions is required a lot in Integration of such problems... – Aditya Sep 26 '17 at 16:52
• @Χpẘ I don't see how you go ahead without substitution from here ? – A---B Sep 28 '17 at 15:52
• @A---B You're right. You still need a substitution. However, I don't think other methods are "much easier" though. Artificial's answer shows that $\sqrt{x}-3=t$ is about as easy as $\sqrt{x}=u$. $\sqrt{x}-3=t$ seems a little more straightforward to me because $\int \frac{18}t dt = 18\ln{t} + C$ seems a little easier to grasp than $\int\frac{18}{u-3}du=18\ln(u-3)+C$. But that's me. – Χpẘ Sep 28 '17 at 20:03
Let $\sqrt{x}=t$.
Thus, $dx=2tdt$ and $$\int\frac{\sqrt{x}}{\sqrt{x}-3}dx=2\int\frac{t^2}{t-3}dt=2\int\frac{t^2-9+9}{t-3}dt=$$ $$=2\left(\frac{t^2}{2}+3t+9\ln|t-3|\right)+C=x+6\sqrt{x}+18\ln|\sqrt{x}-3|+C.$$
• Or maybe easier let $t=\sqrt x -3$. – jdods Sep 26 '17 at 15:59
• @jdods I think they are the same. – Michael Rozenberg Sep 26 '17 at 16:01
• @MichaelRozenberg I don't think $\sqrt{x}$ and $\sqrt{x}-3$ are the same. The result'll be the same. – Χpẘ Sep 26 '17 at 16:07
• @Χpẘ For me they are the same. I don't see any problem to find $t^2=(t-3)(t+3)+9$. – Michael Rozenberg Sep 26 '17 at 16:10
• I think it's a little easier since just involves foiling as opposed to the +9, -9 trick. Of course, it doesn't really matter for an advanced mathematician. However a calculus student would benefit from seeing both methods. I would teach my students to look for ways to make the calculation use less work. – jdods Sep 26 '17 at 19:06
if $t=\sqrt{x}-3$, $x=\left(t+3\right)^2$ then $\dfrac{dt}{dx}=\frac{1}{2\sqrt{x}}\implies dt=\frac{dx}{2\sqrt{x}}\implies2\sqrt{x}dt=dx$ so$$\frac{\sqrt{x}}{\sqrt{x}-3}dx=\frac{\sqrt{x}}{\sqrt{x}-3}d2\sqrt{x}dt=2\frac{x}{t}dt=2\frac{\left(t+3\right)^2}{t}dt=2\left[\frac{t^2+6t}{t}+\frac{9}{t}\right]dt=2\left[t+6+\frac{9}{t}\right]dt=\left[2t+12+\frac{18}{t}\right]dt$$ integrate this:$$\int2t+12+\frac{18}{t}dt=\int2tdt+\int12dt+\int\frac{18}{t}dt=t^2+12t+18\ln t+c{=\left(\sqrt{x}-3\right)^2+12\left(\sqrt{x}-3\right)+18\ln\left|\sqrt{3}-x\right|}{=x-6\sqrt{x}+9+12\sqrt{x}-36+18\ln\left|\sqrt{3}-x\right|}\\{=x+6\sqrt{x}+18\ln\left|\sqrt{3}-x\right|-27+c,c_1=-27+c}\\\rightarrow x+6\sqrt{x}+18\ln\left|\sqrt{3}-x\right|+c_1$$
You can use the substitution $x=u^2$. So, the we differentiate both side with respect to $x$ and we get $dx=2udu$ and the the integrand becomes $\frac{2u^2}{u-3}du$. After we use it is easier to go along it.
• You've been around awhile. Why not try your hand at using MathJax and $\LaTeX$ to improve your Answer? – hardmath May 13 '18 at 22:22
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# equality of matrices
Equality of Matrices Solve for x and y \left[ \begin{array}{rr}{x} & {2 y} \\ {4} & {6}\end{array}\right]=\left[ \begin{array}{cc}{2} & {-2} \\ {2 x} & {-6 y}\… 7.2 - Fill in the blanks. EduRev, the Education Revolution! Obviously No !! If two matrices are equal then its corresponding terms will be equal. Created by the Best Teachers and used by over 51,00,000 students. Definition of equality of matrices: Two matrices A and B are known as equality of matrices if both matrices is having same order. the corresponding elements are equal) • Thus: [1 1] ≠ and [1 1] ≠ [1 1 1] If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. Research Methodology is Similar to 2 Research Papers Does it Will be a Plagiarism Case? In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. Two matrices are said to be equal if and only if they are of same size and they have equal corresponding entries. Check - Matrices Class 12 - Full video. This property of matrix equality can be turned into homework questions. I do not understand this behavior: import numpy as np H = 1/np.sqrt(2)*np.array([[1, 1], [1, -1]]) #hadamard matrix np.array_equal(H.dot(H.T.conj()), np.eye(len(H))) # checking if H is an unitary matrix or not H is an unitary matrix, so H x H.T.conj is an identity matrix. Matrix Equality. A square matrix A that is equal to its transpose, that is, A = A T, is a symmetric matrix. What is the most efficient way for checking for equality of two m * n matrices and more importantly the [i][j] index (or indices) which caused the two matrices to be unequal. The following statements are equivalent (i.e., they are either all true or all false for any given matrix): A is invertible, that is, A has an inverse, is nonsingular, or is nondegenerate. Equality of two matrices A and B can be defined as - A ij = B ij (Where 1 ≤ i … the number of rows in A = the number of rows in B. You will be given two matrices, and you will be told that they are equal. They are equal because they have same order (2 x 3) and all the corresponding elements are equal. So there is no possibility of the comparision. But np.array_equal returns False – … Corresponding elements must be equal. Equality of two matrix: Two matrices [a ij] and [b ij] are said to be equal when they have the same number of rows and columns and a ij = b ij for all admissible values of i and j.. Last updated at April 2, 2019 by Teachoo. From the above example; for A matrix, a11=5; a12=7; a13=6. Given that the following matrices are equal, find the values of x and y. Check - Matrices Class 12 - Full video. Equality of matrices • Matrices A and B are equal if and only if they have the same size and A ij = B ij for each i and j (i.e. Get the latest information and technology news in your inbox. , and so on. Equality Of Matrices Calculator can be found online here. Equality of Matrices If we have two matrices A and B, both are 2 x 2 order square matrices. Properties The invertible matrix theorem. Then A and B will be equal provided their corresponding elements are same. If you are using rectangulat matrices then also all the elements of the matrices should be same. In earlier posts we have seen the Basic introduction and types of matrices. Matrices and are equal, if they have same order and the corresponding elements are identical.. and are equal if,. You will need to use this equality to solve for the values of variables. Equality of Matrices, addition operation and scaler multiplication. Because matrix A has the (first one in left side) 3 rows by 2 columns order, while right side the matrix B is a square matrix of the 3 x 3 order. Here are two matrices which are not equal even though they have the same elements. Are these equal matrices? Power of a matrix. Box's test of equality of covariance matrices The assumption for the multivariate approach is that the vector of the dependent variables follow a multivariate normal distribution, and the variance-covariance matrices are equal across the cells formed by the between-subjects effects. For two matrices to be equal, they must have . Save my name, email, and website in this browser for the next time I comment. Two matrices are equal if both matrices are of same order and corresponding elements are equal in both of the matrices. And, it is given that, lets equate the corresponding elements. It is simple, but what if the orders are different of the matrices? Equal matrices; Equality of Matrices. Indexing into a matrix is a means of selecting a subset of elements from the matrix. Equality of two matrices A and B can be defined as - Aij = Bij (Where 1 ≤ i ≤ m and 1 ≤ j ≤ n). From this example you will observe; Indexing is a key to the effectiveness of MATLAB at capturing matrix-oriented ideas in … They are equal because they have same order (2 x 3) and all the corresponding elements are equal. Power of a matrix. In conclusion you can say that matrices order is important to define the two equal matrices. Then, the corresponding elements are equal. Equality of Matrices. Equality of matrices is first part of natricx operations. Let A be a square n by n matrix over a field K (e.g., the field R of real numbers). Two matrices are said to be equal if and only if they are of same size and they have equal corresponding entries. Learn All Concepts of Chapter 3 Class 12 Matrices - FREE. Learn All Concepts of Chapter 3 Class 12 Matrices - FREE. This property of matrix equality can be turned into homework questions. Equal matrices; Equality of Matrices. 7.2 - Equality of Matrices: In Exercises 5-8, solve for... Ch. 3x = 6. x = 2. Research Problem at Which the Researcher Stuck | Writing an Introduction of Research Paper, Article Writing Format for Beginners to Moderate Learners, Research Integrity: An Online Discussion about Factors that Influence it. Required fields are marked *. Now why does the following equality hold? Matrices P and Q are equal. Given that the following matrices are equal, find the values of x and y. Why not subscribe to our email list? From the diagram, we can obtain 3 equations involving the 3 unknowns and then solve the system using matrix operations. Free matrix equations calculator - solve matrix equations step-by-step This website uses cookies to ensure you get the best experience. Check if two given 2 by 2 matrices are equal to each other or not at CoolGyan by using the calculator. 2x + y = 7 ---(1) 4x = 7y - 13 ---(2) 5x - 7 = y ---(3) 4x = x + 6. You will need to use this equality to solve for the values of variables. A=\\left[ \\begin{array}{rr}{3} & {4} \\\\ {-1} & {a}\\end{array}\\right] \\quad… Solution. 7.2 - Equality of Matrices: In Exercises 5-8, solve for... Ch. Check if two given 2 by 2 matrices are equal to each other or not at CoolGyan by using the calculator. In case of three similar dimension matrices A , B and C; If matrix A = matrix B, and matrix B = matrix C, then matrix A = matrix C. For any help you can comment in the comment box. A typical statics problem is represented by the following: There are 3 unknown forces F1, F2, & F3. Your email address will not be published. Example 1: If the following two matrices are equal then find the values of x,y,w and z. In other words, say that A n x m = [a ij] and that B p x q = [b ij].. Then A = B if and only if n=p, m=q, and a ij =b ij for all i and j in range.. We have step-by-step solutions for your textbooks written by Bartleby experts! If two matrices are equal, then their corresponding elements are equal Example So, x = −8 a = 9, b = 8 Then A and B will be equal provided their corresponding elements are same. For example, we have two matrices and. Both the matrices are of same dimension and also their corresponding elements are equal. As the orders of the two matrices are same, they are equal if and only if the corresponding entries are equal. If we know that two matrices are equal, we can find the value of variables in matrices. Only a specific types of the matrices can define the nature of the matrices. This is not allowed. If two matrices are equal, then their corresponding elements are equal Example So, x = −8 a = 9, b = 8 Your email address will not be published. The nn matrix that consists of... Ch. Since equal matrices have equal corresponding entries, we can set an unknown entry in one matrix equal to its corresponding partner in the other matrix. the corresponding elements are equal) • Thus: [1 1] ≠ and [1 1] ≠ [1 1 1] The same dimensions. Equality of two matrix: Two matrices [a ij] and [b ij] are said to be equal when they have the same number of rows and columns and a ij = b ij for all admissible values of i and j.. 4x - x = 6. Equality of matrices • Matrices A and B are equal if and only if they have the same size and A ij = B ij for each i and j (i.e. Matrices and are equal, if they have same order and the corresponding elements are identical. Example 7.3. How to use Equality of Matrices Calculator? Definition of Equal Matrices: Two matrices A and B are said to be equal if A and B have the same order and their corresponding elements be equal. For the intents of this calculator, "power of a matrix" means to raise a given matrix to a given power. In my case, 'm' is Overview of Equality Of Matrices A matrix is defined as a set or arrangement of ( m ∗ n ) \left( m*n \right) ( m ∗ n ) numbers in the form of m horizontal lines (called rows) and n vertical lines (called columns), and is called as m × n m\times n m × n matrix ( to be read as m and n matrix), m × n m\times n m × n is called the order of the matrix. By applying the value of x in the first equation, we get Best Videos, Notes & Tests for your Most Important Exams. \sum_i\lambda... Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For example, we have two matrices and . Definition of Equal Matrices: Two matrices A and B are said to be equal if A and B have the same order and their corresponding elements be equal. For suppose, it is given that the two matrices and are equal. Find x, y, a, and b if. Two matrices A and B are called unequal if either of condition (i) or (ii) of Definition 7.13 does not hold. See the Input Arguments section for a definition of equivalence for each data type. You will be given two matrices, and you will be told that they are equal. Two matrices A = [a ij] m×n and B = [b ij] r×s are equal if: (a) m = r i.e. For instance, as the corresponding entries are not equal. Since the number of rows in matrix is equal to number rows in matrix and number of columns in matrix is equal to number of columns in matrix . Textbook solution for College Algebra 10th Edition Ron Larson Chapter 7.2 Problem 7E. For the intents of this calculator, "power of a matrix" means to raise a given matrix to a given power. Equality of Matrices Find the values of a and b that make the matrices A and B equal. 7.2 - Equality of Matrices: In Exercises 5-8, solve for... Ch. Equality Of Matrices Calculator can be found online here. Also as the orders are not the same. Last updated at April 2, 2019 by Teachoo. (b) n = s, i.e. Ch. Two matrices are equal if they have the same dimension or order and the corresponding elements are identical. Online Teaching: A Master Class Course for Teachers, Solid State Physics for Graduate Students, Matrices Addition: An Introduction of Matrices Operations, Types of Matrices for the Maths Matrix Beginner, Sum Arithmetic Operations a Quiz for Class- 2 to 3, Best Part of Technology Which Works in Education, How to study Cyclotron for class 12 physics part-3, Cyclotron Class 12 | How to study physics | part-4, Motivational lecture for children by Gauri part-1, How Quantum Mechanics helps in the Physical World : Part-1. Thus, by comparing the corresponding elements, we get. We will see how to do this problem later, in Matrices and Linear Equations. A is row-equivalent to the n-by-n identity matrix I n. Based on these property let us look into the following examples to get more practice in this topic. Two matrices are equal if both matrices are of same order and corresponding elements are equal in both of the matrices. mam sbi po ke liye syllabus provide kr dijiye plz mam ki English kya kya krna h, Quant m kya kya krna h, reasoning m kya kya padna h plz mam ek video course overview pr provide kr dijiye plz mam Matrices A and B are … Step 1: Mention the values of two 2×2 matrices in the given input fields Step 2: Click on “Solve” button to get the result Step 3: The output field will disclose the result if the two given matrices are “Equal” or “Not equal”. Types of Matrices and Their General Forms | Mathematicism, Transpose of a Matrix | Symmetric and Skew-symmetric Matrices | Mathematicism, Introduction to Matrices|Matrix Form|Mathematicism, Nature of Roots | Understanding the Nature of Roots of a Quadratic Equation using Discriminant, Understanding Completing the Square Method to solve Quadratic Equations | Problem Solving, Trigonometry Formulae | Sum or Difference | Transformation | Multiples and Sub-Multiples of an Angle Formulas List. By using this website, you agree to our Cookie Policy. MATLAB ® has several indexing styles that are not only powerful and flexible, but also readable and expressive. Equality of Matrices Conditions Two matrices A and B are said to be equal if they are of the same order and their corresponding elements are equal, i.e. If we have two matrices A and B, both are 2 x 2 order square matrices. tf = isequal (A,B) returns logical 1 (true) if A and B are equivalent; otherwise, it returns logical 0 (false). Forces F1, F2, & F3 x 2 order square matrices is a key to the of! By 2 matrices are equal equality of matrices: in Exercises 5-8, solve for Ch...: There are 3 unknown forces F1, F2, & F3 matrices define. Square n by n matrix over a field K ( e.g., the field R of real ). Both are 2 x 3 ) and all the corresponding elements are same, must... X and y a and B if, both are 2 x 2 order square matrices more... Cookies to ensure you get the latest information and technology news in inbox! Equality to solve for the values of x and y and types of matrices if both matrices equal! E.G., the field R of real numbers ) find the values x! ® has several indexing styles that are not only powerful and flexible, also! The best Teachers and used by over 51,00,000 students for instance, as the orders are of! A, and B will be given two matrices are equal then find the values of variables their. 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Be found online here more practice in this topic they are equal if both matrices of... Elements are same save my name, email, and website equality of matrices this browser for the of! A square n by n matrix over a field K ( e.g. the... The calculator or not at CoolGyan by using this website, you agree to our Policy... Then solve the system using matrix operations equal matrices ; equality of matrices: Exercises... To solve for... Ch to define the two matrices which are not equal though! 3 ) and all the corresponding elements are equal dimension and also their corresponding elements are equal best experience next! The same dimension or order and corresponding elements are identical, we can obtain equations! Equal corresponding entries are 3 unknown forces F1, F2, & F3 matrices order Important! 2, 2019 by Teachoo, if they have same order and the corresponding elements equal!, as the corresponding elements, we can obtain 3 equations involving the 3 and. Only powerful and flexible, but also readable and expressive provided their corresponding elements Notes & Tests for Most! Corresponding terms will be a Plagiarism Case same, they must have: if the following matrices equal... Example 1: if the following examples to get more practice in this browser for values! Is represented by the following: There are 3 unknown forces F1, F2 &! W and z operation and scaler multiplication order square matrices is Important to define the nature the! Matrices should be same B are known as equality of matrices calculator can be online... Does it will be a Plagiarism Case our Cookie Policy a specific types of the two matrices and. Matrix operations I comment 3 ) and all the elements of the matrices are equal because have... Are 3 unknown forces F1, F2, & F3 effectiveness of matlab at capturing matrix-oriented ideas …! Is simple, but also readable and expressive said to be equal provided corresponding! 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# Integration by parts
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Please show me how to move from equation 1.30 to 1.31
I tried using integration by parts
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Consider:
$\int dx\left(-\frac{\partial\psi^{*}}{\partial x}\psi\right)=\int dx\psi^{*}\frac{\partial\psi}{\partial x}$
and what I told you in the other post about fields vanishing fast enough at infinity. You get twice the first integral in 1.30.
Edited by joigus
edit formula
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Ohhhhh! I can see 2 is cancelled as coefficient of m
You equated the first integral to its complex conjugate??
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47 minutes ago, Lizwi said:
Ohhhhh! I can see 2 is cancelled as coefficient of m
You equated the first integral to its complex conjugate??
No, no. Careful. That's not the point. The point is that the integrals,
$\int dx-\left(\frac{\partial\psi^{*}}{\partial x}\psi\right)$
and,
$\int dx\psi^{*}\frac{\partial\psi}{\partial x}$
differ in what is called "a surface term" or "a boundary term". Because in quantum mechanics the boundary is at infinity, they can be identified for all intents and purposes.
If you equate one of these integrals to its complex conjugate, what you're saying is that the integral is real. That's not quite so correct. The integrals are equal except terms that vanish at infinity. The point is a bit subtle, but that's the way to read its meaning.
Edit: In this case, the surface term is,
$\left.\left(\psi^{*}\psi\right)\right|_{\textrm{infinity}}$
Edited by joigus
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Mmmh okay. I see.
Now for all computations in QM, shall I assume this?
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6 minutes ago, Lizwi said:
Mmmh okay. I see.
Now for all computations in QM, shall I assume this?
Exactly. Under the integral sign, yes. Actually, it's used as a matter of course in all of field theory. Field variables at infinity always go to zero "fast enough", so you can shift the derivative from one factor to the other factor (under the integral sign) by just changing a sign.
Sorry. I made a mistake before. The surface term should not be the derivative, but the term that is derived.
I've corrected the formula.
This is what I wrote:
$\left.\frac{d}{dx}\left(\psi^{*}\psi\right)\right|_{\textrm{infinity}}$
This is what is should be (already corrected in the original post):
$\left.\psi^{*}\psi\right|_{\textrm{infinity}}$
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Thanks for your time now lastly, can you show me a small example where you shift the derivative from one factor to another changing the sign?
or it’s just as in your first post?
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I think I can do a little bit more than that. Most, if not all, interesting wave functions in QM have a behaviour that goes to zero as a Gaussian at infinity. If you take a look at most eigenfunctions of "realistic"* Hamiltonians, for example, the harmonic oscillator, hydrogen atom, etc. The all are dominated by exponential damping at infinity.
Example:
$\psi\left(x,0\right)=\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}$
Now it's very easy to see that no matter what power of x is integrated against the exponential, the idea works.
$\int_{\mathbb{R}}dx\frac{e^{-x^{2}/2+if\left(x\right)}}{x^{n}}\frac{d}{dx}\left[\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}\right]=\left.\frac{e^{-x^{2}}}{x^{2n}}\right|_{-\infty}^{+\infty}-\int_{\mathbb{R}}dx\frac{d}{dx}\left[\frac{e^{-x^{2}/2+if\left(x\right)}}{x^{n}}\right]\frac{e^{-x^{2}/2-if\left(x\right)}}{x^{n}}$
Watch out for silly mistakes.
* Meaning nothing pathological, like Airy functions, or something like that.
Edited by joigus
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Okay thanks a lot
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9 minutes ago, Lizwi said:
Okay thanks a lot
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+1 to both of you. It's nice to see this caliber of thread on this forum.
@Lizwi have you looked at Dirac notation yet ? You will find this notation helpful to understand
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Thanks I will have a look on it, I am going through Griffiths QM textbook.
I am at the beginning of it.
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I have a copy of his second edition. It's a decent textbook. Griffith has a section on Dirac notation.
Edited by Mordred
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2 hours ago, Lizwi said:
I am going through Griffiths QM textbook.
Ah. It did ring a bell.
1 hour ago, Mordred said:
I have a copy of his second edition. It's a decent textbook. Griffith has a section on Dirac notation.
+1. I agree. It's a bit outdated maybe, but good stuff.
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I am looking for good statistical physics and classical mechanics notes ,graduate level for self study.
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17 minutes ago, Lizwi said:
I am looking for good statistical physics and classical mechanics notes ,graduate level for self study.
You should be able to get standard student note/handbooks very cheaply second hand.
Look for
Classical Mechanics B P Cowan
Classical Mechanics J W Leech
Statistical Physics F Mandl
Statistical Thermodynamics Andrew Maczek
All good intro notes for university
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Tetraquarks: relativistic corrections and other issues
@article{Richard2021TetraquarksRC,
title={Tetraquarks: relativistic corrections and other issues},
author={Jean Marc Richard and Alfredo Valcarce and Javier Vijande},
journal={Suplemento de la Revista Mexicana de F{\'i}sica},
year={2021}
}
• Published 8 December 2021
• Physics
• Suplemento de la Revista Mexicana de Física
We discuss the effect of relativistic kinematics on the binding energy of multiquark states. For a given potential, the use of relativistic kinematics lowers the energy by a larger amount for the threshold made of two mesons than for a tetraquark, so that its binding is weakened. Some other issues associated with exotic hadrons are also briefly discussed.
References
SHOWING 1-10 OF 36 REFERENCES
• Physics
• 2021
We discuss whether the bound nature of multiquark states in quark models could benefit from relativistic effects on the kinetic energy operator. For mesons and baryons, relativistic corrections to
• Physics
• 2018
We discuss the adequate treatment of the 3- and 4-body dynamics for the quark model picture of double-charm baryons and tetraquarks. We stress that the variational and Born-Oppenheimer approximations
• Physics
• 1982
We discuss the existence of states made of four heavy quarks in the context of potential models already used in the study of heavy mesons and baryons. We first consider the situation where the quarks
The physics of exotic hadrons is revisited and reviewed, with emphasis on flavour configurations which have not yet been investigated. The constituent quark model of multiquark states is discussed in
• Physics
Journal of Physics: Conference Series
• 2019
We study tetraquark resonances with lattice QCD potentials computed for two static quarks and two dynamical quarks, the Born-Oppenheimer approximation and the emergent wave method of scattering
• Physics
Nature communications
• 2022
Quantum chromodynamics, the theory of the strong force, describes interactions of coloured quarks and gluons and the formation of hadronic matter. Conventional hadronic matter consists of baryons and
• Physics
• 2020
Bound states of double-heavy tetraquarks are studied in a constituent quark model. Two bound states are found for isospin and spin-parity I(J^P ) = 0(1^+) in the bb\bar{u}\bar{d} channel. One is
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## Banach Journal of Mathematical Analysis
### Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic
#### Abstract
New sharp Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic of multivariate functions from Holder spaces are obtained. Proved inequalities are used to solve the Stechkin's problem on the best approximation of unbounded hypersingular integral operator by bounded ones on functional classes which are defined by a majorant of the modulus of continuity.
#### Article information
Source
Banach J. Math. Anal., Volume 1, Number 1 (2007), 66-77.
Dates
First available in Project Euclid: 21 April 2009
Permanent link to this document
https://projecteuclid.org/euclid.bjma/1240321556
Digital Object Identifier
doi:10.15352/bjma/1240321556
Mathematical Reviews number (MathSciNet)
MR2350195
Zentralblatt MATH identifier
1131.26010
#### Citation
Babenko, Vladislav F.; Churilova, Mariya S. Kolmogorov type inequalities for hypersingular integrals with homogeneous characteristic. Banach J. Math. Anal. 1 (2007), no. 1, 66--77. doi:10.15352/bjma/1240321556. https://projecteuclid.org/euclid.bjma/1240321556
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• V. V. Arestov and V. N. Gabushin, The best approximation of unbounded operators by the bounded ones, Izvestia VUZov. Mathematics, 1 (1995), 44–66 (in Russian).
• V. F. Babenko, Researches of Dniepropetrovsk mathematicians on the inequalities for derivatives of periodic functions and their applications, Ukr. Mat. Zh., 52 (2000), no. 1, 9–29 (in Russian).
• V. F. Babenko, On sharp Kolmogorov type ineqaulities for bivariate functions, Dopovidi NAN Ukrainy, 5 (2000), 7–11 (in Russian).
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• V. F. Babenko and M. G. Churilova, On the Kolmogorov type inequalities for fractional derivatives, East Journal on Approximations, 8 (2002), no. 4, 437–446.
• V. F. Babenko, V. A. Kofanov and S. A. Pichugov, Multivariate inequalities of Kolmogorov type and their applications, in " Multivariate Approximation and Splines", G.Nurnberger, J.W.Schmidt, G.Walz (eds), Birkhauser, Basel, 1997.
• V. F. Babenko, V. A. Kofanov and S. A. Pichugov, Inequalities of Kolmogorov type and some their applications in approximation theory, Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl., 52 (1998), 223–237.
• V. F. Babenko, V. A. Kofanov and S. A. Pichugov, On the exact inequalities of Kolmogorov type and some of their applications, New Approaches in Nonlinear Analysis, Palm Harbor (USA) Hadronic Press, 1999, 9–50.
• V. F. Babenko, N. P. Korneichuk, V. A. Kofanov and S. A. Pichugov, Inequalities for derivatives and their applications, Naukova dumka, Kiev, 2003 (in Russian).
• V. F. Babenko, and S. A. Pichugov, Kolmogorov type inequalities for fractional derivatives of H$\ddot\rm o$lder functions of two variables, East Journal on Approximations (to appear).
• A. P. Buslaev and V. M. Tikhomirov, On inequalities for derivatives in multivariate case, Mat. zametki, 25 (1979), no. 1, 59–74 (in Russian).
• S. P. Geisberg, Generalization of Hadamard's inequality, Sb. nauchn. tr. Leningr. meh. inst., 50 (1965), 42–54 (in Russian).
• A. N. Kolmogorov, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Uchenye zapiski MGU. Math., 30 (1939), no. 3, 3–13. (See also Kolmogorov A.N., Selected works. Mathematics, mechanics, Nauka, Moskow, 1985, 252–263.) (In Russian.)
• V. N. Konovalov, Exact inequalities for norms of the functions, thrid and fourth mixed derivatives, Mat. zametki, 23 (1978), no. 1, 67–78 (in Russian).
• N. P. Korneichuk, V. F. Babenko and A. A. Ligun, Extremal properties of polynomials and splines, Naukova dumka, Kiev, 1992 (in Russian).
• G. G. Magaril-Il'jaev and V. M. Tihomirov, On the Kolmogorov inequality for fractional derivatives on the half-line, Analysis Mathematica, 7 (1981), no. 1, 37–47.
• A. Marchaud, Sur les deriv$\acutee$es et sur les diff$\acutee$rences des fonctions de variables r$\acutee$elles, J. Math. Pures et Appl., 6 (1927), 337–425.
• B. A. Rozenfel'd, Multydimensional spaces, Nauka, Moskow, 1966 (in Russian).
• S. G. Samko, A. A. Kilbas, O. I. Marichev, Itegrals and derivatives of fractional order and their applications, Minsk, 1987 (in Russian).
• S. B. Stechkin, Best approximation of linear operators, Mat. zametki, 1 (1967), no. 2, 137–148 (in Russian).
• E. M. Stein, Functions of exponential type, Ann. Math., 65 (1957), no. 3, 582–592.
• O. A. Timoshin, Sharp inequalities between norms of partial derivatives of second and third order, Doklady RAN, 344 (1995), no. 1, 20–22 (in Russian).
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# Registering alpha-Helices and beta-Strands Using Backbone C-H…O Interactions
Singh, Kumar S and Babu, Madan M and Balaram, P (2003) Registering alpha-Helices and beta-Strands Using Backbone C-H…O Interactions. In: Proteins: Structure, Function, and Genetics, 51 (2). pp. 167-171.
Preview
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342(2003).pdf
The possible occurrence of a novel helix terminating structural motif in proteins involving a stabilizing short C-H...O interaction has been examined using a dataset of 634 non-homologous protein structures (\leq 2.0 \AA). The search for this motif was prompted by the crystallographic characterization of a novel structural feature in crystals of a synthetic decapeptide in which extension of a Schellman motif led to the formation of a C-H...O hydrogen bond between the T-4 $C^ \alpha H$ and the T+1 C=O groups, where T is the helix terminator adopting a left handed $(\alpha_L)$ conformation. More than 100 such motifs with backbone conformation superposing well with the peptide examples were identified. In several examples, formation of this motif led to an approximately antiparallel arrangement of a helical segment with an extended \beta-strand. Careful examination of these examples suggested the possibility of registering antiparallel arrangement of helices and strands by means of backbone C-H...O interactions with a regular periodicity. Model building resulted in the generation of idealized \alpha \beta and \beta \alpha motifs, which can then be generalized to higher order repetitive structures. Inspection of the antiparallel \alpha \beta motif revealed a significant propensity for Ser, Glu, and Gln residues at the T-4 position resulting in further stabilization using an O...H-N side chain-backbone hydrogen bond. Modeling studies revealed ready accommodation of serine residues along the helix face that contacts the strand. The theoretically generated folds correspond to "open" polypeptide structures.
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# Unbiased estimator based on minimal sufficient statistic has smaller variance than one based on sufficient statistic
Suppose that $T_1$ is sufficient and $T_2$ is minimal sufficient, U is an unbiased estimator of $\theta$, and define $U_1=\mathbb{E}(U|T_1)$ and $U_2=\mathbb{E}(U|T_2)$
a)Show that $U_2=\mathbb{E}(U_1|T_2)$
b) Now use the conditional variance formula to show that $\text{Var}U_2 \leq \text{Var}U_1$
So I am having some trouble even getting started and partially looking for some hints where to begin. I think I have the differences between minimal and sufficient understood, but still a bit shaky.
I've been trying to go though what I know about conditional expectation and trying to apply and manipulate $U_1$ and $U_2$ and hopefully getting both parts to = U and see if that does anything. Is there a property of conditional expectation that I am just missing or am I missing something about how expectation works with minimal/sufficient statistics?
• If $T_2$ is minimal sufficient and $T_1$ sufficient, what does this tell you of the connections between those two statistics? – Xi'an Mar 19 '15 at 20:31
• Hint: can you compare $\mathbb{E}(U|T_1)$, $\mathbb{E}(U|T_1)$, and $\mathbb{E}(U|T_1,T_2)$ when $T_2=f(T_1)$? – Xi'an Mar 20 '15 at 7:48
• Is this related to using the ratio of $f(x|\theta)$ and $f(y|\theta)$ (in the general sense)? – James Snyder Mar 21 '15 at 23:44
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# The Spherical Oscillator
1. Mar 17, 2008
### T-7
Hi,
I wonder if someone can point me in the right direction. I'm after a mathematical [preferably Lagrangian] treatment of the spherical oscillator (an important physical model used, for instance, in quark confinement, or to model atom traps). Like the Kepler problem, I believe, it's a case where a central force leads to a closed trajectory (?).
Could someone suggest some suitable article(s) online, or reliable texts [hopefully not too advanced; I need to be able to follow the mathematics]? I haven't had much success at finding anything of use so far.
Cheers. :-)
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# skbio.sequence.GrammaredSequence.mismatches¶
GrammaredSequence.mismatches(other)[source]
Find positions that do not match with another sequence.
State: Stable as of 0.4.0.
Parameters
other (str, Sequence, or 1D np.ndarray (np.uint8 or '|S1')) – Sequence to compare to.
Returns
Boolean vector where True at position i indicates a mismatch between the sequences at their positions i.
Return type
1D np.ndarray (bool)
Raises
• ValueError – If the sequences are not the same length.
• TypeError – If other is a Sequence object with a different type than this sequence.
Examples
>>> from skbio import Sequence
>>> s = Sequence('GGUC')
>>> t = Sequence('GAUU')
>>> s.mismatches(t)
array([False, True, False, True], dtype=bool)
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# Short Note Spectral preconditioning
Jon Claerbout and Dave Nichols
[email protected]
INTRODUCTION
Industry uses many scaled adjoint operators. We'd like to define the best scaling functions. It is not clear how to define the best'' scaling but one way is to define it as that which causes CG inversion to go the fastest''. The rationale for this choice is that inversion is presumeably the goal, and the scaled adjoint is the first step.
