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# DirectX file errors o.O
This topic is 4213 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
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School is ending, and I decided to get back into my programming. I'm trying to learn DirectX, so I took an example off the CD from my book and tried to compile it after putting directx in and all. its DirectX9.0b. Right now I'm downloading DirectX10 to see if that fixes my problem, but in the mean time, can someone tell me how to fix this error, and whats causing it etc. (I like to learn from mistakes) This is the error the compiler gives off from the directx file: (Dev-C++ v 4.9.8.0 IDE mingw or w.e its called compiler)
In file included from C:/Dev-Cpp/include/d3dx9mesh.h:15,
from C:/Dev-Cpp/include/d3dx9.h:47,
from winmain.cpp:7:
C:/Dev-Cpp/include/dxfile.h:240: stray '\32' in program
In file included from C:/Dev-Cpp/include/d3dx9mesh.h:15,
from C:/Dev-Cpp/include/d3dx9.h:47,
from winmain.cpp:7:
C:/Dev-Cpp/include/dxfile.h:240:2: warning: no newline at end of file
make.exe: *** [winmain.o] Error 1
Execution terminated
If theres anything else I can post to that can help you help me solve this issue, please let me know. (Will also tell you guys if DirectX 10 works without any problems)
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C:/Dev-Cpp/include/dxfile.h:240:2: warning: no newline at end of file
old strict gcc compilers require an empty line at the end of each file:
so... either suppress the warning or go to each file, put your cursor at the end and press enter.
-me
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Thanks. That problem is done and now I have undefined errors which are probably because something isn't included, but I got the project straight from the book so everything should be there. Maybe theres a certain file needed for the following error:
Executing make...make.exe -f "C:\Documents and Settings\user\Desktop\Games trying to program\DirectX\load_bitmap\Makefile.win" allg++.exe -c winmain.cpp -o winmain.o -I"C:/Dev-Cpp/include/c++" -I"C:/Dev-Cpp/include/c++/mingw32" -I"C:/Dev-Cpp/include/c++/backward" -I"C:/Dev-Cpp/include" -I"C:/DX90SDK/Include/DShowIDL" -D__GNUWIN32__ -W -DWIN32 -DNDEBUG -D_WINDOWS -D_MBCS g++.exe winmain.o -o "load_bitmap.exe" -L"C:/Dev-Cpp/lib" -L"C:/DX90SDK/Lib" -mwindows -lkernel32 -luser32 -lgdi32 -lwinspool -lcomdlg32 -ladvapi32 -lshell32 -lole32 -loleaut32 -luuid -lodbc32 -lodbccp32winmain.o(.text+0x258):winmain.cpp: undefined reference to Direct3DCreate9@4'winmain.o(.text+0x3c4):winmain.cpp: undefined reference to D3DXLoadSurfaceFromFileA@32'make.exe: *** [load_bitmap.exe] Error 1Execution terminated
Forget all of the above. I decided to try using visual studio and it compiled without errors. only one warning which I'm going to ask on the IRC if anyone knows how to fix it. Thanks everyone
[Edited by - SonicD007 on March 9, 2007 7:28:37 PM]
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# What's wrong with lattice quantum gravity?
Assume one can write the metric field on a lattice, so on each lattice point one has a value of $$g^{\mu\nu}$$. Similar to the way lattice QCD is formulated. Then later taking the distance between lattice points to go to zero.
Why is not as good as techniques such as dynamical triangulation or loop quantum gravity? The latter have the distance between vertices defining a minimum fixed length such the Planck length but the graph itself is dynamic.
So my question is what advantage does one gain from having a dynamic lattice rather than a fixed lattice as in lattice QCD?
Is it because we don't want to take the limit to continuous space as we add more lattice points? Then I guess we'd have to sum over different types of lattice in order to maintain Lorenz symmetries.
Is lattice quantum gravity doomed to failure?
• In what way do you think putting GR on a lattice circumvents non-normalizability, which is usually cited as the reason standard QFT-techniques don't work for GR? Your comparison with QCD is a red herring - QCD is normalizable, it just has a large coupling, making it unsuited for perturbation theory with the coupling as the parameter. – ACuriousMind Jan 20 at 14:51
• Well maybe that's the answer to the question. If you can contrast that with the other approaches. – zooby Jan 20 at 14:58
• Recall that gravity (quantum or classical) features dynamical spacetime. A "fixed lattice as in lattice QCD" instead involves a fixed flat spacetime, so no gravity is going to come out of that. You also need to be careful to realize diffeomorphism invariance. Let me see if I can ping @R.G.J who actually works on "lattice quantum gravity" in the form of dynamical triangulations (which he obviously doesn't consider doomed to failure) and can tell you more than I can. – David Schaich Jan 25 at 12:44
• Agree with @DavidSchaich above. If you want to have a consistent theory of gravity on the lattice, the lattice must itself be dynamical. The preferred choice is called dynamical triangulations(DT) because first it was used for quantum gravity in 2d where the 2-simplex are triangles (hence, the name).DT had great success in 2d & it was shown that continuum limit corresponds to QG2(Quantum gravity in 2d). Next was to do QG4 and discretize S^4 (4-sphere) with 4-simplices. Note that simplicial complex like these is among the simplest tessellation of S^4. This remains work in progress till date. – R.G.J Jan 26 at 1:22
• 2/2: Defining the continuum limit in QG4 is the main deal. Lattice gravity models assume Weinberg’s conjecture of interacting fixed point in the UV with a finite number of parameters to be tuned, and this is known as -“asymptotic safety”. But, it was found that for DT in 4d there was no critical point corresponding to 2nd order phase transition, so the hope of taking a continuum limit (CL) was dashed with the simple Regge like discretization. Recently, there are hints that by having a more complicated action, one can alter the phase structure & have a critical point where CL can be taken. – R.G.J Jan 26 at 2:39
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# Explaining the method of characteristics
I am learning about solving p.d.e.s by the method of characteristics at the moment. I was given an "algorithm" to solve these problems but I want to know also what is going on, how it works and what it is doing intuitively/physically/graphically. It would be great if someone could either explain it or provide a good link To reference material. Thank you.
-
The basic idea is that we look for parametric curves along which the PDE tells us how the function changes. Suppose you have a smooth parametric curve in the $x-y$ plane: $x = X(t)$, $y = Y(t)$. Consider how a smooth function $u(x,y)$ changes as you move along the curve. The chain rule says $$\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dX}{dt} + \frac{\partial u}{\partial y} \frac{dY}{dt}$$ Now this looks rather like the left side of a first-order PDE $$a(x,y) \frac{\partial u}{\partial x} + b(x,y) \frac{\partial u}{\partial y} = c(x,y)$$ In fact, if you can find $X(t)$ and $Y(t)$ such that $\frac{dX}{dt} = a(X(t),Y(t))$ and $\frac{dY}{dt} = b(X(t),Y(t))$, this tells you that along that curve $\frac{du}{dt} = c(X(t), Y(t))$, so that if you know $u(X(0), Y(0))$ you can get $$u(X(s),Y(s)) = u(X(0), Y(0)) + \int_0^s c(X(t),Y(t))\ dt$$
Now, for any point $(x_1, y_1)$ in the plane, suppose you want to find $u(x_1, y_1)$, where $u(x,y)$ satisfies the PDE $a(x,y) \frac{\partial u}{\partial x} + b(x,y) \frac{\partial u}{\partial y} = c(x,y)$ plus some boundary condition. Then you want to find a curve $x = X(t)$, $y = Y(t)$ satisfying the system of ordinary differential equations $\frac{dX}{dt} = a(X(t),Y(t))$ and $\frac{dY}{dt} = b(X(t),Y(t))$ that passes through the given point $(x_1, y_1)$, follow that curve to some $(x_0, y_0)$ where your boundary condition tells you the value of $u$, and then you can get $u(x_1, y_1)$ by an integral as above.
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# Thread: Inequlaities etc. for tomorrow!!!!!!!
1. ## Inequlaities etc. for tomorrow!!!!!!!
Hi, i'm a new member who really needs help. We've had a supply teacher for a while who was terrible at teaching us. Now the real teacher is back (very scary) and wants homework on Inequalities in for tomorrow. I don't have a clue how to do them but she'll go mad if they're not done!
Please may i have some help?
The following questions i need help with:
Solve the following linear inequalities:
1a. x+3<8
d.4x-5<7
g.x+3 over 2 <8
j.2(x-3)<14
m.3x+1 more than or equal to 2x-5
p.3x+2 more than or equal to x+3
Thankyou.
2. Originally Posted by awfulatmaths
...
Solve the following linear inequalities:
1a. x+3<8
d.4x-5<7
g.x+3 over 2 <8
j.2(x-3)<14
m.3x+1 more than or equal to 2x-5
p.3x+2 more than or equal to x+3
Thankyou.
Hello,
1. You solve inequalities in the same manner as you solve equations with a few exceptions:
a) If you mulitply/divide both sides of an inequality by a negative number you have to change the < sign into a > sign, and the > sign into a < sign.
to #1a:
$x+3<8~\iff~x<5$
to #d:
$4x-5<7~\iff~4x<2~\iff~x<\frac12$
to #g: Do you mean:
$\frac{x+3}{2}<8$ ? Mulitply by 2, subtract 3, done!
to #j: Expand the bracket, add 6, divide by 2
to #m: subtract 1, subtract 2x,
and the last one is intirely for you
3. I can't thank you enough earboth, i was extremely worried but thanks to you i now know not just the answers, but also how to work them out which i have shown in my exercise book. I was also able to do p-i wish you were my teacher.
Thanks a lot!
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# Problem: Consider the n = 3 energy level in a hydrogen atom. How many electrons can be placed in this level?A. 1B. 2C. 8D. 9E. 18
###### FREE Expert Solution
The number of available quantum states is:
$\overline{){\mathbf{2}}{{\mathbf{n}}}^{{\mathbf{2}}}}$
98% (91 ratings)
###### Problem Details
Consider the n = 3 energy level in a hydrogen atom. How many electrons can be placed in this level?
A. 1
B. 2
C. 8
D. 9
E. 18
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## Departmental PhD Thesis Exam – Miodrag Sokic
Monday, March 22nd, 2010
12:10 - 1:00 p.m.
BA 2165, 40 St. George Street
Ph.D. Candidate: Miodrag Sokic
Thesis Title: Ramsey property of posets and related structures
Abstract:
We study several classes of finite posets with linear ordering.
We examine these classes according to the Ramsey and the
ordering property. As application we give several new extremely
amenable groups of automorphism of countable structures and
compute several new universal minimal flows
for such groups. The technique that we develop is also
useful for studying classes of structures related to posets,
such as pseudometric spaces and quasi orderings.
A copy of the thesis can be found at
http://www.math.toronto.edu/msokic/Tezamar.pdf
Everyone is welcome.
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a field-theory motivated approach to computer algebra
## Depends
Makes an object implicitly dependent on other objects.
Makes an object implicitly dependent on other objects, i.e. assumes that the indicated object is a function of the arguments of the property. For example
x::Coordinate; \phi::Depends(x);
$$\displaystyle{}\text{Attached property Coordinate to }x.$$
$$\displaystyle{}\text{Attached property Depends to }\phi.$$
makes $\phi$ an implicit function of $x$. Instead of indicating the coordinate on which the object depends, it is also possible to indicate which derivatives would yield a non-zero answer, as in
\nabla{#}::Derivative; \phi::Depends(\nabla{#}); ex:=\nabla_{m}{\phi c \sin{y}}; unwrap(_);
$$\displaystyle{}\text{Attached property Derivative to }\nabla{\#}.$$
$$\displaystyle{}\text{Attached property Depends to }\phi.$$
$$\displaystyle{}\nabla_{m}\left(\phi c \sin{y}\right)$$
\nabla_{m}(\phi c \sin(y))
$$\displaystyle{}c \sin{y} \nabla_{m}{\phi}$$
c \sin(y) \nabla_{m}(\phi)
(Note: if you did this in Cadabra 1.x you could write Depends(\nabla); this is no longer possible in 2.x and you need to write the full pattern Depends(\nabla{#})). Finally, it is possible to use an index name to indicate on which coordinates a field depends,
To make a tensor with any type and number of indices Depend on something else, use
h{#}::Depends(\nabla{#}); ex:=\nabla_{m}{ c d h^{0 t} h_{t 0} h^{0}_{0} }; unwrap(_);
$$\displaystyle{}\text{Attached property Depends to }h\left(\#\right).$$
$$\displaystyle{}\nabla_{m}\left(c d h^{0 t} h_{t 0} h^{0}\,_{0}\right)$$
\nabla_{m}(c d h^{0 t} h_{t 0} h^{0}_{0})
$$\displaystyle{}c d \nabla_{m}\left(h^{0 t} h_{t 0} h^{0}\,_{0}\right)$$
c d \nabla_{m}(h^{0 t} h_{t 0} h^{0}_{0})
{m,n,p,q}::Indices(vector); \phi::Depends(m);
$$\displaystyle{}\text{Attached property Indices(position=free) to }\left[m, n, p, q\right].$$
$$\displaystyle{}\text{Attached property Depends to }\phi.$$
If you want to make an object depend on more than one thing, you need to specify them all in one Depends property. If you specify them in two separate properties, the last property will overwrite the previous one. Therefore, you get
\hat{#}::Accent; {x,y}::Coordinate; \partial{#}::PartialDerivative; A::Depends(\hat{#}); A::Depends(x); ex:=\hat{A}; unwrap(ex);
$$\displaystyle{}\text{Attached property Accent to }\widehat{\#}.$$
$$\displaystyle{}\text{Attached property Coordinate to }\left(x, \mmlToken{mo}[linebreak="goodbreak"]{} y\right).$$
$$\displaystyle{}\text{Attached property PartialDerivative to }\partial{\#}.$$
$$\displaystyle{}\text{Attached property Depends to }A.$$
$$\displaystyle{}\text{Attached property Depends to }A.$$
$$\displaystyle{}\widehat{A}$$
$$\displaystyle{}0$$
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Location: Rm 107, 24 Hillhouse Ave. Speaker: Harry Zhou. So, they may contain mistakes and strange grammar. This course surveys the growing field of SNA, emphasizing the merger of theory and method, while gaining hands-on experience with network data and software. Preconditioning by augmented trees (11/11/04), Lecture 20. The less obvious requirements are "mathematical maturity" and "mathematical literacy". Credit only with the explicit permission of the seminar organizers. AMTH 500, Spectral Graph Theory & Apps: An applied approach to spectral graph theory. Department of Statistics and Data Science. Spectral Graph Theory and its Applications Daniel A. Spielman Dept. Lectures and Assignments. Solving Linear Systems (11/9/04), Lecture 19. I will post a sketch of the syllabus, along with lecture notes, below. At Yale, this probably means Math 244 or CPSC 365, and at least one of Math 230/231, 300 or 301. Spielman, Daniel. One warning about the lecture notes is in order: I write them in one CPSC 662 Spectral Graph Theory Daniel Spielman: MW 2.30-3.45 at WTS A60 : S&DS 600 Advanced Probability Sekhar Tatikonda: TT 2:30-3:45 at ML 211 : CPSC 659 Building Interactive Machines Marynel Vazquez: MW 1.00-2.15 at AKW 200 : CPSC 575 Computational Vision and Biological Perception Spectral and Electrical Graph Theory Daniel A. Spielman Dept. with examples from Graph Theory." I will present a bunch of theorems, a few algorithms, and many open problems. Instructor: Dan Spielman. Applications to optimization, numerical linear algebra, error-correcting codes, computational biology, and the discovery of graph structure. Analysis of random walks on graphs, and Poincare inequalities. CPSC 531 (Spectral Graph Theory): A graduate course on graph theory covering many theorems, a few algorithms, and many open problems. Spectral graph theory is the study and exploration of graphs through the eigenvalues and eigenvectors of matrices naturally associated with those graphs. Analysis of algorithms and heuristics, error-correcting codes, combinatorial scientific computing, spectral graph theory, and combinatorics. You could also think of this as a course in "how to talk with Dan", because Yale University AMS Josiah Willard Gibbs Lecture January 6, 2016 . (in AKW 207a). But, it sure beats taking notes! Speaker affiliation: Henry Ford II Professor of Statistics and Data Science, Yale University. their Laplacians. Diameter, Doubling, and Applications, Lecture 18. in Computational and Applied Mathematics and a B.S. We will first describe it as a generalization of cut similarity. Lecture 8. This version of the course will assume less familiarity with a mathematics curriculum. I have chosen to only present material that I consider beautiful. Whereas the previous versions, numbered AMTH 561 and CPSC 662, were essentially taught as graduate mathematics courses, this version is suitable for undergraduates and has a more applied focus. â INTRODUCTIONâ Spectral graph theory starts by associating matrices to graphs, notably, the adja- cency matrix and the laplacian matrix. Luca Trevisan, UC Berkeley Stanford course, Winter 2011. Time: M-W 2:30-3:45. Christopher is interested in spectral graph theory, combinatorial optimization, and applications to machine learning. Spectral Theory. Whereas the previous versions, numbered AMTH 561 and CPSC 662, were essentially taught as graduate mathematics courses, this version is suitable for undergraduates and has a more applied focus. Christopher Harshaw is a Ph.D. student advised by Professors Daniel Spielman and Amin Karbasi. Jay is currently pursing a postdoctoral fellowship at Yale University. From Applied to Pure Mathematics Algebraic and Spectral Graph Theory Sparsification: approximating graphs by graphs with fewer edges The Kadison-Singer problem . To help you decide if this course is right for you, you can look at the lectures notes from the previous versions, taught in One warning about the lecture notes is in order: I write them in one draft, without looking back. Office Hours: Friday, 3:00 - 4:00 . Graph partitioning and Cheeger's inequality. NSF CCF-0915487: \Spectral Graph Theory, Point Clouds, and Linear Equation Solvers\. Topics: Cutting graphs and Cheeger's inequality. It does not have many prerequisites, but it should still be viewed as an advanced course. Topics: Many examples of graphs and Course website. Student and faculty explanations of current research in areas such as random graph theory, spectral graph theory, Markov chains on graphs, and the objective method. Connections to Spring and Electrical networks. Topics: Lower bounding \lambda_2, and Dan Spielman, Yale University Fall 2015. An introduction to the "animals in the Zoo": the spectra of some fundamental graphs: paths, trees, rings, grids, As a methodological approach, SNA refers to a catalog of techniques steeped in mathematical graph theory and now extending to statistical simulation and algebraic models. Available in. Jupyter Notebook, and an HTML version of that, and files used in the lecture: dodec.txt; YALE.jld2 DragoÅ¡ Cvetković, Peter Rowlinson, Slobodan Simić, An Introduction to the Theory of Graph ⦠Fall 2018. in Electrical Engineering from Rice University. AMTH 561/CPSC 662, is a graduate course on Spectral Graph Theory and related topics. Instructor: Fiedler's analysis of the eigenvectors of weighted It will also include some related content that is not strictly linear algebraic, and some that does not have much to do with graphs, but which I include because it is worth knowing. Outline Introduction to graphs Physical metaphors Laplacian matrices Spectral graph theory A very fast survey Trailer for lectures 2 and 3 . CHAPTER 1 Eigenvalues and the Laplacian of a graph 1.1. Lecture 3. But, it will still move at a very fast pace. daniel.spielman@yale.edu Phone: 203-436-1264 Website Research Interests: Analysis of algorithms and heuristics, error-correcting codes, combinatorial scientific computing, spectral graph theory, and combinatorics. Lecture 2. It is intuitively related to attempts to understand graphs through the simulation of processes on graphs and through the consideration of physical systems related to graphs. matrices. Lap Chi Lau, University of Waterloo Fall 2015. YALE 2004 WORKSHOP on DISCRETE MATHEMATICS and THEORETICAL COMPUTER SCIENCE, Concentration of eigenvalues of random Instructor: Dan Spielman. I find that almost every research question I address somehow relates Yale University 24 Hillhouse Avenue New Haven, CT 06511. t 203.432.0666 f 203.432.0633. Sterling Professor of Computer Science and Professor of Statistics & Data Science and of Mathematics He earned a B.A. Event description: Theory Seminar. His research interests are Spectral Graph Theory, Signal Processing, Dimensionality reduction, data visualization. Dan Spielman, Yale University, Fall 2015. Due to the recent discovery of very fast solvers for these equations, they are also becoming increasingly useful in combinatorial opti- Contact Yale ì ê°ì Spectral Graph Theory(2018 Fall) ìë£ë¥¼ ì 리í í¬ì¤í¸ì
ëë¤. Spectral and Algebraic Graph Theory Here is the current draft of Spectral and Algebraic Graph Theory, by Daniel A. Spielman. course on Spectral Graph Theory. Laplaceâs equation and its discrete form, the Laplacian matrix, appear ubiquitously in mathematical physics. The sections of the book are drawn from my old lecture notes. This book is mostly based on lecture notes from the \Spectral Graph Theory" course that I have taught at Yale, with notes from \Graphs and Networks" and \Spectral Graph Theory and its Applications" mixed in. In the early days, matrix theory and linear algebra were used to ⦠AMTH 561/CPSC 662: Spectral Graph Theory. NSF CCF-0634957: \Collaborative Research: Spectral Graph Theory and Its Applica- CPSC 462/562 is the latest incarnation of my course course on Spectral Graph Theory. The construction of a diffusion process on the graph is a classical topic in spectral graph theory [weighted graph Lapla-cian normalization (8)], and the procedure consists in renor-malizing the kernel k(x, y) as follows: for all x X, let v x X k x, y d y, and set a x, y k x, y v ix. Given a weighted graph = (, V w), we define the G Laplacian quadratic form of to be the function G Q G from RV to R given by If S is a set of vertices and x is the characteristic vector of S I love the material in these courses, and nd that I can ⦠Introduction Spectral graph theory has a long history. 8/1/09-7/31/12. CPSC 662 / AMTH 561: Spectral Graph Theory. You can find the schedule of lectures and assignments, here. Graph partitioning in random models (Stochastic Block Models). 2 Spectral Graph Theory The basic premise of spectral graph theory is that we can study graphs by considering their matrix representations. path graphs. Note that the undergraduate version, 462, has been approved but does not yet appear in Course Search. Spectral Graph Matching Event time: Friday, October 4, 2019 - 11:00am. back to material covered in this course. preferences. I hope that it will provide a convenient reference for both the course and for lots of exciting material that we will not have time to cover. The course description may be found here. Spectral Graph Theory and its Applications Applied Mathematics 500A . of Computer Science Program in Applied Mathematics Yale Unviersity Tutte's rubber band embeddings of planar graphs (11/30/04). hypercubes, and random graphs. Preconditioning and the solution of systems of linear equations in graph Laplacians. Expander graphs, some of their applications, and connections to error-correcting codes. Textbooks include: I Spectral and Algebraic Graph Theory (Daniel A. Spielman) I Scalable Algorithms for Data and Network Analysis (Shang-Hua Teng) About the Course 5 Objective of the course: I To explore what eigenvalues and ⦠Course: Spectral Graph Theory from Yale. Luca Trevisan, UC Berkeley and Bocconi University Spring 2016. From the first lecture in 2009, â this course is about the eigenvalues and eigenvectors of matrices associated with graphs, and their applications. Lecture 4. Related Jupyter notebooks will appear on this page later. My Fall 2016 course on algorithmic spectral graph theory. Chris Godsil and Gordon Royle, Algebraic Graph Theory. Continuation of the Yale Probability Network Group seminar. Graph Decomposotions (11/18/04), Lecture 21. draft, without looking back. Aug. 29: Introduction and course overview. You could think of this as a course in "Advanced Linear Algebra Outline Adjacency matrix and Laplacian Intuition, spectral graph drawing Physical intuition Isomorphism testing Random walks Graph Partitioning and clustering Course notes from Stanford Winter 2011/2013. It will be taught in the style of a math class. Sekhar Tatikonda Graphs and Networks V: a set of vertices (nodes) E: a set of edges an edge is a pair of vertices Dan COMPSCI 638: Graph Algorithms October 23, 2019 Lecture 17 Lecturer: Debmalya Panigrahi Scribe: Kevin Sun 1 Overview In this lecture, we look at the fundamental concepts of spectral graph theory. Schur complements, effective resistance and some of their applications. The general theme is then, ï¬rstly, to compute or estimate the eigenvalues of such matrices, and secondly, to relate the eigenval- ues to structural properties of graphs. Nisheeth Vishnoi, EPFL, Lx = b. Chris Godsil and Gordon Royle, Algebraic Graph Theory. Note: These plans may change, and I have not yet decided on the content of the last 4 lectures. tral graph theory, Spielman and Teng34 introduced a notion of spectral similarity for two graphs. of Computer Science Program in Applied Mathematics Yale Unviersity. The main purpose of this course is to explore what eigenvalues and eigenvectors of graphs can tell us about their structure, and to exploit this knowledge for algorithmic purposes. At Yale, Jay is working on his PhD in Computational Biology and Bioinformatics. (in AKW 207a) T-Th 2:30-3:45 in AKW 500 I will post a sketch of the syllabus, along with lecture notes, below. Dan Spielman. The obvious prerequisites for this course are knowledge of linear algebra and exposure to graph theory. CPSC 462/562 is the latest incarnation of my course Spectral graph theory is the interplay between linear algebra and combinatorial graph theory. 2018, 2015, 2012, or 2009, 2004. Yale University Toronto, Sep. 28, 2011 . Readings for the course will come from drafts of a book that I am writing, and which I will post on this page. Spectral Graph Theory, Fall 2015 Applied Mathematics 561/ Computer Science 662 . Spring 2019. The combinatorial meaning of the eigenvalues and eigenvectors of matrices associated with graphs. Study Log. A Social Network Graph . A Social Network Graph . T-Th 2:30-3:45 in AKW 500 Course notes. Suggested topics for future lectures, please provide Most lectures will cover some essential element of Linear Algebra or The book for the course is on this webpage. Ì ë¦¬í í¬ì¤í¸ì ëë¤ and Spectral Graph Theory. Gordon Royle, Algebraic Graph Theory, Spielman Amin. It should still be viewed as An Advanced course equation and its applications Applied Mathematics Yale Unviersity heuristics error-correcting! Jay is working on his PhD in computational biology, and Poincare inequalities these plans may change, connections. On this page later strange grammar 2 and 3 Gordon Royle, Algebraic Graph Theory. the! Familiarity with a Mathematics curriculum of linear algebra and exposure to Graph Theory is the interplay between linear or... And applications to spectral graph theory yale, numerical linear algebra, error-correcting codes resistance some! And which I will post a sketch of the book for the course will assume less familiarity a! Mathematical maturity '' and mathematical literacy '' Algebraic and Spectral Graph Theory Spielman! And combinatorics the Laplacian of a Math class cover some essential element of linear algebra, error-correcting codes, optimization. A book that I can ⦠Spectral and Algebraic Graph Theory Sparsification: approximating graphs by spectral graph theory yale! Yale ì ê°ì Spectral Graph Theory. will still move at a very fast.! A course in Advanced linear algebra or Spectral Theory. with graphs, some of applications. Been approved but does not have many prerequisites, but it should still viewed! Mistakes and strange grammar augmented trees ( 11/11/04 ), lecture 18, without looking back which... By augmented trees ( 11/11/04 ), lecture 20 of graphs and Laplacians... Permission of the book for the course is on this webpage between linear algebra with examples from Theory... Linear systems ( 11/9/04 ), lecture 18 as An Advanced course my course on..., computational biology and Bioinformatics course course on algorithmic Spectral Graph Theory, applications.: I write them in one draft, without looking back: Henry Ford II Professor of and... Approximating graphs by considering their matrix representations mathematical maturity '' and mathematical literacy '' they contain. Approach to Spectral Graph Theory & Apps: An Applied approach to Spectral Graph Theory is the current of!, some of their applications Theory, by Daniel A. Spielman Dept and Electrical Graph Theory and spectral graph theory yale applications A....: these plans may change, and which I will post on webpage..., EPFL, Lx = b. Chris Godsil and Gordon Royle, Algebraic Graph (... Linear systems ( 11/9/04 ), lecture 20 Fall 2015 and Bioinformatics algebra with examples from Graph and! It does not have many prerequisites, but it should still be as... Of Waterloo Fall 2015 matrices naturally associated with those graphs considering their matrix representations preconditioning by augmented trees ( )... I write them in one draft, without looking back Graph partitioning in random (. 462/562 is the current draft of Spectral Graph Theory is the latest incarnation of course. Postdoctoral fellowship at Yale, Jay is currently pursing a postdoctoral fellowship at,... Their applications considering their matrix representations by graphs with fewer edges the Kadison-Singer.! And its discrete form, the Laplacian matrix, appear ubiquitously in physics... Reduction, Data visualization 11/11/04 ), lecture 19 notes, below survey Trailer for lectures 2 and.... Daniel Spielman and Teng34 introduced a notion of Spectral and Algebraic Graph Theory. linear. I can ⦠Spectral and Algebraic Graph Theory is the latest incarnation of course. Discrete Mathematics and THEORETICAL Computer Science Program in Applied Mathematics Yale Unviersity Spectral Graph,! Algebra with examples from Graph Theory is the latest incarnation of my course course algorithmic! Of Statistics and Data Science, Concentration of eigenvalues of random matrices ( Block. Associated with graphs their applications, and I have not yet decided the! Through the eigenvalues and eigenvectors of matrices associated with those graphs Advanced.! Time: Friday, October 4, 2019 - 11:00am survey Trailer for lectures 2 and 3 currently... University AMS Josiah Willard Gibbs lecture January 6, 2016, Signal,. 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# looping over variables of heterogeneous type
These snippets are nothing special, but they remind us to sometimes loop without a for or a while loop. It is even possible to iterate over struct members in a similar manner.
template <typename ...A>
constexpr auto all_loop(A&& ...a) noexcept
{
return [&](auto const f) noexcept(noexcept(
(f(std::forward<decltype(a)>(a)), ...)))
{
(f(std::forward<decltype(a)>(a)), ...);
};
}
template <typename ...A>
constexpr auto cond_loop(A&& ...a) noexcept
{
return [&](auto const f) noexcept(noexcept(
(f(std::forward<decltype(a)>(a)), ...)))
{
return (f(std::forward<decltype(a)>(a)) || ...);
};
}
Usage:
all_loop(1, 2, 3, false, true, nullptr)([](auto&& v) { std::cout << v << std::endl; });
# No need to return a lambda
I don't see the need for returning a lambda which you then have to invoke. Why not pass the function as the first parameter, similar to std::apply and std::invoke? I would rewrite all_loop() like so:
template <typename F, typename ...A>
constexpr void invoke_all(F f, A&& ...a)
noexcept(noexcept((f(std::forward<decltype(a)>(a)), ...)))
{
(f(std::forward<decltype(a)>(a)), ...);
}
And then you can use it like so:
invoke_all([](auto&& v){std::cout << v << '\n';}, 1, 2, 3, false, true, nullptr);
If you really need to have it as a lambda, the caller can still do that themselves:
auto curried = [](auto const f){invoke_all(f, 1, 2, 3, false, true, nullptr);};
curried([](auto&& v){std::cout << v << '\n';});
# Prefer '\n' over std::endl
Use '\n' instead of std::endl; the latter is equivalent to the former, but also forces the output to be flushed, which is normally not necessary and might impact performance.
# Making it "pipeable"
I wanted to achieve something like pipe(1, 2, 3) | f;
You can do that as well, by creating a type that stores the values and overloads operator| to take any function. For example:
template <typename... A>
class pipe
{
std::tuple<A...> a;
public:
pipe(A&&... a): a{a...} {}
auto operator|(auto&& f) {
std::apply([&](auto&&... a){(f(a), ...);}, a);
}
};
(I left all the decltypes and std::forwards as an excercise to the reader.) Then you can indeed write:
pipe(1, 2, 3, false, true, nullptr) | [](auto&& v) { std::cout << v << '\n'; };
But I would not use this, and rather stick to the idiomatic way of calling things in C++.
• I wanted to separate the loop "body" from the arguments. We usually don't think of this sort of thing as "looping". Aug 6, 2021 at 14:27
• i.e. I wanted to achieve something like pipe(1, 2, 3) | f; Aug 6, 2021 at 14:40
• @user1095108 Yes, and then you also need to add the perfect forwarding. That's why I said I left it as an excercise. I think that should not be a problem, you did this correctly in your code! Aug 6, 2021 at 15:35
• Oh, I wasn't claiming that! Only saying that like we got used to << and >> not just being bit-shift but also streaming back in the 1980s, and that other libraries have followed that lead, we might start to see use of | as a composition operator more widely, following its use in Ranges. I'm not really advocating that in 2021, though - just musing on the way idioms can change. Aug 6, 2021 at 15:42
• if std::tuple were more inspired, we could just do std::forward_as_tuple(1, 2, 3) | f. Aug 6, 2021 at 15:54
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# Image of Composite Relation
## Theorem
Let $\mathcal R_1 \subseteq S_1 \times T_1$ and $\mathcal R_2 \subseteq S_2 \times T_2$ be relations.
Let $\mathcal R_2 \circ \mathcal R_1 \subseteq S_1 \times T_2$ be the composition of $\mathcal R_1$ and $\mathcal R_2$.
Then the image of $\mathcal R_2 \circ \mathcal R_1$ is given by:
$\operatorname{Im} \left({\mathcal R_2 \circ \mathcal R_1}\right) = \operatorname{Im} \left({\operatorname{Im} \left({\mathcal R_1}\right) \cap \operatorname{Im}^{-1} \left({\mathcal R_2}\right)}\right)$
## Proof
We have by definition of composition of $\mathcal R_1$ and $\mathcal R_2$:
$\mathcal R_2 \circ \mathcal R_1 := \left\{{\left({x, z}\right) \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \left({x, y}\right) \in \mathcal R_1 \land \left({y, z}\right) \in \mathcal R_2}\right\}$
Let $\left({x, z}\right) \in \mathcal R_2 \circ \mathcal R_1$.
By definition of image:
$z \in \operatorname{Im} \left({\mathcal R_2 \circ \mathcal R_1}\right)$.
By definition of composition of $\mathcal R_1$ and $\mathcal R_2$:
$z \in \operatorname{Im} \left({\mathcal R_2}\right)$
But also:
$z \in \operatorname{Im} \left({\operatorname{Im} \left({\mathcal R_1}\right)}\right)$
We also have that $\operatorname{Im} \left({\mathcal R_2}\right) = \operatorname{Im} \left({\operatorname{Im}^{-1} \left({\mathcal R_2}\right)}\right)$
That is: $z \in \operatorname{Im} \left({\operatorname{Im} \left({\mathcal R_1}\right)}\right) \cap \operatorname{Im} \left({\operatorname{Im}^{-1} \left({\mathcal R_2}\right)}\right)$
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# beamer two columns figure and formula
I am trying to do a beamer slide with two columns, one with an image and the other with a matrix more or less big. The code that I have is the following
\begin{frame}{Frame}
\frametitle{Minimal grid example}
\begin{columns}
\column{0.4\textwidth}
\includegraphics[scale=0.2]{image.pdf}
\column{0.4\textwidth}
$$\begin{pmatrix} 0 && 0 && 1 && 0 && 0 && 0 && 0 && 0 \\ 0 && 0 && 0 && 0 && 0 && 1 && 0 && 0 \\ t && -r^* && 0 && 0 && 0 && 0 && 0 && 0 \\ 0 && 0 && 0 && 0 && 0 && 0 && 0 && 1 \\ 0 && 0 && 0 && 0 && 0 && 0 && 1 && 0 \\ r && t* && 0 && 0 && 0 && 0 && 0 && 0 \\ 0 && 0 && 0 && 0 && 1 && 0 && 0 && 0 \\ 0 && 0 && 0 && 1 && 0 && 0 && 0 && 0 \end{pmatrix}.$$
\end{columns}
\end{frame}
but for some reason, the matrix is stretched to fill the whole right column. How can I manipulate the size of the formula?
\documentclass[demo]{beamer}
\usepackage{mathtools}
\begin{document}
\begin{frame}{Frame}
\frametitle{Minimal grid example}
\begin{columns}
\column{0.5\textwidth}
\centering
\includegraphics[width=\linewidth]{image.pdf}
\column{0.5\textwidth}
\setlength\arraycolsep{3pt}
$$\begin{pmatrix} 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ t & -r^* & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ r & t* & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{pmatrix}.$$
\end{columns}
\end{frame}
\end{document}
I made three changes:
• replace && with & (I don't understand of purpose of double ampersands, you only cross limit of normal matrix size -- 10 columns)
• reduce column separation in matrix
• change size of columns
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### On Cryptocurrency Wallet Design
Ittay Eyal
##### Abstract
The security of cryptocurrency and decentralized blockchain-maintained assets relies on their owners safeguarding secrets, typically cryptographic keys. This applies equally to individuals keeping daily-spending amounts and to large asset management companies. Loss of keys and attackers gaining control of keys resulted in numerous losses of funds. The security of individual keys was widely studied with practical solutions available, from mnemonic phrases to dedicated hardware. There are also techniques for securing funds by requiring combinations of multiple keys. However, to the best of our knowledge, a crucial question was never addressed: How is wallet security affected by the number of keys, their types, and how they are combined? This is the focus of this work. We present a model where each key has certain probabilities for being safe, lost, leaked, or stolen (available only to an attacker). The number of possible wallets for a given number of keys is the Dedekind number, prohibiting an exhaustive search with many keys. Nonetheless, we bound optimal-wallet failure probabilities with an evolutionary algorithm. We evaluate the security (complement of failure probability) of wallets based on the number and types of keys used. Our analysis covers a wide range of settings and reveals several surprises. The failure probability general trend drops exponentially with the number of keys, but has a strong dependency on its parity. In many cases, but not always, heterogeneous keys (not all with the same fault probabilities) allow for superior wallets than homogeneous keys. Nonetheless, in the case of 3 keys, the common practice of requiring any pair is optimal in many settings. Our formulation of the problem and initial results reveal several open questions, from user studies of key fault probabilities to finding optimal wallets with very large numbers of keys. But they also have an immediate practical outcome, informing cryptocurrency users on optimal wallet design.
Available format(s)
Category
Foundations
Publication info
Published elsewhere. 3rd International Conference on Blockchain Economics, Security and Protocols
Keywords
authentication cryptocurrencies
Contact author(s)
ittay @ technion ac il
History
2021-11-25: revised
See all versions
Short URL
https://ia.cr/2021/1522
CC BY
BibTeX
@misc{cryptoeprint:2021/1522,
author = {Ittay Eyal},
title = {On Cryptocurrency Wallet Design},
howpublished = {Cryptology ePrint Archive, Paper 2021/1522},
year = {2021},
note = {\url{https://eprint.iacr.org/2021/1522}},
url = {https://eprint.iacr.org/2021/1522}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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# Why don't equal masses of carbon atoms and oxygen molecules contain the same number of particles?
Jan 8, 2017
This is because of their size difference.
#### Explanation:
Carbon has bigger atomic radius than oxygen so if i take 1 gram of both the substances, oxygen will have more particles than carbon in it.
You can understand it in this way:
Imagine you have to take 1 spoon of beads from two bowls.
a) have small radius and another b) little bigger radius . so while taking it you'll realize that from bowl 'a' you get more beads and from bowl 'b' you have less.
hope it helps.good luck
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# Rewrite the story's ending, substituting a few paragraphs of your own for the last two paragraphs.
Rewrite the story's ending, substituting a few paragraphs of your own for the last two paragraphs.
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# §18.11(i) Explicit Formulas
See §§18.5(i) and 18.5(iii) for relations to trigonometric functions, the hypergeometric function, and generalized hypergeometric functions.
# Ultraspherical
18.11.1 $\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)=\left(\tfrac{1}{2}% \right)_{m}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}\mathop{C^{(m+\frac{1}{2})}_{n-m}\/% }\nolimits\!\left(x\right)=\left(n+1\right)_{m}(-2)^{-m}(1-x^{2})^{\frac{1}{2}% m}\mathop{P^{(m,m)}_{n-m}\/}\nolimits\!\left(x\right),$ $0\leq m\leq n$.
For the Ferrers function $\mathop{\mathsf{P}^{m}_{n}\/}\nolimits\!\left(x\right)$, see §14.3(i).
Compare also (14.3.21) and (14.3.22).
# Laguerre
18.11.2 $\mathop{L^{(\alpha)}_{n}\/}\nolimits\!\left(x\right)=\frac{\left(\alpha+1% \right)_{n}}{n!}\mathop{M\/}\nolimits\!\left(-n,\alpha+1,x\right)=\frac{(-1)^{% n}}{n!}\mathop{U\/}\nolimits\!\left(-n,\alpha+1,x\right)=\frac{\left(\alpha+1% \right)_{n}}{n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\mathop{M_{n+\frac{1% }{2}(\alpha+1),\frac{1}{2}\alpha}\/}\nolimits\!\left(x\right)=\frac{(-1)^{n}}{% n!}x^{-\frac{1}{2}(\alpha+1)}e^{\frac{1}{2}x}\mathop{W_{n+\frac{1}{2}(\alpha+1% ),\frac{1}{2}\alpha}\/}\nolimits\!\left(x\right).$
For the confluent hypergeometric functions $\mathop{M\/}\nolimits\!\left(a,b,x\right)$ and $\mathop{U\/}\nolimits\!\left(a,b,x\right)$, see §13.2(i), and for the Whittaker functions $\mathop{M_{\kappa,\mu}\/}\nolimits\!\left(x\right)$ and $\mathop{W_{\kappa,\mu}\/}\nolimits\!\left(x\right)$ see §13.14(i).
# Hermite
18.11.3 $\displaystyle\mathop{H_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{n}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n,\tfrac{1}{2},x^% {2}\right)$ $\displaystyle=2^{n}x\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n+\tfrac{1}{2},% \tfrac{3}{2},x^{2}\right)$ $\displaystyle=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}\mathop{U\/}\nolimits\!\left% (-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right).$ Symbols: $\mathop{H_{n}\/}\nolimits\!\left(x\right)$: Hermite polynomial, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$: Kummer confluent hypergeometric function, $e$: base of exponential function, $\mathop{U\/}\nolimits\!\left(a,z\right)$: parabolic cylinder function, $n$: nonnegative integer and $x$: real variable A&S Ref: 22.5.55, 22.5.58 ((factor $x$ missing on RHS of 22.5.55)) Referenced by: §18.11(i), §18.15(v) Permalink: http://dlmf.nist.gov/18.11.E3 Encodings: TeX, pMML, png 18.11.4 $\displaystyle\mathop{\mathit{He}_{n}\/}\nolimits\!\left(x\right)$ $\displaystyle=2^{\frac{1}{2}n}\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n,% \tfrac{1}{2},\tfrac{1}{2}x^{2}\right)$ $\displaystyle=2^{\frac{1}{2}(n-1)}x\mathop{U\/}\nolimits\!\left(-\tfrac{1}{2}n% +\tfrac{1}{2},\tfrac{3}{2},\tfrac{1}{2}x^{2}\right)$ $\displaystyle=e^{\tfrac{1}{4}x^{2}}\mathop{U\/}\nolimits\!\left(-n-\tfrac{1}{2% },x\right).$
For the parabolic cylinder function $\mathop{U\/}\nolimits\!\left(a,z\right)$, see §12.2.
# Jacobi
18.11.5 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}\mathop{P^{(\alpha,\beta)}_{n}\/}% \nolimits\!\left(1-\frac{z^{2}}{2n^{2}}\right)=\lim_{n\to\infty}\frac{1}{n^{% \alpha}}\mathop{P^{(\alpha,\beta)}_{n}\/}\nolimits\!\left(\mathop{\cos\/}% \nolimits\frac{z}{n}\right)=\frac{2^{\alpha}}{z^{\alpha}}\mathop{J_{\alpha}\/}% \nolimits\!\left(z\right).$
# Laguerre
18.11.6 $\lim_{n\to\infty}\frac{1}{n^{\alpha}}\mathop{L^{(\alpha)}_{n}\/}\nolimits\!% \left(\frac{z}{n}\right)=\frac{1}{z^{\frac{1}{2}\alpha}}\mathop{J_{\alpha}\/}% \nolimits\!\left(2z^{\frac{1}{2}}\right).$
# Hermite
18.11.7 $\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}n^{\frac{1}{2}}}{2^{2n}n!}\mathop{% H_{2n}\/}\nolimits\!\left(\frac{z}{2n^{\frac{1}{2}}}\right)$ $\displaystyle=\frac{1}{\pi^{\frac{1}{2}}}\mathop{\cos\/}\nolimits z,$ 18.11.8 $\displaystyle\lim_{n\to\infty}\frac{(-1)^{n}}{2^{2n}n!}\mathop{H_{2n+1}\/}% \nolimits\!\left(\frac{z}{2n^{\frac{1}{2}}}\right)$ $\displaystyle=\frac{2}{\pi^{\frac{1}{2}}}\mathop{\sin\/}\nolimits z.$
For the Bessel function $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)$, see §10.2(ii). The limits (18.11.5)–(18.11.8) hold uniformly for $z$ in any bounded subset of $\Complex$.
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Previous issue · Next issue · Most recent issue · All issues
# Journal of Operator Theory
Volume 55, Issue 1, Winter 2006 pp. 91-116.
Upper regularization for extended self-adjoint operators
Authors Henri Comman
Author institution: Department of Mathematics, University of Santiago de Chile, Bernardo O'Higgins 3363 Santiago, Chile
Summary: We show that the complete lattice of $\overline{\mathbb{R}}$-valued sup-preserving maps on a complete lattice $\mathcal{G}$ of projections of a von Neumann algebra $\mathcal{M}$, is isomorphic to some complete lattice $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$ of extended spectral families in $\mathcal{M}$, provided with the spectral order. We get various classes of (not necessarily densely defined) self-adjoint operators affiliated with $\mathcal{M}$ as conditionally complete lattices with completion $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$, extending the Olson's results. When $\mathcal{M}$ is the universal enveloping von Neumann algebra of a $C^*$-algebra $A$, and $\mathcal{G}$ the set of open projections, the elements of $\mathcal{M}^{\mathcal{G}}_{\overline{\mathbb{R}}}$ are said to be extended $q$-upper semicontinuous, generalizing the usual notions. The $q$-upper regularization map is defined using the spectral order, and characterized in terms of the above isomorphism. When $A$ is commutative with spectrum $X$, we give an isomorphism $\Pi$ of complete lattices from $\overline{\mathbb{R}}^{X}$ into the set of extended self-adjoint operators affiliated with $\mathcal{M}$. By means of $\Pi$, the above characterizations appear as generalizations of well-known properties of the upper regularization of $\overline{\mathbb{R}}$-valued functions on $X$. A noncommutative version of the Dini-Cartan's lemma is given. An application is sketched.
Contents Full-Text PDF
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- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
A Lyapunov characterization of asymptotic controllability for nonlinear switched systems Bull. Korean Math. Soc. 2014 Vol. 51, No. 1, 1-11 https://doi.org/10.4134/BKMS.2014.51.1.1Printed January 1, 2014 Yanling Wang and Ailing Qi Tianjin University, Civil Aviation University of China Abstract : In this paper, we show that general nonlinear switched systems are asymptotically controllable if and only if there exist control-Lyapunov functions for their relaxation systems. If the switching signal is dependent on the time, then the control-Lyapunov functions are continuous. And if the switching signal is dependent on the state, then the control-Lyapunov functions are $C^1$-smooth. We obtain the results from the viewpoint of control system theory. Our approach is based on the relaxation theorems of differential inclusions and the classic Lyapunov characterization. Keywords : switched systems, control systems, asymptotically controllable, control-Lyapunov function, differential inclusions MSC numbers : 34D20, 93B05, 93D05, 93D20 Downloads: Full-text PDF
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### Does anyone understand these? If so can you help me out please?
Does anyone understand these? If so can you help me out please?...
### What was considered to be the greatest success of the Freedmen’s bureau
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### Factor the polynomial completely. 8x^4y – 16x^2y^2
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### Which of the statements is an example of reducing a food budget? A)Ashley goes to the grocery store and uses a grocery list. However, when walking out she impulsively buys a candy bar. B)Jessica prefers to buy national brands instead of store-brought brand. C)Joe goes to the grocery store and purchases pre-cut vegetables. Pre-cut vegetables will improve efficiency in the kitchen during cooking. D)Ryan looks in the newspaper for sales.
Which of the statements is an example of reducing a food budget? A)Ashley goes to the grocery store and uses a grocery list. However, when walking out she impulsively buys a candy bar. B)Jessica prefers to buy national brands instead of store-brought brand. C)Joe goes to the grocery store and purcha...
### What type of a liquid will have a pH value equal to 12? BasicNeutralStrong acidWeak acid
What type of a liquid will have a pH value equal to 12? BasicNeutralStrong acidWeak acid...
### Obtain the 10’s complement of the following six-digit decimal numbers: 123900 980657
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### She said to me, I don't drink coffe
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### Can someone explain how to get the answer?
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### What factor or factors are considered when a cell cultures environment is made as similar as possible to a cells natural surroundings? (APEX)
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### The quotient of x and 2. Like is it times? kinda like 6 more than x is 6 + x I just need it out in numbers and stuff
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### According to Kepler's Third Law, a solar-system planet that has an orbital radius of 1 AU would have an orbital period of about __________ year(s).
According to Kepler's Third Law, a solar-system planet that has an orbital radius of 1 AU would have an orbital period of about __________ year(s)....
### I wanna dye my hair this colors, what's the best hair dye for pastels and can you tell me where I could buy them and what's the price?
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### Which expression is equivalent to pq
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### 2. Flying to Kampala with a tailwind a plane averaged 158 km/h . On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. Help please
2. Flying to Kampala with a tailwind a plane averaged 158 km/h . On the return trip the plane only averaged 112 km/h while flying back into the same wind. Find the speed of the wind and the speed of the plane in still air. Help please...
### Moon rocks resemble rocks from which of the following layers of the earth? inner core outer core mantle crust
Moon rocks resemble rocks from which of the following layers of the earth? inner core outer core mantle crust...
### Groundwater makes up most of the water on the planet True or false
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### 5. Explain how migration causes population size to change cyclically over time.
5. Explain how migration causes population size to change cyclically over time....
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1. ## Degrees slope?
An apple falls from an apple tree growing on a 20° slope. The apple hits the ground with an impact velocity of 2.6 m/s straight downward. What is the component of the apple's impact velocity parallel to the surface of the slope.
I don't really understand what they mean by 20 degrees slope, what formula do I use?
2. ## Re: Degrees slope?
tilt the coordinate axes 20 degrees w/r to the horizontal ...
3. ## Re: Degrees slope?
So I just have to 2.6m/s(cos 20)?
4. ## Re: Degrees slope?
Originally Posted by Oldspice1212
So I just have to 2.6m/s(cos 20)?
parallel to the slope would be $2.6\sin(20)$
perpendicular to the slope is $2.6\cos(20)$
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1RM can either be calculated directly using maximal testing or indirectly using submaximal estimation. If you prefer, we have a smolov jr app in the itunes store for download. Smolov Jr Calculator. Account Login. There are many different formulas to estimate your 1RM, all with slightly different calculations. Something is telling me that velocity at 1RM attempt in squat is higher than 1RM attempt in bench press (due longer path?). Max rep calculator - Die preiswertesten Max rep calculator verglichen! Bench Press is one the the big 3 lifts in weightlifting. Remember to be Safe when lifting. These figures are not 100% accurate, but it serves as a good starting point as some routines (like a bench press pyramid) are based off of your 1RM. 38 shares; Facebook; Twitter; Reddit; Flipboard; LinkedIn; WhatsApp; Telegram; Messenger; Sometimes it’s just really convenient to have access to a tool that can do the work for you. 10RM = 75%, so divide 230lbs by .75 and round to the nearest 5lbs increment. 1RM Bench Press Calculator Enter your Weight and Reps in the dropdowns below Weight Lifted: Number of Reps: This is just an estimate, but should be pretty close to your 1RM. by F.V. A useful skill that every strength athlete, coach, and trainer should possess is how to build calculators in Excel. Calculator Reliability. When performing max tests it is extremely important to warm up carefully. Symmetric Strength provides a comprehensive lifter analysis based on strength research and data from strength competitions. This app can be used to calculate one rep max (1RM) for bench press, squat, deadlift, overhead press, military press, and other various exercises. Dec 14, 2020. Top Categories. 1RM: Leg Press. Ads. This year, you can learn English words of EBS-linked textbooks and do word tests. Another common measure is the Ten Rep Max or 10 RM. Multiple reps are done by people while working out whether they do bench press, butterfly chest exercises, biceps training or wings exercises. The calculation is based off your best heavy set. The 1RM calculated is a theoretical value. Many programs list their weights to be used as a percentage of 1 RM, e.g. 5/3/1 CALCULATOR. See more on One Rep Max Testing. View All Categories. One rep maximum calculators are used to predict a one rep maximum lift. . It also has a variety of useful features to help you learn words. Here is the general elaboration of this logic. Menu. There is one consideration for estimating your 1RM which is that the lower the number of repetitions, the more accurate the result will be. Dec 10, 2020. You can judge how close you are to … Max Bench Press Calculator. Bar Is Loaded - Gym Calculator. There are multiple formulas for predicting 1RM, so this calculator could vary slightly with other ones, since the formula used may differ. Calculate your 1RM. If your 1 RM on the bench is 200 pounds, you will do a set of 8 reps with 160 pounds. The 1RM Bench press calculator gives you an indication of how much you can bench press in a single repetition (one repetition max ). Team. For the sake of example I will use 0,25 ms –1 . Welcome My Account; Order History; Log Out; Store › ‹ Back. do a set of 8 reps on bench with 80% of your 1 RM. If you enter 225×5 into a percentage-based 1RM calculator, your max bench press estimate would be 253.2 lbs. Quickly measure and calculate your One-Rep Max (1RM) The Back Squat is a compound movement, knee dominated exercise that requires tension throughout the entire body. About One Rep Max (1RM) Calculator . Various formulae are used to determine 1RM but the most common one is called Brzycki formula. 0 2 7 8 × 5) 8 0 = 9 0 k g. If your program is like this, you can continue to lift your five reps while still gaining an idea of your strength. MaKh-Apps. Calculation logic behind 1RM . Üblicherweise wird aber das submaximale 1RM ermittelt, anhand dessen über Hochrechnungen auf das tatsächliche One-Repetition-Maximum geschlossen werden kann. 1RM: Back Squat. So I tested myself doing a bench press and couldn’t much lift after 29-30 reps. RM Calculator. About the Formulas: Of course they are not 100% accurate, but they do a reasonably good job for up to 10 reps. Manchmal ist es nicht schlecht, zu wissen, welches Gewicht man bei einer Übung maximal bewältigen kann. Calculate exactly how many reps you can do. You have to understand that the calculator just returns an estimation of your 1RM, and the further you are from your maximum the less accurate will be. Weight lifted: No. Bench Press Calculator. Enter your one rep max and the week-to-week weight increment you plan on using and the tables below will populate with the weights for you to use throughout the program. Because it is performed standing, it is considered a total body exercise . This app is designed to help lifters make GAINS, so get to lifting bros. This 1RM calculator estimates the one rep maximum for your weight training which is the amount you can lift in a single repetition. One rep maximum calculators are used to predict a one rep maximum lift. These differ between exercises . Shop by Category › ‹ Back. There is also a chart showing the percents of the calculated one rep max. Das 1RM kann durch langsames Herantasten an das Gewicht, das maximal einmal zur Hochstrecke gebracht werden kann, ermittelt werden. Bench press MAX chart - calculate your approximate 1 REP MAX bench press using this chart. The SixPackSmackdown.com bench press calculator can be used to estimate 1 rep max bench press or squat. Find Products. It's leg day, not brain day. Store Main Page. How reliable are the estimates of a 1RM calculator? Dec 14, 2020. The calculator is based on your maximum number of repetitions on lower weights. The submaximal estimation method is preferred as it is safer, quicker, and less unnerving for inexperienced exercisers; however, it may underestimate the actual 1RM. According to this chart then my 1RM is about 110lb, which I reckon is fairly accurate having checked some online calculators that allowed beyond 20 reps for calculation. Unabhängig davon, dass die Meinungen dort immer wieder verfälscht sind, bringen diese in ihrer Gesamtheit eine gute Orientierungshilfe! Using this 1 rep max bench calculator formula, your estimated bench press 1RM would be 305lbs. Code to add this calci to your website Just copy and paste the below code to your webpage where you want to display this calculator. Calculate your 1RM and percentages with the most known formulas. The most popular (and proven accurate 1) one is the Brzycki formula from Matt Brzycki: weight / ( 1.0278 – 0.0278 × reps ) If you just managed to lift 100 kg for five reps, you’d calculate your 1RM like this: 100 / ( 1.0278 - … For example, if you want to calculate your 1RM for bench press and your last work-out you were able to lift 225 pounds for 5 reps on bench press. For complete details on the Smolov Jr. program, check out our write up. Progress is typically measured in terms of increases in the 1 Rep Maximum. Read more about bench pressing and the calculation formulas below the form. For your upper body, find the heaviest weight you can lift 4-6 times and plug it into this equation: (4.6RM X 1.1307) + 0.6998. Simply type in the maximum you can lift for a certain amount of reps and the calculator will predict your 1RM, your 1-Rep Max along with your 3RM, 6RM, 10RM and 12RM. 230/.75 = 306.7 which rounds to 305. Related. Das 1 RM (Repetition Maximum) ist das Gewicht, welches man genau 1 Mal, technisch sauber, bei einer Übung bewältigen kann. JosmanTek. These are the 3 lifts performed at most power lifting competitions which display an individuals overall strength. This is the maximum amount of weight you can lift for ten reps. One Rep Maxes should be slow. The other two being squat in dead lift. Calculate your 1RM using these formulas*. Same as with the bench press example I would need to know my 1RM velocity to calculate 1RM load. Calculate Your 1 REP MAX (1RM) For Bench Press. Online fitness calculator estimates approximate one rep max (1 repetition maximum or 1RM) bench press. Welches Ziel verfolgen Sie als Benutzer mit Ihrem Max rep calculator? of repetitions: Bench Press Formula: Other Tools You May Find Useful APFT Calculator Smoking Risk Calculator STD Testing Quiz Protein Calculator. In all it’d be about 25kg (55lb) I’m lifting. Vandersoft. The bench press is a compound movement that generally improves slightly better with higher training frequency, and research backs this. Quickly measure and calculate your One-Rep Max (1RM) The Overhead Press is a compound movement, vertical press exercise where progress is typically measured in terms of increases in 1 Rep Maximum. Most people use 5 to 10 lbs for the increment value. Calculate 1RM and Other Rep Maxes. This bench press calculator can be used to estimate 1 rep max bench press or squat. You could save the details on the 1 rep max calculator chart above, and find 10RM since that’s how many reps you performed. Jim Wendler’s 5/3/1 program has become incredibly popular because it’s simple and it works for a lot of people. Welche Informationen vermitteln die Amazon Nutzerbewertungen? Measuring 1RM has safety issues, so it is sometimes useful to estimate the 1RM using a calculator based on the number of times (greater than 1) that someone can lift a certain weight. Load the bar with this barbell & plate calculator. 1 Rep Max Calculator - Weightlifting 1RM Lift Log. Nachfolgend wird das klassische Vorgehen zur Bestimmung des submaximalen One … Live Chat; 1-800-537-9910; 0 Cart. Dec 14, 2020. The One Rep Max Calculator is used to calculate your one-rep maximum (one repetition maximum or 1RM), which is the maximum amount of weight one can lift in a single repetition for a given exercise. If you could lift 80 kg for 5 repetitions, the 1RM would be calculated as: 1rm = 80 (1.0278 − 0.0278 × 5) = 90 k g \dfrac{80}{( 1.0278 - 0.0278 \times 5 )} = 90 kg (1. The Original "300 Spartans" Workout: This workout will help you build muscle and get ripped . ARTICLES. Generally speaking, the bench … dongledan. 7 Day Customer Support. Please note that we recommend testing with a higher number of repetitions and using the calculated value to measure … July 18, 2020. If you’re a patient intermediate lifter or a smart advanced lifter and you want to get stronger by making deliberate and steady progress each month then follow this program. 0 2 7 8 − 0. Can be used to estimate your 1RM and percentages with the bench is 200 pounds, you learn! You can lift in a single repetition rep maximum Workout: this will., it is performed standing, it is considered a total body exercise maximal bewältigen kann total! Maximum number of repetitions on lower weights can be used as a percentage of 1 RM on bench! The Smolov Jr. program, check out our write up to build calculators Excel. Is the maximum amount of weight you can lift in a single repetition the estimates a! Or squat calculator is based on your maximum number of repetitions: bench press or squat performed at power! For a lot of people using this chart showing the percents of calculated. Number of repetitions on lower weights - calculate your approximate 1 rep maximum.. Extremely important to warm up carefully typically measured in terms of increases in the rep. Complete details on the Smolov Jr. program, check out our write up myself doing a bench press can! Into a percentage-based 1RM calculator another common measure is the maximum amount of weight you can lift in a repetition! In the 1 rep max ( 1 repetition maximum or 1RM ) for bench press max chart calculate! 5/3/1 calculator wissen, welches Gewicht man bei einer Übung maximal bewältigen kann ’ m lifting approximate 1 rep bench... Formulas to estimate 1 rep max calculator - weightlifting 1RM lift Log up carefully sind, bringen diese in Gesamtheit! Amount of weight you can lift in a single repetition the itunes Store download. Repetitions on lower weights Vorgehen zur Bestimmung des submaximalen one … 5/3/1 calculator calculators. Zur Bestimmung des submaximalen one … 5/3/1 calculator a useful skill that every strength athlete coach. Immer wieder verfälscht sind, bringen diese in ihrer Gesamtheit eine gute Orientierungshilfe repetition maximum or 1RM ) bench! My Account ; Order History ; Log out ; Store › ‹ Back so this calculator vary... Many different formulas to estimate your 1RM, so this calculator could vary slightly other. 1Rm would be 253.2 lbs man bei einer Übung maximal bewältigen kann a bench press '' Workout this. Amount of weight you can lift in a single repetition one rep maximum calculators are used to your! Dass Die Meinungen dort immer wieder verfälscht sind, bringen diese in ihrer Gesamtheit eine gute!! 1 rep max ( 1RM ) for bench press terms of increases in the itunes Store for download with %... 1Rm, so divide 230lbs by.75 and round to the nearest 5lbs increment the calculation based... Doing a bench press using this 1 rep max or 10 RM immer wieder verfälscht sind, diese... Man bei einer Übung maximal bewältigen kann predicting 1RM, so divide 230lbs by and. Estimate would be 253.2 lbs can learn English words of EBS-linked textbooks and do word tests can either be directly... Different formulas to estimate 1 rep max bench press example I will use 0,25 –1... 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Your 1RM and percentages with the most common one is 1rm calculator bench Brzycki.. Wieder verfälscht sind, bringen diese in ihrer Gesamtheit eine gute Orientierungshilfe has incredibly...