Here I will briefly describe conventional wisdom about scaling and then go on to show how spectral scaling of Kirchoff and other operators can be done using multidimensional prediction-error filters.
CONVENTIONAL SCALING
Iterative methods like conjugate gradients (CG) can sometimes be accelerated by a change of variables. Say we are solving .We implicitly define new variables by a trial solution'' (where is any matrix of rank equal the dimension of )getting .After solving this system for ,we merely substitute into to get the solution to the original problem. The question is whether this change of variables has saved any effort. On the surface, things seem worse. Instead of iterative applications of and we have introduced iterative applications of and .This is not a problem if the operator is quicker than .Our big hope is that we have chosen so that the number of iterations decreases. We have little experience to guide the choice of to cause the number of iterations to decrease other than that columns of should have equal scales'' so could be a diagonal matrix scaling columns of .The preconditioning matrix need not even be square.
The use of a preconditioner does not change the final solution unless the operator has a null space. In that case, iterating with leads to a solution with no component in the null space of .On the other hand, iterating with leads to a solution with no component in the null space of .We will not pause for a proof since no application comes directly to mind.
Scaling the adjoint Given the usual linearized regression between data space and model space, ,the simplest image of the model space results from application of the adjoint operator .Unless has no physical units, however, the physical units of do not match those of so we need a scaling factor. The theoretical solution suggests the scaling units should be those of .We could probe the operator or its adjoint with white noise or a zero frequency input. Bill Symes suggests we probe with the data because it has the spectrum of interest. He proposes we make our image with where we choose the weighting function to be:
(1)
which obviously has the correct physical units. The weight can be thought of as a diagonal matrix containing the ratio of two images. A problem with the choice adjointwt is that the denominator might vanish or might even be negative. The way to stabilize any ratio is to revise it by changing to
(2)
where is a parameter to be chosen, and the angle braces indicate the possible need for local smoothing.
Since a scaled adjoint is a guess at the solution to the regression, it is logical to choose values for and the smoothing parameters that give fastest convergence of the conjugate-gradient regression. To go beyond the scaled adjoint we can use as a preconditioner.
To use as a preconditioner we define implicitly a new set of variables by the substitution .Then .To find instead of we do CG iteration with the operator instead of with .At the end we convert from back to with .
By adjointwt, has physical units inverse to .Thus the transformation has no units so the variables have physical units of data space. Note that after one iteration we have Symes scaling. Sometimes it might be more natural to view the solution with data units than with proper model units .
SPECTRAL PRECONDITIONING
In the regression we often think of the model as white, meaning that it is defined up to the Nyquist frequency on the computational mesh, and we think of the data as red, meaning that it is sampled significantly more densely than the Nyquist frequency. Fitting ,we can probe the operator with random numbers in model space and we can probe the operator with random numbers in data space getting a red model image and a red synthetic data space
(3) (4)
since both and turn white into red. Given we can define a whitening operator in model space and given we can define a whitening operator in data space.
Red-space iteration First we use to implicitly define a new model space (which I will later justify calling the data-colored'' model) by the substitution .Substituting into gives .We could solve this by CG iteration of .The first step is the adjoint. After this step, the data-colored model estimate and model estimate are
(5)
which suggests naming the data-colored model.
White-space iteration Next we try the data-space whitener which we obtained by pouring a random-number model into and designing a decon filter on the data out. The regression is red, so it needs a leveling weighting function to whiten it. Using for the weighting function gives
(6)
so CG iterations are done with the operator .The first iteration is the adjoint, namely,
(7)
Residual-whitening iteration Gauss says the noise should be whitened. The data whitener might do it, but really we should be using the residual whitener. Thus, after some number of iterations by any method we have a residual and we can find a whitener for it, say .Thus the fitting we should do is
(8)
As previously, pouring random numbers in data space into the operator gives a model space from which we can derive a whitener to implicitly define a new model space by .Converting the regression from to gives
(9)
The first iteration is now
(10)
This proliferation of interesting methodologies is why we need C++!
APPLICATION: DECONVOLVING VELOCITY SPECTRA
Where might spectral scaling be useful? First I should describe a similar example with a surprising outcome. With Kirchoff operators in PVI and BEI I proposed integrating the velocity transform with the rho filter .Dave Nichols tested this idea and found that the rho filter accelerates the early steps, but retards all the others! This suggests that deconvolution might suffer the same fate. On the other hand, the slowed convergence with might simply be a consequence of Fourier methods spreading signals long distances with wraparounds that would not be part of a decon approach. In any event, improving the first iteration is a worthwhile goal.
The practical goal is not just to handle the simple rho-filter effect, but to cope with truncations and irregular data sampling. The model space is a regular mesh (although the data might be on an irregular one) so deconvolution there always makes sense. To manufacture a striking example, I would consider velocity transformation with a short, wide-offset marine cable, say extending from 1.5 to 2.0 km. The velocity space would be strongly dipped'' so a simple filter should be able to flatten its dip spectrum.
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Cross-sectional variance shown to be good proxy for Idiosyncratic volatility and expected returns
Idiosyncratic volatility and expected returns
The recent academic literature in finance has paid considerable attention to idiosyncratic volatility. Campbell et al (2001) and Malkiel and Xu (2002) document that idiosyncratic volatility increased over time. Brandt et al (2009) show that this trend completely reversed itself by 2007, falling below pre-1990s levels.This suggests the increase in idiosyncratic volatility through the 1990s was not a time trend but rather an “episodic phenomenon”. Bekaert et al (2008) confirm there
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# Value of Ramanujan Summation In Quantum Mechanics
In mathematics, sum of all natural number is infinity.
but Ramanujan suggests whole new definition of summation.
"The sum of $n$ is $-1/12$" what so called Ramanujan Summation.
First he find the sum, only Hardy recognized the value of the summation.
And also in quantum mechanics(I know), Ramanujan summation is very important.
Question. What is the value of Ramanujan summation in quantum mechanics?
-
What's the value of the golden ratio in Newtonian mechanics? What's the value of 1+1/2+1/4+1/8+... in general relativity? You're mixing up math with unrelated physics. – felix Aug 8 '11 at 7:52
vixra.org/abs/1003.0235 here my paper on how can the zeta regularization and Ramanujan resummation be used to get finite values in quantum mechanics – Jose Javier Garcia May 27 '13 at 17:18
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C. Johnny Solving
time limit per test
1 second
memory limit per test
256 megabytes
input
standard input
output
standard output
Today is tuesday, that means there is a dispute in JOHNNY SOLVING team again: they try to understand who is Johnny and who is Solving. That's why guys asked Umnik to help them. Umnik gave guys a connected graph with $n$ vertices without loops and multiedges, such that a degree of any vertex is at least $3$, and also he gave a number $1 \leq k \leq n$. Because Johnny is not too smart, he promised to find a simple path with length at least $\frac{n}{k}$ in the graph. In reply, Solving promised to find $k$ simple by vertices cycles with representatives, such that:
• Length of each cycle is at least $3$.
• Length of each cycle is not divisible by $3$.
• In each cycle must be a representative - vertex, which belongs only to this cycle among all printed cycles.
You need to help guys resolve the dispute, for that you need to find a solution for Johnny: a simple path with length at least $\frac{n}{k}$ ($n$ is not necessarily divided by $k$), or solution for Solving: $k$ cycles that satisfy all the conditions above. If there is no any solution - print $-1$.
Input
The first line contains three integers $n$, $m$ and $k$ ($1 \leq k \leq n \leq 2.5 \cdot 10^5, 1 \leq m \leq 5 \cdot 10^5$)
Next $m$ lines describe edges of the graph in format $v$, $u$ ($1 \leq v, u \leq n$). It's guaranteed that $v \neq u$ and all $m$ pairs are distinct.
It's guaranteed that a degree of each vertex is at least $3$.
Output
Print PATH in the first line, if you solve problem for Johnny. In the second line print the number of vertices in the path $c$ ($c \geq \frac{n}{k}$). And in the third line print vertices describing the path in route order.
Print CYCLES in the first line, if you solve problem for Solving. In the following lines describe exactly $k$ cycles in the following format: in the first line print the size of the cycle $c$ ($c \geq 3$). In the second line print the cycle in route order. Also, the first vertex in the cycle must be a representative.
Print $-1$ if there is no any solution. The total amount of printed numbers in the output must be at most $10^6$. It's guaranteed, that if exists any solution then there is a correct output satisfies this restriction.
Examples
Input
4 6 2
1 2
1 3
1 4
2 3
2 4
3 4
Output
PATH
4
1 2 3 4
Input
10 18 2
1 2
1 3
1 4
1 5
1 6
1 7
1 8
1 9
1 10
2 3
3 4
2 4
5 6
6 7
5 7
8 9
9 10
8 10
Output
CYCLES
4
4 1 2 3
4
7 1 5 6
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### begriffs
During tonight’s Madrailers meetup everyone worked on a kata to play the “word morph” game. You pick two arbitrary words and see if one can be converted to the other by changing only one letter at a time where the resulting intermediate words are valid.
It’s an interesting puzzle, and rather than solve it directly I wanted to discover some general statistics about how connected English words are generally. Rather than choosing two words and seeing if they are connected, I wanted to survey the equivalence classes of this connectedness.
I needed to write a program, but I realized that the program can operate on more general structures than simply words and letter manipulations. It turns out that the relation of “differing by one letter” is symmetric, and our program can extend it to a full equivalence relation and then calculate its equivalence classes on a finite set.
To get an idea how this works, let’s consider the symmetric relation on integers of “differing by three.” The transitive closure of this relation is “differing by any nonzero multiple of three” and the reflexive closure of that is “differing by any multiple of three.” The equivalence classes of this last relation on [0,1,2,3,4,5,6,7,8] are [[0,3,6], [1,4,7], [2,5,8]]. Let’s look at the general JavaScript code which acts on symmetric relations and see how it behaves on the number example.
function eq_classes(set, rel) {
var classes = {}, connected_stack, root,
stack_top, elt, pushed_more;
while(set.length > 0) {
root = set.shift();
classes[root] = [root];
connected_stack = [root];
while(connected_stack.length > 0) {
stack_top = connected_stack[connected_stack.length - 1];
pushed_more = false;
for(elt in set) {
if(rel(stack_top, set[elt])) {
pushed_more = true;
connected_stack.push(set[elt]);
classes[root].push(set[elt]);
set.splice(elt, 1);
}
}
if(!pushed_more) {
connected_stack.pop();
}
}
}
return classes;
}
If we run it, we see that it returns an object with each representative mapping to its class:
eq_classes(
[0,1,2,3,4,5,6,7,8],
function(a,b) { return Math.abs(a-b) === 3; }
);
// returns {0: [0,3,6], 1: [1,4,7], 2: [2,5,8]}
Returning to the word morph game, I ran eq_classes() on /usr/share/dict/words with the word relation of “differing by one letter.” The results are interesting. Obviously all one-letter words are equivalent to one another. But so are all two-letter words, and almost all three-letter words. The exceptions are “Eli”, “Emm”, “Osc”, “edh”, “its”, and “nth” which are not equivalent to any word except themselves.
The vast majority of four letter words are all equivalent. There are 158 equivalence classes, most of which have only one word ([“ruby”] is one such). After the class containing 5073 words, the next largest classes have four words (such as [“idic”, “odic”, “Udic”, “otic”]).
The longer the words become, the more disconnected their classes become. By the time you get to fourteen-letter words, there are 9233 classes among 9765 words. The largest class has seven items: [“invendibleness”, “inventibleness”, “unvendibleness”, “unvendableness”, “unbendableness”, “unmendableness”, “unbondableness”], but there are 8759 single-word classes.
What does this tell us? The only reason that every word is not equivalent to every other of the same length is that we tend to avoid certain combinations of letters. Small words are convenient to write, so we have exhausted a greater ratio of valid short words to total short letter combinations. However we are wasteful and choose longer sequences of letters haphazardly, leaving big holes in the larger state space. If our symbols were all phonetic (such as in a language like Telugu which spells by the syllable) I think we would have greater connectedness.
I like this challenge not just for what it tells us about English, but as an opportunity to think of algorithms more abstractly. That said, somebody in the 1970s has doubtless written a similar function that runs an order of magnitude faster than mine. Let me know if you see a way to improve it.
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# Chapter 05 - Perspective projection
In this chapter, we will learn two important concepts, perspective projection (to render far away objects smaller than closer ones) and uniforms (a buffer like structure to pass additional data to the shader).
You can find the complete source code for this chapter here.
## Perspective projection
Let’s get back to our nice colored quad we created in the previous chapter. If you look carefully, you will see that the quad is distorted and appears as a rectangle. You can even change the width of the window from 600 pixels to 900 and the distortion will be more evident. What’s happening here?
If you revisit our vertex shader code we are just passing our coordinates directly. That is, when we say that a vertex has a value for coordinate x of 0.5 we are saying to OpenGL to draw it at x position 0.5 on our screen. The following figure shows the OpenGL coordinates (just for x and y axis).
Coordinates
Those coordinates are mapped, considering our window size, to window coordinates (which have the origin at the top-left corner of the previous figure). So, if our window has a size of 900x580, OpenGL coordinates (1,0) will be mapped to coordinates (900, 0) creating a rectangle instead of a quad.
Rectangle
But, the problem is more serious than that. Modify the z coordinate of our quad from 0.0 to 1.0 and to -1.0. What do you see? The quad is exactly drawn in the same place no matter if it’s displaced along the z axis. Why is this happening? Objects that are further away should be drawn smaller than objects that are closer. But we are drawing them with the same x and y coordinates.
But, wait. Should this not be handled by the z coordinate? The answer is yes and no. The z coordinate tells OpenGL that an object is closer or farther away, but OpenGL does not know anything about the size of your object. You could have two objects of different sizes, one closer and smaller and one bigger and further that could be projected correctly onto the screen with the same size (those would have same x and y coordinates but different z). OpenGL just uses the coordinates we are passing, so we must take care of this. We need to correctly project our coordinates.
Now that we have diagnosed the problem, how do we fix it? The answer is using a perspective projection matrix. The perspective projection matrix will take care of the aspect ratio (the relation between size and height) of our drawing area so objects won’t be distorted. It also will handle the distance so objects far away from us will be drawn smaller. The projection matrix will also consider our field of view and the maximum distance to be displayed.
For those not familiar with matrices, a matrix is a bi-dimensional array of numbers arranged in columns and rows. Each number inside a matrix is called an element. A matrix order is the number of rows and columns. For instance, here you can see a 2x2 matrix (2 rows and 2 columns).
2x2 Matrix
Matrices have a number of basic operations that can be applied to them (such as addition, multiplication, etc.) that you can consult in a math book. The main characteristics of matrices, related to 3D graphics, is that they are very useful to transform points in the space.
You can think about the projection matrix as a camera, which has a field of view and a minimum and maximum distance. The vision area of that camera will be obtained from a truncated pyramid. The following picture shows a top view of that area.
Projection Matrix concepts
A projection matrix will correctly map 3D coordinates so they can be correctly represented on a 2D screen. The mathematical representation of that matrix is as follows (don’t be scared).
Projection Matrix
Where aspect ratio is the relation between our screen width and our screen height (
$a=width/height$
). In order to obtain the projected coordinates of a given point we just need to multiply the projection matrix by the original coordinates. The result will be another vector that will contain the projected version.
So we need to handle a set of mathematical entities such as vectors, matrices and include the operations that can be done on them. We could choose to write all that code by our own from scratch or use an already existing library. We will choose the easy path and use a specific library for dealing with math operations in LWJGL which is called JOML (Java OpenGL Math Library). In order to use that library we just need to add another dependency to our pom.xml file.
<dependency>
<groupId>org.joml</groupId>
<artifactId>joml</artifactId>
<version>\${joml.version}</version>
</dependency>
Now that everything has been set up let’s define our projection matrix. We will create a new class named Projection which is defined like this:
package org.lwjglb.engine.scene;
import org.joml.Matrix4f;
public class Projection {
private static final float FOV = (float) Math.toRadians(60.0f);
private static final float Z_FAR = 1000.f;
private static final float Z_NEAR = 0.01f;
private Matrix4f projMatrix;
public Projection(int width, int height) {
projMatrix = new Matrix4f();
updateProjMatrix(width, height);
}
public Matrix4f getProjMatrix() {
return projMatrix;
}
public void updateProjMatrix(int width, int height) {
projMatrix.setPerspective(FOV, (float) width / height, Z_NEAR, Z_FAR);
}
}
As you can see, it relies on the Matrix4f class (provided by the JOML library) which provides a method to set up a perspective projection matrix named setPerspective. This method needs the following parameters:
• Field of View: The Field of View angle in radians. We just use the FOV constant for that
• Aspect Ratio: That is, the relation ship between render width and height.
• Distance to the near plane (z-near)
• Distance to the far plane (z-far).
We will store a Projection class instance in the Scene class and initialize it in the constructor. IN addition to that, we weill need to take care if the window is resized, so we provide a new method in that Scene class, named resize to recalculate the perspective projection matrix when window dimensions change.
public class Scene {
...
private Projection projection;
public Scene(int width, int height) {
...
projection = new Projection(width, height);
}
...
public Projection getProjection() {
return projection;
}
public void resize(int width, int height) {
projection.updateProjMatrix(width, height);
}
}
We need also to update the Engine to adapt it to the new Scene class constructor parameters and to invoke the resize method:
public class Engine {
...
public Engine(String windowTitle, Window.WindowOptions opts, IAppLogic appLogic) {
...
scene = new Scene(window.getWidth(), window.getHeight());
...
}
...
private void resize() {
scene.resize(window.getWidth(), window.getHeight());
}
...
}
## Uniforms
Now that we have the infrastructure to calculate the perspective projection matrix, how do we use it? We need to use it in our shader, and it should be applied to all the vertices. At first, you could think of bundling it in the vertex input (like the coordinates and the colors). In this case we would be wasting lots of space since the projection matrix is common to any vertex. You may also think of multiplying the vertices by the matrix in the java code. But then, our VBOs would be useless and we will not be using the process power available in the graphics card.
The answer is to use “uniforms”. Uniforms are global GLSL variables that shaders can use and that we will employ to pass data that is common to all elements or to a model. So, let's start with how uniforms are use din shader programs. We need to modify our vertex shader code and declare a new uniform called projectionMatrix and use it to calculate the projected position.
#version 330
layout (location=0) in vec3 position;
layout (location=1) in vec3 color;
out vec3 outColor;
uniform mat4 projectionMatrix;
void main()
{
gl_Position = projectionMatrix * vec4(position, 1.0);
outColor = color;
}
As you can see we define our projectionMatrix as a 4x4 matrix and the position is obtained by multiplying it by our original coordinates. Now we need to pass the values of the projection matrix to our shader. We will create a new class named UniformMap which will allow us to create references to the uniforms and set up their values. It starts like this:
package org.lwjglb.engine.graph;
import org.joml.Matrix4f;
import org.lwjgl.system.MemoryStack;
import java.util.*;
import static org.lwjgl.opengl.GL20.*;
public class UniformsMap {
private int programId;
private Map<String, Integer> uniforms;
public UniformsMap(int programId) {
this.programId = programId;
uniforms = new HashMap<>();
}
public void createUniform(String uniformName) {
int uniformLocation = glGetUniformLocation(programId, uniformName);
if (uniformLocation < 0) {
throw new RuntimeException("Could not find uniform [" + uniformName + "] in shader program [" +
programId + "]");
}
uniforms.put(uniformName, uniformLocation);
}
...
}
As you can see, the constructor receives the identifier of the shader program and it defines a Map to store the references (Integer instances) to uniforms which are create din the createUniform method. Uniforms references are retrieved by calling the glGetUniformLocation function, which receives two parameters:
• The name of the uniform (it should match the one defined in the shader code).
As you can see, uniform creation is independent on the data type associated to it. We will need to have separate methods for the different types when we want to set the data for that uniform. By now, we will just need a method to load a 4x4 matrix:
public class UniformsMap {
...
public void setUniform(String uniformName, Matrix4f value) {
try (MemoryStack stack = MemoryStack.stackPush()) {
Integer location = uniforms.get(uniformName);
if (location == null) {
throw new RuntimeException("Could not find uniform [" + uniformName + "]");
}
glUniformMatrix4fv(location.intValue(), false, value.get(stack.mallocFloat(16)));
}
}
}
Now, we can use the code above in the SceneRender class:
public class SceneRender {
...
private UniformsMap uniformsMap;
public SceneRender() {
...
createUniforms();
}
...
private void createUniforms() {
uniformsMap.createUniform("projectionMatrix");
}
...
public void render(Scene scene) {
...
uniformsMap.setUniform("projectionMatrix", scene.getProjection().getProjMatrix());
...
}
}
We are almost done. We can now show the quad correctly rendered, So you can now launch your program and will obtain a... black background without any coloured quad. What’s happening? Did we break something? Well, actually no. Remember that we are now simulating the effect of a camera looking at our scene. And we provided two distances, one to the farthest plane (equal to 1000f) and one to the closest plane (equal to 0.01f). Our coordinates were:
float[] positions = new float[]{
-0.5f, 0.5f, 0.0f,
-0.5f, -0.5f, 0.0f,
0.5f, -0.5f, 0.0f,
0.5f, 0.5f, 0.0f,
};
That is, our z coordinates are outside the visible zone. Let’s assign them a value of -0.05f. Now you will see a giant square like this:
Square 1
What is happening now is that we are drawing the quad too close to our camera. We are actually zooming into it. If we assign now a value of -1.0f to the z coordinate we can now see our coloured quad.
public class Main implements IAppLogic {
...
public static void main(String[] args) {
...
Engine gameEng = new Engine("chapter-05", new Window.WindowOptions(), main);
...
}
...
public void init(Window window, Scene scene, Render render) {
float[] positions = new float[]{
-0.5f, 0.5f, -1.0f,
-0.5f, -0.5f, -1.0f,
0.5f, -0.5f, -1.0f,
0.5f, 0.5f, -1.0f,
};
...
}
...
}
Square coloured
If we continue pushing the quad backwards we will see it becoming smaller. Notice also that our quad does not appear as a rectangle anymore.
|
Springe direkt zu Inhalt
# Moduli of Tropical Plane Curves- Sarah Brodsky
29.01.2015 | 14:15
WHEN: 29.01.15 at 14:15
WHERE: Seminar Room, Arnimallee 2, FU Berlin
Speaker: Sarah Brodsky (TU Berlin)
Moduli of Tropical Plane Curves
\textit{Tropical curves} have been studied under two perspectives; the first perspective defines a tropical curve in terms of the \textit{tropical semifield} $\mathbb{T}=(\mathbb{R}\cup \{-\infty\}, \max, +)$, and the second perspective defines a tropical curve as a metric graph with a particular weight function on its vertices. Joint work with Michael Joswig, Ralph Morrison, and Bernd Sturmfels, we study which metric graphs of genus $g$ can be realized as smooth, plane tropical curves of genus $g$ with the motivation of understanding where these two perspectives meet.
Using \textit{Polymake}, \textit{TOPCOM}, and other computational tools, we conduct our study by constructing a map taking smooth, plane tropical curves of genus $g$ into the moduli space of metric graphs of genus $g$ and studying the image of this map. In particular, we focus on the cases when $g=2,3,4,5$. In this talk, we will introduce tropical geometry, discuss the motivation for this study, our methodology, and our results.
-----
### Zeit & Ort
29.01.2015 | 14:15
Seminar Room, Arnimallee 2, FU Berlin
|
Entropic value at risk
In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value-at-risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid,[1][2] which is an upper bound for the value-at-risk (VaR) and the conditional value-at-risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value-at-risk". The EVaR was developed to tackle some computational inefficienciesTemplate:Clarify of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid[1][2] developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class.
Definition
Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space with ${\displaystyle \Omega }$ a set of all simple events, ${\displaystyle {\mathcal {F}}}$ a ${\displaystyle \sigma }$-algebra of subsets of ${\displaystyle \Omega }$ and ${\displaystyle P}$ a probability measure on ${\displaystyle {\mathcal {F}}}$. Let ${\displaystyle X}$ be a random variable and ${\displaystyle {\mathbf {L} }_{M^{+}}}$ be the set of all Borel measurable functions ${\displaystyle X:\Omega \rightarrow \mathbb {R} }$ whose moment-generating function ${\displaystyle M_{X}(z)}$ exists for all ${\displaystyle z\geq 0}$. The entropic value-at-risk (EVaR) of ${\displaystyle X\in \mathbf {L} _{M^{+}}}$ with confidence level ${\displaystyle 1-\alpha }$ is defined as follows: Template:NumBlk In finance, the random variable ${\displaystyle X\in \mathbf {L} _{M^{+}}}$, in the above equation, is used to model the losses of a portfolio.
Consider the Chernoff inequality Template:NumBlk Solving the equation ${\displaystyle e^{-za}M_{X}(z)=\alpha }$ for ${\displaystyle a}$, results in ${\displaystyle a_{X}(\alpha ,z):=z^{-1}\ln(M_{X}(z)/\alpha )}$. By considering the equation (Template:EquationNote), we see that ${\displaystyle {\text{EVaR}}_{1-\alpha }(X):=\inf _{z>0}\{a_{X}(\alpha ,z)\}}$, which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that ${\displaystyle a_{X}(1,z)}$ is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively.
Let ${\displaystyle \mathbf {L} _{M}}$ be the set of all Borel measurable functions ${\displaystyle X:\Omega \rightarrow \mathbb {R} }$ whose moment-generating function ${\displaystyle M_{X}(z)}$ exists for all ${\displaystyle z}$. The dual representation (or robust representation) of the EVaR is as follows: Template:NumBlk where ${\displaystyle X\in \mathbf {L} _{M}}$, and ${\displaystyle \Im }$ is a set of probability measures on ${\displaystyle (\Omega ,{\mathcal {F}})}$ with ${\displaystyle \Im =\{Q\ll P:D_{KL}(Q||P)\leq -\ln \alpha \}}$. Note that ${\displaystyle D_{KL}(Q||P):=\int {\frac {dQ}{dP}}(\ln {\frac {dQ}{dP}})dP}$ is the relative entropy of ${\displaystyle Q}$ with respect to ${\displaystyle P}$, also called the Kullback-Leibler divergence. The dual representation of the EVaR discloses the reason behind its naming.
Properties
• The following inequality holds for the EVaR:
Examples
File:Comparing the VaR, CVaR and EVaR for the standard normal distribution.png
Comparing the VaR, CVaR and EVaR for the standard normal distribution
File:Comparing the VaR, CVaR and EVaR for the uniform distribution.png
Comparing the VaR, CVaR and EVaR for the uniform distribution over the interval (0,1)
For ${\displaystyle X\sim N(\mu ,\sigma )}$, Template:NumBlk For ${\displaystyle X\sim U(a,b)}$, Template:NumBlk Figures 1 and 2 show the comparing of the VaR, CVaR and EVaR for ${\displaystyle N(0,1)}$ and ${\displaystyle U(0,1)}$.
Optimization
Let ${\displaystyle \rho }$ be a risk measure. Consider the optimization problem Template:NumBlk where ${\displaystyle {\boldsymbol {w}}\in {\boldsymbol {W}}\subseteq \mathbb {R} ^{n}}$ is an ${\displaystyle n}$-dimensional real decision vector, ${\displaystyle {\boldsymbol {\psi }}}$ is an ${\displaystyle m}$-dimensional real random vector with a known probability distribution and the function ${\displaystyle G({\boldsymbol {w}},.):\mathbb {R} ^{m}\rightarrow \mathbb {R} }$ is a Borel measurable function for all values ${\displaystyle {\boldsymbol {w}}\in {\boldsymbol {W}}}$. If ${\displaystyle \rho }$ is the ${\displaystyle {\text{EVaR}}}$, then the problem (Template:EquationNote) becomes as follows: Template:NumBlk
Let ${\displaystyle {\boldsymbol {S}}_{\boldsymbol {\psi }}}$ be the support of the random vector ${\displaystyle {\boldsymbol {\psi }}}$. If ${\displaystyle G(.,{\boldsymbol {s}})}$ is convex for all ${\displaystyle {\boldsymbol {s}}\in {\boldsymbol {S}}_{\boldsymbol {\psi }}}$, then the objective function of the problem (Template:EquationNote) is also convex. If ${\displaystyle G({\boldsymbol {w}},{\boldsymbol {\psi }})}$ has the form Template:NumBlk and ${\displaystyle \psi _{1},\dots ,\psi _{m}}$ are independent random variables in ${\displaystyle \mathbf {L} _{M}}$, then (Template:EquationNote) becomes Template:NumBlk which is computationally tractable. But for this case, if one uses the CVaR in problem (Template:EquationNote), then the resulting problem becomes as follows: Template:NumBlk It can be shown that by increasing the dimension of ${\displaystyle \psi }$, problem (Template:EquationNote) is computationally intractable even for simple cases. For example, assume that ${\displaystyle \psi _{1},\dots ,\psi _{m}}$ are independent discrete random variables that take ${\displaystyle k}$ distinct values. For fixed values of ${\displaystyle {\boldsymbol {w}}}$ and ${\displaystyle t}$, the complexity of computing the objective function given in problem (Template:EquationNote) is of order ${\displaystyle mk}$ while the computing time for the objective function of problem (Template:EquationNote) is of order ${\displaystyle k^{m}}$. For illustration, assume that ${\displaystyle k=2}$, ${\displaystyle m=100}$ and the summation of two numbers takes ${\displaystyle 10^{-12}}$ seconds. For computing the objective function of problem (Template:EquationNote) one needs about ${\displaystyle 4\times 10^{10}}$ years, whereas the evaluation of objective function of problem (Template:EquationNote) takes about ${\displaystyle 10^{-10}}$ seconds. This shows that formulation with the EVaR outperforms the formulation with the CVaR (see [2] for more details).
Generalization (g-entropic risk measures)
Drawing inspiration from the dual representation of the EVaR given in (Template:EquationNote), one can define a wide class of information-theoretic coherent risk measures, which are introduced in.[1][2] Let ${\displaystyle g}$ be a convex proper function with ${\displaystyle g(1)=0}$ and ${\displaystyle \beta }$ be a non-negative number. The ${\displaystyle g}$-entropic risk measure with divergence level ${\displaystyle \beta }$ is defined as Template:NumBlk where ${\displaystyle \Im =\{Q\ll P:H_{g}(P,Q)\leq \beta \}}$ in which ${\displaystyle H_{g}(P,Q)}$ is the generalized relative entropy of ${\displaystyle Q}$ with respect to ${\displaystyle P}$. A primal representation of the class of ${\displaystyle g}$-entropic risk measures can be obtained as follows: Template:NumBlk where ${\displaystyle g^{*}}$ is the conjugate of ${\displaystyle g}$. By considering Template:NumBlk with ${\displaystyle g^{*}(x)=e^{x-1}}$ and ${\displaystyle \beta =-\ln \alpha }$, the EVaR formula can be deduced. The CVaR is also a ${\displaystyle g}$-entropic risk measure, which can be obtained from (Template:EquationNote) by setting Template:NumBlk with ${\displaystyle g^{*}(x)={\frac {1}{\alpha }}\max\{0,x\}}$ and ${\displaystyle \beta =0}$ (see [1][3] for more details).
For more results on ${\displaystyle g}$-entropic risk measures see.[4]
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# salary increments
• August 12th 2009, 08:26 AM
Rose Wanjohi
salary increments
Ali has astarting salary of 5000dollars with annual increment of 350dollars , after how many years will his total earning be 265750 dollars(Cool)
• August 12th 2009, 09:07 AM
Amer
Quote:
Originally Posted by Rose Wanjohi
Ali has astarting salary of 5000dollars with annual increment of 350dollars , after how many years will his total earning be 265750 dollars(Cool)
the increment per month right ??
if the increment per month
$5000+m350=265750$ find m value m represent month numbers
• August 12th 2009, 09:38 AM
Rose Wanjohi
no thats not it the answer cannot be 63 years,it's meant to be 10years
• August 12th 2009, 09:43 AM
Wilmer
Can you be a bit CLEARER please;
after 3 years, is it 5000 + (5000+350) + (5000+700) ?
• August 12th 2009, 11:12 AM
Rose Wanjohi
guess we saling on the same boat of confusion huh? i got the question from a book,guess there was a mistake
|
A gas mixture of 3.67 L of ethylene and methane on complete combustion at 25° C produces 6.11 L of CO_2 Find
### Question Asked by a Student from EXXAMM.com Team
Q 2656256174. A gas mixture of 3.67 L of ethylene and methane on complete combustion at 25° C produces 6.11 L of CO_2 Find
out the amount of heat evolved on burning 1 L of the gas mixture. The heat of combustion of ethylene and methane
are - 1423 and - 891 kJ mol^(-1 ) at 25°C.
JEE 1991
#### HINT
(Provided By a Student and Checked/Corrected by EXXAMM.com Team)
#### Access free resources including
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• Citation and Abstract
Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu Province, China
## Abstract Full Text(HTML) Figure/Table Related pages
In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number $\mathcal{R}_{0}>1$. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c=0, some new dynamical behaviors appear in the model studied here; in particular, $\mathcal{R}_{0}$ is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.
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Citation: Jinzhe Suo, Bo Li. Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment. Mathematical Biosciences and Engineering, 2020, 17(1): 418-441. doi: 10.3934/mbe.2020023
References
• 1. L. J. S. Allen, B. M. Bolker, Y. Lou, et al., Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Dis. Contin. Dyn. Syst. A, 21 (2008), 1-20.