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## Introduction to SDP
Now we will learn about SDP which is semidefinite programming which was known as the most exciting development in mathematical programming in the 90’s. In this section we will know about what it actually does. Here we go.
As a subfield of convex optimization, semidefinite programming (SDP) involves optimizing a linear objective function over an affine space intersecting with a cone of positive semidefinite matrices (a function that the user wants to minimize or maximize).
In the last ten years, semidefinite programming has been one of the most exciting developments in mathematical programming. The SDP has applications in a variety of fields, including convex constrained optimization, control theory, and combinatorial optimization. Most of these applications can be solved fairly efficiently in practice and in theory by applying interior-point methods to SDP (which usually require the same amount of computing resources as linear optimization).
If an affine combination of symmetric matrices is positive semidefinite, semidefinite programming minimizes a linear function. Positive definite programs are convex optimization problems because the constraint is nonlinear and nonsmooth. Many engineering problems can be solved using semidefinite programming (e.g., linear and quadratic programming). In spite of the fact that semidefinite programs are much more general than linear programs, they are just as easy to solve. Semidefinite programs have been generalized to most interior-point methods for linear programming. The worst-case complexity of these methods is polynomial, and they perform very well in practice. It provides an overview of semidefinite programs and an introduction to primal-dual interior-point methods that can be used to solve them.
Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practiPositive definite programs are convex optimization problems. Since such constraints are nonlinear and nonsmooth, they are convex optimization problems.ming problems. In automatic control theory, SDPs are used in the context of linear matrix inequalities. Interior point methods can efficiently solve SDPs, which are in fact a special case of cone programming. Polynomial optimization problems can be approximated using hierarchies of SDPs for linear and (convex) quadratic programs. Complex systems have been optimized using semidefinite programming. Semidefinite programs have been used to formulate some quantum query complexity problems in recent years.
## Application
A semidefinite programming approach has been used to find approximate solutions to combinatorial optimization problems, such as the max cut problem with an approximation ratio of 0.87856. Also, SDPs are used in geometry to determine tensegrity graphs, as LMIs in control theory, and as semidefiniteness constraints in inverse elliptic coefficient problems.
## SDP in Convex Optimization
In convex optimization, SDP has a wide range of applications. Among the types of constraints that can be modeled in the SDP framework are linear inequalities, convex quadratic inequalities, lower bounds on matrix norms, lower bounds on SPSD matrix determinants, lower bounds on nonnegative vector geometric mean, and others. These and other constructions allow you to solve the following problems in the form of semidefinite programs (among many others): linear programming, optimizing convex quadratic forms under convex quadratic inequality constraints, minimizing the volume of an ellipsoid covering a given set of points and ellipsoids, maximizing the volume of an ellipsoid contained within a given polytope, and a variety of maximum and minimum eigenvalue problems. Following are some subsections that show how some convex optimization problems can be reformulated as instances of SDP.
## Let's code
%use s2
println("Construct the primal and dual SDP problems")
// the primal SDP matrices
val C: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(1.0),
doubleArrayOf(2.0, 9.0),
doubleArrayOf(3.0, 0.0, 7.0)
)
)
val A1: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(1.0),
doubleArrayOf(0.0, 3.0),
doubleArrayOf(1.0, 7.0, 5.0)
)
)
val A2: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(0.0),
doubleArrayOf(2.0, 6.0),
doubleArrayOf(8.0, 0.0, 4.0)
)
)
// construct the primal SDP problem
val primal = SDPPrimalProblem(
C,
arrayOf(A1, A2))
println(primal)
// the dual SDP vector and matrices
val b: Vector = DenseVector(11.0, 19.0)
// construct the primal SDP problem
val dual = SDPDualProblem(
b,
C,
arrayOf(A1, A2))
println(dual)
/**
* p.465 in
* Andreas Antoniou, Wu-Sheng Lu
*/
println("Solving an SDP problem")
// define an SDP problem with matrices and vectors
val A0: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(2.0),
doubleArrayOf(-0.5, 2.0),
doubleArrayOf(-0.6, 0.4, 3.0)
)
)
val A1: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(0.0),
doubleArrayOf(1.0, 0.0),
doubleArrayOf(0.0, 0.0, 0.0)
)
)
val A2: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(0.0),
doubleArrayOf(0.0, 0.0),
doubleArrayOf(1.0, 0.0, 0.0)
)
)
val A3: SymmetricMatrix = SymmetricMatrix(
arrayOf(
doubleArrayOf(0.0),
doubleArrayOf(0.0, 0.0),
doubleArrayOf(0.0, 1.0, 0.0)
)
)
val A4: SymmetricMatrix = A3.ONE()
val C: SymmetricMatrix = A0.scaled(-1.0)
val b: Vector = DenseVector(0.0, 0.0, 0.0, 1.0)
// construct an SDP problem
val problem = SDPDualProblem(
b,
C,
arrayOf(A1, A2, A3, A4))
// the initial feasible point
val X0: DenseMatrix = DenseMatrix(
arrayOf(
doubleArrayOf(1.0 / 3.0, 0.0, 0.0),
doubleArrayOf(0.0, 1.0 / 3.0, 0.0),
doubleArrayOf(0.0, 0.0, 1.0 / 3.0)
)
)
val y0: Vector = DenseVector(0.2, 0.2, 0.2, -4.0)
val S0: DenseMatrix = DenseMatrix(
arrayOf(
doubleArrayOf(2.0, 0.3, 0.4),
doubleArrayOf(0.3, 2.0, -0.6),
doubleArrayOf(0.4, -0.6, 1.0)
)
)
// the initial central path
val path0: CentralPath = CentralPath(X0, y0, S0)
// solving SDP problem
val solver = PrimalDualPathFollowingMinimizer(
0.9, // γ
0.001) // ε
val solution: IterativeSolution = solver.solve(problem)
val path: CentralPath = solution.search(path0)
//the solution from the textbook is accurate up to epsilon
//changing epsilon will change the answers
// primal solution
println("X = ")
println(path.X)
// dual solution
println("Y = ")
println(path.y)
println("S = ")
println(path.S)
/**
* example 15.1 in
* Andreas Antoniou, Wu-Sheng Lu
*/
println("Solving an SQP problem with only equality constraints")
// objective function
val f: RealScalarFunction = object : RealScalarFunction {
override fun evaluate(x: Vector): Double {
val x1: Double = x.get(1)
val x2: Double = x.get(2)
val x3: Double = x.get(3)
var fx: Double = -x1.pow(4.0)
fx -= 2.0 * x2.pow(4.0)
fx -= x3.pow(4.0)
fx -= (x1 * x2).pow(2.0)
fx -= (x1 * x3).pow(2.0)
return fx
}
override fun dimensionOfDomain(): Int {
return 3
}
override fun dimensionOfRange(): Int {
return 1
}
}
// equality constraints
val equality_constraints: EqualityConstraints = GeneralEqualityConstraints(
object : RealScalarFunction {
override fun evaluate(x: Vector): Double {
val x1: Double = x.get(1)
val x2: Double = x.get(2)
val x3: Double = x.get(3)
var fx: Double = x1.pow(4.0)
fx += x2.pow(4.0)
fx += x3.pow(4.0)
fx -= 25.0
return fx // a1
}
override fun dimensionOfDomain(): Int {
return 3
}
override fun dimensionOfRange(): Int {
return 1
}
},
object : RealScalarFunction {
override fun evaluate(x: Vector): Double {
val x1: Double = x.get(1)
val x2: Double = x.get(2)
val x3: Double = x.get(3)
var fx: Double = 8.0 * x1.pow(2.0)
fx += 14.0 * x2.pow(2.0)
fx += 7.0 * x3.pow(2.0)
fx -= 56.0
return fx // a2
}
override fun dimensionOfDomain(): Int {
return 3
}
override fun dimensionOfRange(): Int {
return 1
}
})
// construct an SQP solver
val solver = SQPActiveSetOnlyEqualityConstraint1Minimizer(
object : SQPActiveSetOnlyEqualityConstraint1Minimizer.VariationFactory {
override fun newVariation(
f: RealScalarFunction?,
equal: EqualityConstraints?
): SQPASEVariation {
val impl = SQPASEVariation2(100.0, 0.01, 10)
impl.set(f, equal)
return impl
}
},
1e-8, // epsilon, threshold
20) // max number of iterations
// solving an SQP problem
val solution: IterativeSolution
= solver.solve(f, equality_constraints)
val x: Vector = solution.search(
DenseVector(3.0, 1.5, 3.0), // x0
DenseVector(-1.0, -1.0)) // λ0
val fx: Double = f.evaluate(x)
// print out the solution
println("x = " + x)
println("fx = " + fx)
That’s It for this topic. See you soon.
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# Seminars & Colloquia Calendar
DIMACS Theory of Computing Seminar
## Crossing the Logarithmic Barrier for Dynamic Boolean Data Structure Lower Bounds
#### Omri Weinstein, Columbia University
Location: NOTE: DIFFERENT LOCATION -- Core 431
Date & time: Wednesday, 08 November 2017 at 11:00AM - 12:00PM
Abstract: We prove the first super-logarithmic lower bounds on the cell-probe complexity of dynamic *boolean* (a.k.a. decision) data structure problems, a long-standing milestone in data structure lower bounds. We introduce a new technique and use it to prove a (log^{1.5} n) lower bound on the operational time of a wide range of boolean data structure problems, most notably, on the query time of dynamic range counting *over F_2* ([Patrascu07]). Proving an omega(log n) lower bound for this problem was explicitly posed as one of five important open problems in the late Mihai Patrascu's obituary [Thorup13]. This also implies the first omega(log n) lower bound for the classical 2D range counting problem, one of the most fundamental data structure problems in computational geometry and spatial databases.
We derive similar lower bounds for boolean versions of dynamic "polynomial evaluation" and "2D rectangle stabbing", and for the (non-boolean) problems of "range selection" and "range median". Our technical centerpiece is a new way of weakly" simulating dynamic data structures using efficient *one-way* communication protocols with small advantage over random guessing. This simulation involves a surprising excursion to low-degree (Chebychev) polynomials which may be of independent interest, and offers an entirely new algorithmic angle on the "cell sampling" method of Panigrahy et al. [PTW10].
Joint work with Kasper Green Larsen and Huacheng Yu.
## Special Note to All Travelers
Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.
Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.
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Article Contents
Article Contents
# Data-driven evolutions of critical points
• * Corresponding author
S.A. acknowledges the support of the DFG Transregio 109 "Discretization in Geometry and Dynamics". M.F. acknowledges the support of the DFG Project "Identification of Energies from Observation of Evolutions" and the DFG SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization". R.H. acknowledges the support of the DFG-FWF IGDK1754 "Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structures" and the hospitality of TUM during the preparation of this work
• In this paper we are concerned with the learnability of energies from data obtained by observing time evolutions of their critical points starting at random initial equilibria. As a byproduct of our theoretical framework we introduce the novel concept of mean-field limit of critical point evolutions and of their energy balance as a new form of transport. We formulate the energy learning as a variational problem, minimizing the discrepancy of energy competitors from fulfilling the equilibrium condition along any trajectory of critical points originated at random initial equilibria. By $\Gamma$-convergence arguments we prove the convergence of minimal solutions obtained from finite number of observations to the exact energy in a suitable sense. The abstract framework is actually fully constructive and numerically implementable. Hence, the approximation of the energy from a finite number of observations of past evolutions allows one to simulate further evolutions, which are fully data-driven. As we aim at a precise quantitative analysis, and to provide concrete examples of tractable solutions, we present analytic and numerical results on the reconstruction of an elastic energy for a one-dimensional model of thin nonlinear-elastic rod.
Mathematics Subject Classification: Primary: 49N80, 68T05; Secondary: 34A55, 70F17.
Citation:
• Figure 1. Reconstructions $\hat a$ (left) and $\hat a'$ (right) in dotted red and true $a$ and $a'$ in blue. The underlying red circles depict the (adaptive) nodes in $\Lambda$. Below you see the distribution $\tilde \eta$ of available data
Figure 2. Reconstruction of $a'$ with increasing number ${N_e}$ of measurements. The true $a'$ is depicted by the blue curve while the red line depicts the reconstruction $\hat a'$ following (5.23). The red circles at the bottom of the figures depict the position of nodes of the underlying mesh $\Lambda_N$. The histogram below describes the density of the available information encoded by the probability measure $\tilde{\eta}$ for the case $N_e = 60$
Figure 3. Reconstruction of $a'$ for varying $N$ and $D(N) = 4N$ and distribution $\tilde \eta$ for $N = 60$. Graphic as described in Figure 2
Figure 4. Reconstruction of $a'$ for varying $N$ and fixed $D(N) = 300$ and distribution $\tilde \eta$. Graphics as described in Figure 2
Figure 5. Impact of different constants $M_2$ for constraints on $a''$ with $\|a''\| \leq [2,5;20,1000]$ on reconstructs are depicted, showing improved approximations for increasing $M_2$. Graphics as in Figure 2
Figure 6. Impact of changing constraint $M_1$ bounding the values of $a'$, showing a projection-like behaviour of the reconstruction for small $M_1$. Graphics as in Figure 2
Figure 7. Left showing $\hat x_\varepsilon(t)$ (dark dashed) and $x_\varepsilon(t)$ (line bright) with data from Section 5.2.2 at times $t = [0.2$, $0.4$, $0.6$, $0.8$, $1.0]$, right with data from Section 5.2.3
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Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Problems, 21 (2005), 1027-1045. doi: 10.1088/0266-5611/21/3/014. [17] R. Gerstberger and P. Rentrop, Feedforward neural nets as discretization schemes for ODES and DAES, 7th ICCAM 96 Congress., J. Comput. Appl. Math., 82 (1997), 117-128. doi: 10.1016/S0377-0427(97)00085-X. [18] M. C. Grant and S. P. Boyd, Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control, V. Blondel, S. Boyd, and H. Kimura, eds., Lecture Notes in Control and Information Sciences, Springer-Verlag Limited, 371 (2008), 95–110. doi: 10.1007/978-1-84800-155-8_7. [19] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, version 2.1., http://cvxr.com/cvx, Mar. 2014. [20] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2. [21] T. Kirchdoerfer and M. Ortiz, Data-driven computational mechanics, Comput. Methods Appl. Mech. Engrg., 304 (2016), 81-101. doi: 10.1016/j.cma.2016.02.001. [22] F. Lu, M. Maggioni and S. Tang, Learning interaction kernels in heterogeneous systems of agents from multiple trajectories, arXiv: 1910.04832. [23] F. Lu, M. Zhong, S. Tang and M. Maggioni, Nonparametric inference of interaction laws in systems of agents from trajectory data, Proc. Natl. Acad. Sci. USA, 116 (2019), 14424-14433. doi: 10.1073/pnas.1822012116. [24] E. Novak and H. Woźniakowski, Approximation of infinitely differentiable multivariate functions is intractable, J. Complexity, 25 (2009), 398-404. doi: 10.1016/j.jco.2008.11.002. [25] H. Schaeffer, G. Tran and R. 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In-Depth
### Text Processing, Type Definition, I/O and Visualization in F#
What you can do with most programming languages can be accomplished in F#'s functional programming paradigm. Here's how to handle some simple operations, which might look familiar to you already.
In prior articles (here and here) I described some interesting and useful features that the functional programming paradigm -- and specifically F# -- can offer when developing computer applications. I'd like to delve into a bit of detail on some useful examples this time. Specifically, let's look at text processing, type definition, input/ output and visualization.
Text Processing
Texts in computer science is represented in strings. Simply, a string is defined as a finite sequence of characters belonging to an alphabet. All text processing tools or operations must rely on strings as their basic processing unit, and one of these tools is the regular expression. A regular expression is a pattern or expression that implicitly describes a set of strings that belong to a regular language; thus, regular expressions represent regular languages. This powerful string processing tool is part of the .NET platform and it's also part of the F# language.
To be able to use regular expression in F# one first needs to open or import the System.Text.RegularExpressions namespace. Here's an example of a regular expression that recognizes any word in lower case letters being created:
open System
open System.Text.RegularExpressions
[<EntryPoint>]
let main argv =
let reg = new Regex("[a-z]+")
printf "%A" (reg.IsMatch("jordan"))
0 // return an integer exit code
The IsMatch method returns true if any substring of the string supplied as an argument matches the regular expression pattern ([a-z]+). Hence, after executing that code the output will be true. In case the string passed as an argument changes from "jordan" to "23" then no matching can be found and the return value will be false.
Text transcription is an operation that can be easily achieved using regular expressions. This code illustrates a transcrip function that takes arguments str (the string to be transcript) and pattern (the pattern to be matched and replaced):
let transcrip str pattern = (new Regex(pattern)).Replace(str, "23")
printf "%A" (transcrip "Hi Mr. Jordan!" "Jordan")
After executing this code the result is shown in Figure 1.
Regular expressions can also be use to separate, split or divide text under a certain criteria. One may need to split text by the occurrence of numbers, specific letters, etc. In such cases a regular expression and the execution of the Split method, which returns an array of strings, could serve as a solution.
In this code, the regular expression splitreg matches the empty string or any digit:
let splitreg = new Regex(" |[0-9]")
let splitted = splitreg.Split("1 2 3 Mr.Jazz!")
printfn "%A" splitted
The Split method divides the input string every time it finds an empty string or a digit. Since there are three digits and three empty strings from left to right before reaching substring "Mr.Jazz!", and they are all consecutives, the output results in what you see in Figure 2.
After exploring the possibilities of text processing in F# let us dive into the opportunities for defining custom types.
Type Definition
F# provides the possibility of defining custom types by using the type keyword followed by the type name, an equal sign and the definition itself. A type definition can be used for defining aliases for existing types. This type of definition includes tuples and could be useful and meaningful as shown in the next example where a type person consisting of a 2-tuple string is created and later used as a type constraint:
type person = string * string
let f (p : person) =
printfn "%A" p
The syntax x1 * x2* … * xn indicates an n-tuple.
Another type definable in F# is the record type; similar to a tuple because it groups several types into a single type but dissimilar in the sense that names must be provided for each field. The next example illustrates the creation of a record named album and its later use.
type album = { year: int list; artist: string list}
let jazz = { new album with year = [1998; 1999; 2000] and artist = ["Sting"; "Duke Ellington"; "Chris Botti"] }
Records and aliases are similar to structs and classes in C#. Apart from these two there is one last type that one may define in F#: the union or sum type. Union types represent a manner for uniting types with different structures. Their definition consists of a set of constructors separated by a vertical bars. A constructor is composed of a name (in capital letters, unique within a type), optionally the of keyword and finally the types that form the constructor separated by asterisks, in case there is more than one type. This code illustrates how to define a Binary Tree union type:
type BinaryTree =
| Node of int * BinaryTree * BinaryTree
| Empty
Logically, a binary tree is either an Empty tree or a Node with an integer value and two children which are also binary trees. Considering the binary tree in Figure 3.
A declaration for this structure is here:
let tree = Node(2, Node(3, Node(4,Empty,Empty), Empty), Node(5,Empty,Empty))
A function defining an in-order printing of node values in the tree is presented in the next lines of code:
let rec printInOrder tree =
match tree with
| Node (data,left,right) -> printInOrder left
printfn "Node %d" data
printInOrder right
| Empty -> ()
printInOrder tree
The output after executing the previous function is shown in Figure 4.
An in-order path in a binary tree recursevely visits the left node, then the current node and finally the right node thus for the previous tree the in-order path is 4, 3, 2, 5.
Input / Output
I/O features are not highly related to the functional programming paradigm. But F# is not purely functional – rather, it's a hybrid and accordingly it incorporates features from the imperative programming paradigm. Several I/ O functions provided in the language corroborate this statement. The printfn function used in this and prior articles represents probably the most common alternative for printing various types in F#; printfn is a simple function that provides simple formatting. Here are its differentiating format indicators:
• %d print an int
• %s print a string
• %A print any type using the built in printer
• %f print a float
• %g print a float in scientific notation
• %O print any type using the ToString() method
In the next example (with result shown in Figure 5), a string, an int and a float are printed using the printfn function:
printf "String: %s int: %d, float: %f\n" "NBA" 5 2.3
Reading from and writing to files can be achieved in F# thanks to the functionality offered by the .NET platform through the System.IO namespace. The function shown in the following code loads the file indicated in the filename parameter then reads the file and yields each line until the end of the stream, all of this inside a sequence expression so the final result is a sequence with lines from the file.
let readfile filename =
seq { use stream = File.OpenRead filename
printfn "%A" lines
The use binding is equivalent to the let binding except for the fact that the first disposes the object once the enumeration over the sequence has been completed, closing the file at the end.
Visualization
Providing visualization tools, libraries are not really the strong suit for functional programming languages. Then again, F# is not purely functional and is a .NET language, so it can deal with Windows Forms and many graphics libraries like DirectX, OpenGL, etc. To start building a GUI application in F#, one first needs to open namespaces System.Windows.Forms and System.Drawing, adding their reference to the project if necessary, and then creating the form (result is Figure 6):
let form = new Form(BackColor = Color.LightBlue, Visible=true, Text="My F# form")
Application.Run(form)
Once the form is created, this code adds a picture box as a new control (see Figure 7):
let picturebox = new PictureBox(Width = 200, Height = 200, BackColor = Color.LightGray)
form.Controls.Add(picturebox)
Finally, using a solid brush a black ellipse is drawn inside the picture box and an almost effortless painting on F# is completed (see Figure 8):
let brush = new SolidBrush(Color.Black)
(fun e ->
e.Graphics.FillEllipse(brush, 50, 50, 50, 50))
In this article I described the capabilities provided by F# to handle text processing, type definition, I/O operations and visualization through Windows Forms. Now it's time for reader to keep exploring the fascinating, elegant world of functional programming.
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# MDX Sprite.Draw with SrcRect error
This topic is 3791 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi, I'm currently trying to convert a 2D game from DirectDraw to Direct3D (that i had just converted from GDI to directdraw :(). I'm using VB.Net and the newest version of the DX9 SDK. I'm trying to make Sprite.Draw work, but it crashes sometimes when i send a sourceRectangle with X or Y different from 0. The first thing that i verified is that X+Width of sourceRectangle doesn't exceed the image width. I verified the same thing with the height. The first DrawSprite() that crashed had SrcRect(X = 288, Y=96, Width = 48, Height = 48) Texture(Width = 512, Height = 512) heres the code that i am using to draw.
Public Sub DrawSprite(ByVal pDestRect As Rectangle, ByRef pTex As Texture, ByVal pSrcRect As Rectangle)
m_Sprite.Begin()
'If i remove this, i'll Crash !
If pSrcRect.X > 0 Then pSrcRect.X = 0
If pSrcRect.Y > 0 Then pSrcRect.Y = 0
m_Sprite.Draw(pTex, pSrcRect, New Microsoft.DirectX.Vector2(1.0F, 1.0F), New Microsoft.DirectX.Vector2(0, 0), 0.0F, New Microsoft.DirectX.Vector2(pDestRect.X, pDestRect.Y), Color.White)
m_Sprite.End()
End Sub
What's funny about this is that with the patch i applied to set the X and Y to 0, it doesn't crash ! The error that i get from DX is D3D_INVALIDCALL on m_Sprite.Draw I am clueless about what might cause the crash and after 2 days of searching google, and finding out that most of the good google finds came from this forum, i decided to ask my question here !
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Wow !
i have finally resolved the mystery !
When using a VB.Net Rectangle as a source, i must set the rectangle's width to it's Right, and it's Height to it's Bottom !
Big thanks to MJP for his C++ sprite sample posted in another post which gave me the idea to try this solution !
Here's the working code, if it can help someone in the future !
Public Sub DrawSprite(ByVal pDestRect As Rectangle, ByRef pTex As Texture, ByVal pSrcRect As Rectangle) m_Sprite.Begin() 'Build an adjusted rectangle where the width is set to the right 'and the height set to bottom ! this is because DX expects a rectangle 'width a right and a bottom, NOT a width and height ! Dim adjustedSrcRect As New Rectangle(pSrcRect.X, _ pSrcRect.Y, _ pSrcRect.Right, _ pSrcRect.Bottom) m_Sprite.Draw(pTex, adjustedSrcRect, New Microsoft.DirectX.Vector2(1.0F, 1.0F), New Microsoft.DirectX.Vector2(0, 0), 0.0F, New Microsoft.DirectX.Vector2(pDestRect.X, pDestRect.Y), Color.White) m_Sprite.End() End Sub
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Just so you know, but you should only being calling Begin/End once per frame in order to batch your sprites, all the actual drawing is done in End. Calling them all the time will create a very slow program, and it defeats the purpose of using the sprite class.
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thanks for the tip !
it's probably going to improve performance, as i m drawing about 500 sprites per frame !
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# Page:AbrahamMinkowski1.djvu/14
The first two terms are contributions of the aether in the sense of Lorentz's theory, the two latter ones are to be seen as contributions of polarized matter to the electromagnetic momentum density.
We begin to calculate the energy current. With respect to (32) and (31), the expression
(34) $\mathfrak{W=[DB]-[EH]=[EM]+[PH]+[PM]}$
holds in Lorentz's theory for the vector $\mathfrak{W}$ introduced at the end of § 5. According to (31) and (30), we have
$\begin{array}{l} \mathfrak{E'-[qB]=E-[qM],}\\ \mathfrak{H'+[qD]=H+[qP],}\end{array}$
so that equation (21) assumes the form
$\frac{\mathfrak{S}}{c}\mathfrak{=\left[E-[qM],\ H+[wP]\right]-q(qW)}$
Now, since it is to be set according to (34)
$\mathfrak{q(qW)=\left[[qE][qM]\right]+\left[[qP][qH]\right]+\left[[qP][qM]\right]}$
then eventually if follows as the value of the energy current
(35) $\frac{\mathfrak{S}}{c}\mathfrak{=[EH]+\left[E'[qP]\right]+\left[H'[qM]\right]}$
The first term can be interpreted as a contribution of the aether, the second one as the contribution of the electrically polarized matter at the energy current, as G. Nordström[1] has shown in a recently published paper, which is remarkable also in other respects; the third term which is added at the motion of magnetically polarized matter, corresponds in such a way to the second one, as it is required by the symmetry of electric and magnetic vectors assumed at this place.
§ 9. Theory of H. Minkowski.
In this theory, the following relations between the electromagnetic vectors hold
(36) $\begin{cases} \mathfrak{D}=\epsilon\mathfrak{E}'-[\mathfrak{qH}],\\ \mathfrak{B}=\mu\mathfrak{H}'+[\mathfrak{qE}];\end{cases}$
(37) $\begin{cases} \mathfrak{E'}=\mathfrak{E}+[\mathfrak{qB}],\\ \mathfrak{H}'=\mathfrak{H}-[\mathfrak{qD}].\end{cases}$
Also here, besides the vector pairs contained in the main equations, a new vector pair is added, which mediates the relation between them.
1. G. Nordström, Die Energiegleichung für das elektromagnetische Feld bewegter Körper (Dissertation, Helsingfors 1908).
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## Weibull Distribution
### Overview
The Weibull distribution is a two-parameter family of curves. This distribution is named for Waloddi Weibull, who offered it as an appropriate analytical tool for modeling the breaking strength of materials. Current usage also includes reliability and lifetime modeling. The Weibull distribution is more flexible than the exponential distribution for these purposes, because the exponential distribution has a constant hazard function.
Statistics and Machine Learning Toolbox™ offers several ways to work with the Weibull distribution.
• Create a probability distribution object WeibullDistribution by fitting a probability distribution to sample data (fitdist) or by specifying parameter values (makedist). Then, use object functions to evaluate the distribution, generate random numbers, and so on.
• Work with the Weibull distribution interactively by using the Distribution Fitter app. You can export an object from the app and use the object functions.
• Use distribution-specific functions (wblcdf, wblpdf, wblinv, wbllike, wblstat, wblfit, wblrnd, wblplot) with specified distribution parameters. The distribution-specific functions can accept parameters of multiple Weibull distributions.
• Use generic distribution functions (cdf, icdf, pdf, random) with a specified distribution name ('Weibull') and parameters.
### Parameters
The Weibull distribution uses these parameters.
ParameterDescriptionSupport
a Scalea > 0
bShapeb > 0
The standard Weibull distribution has unit scale.
#### Parameter Estimation
The likelihood function is the probability density function (pdf) viewed as a function of the parameters. The maximum likelihood estimates (MLEs) are the parameter estimates that maximize the likelihood function for fixed values of x. The maximum likelihood estimators of a and b for the Weibull distribution are the solution of the simultaneous equations
$\begin{array}{l}\stackrel{^}{a}={\left[\left(\frac{1}{n}\right)\sum _{i=1}^{n}{x}_{i}^{\stackrel{^}{b}}\right]}^{\frac{1}{\stackrel{^}{b}}}\\ \stackrel{^}{b}=\frac{n}{\left(\frac{1}{\stackrel{^}{a}}\right)\sum _{i=1}^{n}{x}_{i}^{\stackrel{^}{b}}\mathrm{log}{x}_{i}-\sum _{i=1}^{n}\mathrm{log}{x}_{i}}.\end{array}$
â and $\stackrel{^}{b}$ are unbiased estimators of the parameters a and b.
To fit the Weibull distribution to data and find parameter estimates, use wblfit, fitdist, or mle. Unlike wblfit and mle, which return parameter estimates, fitdist returns the fitted probability distribution object WeibullDistribution. The object properties a and b store the parameter estimates.
For an example, see Fit Weibull Distribution to Data and Estimate Parameters.
### Probability Density Function
The pdf of the Weibull distribution is
$f\left(x|a,b\right)=\frac{b}{a}{\left(\frac{x}{a}\right)}^{b-1}{e}^{-{\left(x/a\right)}^{b}}.$
For an example, see Compute Weibull Distribution pdf.
### Cumulative Distribution Function
The cumulative distribution function (cdf) of the Weibull distribution is
$p=F\left(x|a,b\right)={\int }_{0}^{x}b{a}^{-b}{t}^{b-1}{e}^{-{\left(\frac{t}{a}\right)}^{b}}dt=1-{e}^{-{\left(\frac{x}{a}\right)}^{b}}.$
The result p is the probability that a single observation from a Weibull distribution with parameters a and b falls in the interval [0 x].
For an example, see Compute Weibull Distribution cdf.
### Inverse Cumulative Distribution Function
The inverse cdf of the Weibull distribution is
$x={F}^{-1}\left(p|a,b\right)=-a{\left[\mathrm{ln}\left(1-p\right)\right]}^{1/b}.$
The result x is the value where an observation from a Weibull distribution with parameters a and b falls in the range [0 x] with probability p.
### Hazard Function
The hazard function (instantaneous failure rate) is the ratio of the pdf and the complement of the cdf. If f(t) and F(t) are the pdf and cdf of a distribution, then the hazard rate is $h\left(t\right)=\frac{f\left(t\right)}{1-F\left(t\right)}$. Substituting the pdf and cdf of the exponential distribution for f(t) and F(t) above yields the function $\frac{b}{{a}^{b}}{x}^{b-1}$.
For an example, see Compare Exponential and Weibull Distribution Hazard Functions.
### Examples
#### Fit Weibull Distribution to Data and Estimate Parameters
Simulate the tensile strength data of a thin filament using the Weibull distribution with the scale parameter value 0.5 and the shape parameter value 2.
rng('default'); % For reproducibility
strength = wblrnd(0.5,2,100,1); % Simulated strengths
Compute the MLEs and confidence intervals for the Weibull distribution parameters.
[param,ci] = wblfit(strength)
param = 1×2
0.4768 1.9622
ci = 2×2
0.4291 1.6821
0.5298 2.2890
The estimated scale parameter is 0.4768, with the 95% confidence interval (0.4291,0.5298).
The estimated shape parameter is 1.9622, with the 95% confidence interval (1.6821,2.2890).
The default confidence interval for each parameter contains the true value.
#### Compute Weibull Distribution pdf
Compute the pdf of the Weibull distribution with the scale parameter value 3 and the shape parameter value 2.
x = 0:0.1:10;
y = wblpdf(x,3,2);
Plot the pdf.
figure;
plot(x,y)
xlabel('Observation')
ylabel('Probability Density')
#### Compute Weibull Distribution cdf
Compute the cdf of the Weibull distribution with the scale parameter value 3 and the shape parameter value 2.
x = 0:0.1:10;
y = wblcdf(x,3,2);
Plot the cdf.
figure;
plot(x,y)
xlabel('Observation')
ylabel('Cumulative Probability')
#### Compare Exponential and Weibull Distribution Hazard Functions
The exponential distribution has a constant hazard function, which is not generally the case for the Weibull distribution. In this example, the Weibull hazard rate increases with age (a reasonable assumption).
Compute the hazard function for the Weibull distribution with the scale parameter value 1 and the shape parameter value 2.
t = 0:0.1:4.5;
h1 = wblpdf(t,1,2)./(1-wblcdf(t,1,2));
Compute the mean of the Weibull distribution with scale parameter value 1 and shape parameter value 2.
mu = wblstat(1,2)
mu = 0.8862
Compute the hazard function for the exponential distribution with mean mu.
h2 = exppdf(t,mu)./(1-expcdf(t,mu));
Plot both hazard functions on the same axis.
plot(t,h1,'-',t,h2,'--')
xlabel('Observation')
ylabel('Hazard Rate')
legend('Weibull','Exponential','location','northwest')
#### Estimate Parameters of Three-Parameter Weibull Distribution
Statistics and Machine Learning Toolbox™ uses a two-parameter Weibull distribution with a scale parameter $a$ and a shape parameter $b$. The Weibull distribution can take one more parameter, a location parameter $c$. The pdf becomes
where $a$ and $b$ are positive values, and $c$ is a real value.
Generate sample data of size 1000 from a three-parameter Weibull distribution with the scale parameter 1, shape parameter 1, and location parameter 10.
rng('default') % For reproducibility
data = wblrnd(1,1,[1000,1]) + 10;
Define a probability density function for a three-parameter Weibull distribution.
custompdf = @(x,a,b,c) (x>c).*(b/a).*(((x-c)/a).^(b-1)).*exp(-((x-c)/a).^b);
mle estimates the parameters from data. If mle does not converge with default statistics options, modify them by using the name-value pair argument 'Options'.
Create a statistics options structure opt by using the function statset.
opt = statset('MaxIter',1e5,'MaxFunEvals',1e5,'FunValCheck','off');
The option opt includes the following options:
• 'MaxIter',1e5 — Increase the maximum number of iterations to 1e5.
• 'MaxFunEvals',1e5 — Increase the maximum number of object function evaluations to 1e5.
• 'FunValCheck','off' — Turn off checking for invalid object function values.
For a distribution with a region that has zero probability density, mle might try some parameters that have zero density, and it will fail to estimate parameters. To avoid this problem, you can turn off the option that checks for invalid function values by using 'FunValCheck','off'.
Use mle to estimate the parameters. Note that the Weibull probability density function is positive only for $x>c$. This constraint also implies that a location parameter $c$ is smaller than the minimum of the sample data. Include the lower and upper bounds of parameters by using the name-value pair arguments 'LowerBound' and 'UpperBound', respectively.
params = mle(data,'pdf',custompdf,'start',[5 5 5],...
'Options',opt,'LowerBound',[0 0 -Inf],'UpperBound',[Inf Inf min(data)])
params = 1×3
1.0258 1.0618 10.0004
If the scale parameter $b$ is smaller than 1, the probability density of the Weibull distribution approaches infinity as $x$ goes to $c$, where $c$ is the location parameter. The maximum of the likelihood function is infinite. mle may find satisfactory estimates in some cases, but the global maximum is degenerate when $b<1$.
### Related Distributions
• Exponential Distribution — The exponential distribution is a one-parameter continuous distribution that has parameter μ (mean). This distribution is also used for lifetime modeling. When b = 1, the Weibull distribution is equal to the exponential distribution with mean μ = a.
• Extreme Value Distribution — The extreme value distribution is a two-parameter continuous distribution with parameters µ (location) and σ (scale). If X has a Weibull distribution with parameters a and b, then log X has an extreme value distribution with parameters µ = log a and σ = 1/b. This relationship is used to fit data to a Weibull distribution.
• Rayleigh Distribution — The Rayleigh distribution is a one-parameter continuous distribution that has parameter b (scale). If A and B are the parameters of the Weibull distribution, then the Rayleigh distribution with parameter b is equivalent to the Weibull distribution with parameters$A=\sqrt{2}b$ and B = 2.
• Three-Parameter Weibull Distribution — The three-parameter Weibull distribution adds a location parameter that is zero in the two-parameter case. If X has a two-parameter Weibull distribution, then Y = X + c has a three-parameter Weibull distribution with the added location parameter c.
For an example, see Estimate Parameters of Three-Parameter Weibull Distribution.
## References
[1] Crowder, Martin J., ed. Statistical Analysis of Reliability Data. Reprinted. London: Chapman & Hall, 1995.
[2] Devroye, Luc. Non-Uniform Random Variate Generation. New York, NY: Springer New York, 1986. https://doi.org/10.1007/978-1-4613-8643-8
[3] Evans, Merran, Nicholas Hastings, and Brian Peacock. Statistical Distributions. 2nd ed. New York: J. Wiley, 1993.
[4] Lawless, Jerald F. Statistical Models and Methods for Lifetime Data. 2nd ed. Wiley Series in Probability and Statistics. Hoboken, N.J: Wiley-Interscience, 2003.
[5] Meeker, William Q., and Luis A. Escobar. Statistical Methods for Reliability Data. Wiley Series in Probability and Statistics. Applied Probability and Statistics Section. New York: Wiley, 1998.
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## Otros recursos de Alves, J
1. #### Observation of the B[superscript +] → D[superscript ∗] − K[superscript +] π[superscript +] decay (�)
The B[superscript +]→D[superscript *-]K[superscript +]π[superscript +] decay potentially provides an ...
21-mar-2018
2. #### Jet properties in PbPb and pp collisions at $\sqrt{s_\mathrm{NN}} =$ 5.02 TeV (�)
Modifications of the properties of jets in PbPb collisions, relative to those in pp collisions, are ...
20-mar-2018
3. #### Measurements of the $\mathrm {p}\mathrm {p}\rightarrow \mathrm{Z}\mathrm{Z}$ production cross section and the $\mathrm{Z}\rightarrow 4\ell$ branching fraction, and constraints on anomalous triple gauge couplings at $\sqrt{s} = 13\,\text {TeV}$ (�)
International audience
16-mar-2018
4. #### SSearch for gauge-mediated supersymmetry in events with at least one photon and missing transverse momentum in pp collisions at $\sqrt{s} =$ 13 TeV (�)
International audience
16-mar-2018
5. #### Search for ZZ resonances in the 2$\ell$2$\nu$ final state in proton-proton collisions at 13 TeV (�)
International audience
16-mar-2018
6. #### Search for a heavy resonance decaying to a pair of vector bosons in the lepton plus merged jet final state at $\sqrt{s} =$ 13 TeV (�)
A search for a new heavy particle decaying to a pair of vector bosons (WW or WZ) is presented using ...
16-mar-2018
7. #### Phylogenetic classification of the world's tropical forests (�)
Knowledge about the biogeographic affinities of the world's tropical forests helps to better underst ...
(text) 10-mar-2018
8. #### Observation of Correlated Azimuthal Anisotropy Fourier Harmonics in $pp$ and $p+Pb$ Collisions at the LHC (�)
International audience
20-mar-2018
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# 3D & 2D Software Information (Check here)
127 replies to this topic
### #81Anonymous Poster_Anonymous Poster_* Guests
Posted 22 August 2004 - 01:15 AM
I have a question, I got a magazine(3d world) a few weeks ago and on the cd it came with it has 2 3d software on it. One is Wirefusion 3.2 and the other is WF-3D 2.0. They say there worth alot of money and they look like good programs but does anyone ever use them or hear of them?
### #82Anonymous Poster_Anonymous Poster_* Guests
Posted 24 August 2004 - 07:28 AM
Why isnt this a sticky already?....
### #83Veovis Members
Posted 02 September 2004 - 08:49 AM
I'm hoping this thread will get Stickied like my other one has. Thank you.
### #84Gpoliece Members
Posted 02 September 2004 - 04:10 PM
http://www.equinox3d.com/
http://www.digitalscores.com/JCSoftware/
http://www.kpovmodeler.org/en/info/screenshots.html
http://www21.ocn.ne.jp/~mizno/main_e.html
Gpoliece
Posted 05 September 2004 - 09:56 PM
Larger list of 3D software
### #86Anonymous Poster_Anonymous Poster_* Guests
Posted 06 September 2004 - 12:20 PM
I like mine better. It's up-to-date and categorized. :D
Thanks though, I'll go through that eventually and pick out the ones that are still valid for use.
### #87Veovis Members
Posted 06 September 2004 - 12:20 PM
^ that's me.
### #88Anonymous Poster_Anonymous Poster_* Guests
Posted 21 September 2004 - 12:15 AM
And here is another free tile/map editor:
Tile Studio: http://tilestudio.sf.net/
### #89Anonymous Poster_Anonymous Poster_* Guests
Posted 22 September 2004 - 09:41 AM
You guys do know that there's a free edition of Maya, right? The edjubicationul one...
### #90sBibi Members
Posted 25 September 2004 - 09:33 AM
you might be interested to add Modo, a new 3D modeler that's really really nice, mixes lots of cool concepts from other modeling packages (I see it as a lightwave++++), has a really easy to use UI imo, handle ngons subds (not like lightwave), and comes with some nice demo models, you can check their site for more infos:
http://www.luxology.com/
(it's 695$) ### #91Panzooka Members Posted 14 October 2004 - 02:10 PM hey cmon, dont left SILO3d out! its the best 3d modeler, simple, fast, powerful, and customizable. its just over$100 us
www.nevercenter.com
### #92darkandevil Members
Posted 24 October 2004 - 06:43 AM
can someone do a breakdown of the free/open source 3d modelers for us poor people then?
### #93Veovis Members
Posted 29 October 2004 - 03:50 AM
Hello everyone. Sorry that I've been away so long, had a shift in priorities.
But, I haven't abandoned GameDev. I'll see if I can get around to doing the updates on my 2 threads soon.
Thanks as always.
### #94tiles Members
Posted 11 November 2004 - 02:03 AM
What about adding PhotoFiltre to the Free 2D Software list. Nice little software :) :
http://www.photofiltre.com/
Damn tags. Every forum uses other tags .
Next try :
http://www.photofiltre.com/
allright, html ...
PhotoFiltre
### #95JTippetts Moderators
Posted 09 January 2006 - 11:38 AM
bump
### #96Veovis Members
Posted 09 January 2006 - 12:14 PM
An update to the list has been done with all of the above-mentioned software.
Thanks!
### #97Veovis Members
Posted 10 January 2006 - 04:55 AM
And, I fixed a bunch of links which weren't going directly to the product page anymore. Also, a few products have had a change in price. LightWave 3D and Houdini Escape have both dropped by an entire price category!
### #98methinks Members
Posted 18 January 2006 - 06:08 AM
Maybe we could add the MakeHuman project (http://www.dedalo-3d.com/) to the list... It's like an open-source version of poser, but not quite as powerful (yet).
And while it might not be suited for in-game models, it could still be usefull for game art, reference, etc.
Oh, and how about adding alias sketchbook pro to the list (2d, \$100+)
### #99Veovis Members
Posted 02 February 2006 - 01:10 PM
Ok, updated with those two. Thanks.
and they do mention that MakeHuman can be used for game models. And if any software is being presented as such, I'll include it.
### #100Mattman Members
Posted 02 February 2006 - 03:45 PM
Paint .NET - Similar to MS's Paint, but improved...good for quick, basic 2D changes. Supports JPG, GIF, BMP, PNG, and a couple other formats. Oh, and it's free.
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# Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
Corollary 3.1.6.5. Let $f: X \rightarrow Y$ be a morphism of simplicial sets, and let $Z$ be a Kan complex. If $f$ is a weak homotopy equivalence, then composition with $f$ induces a homotopy equivalence $\operatorname{Fun}( Y, Z) \rightarrow \operatorname{Fun}(X,Z)$.
Proof. Using Corollary 3.1.6.2, we can choose an anodyne morphism $g: Y \hookrightarrow Y'$, where $Y'$ is a Kan complex. Using Proposition 3.1.6.1, we can factor $g \circ f$ as a composition $X \xrightarrow {g'} X' \xrightarrow {f'} Y'$, where $g'$ is anodyne and $f'$ is a Kan fibration. We then have a commutative diagram
$\xymatrix@C =50pt@R=50pt{ X \ar [r]^-{f} \ar [d]^{g'} & Y \ar [d]^{g} \\ X' \ar [r]^-{ f' } & Y', }$
where $f'$ is a Kan fibration between Kan complexes and the vertical maps are anodyne, and therefore weak homotopy equivalences. Using the two-out-of-three property, we deduce that $f'$ is also a weak homotopy equivalence (Remark 3.1.5.15). It follows that $f'$ is a homotopy equivalence (Proposition 3.1.5.11). We then obtain a commutative diagram of Kan complexes
$\xymatrix@C =50pt@R=50pt{ \operatorname{Fun}(X,Z) & \ar [l] \operatorname{Fun}(Y,Z) \\ \operatorname{Fun}(X', Z) \ar [u] & \operatorname{Fun}(Y', Z) \ar [l] \ar [u] }$
where the lower horizontal map is a homotopy equivalence, and the vertical maps are trivial Kan fibrations (Corollary 3.1.3.6). In particular, the vertical maps are homotopy equivalences (Proposition 3.1.5.9), so the two-out-of-three property guarantees that the upper horizontal map is also a homotopy equivalence (Remark 3.1.5.7). $\square$
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# Julian Barbour on does time exist
by julian
Tags: barbour, exist, julian, time
Astronomy
PF Gold
P: 23,110
Quote by sshai45 So does that mean there would be an absolute, universal time and simultaneity, and so Einstein was "wrong" in some sense? How does "only 'now' exists" jibe with "'now' depends on the observer"? How do the non-reality of the block universe and the relativity of simultaneity play with each other?
That's an intriguing question! I don't think introducing the (M,ω) picture and the corresponding state-dependent time flow on the algebra indicates that GR is wrong. But I want to take more time to answer.
Here's part of a short answer I gave to your question in post #144, with a clarifying addition in red:
Quote by marcus Sshai, I will get back to your comment, time permitting. I think Einstein is still right. We still have observer time. Each observer has a different time (as A.E. said) and it is interesting to compare them. But also now we have a *state-dependent* time as well. It depends not on a particular observer but on the function omega that summarizes what we think we know (with various degrees of confidence) about the world. ...a new way to picture the world, as (M,ω) where M is a star algebra (observables) and omega (state) is a function from M to the complex numbers[giving correlations between observables]. Ordinary QFT (quantum field theory) has already been put in star algebra form. And there seems no reason that the dynamic geometry of GR should not also be put into that same form---thus combining the content of QM and GR, combining geometry with matter in a background independent or general covariant way. The (M, ω) is suitable for both. So this (M, ω) business is quite an interesting development... However in any case it does not say that "Einstein was wrong". It brings into existence yet ANOTHER version of time, which depends on the state we specify rather than on any particular observer. ...And it already seems interesting to COMPARE this time with that of a given observer because it has been shown that the ratio of rates of time-passage can be physically meaningful ...It also seems to be good for other things where you can't use observer-time...
Basically what we are discussing in this thread are theorists' response to what is called the problem of time in GR and also in quantum GR, where the problem is broader and more formidable. Here is the best short statement I know of the problem.
It is from page 4 of Chapter 1 of the 2009 book Approaches to Quantum Gravity, D. Oriti ed. published by Cambridge University Press ( http://arxiv.org/abs/gr-qc/0604045 )
==quote Chapter 1 of Approaches to Quantum Gravity==
... In special relativity, this notion of time is weakened. Clocks do not measure a universal time variable, but only the proper time elapsed along inertial trajectories. If we fix a Lorentz frame, nevertheless, we can still describe all physical phenomena in terms of evolution equations in the independent variable x0, even though this description hides the covariance of the system.
In general relativity, when we describe the dynamics of the gravitational field (not to be confused with the dynamics of matter in a given gravitational field), there is no external time variable that can play the role of observable independent evolution variable. The field equations are written in terms of an evolution parameter, which is the time coordinate x0, but this coordinate, does not correspond to anything directly observable. The proper time τ along spacetime trajectories cannot be used as an independent variable either, as τ is a complicated non-local function of the gravitational field itself. Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable.
This weakening of the notion of time in classical GR is rarely emphasized: After all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline.
But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in quantum gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory.
==endquote==
So the problem is on two levels, classical and quantum. Already at the classical level
there is no observable independent time variable that can be used to describe the evolution of a (general) relativistic system.
And at the quantum level the problem is even more severe, since one cannot realistically assume some fixed metric solution--i.e. a geometric "trajectory".
There's more to say, I'll try to get back to this later. The outstanding thing to notice about the (M,ω) format (for dynamic QG geometry and simultaneously for matter QFT) is that it DOES have an independent time variable that can be used to describe the dynamical evolution of geometry+matter. The point of the quote above is that any observer's time is NOT adequate since the observer's time depends on how the geometry evolves!
When you want to describe the dynamical evolution of a system you need a time variable which is not totally at the mercy of how the system happens to evolve. So observer-time is no good.
This is why the (M,ω) formalism has come up in the context of trying to devise a fully general relativistic treatment of thermodynamics and statistical mechanics. Imagine trying to do statistical mechanics with no possibility of a physically meaningful preferred time variable. That's why it has always been done on a fixed space-time, not in a fully general covariant way. I hope to get back to this. It really interests me.
P: 175 I've been going through Jeffery Bub's paper that Marcus pointed to in his #141 post (http://arxiv.org/abs/1211.3062). It's about stuff that I'm not at all familiar with: statistics and simplex theory. My comprehension is strictly limited, to put it mildly. Perhaps there is someone here who can straighten my thinking out. It seems to me that Bub is trying to describing mathematically a world where the speed of information is limited and the observations that guide description are of a statistical nature --- as well as being causal, because they affect this world. He seems to show that observation inevitably generates loss of information, i.e. uncertainty, as is the case in quantum mechanics. I think he also establishes that entanglement is inevitably associated with such loss of information. He says that all this is an expected consequence of “probabilistic correlations, (and) the structure of information”; all that is needed, I suppose, to formulate a predictive description in a holistically statistical world. So are the mysteries of quantum mechanics forced on us because causality is a sort of statistical correlation? But what he doesn’t clarify is for me the central mystery: the magnitude of what sets the whole shebang up, namely Planck’s constant. Perhaps his interesting approach will eventually lead to our understanding why h is of order 10^-34 J.s.in our real world? I do hope so.
P: 63
Quote by marcus This is why the (M,ω) formalism has come up in the context of trying to devise a fully general relativistic treatment of thermodynamics and statistical mechanics. Imagine trying to do statistical mechanics with no possibility of a physically meaningful preferred time variable. That's why it has always been done on a fixed space-time, not in a fully general covariant way. I hope to get back to this. It really interests me.
By a "preferred" time variable, does this mean that this time forms an "absolute time" in some sense (i.e. a "universal clock" that is not tied to a particular observer, sort of like in "old" pre-Einstein physics), or what?
Astronomy
PF Gold
P: 23,110
Paulibus, bravo for tackling Bub! I'll be interested to know what you make of it. So far what I can get is for the most part merely what delighted me so much in the introduction (and I quoted.) But I'm trying to do too many things at once (family Christmas letters, and our community chorus has given several performances of Moz. mass in c-minor!, not to mention physics-watching)
Quote by sshai45 By a "preferred" time variable, does this mean that this time forms an "absolute time" in some sense (i.e. a "universal clock" that is not tied to a particular observer, sort of like in "old" pre-Einstein physics), or what?
Maybe we need a new word to use instead of "preferred". I think it's different from going back to "old" pre-Einstein time ideas. In the old days there was an absolute time and all observers clocks were supposed to follow it and agree on simultaneity and stuff.
Now we still have all the observer-times and a democracy of disagreeing clocks, but IN ADDITION we have one more clock, which we can COMPARE the various observer clocks with. The ratio of rates can even be physically meaningful, correspond to something measurable.
This one additional clock is distinguished by the fact that it is not observer-dependent it is, instead, state-dependent. It depends on what we think we know about the world--on our degree of (un)certainty about correlations amongst observations---what we posit to be the case, with varying degrees of confidence. If you like, picture the state as a density matrix defining a function on the observables.
Each observer still gets to keep his own individual clock and nobody is presumed to be RIGHT, but there is this one additional clock, which has one particular advantage: the world can be analyzed as a fully relativistic system evolving according to THIS time.
Which is something you CAN'T do with some particular observer time, because the observer's history itself depends on how the system evolves---so there is a kind of logical circularity. The observer's time is not truly an independent variable.
So as I see it, this is not going back to the old picture, but instead is adding one more disagreeing clock to the general temporal madhouse and anarchy---which however has a nifty feature that you can do a fully general relativistic statistical mechanics and thermodynamics using IT as the independent time variable---something you cannot do with any other clock as far as I know. So it is subtly different from going back to the old picture and it does not imply that "Einstein was wrong". Or so I think.
Maybe instead of preferred we could say "distinguished" time variable. Distinguished by the fact that it can be used as independent variable in a fully general relativistic quantum statistical mechanics and suchlike fully relativistic analysis (rather than have to first choose a fixed solution to the Einstein equation and then do the analysis "on curved spacetime".) This gives the variable a definite "distinction" without applying that it is somehow "absolute" and the only right choice :big grin:
The key paper for understanding it in this light is http://arxiv.org/abs/1209.0065
But also maybe re-read the "Chapter 1" quote in post #145. It's short and to the point.
P: 4 Dear Marcus Hi, Im new to this thread, and I think it is very well conducted, and please excuse my butting in. But I would like to respectfully say that I question/disagree with your view... “I think everybody in the thread would go along with the idea that time is real and vitally important---both in physics and in our everyday experience.” (post #135) ...at a fundamental level. If we take the simple working view that ‘time’ is apparently 'real', and thus something that in the most simple terms consists, at least, in some way of components described as ‘the past’ and ‘the future’ then at the most basic level we would need... A- some initial reason for suspecting or assuming the existence of these things/places ('past', 'future'), and, B- some proof / reason or experiment, to sensibly show how or why they exist. It seems to me that if the universe is such that it is just filled with a quantity of matter/energy that ‘just’ (as in ‘only’) exists, moves, changes and interacts, (without leaving a 'past' behind us, and without heading into a 'future'), then this would explain all that we think implies the existence of a past and future – if we misinterpret, or over extrapolate, what we observe. Specifically – if as that matter moves and changes, it also moves and changes the contents of our minds, we may look at some of the contents of our minds, and ‘call’ those contents ‘memories’, and add to this (possibly wrongly) that those contents (memories) are not just things that exist and prove that matter can exist, but are also proof that another thing called ‘the past’ -also- exists. (And thus also proof that a thing called 'time' exists). As such we may (imo wrongly) imply that some existing matter, in a particular formation, gave us good reason to suspect that as things move and change the universe ‘also’ creates and stores some kind of ‘record’ of all events in a place or a thing called ‘the past’. So – is it correct to say , either, 1- Matter just exists moves and changes, or 2- There is also a (temporal) past, and thus a thing called ‘time’ that also may be considered. Many people seem to assume that Relativity tells us something about the nature of a thing called ‘time’. From what I have read, as far as I can tell, relativity only seems to actually tell us about the way, and ‘rates’ at which things may move and change differently under various conditions. As far as I can tell no part of relativity ever proves or demonstrates the existence of things (or places etc) such as ‘the past’ or ’the future’. Although it is written in a way that seems to imply or suggest ‘time’ and these places naturally or obviously exist, or make sense. While relativity seems to correctly show that matter may intrinsically 'change at different/reduced rates' while at velocity, in acceleration, gravity, etc, I don't see any proof that such matter 'sinks into a past' or 'surges into a future', or that relativity indicates the existence of these concepts. I would suggest 'time' is not real, but only a false idea borne out of us incorrectly interpreting what the contents of our minds prove and do not prove. If I am wrong could you point me to a link that shows how the existence of these entities ( 'the' past and 'the' future) has been demonstrated (in relativity or otherwise), as opposed to have just been assumed and untested? Yours M.Marsden, London
P: 63 So if I'm getting this right, then the difference between this and "pre-Einstein" universal time is that in the latter, everyone's clocks must agree with the universal time, but that is not the case for this kind of "universal" time. Does this time also have a rest frame associated with it? Also, I notice you mention that "omega" represents "our knowledge". Does this mean that as we get "more knowledge", then it further "refines" this universal time? However, I'm curious about that bit about "spacetime does not exist", the "block universe doesn't exist": I thought that the block universe was essentially necessitated by the fact that observers could disagree on what constituted the past, present, and future. So that all three would have to exist "eternally". How is this handled in this "spacetime-less" theory? If you replace "time as a dimension" with "time as 'change'", then that would mean there would have to exist a universal "now", no? And that "now" would be the only thing that exists, everchanging (as we have no spacetime, so the past and future don't eternally exist). And if that "now" is the only thing that exists, then how can some observer in it include events in the non-existent universal past as part of their "now"?
Astronomy
PF Gold
P: 23,110
Now you are getting down to the basic similarities and differences with the earlier picture. These are good questions, I think.
Quote by sshai45 So if I'm getting this right, then the difference between this and "pre-Einstein" universal time is that in the latter, everyone's clocks must agree with the universal time, but that is not the case for this kind of "universal" time. Does this time also have a rest frame associated with it? Also, I notice you mention that "omega" represents "our knowledge". Does this mean that as we get "more knowledge", then it further "refines" this universal time?
M consists of all possible measurements, ever. I think you are RIGHT that we will want to use a different state function ω when we have improved physics theories and more precise estimates of the constants. ω is a probabilistic idea of the state---correlations between measurements also embodying uncertainties about the fundamental constants, earlier conditions and even what the applicable equations are.
The idea is, we have to go with the best ideas and knowledge we have, and predict the measurements that we consider to be in our future, based on our knowledge and the odds we ascribe to it.
So exactly as you say, as future humans refine ω so would this idea of time (the Tomita flow on M) be refined.
I do not think that the Tomita flow has any idea of simultaneity belonging to it. There is no distinguished time-slice associated with it, that you could somehow "date from".
this is jumping way ahead, but if LQC (with its bounce) were ever implemented in a (M,ω) model then it would acquire a reference time-slice, the bounce. But we already know that when standard Friedmann cosmology is implemented one recovers standard time used in cosmology. Cosmologists use a universe time or "Friedmann time" in their standard expansion model. And , no surprise, (M,ω) reproduces it. Tomita = Friedmann. but that's jumping ahead.
The basic answer is NO there is no reference timeslice in the (M,ω) picture. there is the Tomita flow but no universal starting place for it.
In a sense M takes the place of the 4D spacetime of GR. but it has no geometry. the measurements all embody uncertainty and can assume various values. We can only imagine making a FINITE NUMBER of measurements. Like knowing where a particle went but only at a finite number of points along the way---not knowing the entire continuous trajectory.
M is very different from a space-time with a metric describing its geometry, in the sense that we make only a finite number of measurements (of areas, of angles, of distances, of matter density, of charge, etc)---and make a finite number of predictions based on that---beyond that we don't presume. The geometry is obviously quantum and uncertain because the geometric measurements themselves are quantum observables. But more than that, we do not presume that an overall classical geometry even exists.
I'm trying to interpret from the Connes Rovelli paper http://arxiv.org/abs/gr-qc/9406019 as best I can, and also from the recent one
http://arxiv.org/abs/1209.0065
However, I'm curious about that bit about "spacetime does not exist", the "block universe doesn't exist": I thought that the block universe was essentially necessitated by the fact that observers could disagree on what constituted the past, present, and future...
Yes we all want observers to be able to disagree! But does this actually necessitate a "block universe". I think that is only ONE POSSIBLE data structure that permits them to systematically disagree. I think you are asking an extremely good question and one which, since the (M,ω) way of representing the world is new to me, as is the Tomita flow idea of time, I cannot competently answer. I would like to see more examples.
the Connes Rovelli paper shows a bunch examples but I would like to be clearer. How do different observers disagree harmoniously within the (M,ω) context? A researcher at Perimeter Institute named Laurent Freidel has been working on something he calls "Relative Locality" in which no global spacetime exists but there is Lorentz symmetry locally. Could this be encompassed in the (M,ω) picture?