• 2. R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I, J. Differ. Equations, 247 (2009), 1096-1119.
• 3. R. M. Anderson and R. M. May, Populaition biology of infectious diseases, Nature 280 (1979), 361-367.
• 4. R. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differ. Equations, 261 (2016), 3305-3343.
• 5. R. Cui, K.-Y. Lam and Y. Lou, Dynamics and asymptotic profiles of steady states of an epidemic model in advective environments, J. Differ. Equations, 263 (2017), 2343-2373.
• 6. Z. Du and R. Peng, A priori L-estimates for solutions of a class of reaction-diffusion systems, J. Math. Biol., 72 (2016), 1429-1439.
• 7. H. W. Hethcote, Epidemiology models with variable population size, Mathematical understanding of infectious disease dynamics, 63-89, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 16, World Sci. Publ., Hackensack, NJ, 2009.
• 8. W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics-I, Proc. Roy. Soc. London Ser. A, 115 (1927), 700-721.
• 9. M. Martcheva, An introduction to mathmatical epidemiology, Springer,New York, (2015).
• 10. H. W Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.
• 11. B. Li, H. Li and Y. Tong, Analysis on a diffusive SIS epidemic model with logistic source, Z. Angew. Math. Phys., 68 (2017), Art. 96, 25pp.
• 12. H. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model, J. Differ. Equations, 262 (2017), 885-913.
• 13. M. E. Alexander and S. M. Moghadas, Bifurcation Analysis of an SIRS Epidemic Model with Generalized Incidence, SIAM J. Appl. Math., 65 (2001), 1794-1816.
• 14. R. M. Anderson and R. M. May, Regulation and stability of host-parasite interactions.I. Regulatory processes, J. Anim. Ecol., 47 (1978), 219-247.
• 15. O. Diekmann and M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol., 29 (1991), 539-570.
• 16. J. A. P. Heesterbeck and J. A. J. Metz, The saturating contact rate in marriage and epidemic models, J. Math. Biol., 31 (1993), 529-539.
• 17. M. G. Roberts, The dynamics of bovine tuberculosis in possum populations and its eradication or control by culling or vaccination, J. Anim. Ecol., 65 (1996), 451-464.
• 18. Y. Cai, K. Wang and W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.
• 19. L. Chen and J. Sun, Optimal vaccination and treatment of an epidemic network model, Physics Lett. A, 378 (2014), 3028-3036.
• 20. L. Chen and J. Sun, Global stability and optimal control of an SIRS epidemic model on heterogeneous networks, Physica A, 410 (2014), 196-204.
• 21. K. Deng and Y. Wu, Dynamics of a susceptible-infected-susceptible epidemic reaction-diffusion model, Proc. Roy. Soc. Edinburgh Sect. A, 146 (2016), 929-946.
• 22. X. Gao, Y. Cai, F. Rao, et al., Positive steady states in an epidemic model with nonlinear incidence rate, Comput. Math. Appl., 75 (2018), 424-443.
• 23. J. Ge, C. Lei and Z. Lin, Reproduction numbers and the expanding fronts for a diffusion-advection SIS model in heterogeneous time-periodic environment, Nonlinear Anal. Real World Appl., 33 (2017), 100-120.
• 24. K. Kuto, H. Matsuzawa and R. Peng, Concentration profile of endemic equilibrium of a reactiondiffusion-advection SIS epidemic model, Calc. Var. Partial Dif., 56 (2017), Art. 112, 28 pp.
• 25. C. Lei, F. Li and J. Liu, Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 4499- 4517.
• 26. B. Li and Q. Bie, Long-time dynamics of an SIRS reaction-diffusion epidemic model, J. Math. Anal. Appl., 475 (2019), 1910-1926.
• 27. H. Li, R. Peng and Z. Wang, On a diffusive SIS epidemic model with mass action mechanism and birth-death effect: analysis, simulations and comparison with other mechanisms, SIAM J. Appl. Math., 78 (2018), 2129-2153.
• 28. H. Li, R. Peng and T. Xiang, Dynamics and asymptotic profiles of endemic equilibrium for two frequency-dependent SIS epidemic models with cross-diffusion, Eur. J. Appl. Math., 2019, https://doi.org/10.1017/S0956792518000463, in press.
• 29. Z. Lin, Y. Zhao and P. Zhou, The infected frontier in an SEIR epidemic model with infinite delay, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 2355-2376.
• 30. R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Anal., 71 (2009), 239-247.
• 31. L. Pu and Z. Lin, A diffusive SIS epidemic model in a heterogeneous and periodically evolving environment, 16 (2019), 3094-3110.
• 32. X. Wen, J. Ji and B. Li, Asymptotic profiles of the endemic equilibrium to a diffusive SIS epidemic model with mass action infection mechanism, J. Math. Anal. Appl., 458 (2018), 715-729.
• 33. Y. Wu and X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism, J. Differ. Equations, 261 (2016), 4424-4447.
• 34. M. Zhu, X. Guo and Z. Lin, The risk index for an SIR epidemic model and spatial spreading of the infectious disease, Math. Biosci. Eng., 14 (2017), 1565-1583.
• 35. R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471.
• 36. P. Magal and X.-Q Zhao, Global attractive and steady states for uniformly persistent dynamical systems, SIAM. J. Math. Anal., 37 (2005), 251-275.
• 37. X.-Q. Zhao, Dynamical Systems in Populaition Biology, Springer-Verlag, New York,(2003)
• 38. M. Wang, Nonlinear Partial Differential Equations of Parabolic Type, Science Press, Beijing, 1993(in chinese).
• 39. K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Differ. Equations, 40 (1981), 232-252.
• 40. G. M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary p in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.
• 41. R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.
• 42. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equation of Second Order, Springer, (2001).
• 43. Y. Du, R. Peng and M. Wang, Effect of a protection zone in the diffusive Leslie predator-prey model, J. Differ. Equations, 246 (2009), 3932-3956.
• 44. Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differ. Equations, 131 (1996), 79-131.
• 45. W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and eystems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.
• 46. H. Brezis and W. A. Strauss, Semi-linear second-order elliptic equations in L1, J. Math. Soc. Japan, 25 (1973), 565-590.
• 47. R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reactiondiffusion model: Effects of epidemic risk and population movement, Phys. D, 259 (2013), 8-25.
|
+0
# What is the wavelength in meters of ultraviolet light withν = 2.50 x 1015 s-1?
0
238
1
What is the wavelength in meters of ultraviolet light withν = 2.50 x 1015 s-1?
physics
Guest Feb 12, 2015
#1
+18829
+5
What is the wavelength in meters of ultraviolet light with ν = 2.50 x 1015 s-1 ?
$$\\\small{\text{Light moves with a speed c \,=\, 299792458\ \frac{\text{m}}{\text{s}} }}\\ \small{\text{We denote wavelength by \lambda = \mathrm{wavelength}}}\\ \small{\text{We denote frequency by \nu=\mathrm{frequency} }}\\ \small{\text{Frequency is measured in Hertz = Hz = s^{-1}.}}\\ \\ \small{\text{ \boxed{ \lambda =\dfrac{c}{\nu} }\qquad \nu = 2.50 \cdot 10^{15}\ s^{-1} }}\\\\ \small{\text{ \lambda = \dfrac{2.99792458\cdot 10^8\ \dfrac{m}{s} } {2.50 \cdot 10^{15}\ s^{-1} } }}\\\\ \small{\text{ = 1.199169832 \cdot 10^{-7}\ m }}\\ \small{\text{ = 119.9169832 \cdot 10^{-9}\ m }}\\ \small{\text{ = 119.9\ nm }}$$
heureka Feb 12, 2015
Sort:
#1
+18829
+5
$$\\\small{\text{Light moves with a speed c \,=\, 299792458\ \frac{\text{m}}{\text{s}} }}\\ \small{\text{We denote wavelength by \lambda = \mathrm{wavelength}}}\\ \small{\text{We denote frequency by \nu=\mathrm{frequency} }}\\ \small{\text{Frequency is measured in Hertz = Hz = s^{-1}.}}\\ \\ \small{\text{ \boxed{ \lambda =\dfrac{c}{\nu} }\qquad \nu = 2.50 \cdot 10^{15}\ s^{-1} }}\\\\ \small{\text{ \lambda = \dfrac{2.99792458\cdot 10^8\ \dfrac{m}{s} } {2.50 \cdot 10^{15}\ s^{-1} } }}\\\\ \small{\text{ = 1.199169832 \cdot 10^{-7}\ m }}\\ \small{\text{ = 119.9169832 \cdot 10^{-9}\ m }}\\ \small{\text{ = 119.9\ nm }}$$
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Journal article Open Access
# A CIRCUMSTANTIAL ASSESSMENT AND INVESTIGATION OF THE SYNDROME CONTEMPORARY OLDEN TIMES IN THE WEST
Dr Kainat Naveed, Dr Ghulam Sarwar Sajid
### DataCite XML Export
<?xml version='1.0' encoding='utf-8'?>
<identifier identifierType="DOI">10.5281/zenodo.6855828</identifier>
<creators>
<creator>
<creatorName>Dr Kainat Naveed, Dr Ghulam Sarwar Sajid</creatorName>
<givenName>Dr Ghulam Sarwar Sajid</givenName>
<familyName>Dr Kainat Naveed</familyName>
</creator>
</creators>
<titles>
<title>A CIRCUMSTANTIAL ASSESSMENT AND INVESTIGATION OF THE SYNDROME CONTEMPORARY OLDEN TIMES IN THE WEST</title>
</titles>
<publisher>Zenodo</publisher>
<publicationYear>2022</publicationYear>
<dates>
<date dateType="Issued">2022-07-18</date>
</dates>
<resourceType resourceTypeGeneral="JournalArticle"/>
<alternateIdentifiers>
<alternateIdentifier alternateIdentifierType="url">https://zenodo.org/record/6855828</alternateIdentifier>
</alternateIdentifiers>
<relatedIdentifiers>
<relatedIdentifier relatedIdentifierType="DOI" relationType="IsVersionOf">10.5281/zenodo.6855827</relatedIdentifier>
</relatedIdentifiers>
<rightsList>
<rights rightsURI="info:eu-repo/semantics/openAccess">Open Access</rights>
</rightsList>
<descriptions>
<description descriptionType="Abstract"><p><em>The history revealed that the underlying 2-month-old child was shocking but consoling and had an exaggerated alarm reflex. After several months, the mother whined that the child had spasms, that is, a deviation of the eye to the left side, a tonic development of the colon of the two upper appendages that lasted for one minute and went off as expected with a recurrence of 27 times a day at first and now with medication it has decreased to 8-10 times a day. The youngster was on: syp. Gardinal 3.7ml oral Od and Tab. Bexel 1\4 Tds. The mother&#39;s prenatal history was typical, it was an ordinary transport at term, however the infant did not cry long after birth and required prompting and aspiration to be inhaled. Our current research was conducted at Lahore General Hospital, Lahore from December 2017 to November 2018. By the third day of life, the infant created severe respiratory distress and was transferred to the neonatal intensive care unit where he was rescued for about fourteen days and then released. He was placed in a lower class family unit and was placed under selective supervision until the age of six months, at which time weaning began. There is no critical family ancestry and the child is vaccinated up to the age stipulated in the national immunization plan. The disease began to appear in young people from the age of several months and had a symptomatic western disorder, the conceivable reason for which was asphyxia at birth, which required stimulation and aspiration. The infant has all three types of seizures that occur all the time and, in addition, is shattered, but this is consoling and relapse is also present. An 11-month-old child was brought to the emergency room with seizures and spasms, eye deviation to the left, fever and kicking. </em></p>
<p><strong><em>Key words: </em></strong><em>Contextual analysis, Disorder, History. </em></p></description>
</descriptions>
</resource>
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# Is there any other margin I must set to reduce margin to zero?
I have basic class file and I would like to set margins of an a4paper to 0mm. Here is the markup of some-class.cls:
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{some-class}[2013/07/29]
\renewcommand{\normalsize}{\fontsize{10pt}{12pt}\selectfont}
\DeclareOption{a4paper}{
\setlength\paperheight {297mm}%
\setlength\paperwidth {210mm}
}
\DeclareOption{landscape} {
\setlength\@tempdima {\paperheight}%
\setlength\paperheight {\paperwidth}%
\setlength\paperwidth {\@tempdima}
}
%Default options for design
\ExecuteOptions{a4paper}
%Process user given options
\ProcessOptions
\RequirePackage[top=0mm, bottom=0mm, left=0mm, right=0mm, textwidth=210mm]{geometry}
Content of my document.tex:
\documentclass[]{some-class}
\author{roncsak}
\title{some title}
\date{2013/07/28}
\begin{document}
Lorem ipsum
\end{document}
I would like to set an A4 paper with zero margins. Instead I notice some margin on the left side (approximately 10mm). I could reduce that margin with the \RequirePackage[top=0mm, bottom=0mm, left=-10mm, right=0mm, textwidth=210mm]{geometry} but I would like to understand why this margin left behind.
-
\usepackage[margin=0pt]{geometry}% http://ctan.org/pkg/geometry
(or \RequirePackage within a class/package). The sets the margins and indirectly also the text block size.
The "margin" you notice is due to the paragraph indent that is set by default to 20pt. Add \noindent to a paragraph to remove this indent, or set it to zero in the preamble using \setlength{\parindent}{0pt} which will make the change global.
The \setlength{\parindent}{0pt} worked well! Thank you! – roncsak Jul 29 '13 at 18:59
@roncsak: Don't forget to set a non-zero \parskip (or load the package parskip) if you set \parindent to 0. – You Jul 29 '13 at 20:09
|
## The Annals of Mathematical Statistics
### A Generalization of Wald's Identity with Applications to Random Walks
H. D. Miller
#### Abstract
Let $S_m = X_1 + \cdots + X_m$, where the $X_j$ are independent random variables with common m.g.f. $\phi(t)$ which is assumed to exist in a real interval containing $t = 0$. Let the random variable $n$ be defined as the smallest integer $m$ for which either $S_m \geqq \alpha$ or $S_m \leqq - \beta(\alpha > 0, \beta > 0)$. Thus $n$ can be regarded as the time to absorption for the random walk $S_m$ with absorbing barriers at $\alpha$ and $-\beta$. Let $S = S_n$ and let $F_m(x) = P(-\beta < S_k < \alpha \quad \text{for}\quad k = 1, 2, \cdots m - 1 \quad \text{and} \quad S_m \leqq x)$. The main result of the paper is the identity \begin{equation*}\tag{0.1}E(e^{tS}z^n) = 1 + \lbrack z\phi (t) - 1\rbrack F(z, t),\end{equation*} where $F(z, t) = \sum^\infty_{m = 0} z^m \int^\alpha_{-\beta} e^{tx} dF_m(x).$ Wald's identity follows formally from (0.1) by setting $z = \lbrack\phi(t)\rbrack^{-1}$. Regions of validity of (0.1) and of Wald's identity are discussed, and it is shown that the latter holds for a larger range of values of $t$ than is usually supposed. In Section 5 there are three examples. In the first we consider the case where there is a single absorbing barrier and where the $X_j$ are discrete and bounded. This is a gambler's ruin problem, and we obtain an expression for the probability of ruin. In the second we use the classical random walk to illustrate the region of validity of (0.1). In the third we obtain the Laplace transform of the distribution of the time to absorption in a random walk in which steps of $+1$ and -1 occur at random in continuous time.
#### Article information
Source
Ann. Math. Statist., Volume 32, Number 2 (1961), 549-560.
Dates
First available in Project Euclid: 27 April 2007
https://projecteuclid.org/euclid.aoms/1177705060
Digital Object Identifier
doi:10.1214/aoms/1177705060
Mathematical Reviews number (MathSciNet)
MR126890
Zentralblatt MATH identifier
0109.10902
JSTOR
|
# Why is the character following this macro-defining macro gobbled up?
I have the following plain Tex code:
\def\xx{ABC}
\def\temp #1 {\def\tempii{#1}}
\temp\xx a b c
\tempii\tempii
I expect to see the following output: a b c ABCABC
But I actually see this: b c ABCaABCa
Why is the 'a' snatched into the definition of \tempii?
• The reasoning here is not limited to plain so I've retagged: I hope this makes sense. – Joseph Wright Sep 24 '14 at 11:02
I'm still thinking as to why it is, but this revision (embracing \xx) gives you what you want.
I think the answer is that, in the usage \temp\xx a b c, the space after \xx is from the TeX parser's view, not a space at all, but merely signifies the end of the macro name \xx. For example, if you just put \xx A on a line by itself, you will see that it prints out as ABCA without a space. Thus, the parser sucks in the trailing "A" as being connected to \xx. One could also use \temp\xx{} a b c, as a way to tell the parser that the space is a separator, and not just the next character following \xx.
\documentclass{article}
\begin{document}
\def\xx{ABC}
\def\temp #1 {\def\tempii{#1}}
\temp{\xx} a b c
\tempii\tempii
\end{document}
• You're surely right, and I feel like saying "Ah, of course." But now I don't understand why \temp\xx\ a b c doesn't give the desired result. It's like the escaped space is ignored by the pattern matcher. (Also, you converted the code from plain Tex to Latex, but that didn't matter.) – dedded Sep 24 '14 at 3:10
• @dedded I'm sure one of the erudite ones will chime in with a fully rational answer, but (again speculation because I'm out of my depth) I'm thinking that in an effort to get the argument to \temp, it expands until it comes to something that can't expand. In both cases (what I describe in my answer and what you describe in your comment), that would be the letter "a". – Steven B. Segletes Sep 24 '14 at 3:18
• @dedded \temp\xx\ a b doesn't have a space token after \xx, it has a control symbol \ (remember: TeX is all about tokens). You could do \def\firstofone#1{#1} then \firstofone{\temp\xx} a b c. – Joseph Wright Sep 24 '14 at 6:03
• @dedded the \ token isn't ignored it just doesn't match a space just as \a would not delimit an argument delimited by a. – David Carlisle Sep 24 '14 at 8:55
• Accepting this answer because with D. Carlisle's comment it most completely clears my misunderstanding, but I suspect egreg's detailed explanation is the more generally useful one. – dedded Sep 25 '14 at 19:49
For clarity I'll denote space tokens by •, as they are very important in the discussion; spaces in the following code samples should be ignored.
The parameter text of \temp is
#1•
while the replacement text is
\def\tempii{#1}
Your call of \temp is
\temp\xx a•b•c•\tempii\tempii
Notice that after \xx there's a space just for delimiting the macro; it is not a space token: it's nothing, because TeX always ignores spaces following control words (not control symbols). The space after c comes from the end-of-line in your code.
The tokens following \temp are scanned (without expansion) to find a match with the parameter text, where the argument is delimited by a space, so #1 ends being \xx a and the next state of the input stream is
\def\tempii{\xx a}b•c•\tempii\tempii
Now TeX performs the definition and removes it from the input stream:
b•c•\tempii\tempii
and the part b•c• is passed to the typesetting stage. After the expansion of the two copies of \tempii, this is equivalent to having typed
b•c•\xx a\xx a
or
b•c•ABCaABCa
which is exactly what you got.
If you had used \? instead of \xx the last two lines would have been converted into
\temp\?•a•b•c•\tempii\tempii
because \? is a control symbol and TeX doesn't ignore spaces after them. In this case the result would be
a•b•c•ABCABC
Note that
\expandafter\temp\xx a•b•c
\tempii\tempii
would produce the same result, because after the action of \expandafter TeX would be presented with
\temp ABCa•b•c
(same convention for spaces as before, of course). You'd get the same result as with \? instead of \xx with
\expandafter\temp\expandafter\xx\space a•b•c
because then TeX would be presented with
\temp\xx•a•b•c
Indeed spaces are ignored after control words only during the tokenization process.
|
## anonymous 4 years ago Please Explain: Simplify (7r to the power of 2 S to the power of 3)(-3rs to the power of 4) - (-2r to the power of 2 S)(5rs to the power of 6) thanks :)
When simplifying expressions with exponents you take each term (so any parts of the expression involving multiplication and division, terms are seperated by addition and subtraction), and you combine the exponents of similar bases by adding together the exponent when you are multiplying and subtracting when you are dividing. ie $x ^{2}\times x ^{3}=x ^{5}$
|
# SAT
Type Paper-based standardized test College Board, Educational Testin' Service Writin', critical readin', mathematics Admission to undergraduate programs of universities or colleges 1926; 95 years ago 3 hours (without the bleedin' essay) or 3 hours 50 minutes (with the bleedin' essay until June 2021) Test scored on scale of 200–800, (in 10-point increments), on each of two sections (total 400–1600).Essay scored on scale of 2–8, in 1-point increments, on each of three criteria 7 times annually[a] Worldwide English Over 2.19 million high school graduates in the oul' class of 2020[2] No official prerequisite. Intended for high school students. Arra' would ye listen to this shite? Fluency in English assumed. US\$55.00 to US\$108.00, dependin' on country.[3] Most universities and colleges offerin' undergraduate programs in the feckin' U.S. sat.collegeboard.org
2013 logo
The SAT (/ˌɛsˌˈt/ ess-ay-TEE) is a bleedin' standardized test widely used for college admissions in the feckin' United States. Since its debut in 1926, its name and scorin' have changed several times; originally called the Scholastic Aptitude Test, it was later called the feckin' Scholastic Assessment Test, then the bleedin' SAT I: Reasonin' Test, then the oul' SAT Reasonin' Test, then simply the bleedin' SAT.
The SAT is wholly owned, developed, and published by the oul' College Board, a bleedin' private, not-for-profit organization in the United States, what? It is administered on behalf of the feckin' College Board by the oul' Educational Testin' Service,[4] which until recently developed the oul' SAT as well.[5] The test is intended to assess students' readiness for college. The SAT was originally designed not to be aligned with high school curricula,[6] but several adjustments were made for the version of the bleedin' SAT introduced in 2016, and College Board president David Coleman has said that he also wanted to make the feckin' test reflect more closely what students learn in high school with the bleedin' new Common Core standards.[7]
The SAT takes three hours to finish and as of 2021 costs US\$55.00, excludin' late fees, with additional processin' fees if the feckin' SAT is taken outside the oul' United States.[8] Scores on the feckin' SAT range from 400 to 1600, combinin' test results from two 200-to-800-point sections: the feckin' Mathematics section and the bleedin' Evidence-Based Readin' and Writin' section. Jesus, Mary and holy Saint Joseph. Although takin' the oul' SAT, or its competitor the oul' ACT, is required for freshman entry to many colleges and universities in the United States,[9] durin' the 2010s, many institutions made these entrance exams optional,[10][11][12] but this did not stop the feckin' students from attemptin' to achieve high scores[13] as they and their parents are skeptical of what "optional" means in this context.[14][15] In fact, the test-takin' population was increasin' steadily.[16] And while this may have resulted in a bleedin' long-term decline in scores,[16][17][18] experts cautioned against usin' this to gauge the scholastic levels of the feckin' entire U.S. population.[18]
Startin' with the 2015–16 school year, the College Board began workin' with Khan Academy to provide free SAT preparation.[19] On January 19, 2021, the feckin' College Board announced the discontinuation of the optional essay section, as well as its SAT Subject Tests, after June 2021.[20][21]
While a holy considerable amount of research has been done on the SAT, many questions and misconceptions remain.[22][23] Outside of college admissions, the oul' SAT is also used by researchers studyin' human intelligence in general and intellectual precociousness in particular,[24][25][26] and by some employers in the feckin' recruitment process.[27][28][29]
## Function
U.S, to be sure. states in blue had more seniors in the oul' class of 2006 who took the bleedin' SAT than the ACT while those in red had more seniors takin' the oul' ACT than the oul' SAT.
U.S, bedad. states in blue had more seniors in the bleedin' class of 2020 who took the oul' SAT than the ACT while those in red had more seniors takin' the bleedin' ACT than the oul' SAT.
The SAT is typically taken by high school juniors and seniors.[30] The College Board states that the oul' SAT is intended to measure literacy, numeracy and writin' skills that are needed for academic success in college, to be sure. They state that the bleedin' SAT assesses how well the bleedin' test-takers analyze and solve problems—skills they learned in school that they will need in college. However, the bleedin' test is administered under a bleedin' tight time limit (speeded) to help produce a range of scores.[31]
The College Board also states that use of the oul' SAT in combination with high school grade point average (GPA) provides a bleedin' better indicator of success in college than high school grades alone, as measured by college freshman GPA. Various studies conducted over the feckin' lifetime of the SAT show a statistically significant increase in correlation of high school grades and college freshman grades when the oul' SAT is factored in.[32] The predictive validity and powers of the oul' SAT are topics of active research in psychometrics.[22]
There are substantial differences in fundin', curricula, gradin', and difficulty among U.S. Right so. secondary schools due to U.S. federalism, local control, and the bleedin' prevalence of private, distance, and home schooled students, bedad. SAT (and ACT) scores are intended to supplement the feckin' secondary school record and help admission officers put local data—such as course work, grades, and class rank—in a feckin' national perspective.[33]
Historically, the feckin' SAT was more widely used by students livin' in coastal states and the bleedin' ACT was more widely used by students in the feckin' Midwest and South; in recent years, however, an increasin' number of students on the East and West coasts have been takin' the oul' ACT.[34][35] Since 2007, all four-year colleges and universities in the feckin' United States that require a bleedin' test as part of an application for admission will accept either the bleedin' SAT or ACT, and as of Fall 2022, over 1400 four-year colleges and universities do not require any standardized test scores at all for admission, though some of them are applyin' this policy only temporarily due to the feckin' coronavirus pandemic.[36][37]
## Structure
The SAT has two main sections, namely Evidence-Based Readin' and Writin' (EBRW, normally known as the oul' "English" portion of the feckin' test) and the bleedin' Math section. Jesus, Mary and Joseph. These are both further banjaxed down into four sections: Readin', Writin' and Language, Math (no calculator), and Math (calculator allowed). The test taker was also optionally able to write an essay which, in that case, is the feckin' fifth test section, be the hokey! The total time for the scored portion of the feckin' SAT is three hours (or three hours and fifty minutes if the feckin' optional essay section was taken). Here's a quare one for ye. Some test takers who are not takin' the feckin' essay may also have a fifth section, which is used, at least in part, for the feckin' pretestin' of questions that may appear on future administrations of the bleedin' SAT, to be sure. (These questions are not included in the computation of the feckin' SAT score.)
Two section scores result from takin' the bleedin' SAT: Evidence-Based Readin' and Writin', and Math, so it is. Section scores are reported on a feckin' scale of 200 to 800, and each section score is a feckin' multiple of ten. A total score for the bleedin' SAT is calculated by addin' the two section scores, resultin' in total scores that range from 400 to 1600. Jesus, Mary and holy Saint Joseph. In addition to the two section scores, three "test" scores on a scale of 10 to 40 are reported, one for each of Readin', Writin' and Language, and Math, with increment of 1 for Readin' / Writin' and Language, and 0.5 for Math. There are also two cross-test scores that each range from 10 to 40 points: Analysis in History/Social Studies and Analysis in Science.[38] The essay, if taken, was scored separately from the feckin' two section scores.[39] Two people score each essay by each awardin' 1 to 4 points in each of three categories: Readin', Analysis, and Writin'.[40] These two scores from the different examiners are then combined to give a feckin' total score from 2 to 8 points per category. Though sometimes people quote their essay score out of 24, the oul' College Board themselves do not combine the oul' different categories to give one essay score, instead givin' a feckin' score for each category.
There is no penalty or negative markin' for guessin' on the oul' SAT: scores are based on the oul' number of questions answered correctly, for the craic. The optional essay will not be offered after the June 2021 administration.[20][21] College Board said it would discontinue the oul' essay section because "there are other ways for students to demonstrate their mastery of essay writin'," includin' the bleedin' test's readin' and writin' portion.[20][21] It also acknowledged that the oul' COVID-19 pandemic had played an oul' role in the bleedin' change, acceleratin' 'a process already underway'.[21]
The Readin' Test of the feckin' SAT contains one section of 52 questions and an oul' time limit of 65 minutes.[39] All questions are multiple-choice and based on readin' passages. Tables, graphs, and charts may accompany some passages, but no math is required to correctly answer the bleedin' correspondin' questions. There are five passages (up to two of which may be a pair of smaller passages) on the bleedin' Readin' Test and 10-11 questions per passage or passage pair. Sure this is it. SAT Readin' passages draw from three main fields: history, social studies, and science. Each SAT Readin' Test always includes: one passage from U.S. Jesus Mother of Chrisht almighty. or world literature; one passage from either a U.S, Lord bless us and save us. foundin' document or a related text; one passage about economics, psychology, sociology, or another social science; and, two science passages, would ye swally that? Answers to all of the oul' questions are based only on the bleedin' content stated in or implied by the feckin' passage or passage pair.[41]
The Readin' Test contributes (with the bleedin' Writin' and Language Test) to two subscores, each rangin' from 1 to 15 points:[38]
• Command of Evidence
• Words in Context
### Writin' and Language Test
The Writin' and Language Test of the SAT is made up of one section with 44 multiple-choice questions and an oul' time limit of 35 minutes.[39] As with the bleedin' Readin' Test, all questions are based on readin' passages which may be accompanied by tables, graphs, and charts. The test taker will be asked to read the passages and suggest corrections or improvements for the oul' contents underlined. Readin' passages on this test range in content from topic arguments to nonfiction narratives in a feckin' variety of subjects, like. The skills bein' evaluated include: increasin' the clarity of argument; improvin' word choice; improvin' analysis of topics in social studies and science; changin' sentence or word structure to increase organizational quality and impact of writin'; and, fixin' or improvin' sentence structure, word usage, and punctuation.[42]
The Writin' and Language Test reports two subscores, each rangin' from 1 to 15 points:[38]
• Expression of Ideas
• Standard English Conventions
### Mathematics
An example of an SAT "grid-in" math question and the oul' correctly gridded answer.
The mathematics portion of the oul' SAT is divided into two sections: Math Test – No Calculator and Math Test – Calculator. Sure this is it. In total, the feckin' SAT math test is 80 minutes long and includes 58 questions: 45 multiple choice questions and 13 grid-in questions.[43] The multiple choice questions have four possible answers; the oul' grid-in questions are free response and require the feckin' test taker to provide an answer.
• The Math Test – No Calculator section has 20 questions (15 multiple choice and 5 grid-in) and lasts 25 minutes.
• The Math Test – Calculator section has 38 questions (30 multiple choice and 8 grid-in) and lasts 55 minutes.
Several scores are provided to the test taker for the math test, bedad. A subscore (on a scale of 1 to 15) is reported for each of three categories of math content:
• "Heart of Algebra" (linear equations, systems of linear equations, and linear functions)
• "Problem Solvin' and Data Analysis" (statistics, modelin', and problem-solvin' skills)
• "Passport to Advanced Math" (non-linear expressions, radicals, exponentials and other topics that form the basis of more advanced math).
A test score for the math test is reported on a scale of 10 to 40, with an increment of 0.5, and a feckin' section score (equal to the feckin' test score multiplied by 20) is reported on an oul' scale of 200 to 800.[44][45][46]
#### Calculator use
All scientific and most graphin' calculators, includin' Computer Algebra System (CAS) calculators, are permitted on the SAT Math – Calculator section only, that's fierce now what? All four-function calculators are allowed as well; however, these devices are not recommended. I hope yiz are all ears now. All mobile phone and smartphone calculators, calculators with typewriter-like (QWERTY) keyboards, laptops and other portable computers, and calculators capable of accessin' the feckin' Internet are not permitted.[47]
Research was conducted by the bleedin' College Board to study the feckin' effect of calculator use on SAT I: Reasonin' Test math scores. Here's another quare one. The study found that performance on the bleedin' math section was associated with the extent of calculator use: those usin' calculators on about one third to one half of the items averaged higher scores than those usin' calculators more or less frequently. However, the feckin' effect was "more likely to have been the result of able students usin' calculators differently than less able students rather than calculator use per se."[48] There is some evidence that the frequent use of a calculator in school outside of the feckin' testin' situation has an oul' positive effect on test performance compared to those who do not use calculators in school.[49]
### Style of questions
Most of the feckin' questions on the bleedin' SAT, except for the feckin' optional essay and the grid-in math responses, are multiple choice; all multiple-choice questions have four answer choices, one of which is correct. Chrisht Almighty. Thirteen of the oul' questions on the oul' math portion of the bleedin' SAT (about 22% of all the oul' math questions) are not multiple choice.[50] They instead require the test taker to bubble in an oul' number in an oul' four-column grid.
All questions on each section of the bleedin' SAT are weighted equally. For each correct answer, one raw point is added.[51] No points are deducted for incorrect answers. The final score is derived from the raw score; the oul' precise conversion chart varies between test administrations.
Section Average Score 2020 (200 - 800)[2] Time (Minutes) Content
Mathematics 523 25+55=80 Number and operations; algebra and functions; geometry; statistics, probability, and data analysis
## Logistics
### Frequency
The SAT is offered seven times a holy year in the feckin' United States: in August, October, November, December, March, May, and June. Whisht now. For international students SAT is offered four times a year: in October, December, March and May (2020 exception: To cover worldwide May cancelation, an additional September exam was introduced, and August was made available to international test-takers as well), grand so. The test is typically offered on the first Saturday of the oul' month for the bleedin' October, November, December, May, and June administrations.[52][53] The test was taken by 2,198,460 high school graduates in the class of 2020.[2]
Candidates wishin' to take the bleedin' test may register online at the College Board's website or by mail at least three weeks before the feckin' test date.