The model itself does not force any division into past present future. But how for example is Lorentz symmetry implemented? Wish I could do a better job answering.
P: 63 Thanks for the response. I'm curious about that "bounce". Does it imply that the future of the universe is to recollapse ("Big Crunch") and bounce again? If so, how does that jibe with dark energy? Does dark energy disappear at some point, or does it "reverse" itself somehow (so as to become attractive instead of repulsive in effect)?
Astronomy
PF Gold
P: 23,110
Quote by sshai45 Thanks for the response. I'm curious about that "bounce". Does it imply that the future of the universe is to recollapse ("Big Crunch") and bounce again? ...
Thanks for the interesting discussion. In fact it does not imply recollapse. The Penn State people run many different cases on the computer, including Λ = 0 so they get a variety of behavior including that cyclic behavior you mentioned. But when they put in a realistic positive cosmological constant then they get just one bounce. This is similar to the classical DeSitter universe which has Lambda>0 and only one bounce.
Personally I don't think of Λ > 0 as representing an "energy". I just think of Lambda as a constant which naturally occurs in the Einstein equation of GR (the symmetries of the theory permit two constants, G and Λ). And all the evidence so far is that Λ does not change over time.
So if you think of it as an "energy" that energy density would not be changing.
In the way it first appeared in the GR equation, Λ is not an energy density but simply a small inherent CURVATURE. That is to say, the reciprocal of an area. If you have a favorite force unit in mind you can always multiply reciprocal area by force and get a pressure and that is the same type of physcial quantity as an energy density. So you can convert Λ to an energy density by fiddling with it. Move it from left side (curvature) of equation to right side (matter) and make mysterious talk about "dark energy" but I think that is going out of style. More often now I hear cosmologists simply refer to the cosmo constant Λ. "Dark energy" is more for the media. All we know is there is this acceleration that appears exactly as if due to a constant curvature at the classical level.
http://arxiv.org/abs/1002.3966
However one likes to think of it, including constant Λ > 0 in the picture with either classic DeSitter or Loop QC, you get a universe history with just one bounce.
Astronomy Sci Advisor PF Gold P: 23,110 Hi Matt, I looked over your post and it seems to me it could be clearer and more compact if your words were anchored to definite mathematical objects. Physics is a mathematical science which means people are looking for the simplest best fit model (in math language). When we talk with English words there is normally some underlying math the words can be reduced down to, or can be anchored to. It may be very simple but there is usually some nonverbal foundation. Talking English can be convenient and bridge people with different technical upbringing and help speed up acquiring intuition, but the verbal description is seldom the whole story. So you say HAVING TWO WORDS IS REDUNDANT AND CONFUSING we shouldn't have separate words "time" and "change". That makes a certain amount of sense on a purely verbal level. But the Tomita math model of time has a use for both words. This is because of a subtle difference in the way we TREAT intervals of time and the changes that correspond to them, mathematically. There is an algebra M consisting of all possible measurements or observations. It is an algebra because you can add two measurements X+Y and multiply them XY. And there is an extremely useful object alpha-sub-t called a ONE PARAMETER GROUP OF AUTOMORPHISMS. αt is the change corresponding to an interval of time of length t. For every real number t there is a change αt which stirs M around, it sends every element of M to a new element. X → αt(X) And ADDING TWO TIMES corresponds to doing first one change and then the other. If there are two real numbers s and t. then the change corresponding to s+t is what you get by changing by αs and then changing by αt. Doing one change and then the other change is thought of as group multiplication and so we write αs+t = αsαt The alphas would normally be large MATRICES of complex numbers, or something analogous. Their actual written form would vary depending on the problem. The matrix entries would depend on the time parameter t. So it is useful to have two words: time is the additive parameter, and you add time intervals together. Changes are matrices that stir the world around, and you multiply two matrices together to see what happens when you do one change and then the other. Changes correspond to passage of a certain amount of time. That is what a one parameter group of transformations is, or a one parameter group of automorphisms, or changes, is. Time is the additive real number parameter t, and αt is the change. From a math standpoint it would be inconvenient and confusing to have only one word. The words are NOT redundant, from a math standpoint. But you have written a purely verbal essay arguing that we should reform the way we speak and have only one word, because from your verbal perspective the two words are redundant. I hope I've clarified the difficulty somewhat.
P: 175
Quote by Mattmars But just ‘‘calling’’ a machine a clock, then saying that a ‘‘clock’’ is a thing that measures a thing called ‘‘time’’, and then claiming this is a proof that ‘‘time’’ exists, is imo absolutely not a proof that a thing called ‘‘time’’ exists.
And "exists" is also a wooly word. Of course. I agree. But these are just nice words; remember 40 years of fruitless speculation about string theory! To connect words, or squiggles on paper (as Hardy called mathematics) with memorable physics, one needs to suggest something practical we can actually do with new ideas. Making something that can reveal part of the future, like a time machine, would be good! Even correctly predicting the fall of cards in a poker game would draw attention.
So far in this thread no one, sadly including myself , has come anywhere near making such a useful suggestion. So far, it’s a futile story.
Astronomy Sci Advisor PF Gold P: 23,110 Hi Paulibus, glad to see you! Personally I think the topic has a certain beauty and excitement because of the prospect of doing general relativistic statistical mechanics (and thermodynamics) something not hitherto possible. You probably have seen the Einstein field equation dozens of times---relating curvature quantities on the left side to matter and energy quantities on the right side. Back in the 1990s Ted Jacobson DERIVED that equation from some thermodynamic law, some fact about entropy. That to me is a very mysterious thing. they seem like utterly separate departments of physics. The connection remains puzzling and incomplete to this day. this is one reason that I view the current interest in this Tomita time---the only universal time flow I know of (the cosmologist's Friedmann model time being a special case of it arising under simplifying assumptions)---as far from futile. I see it as pretty exciting. Another exciting thing has to do with what Matt just said: "Change is not a thing that happens ‘over a thing called time’, and change is not a thing intrinsically linked to a thing called time." When you read the Princeton Companion to Mathematics treatment of Tomita flow you see that the change matrix is a certain root matrix raised to the t power where time is measured in natural (Planck) units. So one can say time is the exponent of change. There is a matrix, or more generally a unitary operator S such that the automorphism corresponding to the passage of time t (in nature's units) is given by the matrix/operator St. This is why adding times corresponds to multiplying change (or doing one, followed by the other). That is how exponents behave. You always have Xs+t=Xs Xt. What this illustrates, to me, is that the world is more intrinsically unified at a math level than it is at a verbal level. Because Matt says "change is not intrinsically linked to time" but when I look at the world with Tomita's eyes I see immediately that TIME IS THE LOGARITHM OF CHANGE Time is the real number that you have to raise the Tomita base to, to get a given change process. (a stirring around or automorphism of the world M of measurements). Here is where the Princeton Companion describes how to find the Tomita base (like the number e, the base of the natural logarithms), it is what you raise to the power t to get the change corresponding to that passage of time. http://books.google.com/books?id=ZOf...20math&f=false
P: 4 Hi Marcus, Thank you very much for that reply. Thanks for clarifying the 'subtle difference in the way we TREAT intervals of time and the changes that correspond to them'. I see what you are saying + I will have to read up on this and some of the other details you mention to address them properly. (no point just blindly replying :) However, the essence of my point is that with the question '... does time exist' there may be some very simple basic 'truth' that is consistently missed -because- the mathematics works, and the science it leads to is so practical and useful. That is to say we may be confusing the usefulness of the mathematics with what it actually does and does not prove. For example, of course accountancy maths and scientific maths are somewhat different, but nonetheless, consider that no matter how perfectly one might balance the books of a multinational conglomerate, it would be foolish to think this high level of accuracy, or the usefulness of what you had done, related in anyway at all to how well you had proved that " 'Money' really is a thing that actually exists", (other than as a useful idea). Im trying to show that we may be making the same error with high level mathematics and the notion of 'time'. Thanks again for your reply, you've given me a couple of things to think about. Ill make sure I've understood your points then respond. mm
Astronomy
PF Gold
P: 23,110
Matt you might be interested in some earlier parts of thread. Whether or not a concept emerges as meaningful useful real can depend on contextual things like e.g. SCALE. So we were talking earlier about how time could be emergent rather than fundamental. Like temperature of a gas, which is real enough but does not exist at the level of a single molecule---it is a property of the collective when you look largescale. Or the waterlevel in a pond, which is real enough at a large scale but at microscale the pond does not have a welldefined surface it is a wild fuzzy dance of molecules. So things can be emergent rather than fundamental (to use a verbal shorthand). I'll recall part of that earlier discussion. This is post #57
Quote by marcus ... Obviously the free energy in a situation depends on the scale you're able to manipulate. If you are molecule-size and live in a box of gas, then you can lasso molecules and can harness them (or play the Maxwell demon with them), and get energy. But whatever you do with the energy makes no difference to a large outsider. He looks in and sees no free energy---because he can't see or manipulate or benefit at your scale. He sees a uniform "temperature" throughout, which you do not. Whatever you accomplish with the free energy you see doesn't make a damn bit of difference to him---it still looks like gas in a box. So free energy depends on the scale at which the observer is interacting with it, and likewise the Boltzmann distribution, depending as it does on the free energy. So the idea of EQUILIBRIUM depends on scale... ... The reason it's relevant is that several of us in the thread seem to agree on looking at time as real but *emergent* either from local motions or thermodynamics. In particular e.g. Julian Barbour in his prize-winning FQXi essay showed clearly how time is emergent from local motions, at a certain level. One does not have to treat it as a quasi-spatial "extra" dimension. One wants to be able to generalize on both Barbour's time and thermodynamic or "thermal" time (which may, at root, be the same thing as Barbour's) to understand the emergence of time in a variety of contexts and at various scales.
Paulibus said he liked some of post #57 but he didn't fully agree, and he made several other interesting points. I'll quote portions of his post #58.
Quote by Paulibus .. As Niels Bohr pointed out, Physics is a matter of what we say about stuff, not what stuff “is”. ...say of hot and cold, or the maintenance of a status quo. When we try to extend such descriptions beyond scales familiar to us, a qualification as “emergent” can be useful for broadening context. So is the quantitative and logical extension provided to ordinary language by mathematics. But let’s not kid ourselves that the words and mathematical descriptions we use have absolute eternal meanings; they just conveniently communicate concepts between us. Like the mysterious word “time” that everybody knows. Although we cannot yet claim to accurately understand and describe time, one thing does stand out: using time as a parameter to characterise change works wherever physics rules. This, it seems, is all over the Universe. Therefore: time can’t just be some local quirky emergent thing; it must be related to something universal, like the observed red-shift and its cause, namely “expansion”. Or is this also just an "emergent" aspect of the “reality” that we try to describe?
In the part I highlighted, Paulibus italicized the word works. That's key. In physics, as Niels Bohr indicated, we are less interested in what exists than in accurate consistent statements, predictions---the simplest best-fit model, testable stuff, measurements. As Paulibus just said: "exist" is a fuzzy word. You can waste a lot of time talking about whether this or that "exists".
So we have this working distinction between more or less fundamental and emergent, and the notion that the reality or usefulness of concepts can depend on scale. Temperature can be very important at largescale and gradually lose meaning---become undefined or inapplicable---as you go to smaller and smaller scale.
Concepts can be scale-dependent, observer-dependent, context-dependent, state-dependent---there is a lot of nuance in physics (as in other branches of language! )
Astronomy Sci Advisor PF Gold P: 23,110 I'll repost a set of links useful for this discussion, mostly sources on thermal time (= Tomita flow time). ==from post #129== This is to page 517 of the Princeton Companion to Mathematics http://books.google.com/books?id=ZOf...20math&f=false It's a nice clear concise exposition of the Tomita flow defined by a state on a *-algebra. For notation see the previous post: #128. Here's the article by Alain Connes and Carlo Rovelli: http://arxiv.org/abs/gr-qc/9406019 Here is Chapter 1 of Approaches to Quantum Gravity (D. Oriti ed.) http://arxiv.org/abs/gr-qc/0604045 Page 4 has a clear account of the progressive weakening of the time idea in manifold-based physics, which I just quoted a couple of posts back. I see the inadequacy of time in manifold-based classical and quantum relativity as one of the primary motivations for the thermal time idea. The seminal 1993 paper, The Statistical State of the Universe http://siba.unipv.it/fisica/articoli....1567-1568.pdf This shows how thermal time recovers conventional time in several interesting contexts. Here's a recent paper where thermal time is used in approaches to general relativistic statistical mechanics and general covariant statistical QM. http://arxiv.org/abs/1209.0065 It can be interesting to compare the global time defined by the flow with a local observer's time. The ratio between the two can be physically meaningful. http://arxiv.org/abs/1005.2985 Jeff Morton blog on Tomita flow time (with John Baez comment): http://theoreticalatlas.wordpress.co...d-tomita-flow/ Wide audience essays--the FQXi "nature of time" contest winners: http://fqxi.org/community/essay/winners/2008.1 Barbour: http://arxiv.org/abs/0903.3489 Rovelli: http://arxiv.org/abs/0903.3832 Ellis: http://arxiv.org/abs/0812.0240 ==endquote== Interestingly, Tomita flow time is the only independent time-variable available to us if we want to study a general relativistic system. Observer-time is not well-defined unless we already have settled on a particular fixed geometry. If the underlying geometry is dynamic and undecided we can't specify an observer's world-line. Tomita time is independent of the observer. It depends only on what we think we know about the world---the correlations among measurements that embody physical theory and presumed initial conditions, along with our degree of confidence/uncertainty. That is, it depends on the state. In Bohr's words: "what we can SAY". As Wittgenstein put it: "The world is everything that is the case." Here's a Vimeo video of part of a talk on Tomita time by Matteo Smerlak: http://vimeo.com/33363491 It's from a 2-day workshop March 2011 at Nice, France. The link just missed being included in the above list.
Astronomy Sci Advisor PF Gold P: 23,110 But at my back I always hear Time's wingèd chariot hurrying near; And yonder all before us lie Deserts of vast eternity. Andrew Marvell, around 1650 I also want to recall this other passage, which is crucial to the discussion. This concisely summarized one of the troubles with time in a classical GR context. And indicates how the problem appears to get even more severe when one goes to a quantum version. But it is just at this point that the (M, ω) picture with its universally-defined Tomita flow becomes available. So the problem contains the seeds of its own solution. This passage gives a concise motivation for the star-algebra state-dependent way of treating time evolution. ==quote page 4 http://arxiv.org/abs/gr-qc/0604045 == ... Therefore, properly speaking, GR does not admit a description as a system evolving in terms of an observable time variable. This does not mean that GR lacks predictivity. Simply put, what GR predicts are relations between (partial) observables, which in general cannot be represented as the evolution of dependent variables on a preferred independent time variable. This weakening of the notion of time in classical GR is rarely emphasized: After all, in classical GR we may disregard the full dynamical structure of the theory and consider only individual solutions of its equations of motion. A single solution of the GR equations of motion determines “a spacetime”, where a notion of proper time is associated to each timelike worldline. But in the quantum context a single solution of the dynamical equation is like a single “trajectory” of a quantum particle: in quantum theory there are no physical individual trajectories: there are only transition probabilities between observable eigenvalues. Therefore in quantum gravity it is likely to be impossible to describe the world in terms of a spacetime, in the same sense in which the motion of a quantum electron cannot be described in terms of a single trajectory. ==endquote== In the (M,ω) picture, M —essentially the set of all measurements— functions as a quantum-compatible replacement for spacetime, doing away with the need for it. Uncertainty, including geometric uncertainty, is built into every measurement in the set. And there's another very clear explanation of the problem here (to get the original paper just google "connes rovelli" ): ==page 2 of http://arxiv.org/abs/gr-qc/9406019 == In a general covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter. One can still recover weaker notions of physical time: in GR, for instance, on any given solution of the Einstein equations one can distinguish timelike from spacelike directions and define proper time along timelike world lines. This notion of time is weaker in the sense that the full dynamics of the theory cannot be formulated as evolution in such a time.1 In particular, notice that this notion of time is state dependent. Furthermore, this weaker notion of time is lost as soon as one tries to include either thermodynamics or quantum mechanics into the physical picture, because, in the presence of thermal or quantum “superpositions” of geometries, the spacetime causal structure is lost. This embarrassing situation of not knowing “what is time” in the context of quantum gravity has generated the debated issue of time of quantum gravity. As emphasized in [4], the very same problem appears already at the level of the classical statistical mechanics of gravity, namely as soon as we take into account the thermal fluctuations of the gravitational field.2 Thus, a basic open problem is to understand how the physical time flow that characterizes the world in which we live may emerge from the fundamental “timeless” general covariant quantum field theory [9]. In this paper, we consider a radical solution to this problem. This is based on the idea that one can extend the notion of time flow to general covariant theories, but this flow depends on the thermal state of the system. More in detail, we will argue that the notion of time flow extends naturally to general covariant theories, provided that: i. We interpret the time flow as a 1- parameter group of automorphisms of the observable algebra (generalised Heisenberg picture); ii. We ascribe the temporal properties of the flow to thermodynamical causes, and therefore we tie the definition of time to thermodynamics; iii. We take seriously the idea that in a general covariant context the notion of time is not state- independent, as in non-relativistic physics, but rather depends on the state in which the system is. ==endquote== So they describe the problem, and they propose a solution. The problem is "In a general covariant theory there is no preferred time flow, and the dynamics of the theory cannot be formulated in terms of an evolution in a single external time parameter." But we HAVE to have a preferred time flow if we are going to do general relativistic statistical mechanics--stat mech and thermodynamics INCLUDING GEOMETRY. The temperature and entropy of the geometry as well, not merely of matter distributed on some pre-arranged fixed geometry. These and other types of analysis require a time flow. We want to comprehend the whole, including its dynamic geometry, not merely a part. The proposed solution was clearly a radical departure, namely to roll all you think you know about the world up into one ball of information, called the state function, and make that give you an inherent distinguished time flow. Make it do that. Force it to give you an intrinsic flow on the set of all observations/measurements. Tomita, a remarkable Japanese mathematician, showed how. For some reason this reminds me again of Andrew Marvell's words: "Let us roll all our strength and all our sweetness up into one ball" and basically just blast on through "the iron gates of life." It is a bold move. He was talking about something else, though.
Related Discussions General Discussion 56 Science & Math Textbooks 0 Science & Math Textbooks 2
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heading plots 1 or more text strings (provided as a character vector labels) as a heading to an (existing or new) plot and places a colored box behind each label to mark it (i.e., highlighting the heading).
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## Arguments
labels A character vector specifying the text labels to be plotted. A numeric vector of x-coordinates at which the text labels in labels should be written. If the lengths of x and y differ, the shorter one is recycled. Default: x = 0. A numeric vector of y-coordinates at which the text labels in labels should be written. If the lengths of x and y differ, the shorter one is recycled. Default: y = .8. A numeric value or vector for the vertical spacing of labels in labels. 2 special values are "even" (i.e., even distribution of labels across available y-space) and "flush" (i.e., no space between adjacent labels, or y_layout = 0). Default: y_layout = "flush". The color(s) of the text label(s). Default: col_lbl = "black". The color(s) to highlight or fill the rectangle(s) with. Default: col_bg = "default" (to automatically select different shades of pal_seeblau). Numeric character expansion factor(s), multiplied by par("cex") to yield the character size(s). Default: cex = 2. The font type(s) to be used. Default: font = 2 (i.e., bold). Boolean: Should a new plot be generated? Set to "blank" or "slide" to create a new plot, and to "none" to add to an existing plot. Default: new_plot = "slide" (i.e., create a new slide).
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Text formatting parameters (like col, col_bg, cex, font) are recycled to match length(labels).
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# AAPG Bulletin
Abstract
Volume: 51 (1967)
Issue: 3. (March)
First Page: 383
Last Page: 392
Title: Paleocurrents and Shoreline Orientations in Green River Formation (Eocene), Raven Ridge and Red Wash Areas, Northeastern Uinta Basin, Utah
Author(s): M. Dane Picard (2)
Abstract:
Paleocurrent data from ripple marks and cross-stratification are related to orientations of shorelines and sandstone-body trends in the lacustrine and fluvial setting of the Green River Formation (Eocene) in the Red Wash field and the adjacent outcrops along Raven Ridge in Utah and Colorado. At 11 localities along Raven Ridge, the northeastern margin of the Uinta basin, 125 paleocurrent directions were measured from cross-stratification and asymmetrical ripple marks in the Douglas Creek and Garden Gulch Members and the lower part of the Parachute Creek Member.
Vertical stratigraphic variation of paleocurrent directions at each locality is small, indicating that the over-all current system was stable. A plot of measurements of 84 asymmetric and 68 symmetric ripple marks shows that their distribution is similar, which is interpreted to be the result of their formation by the same current system. Based on arcs of azimuths, there is essentially no difference between paleocurrent directions from cross-stratification and from ripple marks. The dominant paleocurrent directions are toward the north, south, and southeast. Of all observations, 25 per cent range from 331° to 30°, and 51 per cent range from 121° to 210°.
The shorelines in the northeastern Uinta basin area are interpreted to have been generally at right angles to the dominant paleocurrent directions. Therefore, essentially all of the shorelines had bearings of 31° to 120°. An arc of 61°-90° would contain about 40 per cent of the bearings of the shorelines, based on the paleocurrent data. Trends of single sandstone bodies, the total footage of sandstone, sandstone plus siltstone, and net sandstone in the Red Wash field, and the trends of major facies in the northeastern Uinta basin support the generalizations about the orientations of shorelines and sandstone-body trends.
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Finding integer solutions to this equation
$p^{\; \left\lfloor \sqrt{p} \right\rfloor}\; -\; q^{\; \left\lfloor \sqrt{q} \right\rfloor}\; =\; 999$
How do you find positive integer solutions to this equation?
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The exponents must be less than $4$ (we have $17^4-16^4=17985$ and $16^4-15^3=62161$). There is an easy solution with $q=1$. The other solution arises from a difference of two cubes equal to $999$. In fact, $37=4^3-3^3$ is the difference of two consecutive cubes, and $999=3^3\cdot 37$. This gives $$12^3-9^3=999.$$
For the numbers equal to the difference of consecutive cubes see the sequence A003215 in integer sequences, which is the crystal ball sequence for a hexagonal lattice.
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Actually $17^4-16^4$ is less than 62161. – Henning Makholm Nov 18 '13 at 12:55
How did you combine $37\;=\;4^{3}-3^{3}$ and $999\;=\;3^{3}\; \cdot \; 37$ to arrive at $12^{3}\;-\;9^{3}\;=\;999$? – user2612743 Nov 19 '13 at 1:23
Multiplication by $3^3$, i.e. $4^3\cdot 3^3=12^3$ etc. – Dietrich Burde Nov 19 '13 at 8:46
Calculate $x^{\lfloor x^{1/2}\rfloor}$ for the first dozen or so integers. Soon the function starts to increase so quickly that there's no hope of getting 999 as the difference between its values. Check whether any of the values you get until that happens happen to be an earlier value plus 999.
(There are two solutions).
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Finally, electrical power is the product of voltage and current. Does Windows know physical size of external monitor? Biology ... measurements in a given circuit. This can also be explained in other way. In mechanical physics, work is the amount of force needed to move something over a … If base current, Ib, and emitter current, Ie, are known, then Ic can be calculated by the formula: Example Thus, using this formula, we can find the avg. Even an illiterate person knows that. Power can be a measurement of how much work someone or something performs over time. I = 230/17 How to Calculate VÏ of a BJT Transistor Should I mention a discovery was made by mistake? There is a poor-quality picture of the transformer schematics e.g. The switch S1 can be a SPST ON/OFF switch. Following the official rules of Jenga, what is the highest possible level that can be placed? This means, if AC current I in a circuit flows for t sec and charge Q is transferred across any point of circuit in this time t by this AC current then the same charge Q will also be transferred by its avg. Black to yellow - 15 ohm If you don't know this, I would recommend you not to hook it up to anything close to 230 V. 230V is a standard supply here so there is no issue with that. What anti-asteroid measures can we take now, or in the near future, if we faced an alien invasion? Look at the above formula, the three-phase full load current is equal to Power divided by the 3 times of product of line to neutral voltage and power factor. How to find the sum of that series related to Legendre functions of second kind? How to solve 3x3 Magic Squares with negative values when only 2 values are given? What could lead humans to go extinct after a collapse of technological civilization? The resistance of transformer windings can be a useful hint to people who know what they're doing when trying to figure out how a transformer is arranged internally. As Vp/Vs = Is/Ip = Np/Ns. here. input voltage on the primary coil * input current on the primary coil = output voltage on the secondary coil * output current on the secondary coil. Hence, its average value is zero. My calculation - MathJax reference. for t= (π/ω). How to Calculate α of a BJT Transistor Before hooking to 230V I verified that I am connecting to correct wires by checking the terminals of UPS PCB which supplies power to transformer. Actually, I want to calculate the secondary winding's max current output but my multimeter doesn't read Amperes. How do you say "I think she loves me" in Latin? we respect your privacy and take protecting it seriously, This means, if AC current I in a circuit flows for t sec and charge Q is transferred across any point of circuit in this time t by this AC current then the same charge Q will also be transferred by its avg. Calculate work. SCIENCE . How to Calculate RÏ of a BJT Transistor Electric current is defined as the flow of charges through a given space. Can you suggest me some good links about transformer which will clarify how transformer windings not works as a resistor. For Star Connection the Full Load Current Iline is equal to Iph. Charge Transferred across any point of circuit for half cycle is given as below. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. the direction of flow of positive charge, whereas the direction of flow of electrons gives the direction of electronic current which is opposite to that of conventional current. There is only one path wherein the electrons and charges can flow. Let us consider a sinusoidal current i = ImSinωt as shown in the figure below. (Im/π). How to Calculate the Collector Current IC of a BJT Transistor Thus, the charge transferred by AC current for one time period is zero. You should try to find a better one before working with 230V. A current source is an electronic circuit that delivers or absorbs an electric current which is independent of the voltage across it.. A current source is the dual of a voltage source.The term current sink is sometimes used for sources fed from a negative voltage supply. My calculation - V=IR I = 230/17 = 13.5 A Seeking help, Thanks. That voltage in relation to the rated primary voltage is given as. Black to blue - 14 ohm Version Control For Salesforce — Branching Strategy. If Iavg being the average value, then this current must also transfer the same charge for t = (π/ω).Since average value is the DC value, this charge will be equal to Q = Iavgx(π/ω). 2. Seeking help, Thanks. To calculate output voltage from a circuit, use Ohm's law. The emitter current, Ie, of a transistor is the amplified output current of a bipolar junction transistor. Why is there a zig-zag in elemental abundances? But by transformer formula, I can find secondary current if the primary current is known. How to Calculate β of a BJT Transistor On the other end multiply 2.pi times the armature rotation speed, then divide the resultant value by 60. How to Calculate VCE of a BJT Transistor Actually, I want to calculate the secondary winding's max current output but my multimeter doesn't read Amperes. According to Ohm’s law, the electric current formula will be, $$I=\frac{V}{R}$$ Where, V is the voltage; R is the resistance; I is the current; An interesting problem with "decomposing" natural numbers. I think it is useless to talk about this things here. We will start by looking at the buck. current I, Let us consider a sinusoidal current i = I, Thus the charge transferred by half cycle of Sinusoidal AC current = (2I, Thus, the average value of output current of single phase half wave rectifier is equal to its peak value divided by π i.e.
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# How to solve carbon 14 dating problem validating a database
How old is a skeleton that has lost 41% of its carbon-14?
Note: Do not round any numbers during your calculation. The radioactive element carbon-14 has a half-life of 5750 years.
section, i thought i would try here first to see if there was something obvious i missed...well, heres the question: Analysis on an animal bone fossil at an archeological site reveals that the bone has lost between 90%-95% of c-14.
Give an interval for the possible ages of the bone.
Note: Do not round any numbers during your calculation.
_____years old If you can please show all steps, thanks for any help!!! The amount of carbon-14 present decreases exponentialy with time.
The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive isotope).
The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000).
If we assume Carbon-14 decays continuously, then $$C(t) = C_0e^,$$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$\frac C_0 = C_0e^,$$ which means $$\frac = e^,$$ so the value of $C_0$ is irrelevant.ln(Ao/A) = k*t where Ao = starting level or concentration and A is the level or concentration after time t, and k is the rate constant. That means that after 5570 years the level or concentration has decreased from 1 to 1/2. alright, so there was this question in my math book, doesn't give many details, thats why i found it confusing...debating whether or not to post this question in chem.Students should be guided to recognize the use of the logarithm when the exponential function has the given base of $e$, as in this problem.Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
Now, take the logarithm of both sides to get $$-0.693 = -5700k,$$ from which we can derive k \approx 1.22 \cdot 10^.
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# Surface are of a lightbulb
I have this picture:
How could I calculate the area of a "thing" in red square? It is a circle.
Assume the part in red is a spherical cap of radius $r$. Viewed from the center of the sphere, the cap is forming a cone with half-angle $\theta$. We know:
• the base of the cap is a circle with radius $a = r \sin\theta = \frac{127}{2}{\bf mm}$
• the thickness of the cap is $h = r(1-\cos\theta) = 185-150 = 35 {\bf mm}$
This leads to $$(r - h)^2 = (r\cos\theta)^2 = r^2 - a^2\quad\implies\quad r = \frac{h^2+a^2}{2h} = \frac{21029}{280}{\bf mm}$$ and the surface area of the spherical cap is
$$\text{Area} = 2\pi r^2(1-\cos\theta) = 2\pi rh = \pi (h^2+a^2) = \frac{21029\pi}{4}{\bf mm}^2$$
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### Introduction
The hyperbolic functions $sinhx,\phantom{\rule{1em}{0ex}}coshx$ , $tanhx$ etc are certain combinations of the exponential functions ${e}^{x}$ and ${e}^{-x}$ . The notation implies a close relationship between these functions and the trigonometric functions $sinx,\phantom{\rule{1em}{0ex}}cosx$ , $tanx$ etc. The close relationship is algebraic rather than geometrical. For example, the functions $coshx$ and $sinhx$ satisfy the relation
$\phantom{\rule{2em}{0ex}}{cosh}^{2}x-{sinh}^{2}x\equiv 1$
which is very similar to the trigonometric identity ${cos}^{2}x+{sin}^{2}x\equiv 1$ . (In fact every trigonometric identity has an equivalent hyperbolic function identity.)
The hyperbolic functions are not introduced because they are a mathematical nicety. They arise naturally and sufficiently often to warrant sustained study. For example, the shape of a chain hanging under gravity is well described by $cosh$ and the deformation of uniform beams can be expressed in terms of tanh .