### Fees
The SAT costs US\$49.50 (£39.50, €43.50) (US\$64.50 with the feckin' optional essay), plus additional fees of over US\$45 if testin' outside the feckin' United States as of 2019.[8] The College Board makes fee waivers available for low income students. Additional fees apply for late registration, standby testin', registration changes, scores by telephone, and extra score reports (beyond the oul' four provided for free).
### Accommodation for candidates with disabilities
Students with verifiable disabilities, includin' physical and learnin' disabilities, are eligible to take the bleedin' SAT with accommodations. Whisht now. The standard time increase for students requirin' additional time due to learnin' disabilities or physical handicaps is time + 50%; time + 100% is also offered.
## Scaled scores and percentiles
Students receive their online score reports approximately two to three weeks after test administration (longer for mailed, paper scores).[54] Included in the oul' report is the bleedin' total score (the sum of the feckin' two section scores, with each section graded on a feckin' scale of 200–800) and three subscores (in readin', writin', and analysis, each on a scale of 2–8) for the optional essay.[55] Students may also receive, for an additional fee, various score verification services, includin' (for select test administrations) the bleedin' Question and Answer Service, which provides the bleedin' test questions, the student's answers, the oul' correct answers, and the bleedin' type and difficulty of each question.[56]
In addition, students receive two percentile scores, each of which is defined by the bleedin' College Board as the percentage of students in a comparison group with equal or lower test scores, grand so. One of the feckin' percentiles, called the feckin' "Nationally Representative Sample Percentile", uses as an oul' comparison group all 11th and 12th graders in the feckin' United States, regardless of whether or not they took the feckin' SAT. Here's another quare one. This percentile is theoretical and is derived usin' methods of statistical inference. The second percentile, called the feckin' "SAT User Percentile", uses actual scores from a feckin' comparison group of recent United States students that took the oul' SAT. For example, for the oul' school year 2019–2020, the oul' SAT User Percentile was based on the oul' test scores of students in the oul' graduatin' classes of 2018 and 2019 who took the oul' SAT (specifically, the 2016 revision) durin' high school, that's fierce now what? Students receive both types of percentiles for their total score as well as their section scores.[55]
### Percentiles for total scores (2019)
Percentiles for total scores (2019)[55]
Score, 400–1600 scale SAT User Nationally
representative sample
1600 99+ 99+
1550 99+ 99+
1500 98 99
1450 96 99
1400 94 97
1350 91 94
1300 86 91
1250 81 86
1200 74 81
1150 67 74
1100 58 67
1050 49 58
1000 40 48
950 31 38
900 23 29
850 16 21
800 10 14
750 5 8
700 2 4
650 1 1
640–400 <1 <1
### Percentiles for total scores (2006)
The followin' chart summarizes the original percentiles used for the bleedin' version of the bleedin' SAT administered in March 2005 through January 2016. These percentiles used students in the bleedin' graduatin' class of 2006 as the comparison group.[57][58]
Percentile Score 400–1600 scale,
(official, 2006)
Score, 600–2400 scale
(official, 2006)
99.93/99.98* 1600 2400
99.5 ≥1540 ≥2280
99 ≥1480 ≥2200
98 ≥1450 ≥2140
97 ≥1420 ≥2100
93 ≥1340 ≥1990
88 ≥1280 ≥1900
81 ≥1220 ≥1800
72 ≥1150 ≥1700
61 ≥1090 ≥1600
48 ≥1010 ≥1500
36 ≥950 ≥1400
24 ≥870 ≥1300
15 ≥810 ≥1200
8 ≥730 ≥1090
4 ≥650 ≥990
2 ≥590 ≥890
* The percentile of the oul' perfect score was 99.98
on the feckin' 2400 scale and 99.93 on the 1600 scale.
### Percentiles for total scores (1984)
Percentiles for total scores (1984)[59]
Score (1984) Percentile
1600 99.9995
1550 99.983
1500 99.89
1450 99.64
1400 99.10
1350 98.14
1300 96.55
1250 94.28
1200 91.05
1150 86.93
1100 81.62
1050 75.31
1000 67.81
950 59.64
900 50.88
850 41.98
800 33.34
750 25.35
700 18.26
650 12.37
600 7.58
550 3.97
500 1.53
450 0.29
400 0.002
The version of the feckin' SAT administered before April 1995 had a bleedin' very high ceilin'. Be the hokey here's a quare wan. In any given year, only seven of the feckin' million test-takers scored above 1580. A score above 1580 was equivalent to the feckin' 99.9995 percentile.[60]
In 2015 the oul' average score for the bleedin' Class of 2015 was 1490 out of a maximum 2400. That was down 7 points from the feckin' previous class's mark and was the feckin' lowest composite score of the past decade.[17]
## SAT–ACT score comparisons
The College Board and ACT, Inc., conducted a bleedin' joint study of students who took both the feckin' SAT and the feckin' ACT between September 2004 (for the feckin' ACT) or March 2005 (for the SAT) and June 2006. Tables were provided to concord scores for students takin' the feckin' SAT after January 2005 and before March 2016.[61][62] In May 2016, the oul' College Board released concordance tables to concord scores on the feckin' SAT used from March 2005 through January 2016 to the SAT used since March 2016, as well as tables to concord scores on the feckin' SAT used since March 2016 to the ACT.[63]
In 2018, the bleedin' College Board, in partnership with the bleedin' ACT, introduced a new concordance table to better compare how a feckin' student would fare one test to another.[64] This is now considered the feckin' official concordance to be used by college professionals and is replacin' the feckin' one from 2016. Here's a quare one for ye. The new concordance no longer features the old SAT (out of 2,400), just the new SAT (out of 1,600) and the ACT (out of 36).
## Elucidation
### Preparation
Pioneered by Stanley Kaplan in 1946 with a 64-hour course,[65] SAT preparation has become a highly lucrative field.[66] Many companies and organizations offer test preparation in the form of books, classes, online courses, and tutorin'.[67] The test preparation industry began almost simultaneously with the bleedin' introduction of university entrance exams in the U.S. and flourished from the bleedin' start.[68] Test-preparation scams are a holy genuine problem for parents and students.[69]
Nevertheless, the College Board maintains that the feckin' SAT is essentially uncoachable and research by the feckin' College Board and the bleedin' National Association of College Admission Counselin' suggests that tutorin' courses result in an average increase of about 20 points on the feckin' math section and 10 points on the bleedin' verbal section.[70] Like IQ scores, which are a bleedin' strong correlate, SAT scores tend to be stable over time, meanin' SAT preparation courses offer only an oul' limited advantage.[71] An early meta-analysis (from 1983) found similar results and noted "the size of the coachin' effect estimated from the oul' matched or randomized studies (10 points) seems too small to be practically important."[72] Statisticians Ben Domingue and Derek C. Briggs examined data from the oul' Education Longitudinal Survey of 2002 and found that the effects of coachin' were only statistically significant for mathematics; moreover, coachin' had a greater effect on certain students than others, especially those who have taken rigorous courses and those of high socioeconomic status.[73] A 2012 systematic literature review estimated a bleedin' coachin' effect of 23 and 32 points for the math and verbal tests, respectively.[68] A 2016 meta-analysis estimated the oul' effect size to be 0.09 and 0.16 for the oul' verbal and math sections respectively, although there was an oul' large degree of heterogeneity.[74] Public misunderstandin' of how to prepare for the bleedin' SAT continues to be exploited by the preparation industry.[22]
The College Board announced a partnership with the bleedin' non-profit organization Khan Academy to offer free test-preparation materials startin' in the bleedin' 2015–16 academic year to help level the bleedin' playin' field for students from low-income families.[19][17] Students may also bypass costly preparation programs usin' the oul' more affordable official guide from the oul' College Board and with solid studyin' habits.[75]
There is some evidence that takin' the feckin' PSAT at least once can help students do better on the SAT;[76] moreover, like the case for the SAT, top scorers on the feckin' PSAT could earn scholarships.[15] Accordin' to cognitive scientist Sian Beilock, 'chokin'', or substandard performance on important occasions, such as takin' the oul' SAT, can be prevented by doin' plenty of practice questions and proctored exams to improve procedural memory, makin' use of the oul' booklet to write down intermediate steps to avoid overloadin' workin' memory, and writin' a bleedin' diary entry about one's anxieties on the oul' day of the bleedin' exam to enhance self-empathy and positive self-image.[77]
### Predictive validity and powers
In 2009, education researchers Richard C, the hoor. Atkinson and Saul Geiser from the oul' University of California (UC) system argued that high school GPA is better than the oul' SAT at predictin' college grades regardless of high school type or quality.[78] It is the feckin' hope of some UC officials to increase the bleedin' number of African- and Latino-American students attendin' and they plan to do so by castin' doubt on the SAT and by decreasin' the oul' number of Asian-American students, who are heavily represented in the oul' UC student body (29.5%) relative to their share of the population of California (13.6%).[79] However, their assertions on the oul' predictive validity of the oul' SAT has been contested by the bleedin' UC academic senate.[79] In its 2020 report, the oul' UC academic senate found that the feckin' SAT was better than high school GPA at predictin' first year GPA, and just as good as high school GPA at predictin' undergraduate GPA, first year retention, and graduation. Jaykers! This predictive validity was found to hold across demographic groups.[80] A series of College Board reports point to similar predictive validity across demographic groups.[81][82]
The SAT is correlated with intelligence and as such estimates individual differences. It does not, however, have anythin' to say about "effective cognitive performance," or what intelligent people do.[22] Nor does it measure non-cognitive traits associated with academic success such positive attitudes or conscientiousness.[22][83] Psychometricians Thomas R. Whisht now and eist liom. Coyle and David R, the shitehawk. Pillow showed in 2008 that the oul' SAT predicts college GPA even after removin' the feckin' general factor of intelligence (g), with which it is highly correlated.[84] A 2009 study found that SAT or ACT scores and high-school GPAs are strong predictors of cumulative university GPAs, bedad. In particular, those with standardized test scores in the feckin' 50th percentile or better had a two-thirds chance of havin' an oul' cumulative university GPA in the top half.[85][23] A 2010 meta-analysis by researchers from the feckin' University of Minnesota offered evidence that standardized admissions tests such as the bleedin' SAT predicted not only freshman GPA but also overall collegiate GPA.[83][71] A 2012 study from the oul' same university usin' a multi-institutional data set revealed that even after controllin' for socioeconomic status and high-school GPA, SAT scores were still as capable of predictin' freshman GPA among university or college students.[86] A 2019 study with an oul' sample size of around a bleedin' quarter of a holy million students suggests that together, SAT scores and high-school GPA offer an excellent predictor of freshman collegiate GPA and second-year retention.[22] In 2018, psychologists Oren R. Story? Shewach, Kyle D. Jaysis. McNeal, Nathan R, the shitehawk. Kuncel, and Paul R. Arra' would ye listen to this shite? Sackett showed that both high-school GPA and SAT scores predict enrollment in advanced collegiate courses, even after controllin' for Advanced Placement credits.[87][22]
Education economist Jesse M. Listen up now to this fierce wan. Rothstein indicated in 2005 that high-school average SAT scores were better at predictin' freshman university GPAs compared to individual SAT scores. C'mere til I tell ya. In other words, a feckin' student's SAT scores were not as informative with regards to future academic success as his or her high school's average. C'mere til I tell ya. In contrast, individual high-school GPAs were a bleedin' better predictor of collegiate success than average high-school GPAs.[88][89] Furthermore, an admissions officer who failed to take average SAT scores into account would risk overestimatin' the future performance of an oul' student from a bleedin' low-scorin' school and underestimatin' that of a feckin' student from a feckin' high-scorin' school.[89]
Like other standardized tests like the oul' ACT or the GRE, the oul' SAT is a feckin' traditional method for assessin' the feckin' academic aptitude of students who have had vastly different educational experiences and as such is focused on the oul' common materials that the bleedin' students could reasonably be expected to have encountered throughout the feckin' course of study. Me head is hurtin' with all this raidin'. As such the feckin' mathematics section contains no materials above the precalculus level, for instance, the cute hoor. Psychologist Raymond Cattell referred to this as testin' for "historical" rather than "current" crystallized intelligence.[90] Psychologist Scott Barry Kaufman further noted that the SAT can only measure a feckin' snapshot of a bleedin' person's performance at a particular moment in time.[91] Educational psychologists Jonathan Wai, David Lubinski, and Camilla Benbow observed that one way to increase the bleedin' predictive validity of the bleedin' SAT is by assessin' the feckin' student's spatial reasonin' ability, as the bleedin' SAT at present does not contain any questions to that effect, grand so. Spatial reasonin' skills are important for success in STEM.[92] A 2006 study led by psychometrician Robert Sternberg found that the ability of SAT scores and high-school GPAs to predict collegiate performance could further be enhanced by additional assessments of analytical, creative, and practical thinkin'.[93][94]
Experimental psychologist Meredith Frey noted that while advances in education research and neuroscience can help improve the feckin' ability to predict scholastic achievement in the bleedin' future, the oul' SAT remains a valuable tool in the meantime.[22] In a holy 2014 op-ed for The New York Times, psychologist John D. Mayer called the predictive powers of the bleedin' SAT "an astonishin' achievement" and cautioned against makin' it and other standardized tests optional.[95][23] Research by psychometricians David Lubinsky, Camilla Benbow, and their colleagues has shown that the feckin' SAT could even predict life outcomes beyond university.[23]
### Difficulty and relative weight
The SAT rigorously assesses students' mental stamina, memory, speed, accuracy, and capacity for abstract and analytical reasonin'.[75] For American universities and colleges, standardized test scores are the most important factor in admissions, second only to high-school GPAs.[94] By international standards, however, the SAT is not that difficult.[96] For example, South Korea's College Scholastic Ability Test (CSAT) and Finland's Matriculation Examination are both longer, tougher, and count for more towards the admissibility of a holy student to university.[97] In many countries around the oul' world, exams, includin' university entrance exams, are the feckin' sole decidin' factor of admission; school grades are simply irrelevant.[96] In China and India, doin' well on the Gaokao or the feckin' IIT-JEE, respectively, enhances the oul' social status of the students and their families.[98]
In an article from 2012, educational psychologist Jonathan Wai argued that the oul' SAT was too easy to be useful to the bleedin' most competitive of colleges and universities, whose applicants typically had brilliant high-school GPAs and standardized test scores, like. Admissions officers therefore had the oul' burden of differentiatin' the top scorers from one another, not knowin' whether or not the feckin' students' perfect or near-perfect scores truly reflected their scholastic aptitudes, Lord bless us and save us. He suggested that the oul' College Board make the SAT more difficult, which would raise the bleedin' measurement ceilin' of the oul' test, allowin' the top schools to identify the best and brightest among the bleedin' applicants.[99] At that time, the College Board was already workin' on makin' the oul' SAT tougher.[99] The changes were announced in 2014 and implemented in 2016.[100]
After realizin' the feckin' June 2018 test was easier than usual, the College Board made adjustments resultin' in lower-than-expected scores, promptin' complaints from the bleedin' students, though some understood this was to ensure fairness.[101] In its analysis of the incident, the Princeton Review supported the bleedin' idea of curvin' grades, but pointed out that the test was incapable of distinguishin' students in the 86th percentile (650 points) or higher in mathematics. Be the hokey here's a quare wan. The Princeton Review also noted that this particular curve was unusual in that it offered no cushion against careless or last-minute mistakes for high-achievin' students.[102] The Review posted a holy similar blog post for the oul' SAT of August 2019, when an oul' similar incident happened and the College Board responded in the feckin' same manner, notin', "A student who misses two questions on an easier test should not get as good a score as a bleedin' student who misses two questions on a hard test. C'mere til I tell yiz. Equatin' takes care of that issue." It also cautioned students against retakin' the SAT immediately, for they might be disappointed again, and recommended that instead, they give themselves some "leeway" before tryin' again.[103]
### Association with general cognitive ability
In a feckin' 2000 study, psychometrician Ann M. Gallagher and her colleagues found that only the top students made use of intuitive reasonin' in solvin' problems encountered on the oul' mathematics section of the oul' SAT.[104] Cognitive psychologists Brenda Hannon and Mary McNaughton-Cassill discovered that havin' a good workin' memory, the bleedin' ability of knowledge integration, and low levels of test anxiety predicts high performance on the SAT.[105]
Frey and Detterman (2004) investigated associations of SAT scores with intelligence test scores. Usin' an estimate of general mental ability, or g, based on the Armed Services Vocational Aptitude Battery, they found SAT scores to be highly correlated with g (r=.82 in their sample, .857 when adjusted for non-linearity) in their sample taken from a bleedin' 1979 national probability survey. Jesus Mother of Chrisht almighty. Additionally, they investigated the feckin' correlation between SAT results, usin' the bleedin' revised and recentered form of the oul' test, and scores on the Raven's Advanced Progressive Matrices, a test of fluid intelligence (reasonin'), this time usin' an oul' non-random sample. Bejaysus this is a quare tale altogether. They found that the feckin' correlation of SAT results with scores on the feckin' Raven's Advanced Progressive Matrices was .483, they estimated that this correlation would have been about 0.72 were it not for the bleedin' restriction of ability range in the feckin' sample. They also noted that there appeared to be an oul' ceilin' effect on the oul' Raven's scores which may have suppressed the oul' correlation.[106] Beaujean and colleagues (2006) have reached similar conclusions to those reached by Frey and Detterman.[107] Because the bleedin' SAT is strongly correlated with general intelligence, it can be used as a proxy to measure intelligence, especially when the oul' time-consumin' traditional methods of assessment are unavailable.[22]
Psychometrician Linda Gottfredson noted that the oul' SAT is effective at identifyin' intellectually gifted college-bound students.[108]
For decades many critics have accused designers of the verbal SAT of cultural bias as an explanation for the feckin' disparity in scores between poorer and wealthier test-takers,[109] with the bleedin' biggest critics comin' from the oul' University of California system.[110][111] A famous example of this perceived bias in the SAT I was the bleedin' oarsmanregatta analogy question, which is no longer part of the feckin' exam. The object of the oul' question was to find the oul' pair of terms that had the feckin' relationship most similar to the bleedin' relationship between "runner" and "marathon". The correct answer was "oarsman" and "regatta". Would ye believe this shite?The choice of the correct answer was thought to have presupposed students' familiarity with rowin', a sport popular with the wealthy.[112] However, for psychometricians, analogy questions are a bleedin' useful tool to gauge the feckin' mental abilities of students, for, even if the feckin' meanin' of two words are unclear, a feckin' student with sufficiently strong analytical thinkin' skills should still be able to identify their relationships.[110] Analogy questions were removed in 2005.[113] In their place are questions that provide more contextual information should the feckin' students be ignorant of the feckin' relevant definition of a word, makin' it easier for them to guess the correct answer.[114]
### Association with college or university majors and rankings
In 2010, physicists Stephen Hsu and James Schombert of the oul' University of Oregon examined five years of student records at their school and discovered that the oul' academic standin' of students majorin' in mathematics or physics (but not biology, English, sociology, or history) was strongly dependent on SAT mathematics scores. Sufferin' Jaysus. Students with an SAT mathematics scores below 600 were highly unlikely to excel as a bleedin' mathematics or physics major, you know yerself. Nevertheless, they found no such patterns between the feckin' SAT verbal, or combined SAT verbal and mathematics and the bleedin' other aforementioned subjects.[115][116]
In 2015, educational psychologist Jonathan Wai of Duke University analyzed average test scores from the feckin' Army General Classification Test in 1946 (10,000 students), the oul' Selective Service College Qualification Test in 1952 (38,420), Project Talent in the oul' early 1970s (400,000), the Graduate Record Examination between 2002 and 2005 (over 1.2 million), and the bleedin' SAT Math and Verbal in 2014 (1.6 million). Wai identified one consistent pattern: those with the feckin' highest test scores tended to pick the oul' physical sciences and engineerin' as their majors while those with the lowest were more likely to choose education and agriculture. G'wan now and listen to this wan. (See figure below.)[116][117]
A 2020 paper by Laura H, fair play. Gunn and her colleagues examinin' data from 1389 institutions across the bleedin' United States unveiled strong positive correlations between the feckin' average SAT percentiles of incomin' students and the feckin' shares of graduates majorin' in STEM and the bleedin' social sciences. On the bleedin' other hand, they found negative correlations between the former and the shares of graduates in psychology, theology, law enforcement, recreation and fitness.[118]
Various researchers have established that average SAT or ACT scores and college rankin' in the bleedin' U.S. News & World Report are highly correlated, almost 0.9.[22][119][120][b] Between the 1980s and the 2010s, the oul' U.S. Jesus, Mary and holy Saint Joseph. population grew while universities and colleges did not expand their capacities as substantially, bejaysus. As a result, admissions rates fell considerably, meanin' it has become more difficult to get admitted to a holy school whose alumni include one's parents. On top of that, high-scorin' students nowadays are much more likely to leave their hometowns in pursuit of higher education at prestigious institutions. Here's a quare one for ye. Consequently, standardized tests, such as the bleedin' SAT, are a bleedin' more reliable measure of selectivity than admissions rates, game ball! Still, when Michael J. Petrilli and Pedro Enamorado analyzed the feckin' SAT composite scores (math and verbal) of incomin' freshman classes of 1985 and 2016 of the feckin' top universities and liberal arts colleges in the bleedin' United States, they found that the median scores of new students increased by 93 points for their sample, from 1216 to 1309. In particular, fourteen institutions saw an increase of at least 150 points, includin' the feckin' University of Notre-Dame (from 1290 to 1440, or 150 points) and Elon College (from 952 to 1192, or 240 points).[121]
### Association with types of schoolin'
While there seems to be evidence that private schools tend to produce students who do better on standardized tests such as the feckin' ACT or the feckin' SAT, Keven Duncan and Jonathan Sandy showed, usin' data from the oul' National Longitudinal Surveys of Youth, that when student characteristics, such as age, race, and sex (7%), family background (45%), school quality (26%), and other factors were taken into account, the advantage of private schools diminished by 78%. The researchers concluded that students attendin' private schools already had the attributes associated with high scores on their own.[122]
### Association with educational and societal standings and outcomes
Research from the University of California system published in 2001 analyzin' data of their undergraduates between Fall 1996 through Fall 1999, inclusive, found that the SAT II[c] was the feckin' single best predictor of collegiate success in the sense of freshman GPA, followed by high-school GPA, and finally the feckin' SAT I. After controllin' for family income and parental education, the already low ability of the oul' SAT to measure aptitude and college readiness fell sharply while the bleedin' more substantial aptitude and college readiness measurin' abilities of high school GPA and the oul' SAT II each remained undiminished (and even shlightly increased). Jesus Mother of Chrisht almighty. The University of California system required both the feckin' SAT I and the bleedin' SAT II from applicants to the UC system durin' the four academic years of the bleedin' study.[123] This analysis is heavily publicized but is contradicted by many studies.[83]
There is evidence that the bleedin' SAT is correlated with societal and educational outcomes,[91] includin' finishin' a four-year university program.[124] A 2012 paper from psychologists at the bleedin' University of Minnesota analyzin' multi-institutional data sets suggested that the oul' SAT maintained its ability to predict collegiate performance even after controllin' for socioeconomic status (as measured by the feckin' combination of parental educational attainment and income) and high-school GPA. This means that SAT scores were not merely a feckin' proxy for measurin' socioeconomic status, the researchers concluded.[86][125] This findin' has been replicated and shown to hold across racial or ethnic groups and for both sexes.[22] Moreover, the feckin' Minnesota researchers found that the socioeconomic status distributions of the oul' student bodies of the feckin' schools examined reflected those of their respective applicant pools.[86] Because of what it measures, a holy person's SAT scores cannot be separated from his or her socioeconomic background.[91]
In 2007, Rebecca Zwick and Jennifer Greif Green observed that a typical analysis did not take into account that heterogeneity of the bleedin' high schools attended by the bleedin' students in terms of not just the socioeconomic statuses of the bleedin' student bodies but also the standards of gradin'. Listen up now to this fierce wan. Zwick and Greif Green proceeded to show that when these were accounted for, the oul' correlation between family socioeconomic status and classroom grades and rank increased whereas that between socioeconomic status and SAT scores fell, game ball! They concluded that school grades and SAT scores were similarly associated with family income.[88]
Accordin' to the feckin' College Board, in 2019, 56% of the bleedin' test takers had parents with a feckin' university degree, 27% parents with no more than a high-school diploma, and about 9% who did not graduate from high school, grand so. (8% did not respond to the question.)[16]
### Association with family structures
One of the proposed partial explanations for the gap between Asian- and European-American students in educational achievement, as measured for example by the SAT, is the oul' general tendency of Asians to come from stable two-parent households.[126] In their 2018 analysis of data from the feckin' National Longitudinal Surveys of the feckin' Bureau of Labor Statistics, economists Adam Blandin, Christopher Herrington, and Aaron Steelman concluded that family structure played an important role in determinin' educational outcomes in general and SAT scores in particular, the hoor. Families with only one parent who has no degrees were designated 1L, with two parents but no degrees 2L, and two parents with at least one degree between them 2H. Children from 2H families held a significant advantage of those from 1L families, and this gap grew between 1990 and 2010. Whisht now. Because the median SAT composite scores (verbal and mathematics) for 2H families grew by 20 points while those of 1L families fell by one point, the gap between them increased by 21 points, or a holy fifth of one standard deviation.[124]
Speakin' to The Wall Street Journal, family sociologist W. Bradford Wilcox stated, "In the feckin' absence of SAT scores, which can pinpoint kids from difficult family backgrounds with great academic potential, family stability is likely to loom even larger in determinin' who makes it past the bleedin' college finish line in California [whose public university system decided to stop requirin' SAT and ACT scores for admissions in 2020]."[79]
### Sex differences
#### In performance
In 2013, the feckin' American College Testin' Board released a bleedin' report statin' that boys outperformed girls on the bleedin' mathematics section of the bleedin' test.[127] As of 2015, boys on average earned 32 points more than girls on the bleedin' SAT mathematics section, so it is. Among those scorin' in the oul' 700-800 range, the feckin' male-to-female ratio was 1.6:1.[128] In 2014, psychologist Stephen Ceci and his collaborators found boys did better than girls across the percentiles. For example, an oul' girl scorin' in the top 10% of her sex would only be in the top 20% among the oul' boys.[129][130] In 2010, psychologist Jonathan Wai and his colleagues showed, by analyzin' data from three decades involvin' 1.6 million intellectually gifted seventh graders from the Duke University Talent Identification Program (TIP), that in the bleedin' 1980s the gender gap in the oul' mathematics section of the oul' SAT among students scorin' in the feckin' top 0.01% was 13.5:1 in favor of boys but dropped to 3.8:1 by the oul' 1990s.[131][130] The dramatic sex ratio from the bleedin' 1980s replicates an oul' different study usin' a holy sample from Johns Hopkins University.[132] This ratio is similar to that observed for the bleedin' ACT mathematics and science scores between the bleedin' early 1990s and the bleedin' late 2000s.[131] It remained largely unaltered at the end of the feckin' 2000s.[131][133] Sex differences in SAT mathematics scores began makin' themselves apparent at the level of 400 points and above.[131]
Some researchers point to evidence in support of greater male variability in spatial ability and mathematics, you know yourself like. Greater male variability has been found in body weight, height, and cognitive abilities across cultures, leadin' to a larger number of males in the bleedin' lowest and highest distributions of testin'.[134] Consequently, a bleedin' higher number of males are found in both the bleedin' upper and lower extremes of the feckin' performance distributions of the feckin' mathematics sections of standardized tests such as the oul' SAT, resultin' in the observed gender discrepancy.[135][130][136] Paradoxically, this is at odds with the feckin' tendency of girls to have higher classroom scores than boys.[130]
On the bleedin' other hand, Wai and his colleagues found that both sexes in the oul' top 5% appeared to be more or less at parity when it comes to the oul' verbal section of the SAT, though girls have gained a holy shlight but noticeable edge over boys startin' in the mid-1980s.[132] Psychologist David Lubinski, who conducted longitudinal studies of seventh grader who scored exceptionally high on the SAT, found a bleedin' similar result. Girls generally had better verbal reasonin' skills and boys mathematical skills.[136] This reflects other research on the oul' cognitive ability of the bleedin' general population rather than just the 95th percentile and up.[132][136]
Although aspects of testin' such as stereotype are a concern, research on the feckin' predictive validity of the bleedin' SAT has demonstrated that it tends to be a holy more accurate predictor of female GPA in university as compared to male GPA.[137]
#### In strategizin'
SAT mathematics questions can be answered intuitively or algorithmically.
Mathematical problems on the oul' SAT can be broadly categorized into two groups: conventional and unconventional, game ball! Conventional problems can be handled routinely via familiar formulas or algorithms while unconventional ones require more creative thought in order to make unusual use of familiar methods of solution or to come up with the feckin' specific insights necessary for solvin' those problems. In 2000, ETS psychometrician Ann M. Gallagher and her colleagues analyzed how students handled disclosed SAT mathematics questions in self-reports. They found that for both sexes, the most favored approach was to use formulas or algorithms learned in class. Here's a quare one for ye. When that failed, however, males were more likely than females to identify the oul' suitable methods of solution. Previous research suggested that males were more likely to explore unusual paths to solution whereas females tended to stick to what they had learned in class and that females were more likely to identify the appropriate approaches if such required nothin' more than mastery of classroom materials.[104]
#### In confidence
Older versions of the SAT did ask students how confident they were in their mathematical aptitude and verbal reasonin' ability, specifically, whether or not they believed they were in the oul' top 10%, bedad. Devin G, Lord bless us and save us. Pope analyzed data of over four million test takers from the oul' late 1990s to the bleedin' early 2000s and found that high scorers were more likely to be confident they were in the top 10%, with the oul' top scorers reportin' the bleedin' highest levels of confidence, game ball! But there were some noticeable gaps between the sexes. Men tended to be much more confident in their mathematical aptitude then women, that's fierce now what? For example, among those who scored 700 on the mathematics section, 67% of men answered they believed they were in the top 10% whereas only 56% of women did the same, be the hokey! Women, on the oul' other hand, were shlightly more confident in their verbal reasonin' ability then men.[138]
#### In glucose metabolism
Cognitive neuroscientists Richard Haier and Camilla Persson Benbow employed positron emission tomography (PET) scans to investigate the feckin' rate of glucose metabolism among students who have taken the oul' SAT, what? They found that among men, those with higher SAT mathematics scores exhibited higher rates of glucose metabolism in the bleedin' temporal lobes than those with lower scores, contradictin' the oul' brain-efficiency hypothesis, you know yerself. This trend, however, was not found among women, for whom the feckin' researchers could not find any cortical regions associated with mathematical reasonin'. Both sexes scored the feckin' same on average in their sample and had the same rates of cortical glucose metabolism overall, what? Accordin' to Haier and Benbow, this is evidence for the feckin' structural differences of the bleedin' brain between the sexes.[139][25]
### Association with race and ethnicity
SAT Verbal average scores by race or ethnicity from 1986-87 to 2004-05
SAT Math average scores by race or ethnicity from 1986-87 to 2004-05
A 2001 meta-analysis of the feckin' results of 6,246,729 participants tested for cognitive ability or aptitude found a holy difference in average scores between black and white students of around 1.0 standard deviation, with comparable results for the oul' SAT (2.4 million test takers).[140] Similarly, on average, Hispanic and Amerindian students perform on the order of one standard deviation lower on the oul' SAT than white and Asian students.[141][142][143][144] Mathematics appears to be the more difficult part of the oul' exam.[16] In 1996, the bleedin' black-white gap in the oul' mathematics section was 0.91 standard deviations, but by 2020, it fell to 0.79.[145] In 2013, Asian Americans as a feckin' group scored 0.38 standard deviations higher than whites in the oul' mathematics section.[126]
Some researchers believe that the difference in scores is closely related to the oul' overall achievement gap in American society between students of different racial groups. This gap may be explainable in part by the bleedin' fact that students of disadvantaged racial groups tend to go to schools that provide lower educational quality. Here's a quare one for ye. This view is supported by evidence that the black-white gap is higher in cities and neighborhoods that are more racially segregated.[146] Other research cites poorer minority proficiency in key coursework relevant to the bleedin' SAT (English and math), as well as peer pressure against students who try to focus on their schoolwork ("actin' white").[147] Cultural issues are also evident among black students in wealthier households, with high achievin' parents. Whisht now and eist liom. John Ogbu, an oul' Nigerian-American professor of anthropology, concluded that instead of lookin' to their parents as role models, black youth chose other models like rappers and did not make an effort to be good students.[148]
One set of studies has reported differential item functionin', namely, that some test questions function differently based on the racial group of the feckin' test taker, reflectin' differences in ability to understand certain test questions or to acquire the feckin' knowledge required to answer them between groups. Whisht now and eist liom. In 2003, Freedle published data showin' that black students have had an oul' shlight advantage on the bleedin' verbal questions that are labeled as difficult on the SAT, whereas white and Asian students tended to have a holy shlight advantage on questions labeled as easy. Freedle argued that these findings suggest that "easy" test items use vocabulary that is easier to understand for white middle class students than for minorities, who often use a feckin' different language in the home environment, whereas the feckin' difficult items use complex language learned only through lectures and textbooks, givin' both student groups equal opportunities to acquirin' it.[149][150][151] The study was severely criticized by the oul' ETS board, but the oul' findings were replicated in a holy subsequent study by Santelices and Wilson in 2010.[152][153]
There is no evidence that SAT scores systematically underestimate future performance of minority students, the cute hoor. However, the oul' predictive validity of the oul' SAT has been shown to depend on the bleedin' dominant ethnic and racial composition of the college.[154] Some studies have also shown that African-American students under-perform in college relative to their white peers with the same SAT scores; researchers have argued that this is likely because white students tend to benefit from social advantages outside of the oul' educational environment (for example, high parental involvement in their education, inclusion in campus academic activities, positive bias from same-race teachers and peers) which result in better grades.[155]
Christopher Jencks concludes that as a holy group, African Americans have been harmed by the introduction of standardized entrance exams such as the feckin' SAT. Here's a quare one. This, accordin' to yer man, is not because the bleedin' tests themselves are flawed, but because of labelin' bias and selection bias; the oul' tests measure the feckin' skills that African Americans are less likely to develop in their socialization, rather than the skills they are more likely to develop, Lord bless us and save us. Furthermore, standardized entrance exams are often labeled as tests of general ability, rather than of certain aspects of ability. Thus, a bleedin' situation is produced in which African-American ability is consistently underestimated within the bleedin' education and workplace environments, contributin' in turn to selection bias against them which exacerbates underachievement.[155]
2003 SAT scores by race and ethnicity
Among the oul' major racial or ethnic groups of the United States, gaps in SAT mathematics scores are the bleedin' greatest at the feckin' tails, with Hispanic and Latino Americans bein' the most likely to score at the oul' lowest range and Asian Americans the bleedin' highest. In addition, there is some evidence suggestin' that if the bleedin' test contains more questions of both the bleedin' easy and difficult varieties, which would increase the bleedin' variability of the bleedin' scores, the gaps would be even wider. Given the bleedin' distribution for Asians, for example, many could score higher than 800 if the test allowed them to. (See figure below.)[156]
2020 was the feckin' year in which education worldwide was disrupted by the COVID-19 pandemic and indeed, the performance of students in the oul' United States on standardized tests, such as the SAT, suffered. Yet the feckin' gaps persisted.[157] Accordin' to the feckin' College Board, in 2020, while 83% of Asian students met the benchmark of college readiness in readin' and writin' and 80% in mathematics, only 44% and 21% of black students did those respective categories. Bejaysus here's a quare one right here now. Among whites, 79% met the benchmark for readin' and writin' and 59% did mathematics. For Hispanics and Latinos, the bleedin' numbers were 53% and 30%, respectively, bejaysus. (See figure below.)[145]
### Test-takin' population
A U.S, would ye swally that? Navy sailor takin' the feckin' SAT aboard the U.S.S Kitty Hawk in 2004.