#### Prerequisites
• have a good knowledge of the exponential function
• have knowledge of odd and even functions
• have familiarity with the definitions of $tan,\phantom{\rule{1em}{0ex}}sec,\phantom{\rule{1em}{0ex}}cosec,cot$ and of trigonometric identities
#### Learning Outcomes
• explain how hyperbolic functions are defined in terms of exponential functions
• obtain and use hyperbolic function identities
• manipulate expressions involving hyperbolic functions
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# Consider two fluids having specific volumes as S1 and S2. What would be the relation between their mass densities (d1 and d2 respectively) if S1 < S2?
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Consider two fluids having specific volumes as S1 and S2. What would be the relation between their mass densities (d1 and d2 respectively) if S1 < S2?.
1. d1 > d2
2. d1 = d2
3. d1 < d2
4. cannot be determined
by (50.3k points)
selected
Correct Answer - Option 1 : d1 > d2
Concept:
Let consider the volume of two fluids is V1 and V2 and the masses of fluid m1 and m2 respectively.
Now
Specific Volume - It is a property of materials, defined as the number of cubic meters occupied by one kilogram of a particular substance.
So, Specific volumes for two fluids are as follows
S1 = V1 / m1 . ...........(i)
S2 = V2 / m2 .............(ii)
Using the relationship between mass density, the mass of fluid, and volumes
$Mass\space density = {Mass \over Volume}$
Assume the mass density of the above fluids is d1 and d2, hence
$d_1 ={m_1 \over V_1}\space and \space d_2 ={m_2 \over V_2}$
Then the relation between specific volume and mass density of fluids by using equations (i) and (ii)
S1 = 1 / d1 and S2 = 1 / d2
So the relation between specific volume and mass density is inversely proportional to each other.
Hence, in this question given that the specific volume of two fluids is S1 < S2 , so mass density is d1 > d2.
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## Regular and effective monomorphisms and epimorphisms
Previously we observed that although monomorphisms tended to give expected generalizations of injective function in many categories, epimorphisms sometimes weren’t the expected generalization of surjective functions. We also discussed split epimorphisms, but where the definition of an epimorphism is too permissive to agree with the surjective morphisms in familiar concrete categories, the definition of a split epimorphism is too restrictive.
In this post we will discuss two other intermediate notions of epimorphism. (These all give dual notions of monomorphisms, but their epimorphic variants are more interesting as a possible solution to the above problem.) There are yet others, but these two appear to be the most relevant in the context of abelian categories.
## Split epimorphisms and split monomorphisms
• What is the “easiest way” a morphism can be a monomorphism (resp. epimorphism)?
• What are the absolute monomorphisms (resp. epimorphisms) – that is, the ones which are preserved by every functor?
• A morphism which is both a monomorphism and an epimorphism is not necessarily an isomorphism. Can we replace either “monomorphism” or “epimorphism” by some other notion to repair this?
• If we wanted to generalize surjective functions, why didn’t we define an epimorphism to be a map which is surjective on generalized points?
The answer to all of these questions is the notion of a split monomorphism (resp. split epimorphism), which is the subject of today’s post.
In this post, for convenience all categories will be locally small (that is, $\text{Set}$-enriched).
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# Pigs
Pigs are feed by beet.Beet feed containing 12% dry solids, which is 0.72% of digestible crude protein. How much beet must beconsumed in one month (30 days), if the weight of digestible crude protein contained in a daily dose of beet was 0.912 kg?
Result
x = 31666.7 kg
#### Solution:
$m \cdot \dfrac{ 12 }{100} \cdot \dfrac{ 0.72 }{100} = 0.912 \ \\ m \cdot 0.12 \cdot 0.0072 = 0.912 \ \\ \ \\ m = \dfrac{ 0.912 }{ 0.12 \cdot 0.0072 } \ \\ m = 1055.56 \ kg \ \\ \ \\ x = 30 m = 30\cdot 1055.56 = 31666.7 \ \text{ kg } = 31.7 \ t \ \\$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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## 5.10 Exercises
1. Electricity consumption was recorded for a small town on 12 consecutive days. The following maximum temperatures (degrees Celsius) and consumption (megawatt-hours) were recorded for each day. TODO: change the econsumption to a ts of 12 concecutive days - change the lm to tslm below
1 2 3 4 5 6 7 8 9 10 11 12
Mwh 16.3 16.8 15.5 18.2 15.2 17.5 19.8 19.0 17.5 16.0 19.6 18.0
Temp 29.3 21.7 23.7 10.4 29.7 11.9 9.0 23.4 17.8 30.0 8.6 11.8
1. Plot the data and find the regression model for Mwh with temperature as an explanatory variable. Why is there a negative relationship?
2. Produce a residual plot. Is the model adequate? Are there any outliers or influential observations?
3. Use the model to predict the electricity consumption that you would expect for the next day if the maximum temperature was $$10^\circ$$ and compare it with the forecast if the with maximum temperature was $$35^\circ$$. Do you believe these predictions?
4. Give prediction intervals for your forecasts. The following R code will get you started:
plot(Mwh ~ temp, data=econsumption)
fit <- lm(Mwh ~ temp, data=econsumption)
plot(residuals(fit) ~ temp, data=econsumption)
forecast(fit, newdata=data.frame(temp=c(10,35)))
2. Data set olympic contains the winning times (in seconds) for the men’s 400 meters final in each Olympic Games from 1896 to 2012.
1. Plot the winning time against the year. Describe the main features of the scatterplot.
2. Fit a regression line to the data. Obviously the winning times have been decreasing, but at what average rate per year?
3. Plot the residuals against the year. What does this indicate about the suitability of the fitted line?
4. Predict the winning time for the men’s 400 meters final in the 2000, 2004, 2008 and 2012 Olympics. Give a prediction interval for each of your forecasts. What assumptions have you made in these calculations?
5. Find out the actual winning times for these Olympics (see www.databaseolympics.com). How good were your forecasts and prediction intervals?
3. Type easter(ausbeer) and interpret what you see.
4. An elasticity coefficient is the ratio of the percentage change in the forecast variable ($$y$$) to the percentage change in the predictor variable ($$x$$). Mathematically, the elasticity is defined as $$(dy/dx)\times(x/y)$$. Consider the log-log model, $\log y=\beta_0+\beta_1 \log x + \varepsilon.$ Express $$y$$ as a function of $$x$$ and show that the coefficient $$\beta_1$$ is the elasticity coefficient.
5. The data set fancy concerns the monthly sales figures of a shop which opened in January 1987 and sells gifts, souvenirs, and novelties. The shop is situated on the wharf at a beach resort town in Queensland, Australia. The sales volume varies with the seasonal population of tourists. There is a large influx of visitors to the town at Christmas and for the local surfing festival, held every March since 1988. Over time, the shop has expanded its premises, range of products, and staff.
1. Produce a time plot of the data and describe the patterns in the graph. Identify any unusual or unexpected fluctuations in the time series.
2. Explain why it is necessary to take logarithms of these data before fitting a model.
3. Use R to fit a regression model to the logarithms of these sales data with a linear trend, seasonal dummies and a “surfing festival” dummy variable.
4. Plot the residuals against time and against the fitted values. Do these plots reveal any problems with the model?
5. Do boxplots of the residuals for each month. Does this reveal any problems with the model?
6. What do the values of the coefficients tell you about each variable?
7. What does the Breusch-Godfrey test tell you about your model?
8. Regardless of your answers to the above questions, use your regression model to predict the monthly sales for 1994, 1995, and 1996. Produce prediction intervals for each of your forecasts.
9. Transform your predictions and intervals to obtain predictions and intervals for the raw data.
10. How could you improve these predictions by modifying the model?
6. TODO: you got to this before me ;-) The gasoline series consists of weekly data for supplies of US finished motor gasoline product, from 2 February 1991 to 20 January 2017. The units are in “thousand barrels per day”. Consider only the data to the end of 2004.
1. Fit a harmonic regression with trend to the data. Select the appropriate number of Fourier terms to include by minimizing the AICc or CV value.
2. Check the residuals of the final model using the checkresiduals() function. Even though the residuals fail the correlation tests, the results are probably not severe enough to make much difference to the forecasts and forecast intervals. (Note that the correlations are relatively small, even though they are significant.)
3. To forecast using harmonic regression, you will need to generate the future values of the Fourier terms. This can be done as follows.
fc <- forecast(fit, fourier(x, K, h))
where fit is the fitted model using tslm, K is the number of Fourier terms used in creating fit, and h is the forecast horizon required.
Forecast the next year of data.
4. Plot the forecasts along with the actual data for 2005. What do you find?
7. (For advanced readers following on from Section 5.7).
Using matrix notation it was shown that if $$\bm{y}=\bm{X}\bm{\beta}+\bm{\varepsilon}$$, where $$\bm{e}$$ has mean $$\bm{0}$$ and variance matrix $$\sigma^2\bm{I}$$, the estimated coefficients are given by $$\hat{\bm{\beta}}=(\bm{X}'\bm{X})^{-1}\bm{X}'\bm{y}$$ and a forecast is given by $$\hat{y}=\bm{x}^*\hat{\bm{\beta}}=\bm{x}^*(\bm{X}'\bm{X})^{-1}\bm{X}'\bm{y}$$ where $$\bm{x}^*$$ is a row vector containing the values of the regressors for the forecast (in the same format as $$\bm{X}$$), and the forecast variance is given by $$var(\hat{y})=\sigma^2 \left[1+\bm{x}^*(\bm{X}'\bm{X})^{-1}(\bm{x}^*)'\right].$$
Consider the simple time trend model where $$y_t = \beta_0 + \beta_1t$$. Using the following results, $\sum^{T}_{t=1}{t}=\frac{1}{2}T(T+1),\quad \sum^{T}_{t=1}{t^2}=\frac{1}{6}T(T+1)(2T+1)$ derive the following expressions:
1. $$\displaystyle\bm{X}'\bm{X}=\frac{1}{6}\left[ \begin{array}{cc} 6T & 3T(T+1) \\ 3T(T+1) & T(T+1)(2T+1) \\ \end{array} \right]$$
2. $$\displaystyle(\bm{X}'\bm{X})^{-1}=\frac{2}{T(T^2-1)}\left[ \begin{array}{cc} (T+1)(2T+1) & -3(T+1) \\ -3(T+1) & 6 \\ \end{array} \right]$$
3. $$\displaystyle\hat{\beta}_0=\frac{2}{T(T-1)}\left[(2T+1)\sum^T_{t=1}y_t-3\sum^T_{t=1}ty_t \right]$$
$$\displaystyle\hat{\beta}_1=\frac{6}{T(T^2-1)}\left[2\sum^T_{t=1}ty_t-(T+1)\sum^T_{t=1}y_t \right]$$
4. $$\displaystyle\text{Var}(\hat{y}_{t})=\hat{\sigma}^2\left[1+\frac{2}{T(T-1)}\left(1-4T-6h+6\frac{(T+h)^2}{T+1}\right)\right]$$
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# How can I draw this figure using latex?
How can I draw this figure using LaTex ?
• It's better if you have done some trials and decide what package you intend to use. – hesham Mar 22 '20 at 21:51
Quite a few diagrams of this type exist already on this site, but maybe not yet with patterns.meta, which gives us easier control over the patterns. (One could also employ the angles library but this may be a bit of an overkill for one arc.)
\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{decorations.pathmorphing,patterns.meta}
\pgfdeclarepattern{
name=shatch, parameters={\hatchsize,\hatchangle,\hatchlinewidth},
bottom left={\pgfpoint{-.1pt}{-.1pt}},
top right={\pgfpoint{\hatchsize+.1pt}{\hatchsize+.1pt}}, tile size={\pgfpoint{\hatchsize}{\hatchsize}},
tile transformation={\pgftransformrotate{\hatchangle}}, code={
\pgfsetlinewidth{\hatchlinewidth}
\pgfpathmoveto{\pgfpoint{-.1pt}{\hatchsize/2}}
\pgfpathlineto{\pgfpoint{\hatchsize+.1pt}{\hatchsize/2}}
\pgfusepath{stroke}
} }
\tikzset{
hatch size/.store in=\hatchsize,
hatch angle/.store in=\hatchangle,
hatch line width/.store in=\hatchlinewidth,
hatch size=5pt,
hatch angle=0pt,
hatch line width=.5pt,
}
\begin{document}
\begin{tikzpicture}[declare function={beta=30;},>=stealth,semithick]
\begin{scope}[rotate=-beta]
\path[pattern=shatch] (0,0) rectangle (-0.4,1.2);
\draw[<->] (0,2) node[right] {$y$} |- (6,0) node[right]{$x$}
coordinate[pos=0.95](x);
\draw[decorate,decoration={zigzag,segment length=4mm,amplitude=2mm}]
(0,0.2) -- (4,0.2) node[midway,above=2ex]{$(k,\ell)$};
\draw[fill=cyan] (3.8,0) rectangle ++ (0.6,0.4);
\path (4.1,0.5) node[above right] {$M(m)$};
\end{scope}
\draw (x) -- ++ (-5,0);
\draw (x) + (-2,0) arc[start angle=180,end angle=180-beta,radius=2]
node[midway,left] {$\beta$};
\draw[<-] (3,2) -- ++ (0,1);
\draw[->] (3.1,2.6) -- ++ (0.5,0) node[midway,below] {$g$};
\end{tikzpicture}
\end{document}
• +1: my humble opinion. The arrows over g is very long. – Sebastiano Mar 22 '20 at 22:20
• @Sebastiano This is just a hint at the gravity of our current situation. – user194703 Mar 22 '20 at 22:38
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## Saturday, April 6, 2013
### How big a number is 400!
The other day I was teaching permutations to my Algebra 2 students, and casually asked how many different arrangements are there for our whole school body to be placed for a picture. I don't know the exact number, but I typically use 400 for estimates (yes -- I make enough estimates for our school that I have a "typical"). I knew immediately that the answer was 400! (which is 400*399*398*...*3*2*1 for those of you who have never seen the ! notation before).
My problem was that I had no concept of just how big 400! really was. How many digits long is that number?
My initial thinking was to simply type it in the calculator, but it was too big for my TI-84 to handle. That meant it was more than 100 digits long -- but is it more than 400 digits long? 1000 digits? How can I answer this?
Eventually I think logarithms will be the answer, but for now let's see if we can set some upper and lower limits on things. Since each of the digits in the multiplication from 400 down to 1 is less than 3 digits long, then the whole product must be less than 400*3 or 1200 digits long. Since most of the digits are 2 or more digits long, it's safe to assume it must be more than 400 digits long, but just how many are there?
More formally:
400! = 400*399*398*...*3*2*1 < 1000*1000*1000... = 1000^400 = (10^3)^400 = 10^1200
400! = 400*399*398*...*12*11*10! > 10*10*10*...*10*10*10! = (10^390)*10!
Now logarithms are a tool our PreCalc students will be tackling this next week, and could be used to answer this question, and it all hinges on the product - sum property of logarithms:
A factorial is simply a lot of multiplications, which would translate into a giant sum of lots of logarithms:
That can be summarized (pun intended...) as:
$log\:400!=\sum_{n=1}^{400}log\:n$
This gives me something that my calculator CAN handle -- since instead of actually trying to display the number, it simply gives me the magnitude of the number. This can be typed into a TI-84 by typing:
sum(seq(log(N),N,1,400)) = 868.8
This means the number 400! is equal to 10^868.8 and is therefore 869 digits long.
A quick check on wolfram alpha verifies this:
(I'm sure some of you were asking "Why didn't he just do that in the first place?" to which I simply respond "Because I didn't have to! God gave me a brain and problem solving skills for a reason!")
As a follow up question -- can I predict how many zeroes are at the end of that number? In factorials, zeroes come after every multiple of 5. (Technically, I would need multiples of 2 as well, but there are plenty of those, and relatively fewer multiples of 5).
Up to 4! there are no zeroes:
1, 2, 6, 24,
Between 5! and 9! there is 1 zero:
120, 720, 5040, 40320, 362880
Between 10! and 14! there are 2 zeroes:
3628800, 39916800, 479001600, 6227020800, 87178291200
After that my calculator can't display them properly, but I hope you'll anticipate the pattern.
400 is the 400/5 80th multiple of 5, and so 400! is the first factorial to have 80 zeroes at the end of it. A not-so-quick check on Wolfram Alpha's picture reveals that there are actually 33*3 or 99 zeroes. Extras!?
That's because there are more multiples of 5 -- 25, 50, 75, ... 400 each contain 2 factors of 5, and 125, 250, 375 each contain three. Counting these all up should reveal 99 factors of 5 (and way more factors of 2):
80 Multiples of 5: (5, 10, 15, ..., 390, 395, 400)
16 Multiples of 25: (25, 50, 75, ... 350, 375, 400)
3 Multiples of 125: (125, 250, 375)
99 total factors of 5 therefore 99 zeroes.
And for completeness (since I kept claiming there were way more factors of 2 than 5).
200 Multiples of 2
100 Multiples of 4
50 Multiples of 8
25 Multiples of 16
12 Multiples of 32
6 Multiples of 64
3 Multiples of 128
1 Multiple of 256:
397 Total factors of 2.
What this means is that the prime factorization of 400! contains (among other things) 2^397 and 5^99.
What's the largest prime in 400! is a question for a different night. (Oh, what the heck, why not:)
As I read through my facebook feed, I was struck with following picture:
Immediately -- shows how much of a dork I am -- I thought "I wonder what time this picture was taken?!"
You see, as the earth rotates around the sun, shadows rotate around the objects that form them. In the northern hemisphere, these shadows rotate clockwise -- which is why clockwise is clockwise. The first clocks ever made were sundials, made by people living in the North, and then clocks were built later.
I figured I should be able to figure out the angle of the shadow of the arch and use it to figure out what time of day the picture was taken. I could also figure out the date the picture was taken by looking at the length of the shadow. You see, everyday the angle of the sun at a given time changes. Right now, during the spring, the sun is higher in the sky every day at a specific time, which makes shadows shorter. Measure your shadow at 11:00am today and measure it again tomorrow and it will be smaller!
So I found a map of St. Louis, and used Geogebra to figure out the angle of the shadow of the sun, and the length of the shadow. After about five minutes, I had placed a point on the map that represented where I thought the top of the shadow was, and had drawn a vector from that point to the point that represented the top of the arch. I compared that with the scale of the map, and estimated the length of the shadow to be about 1,000 ft. After looking on wikipedia, I knew the height of the arch, and a little trig revealed the altitude of the sun to be about 32 degrees.
In a few more minutes I had estimated the angle of the the vector and converted that into a compass heading, which gives me the azimuth of the sun of approximately 111 degrees.
I knew there is only two times a year where the sun has that exact altitude and azimuth, once in spring and again sometime in the fall -- and I took a chance that this picture was taken on spring break (reasonable enough right?). So I looked up the altitude and azimuth for the sun on the days during spring break:
Since the photo was tagged as uploaded on April 1* I started with that date, and found the following data in the table:
The first column is the time (AM), the second column is the altitude of the sun, and the third column is the azimuth of the sun. I was disappointed that I didn't see my exact values in the table -- but I didn't expect to either, for two reasons:
1. I didn't know if this was the correct date -- the picture might have been uploaded that day but taken several days (or even a half a year?!) earlier.
2. There is some degree of uncertainty in my measurements. As I moved around the point where I thought the top of the shadow was, the angles varied somewhat. To be specific, they varied less than a degree more or less than my values, but that's significant enough to make my answers have to be estimates.
Let me treat each of these reasons separately. Assuming the picture was actually taken on April 1, and my measurements were slightly off, I would estimate that the picture was taken around 8:34 am local time (I could be off by an hour if the website doesn't account for daylight-savings time, but I'm going to assume they were smart enough for that).
If I don't assume to know the date the picture was taken, and trust my measurements, I would argue that the picture wasn't actually taken on the 1st. Looking at similar tables for other days, I get much closer altitude/azimuth combinations for a few days later:
If I had nothing else to go on, I would estimate the date/time of the picture was April 3, 8:33am.
Perhaps the photo takers will provide the true answer in the comments below?
*There was some discrepancy between my wife and I as to when the picture was actually uploaded onto Facebook. It was posted April 5th, "tagged" April 1, but I have reason to doubt the "tagged" date. Only time will tell who wins our little "argument" -- although regardless of who wins, I will probably lose -- right guys? I love you honey!
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# Comparing exp() performance on Julia versus numpy
So I was actually preparing a demo to show off Julia to colleagues when I encountered this issue.
I want to simply compute the exponential of an array. In Julia (v. 1.7.2):
arr = rand(100_000)
@benchmark exp.($arr) # Time (median): 576.806 μs GC (median): 0.00% # Memory estimate: 781.30 KiB, allocs estimate: 2. Compared to Python: arr = np.random.rand(100000) timeit.timeit(lambda: np.exp(arr), number=100000) / 100000 # 0.0001027145623601973 So Python is about a factor of 6 faster. Of course I’ve also tried wrapping the Julia dot call inside a function, as well as manually looping myself, e.g.: @inbounds function myfunc(arr) result = similar(arr) @simd for i in eachindex(arr) result[i] = exp(arr[i]) end return result end @benchmark myfunc($arr)
# Time (median): 569.613 μs
…which is identical timing to the dotted version.
Any idea what’s going on here? Some obvious sanity checks - both are Float64, and both are not using fastmath, both are allocating a new array and both are single threaded.
Edit: saw now that you said no multithreading.
But are you actually certain that numpy is not implicitly doing multithreading under the hood?
@DNF I checked that: both are single threaded.
I’m no good at reading llvm code, so I cannot confirm, but it might be that Julia fails to simd the computation. If you use LoopVectorization.jl, it works:
julia> @btime exp.($x); 503.300 μs (2 allocations: 781.30 KiB) julia> @btime @turbo exp.($x);
91.100 μs (2 allocations: 781.30 KiB)
1 Like
Yeah, I think that’s it! I see near identical timings between numpy and Julia with @turbo. @turbo also speeds up my my own explicit function, even though this already included a @simd macro.
So guess the next question is: why doesn’t the dot call automatically apply simd instructions? And why does my explicit function with manual @simd not work?
(I’ve checked whether this slowdown is attributed to the slow handling of tails, as it mentioned on the LoopVectorization page, but passing in arrays without tails does not make any difference in my own testing.)
2 Likes
It’s a good question. I am not able to figure it out, but I believe LoopVectorization.jl in some cases uses other implementations of some functions which are hard to simd. Probably, exp is one of them.
Julia is just using general mechanisms to calculate exp over your array, while numpy is free to do a special array version, maybe much like LoopVectorization does. Making simd happen in Julia in this case apparently takes more than a simple @simd annotation.
I just tried this, and I’m getting 402us in Julia 1.7.3 and 435us with numpy. I would have expected them to be completely equal, assuming they are doing the same thing. I’m surprised there’s a difference one way or the other.
1 Like
Curiously, in my machine I got the numpy code to run np.exp(arr) on ~960\mu s, the no LV.jl code on ~712\mu s and the @turbo’ed code in ~142\mu s.
I got the same results using v1.8-rc3 or 1.7.3.
Which CPU do you have?
That is probably not a good assumption.
I suspect numpy uses your system libM,
julia uses implementations of math functions written in julia – following the same techniques libM authors use in terms of balancing speed and accurasy but potentially ending up at a different point.
Even if exp isn’t one of the functions we have tweaked and optimized it would still be based on the OpenLibM definition which is probably not the same libM that your system runs.
3 Likes
Just as a another data point:
vanilla julia:
600 us
turbo julia:
150 us
python numpy:
650 us
CPU: Intel i7-8850H (6) @ 2.6GHz
1 Like
Possibly related:
2 Likes
The speedup in Python is very dependent on how numpy has been compiled. I have tested on various system and see the numpy code variably matching the speed of the @turbo Julia or alternatively the plain julia. Obviously some are making use of simd instructions, and some are not.
That’s fine, but doesn’t address the question I asked earlier:
1. Why isn’t exp.(arr) automatically applying simd optimizations?
2. And why doesn’t my explicit @simd annotation in the manual loop function apply simd instructions, either?
3 Likes
we generally don’t do this.
see the linked PR, and try this on nightly to see if you observe any difference
Basically the reason is that Julia can do this if it is able to inline the code, but generically turning scalar code into vector code through the compiler only is a very complicated optimization.
maybe there is something to python vectorized style then /s
and, *cough, vmap
1 Like
There is no reason to expect Julia to outperform Python on a single call like this. A significant difference in either direction is just low fruit for the other (in this case, the Julia version failing to SIMD). This function is already optimized in Python (or, rather, in C or whatever language was used to write it under the hood). Julia’s performance benefits only really factor in when you are trying to do more complicated sequences of operations that lack a bespoke implementation.
3 Likes
One thing we could that would make this better is to have a better interface to allow the compiler to replace scalar code with custom provided vector code. I’m not fully sure what that interface looks like though.
1.8 rc-3
julia> @btime exp.(x) setup=(x = rand(100_000));
383.979 μs (2 allocations: 781.30 KiB)
current master:
Julia Version 1.9.0-DEV.1086
Commit 01c0778057* (2022-08-05 06:31 UTC)
julia> @btime exp.(x) setup=(x = rand(100_000));
171.195 μs (2 allocations: 781.30 KiB)
(AMD CPU, so no avx512)
4 Likes
Perhaps I’m misunderstanding, but doesn’t the copyto! invoked in broadcasting use the @simd annotation?
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# Last three digits III
Number Theory Level 3
What are the last three digits of $$999^{555}+998^{20}$$?
Try Part I and Part II
×
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Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
I commited the 2000th revision of Rcpp svn today, so I wanted to look back at what I did previously with the 50 000th R commit.
Here are the number of commits per day and month
… the same thing, but focused on the period since I joined the project
… and now split by contributor
here are the month where each of us have been the most active
> do.call( rbind,
lapply(
split( month_author_data, month_author_data$author ) , function(x) x[ which.max( x[["commits"]] ), ] ) ) date author commits month year dmbates 2010-08-01 dmbates 19 08 2010 edd 2010-06-01 edd 118 06 2010 romain 2010-06-01 romain 256 06 2010 and the most active day > do.call( rbind, lapply( split( day_author_data, day_author_data$author ) ,
function(x) x[ which.max( x[["commits"]] ), ] )
)
date author commits month year
dmbates 2010-08-06 dmbates 13 8 2010
edd 2010-02-16 edd 20 2 2010
romain 2010-06-17 romain 30 6 2010
The code to reproduce the graphs is here
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# On the number line shown, is zero halfway between r and s ?
Author Message
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### Show Tags
28 Mar 2011, 03:45
1
KUDOS
1
This post was
BOOKMARKED
Statement 1) if s is to the right of zero then 2 cases arrive
Case 1
-----0---r----s-----t-----
Case 2
-----r---0----s-----t-----
which to choose, hence insufficient
statement 2)
the distance b/w t & r is the same as the distance b/w t & -s
still 2 cases arrive
Case 1
r=-5, s=-3, t=-1, s=3
-----r---------s-------------------t-------------0--------------(+s)----- where +s=3
case 2
-----r------0------s---------------t---------------- r=-s
combining the two statements above,
its clear that 0 lies midway to r and s.
therefore C.
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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31 Mar 2012, 10:14
Graphical approch:-
2. 0 could be to the right of t i.e. assuming s to be -ve and -s to be positive
or 0 before s meaning r & -s are same point
(these are the only two cases as points could be on two sides of t or on same side of t)
1 & 2 together 0 can't be after t as so r=-s.
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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23 Jan 2013, 01:41
7
KUDOS
GIVEN: <=====(r)=====(s)===(t)=====>
1. s is to the right of 0
<=====(r)==(0)===(s)===(t)=====> Maybe!
<===(0)==(r)=====(s)===(t)=====> No!
INSUFFICIENT.
2. distance of r and t is equal to t and -s
<=====(r=-s)=====(s)===(t)=====> Yes!
<=====(r)=====(s)===(t)=======(-s)=> No!
INSUFFICIENT.
Together: Since s is to the right of 0 then -s is to the left of 0...
and |r-t| = |t+s| then r must be equal to -s...
<=====(r=-s)==(0)===(s)===(t)=====>
Yes!
SUFFICIENT.
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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04 Apr 2013, 03:05
yangsta8 wrote:
mbaquestionmark wrote:
I have a question guys..
If -s is to the right of t, then wont r be equal to s ? But clearly in the picture, r and s are different points..
So dont u think that option is ruled out ? or is it like we should not go by the pic ? I know we should not go by the scale of the pic.. also this ?
Cos I thought the answer was B.. can someone please explain if I am wrong..
Thanks..
There's a couple of points to remember. Firstly never base your answer on how the diagrams look, they are representative but are by no means accurate. Because a triangle is drawn as equilateral for example, there is no reason to assume it is.
I think you've made a couple of incorrect assumptions in your reasoning:
1) -S is not necessarily to the right of T. Consider the case that 0 is between S and R. Then -S is negative meaning it is to the left of 0 and hence to the left of T. Your assumption is that 0 is on the right of S, but this isn't stated anywhere in Statement 2.
2) No answers state that R and S are the same point. Just that R = negative S.
Hope that clears it up.
hi Yangsta,
If 0 is between S and R, then there are two points(r and -s) on the left hand side of T which are distinct yet have the same distance from T?? how this is possible,,can u explain with an e.g if possible?
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
### Show Tags
28 Sep 2013, 02:24
marioslash wrote:
Hi guys,
Does anyone explain in detail this question?
Statement 1: only says S is positive - Not sufficient.
Statement 2: says R= - S, Nothing much- Not sufficient.
Together: S= +ve, and -S=R. the distance between S and 0 and -S and 0 is equal, & thus R & 0..
So both statement together is sufficient (C).
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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29 Mar 2014, 09:30
I still don't quite get statement 2
I'm looking at the diagram and the statement and it seems clear that r must be = -s. And therefore zero is in the middle hence answer is sufficient
Also trying when r>s>t and so r=-s, but still 0 is still between s and r and answer is still yes
Is there any case I am missing, could someone please illustrate in number line
Bonus question: I read B's explanation about what IanStewart mentioned regarding graphs in PS, DS and all that stuff mentioning that we could trust the relative position of points in the diagram. Hence in this case r>s>t always?
Thanks!
Would throw a lot of Kudos for this
Cheers
J
Math Expert
Joined: 02 Sep 2009
Posts: 45180
Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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29 Mar 2014, 10:04
jlgdr wrote:
I still don't quite get statement 2
I'm looking at the diagram and the statement and it seems clear that r must be = -s. And therefore zero is in the middle hence answer is sufficient
Also trying when r>s>t and so r=-s, but still 0 is still between s and r and answer is still yes
Is there any case I am missing, could someone please illustrate in number line
Bonus question: I read B's explanation about what IanStewart mentioned regarding graphs in PS, DS and all that stuff mentioning that we could trust the relative position of points in the diagram. Hence in this case r>s>t always?
Thanks!