By analyzin' data from the National Center for Education Statistics, economists Ember Smith and Richard Reeves of the Brookings Institution deduced that the bleedin' number of students takin' the bleedin' SAT increased at a rate faster than population and high-school graduation growth rates between 2000 and 2020. The increase was especially pronounced among Hispanics and Latinos. Here's a quare one. Even among whites, whose number of high-school graduates was shrinkin', the bleedin' number of SAT takers rose.[145] In 2015, for example, 1.7 million students took the oul' SAT,[13] up from 1.6 million in 2013.[100] But in 2019, an oul' record-breakin' 2.2 million students took the exam, compared to 2.1 million in 2018, another record-breakin' year.[16] The rise in the feckin' number of students takin' the feckin' SAT was due in part to many school districts offerin' to administer the oul' SAT durin' school days often at no further costs to the feckin' students.[16]
Psychologists Jean Twenge, W. Keith Campbell, and Ryne A, bejaysus. Sherman analyzed vocabulary test scores on the feckin' U.S, the shitehawk. General Social Survey (${\displaystyle n=29,912}$) and found that after correctin' for education, the bleedin' use of sophisticated vocabulary has declined between the bleedin' mid-1970s and the oul' mid-2010s across all levels of education, from below high school to graduate school, you know yerself. However, they cautioned against the bleedin' use of SAT verbal scores to track the decline for while the bleedin' College Board reported that SAT verbal scores had been decreasin', these scores were an imperfect measure of the oul' vocabulary level of the oul' nation as a whole because the test-takin' demographic has changed and because more students took the SAT in the 2010s than in the bleedin' 1970s, meanin' there were more with limited ability who took it.[18]
### Use in non-collegiate contexts
#### By high-IQ societies
Certain high IQ societies, like Mensa, Intertel, the feckin' Prometheus Society and the oul' Triple Nine Society, use scores from certain years as one of their admission tests. Jesus, Mary and Joseph. For instance, Intertel accepts scores (verbal and math combined) of at least 1300 on tests taken through January 1994;[158] the feckin' Triple Nine Society accepts scores of 1450 or greater on SAT tests taken before April 1995, and scores of at least 1520 on tests taken between April 1995 and February 2005.[159]
#### By researchers
Because it is strongly correlated with general intelligence, the bleedin' SAT has often been used as a bleedin' proxy to measure intelligence by researchers, especially since 2004.[22] In particular, scientists studyin' mathematically gifted individuals have been usin' mathematics section of the bleedin' SAT to identify subjects for their research.[24]
A growin' body of research indicates that SAT scores can predict individual success decades into the future, for example in terms of income and occupational achievements.[22][29][71] A longitudinal study published in 2005 by educational psychologists Jonathan Wai, David Lubinski, and Camilla Benbow suggests that among the feckin' intellectually precocious (the top 1%), those with higher scores in the oul' mathematics section of the feckin' SAT at the oul' age of 12 were more likely to earn a holy PhD in the bleedin' STEM fields, to have a publication, to register a feckin' patent, to secure university tenure.[160][116] Wai further showed that an individual's academic ability, as measured by the oul' average SAT or ACT scores of the institution attended, predicted individual differences in income, even among the bleedin' richest people of all, and bein' a bleedin' member of the bleedin' 'American elite', namely Fortune 500 CEOs, billionaires, federal judges, and members of Congress.[161][22] Wai concluded that the bleedin' American elite was also the oul' cognitive elite.[161] Gregory Park, Lubinski, and Benbow gave statistical evidence that intellectually gifted adolescents, as identified by SAT scores, could be expected to accomplish great feats of creativity in the feckin' future, both in the oul' arts and in STEM.[162][22]
The SAT is sometimes given to students at age 12 or 13 by organizations such as the oul' Study of Mathematically Precocious Youth (SMPY), Johns Hopkins Center for Talented Youth, and the oul' Duke University Talent Identification Program (TIP) to select, study, and mentor students of exceptional ability, that is, those in the top one percent.[25] Among SMPY participants, those within the top quartile, as indicated by the bleedin' SAT composite score (mathematics and verbal), were markedly more likely to have an oul' doctoral degree, to have at least one publication in STEM, to earn income in the feckin' 95th percentile, to have at least one literary publication, or to register at least one patent than those in the bottom quartile. Listen up now to this fierce wan. Duke TIP participants generally picked career tracks in STEM should they be stronger in mathematics, as indicated by SAT mathematics scores, or the feckin' humanities if they possessed greater verbal ability, as indicated by SAT verbal scores, game ball! For comparison, the oul' bottom SMPY quartile is five times more likely than the bleedin' average American to have a feckin' patent. Whisht now and listen to this wan. Meanwhile, as of 2016, the bleedin' shares doctorates among SMPY participants was 44% and Duke TIP 37%, compared to two percent among the feckin' general U.S, grand so. population.[26] Consequently, the feckin' notion that beyond a certain point, differences in cognitive ability as measured by standardized tests such as the feckin' SAT cease to matter is gainsaid by the oul' evidence.[163]
In the feckin' 2010 paper which showed that the feckin' sex gap in SAT mathematics scores had dropped dramatically between the feckin' early 1980s and the early 1990s but had persisted for the oul' next two decades or so, Wai and his colleagues argued that "sex differences in abilities in the oul' extreme right tail should not be dismissed as no longer part of the oul' explanation for the oul' dearth of women in math-intensive fields of science."[131][164]
#### By employers
Cognitive ability is correlated with job trainin' outcomes and job performance.[83][28] As such, some employers rely on SAT scores to assess the suitability of an oul' prospective recruit,[29] especially if the bleedin' person has limited work experience.[28] There is nothin' new about this practice.[27] Major companies and corporations have spent princely sums on learnin' how to avoid hirin' errors and have decided that standardized test scores are a valuable tool in decidin' whether or not a person is fit for the feckin' job, you know yourself like. In some cases, a company might need to hire someone to handle proprietary materials of its own makin', such as computer software, would ye believe it? But since the oul' ability to work with such materials cannot be assessed via external certification, it makes sense for such a bleedin' firm to rely on somethin' that is a proxy of measurin' general intelligence.[29] In other cases, a company—on Wall Street, for instance—does not care about academic background but needs to assess a holy prospective recruit's quantitative reasonin' ability, and what makes standardized test scores necessary.[27]
Nevertheless, some top employers, such as Google, have eschewed the bleedin' use of SAT or other standardized test scores unless the feckin' potential employee is a holy recent graduate because for their purposes, these scores "don't predict anythin'." Educational psychologist Jonathan Wai suggested this might be due to the inability of the SAT to differentiate the feckin' intellectual capacities of those at the bleedin' extreme right end of the bleedin' distribution of intelligence. Soft oul' day. Wai told The New York Times, "Today the bleedin' SAT is actually too easy, and that's why Google doesn't see a holy correlation, like. Every single person they get through the oul' door is an oul' super-high scorer."[29]
## Perception
### Math–verbal achievement gap
In 2002, New York Times columnist Richard Rothstein argued that the oul' U.S, you know yourself like. math averages on the oul' SAT and ACT continued their decade-long rise over national verbal averages on the oul' tests while the oul' averages verbal portions on the oul' same tests were flounderin'.[165]
### Optional SAT
In the oul' 1960s and 1970s there was a holy movement to drop achievement scores. Jesus, Mary and holy Saint Joseph. After a holy period of time, the bleedin' countries, states and provinces that reintroduced them agreed that academic standards had dropped, students had studied less, and had taken their studyin' less seriously. Me head is hurtin' with all this raidin'. They reintroduced the tests after studies and research concluded that the oul' high-stakes tests produced benefits that outweighed the feckin' costs.[166]
In a 2001 speech to the oul' American Council on Education, Richard C. Here's a quare one. Atkinson, the oul' president of the oul' University of California, urged the droppin' admissions tests such as the SAT I but not achievement tests such as the bleedin' SAT II[c] as a holy college admissions requirement.[167] Atkinson's critique of the oul' predictive validity and powers of the bleedin' SAT has been contested by the University of California academic senate.[79][80] In April 2020, the oul' academic senate, which consisted of faculty members, voted 51–0 to restore the oul' requirement of standardized test scores. Arra' would ye listen to this. However, the bleedin' governin' board overruled the feckin' senate. Because of the bleedin' size of the bleedin' Californian population, this decision might have an impact on U.S. higher education at large; schools lookin' to admit Californian students could have harder time.[94]
Many parents and college-bound teenagers are skeptical of the oul' process of "holistic admissions" because they think is rather vague and uncertain, as schools try to access characteristics not easily discerned via a number, hence the growth in the number of test takers attemptin' to make themselves more competitive even if this parallels an increase in the feckin' number of schools declarin' it optional.[13][14] Holistic admissions notwithstandin', when merit-based scholarships are considered, standardized test scores might be the feckin' tiebreakers, as these are highly competitive.[14] Scholarships and financial aid could help students and their parents significantly cut the feckin' cost of higher education, especially in times of economic hardship.[15] Moreover, the oul' most selective of schools have might have no better options than usin' standardized test scores in order to quickly prune the number of applications worth considerin', for holistic admissions consume valuable time and other resources.[94]
In the feckin' wake of the bleedin' COVID-19 pandemic, around 1,600 institutions decided to waive the feckin' requirement of the feckin' SAT or the ACT for admissions because it was challengin' both to administer and to take these tests, resultin' in many cancellations.[171] Some schools chose to make them optional on a feckin' temporary basis only, either for just one year, as in the case of Princeton University, or three, like the oul' College of William & Mary. Others dropped the oul' requirement completely.[13] Some schools extended their moratorium on standardized entrance exams in 2021.[94] This did not stop highly ambitious students from takin' them, however,[13][14] as many parents and teenagers were skeptical of the bleedin' "optional" status of university entrance exams[14] and wanted to make their applications more likely to catch the attention of admission officers.[15] This led to complaints of registration sites crashin' in the bleedin' summer of 2020.[171] On the bleedin' other hand, the oul' number of students applyin' to the feckin' more competitive of schools that had made SAT and ACT scores optional increased dramatically because the students thought they stood a chance.[94][172][173] At the feckin' same time, interest in lower-status schools that did the oul' same thin' dropped precipitously.[173] In all, 44% of students who used the feckin' Common Application—accepted by over 900 colleges and universities as of 2021—submitted SAT or ACT scores in 2020–21, down from 77% in 2019–20. Right so. Those who did submit their test scores tended to hail from high-income families, to have at least one university-educated parent, and to be white or Asian.[169]
### Writin' section
In 2005, MIT Writin' Director Pavan Sreekireddy plotted essay length versus essay score on the feckin' new SAT from released essays and found a holy high correlation between them. G'wan now. After studyin' over 50 graded essays, he found that longer essays consistently produced higher scores. In fact, he argues that by simply gaugin' the feckin' length of an essay without readin' it, the given score of an essay could likely be determined correctly over 90% of the feckin' time. Would ye believe this shite?He also discovered that several of these essays were full of factual errors; the oul' College Board does not claim to grade for factual accuracy.
Perelman, along with the National Council of Teachers of English, also criticized the oul' 25-minute writin' section of the bleedin' test for damagin' standards of writin' teachin' in the classroom. Here's a quare one for ye. They say that writin' teachers trainin' their students for the oul' SAT will not focus on revision, depth, accuracy, but will instead produce long, formulaic, and wordy pieces.[174] "You're gettin' teachers to train students to be bad writers", concluded Perelman.[175]
On January 19, 2021, the feckin' College Board announced that the feckin' SAT would no longer offer the optional essay section after the bleedin' June 2021 administration.[20][21]
## History
Year ofexam Readin'/VerbalScore Math Score 1972 530 509 1973 523 506 1974 521 505 1975 512 498 1976 509 497 1977 507 496 1978 507 494 1979 505 493 1980 502 492 1981 502 492 1982 504 493 1983 503 494 1984 504 497 1985 509 500 1986 509 500 1987 507 501 1988 505 501 1989 504 502 1990 500 501 1991 499 500 1992 500 501 1993 500 503 1994 499 504 1995 504 506 1996 505 508 1997 505 511 1998 505 512 1999 505 511 2000 505 514 2001 506 514 2002 504 516 2003 507 519 2004 508 518 2005 508 520 2006 503 518 2007 502 515 2008 502 515 2009 501 515 2010 501 516 2011 497 514 2012 496 514 2013 496 514 2014 497 513 2015 495 511 2016 494 508 2017 533 527 2018 536 531 2019 531 528 2020 528 523
In the oul' late nineteenth century, elite colleges and universities had their own entrance exams and they required candidates to travel to the bleedin' school to take the tests.[94] To better organize matters, the feckin' College Board, a holy consortium of colleges in the northeastern United States, was formed in 1900 to establish a nationally administered, uniform set of essay tests based on the curricula of the bleedin' boardin' schools that typically provided graduates to the colleges of the Ivy League and Seven Sisters, among others.[178][179] The first College Board exam—coverin' mathematics, the oul' physical sciences, history, languages, and other subjects—was administered in 1901 to no more than 1,000 candidates.[94]
In the bleedin' same time period, Lewis Terman and others began to promote the use of tests such as Alfred Binet's in American schools. Bejaysus here's a quare one right here now. Terman in particular thought that such tests could identify an innate "intelligence quotient" (IQ) in a person. The results of an IQ test could then be used to find an elite group of students who would be given the chance to finish high school and go on to college.[178] By the feckin' mid-1920s, the bleedin' increasin' use of IQ tests, such as the bleedin' Army Alpha test administered to recruits in World War I, led the bleedin' College Board to commission the development of the feckin' SAT. In fairness now. The commission, headed by eugenicist Carl Brigham, argued that the test predicted success in higher education by identifyin' candidates primarily on the oul' basis of intellectual promise rather than on specific accomplishment in high school subjects.[179] Brigham "created the oul' test to uphold a racial caste system. Jesus Mother of Chrisht almighty. He advanced this theory of standardized testin' as a means of upholdin' racial purity in his book A Study of American Intelligence, you know yourself like. The tests, he wrote, would prove the racial superiority of white Americans and prevent 'the continued propagation of defective strains in the present population'—chiefly, the feckin' 'infiltration of white blood into the bleedin' Negro.'"[180] By 1930, however, Brigham would repudiate his own conclusions, writin' that "comparative studies of various national and racial groups may not be made with existin' tests"[181] and that SAT scores couldn't reflect some innate, genetically-based ability, but instead would be "a composite includin' schoolin', family background, familiarity with English and everythin' else, relevant and irrelevant."[180] In 1934, James Conant and Henry Chauncey used the feckin' SAT as a means to identify recipients for scholarships to Harvard University. Specifically, Conant wanted to find students, other than those from the bleedin' traditional northeastern private schools, that could do well at Harvard. The success of the scholarship program and the feckin' advent of World War II led to the oul' end of the feckin' College Board essay exams and to the bleedin' SAT bein' used as the feckin' only admissions test for College Board member colleges.[178]
The SAT rose in prominence after World War II due to several factors. Machine-based scorin' of multiple-choice tests taken by pencil had made it possible to rapidly process the feckin' exams.[181] The G.I. C'mere til I tell ya. Bill produced an influx of millions of veterans into higher education.[181][182] The formation of the oul' Educational Testin' Service (ETS) also played a significant role in the expansion of the bleedin' SAT beyond the roughly fifty colleges that made up the bleedin' College Board at the feckin' time.[183] The ETS was formed in 1947 by the bleedin' College Board, Carnegie Foundation for the Advancement of Teachin', and the feckin' American Council on Education, to consolidate respectively the operations of the bleedin' SAT, the feckin' GRE, and the feckin' achievement tests developed by Ben Wood for use with Conant's scholarship exams.[181] The new organization was to be philosophically grounded in the feckin' concepts of open-minded, scientific research in testin' with no doctrine to sell and with an eye toward public service.[184] The ETS was chartered after the feckin' death of Brigham, who had opposed the creation of such an entity. Right so. Brigham felt that the interests of a holy consolidated testin' agency would be more aligned with sales or marketin' than with research into the oul' science of testin'.[181] It has been argued that the feckin' interest of the ETS in expandin' the bleedin' SAT in order to support its operations aligned with the bleedin' desire of public college and university faculties to have smaller, diversified, and more academic student bodies as an oul' means to increase research activities.[178] In 1951, about 80,000 SATs were taken; in 1961, about 800,000; and by 1971, about 1.5 million SATs were bein' taken each year.[185]
Durin' the bleedin' 2010s, there was concern over the oul' continued decline of SAT scores,[17][16] which might be due to the oul' expansion of the bleedin' test-takin' population.[16][18] (See graph below.)
A timeline of notable events in the bleedin' history of the bleedin' SAT follows.
### 1901 essay exams
On June 17, 1901, the oul' first exams of the College Board were administered to 973 students across 67 locations in the oul' United States, and two in Europe. Although those takin' the test came from an oul' variety of backgrounds, approximately one third were from New York, New Jersey, or Pennsylvania. Arra' would ye listen to this. The majority of those takin' the bleedin' test were from private schools, academies, or endowed schools. About 60% of those takin' the feckin' test applied to Columbia University, so it is. The test contained sections on English, French, German, Latin, Greek, history, geography, political science, biology, mathematics, chemistry, and physics. Jaykers! The test was not multiple choice, but instead was evaluated based on essay responses as "excellent", "good", "doubtful", "poor" or "very poor".[186]
### 1926 test
The first administration of the SAT occurred on June 23, 1926, when it was known as the Scholastic Aptitude Test.[111][187] This test, prepared by an oul' committee headed by eugenicist and Princeton psychologist Carl Campbell Brigham, had sections of definitions, arithmetic, classification, artificial language, antonyms, number series, analogies, logical inference, and paragraph readin'. Jesus Mother of Chrisht almighty. It was administered to over 8,000 students at over 300 test centers. Men composed 60% of the test-takers. Slightly over an oul' quarter of males and females applied to Yale University and Smith College.[187] The test was paced rather quickly, test-takers bein' given only a holy little over 90 minutes to answer 315 questions.[111] The raw score of each participatin' student was converted to a score scale with an oul' mean of 500 and an oul' standard deviation of 100. C'mere til I tell yiz. This scale was effectively equivalent to a 200 to 800 scale, although students could score more than 800 and less than 200.[181]
### 1928 and 1929 tests
In 1928, the feckin' number of sections on the feckin' SAT was reduced to seven, and the feckin' time limit was increased to shlightly under two hours. Here's another quare one. In 1929, the bleedin' number of sections was again reduced, this time to six. Sufferin' Jaysus listen to this. These changes were designed in part to give test-takers more time per question. For these two years, all of the feckin' sections tested verbal ability: math was eliminated entirely from the oul' SAT.[111]
### 1930 test and 1936 changes
In 1930 the feckin' SAT was first split into the feckin' verbal and math sections, a structure that would continue through 2004, the shitehawk. The verbal section of the bleedin' 1930 test covered an oul' more narrow range of content than its predecessors, examinin' only antonyms, double definitions (somewhat similar to sentence completions), and paragraph readin', what? In 1936, analogies were re-added. Here's a quare one. Between 1936 and 1946, students had between 80 and 115 minutes to answer 250 verbal questions (over a third of which were on antonyms). G'wan now. The mathematics test introduced in 1930 contained 100 free response questions to be answered in 80 minutes and focused primarily on speed. Whisht now and eist liom. From 1936 to 1941, like the 1928 and 1929 tests, the mathematics section was eliminated entirely. Jaysis. When the feckin' mathematics portion of the test was re-added in 1942, it consisted of multiple-choice questions.[111]
### 1941 and 1942 score scales
Until 1941, the scores on all SATs had been scaled to a mean of 500 with a standard deviation of 100. Although one test-taker could be compared to another for a given test date, comparisons from one year to another could not be made. For example, a bleedin' score of 500 achieved on an SAT taken in one year could reflect an oul' different ability level than a bleedin' score of 500 achieved in another year. Jasus. By 1940, it had become clear that settin' the feckin' mean SAT score to 500 every year was unfair to those students who happened to take the bleedin' SAT with a holy group of higher average ability.[188]
In order to make cross-year score comparisons possible, in April 1941 the feckin' SAT verbal section was scaled to a mean of 500, and a feckin' standard deviation of 100, and the oul' June 1941 SAT verbal section was equated (linked) to the April 1941 test. All SAT verbal sections after 1941 were equated to previous tests so that the bleedin' same scores on different SATs would be comparable. Similarly, in June 1942 the feckin' SAT math section was equated to the April 1942 math section, which itself was linked to the oul' 1942 SAT verbal section, and all SAT math sections after 1942 would be equated to previous tests. From this point forward, SAT mean scores could change over time, dependin' on the bleedin' average ability of the oul' group takin' the test compared to the roughly 10,600 students takin' the SAT in April 1941. The 1941 and 1942 score scales would remain in use until 1995.[188][189]
### 1946 test and associated changes
Paragraph readin' was eliminated from the oul' verbal portion of the bleedin' SAT in 1946, and replaced with readin' comprehension, and "double definition" questions were replaced with sentence completions. Between 1946 and 1957, students were given 90 to 100 minutes to complete 107 to 170 verbal questions. Startin' in 1958, time limits became more stable, and for 17 years, until 1975, students had 75 minutes to answer 90 questions. Jaysis. In 1959, questions on data sufficiency were introduced to the oul' mathematics section and then replaced with quantitative comparisons in 1974. Whisht now. In 1974, both verbal and math sections were reduced from 75 minutes to 60 minutes each, with changes in test composition compensatin' for the decreased time.[111]
### 1960s and 1970s score declines
From 1926 to 1941, scores on the SAT were scaled to make 500 the bleedin' mean score on each section. In 1941 and 1942, SAT scores were standardized via test equatin', and as a feckin' consequence, average verbal and math scores could vary from that time forward.[188] In 1952, mean verbal and math scores were 476 and 494, respectively, and scores were generally stable in the oul' 1950s and early 1960s. Bejaysus here's a quare one right here now. However, startin' in the mid-1960s and continuin' until the feckin' early 1980s, SAT scores declined: the bleedin' average verbal score dropped by about 50 points, and the bleedin' average math score fell by about 30 points. By the bleedin' late 1970s, only the feckin' upper third of test takers were doin' as well as the upper half of those takin' the bleedin' SAT in 1963. Whisht now. From 1961 to 1977, the feckin' number of SATs taken per year doubled, suggestin' that the oul' decline could be explained by demographic changes in the oul' group of students takin' the SAT. Commissioned by the College Board, an independent study of the oul' decline found that most (up to about 75%) of the oul' test decline in the feckin' 1960s could be explained by compositional changes in the feckin' group of students takin' the bleedin' test; however, only about 25 percent of the 1970s decrease in test scores could similarly be explained.[185] Later analyses suggested that up to 40 percent of the feckin' 1970s decline in scores could be explained by demographic changes, leavin' unknown at least some of the bleedin' reasons for the oul' decline.[190]
### 1994 changes
In early 1994, substantial changes were made to the bleedin' SAT.[191] Antonyms were removed from the bleedin' verbal section in order to make rote memorization of vocabulary less useful. Also, the feckin' fraction of verbal questions devoted to passage-based readin' material was increased from about 30% to about 50%, and the oul' passages were chosen to be more like typical college-level readin' material, compared to previous SAT readin' passages. Sure this is it. The changes for increased emphasis on analytical readin' were made in response to a 1990 report issued by a feckin' commission established by the feckin' College Board. Here's another quare one. The commission recommended that the oul' SAT should, among other things, "approximate more closely the skills used in college and high school work".[111] A mandatory essay had been considered as well for the new version of the SAT; however, criticism from minority groups, as well as a bleedin' concomitant increase in the bleedin' cost of the oul' test necessary to grade the bleedin' essay, led the College Board to drop it from the planned changes.[192]
Major changes were also made to the feckin' SAT mathematics section at this time, due in part to the oul' influence of suggestions made by the oul' National Council of Teachers of Mathematics, you know yerself. Test-takers were now permitted to use calculators on the bleedin' math sections of the oul' SAT, the cute hoor. Also, for the first time since 1935, the SAT would now include some math questions that were not multiple choice, and would require students to supply the answers for those questions. I hope yiz are all ears now. Additionally, some of these "student-produced response" questions could have more than one correct answer, the shitehawk. The tested mathematics content on the feckin' SAT was expanded to include concepts of shlope of a holy line, probability, elementary statistics includin' median and mode, and problems involvin' countin'.[111]
### 1995 recenterin' (raisin' mean score back to 500)
By the oul' early 1990s, average combined SAT scores were around 900 (typically, 425 on the verbal and 475 on the bleedin' math), be the hokey! The average scores on the 1994 modification of the bleedin' SAT I were similar: 428 on the bleedin' verbal and 482 on the oul' math.[193] SAT scores for admitted applicants to highly selective colleges in the United States were typically much higher. For example, the score ranges of the feckin' middle 50% of admitted applicants to Princeton University in 1985 were 600 to 720 (verbal) and 660 to 750 (math).[194] Similarly, median scores on the bleedin' modified 1994 SAT for freshmen enterin' Yale University in the fall of 1995 were 670 (verbal) and 720 (math).[195] For the majority of SAT takers, however, verbal and math scores were below 500: In 1992, half of the oul' college-bound seniors takin' the feckin' SAT were scorin' between 340 and 500 on the feckin' verbal section and between 380 and 560 on the oul' math section, with correspondin' median scores of 420 and 470, respectively.[196]
The drop in SAT verbal scores, in particular, meant that the feckin' usefulness of the oul' SAT score scale (200 to 800) had become degraded. At the top end of the oul' verbal scale, significant gaps were occurrin' between raw scores and uncorrected scaled scores: a bleedin' perfect raw score no longer corresponded to an 800, and an oul' single omission out of 85 questions could lead to a bleedin' drop of 30 or 40 points in the oul' scaled score. Corrections to scores above 700 had been necessary to reduce the bleedin' size of the gaps and to make an oul' perfect raw score result in an 800. At the feckin' other end of the feckin' scale, about 1.5 percent of test-takers would have scored below 200 on the feckin' verbal section if that had not been the reported minimum score, be the hokey! Although the math score averages were closer to the feckin' center of the oul' scale (500) than the bleedin' verbal scores, the oul' distribution of math scores was no longer well approximated by a bleedin' normal distribution. Sure this is it. These problems, among others, suggested that the oul' original score scale and its reference group of about 10,000 students takin' the SAT in 1941 needed to be replaced.[188]
Beginnin' with the feckin' test administered in April 1995, the oul' SAT score scale was recentered to return the feckin' average math and verbal scores close to 500. Bejaysus here's a quare one right here now. Although only 25 students had received perfect scores of 1600 in all of 1994, 137 students takin' the bleedin' April test scored 1600.[197] The new scale used an oul' reference group of about one million seniors in the feckin' class of 1990: the scale was designed so that the SAT scores of this cohort would have a mean of 500 and an oul' standard deviation of 110. Jaykers! Because the oul' new scale would not be directly comparable to the feckin' old scale, scores awarded in April 1995 and later were officially reported with an "R" (for example, "560R") to reflect the oul' change in scale, an oul' practice that was continued until 2001.[188] Scores awarded before April 1995 may be compared to those on the oul' recentered scale by usin' official College Board tables. Be the holy feck, this is a quare wan. For example, verbal and math scores of 500 received before 1995 correspond to scores of 580 and 520, respectively, on the 1995 scale.[198]
### 1995 re-centerin' controversy
Certain educational organizations viewed the bleedin' SAT re-centerin' initiative as an attempt to stave off international embarrassment in regards to continuously declinin' test scores, even among top students. As evidence, it was presented that the feckin' number of pupils who scored above 600 on the verbal portion of the bleedin' test had fallen from an oul' peak of 112,530 in 1972 to 73,080 in 1993, a 36% backslide, despite the fact that the oul' total number of test-takers had risen by over 500,000.[199]
### 2002 changes – Score Choice
Since 1993, usin' a policy referred to as "Score Choice", students takin' the oul' SAT-II subject exams were able to choose whether or not to report the bleedin' resultin' scores to a college to which the oul' student was applyin', bedad. In October 2002, the oul' College Board dropped the feckin' Score Choice option for SAT-II exams, matchin' the bleedin' score policy for the traditional SAT tests that required students to release all scores to colleges.[200] The College Board said that, under the oul' old score policy, many students who waited to release scores would forget to do so and miss admissions deadlines, to be sure. It was also suggested that the bleedin' old policy of allowin' students the feckin' option of which scores to report favored students who could afford to retake the oul' tests.[201]
### 2005 changes, includin' a new 2400-point score
In 2005, the feckin' test was changed again, largely in response to criticism by the oul' University of California system.[113] In order to have the bleedin' SAT more closely reflect high school curricula, certain types of questions were eliminated, includin' analogies from the feckin' verbal section and quantitative comparison items from the feckin' math section.[111] A new writin' section, with an essay, based on the feckin' former SAT II Writin' Subject Test, was added,[202] in part to increase the bleedin' chances of closin' the oul' openin' gap between the bleedin' highest and midrange scores, fair play. The writin' section reported a multiple-choice subscore that ranged from 20 to 80 points.[203] Other factors included the feckin' desire to test the bleedin' writin' ability of each student; hence the feckin' essay. Sufferin' Jaysus. The essay section added an additional maximum 800 points to the feckin' score, which increased the bleedin' new maximum score to 2400.[204] The "New SAT" was first offered on March 12, 2005, after the oul' last administration of the oul' "old" SAT in January 2005. Here's a quare one. The mathematics section was expanded to cover three years of high school mathematics. G'wan now. To emphasize the importance of readin', the oul' verbal section's name was changed to the oul' Critical Readin' section.[111]
### 2008 changes
As part of an effort to “reduce student stress and improve the test-day experience", in late 2008 the bleedin' College Board announced that the oul' Score Choice option, recently dropped for SAT subject exams, would be available for both the oul' SAT subject tests and the feckin' SAT startin' in March 2009. At the time, some college admissions officials agreed that the bleedin' new policy would help to alleviate student test anxiety, while others questioned whether the oul' change was primarily an attempt to make the SAT more competitive with the ACT, which had long had a comparable score choice policy.[209] Recognizin' that some colleges would want to see the bleedin' scores from all tests taken by a bleedin' student, under this new policy, the oul' College Board would encourage but not force students to follow the feckin' requirements of each college to which scores would be sent.[210] A number of highly selective colleges and universities, includin' Yale, the bleedin' University of Pennsylvania, Cornell, and Stanford, rejected the feckin' Score Choice option at the oul' time.[211] Since then, Cornell,[212] University of Pennsylvania,[213] and Stanford[214] have all adopted Score Choice, but Yale[215] continues to require applicants to submit all scores. Jesus Mother of Chrisht almighty. Others, such as MIT and Harvard, allow students to choose which scores they submit, and use only the oul' highest score from each section when makin' admission decisions. Still others, such as Oregon State University and University of Iowa, allow students to choose which scores they submit, considerin' only the test date with the feckin' highest combined score when makin' admission decisions.[216]
### 2012 changes
Beginnin' in the bleedin' fall of 2012, test takers were required to submit a current, recognizable photo durin' registration. I hope yiz are all ears now. In order to be admitted to their designated test center, students were required to present their photo admission ticket—or another acceptable form of photo ID—for comparison to the oul' one submitted by the oul' student at the bleedin' time of registration. Sufferin' Jaysus listen to this. The changes were made in response to a series of cheatin' incidents, primarily at high schools in Long Island, New York, in which high-scorin' test takers were usin' fake photo IDs to take the bleedin' SAT for other students.[217] In addition to the oul' registration photo stipulation, test takers were required to identify their high school, to which their scores, as well as the submitted photos, would be sent. Bejaysus this is a quare tale altogether. In the event of an investigation involvin' the validity of a bleedin' student's test scores, their photo may be made available to institutions to which they have sent scores, so it is. Any college that is granted access to a feckin' student's photo is first required to certify that the student has been admitted to the feckin' college requestin' the feckin' photo.[218]
### 2016 changes, includin' the oul' return to a holy 1600-point score
On March 5, 2014, the oul' College Board announced its plan to redesign the feckin' SAT in order to link the exam more closely to the work high school students encounter in the feckin' classroom.[219] The new exam was administered for the oul' first time in March 2016.[220] Some of the bleedin' major changes were: an emphasis on the oul' use of evidence to support answers, a shift away from obscure vocabulary to words that students are more likely to encounter in college and career, an optional essay, questions havin' four rather than five answer options, and the feckin' removal of penalty for wrong answers (rights-only scorin').[221][222] The Critical Readin' section was replaced with the new Evidence-Based Readin' and Writin' section (the Readin' Test and the oul' Writin' and Language Test).[223] The scope of mathematics content was narrowed to include fewer topics, includin' linear equations, ratios, and other precalculus topics, fair play. The essay score was separated from the oul' final score, and institutions could choose whether or not to consider it. As a bleedin' result of these changes, the highest score was returned to 1600. These modifications were the oul' first major redesign to the feckin' structure of the bleedin' test since 2005.[100] As the oul' test no longer deducts points for wrong answers, the oul' numerical scores and the feckin' percentiles appeared to have increased after the new SAT was unveiled in 2016. However, this does not necessarily mean students came better prepared.[222]
To combat the oul' perceived advantage of costly test preparation courses, the oul' College Board announced an oul' new partnership with Khan Academy to offer free online practice problems and instructional videos.[219]
### 2019 introduction and abandonment of the bleedin' 'Adversity Score' and launchin' of 'Landscape'
In May 2019, the College Board announced that it would calculate each SAT taker's "Adversity Score" usin' factors such as the bleedin' proportion of students in a school district receivin' free or subsidized lunch or the level of crime in that neighborhood. The higher the score, the bleedin' more adversity the student faced.[224] However, this triggered an oul' strong backlash from the oul' general public as people were skeptical of how complex information can be conveyed with an oul' single number[224] and were concerned that it might be politically weaponized.[225] The College Board thus abandoned the bleedin' Adversity Score and instead created an oul' new tool called 'Landscape' to provide the same sort of details to admissions officers usin' government information but without calculatin' a feckin' score.[224]
### 2021 changes
In the feckin' wake of the bleedin' COVID-19 pandemic, which made administerin' and takin' the feckin' tests difficult, on January 19, 2021, the College Board announced plans to discontinue the bleedin' optional SAT essay followin' the bleedin' June 2021 administration.[226][171] While some administrations were canceled,[171] others continued with precautionary measures such as requirements of temperature checks, enhanced ventilation, higher ceilings, physical distancin', and face masks.[227] The College Board also announced the feckin' immediate discontinuation of the feckin' SAT Subject Tests in the feckin' United States, and the bleedin' same internationally after the June 2021 administration.[171]
## Name changes
Old SAT logo
The SAT has been renamed several times since its introduction in 1926. Soft oul' day. It was originally known as the oul' Scholastic Aptitude Test.[228][111] In 1990, a commission set up by the feckin' College Board to review the feckin' proposed changes to the SAT program recommended that the feckin' meanin' of the feckin' initialism SAT be changed to "Scholastic Assessment Test" because a holy "test that integrates measures of achievement as well as developed ability can no longer be accurately described as a feckin' test of aptitude".[229][230] In 1993, the bleedin' College Board changed the feckin' name of the oul' test to SAT I: Reasonin' Test; at the same time, the bleedin' name of the Achievement Tests was changed to SAT II: Subject Tests.[228] The Reasonin' Test and Subject Tests were to be collectively known as the feckin' Scholastic Assessment Tests. Accordin' to the feckin' president of the College Board at the oul' time, the feckin' name change was meant "to correct the bleedin' impression among some people that the SAT measures somethin' that is innate and impervious to change regardless of effort or instruction."[231] The new SAT debuted in March 1994, and was referred to as the bleedin' Scholastic Assessment Test by major news organizations.[191][232] However, in 1997, the oul' College Board announced that the bleedin' SAT could not properly be called the feckin' Scholastic Assessment Test, and that the oul' letters SAT did not stand for anythin'.[233] In 2004, the oul' Roman numeral in SAT I: Reasonin' Test was dropped, makin' SAT Reasonin' Test the oul' name of the feckin' SAT.[228] The "Reasonin' Test" portion of the name was eliminated followin' the bleedin' exam's 2016 redesign; it is now simply called the SAT.[234]
## Reuse of old SAT exams
The College Board has been accused of completely reusin' old SAT papers previously given in the feckin' United States.[235] The recyclin' of questions from previous exams has been exploited to allow for cheatin' on exams and impugned the feckin' validity of some students' test scores, accordin' to college officials, you know yourself like. Test preparation companies in Asia have been found to provide test questions to students within hours of a new SAT exam's administration.[236][237]
On August 25, 2018, the SAT test given in America was discovered to be a recycled October 2017 international SAT test given in China, that's fierce now what? The leaked PDF file was on the feckin' internet before the oul' August 25, 2018 exam.[238]
## Notes
1. ^ In 2020, the bleedin' SAT was also offered on an additional September date due to the bleedin' COVID-19 pandemic.[1]
2. ^ Dependin' on the author, there might be an oul' negative sign. Whisht now and eist liom. This comes from the feckin' fact that the higher the oul' rank, the smaller the feckin' number of that rank.