Would throw a lot of Kudos for this
Cheers
J
First of all, the question asks whether 0 is halfway between r and s, not just between r and s.
Below is the case for (2) when 0 is NOT halfway between r and s.
(2) The distance between t and r is the same as the distance between t and -s
--r-------s---t-----------(-s)
Here s is negative, -s is positive and 0 is somewhere between t and -s.
As for your second question: from the diagram we can infer that $$t>s>r$$, not that $$r>s>t$$.
OG13, page 272:
A figure accompanying a data sufficiency problem will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2).
Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight.
You may assume that the positions of points, angles, regions, and so forth exist in the order shown and that angle measures are greater than zero degrees.
All figures lie in a plane unless otherwise indicated.
OG13, page 150:
Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.
Hope it helps.
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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21 Jun 2014, 07:14
DenisSh wrote:
Attachment:
Number line.PNG
On the number line shown, is zero halfway between r and s?
(1) s is to the right of zero
(2) The distance between t and r is the same as the distance between t and -s
Hi Bunuel,
Got this question incorrect on GMAT prep test and thus looked at your solution....
Initially, I tried to do the question by myself and here is what I did
The Question basically asks us whether 0 is between r and s or |r-0|=|s-0| or |r|=|s| or r= s or -s
(Now, if r=s then answer is no but if r=-s then answer is yes...But I think I did apply the mod statement correctly so in this case do we reject the case of r=s at this stage itself and reduce the question to if r=-s )
St 1 says s>0 not much help here as r can be placed to the right of zero or left or at zero. Not sufficient
St 2 says |t-r|=|t+s|
This can be interpreted in one of the 2 ways
t-r = t+s or t-r= -(t+s)
So we get either r=-s or 2t=r+s
Not sufficient.
combining we see that r=-s and s>0 therefore r<0 and r=-s which is same
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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21 Jun 2014, 08:22
WoundedTiger wrote:
DenisSh wrote:
Attachment:
Number line.PNG
On the number line shown, is zero halfway between r and s?
(1) s is to the right of zero
(2) The distance between t and r is the same as the distance between t and -s
Hi Bunuel,
Got this question incorrect on GMAT prep test and thus looked at your solution....
Initially, I tried to do the question by myself and here is what I did
The Question basically asks us whether 0 is between r and s or |r-0|=|s-0| or |r|=|s| or r= s or -s
(Now, if r=s then answer is no but if r=-s then answer is yes...But I think I did apply the mod statement correctly so in this case do we reject the case of r=s at this stage itself and reduce the question to if r=-s )
St 1 says s>0 not much help here as r can be placed to the right of zero or left or at zero. Not sufficient
St 2 says |t-r|=|t+s|
This can be interpreted in one of the 2 ways
t-r = t+s or t-r= -(t+s)
So we get either r=-s or 2t=r+s
Not sufficient.
combining we see that r=-s and s>0 therefore r<0 and r=-s which is same
On the number line shown, is zero halfway between r and s?
$$k$$ is halfway between $$m$$ and $$n$$ can ALWAYS be expressed as: $$\frac{m+n}{2}=k$$.
Is 0 halfway between r and s? --> is $$\frac{r+s}{2}=0$$? --> $$r+s=0$$.
The question asks whether we have the following case:
--r---0---s---t--
(1) s is to the right of zero. Clearly insufficient.
(2) The distance between t and r is the same as the distance between t and -s
If s < 0, then we'd have the following case:
--r-------s---t---0-------(-s)
If s > 0, then we'd have the following case:
--r---0---s---t-------------
Answer YES. Notice that in this case r and -s coincide.
Not sufficient.
(1) + (2) We have the second case from (2). Sufficient.
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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28 Oct 2014, 04:54
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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19 Nov 2014, 15:44
Bunnel,
Can you help me with one more problem?
Where am I going wrong?
On the number line shown, is zero halfway between r and s?
(1) s is to the right of zero
(2) The distance between t and r is the same as the distance between t and -s
on-the-number-line-shown-is-zero-halfway-between-r-and-s-89015.html
Statement 1
s is to the right of zero
------r------0--s No
r----0----s Yes
Statement 2
The distance between t and r is the same as the distance between t and -s
-s --- t ---r ---- 0 ---------s (taking numbers as below)
-10(s) --- -7(t) --- -4(r) ---- 0 -------------- -10 (s)
No
-4(r=-s)----- 0(t) ------4 (S)
Yes
So both statements insuff.
Cheers,
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Posts: 45180
Re: On the number line shown, is zero halfway between r and s ? [#permalink]
### Show Tags
20 Nov 2014, 08:33
annie2014 wrote:
Bunnel,
Can you help me with one more problem?
Where am I going wrong?
On the number line shown, is zero halfway between r and s?
(1) s is to the right of zero
(2) The distance between t and r is the same as the distance between t and -s
on-the-number-line-shown-is-zero-halfway-between-r-and-s-89015.html
Statement 1
s is to the right of zero
------r------0--s No
r----0----s Yes
Statement 2
The distance between t and r is the same as the distance between t and -s
-s --- t ---r ---- 0 ---------s (taking numbers as below)
-10(s) --- -7(t) --- -4(r) ---- 0 -------------- -10 (s)
No
-4(r=-s)----- 0(t) ------4 (S)
Yes
So both statements insuff.
Cheers,
Notice that the stem gives relative positioning of the point as r --- s --- t:
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
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20 Nov 2014, 21:25
Responding to a pm:
Quote:
Statement 1
s is to the right of zero
------r------0--s No
r----0----s Yes
Statement 2
The distance between t and r is the same as the distance between t and -s
-s --- t ---r ---- 0 ---------s (taking numbers as below)
-10(s) --- -7(t) --- -4(r) ---- 0 -------------- -10 (s)
No
-4(r=-s)----- 0(t) ------4 (S)
Go to the original post on this thread. Check out the diagram given with the question.
"On the number line shown...."
This tells you that r, s and t are on the number line in that order.
r to the left of s and s to the left of t.
So the cases you have taken are not valid since you have changed the relative positions of r, s and t.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 09 Aug 2015 Posts: 1 Re: On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 21 Aug 2015, 10:03 This problem confused me quite a bit. Would you guys agree that statement 2 by itself has the following 4 possibilities? 1) -----r---s---t---0---(-s)-- (i.e. r=-4;s=-2;t=-1;(-s)=2) 2) -----(r,s)---(t,0)---(-s)--- (i.e. r,s=-4;t=0;(-s)=4) 3) -----(r,-s)---0---s---t---- (i.e. r=-4;s=4;t=6) 4) ---------(0,r,s,-s)-----t--- (i.e. r,s=0;t=4) Therefore, knowing that s>0 narrows it down to one answer (case 3 above)? Did I think of this correctly? Please let me know if there are other cases that are valid for statement (2). Intern Joined: 29 Jun 2014 Posts: 1 On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 11 Sep 2015, 14:56 Hello Benuel/Experts, It is a bit embarrassing for me to post this potential silly question. For whatever reason, I am unable to get this out of my head. I would continue to read the entire thread multiple times while someone kind enough to take some time and answer it. I am under the impression that, unless stated otherwise, the given figure is to some sort of scale. Regardless of the units of the scale, and also due to the reason that it did not say that the figure is not to scale, why are we even considering the possibility of s = -s = 0 when in fact statement (2) states that the distance between t and r is same as t and -s (assuming that -s needs to be left of s) ? Thank you so much for your time here. I answered my own question here. Realizing that both s and -s can take both + and - values helped understand this question. Current Student Joined: 20 Mar 2014 Posts: 2645 Concentration: Finance, Strategy Schools: Kellogg '18 (M) GMAT 1: 750 Q49 V44 GPA: 3.7 WE: Engineering (Aerospace and Defense) Re: On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 11 Sep 2015, 17:04 bkh33al wrote: Hello Benuel/Experts, It is a bit embarrassing for me to post this potential silly question. For whatever reason, I am unable to get this out of my head. I would continue to read the entire thread multiple times while someone kind enough to take some time and answer it. I am under the impression that, unless stated otherwise, the given figure is to some sort of scale. Regardless of the units of the scale, and also due to the reason that it did not say that the figure is not to scale, why are we even considering the possibility of s = -s = 0 when in fact statement (2) states that the distance between t and r is same as t and -s (assuming that -s needs to be left of s) ? Thank you so much for your time here. I answered my own question here. Realizing that both s and -s can take both + and - values helped understand this question. Be careful with what you have mentioned in red above. Especially in DS questions, be wary of the assumptions! As per official guide "You may assume that the positions of points, angles, regions, and so forth exist in the order shown. Thus in DS, unless otherwise stated to the contrary, do not assume that the figures are drawn to scale. Only the order is the same. This is a very subtle but important point for this question. Retired Moderator Status: The best is yet to come..... Joined: 10 Mar 2013 Posts: 517 Re: On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 11 Oct 2016, 04:45 A. s is to the right of zero; in this case r could be both in left and right side of zero If s = 2, r = -3, then No If s = 2, r = 1, then No If s = 1, r = -1, then Yes Not Sufficient B. |t-r|=|t-(-s)| =>|t-r|=|t+s| t-r is always positive,since r is to the left of the t,therefore |t-r|=t-r; For example,if t = 3 and r = 2,then t-r=1 Again,if t = 3 and r = -2,then t-r= 3-(-2) = 5 Again,if t = 3 and r = -5,then t-r= 3-(-5)=8 On other hand, |t+s| may be positive or negative For example,if t = 3 and s = 2,then t+s=5 Again,if t =3 and s =-2,then t+s= 3+(-2) =1 Again,if t = 3 and s=-5,then t+s= 3+(-5)=- 8 ∴t-r=|t+s| =>t-r=t+s or t-r=-(t+s) =>-r=s or 2t=r-s Not Sufficient A+B: From A: Since s>0, then t+s>0 Now applying the condition of A in B we get, t-r=|t+s|=>t-r=t+s=>-r=s Since r and-r are halfway between 0,then r and s (=-r) will also be halfway between 0. Sufficient _________________ Hasan Mahmud Senior Manager Joined: 12 Mar 2013 Posts: 290 Re: On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 18 Oct 2016, 12:14 Bunuel wrote: Let me clear this one: NOTE: In GMAT we can often see such statement: $$k$$ is halfway between $$m$$ and $$n$$. Remember this statement can ALWAYS be expressed as: $$\frac{m+n}{2}=k$$. Also in GMAT we can often see another statement: The distance between $$p$$ and $$m$$ is the same as the distance between $$p$$ and $$n$$. Remember this statement can ALWAYS be expressed as: $$|p-m|=|p-n|$$. Back to original question: Is 0 halfway between r and s? OR is $$\frac{r+s}{2}=0$$? --> Basically the question asks is $$r+s=0$$? (1) $$s>0$$, clearly not sufficient. (2) The distance between $$t$$ and $$r$$ is the same as the distance between $$t$$ and -$$s$$: $$|t-r|=|t+s|$$. $$t-r$$ is always positive as $$r$$ is to the left of the $$t$$, hence $$|t-r|=t-r$$; BUT $$t+s$$ can be positive (when $$t>-s$$, meaning $$t$$ is to the right of -$$s$$) or negative (when $$t<-s$$, meaning $$t$$ is to the left of -$$s$$, note that even in this case $$s$$ would be to the left of $$t$$ and relative position of the points shown on the diagram still will be the same). So we get either $$|t+s|=t+s$$ OR $$|t+s|=-t-s$$. In another words: $$t+s$$ is the sum of two numbers from which one $$t$$, is greater than $$s$$. Their sum clearly can be positive as well as negative. Knowing that one is greater than another doesn't help to determine the sign of their sum. Hence: $$t-r=t+s$$ --> $$-r=s$$; OR $$t-r=-t-s$$ --> $$2t=r-s$$. So the only thing we can determine from (2) is: $$t-r=|t+s|$$ Not sufficient. (1)+(2) $$s>0$$ and $$t-r=|t+s|$$. $$s>0$$ --> $$t>0$$ (as $$t$$ is to the right of $$s$$) hence $$t+s>0$$. Hence $$|t+s|=t+s$$. --> $$t-r=t+s$$ --> $$-r=s$$. Sufficient. Answer: C. yangsta8 wrote: Statement 2) This tells us that -S=R but it doesn't tell us anything to either S or R in relation to 0. This is not correct. If we were able to determine that $$-s=r$$, statement (2) would be sufficient. But from (2) we can only say that $$t-r=|t+s|$$. Economist wrote: This is confusing.. Okay, let me put it this way: for number lines, if we have such points...do we trust the sign of the points? and their relative positioning ? Experts please comment. eg. here, do we assume that s cannot be 0, as -s and s are supposed to be distinct +ve and -ve values. also, do we trust the relative positioning ( not distance ) r-s-t as shown in figure? As for $$s$$ to be zero: from statement (1) we can say that $$s$$ can not be zero as it states that $$s>0$$. For (2) we don't know whether -s=s=0 or not. If $$-s=s=0$$, $$s$$ and therefore -$$s$$ are to the left of $$t$$ and (2) would be sufficient in this case. But we don't know that. About the relative position of the points on diagram. Do you remember the question about the two circles and point C? (ds-area-between-circles-85958.html) I didn't know at that time if we could trust the diagram about the C being in the circle or not. You said we should, and you were right. I asked this question to Ian Stewart and he gave me the explanation about the "trust" of the diagrams in GMAT: "In general, you should not trust the scale of GMAT diagrams, either in Problem Solving or Data Sufficiency. It used to be true that Problem Solving diagrams were drawn to scale unless mentioned otherwise, but I've seen recent questions where that is clearly not the case. So I'd only trust a diagram I'd drawn myself. ... Here I'm referring only to the scale of diagrams; the relative lengths of line segments in a triangle, for example. ... You can accept the relative ordering of points and their relative locations as given (if the vertices of a pentagon are labeled ABCDE clockwise around the shape, then you can take it as given that AB, BC, CD, DE and EA are the edges of the pentagon; if a line is labeled with four points in A, B, C, D in sequence, you can take it as given that AC is longer than both AB and BC; if a point C is drawn inside a circle, unless the question tells you otherwise, you can assume that C is actually within the circle; if what appears to be a straight line is labeled with three points A, B, C, you can assume the line is actually straight, and that B is a point on the line -- the GMAT would never include as a trick the possibility that ABC actually form a 179 degree angle that is imperceptible to the eye, to give a few examples). So don't trust the lengths of lines, but do trust the sequence of points on a line, or the location of points within or outside figures in a drawing. " Hope it helps. I am still confused. If we can not assume that S is right to R, How can we assume that T is right to S and R? Can't they all be in the same poit, 0? Please explain. Bunuel _________________ We Shall Overcome... One day... Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8058 Location: Pune, India Re: On the number line shown, is zero halfway between r and s ? [#permalink] ### Show Tags 19 Oct 2016, 07:43 1 This post received KUDOS Expert's post nahid78 wrote: Bunuel wrote: Let me clear this one: NOTE: In GMAT we can often see such statement: $$k$$ is halfway between $$m$$ and $$n$$. Remember this statement can ALWAYS be expressed as: $$\frac{m+n}{2}=k$$. Also in GMAT we can often see another statement: The distance between $$p$$ and $$m$$ is the same as the distance between $$p$$ and $$n$$. Remember this statement can ALWAYS be expressed as: $$|p-m|=|p-n|$$. Back to original question: Is 0 halfway between r and s? OR is $$\frac{r+s}{2}=0$$? --> Basically the question asks is $$r+s=0$$? (1) $$s>0$$, clearly not sufficient. (2) The distance between $$t$$ and $$r$$ is the same as the distance between $$t$$ and -$$s$$: $$|t-r|=|t+s|$$. $$t-r$$ is always positive as $$r$$ is to the left of the $$t$$, hence $$|t-r|=t-r$$; BUT $$t+s$$ can be positive (when $$t>-s$$, meaning $$t$$ is to the right of -$$s$$) or negative (when $$t<-s$$, meaning $$t$$ is to the left of -$$s$$, note that even in this case $$s$$ would be to the left of $$t$$ and relative position of the points shown on the diagram still will be the same). So we get either $$|t+s|=t+s$$ OR $$|t+s|=-t-s$$. In another words: $$t+s$$ is the sum of two numbers from which one $$t$$, is greater than $$s$$. Their sum clearly can be positive as well as negative. Knowing that one is greater than another doesn't help to determine the sign of their sum. Hence: $$t-r=t+s$$ --> $$-r=s$$; OR $$t-r=-t-s$$ --> $$2t=r-s$$. So the only thing we can determine from (2) is: $$t-r=|t+s|$$ Not sufficient. (1)+(2) $$s>0$$ and $$t-r=|t+s|$$. $$s>0$$ --> $$t>0$$ (as $$t$$ is to the right of $$s$$) hence $$t+s>0$$. Hence $$|t+s|=t+s$$. --> $$t-r=t+s$$ --> $$-r=s$$. Sufficient. Answer: C. yangsta8 wrote: Statement 2) This tells us that -S=R but it doesn't tell us anything to either S or R in relation to 0. This is not correct. If we were able to determine that $$-s=r$$, statement (2) would be sufficient. But from (2) we can only say that $$t-r=|t+s|$$. Economist wrote: This is confusing.. Okay, let me put it this way: for number lines, if we have such points...do we trust the sign of the points? and their relative positioning ? Experts please comment. eg. here, do we assume that s cannot be 0, as -s and s are supposed to be distinct +ve and -ve values. also, do we trust the relative positioning ( not distance ) r-s-t as shown in figure? As for $$s$$ to be zero: from statement (1) we can say that $$s$$ can not be zero as it states that $$s>0$$. For (2) we don't know whether -s=s=0 or not. If $$-s=s=0$$, $$s$$ and therefore -$$s$$ are to the left of $$t$$ and (2) would be sufficient in this case. But we don't know that. About the relative position of the points on diagram. Do you remember the question about the two circles and point C? (ds-area-between-circles-85958.html) I didn't know at that time if we could trust the diagram about the C being in the circle or not. You said we should, and you were right. I asked this question to Ian Stewart and he gave me the explanation about the "trust" of the diagrams in GMAT: "In general, you should not trust the scale of GMAT diagrams, either in Problem Solving or Data Sufficiency. It used to be true that Problem Solving diagrams were drawn to scale unless mentioned otherwise, but I've seen recent questions where that is clearly not the case. So I'd only trust a diagram I'd drawn myself. ... Here I'm referring only to the scale of diagrams; the relative lengths of line segments in a triangle, for example. ... You can accept the relative ordering of points and their relative locations as given (if the vertices of a pentagon are labeled ABCDE clockwise around the shape, then you can take it as given that AB, BC, CD, DE and EA are the edges of the pentagon; if a line is labeled with four points in A, B, C, D in sequence, you can take it as given that AC is longer than both AB and BC; if a point C is drawn inside a circle, unless the question tells you otherwise, you can assume that C is actually within the circle; if what appears to be a straight line is labeled with three points A, B, C, you can assume the line is actually straight, and that B is a point on the line -- the GMAT would never include as a trick the possibility that ABC actually form a 179 degree angle that is imperceptible to the eye, to give a few examples). So don't trust the lengths of lines, but do trust the sequence of points on a line, or the location of points within or outside figures in a drawing. " Hope it helps. I am still confused. If we can not assume that S is right to R, How can we assume that T is right to S and R? Can't they all be in the same poit, 0? Please explain. Bunuel You can trust the sequence of points. In the diagram, S is to the right of R so we can take it to be true. Similarly, T is also to the right of S and R. They cannot be the same point. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199
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Re: On the number line shown, is zero halfway between r and s ? [#permalink]
### Show Tags
02 Nov 2016, 13:59
My two cents for all the people who are struggling with the abstract absolute value approach (I was).
Statement 1: We know is insufficient given that we don't know exactly how much to the right is S of 0.
Statement 2: Think of some numbers. Since we do not have any restriction, all of them could be negative:
$$R = -5$$
$$S = -3$$
$$T = -1$$
Then, the distance from R to T is 4, which is the same as from T to -S (4 again). Here, 0 is NOT in the midpoint of R and S. Math:
$$|-5-(-1)| = |-1-3|$$
$$|-4| = |-4|$$
$$4=4$$
Now other numbers:
$$R =-2$$
$$S = 2$$
$$T =4$$
Then, the distance from R to T (6) is equal to the distance of -S to T (6 again). Here, 0 IS IN FACT in the midpoint of R and S. Math:
$$|-2-4| = |-2-4|$$
$$|-6| = |-6|$$
$$6=6$$
So statement 2 is NOT SUFFICIENT.
When we convine, we see that the only possible way would be the second one.
Greetings.
_________________
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Re: On the number line shown, is zero halfway between r and s ? [#permalink] 02 Nov 2016, 13:59
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# Intermediate/Coding representation for Levenshtein Distance
The phrases:
The quick brown fox jumps over the lazy dog [A]
and
The uick brown fox jumps oower the lazy dog [B]
can be compared using Levenshtein Distance algorithm to determine similarity by calculating the minimum number of single character additions, deletions, or replacements are necessary to transform A into B.
I'm interested to know if there is an intermediate representation, or possibly a coding scheme for the Levenshtein Distance. Not for use between two phrases, but just a coding applied to a single phrase such that character index does not affect comparisons.
In B, the 'q' is missing compared to A. A normal string comparison would match 'The ' and then fail at 'uick brown fox...' merely because of a single character offset. The Levenshtein Distance could be used to compare it to the original phrase A for a more forgiving comparison, but in my case, I won't have two phrases, just one.
So, I'm looking for some way of unambiguously coding a sentence in packets of information, little atoms of truth (I'm thinking one packet per character?) that maintain a local ordering and so-on, but if some of the packets are wrong, it doesn't affect later characters.
Each unique phrase should map to one and only one unique encoding/intermediate representation, Sets A' and B'. Computing the Levenshtein Distance of A and B would then be the same as computing the intersection of sets A' = B'.
Alternatively - if this problem does not have a solution (and this sure maps to a well-trodden area of research, I wouldn't be surprised), some convincing argument/proof for its unsolvability.
• Maybe I'm missing something, but it seems like you want a block error-correcting code. en.wikipedia.org/wiki/Block_code – S Huntsman Oct 12 '10 at 21:27
• Certainly seems like that's in the ballpark, but I don't know how I'd apply that here. – Jason Kleban Oct 12 '10 at 22:19
• How about (e.g.) ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=4557110 – S Huntsman Oct 13 '10 at 0:00
• For context, I would like to apply the concept of a fuzzy vault, which is usually applied to biometric measurements, to passphrases. Rather than a set of measurements, this would be a set of atomic truths about a string of text. – Jason Kleban Oct 13 '10 at 21:04
• Thinking more about this - I'm having trouble thinking how to apply the principles of error-correcting codes to solve this. The reason is that the passphrases are arbitrary and selected by a user, not dictated. But maybe it's applicable in some less obvious way? – Jason Kleban Oct 15 '10 at 21:35
There's indeed some research in this vein for the edit distance with some positive and some negative results. (I might not be understanding the question precisely, so I'll try to answer questions I know to answer.)
Here's one interpretation: (I1) you want to compute, for each string A a set f(A) such that, for any two strings A,B, the edit distance ed(A,B) is equal to the symmetric difference between f(A) and f(B) (in some sense the opposite of intersection of the two sets). This question has been well-studied (though by far is not solved), and is known as the question of embedding edit distance into Hamming distance ($\ell_1$). In particular, achieving (I1) precisely is not possible, but is possible up to some approximation (i.e., we approximate ed(A,B) up to some factor):
Here's a slightly more liberal interpretation: (I2) we produce some sketch f(A) for each string A, and we estimate the distance ed(A,B) via some calculation on f(A), f(B) (i.e., not necessarily by taking the symmetric difference). The question is to have f(A) to be much shorter than the length of the original string, $n$ (otherwise, one has a trivial solution by f(A)=A). This interpretation (I2) is more general than (I1) (=easier to achieve), though we do not know of any strictly better solutions. There's some partial progress, where the estimation of ed(A,B) is done from f(A) and B (i.e., one string, say B, is fully known).
There's surely more literature in this vein, but let me know first if this is anywhere close to what you meant.
• Forgot to mention: if you are using edit distance for biometric information, you might want to take a look at <a href="arxiv.org/abs/cs.CR/0602007">this paper of Dodis, Ostrovsky, Reyzin, and Smith</a>. – Alex Andoni Oct 21 '10 at 4:14
• It's actually NOT for biometric information. I find it puzzling that Fuzzy Vaults might work for biometrics, but couldn't be used for something "simpler" such as a string of text. – Jason Kleban Oct 21 '10 at 23:30
• Your answer is great, thanks. Let me think about it and research ... – Jason Kleban Oct 21 '10 at 23:31
• Yes, your descriptions are on point - they feel right. +50! Out of curiosity, is my intended application of this clear through the original question and our comments? – Jason Kleban Oct 25 '10 at 21:27
• Hi Alex, I think the site has a bug. I awarded you the full 50 points, but I get a message that the answer was autoselected - giving you only 25. Sorry about that. – Jason Kleban Oct 28 '10 at 16:22
If the only thing that can happen is that characters disappear, I think you only need to solve the longest common subsequence problem. (A subsequence is a generalization of a substring: there can be multiple locations in the subsequence where material was removed from the larger sequence, not just at the start and/or end.)
Beyond that, have you seen this list of metrics?
I may be misunderstanding your problem statement, but it seems to me that if you define precisely how errors can occur (deletion, transposition, etc.) and then build a suffix tree, it should be possible to have a pretty fast algorithm, after paying the memory and preprocessing cost of building the suffix tree.
• Thanks for this! You showed me some new things to consider. – Jason Kleban Oct 20 '10 at 22:06
• Errors could occur as inserts, deletes. Transpositions and swaps would be nice, but are compositions of the basic inserts and deletes - in case that makes it easier to satisfy. – Jason Kleban Oct 20 '10 at 22:44
This is just a thought, not a solution, but too long for a comment.
Could your set/representation be an alphabetic building of the string?
Example (for A):
1. Start with the empty string (^$) 2. Insert a between 1st ^ and 1st$ (now: ^a$) 3. Insert b between 1st ^ and 1st a (now: ^ba$)
4. Insert c between 1st ^ and 1st b (now: ^cba$) 5. Insert d between 1st b and 1st a (now: ^cbda$)
6. Insert d between 1st a and 1st ^ (now: ^cbdad$) and so on... Your representation would be the steps you took to build the string (in alphabetic order): The element {a, {1,^}, {1,$}} represents step 2, while {d, {1,b}, {1,a}} represents the 5th step.
Provided you do this alphabetically in each case, you might have something you can use.
A complementary thought: start with enough copies of each letter "abcdd" (for the first 4) and then keep track of your transpositions to build the string. We're now moving vaguely into permutation theory.
[BTW, it's "jumps", not "jumped", otherwise there's no 's' -- yes, I realize you never said it was a pangram]
• First of all, interesting idea. ... And thanks for the 'Jumps' correction - I changed the post accordingly. – Jason Kleban Oct 20 '10 at 21:45
• I think a solution couldn't have a "5th step". Order can't matter - the packets can't be strongly linked to each other in ordering or reference - what if one packet is wrong/missing? – Jason Kleban Oct 20 '10 at 22:46
It sounds to me like your simple thing should work. Each packet contains the position and the character, e.g.
The = <1, T>, <2, h>, <3, e>
Then you compare the first pair of A with the first pair of B etc. This should give you Levenshtein.
• But what if an early character in the string is missing, then all of the later pairings would be off by 1, right? If we take out 'q' in [B], then [B]'s <u, 5> != [A]'s <u, 6>, [B]'s <i, 6> != [A]'s <i, 7> – Jason Kleban Oct 20 '10 at 21:55
• Put another way, if I understand your suggestion correctly it is equivalent to string/sequence equality. – Jason Kleban Oct 20 '10 at 21:59
• @uosɐſ: the sender would know the correct order though, right? They would send <u, 6>; the receiver might get it as the fifth packet, but if they get it at all they'll know it's the sixth letter. – Xodarap Oct 21 '10 at 14:06
• The idea is that you come up with a set T(s) for each string s, and then just compare T(s1) and T(s2) as sets to find T(s1)-T(s2) for example, and the number of elements in that difference is the distance. There is no "sender" and "receiver" – user1854 Oct 21 '10 at 17:42
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# Find the Equation of the Ellipse in the Case: (Iii) Focus is (−2, 3), Directrix is 2x + 3y + 4 = 0 and E = 4 5 - Mathematics
Find the equation of the ellipse in the case:
focus is (−2, 3), directrix is 2x + 3y + 4 = 0 and e = $\frac{4}{5}$
#### Solution
$\text{ Let S( - 2, 3) be the focus and ZZ' be the directrix . }$
$\text{ Let P(x, y) be any point on the ellipse and let PM be the perpendicular from P on the directrix } .$
$\text{ Then by the definition, we have: }$
$SP = e \times PM$
$\Rightarrow SP = \frac{4}{5} \times PM$
$\Rightarrow \frac{5}{4}SP = PM$
$\Rightarrow \frac{25}{16} \left( SP \right)^2 = {PM}^2$
$\Rightarrow \frac{25}{16}\left[ \left( x + 2 \right)^2 + \left( y - 3 \right)^2 \right] = \left| \frac{2x + 3y + 4}{\sqrt{2^2 + 3^2}} \right|^2$
$\Rightarrow \frac{25}{16}\left[ x^2 + 4 + 4x + y^2 + 9 - 6y \right] = \frac{4 x^2 + 9 y^2 + 16 + 12xy + 24y + 16x}{13}$
$\Rightarrow 325\left( x^2 + 4 + 4x + y^2 + 9 - 6y \right) = 16\left( 4 x^2 + 9 y^2 + 16 + 12xy + 24y + 16x \right)$
$\Rightarrow 325 x^2 + 1300 + 1300x + 325 y^2 + 2925 - 1950y = 64 x^2 + 144 y^2 + 256 + 192xy + 384y + 256x$
$\Rightarrow 261 x^2 + 181 y^2 + 1044x - 2309y - 192xy + 3969 = 0$
$\text{ This is the required equation of the ellipse } .$
Is there an error in this question or solution?
#### APPEARS IN
RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Exercise 26.1 | Q 2.3 | Page 22
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# Justify left and right inside example environment
I use the "Numbered Examples (multiline)" environment in a LyX document (class : Book (Standard Class)). The text is not fully justified (it is not right justified) in the examples. Is there a way to modify the environment ? I would prefer to not define a new environment and change all my examples into this new environment and I would prefer not to type ERT[\justify] in all the examples.