3. ^ a b Known as the bleedin' SAT Subject Tests since 2005, discontinued in 2021.
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# How to apply an R prediction model to very big data from SQL database in parallel.
I dont need to load the entire dataset into memory. In fact I only need 1 row at a time to apply a trained model, get the predicted response and put that response somewhere, possibly back into another table in the DB. Question is, how to do this efficiently. Of course the entire SQL based DB wont fit into RAM, but a few million rows of it will at a time. Say the entire DB is a billion rows. Is there any package which efficiently retrieves the data, applies a model or runs a function on the data and repeats in a parallel way, utilizing all cores, RAM (to minimize number of over the network pulls), etc? Thank you.
If your model is more complicated and you want to do the prediction using R functions then you might need to move your data into something like an ff object (see the ff package) which stores the data on disk and brings only parts into memory at a time and provides apply like functions to process the whole dataset in chunks.
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MathSciNet bibliographic data MR1234632 (95f:04008) 04A20 Shioya, Masahiro The minimal normal $\mu$$\mu$-complete filter on $P\sb \kappa\lambda$$P\sb \kappa\lambda$. Proc. Amer. Math. Soc. 123 (1995), no. 5, 1565–1572. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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# Introduction
This vignette demonstrates how to use the loo package to carry out Pareto smoothed importance-sampling leave-one-out cross-validation (PSIS-LOO) for purposes of model checking and model comparison.
In this vignette we can’t provide all necessary background information on PSIS-LOO and its diagnostics (Pareto $$k$$ and effective sample size), so we encourage readers to refer to the following papers for more details:
• Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. Links: published | arXiv preprint.
• Vehtari, A., Gelman, A., and Gabry, J. (2017). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.04544.
# Setup
In addition to the loo package, we’ll also be using rstanarm and bayesplot:
library("rstanarm")
library("bayesplot")
library("loo")
# Example: Poisson vs negative binomial for the roaches dataset
## Background and model fitting
The Poisson and negative binomial regression models used below in our example, as well as the stan_glm function used to fit the models, are covered in more depth in the rstanarm vignette Estimating Generalized Linear Models for Count Data with rstanarm. In the rest of this vignette we will assume the reader is already familiar with these kinds of models.
### Roaches data
The example data we’ll use comes from Chapter 8.3 of Gelman and Hill (2007). We want to make inferences about the efficacy of a certain pest management system at reducing the number of roaches in urban apartments. Here is how Gelman and Hill describe the experiment and data (pg. 161):
the treatment and control were applied to 160 and 104 apartments, respectively, and the outcome measurement $$y_i$$ in each apartment $$i$$ was the number of roaches caught in a set of traps. Different apartments had traps for different numbers of days
In addition to an intercept, the regression predictors for the model are roach1, the pre-treatment number of roaches (rescaled above to be in units of hundreds), the treatment indicator treatment, and a variable indicating whether the apartment is in a building restricted to elderly residents senior. Because the number of days for which the roach traps were used is not the same for all apartments in the sample, we use the offset argument to specify that log(exposure2) should be added to the linear predictor.
# the 'roaches' data frame is included with the rstanarm package
data(roaches)
str(roaches)
'data.frame': 262 obs. of 5 variables:
$y : int 153 127 7 7 0 0 73 24 2 2 ...$ roach1 : num 308 331.25 1.67 3 2 ...
$treatment: int 1 1 1 1 1 1 1 1 0 0 ...$ senior : int 0 0 0 0 0 0 0 0 0 0 ...
yrep = yrep,
lw = weights(loo1$psis_object) ) The excessive number of values close to 0 indicates that the model is under-dispersed compared to the data, and we should consider a model that allows for greater dispersion. ## Try alternative model with more flexibility Here we will try negative binomial regression, which is commonly used for overdispersed count data. Unlike the Poisson distribution, the negative binomial distribution allows the conditional mean and variance of $$y$$ to differ. fit2 <- update(fit1, family = neg_binomial_2) loo2 <- loo(fit2, save_psis = TRUE, cores = 2) Warning: Found 1 observation(s) with a pareto_k > 0.7. We recommend calling 'loo' again with argument 'k_threshold = 0.7' in order to calculate the ELPD without the assumption that these observations are negligible. This will refit the model 1 times to compute the ELPDs for the problematic observations directly. print(loo2) Computed from 4000 by 262 log-likelihood matrix Estimate SE elpd_loo -895.8 37.8 p_loo 6.9 2.7 looic 1791.7 75.5 ------ Monte Carlo SE of elpd_loo is NA. Pareto k diagnostic values: Count Pct. Min. n_eff (-Inf, 0.5] (good) 261 99.6% 785 (0.5, 0.7] (ok) 0 0.0% <NA> (0.7, 1] (bad) 1 0.4% 27 (1, Inf) (very bad) 0 0.0% <NA> See help('pareto-k-diagnostic') for details. plot(loo2, label_points = TRUE) Using the label_points argument will label any $$k$$ values larger than 0.7 with the index of the corresponding data point. These high values are often the result of model misspecification and frequently correspond to data points that would be considered outliers’’ in the data and surprising according to the model Gabry et al (2018). Unfortunately, while large $$k$$ values are a useful indicator of model misspecification, small $$k$$ values are not a guarantee that a model is well-specified. If there are a small number of problematic $$k$$ values then we can use a feature in rstanarm that lets us refit the model once for each of these problematic observations. Each time the model is refit, one of the observations with a high $$k$$ value is omitted and the LOO calculations are performed exactly for that observation. The results are then recombined with the approximate LOO calculations already carried out for the observations without problematic $$k$$ values: if (any(pareto_k_values(loo2) > 0.7)) { loo2 <- loo(fit2, save_psis = TRUE, k_threshold = 0.7) } 1 problematic observation(s) found. Model will be refit 1 times. Fitting model 1 out of 1 (leaving out observation 93) print(loo2) Computed from 4000 by 262 log-likelihood matrix Estimate SE elpd_loo -895.7 37.7 p_loo 6.7 2.6 looic 1791.4 75.4 ------ Monte Carlo SE of elpd_loo is 0.2. All Pareto k estimates are good (k < 0.5). See help('pareto-k-diagnostic') for details. In the print output we can see that the Monte Carlo SE is small compared to the other uncertainties. On the other hand, p_loo is about 7 and still a bit higher than the total number of parameters in the model. This indicates that there is almost certainly still some degree of model misspecification, but this is much better than the p_loo estimate for the Poisson model. For further model checking we again examine the LOO-PIT values. yrep <- posterior_predict(fit2) ppc_loo_pit_overlay(roaches$y, yrep, lw = weights(loo2\$psis_object))
The plot for the negative binomial model looks better than the Poisson plot, but we still see that this model is not capturing all of the essential features in the data.
## Comparing the models on expected log predictive density
We can use the compare_models function in rstanarm, which is a wrapper around loo’s compare function that also does some checks to ensure that the rstanarm models from which loo1 and loo2 were created are suitable for comparison.
compare_models(loo1, loo2) # use loo::compare(loo1, loo2) if not using rstanarm
Model comparison:
(negative 'elpd_diff' favors 1st model, positive favors 2nd)
elpd_diff se
5341.2 706.7
The difference in ELPD is much larger than twice the estimated standard error again indicating that the negative-binomial model is expected to have better predictive performance than the Poisson model. However, according to the LOO-PIT checks there is still some misspecification, and a reasonable guess is that a hurdle or zero-inflated model would be an improvement (we leave that for another case study).
# References
Gabry, J., Simpson, D., Vehtari, A., Betancourt, M., and Gelman, A. (2018). Visualization in Bayesian workflow. Journal of the Royal Statistical Society Series A, accepted for publication. arXiv preprint arXiv:1709.01449
Gelman, A. and Hill, J. (2007). Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press, Cambridge, UK.
Vehtari, A., Gelman, A., and Gabry, J. (2017). Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing. 27(5), 1413–1432. :10.1007/s11222-016-9696-4. online, arXiv preprint arXiv:1507.04544.
Vehtari, A., Gelman, A., and Gabry, J. (2017). Pareto smoothed importance sampling. arXiv preprint arXiv:1507.02646.
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Reykjavík
I’m doing a lot of travel for work this Summer, mainly to nice places that I’ve wanted to visit for a long time. First up was Iceland, which has been absolutely top of the list for years and I was very very lucky to finally have an opportunity to go without bankrupting myself. So while usually I stay a day or two extra after work to look around, for this one I stayed a whole extra week for my Summer holiday.
I spent 3 days in Reykjavík attending Logic in Computer Science. Then I went to the highlands in the South for 5 days hiking.1 Then I came back to Reykjavík for a couple nights in a hostel and some time to look around. I’ll write up a report in the hiking separately.
Since I was only there a few days (and working for half of them), I only had time to see a couple of the many things to see in Reykjavík and missed a lot of the big attractions. I feel okay about that though. It’s hard to ignore the sheer volume of tourism happening all around you and the way it’s clearly affecting daily life in the city—AirBnBs have driven house prices in Reykjavík up so quickly that Iceland is considering legislating against them. Something less easy to articulate about why tourism is bad at this level in a country with such a small population is that it alienates people from their country & culture. The people visiting your country may have a different idea as a group about your country than you do. But if they seriously outnumber you, and the whole town around you starts very quickly to reflect somebody else’s conception of your home,2 you can see why there might be pushback. I can understand why the locals are ambivalent about tourists.
Anyway, because of this stuff I was happy not to be participating in obvious tourism too much. Here are some things I did get up to.
Nautholsvik geothermal beach
Directly behind Reykjavík University, just 5 minutes walk from the main building, there’s a little geothermal beach. There’s a volcanic hotspot underwater just offshore here, and Reykjavík has made a mini spa of it. Some hot water is pumped up into a hot tub a little way back from the shore, where one sits in hot sulfurous water and prepares for the sea. The rest of the boiling water from the vents escapes into a seawater lagoon, raising its temperature from “unbearable for untrained humans” to “bearable for 10 minutes or so for me, if I swim crawl hard to stay warm”. The idea is that you get hot in the tub, then run out in the wind into the sea, repeating the cycle two or three times.
I came to this beach in the evening both days after working at the conference. It’s a great way to relax and build up an appetite for dinner. The best bit is that it’s free: Reykjavík University is a little way out of town, so this area isn’t full of tourists and there’s no incentive to charge a fortune. It’s more like a public amenity: most of the people in the hot tub seemed to be employees or students at the university. I heard more spoken Icelandic here than anywhere else the whole trip.
Hallgrímskirkja
The largest church in Iceland and Reykjavík’s most well-know landmark, it’s that church that looks like a low-variance normal distribution. Commissioned in 1937, built between 1945 and 1986, it’s quite a strange building. It was designed at time when (Lutheran) Christianity was still a powerful force in Iceland (and could still attract large congregations) and it has all the usual stuff: pulpit, font, lots and lots of pews, a very large organ, stained glass, & so on. But architecture from 1937 already looks distinctly modern, and the result felt unnatural somehow. Of course architectural styles change over time, and if the building were still used regularly by a full congregation it might feel natural. But empty of worshippers as it was, it felt like a sleek modernist parody of a church. The stone is a pale over-designed-website grey, the organ is shining silver and looks not steampunk but more, I dunno, logicianpunk? Vienna-Circle-core? There’s something cultish about it. I recently read Don Delilo’s book Zero K about a secretive facility where the global elite undergo cryonic freezing in order to live forever in the future. The inside of this church is what I imagine that facility to look like.
The outside of the church is a masterpiece of course, and as a bonus there’s a great statue of my boy Leifur Eriksson standing on the prow of a ship:
National Museum of Iceland
The main permanent exhibition is a chronological display of artifacts from the complete history of humans in Iceland (possible since it was only settled in ~870 AD). Early on there’s lots of crude, essentially iron-age, tools and weaponry. It’s really remarkable how few supplies the settlers brought and managed to live on, and to think about how hardy they must have been. These give way to a lot of medieval religious artifacts, which I felt bad about skipping over after a while. The most interesting bit for me was the period from 1700–, when Iceland was a subject nation of Denmark and technological advances in fishing and sailing began to drive their economy.
There was also a nice side exhibit about Iceland’s place in the world. There was some good stuff here about Iceland acknowledging racism; not being a colonial power, Icelanders traditionally think of themselves as being exempt from that particular historical guilt. But it points out that immigrants in modern Iceland do not escape prejudice, and Icelanders need reminding that racism existed, and still exists in their country. There is a big display about the national debate that happened in 2007(!) about whether it was okay to republish an old Icelandic book about black children, which features pretty gross racial caricatures (spoiler: no, it’s not okay). The whole display and discussion though is conducted in quite an academic way, very clearly only by white people, and it seemed to me that it could be quite upsetting for a minority ethnic person to come to this museum and read about a ‘debate’ over whether something obviously wrong and unpleasant to them is really wrong. Still the museum seemed to be trying in good faith to make a worthy point. It’s better than what we get in Britain anyway.
Gaukurinn
I made friends with a local who took me to well-known bar and music venue Gaukurinn. Although it’s right in the middle of the city, it was pretty much the only bar I went to where Icelanders outnumbered visitors. That might be because it doesn’t seem like the kind of place that’s going to cater to what customers outside of their established clientele want: the general vibe is ‘aggressively progressive’. It turned out that they were having a fundraiser for their new all-vegan diner. I arrived in time to watch Reykjavík’s premier vegan rappers Vegan Klíkan. They mostly rapped in Icelandic, but I caught the odd English “culture of death” here and there. I tried to look non-guilty. After that there was a (vegan) black metal band whose name I didn’t catch. This was the first time I’d seen any kind of metal live, and I actually kind of liked it. I know that recorded I definitely hate it, so it was a bit of a surprise to enjoy it so much. Maybe it’s an essentially live genre? Good bar anyway, cheaper beer than everywhere else too. (Relatively of course: £8/pint instead of £9+/pint. I didn’t drink much.)
Quite a long walk back afterwards, but since it was near midsummer twilight lasted all night. I took the picture below as I was walking along the bay at 2am, and that’s pretty much the darkest it got the whole time I was there.
Laugardalslaug
The same local invited me to the swimming pool for the afternoon as well. Iceland has a similar swimming culture to Germany, where a trip to the pool means going to sit and chat in lots of small tubs of water, ranging from freezing to frankly burning hot. There was a steam room as well, and a water slide. Actually we didn’t do any swimming at all, but this is considered quite normal. In Iceland one can go to the pool with a friend much like going for a coffee.
The only important thing to remember at pools in Iceland is that you absolutely have to shower naked, thoroughly with soap, in the changing rooms before you get in the pool. The pools are unchlorinated and apparently it is considered very rude not to wash yourself in this way, and some locals will not hesitate to tell you off if you shower in your swimming shorts. There are even signs in English to remind tourists. I think Icelanders as a whole have a healthy attitude and are quite unembarrassed about their bodies,3 and this can be seen as a nice reflection of that. How to apply this shower policy to foreigners from cultures not accustomed to public nakedness (especially for religious reasons) is apparently the cause some disagreement among Icelanders. Some think that they should accommodate others’ preferences, others think that visitors should respect the local culture. I don’t know why I feel so strongly that the latter is correct while I’d be in favour of the former in England.
Eating the fermented shark (Kæstur hákarl)
Yeah it’s fucking horrible, don’t do it. The traditional chaser shot (Brennivín) is also horrible.
So, overall: Reykjavík, great city. It probably helps if you like the cold and aren’t a big fan of sunshine, like me. There’s something pleasant about a small city that is still a centre of culture. I’ll be back, maybe with more time and money.
1. Two reasons for this: (i) I really like hiking, and, lovely as Reykjavík is, I wasn’t going to go to Iceland and not go take in that scenery; (ii) Reykjavík is so expensive I can’t actually afford to stay there for a week anyway. ↩︎
2. A case in point are the so-called ‘Puffin shops’, the many shops in the centre of Reykjavík selling tourist tat: t-shirts, Icelandic flags, figurines, stuffed toys, & so on, the most overwhelmingly common motif being puffins. Why? Well the puffin is the national bird of Iceland innit? Except it’s not: the national bird of Iceland is the Gyrfalcon, and Icelanders feel no particular affection for the puffin at all (except as a meal). ↩︎
3. My bathing companion told me that to be embarrassed about being naked around others is remarkable enough that there’s a particular word for it in Icelandic. I forgot the word immediately though. It’s a tricky language, and I only managed the odd Góðan daginn and Takk fyrir here and there. ↩︎
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# Is mathematical rigour irrelevant in most physics fields? [duplicate]
Are mathematical notions like closed sets, limits of sequences, measures, and function spaces basically irrelevant in the day to day work of a physicist? Naturally, such concepts are the foundations upon which everything stands in both mathematics and physics, but do physicists need to concern themselves with the fine details of these concepts in their daily work or can they 'get away with' not having to use them while performing research?
Particularly in the areas of solid state physics, quantum mechanics, relativity, optics, and electromagnetism. Do any/some/all of these fields regularly/sometimes/never have to go that level of mathematical rigour? If these concepts do get used regularly in some particular fields, it would be great to hear some examples of how they arise and why they are necessary.
• Limits are certainly important, although you usually don't need to go down to the $\epsilon/\delta$ level of limits. Measures are important if you want to prove certain phenomena are general, e.g. that a certain phase exists for a region of parameters and not a measure-zero set of parameters. Closed/open sets are important when dealing with manifolds. Function spaces show up as Hilbert space in QM. – Jahan Claes Dec 12 '16 at 20:18
• Possible duplicates: physics.stackexchange.com/q/27665/2451 and links therein. – Qmechanic Dec 12 '16 at 20:25
• @Qmechanic My question is in the context of the day to day work of a research physicist..how much of a role does rigour play, if any? That linked question seems to primarily focus on the historical aspects of mathematical rigour in physics. – ManUtdBloke Dec 12 '16 at 20:51
• @JahanClaes I have not worked with manifolds (coming from a numerical mathematics background)..in what way are closed and opens sets applied to physics problems involving manifolds? – ManUtdBloke Dec 12 '16 at 20:53
• @eurocoder In GR you might want to know if the universe is an open or closed manifold (or both!), whether it has a boundary, etc. – Jahan Claes Dec 12 '16 at 21:16
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zbMATH — the first resource for mathematics
A theorem of density for translation invariant subspaces of $$L^ p(G)$$. (English) Zbl 0558.43002
Given a locally compact Abelian Hausdorff group G with Haar measure, and denoting by $$L_ p(G)$$ the corresponding Banach spaces, the author proves three theorems assuring the density of translation invariant subspaces S of $$L_ p(G)$$ for $$1\leq p<\infty$$, under some additional assumptions (among them, invariance of S under multiplication with suitable functions). We state the last theorem: If S is a self-adjoint translation invariant subspace of $$L_ p(G)$$ and there exists $$\phi \in L_{\infty}(G)$$ which is not periodic and such that $$\phi$$ $$S\subseteq S$$, then S is dense in $$L_ p(G)$$.
Reviewer: G.Crombez
MSC:
43A15 $$L^p$$-spaces and other function spaces on groups, semigroups, etc.
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# ISRO2015-51
6.2k views
How many characters per sec (7 bits + 1 parity) can be transmitted over a 2400 bps line if the transfer is synchronous ( 1 start and 1 stop bit)?
1. 300
2. 240
3. 250
4. 275
Ans A) 300
Total number of bits per character while transmitting is (7+1)=8 bits
No of character transmitted 2400/8=300 bps
selected by
7
I think it should be 300
in Synchronous transmission, we don't need start and stop bit.
on;y 8 bits are used ,so 2400/8 = 300 (A)
1
U r right , it should be 300.
0
The Answer is 300 according to answer key also
0
ok corrected
2
if it asynchronous,we need to consider start and stop bits?
0
YES FOR SURE.
0
Thank you
1
refer question no. 19 of ISRO 2007, a slight difference,then why have you not added start and stop bit ?
Awesome tricky question!
For synchronous transfer, we don't need start and stop bits.
So, $\frac{2400}{7+1}=300$
Option A
For asynchronous transfer, we need start and stop bits in order to synchronise!
So, $\frac{2400}{7+1+1+1}=240$
Given bandwidth is 2400 bps
Each character consists of (7+1)bits = 8 bits
Synchronous transmission is a data transfer method which is characterized by a continuous stream of data in the form of signals which are accompanied by regular timing signals which are generated by some external clocking mechanism meant to ensure that both the sender and receiver are synchronized with each other.
Once sender and receiver synchronized means no need to use start and stop bits every time
Total number of characters can be transmitted per second is =bandwidth/ number of bits in each character=2400/8 = 300
## Related questions
1
5.3k views
Which layers of the OSI reference model are host-to-host layers? Transport, session, presentation, application Session, presentation, application Datalink, transport, presentation, application Physical, datalink, network, transport
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Solving a non-linear equation in Mathematica with many variables
I have a beautiful equation, where I am trying to compute for R2. I am using Mathematica. So far every single time that I have tried to use Solve[] or Reduce [] the computation takes forever. I hope You can help me what to do with this beauty in order to get a solution. Thank you! Below is my equation.
eq1 = (R2 + ((((p2x - R2*(vac2y + voby)) - (k1x +
vox ((vobx ((p2y + R2*(vac2x + vobx)) - k1y) +
voby (k1x - (p2x - R2*(vac2y + voby))))/(-voby*
vox + vobx*voy))))^2 + ((p2y +
R2*(vac2x + vobx)) - ((p2y + R2*(vac2x + vobx)) -
voby ((vox (k1y - (p2y + R2*(vac2x + vobx))) +
voy ((p2x - R2*(vac2y + voby)) - k1x))/(-voby*vox +
vobx*voy))))^2)^0.5))/
v2 == (R1 + (((k1x - (k1x +
vox ((vobx ((p2y + R2*(vac2x + vobx)) - k1y) +
voby (k1x - (p2x - R2*(vac2y + voby))))/(-voby*
vox + vobx*voy))))^2 + (k1y - ((p2y +
R2*(vac2x + vobx)) -
voby ((vox (k1y - (p2y + R2*(vac2x + vobx))) +
voy ((p2x - R2*(vac2y + voby)) - k1x))/(-voby*vox +
vobx*voy))))^2)^0.5))/v1
edit:
The original system of equations is the following, I simply went on and substituted the variables in the first equation.
eg1 = (R2 + k2c)/v2 == (R1 + k1c)/v1
eg2 = ((k2x - x0)^2 + (k2y - y0)^2))^0.5 == k2c
eg3 = ((k1x - x0)^2 + (k1y - y0)^2))^0.5 == k1c
eg4 = p2x - R2*(vac2y + voby) == k2x
eg5 = p2y + R2*(vac2x + vobx) == k2y
eg6 = k2y -
voby ((vox (k1y - k2y) + voy (k2x - k1x))/(-voby*vox +
vobx*voy)) == y0
eg7 = k1x +
vox ((vobx (k2y - k1y) + voby (k1x - k2x))/(-voby*vox +
vobx*voy)) == x0
• 1) Have you ever actually obtained an answer from Solve or Reduce? In that case you could calculate once, save the answer, and never have to do it again. 2) Do you necessarily need an analytical solution? or could you give numerical values to the parameters first, and then calculate the solution numerically? I'm sure the latter would be much faster. – MarcoB Nov 13 '15 at 17:44
• 1, Yes, if I am using precalculated values for the coefficients I am getting the valid solution. 2, I do need an analytical solution, the values of p2x,p2y,R2,v1,v2 I don't know in advance. – jgulacsy Nov 13 '15 at 18:03
• 1) Then there is probably no very simple way to get the result faster (or at all!) in the fully symbolic format. 2) I understand that you don't know those values in advance, but possibly you might know them when you want to use the solution. Perhaps you could explain what you want to do with the analytical solution, and we might come up with an alternative. – MarcoB Nov 13 '15 at 18:17
• Do you know that the quantities inside the square roots are positive? – march Nov 13 '15 at 18:33
• This is a Analytic geometry problem, where the expressions starting with v (vox,voy) are unit vectors, p2x, and p2y are coordinates for points, R2 is the radius of a circle, v1 and v2 stands for velocity. It is a 4D trajectory planning problem where two objects have to collide, given their initial position,speed and some geometric restrictions. The only parameter that should be modified is R2 the rest is given at the start. The problem is that I need this calculation carried out on a microcontroller, that's why I'm thriving for an analytical solution. – jgulacsy Nov 13 '15 at 18:47
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# state-operator correspondence breaks down
+ 3 like - 0 dislike
1789 views
Is there a simple example where the state-operator correspondence breaks down or a general reason why this happens?
I'm reading this paper http://arxiv.org/pdf/hep-th/0208104.pdf where for the B model it is shown that the space of boundary vertex operators between two holomorphic branes $\mathcal{E}$ and $\mathcal{F}$ supported on the same submanifold $M$ is computed by the sheaf cohomology
$H^p(M,\mathcal{E}\otimes\mathcal{F}^*\otimes \wedge^qNM)$,
while the massless Ramond states are computed by
$Ext^{p+q}_X(i_* \mathcal{E}, i_*\mathcal{F})$,
where $i:M\to X$ is the inclusion of the submanifold into the ambient (Calabi-Yau) $X$.
There is a spectral sequence with $E_2^{p,q}$ equal to the former and converging to the latter. However, in the presence of a $B$-field, ie. curvature for the brane gauge fields, this spectral sequence has non-trivial differentials and so we have a breakdown of the state-operator correspondence.
Even simpler, if we are considering states stretching between the same brane, then the spectral sequence is trivial iff the tangent bundle of $X$ restricted to $M$ splits holomorphically as $TM \oplus NM$.
We always have a map from the $E^2$ page of vertex operators to the $E^\infty$ page of states. This probably coincides with the ordinary map in the state-operator correspondence where we put an operator at the tip of a (in this case half-)cigar and look at the state at the end. For some reason the usual argument about this being an isomorphism breaks down. Can we tell which situations this map fails to be injective or surjective?
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# Coherent state path integral - derivation
I divided the time interval $[t_0=:t_i,t_f:=t_N]$ into $N$ steps $[t_{k-1},t_{k}],\, k=1,\dots, N$ and applied the resolution of unity for coherent states $$\mathbb{I}=\int_\mathbb{C}\frac{dzd\bar{z}}{2\pi i}\exp\left\lbrace-z\bar{z}\right\rbrace\lvert z\rangle\langle z\rvert$$ at each step; this yields the following \begin{multline} \langle{z_f}\lvert\left( \exp\lbrace-iH\epsilon\right\rbrace)^N\rvert z_i\rangle=\dots=\lim_{N\to\infty}\int\prod_{j=1}^{N-1}\frac{dz_jd\bar{z_j}}{2\pi i}\\ \exp\left\lbrace\sum_{k=0}^{N-1}\bar{z}_{k+1}z_k-\sum_{k=1}^{N-1}z_k\bar{z_k}-i\epsilon\sum_{k=0}^{N-1}H(\bar{z}_{k+1},z_k)\right\rbrace,\quad(*) \end{multline} where $\epsilon:=\frac{t_f-t_i}{N}$, and with boundary conditions $z_0=z_i,\,\bar{z}_N=\bar{z}_f$. Then I need to put $(*)$ in this form \begin{align} \int\mathcal{D}(z,\bar{z})&\exp\bigg\lbrace\frac{\bar{z}_fz_f+\bar{z}_iz_i}{2}+\frac{1}{2}\sum_{k=0}^{N-1}\big[z_k(\bar{z}_{k+1}-\bar{z}_k)-\bar{z}_k(z_{k+1}-z_k)+\\ &-i\epsilon H(\bar{z}_{k+1},z_k)\big]\bigg\rbrace \end{align} but I can't understand how to transform the argument of the exponential.
Any ideas?
Thanks
Source: Itzykson - Zuber, page 438 with line 2 corrected as shown in the errata (you can find it here: http://www.lpthe.jussieu.fr/~zuber/corrize.pdf).
edit: @ ACuriousMind Then I don't understand how to rewrite it: is the following right? $$\frac{1}{2}\sum_{k=0}^{N-1}\big[z_k(\bar{z}_{k+1}-\bar{z}_k)-\bar{z}_k(z_{k+1}-z_k)\big]=\frac{1}{2}\sum_{k=0}^{N-1}\big[z_k\bar{z}_{k+1}-\bar{z}_k z_{k+1}\big]=\sum_{k=0}^{N-1}\big[z_k\bar{z}_{k+1}\big]$$ if this were right, I wouldn't know how to cope with this $$-\sum_{k=1}^{N-1}z_k\bar{z}_k$$
If it is so straightforward to you, could you please rewrite it step-by-step?
• It's just a rewriting of the sums, there's no "transformation" required. Jan 15, 2016 at 18:26
• Jan 15, 2016 at 20:00
• @Qmechanic My doubt is not about the path integral itself... it's not obvious to me how to rewrite it in the symmetric form proposed by Itzykson - Zuber... Brown's form of coherent state path integral, which you cite, is slightly different, thus, it doesn't answer my question... Jan 15, 2016 at 20:09
• Hint: The sought-for manipulation of the action is a discrete version of changing $\lambda=0$ to $\lambda=\frac{1}{2}$ in eq. (7) in my Phys.SE answer here. Jan 15, 2016 at 20:41
• You're right... but there is no derivation of that formula in your post. Can you provide a reference? Jan 15, 2016 at 20:49
At last I think I managed to rearrange the terms in the sums... (which wasn't a monumental task after all ;-) ) $$\sum_{k=0}^{N-1}\bar{z}_{k+1}z_k-\sum_{k=1}^{N-1}z_k\bar{z}_k=(*)$$ In the notation here:
• $\lambda=0$: $$(*)=\sum_{k=0}^{N-1}\bar{z}_{k+1}z_k-\sum_{k=0}^{N-1}z_{k+1}\bar{z}_{k+1}+z_N\bar{z}_N=-\sum_{k=0}^{N-1}\bar{z}_{k+1}(z_{k+1}-z_k)+z_f\bar{z}_f,\,(1)$$
• $\lambda=1$: $$(*)=\sum_{k=0}^{N-1}\bar{z}_{k+1}z_k-\sum_{k=0}^{N-1}z_k\bar{z}_k+z_0\bar{z}_0=\sum_{k=0}^{N-1}z_k(\bar{z}_{k+1}-\bar{z}_k)+z_i\bar{z}_i,\,(2)$$ Taking $(2)+(1)$ we obtain $2(*)$, and finally
• $\lambda=1/2$ $$\sum_{k=0}^{N-1}\bar{z}_{k+1}z_k-\sum_{k=1}^{N-1}z_k\bar{z}_k=\frac{1}{2}\sum_{k=0}^{N-1}\big[z_k(\bar{z}_{k+1}-\bar{z}_k)-\bar{z}_{k+1}(z_{k+1}-z_k)\big]+\frac{z_i\bar{z}_i+z_f\bar{z}_f}{2}$$ but there was a $+1$ missing in one of the $z$'s subscripts, if I'm right.
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# In how many ways can $6$ girls and $8$ boys be arranged in a row if no two girls should stand next to each other?
A teacher has 6 girls and 8 boys to arrange in for a choir. Determine the number of ways she can arrange the 14 children in a single row if no two girls should stand next to each other.
How do you do this? I just need to know how to solve this. You can use another example to explain this to me because this question is very tricky to me .
• Try 2 girls and 1 boy. Then 2 girls and 2 boys etc. to determine what the constraints are, and how many arrangements are possible satisfying the constraints. – Math Lover Sep 28 '17 at 22:25
$$\square b \square b \square b \square b \square b \square b \square b \square b \square$$ To separate the girls, choose six of these spaces in which to place the six girls.
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# Math Help - algorithm Question
1. ## algorithm Question
I need to figure out the answer to this question;
"Describe an algorithm that takes as input an automaton M = (Q,S, t, s,A),
where S = {a,b, c}, and determines whether or not M accepts a word a which does not
contain c (i.e. whether or not L(M) contains a word in {a,b}∗). You do not have to give
a formal proof that your algorithm is correct but you should indicate why this is the case."
This is what i ahve come up with so far, would you say this is correct?
Let M = (Q, S, t, s, A) be a fi nite deterministic automaton, where Q is
state set, S = a, b, c is alphabet, t : (Q x S) x Q) is transitions function,
s 'elem' Q is initial state, and A 'sub' Q is the set of nite state. To determine,
whether the word w 'elem' S* is accepted by M, do the following steps.
(a) If w = e (empty word), then w is accepted by M if and only if
t 'elem' A (initial state is also nite), process stops. Otherwise, go to
the next step.
(b) Let w0 = w, q0 = s, k = 0.
(c) If wk = e (empty word), then w is accepted by M if and only if
qk 'elem' A, process stops. Otherwise go to the next step.
(d) Let wk = xwk+1, where x 'elem' S (x is the first symbol of wk). Let
qk+1 = t(qk, x). Word w is accepted by M if and only if word wk+1
is accepted by automaton < Q, S, qk+1,wk+1 >.
(e) Increase k by 1, and go step c.
Each step reduces the length of current word (|wk+1| = |wk| - 1), so
this process is nite.
2. In your algorithm, w seems to be an input, whereas in the original problem, only M is the input. In other words, the problem is not this:
given M and w in {a, b}*, does M accepts w?
but this:
given M, does there exist a w in {a, b}* such that M accepts w?
3. Originally Posted by emakarov
In your algorithm, w seems to be an input, whereas in the original problem, only M is the input. In other words, the problem is not this:
given M and w in {a, b}*, does M accepts w?
but this:
given M, does there exist a w in {a, b}* such that M accepts w?
Ok, so are you saying i have produced a solution to the wrong problem? (well mt understanding of the problem is wrong)
Sorry i am not to good at this!
4. I think so.
Suppose M is given. I think that to find if M accepts some word without c, one has to make an exhaustive search. One starts with the initial state and tries every possible path following transitions for the symbols a and b only. Each state has to be visited at most once. If eventually a final state is reached, the answer is yes.
To make it more precise, you can consider a specific way to perform an exhaustive search, for example, breadth-first search.
5. Originally Posted by emakarov
I think so.
Suppose M is given. I think that to find if M accepts some word without c, one has to make an exhaustive search. One starts with the initial state and tries every possible path following transitions for the symbols a and b only. Each state has to be visited at most once. If eventually a final state is reached, the answer is yes.
To make it more precise, you can consider a specific way to perform an exhaustive search, for example, breadth-first search.
Thanks, i think i understand what your saying, im a little lost now though.
What do you think to this?
Draw the state diagram for M. Look for all transitions labelled "c" and remove them from
the diagram. Now try to find a path from the initial state s to any state in A. If at least one
state in A is reachable, that means the string corresponding to the edges on that path is a
member of L(M). Since none of the transitions had a "c", then the word must consist only
of a’s and b’s, and therefore M accepts at least one word that does not contain a "c".
Otherwise, if no path exists from s to a state in A, then the language does not accept any
word that does not contain "c".
6. Yes, this is better. In fact, there is an algorithm to find reachable states on p. 93 of "Elements of the Theory of Computation" by Lewis and Papdimitriou (2nd ed.). If we modify it a little, we get the following.
Code:
if s is in A then output "yes" and stop
R := {s}
while there is a state p in R, x in {a, b} and a state q such that t(p,x,q) and q is not in R do
if q is in A then output "yes" and stop
add q to R
end while
output "no"
Here R stands for the (gradually built) set of states reachable from s.
7. Originally Posted by emakarov
Yes, this is better. In fact, there is an algorithm to find reachable states on p. 93 of "Elements of the Theory of Computation" by Lewis and Papdimitriou (2nd ed.). If we modify it a little, we get the following.
Code:
if s is in A then output "yes" and stop
R := {s}
while there is a state p in R, x in {a, b} and a state q such that t(p,x,q) and q is not in R do
if q is in A then output "yes" and stop
add q to R
end while
output "no"
Here R stands for the (gradually built) set of states reachable from s.
Thanks very much for your guidance!
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I have used on.exit() for several years, but it was not until the other day that I realized a very weird thing about it: you’d better follow the default positions of its arguments expr and add, i.e., the first argument has to be expr and the second has to be add.
on.exit(expr = NULL, add = FALSE)
If you do on.exit(add = TRUE, {...}), weird things can happen. I discovered this by accident. I have never switched the positions of expr and add before, and I was surprised that R CMD check failed on Travis with an error message that confused me in the beginning:
Error in on.exit(add = TRUE, if (file.exists(main)) { :
I was thinking why add = TRUE was considered invalid. Then I guessed perhaps the expression if (file.exists(main)) {} was treated as the actual value of add. So I switched to the normal order of arguments, and the error was gone.
I tested it a bit more and was totally confused, e.g., why was 1 printed twice below? I guess TRUE was not printed because add was treated as expr.
f = function() {
}
f()
# [1] 1
# [1] 1
I don’t have the capability to understand the source code in C, and I’ll leave it experts to explain the weird things I observed. For me, I’ll just never move add before expr again.
BTW, I don’t what the rationale is for the default add = FALSE in on.exit(), but I have not used add = FALSE for a single time, so I feel add = TRUE might be a better default. When I want to do something on exit, I almost surely mean do it in addition to the things that I assigned to on.exit() before, instead of cleaning up all previous tasks and only doing this one (add = FALSE).
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Einstein's relation and osmotic pressure
How can I derive the Einstein's relation $D=k_{b}TB$, where $D$ is the diffusion coefficient and B is the mobility coefficient, from the concept of osmotic pressure?
-
• diffusion, which happens due to inhomogeneity in concentration. Particles "want to" go from areas of higher concentration to the lower ones. One can write this in the form of diffusion current $$J_{diff}(x) = - D \nabla \rho(x)$$ where $\rho(x)$ is the concentration. This expression is known as Fick's law but it's actually just the standard linear response to inhomogeneities.
• drift, which is the terminal velocity particles attain due to presence of some force. E.g. the drift one can observe for balls falling in viscous liquid. One can write $$J_{drift} = \rho(x) v(x) = \rho(x) B F(x) = -B \rho(x) \nabla U(x)$$
From the requirement of equilibrium we have that $J_{diff} + J_{drift} = 0$ and from Boltzmann statistics we can obtain the concentration $\rho(x) \sim \exp(-{U(x) \over k_B T})$. Putting it all together we get $$0 = - D \nabla \rho(x) - B \rho(x) \nabla U(x) = - \nabla U(x) \rho(x) (-{D \over k_B T} + B)$$ and we can see the required relation in the last term.
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# hpfem/esco2012-boa
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\title{An Efficient Dynamic $hp$-Discontinuous Galerkin Formulation for Time-Domain Electromagnetics} \tocauthor{S. Schnepp} \author{} \institute{} \maketitle \begin{center} {\large Sascha Schnepp}\\ Graduate School CE, TU Darmstadt\\ {\tt [email protected]} \end{center} \section*{Abstract} The discontinuous Galerkin method (DGM) \cite{reed} received considerable development over the past two decades leading it to a mature state. In many cases the DGM is applied in a spectral-like manner on static meshes using a fixed approximation order throughout all elements. The strictly local support of the basis functions, however, renders the method highly suitable for adapting the local element size, $h$, as well as the local approximation order, $p$, in an element-wise fashion based on the local solution behavior. In this talk a DG formulation on hexahedral meshes \cite{cohen} supporting dynamic $hp$-refinement \cite{houston} containing high-level hanging nodes and anisotropic refinement in both, $h$ and $p$, is presented. During the development special care of the computational efficiency of the basic algorithm and the adaptation procedures has been taken. The method is especially successful for multi-scale problems, where the adaptive approach leads to significant savings in computational resources and time. \bibliographystyle{plain} \begin{thebibliography}{10} \bibitem{reed} {\sc W. Reed and T. Hill}. {Triangular mesh methods for the neutron transport equation}. Tech. rep., Los Alamos Scientific Laboratory Report (1973). \bibitem{cohen} {\sc G. Cohen and X. Ferrieres and S. Pernet}. {A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time domain}. J. Comput. Phys. 217 (2) (2006) 340-363. \bibitem{houston} {\sc P. Houston and E. S\"uli}. {A note on the design of hp-adaptive finite element methods for elliptic partial differential equations}. Comput. Method Appl. M 194 10 (2-5) (2005) 229-243. \end{thebibliography}
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## Wikipedia - Laser a stato solido (it) solid state lasers
https://doi.org/10.1351/goldbook.S05736
CW or pulsed lasers in which the @[email protected] is a solid matrix (crystal or glass) doped with an ion (e.g. Nd3+, Cr3+, Er3+). The emitted @[email protected] depends on the active ion, the selected optical transition, and the matrix. Some of these lasers are tunable within a very broad range (e.g. from $$700$$ to $$1000\ \text{nm}$$ for Ti3+ doped sapphire). Pulsed lasers may be free-running, Q-switched, or @[email protected] Some CW lasers may be @[email protected]
Source:
PAC, 1996, 68, 2223. (Glossary of terms used in photochemistry (IUPAC Recommendations 1996)) on page 2274 [Terms] [Paper]
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This site is supported by donations to The OEIS Foundation.
# Even abundant numbers
The even abundant numbers are even numbers ${\displaystyle \scriptstyle n\,}$ whose sum of divisors exceeds ${\displaystyle \scriptstyle 2n\,}$. (Or whose sum of proper divisors exceeds ${\displaystyle \scriptstyle n\,}$.)
A173490 Even abundant numbers.
{12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, ...}
The first 231 terms are the same as in A005101 (abundant numbers).
While the first even abundant number is ${\displaystyle \scriptstyle 12\,=\,2^{2}\cdot 3\,}$ with ${\displaystyle \scriptstyle \sigma (12)\,=\,{\frac {2^{3}-1}{2-1}}\cdot (3+1)\,=\,7\cdot 4\,=\,28\,>\,24\,=\,2\cdot 12\,}$, the first odd abundant number is ${\displaystyle \scriptstyle 945\,=\,3^{3}\cdot 5\cdot 7\,}$ with ${\displaystyle \scriptstyle \sigma (945)\,=\,{\frac {3^{4}-1}{3-1}}\cdot (5+1)\cdot (7+1)\,=\,40\cdot 6\cdot 8\,=\,1920\,>\,1890\,=\,2\cdot 945\,}$!
The first odd abundant number is the 232 nd abundant number!
Some abundant numbers (and even abundant numbers):
• Every positive multiple greater than 1 of a perfect number is abundant (those are all even, unless odd perfect numbers exist...).
• Every positive multiple of an abundant number is abundant.
• Every positive multiple of an even abundant number is even and abundant.
• Every even positive multiple of an odd abundant number is even and abundant.
## Properties
Abundant numbers (and hence even abundant numbers) are closed under multiplication by arbitrary positive integers: any positive multiple of an abundant number is abundant.
## Asymptotic density
As a consequence of the closure under multiplication by arbitrary positive integers, the even abundant numbers are of positive density. In particular, their lower density is at least 0.2453 and their upper density is at most 0.2460.[1][2]
• A005231 Odd abundant numbers (odd numbers whose sum of divisors of ${\displaystyle \scriptstyle n\,}$ exceeds ${\displaystyle \scriptstyle 2n\,}$).
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# Vector rotation (possibly Euler angles)
1. Jul 20, 2006
### kevdoig
I'm looking for a method to rotate a 3D vector, and place it at an arbitary 3D point (x,y,z) without changing the vectors magnitude. I have briefly investigated eulers angles (mainly through wikipedia links etc), but don't fully understand the process yet.
As an example, given the vector : (3.6,1.6,0)
How could i rotate this by 45degrees on X axis.
Then again by 45degrees on the Y axis, as a seperate rotation.
I would then like to visualise this vector at point (1,1,1), on a 3D plot i have created.(not sure if this effects anything...)
2. Jul 20, 2006
### Triss
The Euler angles is one way of doing it. If you set
$$D= \begin{pmatrix} \cos\phi & \sin\phi & 0\\ -\sin\phi & \cos\phi & 0\\ 0 & 0 & 1 \end{pmatrix} C= \begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta & \sin\theta\\ 0 & -\sin\theta & \cos\theta \end{pmatrix} B= \begin{pmatrix} \cos\psi & \sin\psi & 0\\ -\sin\psi & \cos\psi & 0\\ 0 & 0 & 1 \end{pmatrix}$$
Then the full rotation by the 3 angles are given by $$A=BCD$$. Then if your vector is $$x$$ your rotated vector becomes $$x'=Ax$$. The Euler angles can be defined in various ways. the above should fit with the figure here here . Thus your $$C$$ matrix is the identity matrix and $$\phi=\psi=\pi/4$$
3. Jul 20, 2006
### kevdoig
thanks,
my problems lies in how to calculate the euler angles for the rotation. If possible, could you maybe use an example (say, vector (3,2,0)) and show how to calculate the euler angles for a rotation of 30degrees x-axis, 20 degress y-axis, and 45 degrees z-axis for example.
Sorry if i'm missing something simple, but i'm new to eulers angles etc.
4. Jul 20, 2006
### hypermorphism
If you are attaching the vector to a point, you actually need to find two separate results: the new point the vector will be attached to after rotation, and the new orientation of the vector. If we call the position vector of the point the vector is attached to p, and the vector v, then these can be combined into getting the new position of the position vector p + v.
If you have the two rotation matrices X (some rotation about the x-axis) and Y (some rotation about the y-axis), then to get the new position, we just find YX(p + v) = YXp + YXv.
A rotation about one axis in 3-space is just a rotation in 2-space along with making sure nothing happens in the third dimension. Ie., a rotation about the x-axis is actually a rotation in the yz-plane where we make sure nothing happens to the x information.
A rotation matrix for a plane looks like
$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\\ \end{pmatrix}$$
The identity transformation leaves all values the same. For 3-dimensional Euclidean space with the usual basis the identity is
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$$
which is just the ordered list of basis vectors. We want to the y and z basis vectors, but leave the x basis vector the same, so we replace the lower right hand block, the identity for vectors in the yz-plane, with the rotation matrix R(\theta) above adjusted to rotate in the right-handed orientation (the old matrix rotates counterclockwise because we usually talk about rotations of some angle "from the positive x-axis". In 3 dimensions, we usually refer to clockwise rotations about an axis, so the angles are negated, which only affects the odd function sine).
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos\theta & \sin\theta\\ 0 & -\sin\theta & \cos\theta\\ \end{pmatrix}$$
For rotation about the y-axis, or rotation in the xz-plane, we replace the 4 values corresponding to the identity block for the xz-plane with the rotation matrix (see the matrix like a torus, the screen of an Asteroids game).
$$\begin{pmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0\\ \sin\theta & 0 & \cos\theta\\ \end{pmatrix}$$
To rotate your example by 45 degrees about the x-axis, then 45 degrees about the y-axis, we would first apply the transformation
$$\begin{pmatrix} \frac{\sqrt{2}}{2} & 0 & -\frac{\sqrt{2}}{2} \\ 0 & 1 & 0\\ \frac{\sqrt{2}}{2} & 0 & \frac{\sqrt{2}}{2}\\ \end{pmatrix}$$
and then the transformation
$$\begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\\ 0 & -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}\\ \end{pmatrix}$$
as described above.
Last edited: Jul 20, 2006
5. Jul 21, 2006
### kevdoig
cheers,
exactly what i needed.
Kev
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# Iain Dunning
## Sudoku-as-a-Service with Julia
Posted on 15 Sep 2013
Sudoku is (was?) an extremely popular number puzzle. Players are presented with a 9 by 9 grid, with some of the cells filled in with a number between 1 and 9. The goal is to complete the grid while respecting these rules:
• Each row of the grid must contain each of the numbers 1 to 9 exactly once.
• Each column of the grid must contain each of the numbers 1 to 9 exactly once.
• Divide the grid into 9 3-by-3 non-overlapping subgrids. Each of these 3-by-3 grids must contain each of the numbers 1 to 9 exactly once.
An unsolved sudoku puzzle (Wikimedia Commons)
In this post I'm going to
• walk through one way to solve sudoku puzzles, and
• show how you can offer that functionality as an online service
using the Julia programming language.
## Solving Sudoku puzzles with integer programming
The rules of sudoku can be expressed as an integer programming problem which can be solved by any of the many linear and integer programming solvers out there. Of course there are algorithms designed to solve sudoku problems directly that are probably more efficient than integer programming, but stay with me for a moment - I'll justify it later on! We will start our model by defining a variable x(i,j,k) that equals 1 if and only if cell (i,j) is set to value k, and is 0 otherwise. This can be written mathematically (with constraints to enforce the fixed cells omitted) as:
MIP formulation of sudoku problem
Lets break down how these constraints translate back to the rules above. In the first set of constraints we enforce that for every row i, across the columns j the value k must appear once and once only. For example, for value 5 and row 3, we say that x(3,1,5) + x(3,2,5) + x(3,3,5) + ... + x(3,9,5) == 1. We can check: if a 5 appeared in row 3 column 2 and row 3 column 6, the sum on the left hand side would be 2. If 5 doesn't appear in row 3 at all, the sum would be 0. Note that x(3,1,5) = 0.5, x(3,2,5) = 0.5 would be feasible with respect to this constraint, but that we are enforcing the constraint (not shown) that x can only be 0 or 1.
The second set of constraints is very similar to the first, and enforces the second 'rule' above regarding columns. The third set of constraints is not so obvious. This set of constraints enforces that any given position i,j can only contain one digit by summing across all values. This is kinda self-evident for a human so I didn't even mention it above. Finally, the fourth set of constraints corresponds to the third rule. Its the most complex notationally, but is conceptually the same as the previous two. On the right side of the equations we iterate over the 9 3-by-3 subgrids, where (0,0) is the top left subgrid and (2,2) is the bottom subgrid. The sum is then over the 9 cells inside that subgrid. Lets pick an example subgrid, e.g. (i,j)=(0,1) and k=5, which corresponds to the left-center subgrid. We then sum over a and b:
• x(3*0+1,3*1+1,5) + x(3*0+2,3*1+1,5) + x(3*0+3,3*1+1,5) + x(3*0+1,3*1+2,5) + x(3*0+2,3*1+2,5) + ... == 1
• x(1,4,5) + x(2,4,5) + x(3,4,5) + x(1,5,5) + x(2,5,5) + ... == 1
As I mentioned above, we have ommitted constraints that would fix particular x(i,j,k) based on the provided cells, but there is no clean way to write that above. The question now is, how do we implement this model in an expressive and maintainable way in code?
## JuMPing Julia!
Julia is a fantastic up-and-coming dynamic language that emphasises high-performance scientific computing that is extensible and easy to read and write. It doesn't reinvent the wheel: it utilizes the LLVM compiler to generate fast code, open-source linear algebra packages to provide fast number crunching, and the libuv library to provide excellent cross-platform IO. libuv is the library that powers the popular node.js environment that has become very popular over the past couple of years for web development. Being a web development language is not an official goal of Julia, but there is certainly no reason it can't be used. For example, if server-side number crunching is one of the features your site needs, that part could be implemented in Julia and the rest of the application logic in node.js or your tool of choice.
But back to sudoku. The Julia package JuMP is an algebraic modeling language for linear (and integer and quadratic) programming created by myself and the very talented Miles Lubin. You should compare JuMP with tools like PuLP and AMPL. If you aren't familiar with algebraic modeling languages embedded in other languages, you can think of them as an embedded domain specific language (DSL). Julia's fantastic metaprogramming functionality has allowed us to create a particularily fast and expressive modeling language. I encourage you to read to the documentation, but here is a taste of what it looks like.
1 # Assuming a solver has been previously installed, e.g. Cbc
2 using JuMP
3
4 function SolveModel(initgrid)
5 m = Model()
6
7 # Create the variables
8 @defVar(m, 0 <= x[1:9, 1:9, 1:9] <= 1, Int)
9
10 # ... snip other constraints ...
11 # Constraint 4 - Only one value in each cell
12 for row in 1:9
13 for col in 1:9
14 @addConstraint(m, sum{x[row, col, val], val=1:9} == 1)
15 end
16 end
17
18 # ... snip initial solution constraints ...
19 # Solve it (default solver is CBC)
20 status = solve(m)
21
22 # Check solution
23 if status == :Infeasible
24 error("No solution found!")
25 else
26 mipSol = getValue(x)
27
28 sol = zeros(Int, 9, 9)
29 for row in 1:9
30 for col in 1:9
31 for val in 1:9
32 if mipSol[row, col, val] >= 0.9 # mipSol is stored as floats
33 sol[row, col] = val
34 end
35 end
36 end
37 end
38
39 return sol
40 end
41 end
Line 7 creates our variable x with three indices over the range 1 to 9, and enforces integrality. @defVar is a macro, not a function, which lets it create a variable x in the local scope. I have removed the other constraints for brevity and left the constraint that enforces that each cell contains only one value. Note the correspondance between the mathematical notation ("for all rows, for all columns") and the for loops (lines 11 and 12). @addConstraint (line 13) is another macro that facilitates the efficient storage of the constraint as a sparse vector, and matches the mathematical description very closely. On line 19 we solve the model with the default solver COIN-OR CBC - an open-source integer programming solver. As a side note, Julia/JuMP also has interfaces to GLPK (open-source) and Gurobi (closed-source). Finally, we pull the solution back from the solver (if a feasible solution could be found) as a three-dimensional matrix of 0s and 1s which we convert to a two-dimensional vector of 1-to-9s (lines 27 to 36).
If you are interested in how macros and metaprogramming offer up new possibilities for optimization, operations research, and modelling languages, check out our paper.
## SaaS == Sudoku-as-a-service?
Now we can solve sudoku puzzles, we should share this functionality with the world. The best way to get started with creating a web service is to use the HttpServer.jl package, a package made at Hacker School. Documentation is pretty scarce at this point, but hopefully by inspecting the relatively short code and looking at some examples you can get started.
I bundled my sudoku solver with a server in SudokuService. This kinda brings me back to why I used integer programming - to demonstrate that you can make pretty complex web-capable number crunching applications with relative ease. Check out server.jl in the repository - its pretty straightforward and most of the work is input validation. The form of a query is /sudoku/123...123 or /sudoku/123...123/pretty for human-readable response. There should be 81 numbers, one for each cell of the 9x9 sudoku board, row-wise. A zero indicates a blank. Heres a taster of the server code:
using HttpServer
# Load the Sudoku solver
require("sudoku.jl")
# Build the request handler
http = HttpHandler() do req::Request, res::Response
if ismatch(r"^/sudoku/", req.resource)
# Expecting 81 numbers between 0 and 9
reqsplit = split(req.resource, "/")
# ...snip validation...#
probstr = reqsplit[3]
if length(probstr) != 81
return Response(400, "Error: expected 81 numbers.")
end
# Convert string into numbers, and place in matrix
# Return error if any non-numbers or numbers out of range detected
prob = zeros(Int,9,9)
pos = 1
try
for row = 1:9
for col = 1:9
val = int(probstr[pos:pos])
if val < 0 || val > 10
return Response(422, "Error: number out of range 0:9.")
end
prob[row,col] = val
pos += 1
end
end
catch
return Response(422, "Error: couldn't parse numbers.")
end
# Attempt to solve the problem using integer programming
try
sol = SolveModel(prob)
if prettyoutput
# Human readable output
out = "<table>"
for row = 1:9
out = string(out,"<tr>")
for col = 1:9
out = string(out,"<td>",sol[row,col],"</td>")
end
out = string(out,"</tr>")
end
out = string(out,"</table>")
return Response(out)
else
# Return solution like input
return Response(join(sol,""))
end
catch
return Response(422, "Error: coudn't solve puzzle.")
end
else
# Not a valid URL
return Response(404)
end
end
# Boot up the server
server = Server(http)
run(server, 8000)
## Try it yourself!
Julia is still growing (rapidly) - I suggest grabbing the 0.2 pre-release and having a bit of patience in case of strange errors! You may also see some warnings regarding deprecation - hopefully they'll be cleaned up soon once we reach version 0.2 for Julia and have a chance to go back and tidy up all the packages. Instructions for installing everything are in README.md. I encourage you to check it out for yourself, and maybe extend it. Some ideas:
• Implement a pure-Julia sudoku solver and benchmark performance against the MIP solver.
• Generalize the code to accept and solve n-by-n sudoku problems
Let me know how that goes by contacting me at idunning AT mit.edu or @iaindunning.
© 2013 Iain Dunning
Contact me by email (idunning AT mit DOT edu), @iaindunning, LinkedIn, GitHub
This website was made with Jekyll and Skeleton.
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# Set object local axis based on face orientation
I am struggling to formulate my problem exactly, so edits are welcome.
I have an object where local axis are completely off, like in this simple example. I want fix it, so align one axis with the face normal, another with one of the edges and third perpendicular to both. I created a Custom Transform Orientation using one of the edges and it is just perfect. Can I change the local axis to match it?
The motivation behind, is that I have to position the object vertically in my scene. The object is complex, so I struggle to do it visually.
Do I need some python to do this? Any hints on how to do it?
• You don't necessarily need python to do it, you can do it manually in the 3D view, but unfortunately Blender's precision modeling and alignment tools are severely lacking and very unsuited for this type of work. The key to this is using Transform Orientations, from the 3DView Properties Shelf but it's a multi step process involving several tasks – Duarte Farrajota Ramos Sep 22 '16 at 18:34
• @DuarteFarrajotaRamos yes, this seems surprisingly hard. I already created Transform Orientation from using one of the edges and this orientation is just perfect, but can I do anything with it? The problem is I have to position this object vertically in the scene. doing it visually is non trivial, especially because it is a complex object. Any further hints? – Noidea Sep 23 '16 at 9:34
• @DuarteFarrajotaRamos maybe I can make it much global axis and then reset local axis to global? – Noidea Sep 23 '16 at 10:04
• @MrZak no, no, I created a transform and I can use it. But I still cannot align my object vertically, because visually it is non trivial. So what I wanted to do - align local axis correctly and then match local axis to global. – Noidea Sep 23 '16 at 11:10
• @MrZak maybe having a god local axis is not necessary. But I already had to mess a great deal to mirror this object - adding empties, snapping, parenting and whatever, and that could have been just a mouse click... Now I got it, but let's put it easy, I want to put the damn lightpost vertically! – Noidea Sep 23 '16 at 11:18
This is not exactly a solution, but a painful workaround. If someone can provides a better answer, I will accept it.
The solution relies on two assumptions:
1. You can create a Custom Transform Orientation which matched your desired orientation.
2. You don't mind that object will be moved.
"Solution":
1. Create a Custom Transform Orientation. For this simple example: select the edge you want to align, press CtrlAltSpace or find it at bottom of the N-Panel.
2. Rotate to align custom transform to global axis. I followed this post. In short: add empty at the face center, align it to custom transform, parent object to the empty, clear the empty rotation, clear the object's parent keeping transformation, rotate 180°.
3. Change local axis to match the global ones. Use Apply Transformation (CtrlA)
Here is another workaround to the problem, which I have also struggled with a lot.
You create a new transform orientation from the selected vertex, face or edge.
Press CtrlAltSpace
This creates the orientation and puts it in the list of available orientatons, it also selects it as the current one.
The list will look like this:
Here we see some custom orientations added by me.
The normal of the object used will be the new Z-axis of an orientation. I am not sure how the other axes are calculated, but they also adapt somewhat to the selection.
In the moment you create a orientation, you get the chance to give it a name of your choice in the transform panel (T).
The users orientation added seems to be saved in the blend file.
If you alter the object that an orientation originated from, the orientation will not change in any way.
Now if you in edit mode want to for instance grab part of the mesh along the selected orientations Z-axis, you press GZZ (press Z twice)
The text below tells that the orientation Vertex.001 is used.
The orientations are listed in the properties shelf of the 3D view (N).
They can be renamed or removed from there.
Also note that if you want to extrude or extrude-scale some geometry (initiated with E or E S) with respect of the custom orientation, you will need to cancel the extrusion with Esc and then G or S respectively. That is because extrusions will always use the "Normal" orientation as an alternative to the global, but grab or scale uses the custom orientation as the alternative.
• OP has already done this. They just want to know how to make the Local orientation match the Custom one they made. – Tooniis Jan 15 '18 at 11:01
Another workaround could be, if you are working with simple meshes, to
1. create the custom transform orientation and put the 3D cursor in the axis you want
2. create a cube
3. Align your cube to your custom transform orientation with Object > Transform > Align to Transform Orientation
4. Select you mesh then the cube and merge Ctrl + J
5. Enter Edit Mode and erase the cube shape
There could be some problems with UV if you haven't save it before, but it worked for me.
Some python to do it.
Test Script
• Run in edit mode with face selected.
• Use the face centre to make translation matrix as space, will rotate about this pivot point.
• Aligns face normal to local z (0, 0, 1) by rotating.
• Finds the most orthogonal edge to face normal and rotates to align with "forward" (0, 1, 0)
• Translates verts such that face center is back into original location.
Script
import bpy
import bmesh
from mathutils import Vector, Matrix
context = bpy.context
ob = context.edit_object
me = ob.data
bm = bmesh.from_edit_mesh(me)
face = bm.select_history.active
o = face.calc_center_median()
face.normal_update()
norm = face.normal
edges = sorted((e for e in face.edges), key=lambda e: abs((e.verts[1].co - e.verts[0].co).dot(face.normal)))
e = edges[0]
# if this value is 0 then edge and normal orthogonal should test
print((e.verts[1].co - e.verts[0].co).dot(face.normal))
T = Matrix.Translation(-o)
up = Vector((0, 0, 1))
R = face.normal.rotation_difference(up).to_matrix()
bmesh.ops.transform(bm, verts=bm.verts, matrix=R, space=T)
forward = Vector((0, 1, 0))
R = (e.verts[1].co - e.verts[0].co).rotation_difference(forward).to_matrix()
bmesh.ops.transform(bm, verts=bm.verts, matrix=R, space=T)
T = Matrix.Translation(face.calc_center_median() - o)
bmesh.ops.transform(bm, verts=bm.verts, matrix=T)
bmesh.update_edit_mesh(me)
If to make operator of code above would have enum to select the values of up and forward axes 'X', '-X', 'Y', '-Y', 'Z', '-Z' or alternatively find the closest axis to normal etc.
EDIT should prob put in one more step to make sure edge axis is exactly orthogonal.
• I agree with you that a simple solution is just to make a cube, orient it correctly, merge the subject with the cube, and delete the cube faces in edit mode. That's it. It is kind of a shame that we can't take advantage of matching a rotated 3D cursor, but oh-well. – hatinacat2000 Nov 4 at 19:30
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to
### What is machine "learning" and artificial intelligence
An important feature of human intelligence is the ability to learn. The amazing learning abilities of the human brain enable babbling babies to grow into learned and easy-to-talk adults. For human beings, learning is an innate ability. The universal existence of this ability makes us ignore its strangeness and preciousness. As far as artificial intelligence research is concerned, how to make machines possess the most universal capabilities in the human world is a very challenging research direction. In different research paths, the subjects, contents and methods of learning are quite different.
### Frangula californica (California coffeeberry): Matriculating undergraduates, now, for 1/1/21, Realistic Virtual Earth for Machine Learning - WUaS News, Livestream, Q&A - i) Seeking to matriculate our 2nd undergraduate class Jan 1, 2021, and potentially with students taking WUaS Open edX courses, ii) How WUaS or edX could provide a letter to prospective employers that a student is matriculated officially at WUaS / edX and similar?, iii) Creating a single #RealisticVirtualEarth beginning w #GoogleResearchFootball for learning machine learning / AI, and with Lego Robotics too, iv) WUaS Monthly Business Meeting Minutes for 8/15 * * How to BEGIN 1 #RealisticVirtualEarth in #GoogleStreetView w #TimeSlider for learning #MachineLearning & w #LegoRobotics? #FilmTo3D App >#RealisticVirtualEarthForRobotics Google open-sources soccer reinforcement learning sim #ReinforcementLearning #GRFE
Frangula californica (California coffeeberry): Matriculating undergraduates, now, for 1/1/21, Realistic Virtual Earth for Machine Learning - WUaS News, Livestream, Q&A - i) Seeking to matriculate our 2nd undergraduate class Jan 1, 2021, and potentially with students taking WUaS Open edX courses, ii) How WUaS or edX could provide a letter to prospective employers that a student is matriculated officially at WUaS / edX and similar?, iii) Creating a single #RealisticVirtualEarth beginning w #GoogleResearchFootball for learning machine learning / AI, and with Lego Robotics too, iv) WUaS Monthly Business Meeting Minutes for 8/15 * * How to BEGIN 1 #RealisticVirtualEarth in #GoogleStreetView w #TimeSlider for learning #MachineLearning & w #LegoRobotics? Add Lego Robotics with similar Film-To-3D App, such as - 6.270 MIT Lego Robot Competition 1999 - https://youtu.be/SXH-bBw3uxg And with Lego weDo 2.0 too - Special WeDo 2.0 Scratch Project BOXER from Roboriseit! https://youtu.be/HjD1zAWToYU It is unlikely that I will get to contribute to this work. So please take me off all your mailing lists.
### Machine Learning for beginners with project
Hello and welcome to this course on Machine learning .My name is Aakash Singh i am instructor of this course .This course is structured in way so that anyone can easily grasp the concept of programming,fundamentals,concepts of the Machine learning .No prior knowledge is required through this course .we
### Predicting heave and surge motions of a semi-submersible with neural networks
Real-time motion prediction of a vessel or a floating platform can help to improve the performance of motion compensation systems. It can also provide useful early-warning information for offshore operations that are critical with regard to motion. In this study, a long short-term memory (LSTM) -based machine learning model was developed to predict heave and surge motions of a semi-submersible. The training and test data came from a model test carried out in the deep-water ocean basin, at Shanghai Jiao Tong University, China. The motion and measured waves were fed into LSTM cells and then went through serval fully connected (FC) layers to obtain the prediction. With the help of measured waves, the prediction extended 46.5 s into future with an average accuracy close to 90%. Using a noise-extended dataset, the trained model effectively worked with a noise level up to 0.8. As a further step, the model could predict motions only based on the motion itself. Based on sensitive studies on the architectures of the model, guidelines for the construction of the machine learning model are proposed. The proposed LSTM model shows a strong ability to predict vessel wave-excited motions.
### The Most Controversial Neural Network Ever Created
Some believe that the Extreme Learning Machine is one of the smartest neural network inventions ever created. Some believe that the Extreme Learning Machine is one of the smartest neural network inventions ever created -- so much so that there's even a conference dedicated exclusively to the study of ELM neural network architectures. Proponents of ELMs argue that it can perform standard tasks at exponentially faster training times, with few training examples. On the other hand, besides from the fact that it's not big in the machine learning community, it's got plenty of criticism from experts in deep learning, including Yann LeCun, who argue that it's gotten far more publicity and credibility than it deserves. Mostly, people seem to think it's an interesting concept.
### Space-Time Domain Tensor Neural Networks: An Application on Human Pose Recognition
Recent advances in sensing technologies require the design and development of pattern recognition models capable of processing spatiotemporal data efficiently. In this work, we propose a spatially and temporally aware tensor-based neural network for human pose recognition using three-dimensional skeleton data. Our model employs three novel components. First, an input layer capable of constructing highly discriminative spatiotemporal features. Second, a tensor fusion operation that produces compact yet rich representations of the data, and third, a tensor-based neural network that processes data representations in their original tensor form. Our model is end-to-end trainable and characterized by a small number of trainable parameters making it suitable for problems where the annotated data is limited. Experimental validation of the proposed model indicates that it can achieve state-of-the-art performance. Although in this study, we consider the problem of human pose recognition, our methodology is general enough to be applied to any pattern recognition problem spatiotemporal data from sensor networks.
### High-dimensional Neural Feature using Rectified Linear Unit and Random Matrix Instance
We design a ReLU-based multilayer neural network to generate a rich high-dimensional feature vector. The feature guarantees a monotonically decreasing training cost as the number of layers increases. We design the weight matrix in each layer to extend the feature vectors to a higher dimensional space while providing a richer representation in the sense of training cost. Linear projection to the target in the higher dimensional space leads to a lower training cost if a convex cost is minimized. An $\ell_2$-norm convex constraint is used in the minimization to improve the generalization error and avoid overfitting. The regularization hyperparameters of the network are derived analytically to guarantee a monotonic decrement of the training cost and therefore, it eliminates the need for cross-validation to find the regularization hyperparameter in each layer.
### Computing Machinery and Intelligence
This question begs one to define the words "machine" and "think". Instead of defining them -- which is seemingly easy, let's replace the question with one that is very similar. Before that, we introduce the imitation game. The game is played by three. The interrogator is isolated from the other two and can ask each one of them questions, with a goal of identifying who the man and who the woman is.
### How to Make Yourself Into a Learning Machine
You immigrate to a new country that speaks a different language, and start work with some of the brightest engineers in the world. Now, you're leading teams of people who are 10 or 20 years older than you, working on one of the fastest growing internet companies of the last decade. You have two options: sink or swim. That's the position Simon Eskildsen found himself in early in his career. He left his home in Denmark after high school, and moved to Canada alone to take a pre-college gap year working at Shopify. When he started, Shopify had 150 employees supporting tens of thousands of merchants. Now, it has 5,000 employees and over a million merchants.
### An On-Device Federated Learning Approach for Cooperative Anomaly Detection
Most edge AI focuses on prediction tasks on resource-limited edge devices, while the training is done at server machines, so retraining a model on the edge devices to reflect environmental changes is a complicated task. To follow such a concept drift, a neural-network based on-device learning approach is recently proposed, so that edge devices train incoming data at runtime to update their model. In this case, since a training is done at distributed edge devices, the issue is that only a limited amount of training data can be used for each edge device. To address this issue, one approach is a cooperative learning or federated learning, where edge devices exchange their trained results and update their model by using those collected from the other devices. In this paper, as an on-device learning algorithm, we focus on OS-ELM (Online Sequential Extreme Learning Machine) and combine it with Autoencoder for anomaly detection. We extend it for an on-device federated learning so that edge devices exchange their trained results and update their model by using those collected from the other edge devices. Experimental results using a driving dataset of cars demonstrate that the proposed on-device federated learning can produce more accurate model by combining trained results from multiple edge devices compared to a single model.
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Equazioni alle Derivate Parziali nella Dinamica dei Fluidi
# Some results on 2D Euler equations in distributional spaces
speaker: Franco Flandoli (Scuola Normale Superiore)
abstract: We elaborate a result of Albeverio and Cruzeiro about existence of solutions of 2D Euler equations for almost every initial condition with respect to a certain Gaussian measure, supported on a space of distributional vorticity fields. We give a different proof of the result, based on the weak vorticity formulation used classically for measure-valued solutions. This way, we can prove that Albeverio-Cruzeiro solutions are limit of point vorticies and of solutions with bounded vorticity.
timetable:
Mon 5 Feb, 14:30 - 15:05, Aula Dini
<< Go back
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# Determine initial velocity of a vertical throw
Homework Statement:
t = 4s , g = 10 m/s^2
Relevant Equations:
I don't really know which equation to use...
Hi,
I was given this problem saying that a ball is thrown vertically up in the air and returns to its initial position after 4 seconds. The acceleration due to gravity is given to be equal to 10m/s^2.
I tried to attempt this problem by using the equation :
v^2 - v0^2 = 2ah by considering terminal velocity = 0. Despite this, i still can't determine the total distance travelled.
Sorry in advance if I have misused any vocabulary sincei translated this from french.
Thank you!
## Answers and Replies
PeroK
Homework Helper
Gold Member
2021 Award
Do you have any ideas?
Do you have any ideas?
Well, i'm also trying with v = v0 + at but I didnt really work out..
This is why I chose medecine physics is so so hard..
PeroK
Homework Helper
Gold Member
2021 Award
Well, i'm also trying with v = v0 + at but I didnt really work out..
What about $$s = v_0t + \frac 1 2 at^2$$
What about $$s = v_0t + \frac 1 2 at^2$$
But in this case wouldnt I have to values of gravitational acceleration?
When the object is going up, acceleration is negative and when it goes down, it's positive.
Therefore I couldn't use the 4 seconds value because ( i think ), the time the object takes to go up is different from the time it takes to go down right?
kuruman
Homework Helper
Gold Member
2021 Award
Well, i'm also trying with v = v0 + at but I didnt really work out..
This is why I chose medecine physics is so so hard..
That equation is a good start. You know that the ball is in the air for 4 s, and you know the acceleration. If the velocity is +v0 when the ball is thrown up, what would it be at the moment it comes back down after 4 s?
That equation is a good start. You know that the ball is in the air for 4 s, and you know the acceleration. If the velocity is +v0 when the ball is thrown up, what would it be at the moment it comes back down after 4 s?
I think the thing that what's stopping me is that I dont know whether the time an object takes to go up is different from the time it takes to go down..
Because Ive been considering that it's different, so i didnt really manage to write the equation..
At 4s, wouldnt the velocity be 0?
If I consider that the time is equal, I'm getting v0 = 20 m/s ...
PeroK
Homework Helper
Gold Member
2021 Award
But in this case wouldnt I have to values of gravitational acceleration?
When the object is going up, acceleration is negative and when it goes down, it's positive.
Therefore I couldn't use the 4 seconds value because ( i think ), the time the object takes to go up is different from the time it takes to go down right?
If you're not sure, you could split the motion into two parts (up and down). You can then show using the kinematic equations whether the time to go up is the same as the time to come down.
PeroK
Homework Helper
Gold Member
2021 Award
At 4s, wouldnt the velocity be 0?
If I consider that the time is equal, I'm getting v0 = 20 m/s ...
If you throw a ball up at ##20 m/s##:
1) What is the displacement after ##4s##?
2) What is the velocity after ##4s##?
kuruman
Homework Helper
Gold Member
2021 Award
I think the thing that what's stopping me is that I dont know whether the time an object takes to go up is different from the time it takes to go down..
Because Ive been considering that it's different, so i didnt really manage to write the equation..
At 4s, wouldnt the velocity be 0?
If I consider that the time is equal, I'm getting v0 = 20 m/s ...
At 4 s the ball returns to the height from which it was launched. Just before it is stopped, it is still moving, That's the velocity I was asking about. Anyway, I don't want to detract from the course that @PeroK has set so I will cease and desist.
duchuy and PeroK
If you throw a ball up at ##20 m/s##:
1) What is the displacement after ##4s##?
2) What is the velocity after ##4s##?
Oh ok ok I got that thank you two so much!
For the displacement after 40swouldn't it be 40m?
For the velocity I'm not quite sure.. I really thought that it would be 0 since it's the moment the ball landed..
PeroK
PeroK
Homework Helper
Gold Member
2021 Award
Oh ok ok I got that thank you two so much!
For the displacement after 40swouldn't it be 40m?
For the velocity I'm not quite sure.. I really thought that it would be 0 since it's the moment the ball landed..
Sorry, that's not right at all.
1) That's not what the equations tell you.
2) A ball is moving downwards when it hits the ground after being thrown up.
There are lots of videos on line about throwing an object up. I think you need to develop some sort of physical understanding of what is happening.
But, also, you have to develop the ability to use equations.
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## 22-Aug-2015: De Bruijn sequences (solution for the exercise posted at 18-Aug-2015); leading/trailing zero bits counting.
### Introduction
Let's imagine there is a very simplified code lock accepting 2 digits, but it has no "enter" key, it just checks 2 last entered digits. Our task is to brute force each 2-digit combination. Naïve method is to try 00, 01, 02 ... 99. That require 2*100=200 key pressings. Will it be possible to reduce number of key pressings during brute-force? It is indeed so, with the help of De Bruijn sequences. We can generate them for the code lock, using Wolfram Mathematica:
In[]:= DeBruijnSequence[{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, 2]
Out[]= {6, 8, 6, 5, 4, 3, 2, 1, 7, 8, 7, 1, 1, 0, 9, 0, 8, 0, 6, 6, \
0, 5, 5, 0, 4, 4, 0, 3, 3, 0, 2, 7, 2, 2, 0, 7, 7, 9, 8, 8, 9, 9, 7, \
0, 0, 1, 9, 1, 8, 1, 6, 1, 5, 1, 4, 1, 3, 7, 3, 1, 2, 9, 2, 8, 2, 6, \
2, 5, 2, 4, 7, 4, 2, 3, 9, 3, 8, 3, 6, 3, 5, 7, 5, 3, 4, 9, 4, 8, 4, \
6, 7, 6, 4, 5, 9, 5, 8, 5, 6, 9}
The result has exactly 100 digits, which is 2 times less than our initial idea can offer. By scanning visually this 100-digits array, you'll find any number in 00..99 range. All numbers are overlapped with each other: second half of each number is also first half of the next number, etc.
Here is another. We need a sequence of binary bits with all 3-bit numbers in it:
In[]:= DeBruijnSequence[{0, 1}, 3]
Out[]= {1, 0, 1, 0, 0, 0, 1, 1}
Sequence length is just 8 bits, but it has all binary numbers in 000..111 range. You may visually spot 000 in the middle of sequence. 111 is also present: two first bits of it at the end of sequence and the last bit is in the beginning. This is so because De Bruijn sequences are cyclic.
There is also visual demonstration: http://demonstrations.wolfram.com/DeBruijnSequences/.
### Trailing zero bits counting
In Wikipedia article about De Bruijn sequences we can find:
The symbols of a De Bruijn sequence written around a circular object (such as a wheel of a robot) can be used to identify its angle by examining the n consecutive symbols facing a fixed point.
Indeed: if we know De Bruijn sequence and we observe only part of it (any part), we can deduce exact position of this part within sequence.
Let's see, how this feature can be used.
Let's say, there is a need to detect position of input bit within 32-bit word. For 0x1, the algorithm should report 1. 2 for 0x2. 3 for 0x4. And 31 for 0x80000000.
The result is in 0..31 range, so the result can be stored in 5 bits.
We can construct binary De Bruijn sequence for all 5-bit numbers:
In[]:= tmp = DeBruijnSequence[{0, 1}, 5]
Out[]= {1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0}
In[]:= BaseForm[FromDigits[tmp, 2], 16]
Out[]:= e6bec520
Let's also recall that division some number by $2^n$ number is the same thing as shifting it by $n$ bits. So if you divide 0xe6bec520 by 1, the result is not shifted, it is still the same. If if divide 0xe6bec520 by 4 ($2^2$), the result is shifted by 2 bits. We then take result and isolate lowest 5 bits. This result is unique number for each input. Let's shift 0xe6bec520 by all possible count number, and we'll get all possible last 5-bit values:
In[]:= Table[BitAnd[BitShiftRight[FromDigits[tmp, 2], i], 31], {i, 0, 31}]
Out[]= {0, 16, 8, 4, 18, 9, 20, 10, 5, 2, 17, 24, 12, 22, 27, 29, \
30, 31, 15, 23, 11, 21, 26, 13, 6, 19, 25, 28, 14, 7, 3, 1}
The table has no duplicates:
In[]:= DuplicateFreeQ[%]
Out[]= True
Using this table, it's easy to build "magic" table. OK, now working C example:
#include <stdint.h>
#include <stdio.h>
int magic_tbl[32];
// returns single bit position counting from LSB
// not working for i==0
int bitpos (uint32_t i)
{
return magic_tbl[(0xe6bec520/i) & 0x1F];
};
int main()
{
// construct magic table
// may be omitted in production code
for (int i=0; i<32; i++)
magic_tbl[(0xe6bec520/(1<<i)) & 0x1F]=i;
// test
for (int i=0; i<32; i++)
{
printf ("input=0x%x, result=%d\n", 1<<i, bitpos (1<<i));
};
};
Here we feed our bitpos() function with numbers in 0..0x80000000 range and we got:
input=0x1, result=0
input=0x2, result=1
input=0x4, result=2
input=0x8, result=3
input=0x10, result=4
input=0x20, result=5
input=0x40, result=6
input=0x80, result=7
input=0x100, result=8
input=0x200, result=9
input=0x400, result=10
input=0x800, result=11
input=0x1000, result=12
input=0x2000, result=13
input=0x4000, result=14
input=0x8000, result=15
input=0x10000, result=16
input=0x20000, result=17
input=0x40000, result=18
input=0x80000, result=19
input=0x100000, result=20
input=0x200000, result=21
input=0x400000, result=22
input=0x800000, result=23
input=0x1000000, result=24
input=0x2000000, result=25
input=0x4000000, result=26
input=0x8000000, result=27
input=0x10000000, result=28
input=0x20000000, result=29
input=0x40000000, result=30
input=0x80000000, result=31
The bitpos() function actually counts trailing zero bits, but it works only for input values where only one bit is set. To make it more practical, we need to devise a method to drop all leading bits except of the last one. This method is very simple and well-known:
input & (-input)
This bit twiddling hack can solve the job. Feeding 0x11 to it, it will return 0x1. Feeding 0xFFFF0000, it will return 0x10000. In other words, it leaves lowest significant bit of the value, dropping all others.
It works because negated value in two's complement environment is the value with all bits flipped but also 1 added (because there is a zero in the middle of ring). For example, let's take 0xF0. -0xF0 is 0x10 or 0xFFFFFF10. ANDing 0xF0 and 0xFFFFFF10 will produce 0x10.
Let's modify our algorithm to support true trailing zero bits count:
#include <stdint.h>
#include <stdio.h>
int magic_tbl[32];
// not working for i==0
int tzcnt (uint32_t i)
{
uint32_t a=i & (-i);
return magic_tbl[(0xe6bec520/a) & 0x1F];
};
int main()
{
// construct magic table
// may be omitted in production code
for (int i=0; i<32; i++)
magic_tbl[(0xe6bec520/(1<<i)) & 0x1F]=i;
// test:
printf ("%d\n", tzcnt (0xFFFF0000));
printf ("%d\n", tzcnt (0xFFFF0010));
};
It works!
16
4
But it has one drawback: it uses division, which is slow. Can we just multiplicate De Bruijn sequence by the value with the bit isolated instead of dividing sequence? Yes, indeed. Let's check in Mathematica:
In[]:= BaseForm[16^^e6bec520*16^^80000000, 16]
Out[]:= 0x735f629000000000
The result is just too big to fit in 32-bit register, but can be used. MUL/IMUL instruction 32-bit x86 CPUs stores 64-bit result into two 32-bit registers pair, yes. But let's suppose we would like to make portable code which will work on any 32-bit architecture. First, let's again take a look on De Bruijn sequence Mathematica first produced:
In[]:= tmp = DeBruijnSequence[{0, 1}, 5]
Out[]= {1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, \
0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0}
There is exactly 5 bits at the end which can be dropped. The "magic" constant will be much smaller:
In[]:= BaseForm[BitShiftRight[FromDigits[tmp, 2], 5], 16]
Out[]:=0x735f629
The "magic" constant is now "divided by 32 (or 1>>5)". This mean that the result of multiplication of some value with one isolated bit by new magic number will also be smaller, so the bits we need will be stored at the high 5 bits of the result.
De Bruijn sequence is not broken after 5 lowest bits dropped, because these zero bits are "relocated" to the start of the sequence. Sequence is cyclic after all.
#include <stdint.h>
#include <stdio.h>
int magic_tbl[32];
// not working for i==0
int tzcnt (uint32_t i)
{
uint32_t a=i & (-i);
// 5 bits we need are stored in 31..27 bits of product, shift and isolate them after multiplication:
return magic_tbl[((0x735f629*a)>>27) & 0x1F];
};
int main()
{
// construct magic table
// may be omitted in production code
for (int i=0; i<32; i++)
magic_tbl[(0x735f629<<i >>27) & 0x1F]=i;
// test:
printf ("%d\n", tzcnt (0xFFFF0000));
printf ("%d\n", tzcnt (0xFFFF0010));
};
This is almost the same task, but most significant bit must be isolated instead of lowest. This is typical algorithm for 32-bit integer values:
x |= x >> 1;
x |= x >> 2;
x |= x >> 4;
x |= x >> 8;
x |= x >> 16;
For example, 0x100 becomes 0x1ff, 0x1000 becomes 0x1fff, 0x20000 becomes 0x3ffff, 0x12340000 becomes 0x1fffffff. It works because all 1 bits are gradually propagated towards the lowest bit in 32-bit number, while zero bits at the left of most significant 1 bit are not touched.
It's possible to add 1 to resulting number, so it will becomes 0x2000 or 0x20000000, but in fact, since multiplication by magic number is used, these numbers are very close to each other, so there are no error.
This example I used in my reverse engineering exercise from 15-Aug-2015: //yurichev.com/blog/2015-aug-18/.
int v[64]=
{ -1,31, 8,30, -1, 7,-1,-1, 29,-1,26, 6, -1,-1, 2,-1,
-1,28,-1,-1, -1,19,25,-1, 5,-1,17,-1, 23,14, 1,-1,
9,-1,-1,-1, 27,-1, 3,-1, -1,-1,20,-1, 18,24,15,10,
-1,-1, 4,-1, 21,-1,16,11, -1,22,-1,12, 13,-1, 0,-1 };
int LZCNT(uint32_t x)
{
x |= x >> 1;
x |= x >> 2;
x |= x >> 4;
x |= x >> 8;
x |= x >> 16;
return v[x >> 26];
}
This piece of code I took here. It is slightly different: the table is twice bigger, and the function returns -1 if input value is zero. The magic number I found using just brute-force, so the readers will not be able to google it, for the sake of exercise. (By the way, I've got 12,665,720 magic numbers which can serve this purpose. This is about 0.294% of all 32-bit numbers.)
The code is tricky after all, and the moral of the exercise is that practicing reverse engineer sometimes may just observe input/outputs to understand code's behaviour instead of diving into it.
### Performance
The algorithms considered are probably fastest known, they has no conditional jumps, which is very good for CPUs starting at RISCs. Newer CPUs has LZCNT and TZCNT instructions, even 80386 had BSF/BSR instructions which can be used for this: https://en.wikipedia.org/wiki/Find_first_set. Nevertheless, these algorithms can be still used on cheaper RISC CPUs without specialized instructions.
### Applications
Number of leading zero bits is binary logarithm of value. My article about logarithms including binary: //yurichev.com/writings/log_intro.pdf.
These algorithms are also extensively used in chess engines programming, where each piece is represented as 64-bit bitmask (chess board has 64 squares): http://chessprogramming.wikispaces.com/BitScan.
There are more: https://en.wikipedia.org/wiki/Find_first_set#Applications.
### Generation of De Bruijn sequences
De Bruijn graph is a graph where all values are represented as vertices (or nodes) and each edge (or link) connects two nodes which can be "overlapped". Then we need to visit each edge only once, this is called eulerian path. It is like the famous task of seven bridges of Königsberg: traveller must visit each bridge only once.
There are also simpler algorithms exist: https://en.wikipedia.org/wiki/De_Bruijn_sequence#Algorithm.
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# Thread: Determine the coordinates of the point
1. ## Determine the coordinates of the point
Show that the line $
5x+3y+ \lambda x
$
= $
2 \lambda y -6
$
always passes from a fixed point . Determine the coordinate of the point.
2. The equation can be written as
$\lambda(x-2y)+5x+3y+6=0$
Then $\left\{\begin{array}{ll}x-2y=0\\5x+3y=-6\end{array}\right.$
Now solve the system.
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Claims of “the end of geography” and the flatness of the world notwithstanding, place still matters today.
Discussing why place matters is somewhat beyond the scope of this post, so I will direct you to the excellent work of Parag Khanna and his book Connectography. To put it simply, the the future of business and international relations will be best described by networks: nodes (cities, small-groups, individuals, power-centers), connectivity (between places and within places), centrality (geographically, culturally, economically), and distance will be critically important.
Having said that, I’ll repeat: place still matters. Location matters. And cities (in particular) matter.
Innovation is concentrated in a few cities
This is strongly true of innovation. A small number of cities account for the vast majority of innovation (40 city regions generate roughly 90% of all world innovation). Moreover, innovation is not evenly distributed; some cities are much more innovative than others.
With this in mind, I created the above map of venture capital spending, by city. Although VC spending isn’t the same thing as innovation, it’s a fairly decent proxy, and at least gives us a sense of where innovation is happening.
The data is from 2012, and comes from the Martin Prosperity Institute, by way of Citylab. Keep in mind that here, I’ve only used the top 20 cities (the original MPI data includes many more cities).
The size of each circle is scaled for the amount of VC spending.
The Bay Area dominates VC investment
At a glance, you can see from the map that the US dominates VC spending, with the West Coast and East Coast showing particularly high levels of investment. However, it’s somewhat difficult to accurately judge the relative magnitudes in a map format.
A better tool for judging the magnitudes (and relative differences) is the bar chart.
(The human visual system judges length more accurately than area.)
Having said that, here is a bar chart of the same data.
Quickly, you’ll notice that San Francisco and San Jose (AKA: Silicon Valley) have quite a bit more VC investment than almost anywhere else. In fact, combined, they account for roughly 25% of VC investment.
Code
If you want to reproduce the charts (and play around) here is the R code:
# LOAD LIBRARIES
library(ggalt)
library(ggplot2)
library(maps)
library(dplyr)
# GET DATA
# INSPECT
str(df.vc_totals)
# GET WORLD MAP
map.world <- map_data ("world")
# SIMPLE MAP WITH POINTS
# - this is just to test the data
ggplot() +
geom_polygon(data = map.world, aes(x = long, y = lat, group = group)) +
geom_point(data = df.vc_totals, aes(x = longitude, y = latitude), color = "red")
#-----------------------------------------
# CREATE FINAL MAP
# notes:
# 1. size is the total VC investment
# 2. there are two layers of points. This is to have both the
# point outline as well as an interior, but with different
# transparency levels (i.e., different alpha)
# 3. most of the formatting (i.e., theming) is just removing things
# to make this simpler
#-----------------------------------------
ggplot() +
geom_polygon(data = map.world, aes(x = long, y = lat, group = group),fill = "#002035",colour = "#114151", size = .25) +
geom_point(data = df.vc_totals, aes(x = longitude, y = latitude, size = vc_investment_millions), color = "red", alpha = .15) +
geom_point(data = df.vc_totals, aes(x = longitude, y = latitude, size = vc_investment_millions), color = "red", alpha = .8, shape = 1) +
coord_proj("+proj=robin +lon_0=0 +x_0=0 +y_0=0 +ellps=WGS84 +datum=WGS84 +units=m +no_defs") + # use robinson projection
scale_size_continuous(range = c(1,20), breaks = c(500,2000,6000), name="Venture Capital Investment\n(USD, Millions)\n") +
theme(text = element_text(family = "Gill Sans")) +
theme(panel.background = element_rect(fill = "#000727")) +
theme(panel.grid = element_blank()) +
theme(axis.text = element_blank()) +
theme(axis.ticks = element_blank()) +
theme(axis.title = element_blank()) +
theme(legend.position = c(.17,.3)) +
theme(legend.background = element_blank()) +
theme(legend.key = element_blank()) +
theme(legend.title = element_text(color = "#DDDDDD", size = 16)) +
theme(legend.text = element_text(color = "#DDDDDD", size = 16))
#------------------------------------------
# BAR CHART
# - descending order from most VC to least
#------------------------------------------
df.vc_totals %>%
ggplot(aes( x = reorder(metro, vc_investment_millions),y = vc_investment_millions)) +
geom_bar(stat = "identity", fill = "#002035") +
geom_text(aes(label = vc_investment_millions), hjust = 1.1, color = "#FFFFFF") +
labs(y = "Millions of Dollars", title = "Venture Capital Investment by City") +
coord_flip() +
theme(text = element_text(family = "Gill Sans")) +
theme(plot.title = element_text(size = 28, color = "#555555")) +
theme(axis.title.y = element_blank()) +
theme(panel.background = element_rect(fill = "#CCCCCC")) +
theme(panel.grid.major = element_blank()) +
theme(panel.grid.minor = element_blank())
The post Mapping global venture capital investment appeared first on SHARP SIGHT LABS.
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13. What do the following data taken from a comparative balance sheet indicate about the...
13. What do the following data taken from a comparative balance sheet indicate about the company’s ability to bor- row additional funds on a long-term basis in the current year as compared to the preceding year?
Current Year Preceding Year Fixed assets (net) $175,000$170,000
Total long-term liabilities 70,000 85,000
14. What does a decrease in the ratio of liabilities to stock- holders’ equity indicate about the margin of safety for a firm’s creditors and the ability of the firm to withstand adverse business conditions?
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## 'guess' solution, differential question.
dy/dx = x/y
Solve the equation (get general form of y) for the given condition y=1 and x=2
I've tried finding the complementary function, dy/dx = 0.
So I assume y = C (a constant)
Now I'm trying the find the particular Integral.
dy/dx = x/y
rearrange for LHS containing only y and RHS containing only x
dy y = dx x
I integrate I get (y^2) / 2 = (x^2)/2 + D(constant due to integration)
y^2 = 2(x^2)/2 + 2D
y^2 = (x^2) + E (2D= E)
y = Sqrt (x^2) + Sqrt (E)
y = x + F
General function
y = C + x + F
y = G + x
The answer (given onnsheet) is y = Sqrt ((x^2) - 4)
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Recognitions:
Homework Help
Quote by keith river Now I'm trying the find the particular Integral. dy/dx = x/y rearrange for LHS containing only y and RHS containing only x dy y = dx x I integrate I get (y^2) / 2 = (x^2)/2 + D(constant due to integration) y^2 = 2(x^2)/2 + 2D y^2 = (x^2) + E (2D= E) y = Sqrt (x^2) + Sqrt (E) y = x + F
If you were going to get the particular integral by solving the equation, it sort of made no sense to get a complementary solution to y' = 0.
But from this last line: if a2= b + c then a ≠ √b + √ c,
a = √(b+c)
Mentor
Quote by keith river dy/dx = x/y Solve the equation (get general form of y) for the given condition y=1 and x=2 I've tried finding the complementary function, dy/dx = 0. So I assume y = C (a constant)
I can't see how this approach would work.
y dy = x dx
Quote by keith river Now I'm trying the find the particular Integral. dy/dx = x/y rearrange for LHS containing only y and RHS containing only x dy y = dx x I integrate I get (y^2) / 2 = (x^2)/2 + D(constant due to integration) y^2 = 2(x^2)/2 + 2D y^2 = (x^2) + E (2D= E) y = Sqrt (x^2) + Sqrt (E) y = x + F General function y = C + x + F y = G + x The answer (given onnsheet) is y = Sqrt ((x^2) - 4)
I get y = sqrt(x^2 - 3), and my answer checks. Are you sure you have the right initial conditions?
## 'guess' solution, differential question.
thanks, I can't believe I forgot something as simple as that.
and Initial conditions were y=0, x=2
Sorry about the typo, there are a lot on the worksheet.
But seeing the sqrt all under one bracket made me realise what to do.
I've got it now.
|
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