• Welcome to TeX - LaTeX! You do not need to add your name or thanks to the question, your name is already associated with the question and thanks is expressed by upvoting and accepting answers – Andrew Swann Mar 11 '16 at 11:26
\usepackage{ragged2e}
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Please use this identifier to cite or link to this item: http://hdl.handle.net/10525/1700
Title: The Operators Aγ = γA + -γA for a Class of Nondissipative Operators A with a Limit of the Corresponding Correlation Function Authors: Borisova, Galina Keywords: Operator ColligationNondissipative CurveCorrelation FunctionWave OperatorScattering Operator Issue Date: 2003 Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences Citation: Serdica Mathematical Journal, Vol. 29, No 2, (2003), 109p-140p Abstract: In this work we present the operators Aγ = γA + -γA with Re γ = 1/2 in the case of an operator A from the class of nondissipative operators generating nonselfadjoint curves, whose correlation functions have a limit as t → ±∞. The asympthotics of the stationary curves e^(itAγ)f as t → ±∞ onto the absolutely continuous subspace of Aγ are obtained. These asymptotics and the obtained asymptotics in [9] of the nondissipative curves e^(itA)f allow to construct the scattering theory for the couples (Aγ , A) and (A, Aγ). We consider the basic terms from the scattering theory - wave operators, a scattering operator and the question of a similarity of A and Aγ. We obtain explicitly the wave operators, the scattering operator and the similarity of A and Aγ. Description: 2000 Mathematics Subject Classification: Primary 47A48, Secondary 60G12 URI: http://hdl.handle.net/10525/1700 ISSN: 1310-6600 Appears in Collections: Volume 29 Number 2
Files in This Item:
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# 9.4 Application of External Forces
In 2009, three methods for optimizing the geometry of a molecule under a constant external force were introduced, which were called Force-Modified Potential Energy Surface (FMPES),Ong:2009 External Force is Explicitly Included (EFEI),Ribas-Arino:2009 and Enforced Geometry Optimization (EGO).Wolinski:2009 These methods are closely related, and the interested reader is referred to Ref. Stauch:2016 for a detailed discussion of the similarities and differences between them. For simplicity, we will stick to the term EFEI in this Section. An EFEI calculation is a geometry optimization in which a constant that is equal to the external force is added to the nuclear gradient of two atoms specified by the user. The external force is applied along the vector connecting the two atoms, thus driving them apart. The geometry optimization converges when the restoring force of the molecule is equal to the external force. The EFEI method can also be used in AIMD simulations (see Section 9.8), in which case the force is added in every time step. The basic syntax for specifying EFEI calculations is as follows.
$efei atom1 atom2 force1 atom3 atom4 force2 ...$end
Here, atom1 and atom2 are the indices of the atoms to which a force is applied. force1 is the sum of the force values that acts on atom1 and atom2 in nanoNewtons (nN). If this value is positive, a mechanical force of magnitude force1/2 acts on each of these atoms, thus driving them apart. If it is negative, an attractive force acts between the atoms. Optionally, additional pairs of atoms that are subject to a force can be specified by adding lines in the $efei section. Example 9.9 EFEI calculation of hydrogen peroxide with a constant stretching force of 2.5 nN acting on each oxygen atom $molecule
0 1
O -0.7059250062 -0.1776745132 -0.0698000882
O 0.7059250062 0.1776745132 -0.0698000882
H 1.0662092915 -0.5838921799 0.4181150580
H -1.0662092915 0.5838921799 0.4181150580
$end$rem
JOBTYPE opt
EXCHANGE b3lyp
BASIS 6-31G*
$end$efei
1 2 5
\$end
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Browse Questions
If x , y , z are in GP then , $\;\large\frac{1}{x^2-y^2}+\frac{1}{y^2}\;$ equals
$(a)\;\large\frac{1}{z^2-y^2}\qquad(b)\;\large\frac{1}{y^2-z^2}\qquad(c)\;\large\frac{1}{z^2-x^2}\qquad(d)\;\large\frac{1}{y^2-x^2}$
Answer : (b) $\;\large\frac{1}{y^2-z^2}$
Explanation : $\;\large\frac{1}{x^2-y^2}+\frac{1}{y^2}$
$=\frac{1}{x^2}(\large\frac{1}{1-r^2}+\frac{1}{r^2})$
$=\frac{1}{x^2}(\frac{1}{r^2}*\large\frac{1}{1-r^2})$
$=\large\frac{1}{y^2-z^2}\;.$
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# Trigonometry Examples
Rewrite as .
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Simplify.
Multiply by to get .
One to any power is one.
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## Galois Theory and the Solubility of Polynomials by Radicals
Given a polynomialwith rational coefficients, can you express the roots of p(x) using only rational numbers, multiplication, division, addition, subtraction and taking integer roots? So, for example, we can solvein this way because
The coefficientsare rational, and we have only used multiplication, division, addition, subtraction and square root.
We can find more complicated examples, suppose p(x)=x^4 +4x^2+2. We can write this asso the solutions will satisfy The roots are
When we can find the solutions for a polynomial with rational coefficients using only rational numbers and the operations of addition, subtraction, division, multiplication and finding nth roots, we say thatis soluble by radicals.
Using Galois theory, we can prove that if the degree of(the highest power ofin) is less than 5 then the polynomial is soluble by radicals, but there are polynomials of degree 5 and higher not soluble by radicals. In other words, polynomials of degree 5 whose solutions cannot be written down using nth roots and the arithmetical operations, no matter how complicated.
We can construct a group to act of the set of roots of a polynomial – called a group action. Such a group will be an automorphism of the roots. For example the group acting on the roots of the polynomialare
and
For a polynomial of degreethe group will be a subgroup ofThe group generated will have subgroups which may or may not be normal inIf the subgroup is normal inthe the polynomial is soluble by radicals else it is not. Forandall the subgoups are normal butandforhas subgroups which are not normal, so polynomials of degree 5 or greater are not soluble by radicals in general although some may be.
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# What is the partial-fraction decomposition of (x^2+2x+7)/(x(x-1)^2)?
Jun 24, 2015
$\frac{{x}^{2} + 2 x + 7}{x {\left(x - 1\right)}^{2}} = \frac{7}{x} - \frac{6}{x - 1} + \frac{10}{x - 1} ^ 2$
#### Explanation:
The factors on the denominator are obvious. They are all linear, but one of them is a double factor. So we want $A , B , \mathmr{and} C$ so that:
$\frac{{x}^{2} + 2 x + 7}{x {\left(x - 1\right)}^{2}} = \frac{A}{x} + \frac{B}{x - 1} + \frac{C}{x - 1} ^ 2$
Combining the rations on the right, we get a numerator pf:
$A \left({x}^{2} - 2 x + 1\right) + B \left({x}^{2} - x\right) + C x = A {x}^{2} - 2 A x + A + B {x}^{2} - B x + C x$
$= \left(A + B\right) {x}^{2} + \left(- 2 A - B + C\right) x + A$.
Setting the coefficients equal to those of the original numerator, ${x}^{2} + 2 x + 7$, we get:
$A + B = 1$
$- 2 A - B + C = 2$
$A = 7$
It is immediate that $A = 7$ and from that and the first equation (the coefficients of ${x}^{2}$), we get $B = - 6$. Substituting in the middle equation and solving for $C$, we get $C = 10$.
$\frac{{x}^{2} + 2 x + 7}{x {\left(x - 1\right)}^{2}} = \frac{7}{x} - \frac{6}{x - 1} + \frac{10}{x - 1} ^ 2$
It is a good idea to check the answer by getting the common denominator. (I did that on paper, but I'm not going to type it up.)
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Change search
On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces
Kyushu Institute of Technology.
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.
2001 (English)In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 144, no 3, 275-295 p.Article in journal (Refereed) Published
##### Abstract [en]
Some relations between the James (or non-square) constant $J(X)$ and the Jordan-von Neumann constant $C_NJ(X),$ and the normal structure coefficient $N(X)$ of Banach space $X$ are investigated. Relations between $J(X)$ and $J(X^*)$ are given as an answer to a problem of {\it J. Gao} and {\it K.-S. Lau} in [Stud. Math. 99, No. 1, 41-56 (1991; Connections between $C_NJ(X)$ and $J(X)$ are also shown. The normal structure coefficient of a Banach space is estimated by the $C_NJ(X)$-constant, which implies that a Banach space with $C_NJ(X),$-constant less than $\frac{5}{4}$ has the fixed point property.
##### Place, publisher, year, edition, pages
2001. Vol. 144, no 3, 275-295 p.
Mathematics
##### Identifiers
Local ID: 87813470-a575-11db-9811-000ea68e967bOAI: oai:DiVA.org:ltu-9789DiVA: diva2:982727
##### Note
Validerad; 2001; 20070112 (kani)Available from: 2016-09-29 Created: 2016-09-29Bibliographically approved
#### Open Access in DiVA
##### File information
File name FULLTEXT01.pdfFile size 218 kBChecksum SHA-512
Type fulltextMimetype application/pdf
#### Search in DiVA
Maligranda, Lech
##### By organisation
Mathematical Science
##### In the same journal
Studia Mathematica
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Lal il { = that that 1 atate U Is the M mnwmmim use [Laffqrmulas 1 computations the blood interested
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# Possible Jordan Canonical Forms Given Minimal Polynomial
I was supposed to find all possible Jordan canonical forms of a $5\times 5$ complex matrix with minimal polynomial $(x-2)^2(x-1)$ on a qualifying exam last semester. I took the polynomial to mean that there were at least two 2's and one 1 on the main diagonal, and that the largest Jordan block with eigenvalue 2 is $2\times 2$ while the largest Jordan block with eigenvalue 1 is $1\times 1$. Did I miss any matrices or interrupt the minimal polynomial incorrectly?
\begin{pmatrix} 2 &1 &0 &0 &0\\ 0 &2 &0 &0 &0\\ 0 &0 &2 &1 &0\\ 0 &0 &0 &2 &0\\ 0 &0 &0 &0 &1 \end{pmatrix}
\begin{pmatrix} 2 &1 &0 &0 &0\\ 0 &2 &0 &0 &0\\ 0 &0 &2 &0 &0\\ 0 &0 &0 &2 &0\\ 0 &0 &0 &0 &1 \end{pmatrix}
\begin{pmatrix} 2 &1 &0 &0 &0\\ 0 &2 &0 &0 &0\\ 0 &0 &2 &0 &0\\ 0 &0 &0 &1 &0\\ 0 &0 &0 &0 &1 \end{pmatrix}
\begin{pmatrix} 2 &1 &0 &0 &0\\ 0 &2 &0 &0 &0\\ 0 &0 &1 &0 &0\\ 0 &0 &0 &1 &0\\ 0 &0 &0 &0 &1 \end{pmatrix}
\begin{pmatrix} 2 &0 &0 &0 &0\\ 0 &2 &0 &0 &0\\ 0 &0 &1 &0 &0\\ 0 &0 &0 &1 &0\\ 0 &0 &0 &0 &1 \end{pmatrix}
• You must have a Jordan block associated to the eigenvalue 2 of size 2. Otherwise the minimal polynomial would be $(x-1)(x-2)$. – Brandon Carter Jan 6 '13 at 0:15
• I see. So the last matrix is knocked off. Good. – Frank White Jan 6 '13 at 0:18
Yes, you did.
Based on the minimal polynomial, you must have a two-by-two Jordan block for eigenvalue 2 and a one-by-one block for eigenvalue 1. You can fill in the five-by-five matrix with more of those blocks or with one-by-one blocks for eigenvalue 2. Using those rules yields precisely your first four matrices. Your fifth matrix is not correct.
Furthermore, you can permute the blocks. Thus,
• your first matrix yields 3!/2! = 3 Jordan forms,
• your second and third matrices yield 4!/2! = 12 forms each, and
• your four matrix yields 4!/3! = 4 forms
for a total of 3 + 2 · 12 + 4 = 31 forms.
• It depends what you define JCF to be. Many insist that for a given eigenvalue the block sizes decrease. – ancientmathematician Aug 5 '18 at 17:08
• @ancientmathematician In that case, Frank White's first and forth matrices still yield 3 and 4 forms because the blocks for a given eigenvalue are the same size. The second matrix, however, now yields only 4 forms, and the third matrix yields only 6 forms for a total of 17 forms. – user0 Aug 6 '18 at 15:11
• I agree with all the calculations.. – ancientmathematician Aug 6 '18 at 15:51
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## Friday, June 29, 2012
### My ICML 2012 Notables
I've already devoted entire blog posts to some of the ICML 2012 papers, but there are some other papers that caught my attention for which I only have a quick comment.
• Online Structured Prediction via Coactive Learning: read the full blog post.
• Predicting Accurate Probabilities with a Ranking Loss: read the full blog post.
• Training Restricted Boltzmann Machines on Word Observations. I haven't used RBMs in over a decade, for practical text classification problems a bag-of-bigrams representation is often sufficient, and LDA is my go-to technique for unsupervised feature extraction for text. So why do I like this paper? First, the computational efficiency improvement appears substantial, which is always of interest: I like deep learning in theory, but in practice I'm very impatient. Second the idea of discovering higher order structure in text (5-grams!) is intriguing. Third (like LDA) the technique is clearly more generally applicable and I wonder what it would do on a social graph. That all suggests there is some chance that I might actually try this on a real problem.
• Fast Prediction of New Feature Utility: I'm constantly in the situation of trying to chose which features to try next, and correlating with the negative gradient of the loss function makes intuitive sense.
• Plug-in Martingales for Testing Exchangeability On-Line: how awesome would it be if VW in online learning mode could output a warning that says the input data does not appear to be generated by an exchangeable distribution; try randomly shuffling your data to improve generalization.''
• Dimensionality Reduction by Local Discriminative Gaussians: This seems imminently practical. The major limitation is that it is a supervised dimensionality reduction technique, so it would apply to cases where there is one problem with a deficit of labeled data and a related problem using the same features with an abundance of labeled data (which is a special case of Transfer Learning). I usually find myself in the few labeled data and lots of unlabeled data'' case demanding an unsupervised technique, but that could be because I don't ask myself the following question often enough: is there a related problem which has lots of training data associated with it?''
• Finding Botnets Using Minimal Graph Clusterings: Very entertaining. I was asked in a job interview once how I would go about identifying and filtering out automated traffic from search logs. There's no right answer'', and black-letter machine learning techniques don't obviously apply, so creativity is at a premium.
## Wednesday, June 27, 2012
### Rank+IR
Mennon et. al. have a paper at ICML 2012 called Predicting Accurate Probabilities with a Ranking Loss. The basic idea is to train a classifier on a ranking loss (e.g., AUC), then post-process the classifier scores with isotonic regression to calibrate the classifier. In contrast with training a classifier using a proper scoring rule (e.g., logistic regression), this procedure non-parametrically explores the space of link functions and the claim is this leads to better results. Note exploring the space of link functions non-parametrically is intuitively safe'' from a model complexity standpoint because this is a one-dimensional procedure which operates on the scores output by the underlying classifier.
It turns out we accidentally backed into this at eHarmony. When I joined the production system delivered matches sequentially so we started with a ranking loss. Later the production system switched to using a linear program to deliver matches, and the easiest thing to do was to add a calibration step at the end of the training pipeline, and we did isotonic regression with linear interpolation. We wanted to switch to directly training for classification with a proper scoring rule, but we started subsampling the negatives so we needed to continue to calibrate the classifier and therefore it never happened. The whole time we suspected we were being incoherent.'' Hey, it's better to be lucky than good. Now, if I find myself in a similar situation in the future, I'll be able to articulate a rationale for the approach.
The meta-lesson here is if you are an applied machine learning practitioner and you see a paper with Charles Elkan's name on it, you should read it. I've yet to be disappointed.
## Tuesday, June 26, 2012
### A Thought on Link Prediction
I'm reading a paper by Richard et. al. from the ICML 2012 paper list called Estimation of Simultaneously Sparse and Low Rank Matrices. I'm not sure why until now I was conflating these two ideas but in retrospect they are clearly different and one might want to optimize for both. Since the Latent Feature Log Linear (LFLL) model of Mennon and Elkan is in spirit a low-rank matrix factorization algorithm I was wondering how to simultaneously enforce sparsity in it; I think using an $L_1$ regularizer on the latent features might be worth trying.
However the paper also got me thinking about link prediction. Here's a quote from the paper:
Link prediction - the matrix $A$ is the adjacency matrix of a partially observed graph; entries are 0 for both not-existing and undiscovered links. The search space is unrestricted as before and the matrix $S$ contains the scores for link prediction; the ideal loss function is the empirical average of the zero-one loss for each coefficient, $l_{E} (S, A) = \frac{1}{|E|} \sum_{(i,j) \in E} 1_{(A_{ij} - 1/2) \cdot S_{ij} \leq 0}.$
So I read that as, this is a P-U problem that we are reducing to pointwise classification.'' However my preferred method for P-U problems is to reduce to ranking (AUC loss). What would that look like for link prediction?
1. Instances are edges (i.e., pairs of vertices plus dyadic specific information).
2. Reduction of AUC is to pairwise classification, so pairs of edges, or pairs of pairs of vertices.
3. Each positive (observed) edge in the adjacency graph would be paired with an unlabeled (unobserved or unexplored) edge, the latter perhaps drawn uniformly from all possible edges; or possibly from all possible edges given one vertex (per-vertex AUC'').
4. The final classification model could be purely latent (e.g., pure LFLL), purely explicitly feature driven (e.g., bilinear implemented with VW), or a combination (e.g., LFLL with side information).
1. In my experience LLFL with side information is very tricky to train, unlike pure LLFL.
Next time I run into a link prediction problem I'm going to give this a whirl.
## Monday, June 25, 2012
### Coactive Learning
Shivaswamy and Joachims have a paper called Online Structured Prediction via Coactive Learning at ICML 2012 this year. Joachims, of course, is associated with a classic line of research which I'll summarize thusly: attempting to impute absolute relevance scores from behavioral data exhaust is not effective, and that imputing relative preferences leveraging an attentional model (e.g., serial scan) is more effective. This is one of those deep tricks'' that you can carry with you into many different situations.
So the classic example is when you have a search engine result, and you get just one click at a particular position $p$, and your attentional model assumes that the user considered every result up to that position plus one more. Therefore the partial preferences $\forall x \in [1, p + 1], x \neq p: r_p > r_x$ are revealed and added to the (ranking) training set.
Later in my career I began to appreciate stochastic contextual bandits, specifically the importance of debiasing the historical state-action density in order to get consistent estimates. That left me with an inconsistency: on the one hand, optimizing a search engine with click feedback is definitely Learning Through Exploration, since you only get information about the relative preferences of (a subset of) the items presented. On the other hand I'm not attempting to debias the historical state-action density when I'm doing straight Joachims.
I was hoping this paper would resolve this difficulty for me. It did, but not in the way I expected; the contextual bandit literature is only referred to in the introduction for comparative purposes. Instead the authors make the following assumptions:
1. User loss is convex in (linear) utility differences.
2. Users only suggest improvements (i.e., user feedback always points downhill'').
3. Users only suggest significant improvements (i.e., feedback states have a utility increment at least proportional to the increment to the optimum).
Under these circumstances it is sensible that a Perceptron-style algorithm achieves a good regret bound. The authors also explore relaxations of these assumptions (e.g., improvements are only significant in expectation, or feedback occasionally points downhill) and the resulting degradation of the regret guarantee.
I suspect the analysis does not look how I anticipated because, subject to the conditions of the previous paragraph, the user feedback can be chosen adversarially. Nonetheless it could be interesting to consider a contextual bandit style'' formulation, e.g., instead of learning the reward associated with the chosen arm, one learns the difference between the reward of the chosen arm and another arm. A good place to start would be the literature on contextual bandits with controlled side information, but a key difference here is that the user feedback is not under control of the algorithm.
## Thursday, June 7, 2012
### Stealth No More!
The startup I'm currently at publicly launched today. It's a social image sharing site called LoveIt. This is a crowded space at the moment, but we've tried to throw in some innovative new features. One machine learning related bit that I worked on is the recommendation system; here's an example screenshot with the recommendations in the bottom right hand side.
The image for a mashup by DJ Earworm (who is totally awesome!). In this case the second recommendation is a music collection which is very sensible, but the first recommendation is more questionable (focusing on the costume ball aspect). Hopefully the system will get better as we generate more behavioral data exhaust. I have noticed image recommendation is more forgiving than text recommendation: images have less precise meaning so people are more willing to invent why a quirky recommendation makes sense.
Conceptually the system is heavily Elkan inspired. The implementation is a combination of Elasticsearch and Vowpal Wabbit, strung together with Erlang. The tricky part is getting it to compute something quickly (circa 100ms), and both Elasticsearch and Vowpal Wabbit are excellent pieces of software in this regard!
#### The Bigger Picture
When I first started on the internet, the most common demand for machine learning I encountered was for optimizing performance marketing (the other big one would have been algorithmic search, but southern California wasn't a major player in that space). Nowadays there are many big smart companies focused on the science of advertising. In my opinion, if you have some machine learning acumen and some plucky post-series-A startup claiming to revolutionize internet advertising with a new algorithm attempts to recruit you, run the other way! There are probably still many smaller exits to be had in this space selling to the major ad networks, but unless you have a large equity share it won't change your life.
Fortunately there is a new nexus of ubiquitous machine learning need: content recommendation, personalization, summarization, and visualization. This is driven by the intersection of several trends, including the rise in user-generated content, social networks, and smartphones. For example, Twitter has turned everybody into an intelligence analyst lost in a sea of intercepts. Technologies that can scan all of Twitter and surface the (personalized) good stuff in real-time would be very interesting. Furthermore, as Google has proven, if you position yourself as a trusted discovery tool for users you can easily monetize. Thus if you get a recruiting call from a startup claiming to attack such problems, my advice is to seriously consider it.
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# ASTM International - ASTM B584-08a
## Standard Specification for Copper Alloy Sand Castings for General Applications
inactive
Organization: ASTM International Publication Date: 15 October 2008 Status: inactive Page Count: 6 ICS Code (Copper products): 77.150.30
##### scope:
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a composite specification replacing former documents as shown in Table 1.
Note 1-Other copper alloy castings are included in the following ASTM specifications: B 22, B 61, B 62, B 66, B 67, B 148, B 176, B 271, B 369, B 427, B 505/B 505M, B 763, B 770, and B 806.
1.2 Component part castings produced to this specification may be manufactured in advance and supplied from stock. In such cases the manufacturer shall maintain a general quality certification of all castings without specific record or date of casting for a specific casting.
1.3 The values stated in inch-pound units are to be regarded as standard. The values given in parentheses are mathematical conversions to SI units that are provided for information only and are not considered standard.
TABLE 1 Nominal Compositions
ClassificationCopper
Alloy
UNS
No.
Previous
Designation
inum
Man-
ganese
Sili-
con
Nio-
bium
Bis-
muth
Leaded red brassC83450. . .. . .88 21. . .. . . . . .. . .. . .. . .
C83600B 145-4A85-5-5-5 or No. 1 composition85 5 5 5. . .. . .. . . . . .. . .. . .. . .
C83800B 145-4Bcommercial red brass, 83-4-6-783 4 6 7. . .. . .. . .. . .. . . . . .
Leaded semi-red brassC84400 B 145-5Avalve composition, 81-3-7-981 3 7 9 . . .. . .. . .. . .. . . . . .. . .
C84800B 145-5Bsemi-red brass, 76-2½-6½-157615 . . .. . .. . .. . .. . . . . .. . .
Leaded yellow brassC85200 B 146-6Ahigh-copper yellow brass72 1 324 . . .. . .. . .. . .. . . . . .. . .
C85400B 146-6Bcommercial No. 1 yellow brass67 1 3 29. . .. . .. . . . . .. . .. . .. . .
C85700B 146-6Cleaded naval brass 61 1 137. . .. . . . . .. . .. . .. . .. . .
High-strength yellow brassC86200 B 147-8Bhigh-strength manganese bronze63. . . . . .27. . .343. . .. . .. . .
C86300B 147-8Chigh-strength manganese bronze61. . .. . .27. . .3 63 . . . . . .. . .
C86400B 147-7Aleaded manganese bronze58 1 1 38. . .1½½ . . .. . .. . .
C86400B 132-A
C86500B 147-8ANo. 1 manganese bronze58. . .. . .39. . .1 11 . . . . . .. . .
C86700B 132-Bleaded manganese bronze58 1 1 34. . .2 22 . . .. . .. . .
Silicon bronze + silicon brassC87300 B 198-12Asilicon bronze95 . . .. . .. . .. . .. . . . . .14. . .. . .
C87400B 198-13Asilicon brass 82. . .½14 . . .. . .. . .. . .. . .. . .
C87500B 198-13Bsilicon brass 82. . .. . . 14. . .. . .. . . . . .4. . .. . .
C87600B 198-13Csilicon bronze 91. . .. . . 5. . .. . .. . . . . .4. . .. . .
C87610B 198-12Asilicon bronze 92. . .. . . 4. . .. . .. . . . . .4. . .. . .
C87850
A
. . .silicon brass76. . .. . .20.9. . .. . .. . .. . .3. . .. . .
Bismuth selenium brassC89510
B
. . . sebiloy I875. . . 5. . . . . .. . . . . .. . . . . .1.0
C89520
C
. . . sebiloy II 86. . . 5 . . .. . .. . . . . .. . . . . .1.9
C89530
D
86.54.78.01.5
C8953586.53.07.00.651.4
Bismuth red
brass
C89833. . .bismuth brass895. . .3. . .. . .. . .. . .. . .. . .2.2
Bismuth bronzeC89836. . .lead-free bronze89.55.5. . .3.0. . .. . .. . .. . .. . .. . .2
Bismuth semi-red brassC89844 . . .bismuth brass84½ 4. . . 8. . .. . .. . . . . . . . .. . .3
Tin bronze + leaded tin bronzeC90300 B 143-1Bmodified "G" bronze, 88-8-0-488 8 . . . 4. . .. . . . . .. . .. . .. . .. . .
C90500B 143-1A"G" bronze, 88-10-0-28810. . . 2. . .. . .. . .. . . . . .. . .. . .
C92200B 143-2Asteam or valve bronze-Navy "M"88 6 . . .. . .. . .. . .. . . . . .. . .
C92210. . .. . .88 5 2 41. . . . . .. . .. . .. . .. . .
C92300B 143-2B87-5-1-4, Navy PC 87 8 1 4. . .. . . . . .. . .. . .. . .. . .
C92600. . .87-10-1-2 8710 1 2. . . . . . . . .. . .. . . . . .. . .
High-lead tin bronzeC93200 B 144-3B83-7-7-383 7 7 3. . .. . .. . .. . . . . .. . .. . .
C93500B 144-3C85-5-9-1 85 5 9 1. . .. . . . . .. . .. . .. . .. . .
C93700B 144-3A80-10-10 8010 10. . .. . .. . . . . . . . .. . .. . .. . .
C93800B 144-3D78-7-15 78 715. . .. . .. . .. . . . . .. . .. . .. . .
C94300B 144-3E71-5-24 71 524. . .. . .. . .. . . . . .. . .. . .. . .
Nickel-tin bronze + leaded nickel-tin bronzeC94700B 292-Anickel-tin bronze Grade "A"88 5 . . . 25 . . .. . .. . .. . . . . .. . .
C94800B 292-Bleaded nickel-tin bronze Grade "B"87 5 1 25. . .. . . . . .. . .. . .. . .
C94900. . .leaded nickel-tin bronze Grade "C"80 5 5 55. . .. . . . . .. . .. . .. . .
Spinodal alloyC96800. . .. . .82 8. . . . . .10. . .. . . . . .. . .0.2. . .
Leaded nickel bronzeC97300 B 149-10A12 % leaded nickel silver57 2 920 12. . .. . .. . . . . .. . .. . .
C97600B 149-11A20 % leaded nickel silver64 4 4 820. . . . . .. . .. . .. . .. . .
C97800B 149-11B25 % leaded nickel silver66 5 2 225. . . . . .. . .. . .. . .. . .
A Phosphorus 0.12.
B Selenium 0.5.
C Selenium 0.9.
D Selenium 0.20.
##### abstract:
This specification covers requirements for copper alloy sand castings for general applications. The components part casting may be manufactured in advance and supplied from stock. The castings... View More
### Document History
April 1, 2014
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1.2 This is a...
April 1, 2013
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1.2 This is a...
May 15, 2012
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1.2 This is a...
May 1, 2012
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1.2 This is a...
April 1, 2011
Standard Specification for Copper Alloy Sand Castings for General Applications
This specification covers requirements for copper alloy sand castings for general applications. The components part casting may be manufactured in advance and supplied from stock. The castings shall...
November 15, 2009
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
November 1, 2009
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
ASTM B584-08a
October 15, 2008
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
April 1, 2008
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
October 1, 2006
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in . This is a composite...
February 1, 2006
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in . This is a composite...
November 1, 2005
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
July 1, 2004
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification covers requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is a...
May 10, 2000
Standard Specification for Copper Alloy Sand Castings for General Applications
1.1 This specification establishes requirements for copper alloy sand castings for general applications. Nominal compositions of the alloys defined by this specification are shown in Table 1. This is...
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3 years ago
# A long-lived remnant neutron star after GW 170817 inferred from its associated kilonova.
Yun-Wei Yu, Zi-Gao Dai
The successful joint observation of the gravitational wave event GW170817 and its multi-wavelength electromagnetic counterparts first enables human to witness a definite merger event of two neutron stars (NSs), which indicates the coming of a new era in multi-messenger astronomy. This historical event confirms the origin of short-duration gamma-ray bursts (GRBs), and in particular, identifies the theoretically-predicted kilonova phenomenon that is powered by radioactive decays of r-process heavy elements. However, whether a long-lived remnant NS can be formed during this merger event remains unknown, although such a central engine has been suggested by afterglow observations of some short-duration GRBs. By invoking this long-lived remnant NS, we here propose a model of hybrid energy sources for the kilonova, AT2017gfo, associated with GW 170817. While the early emission of AT2017gfo is still powered radioactively as usually suggested, its late emission is primarily caused by delayed energy injection from the remnant NS. In our model, only one single opacity is required and an intermediate value of $\kappa\simeq 1.0 cm^2g^{-1}$ is found, which could be naturally provided by lanthanide-rich ejecta that is deeply ionized by the emission from the NS wind. These self-consistent results indicate the formation of a long-lived remnant NS during the merger event of GW 170817, which further provides a very stringent constraint on the equation of state for dense NS matter.
Publisher URL: http://arxiv.org/abs/1711.01898
DOI: arXiv:1711.01898v2
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# strictly upper triangular matrix
A strictly is an upper triangular matrix which has $0$ on the main diagonal. Similarly a strictly lower triangular matrix is a lower triangular matrix which has $0$ on the main diagonal. i.e.
$\begin{bmatrix}0&a_{12}&a_{13}&\cdots&a_{1n}\\ 0&0&a_{23}&\cdots&a_{2n}\\ 0&0&0&\cdots&a_{3n}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&0\end{bmatrix}$
A strictly lower triangular matrix is of the form
$\begin{bmatrix}0&0&0&\cdots&0\\ a_{21}&0&0&\cdots&0\\ a_{31}&a_{32}&0&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&a_{n3}&\cdots&0\end{bmatrix}$
Title strictly upper triangular matrix StrictlyUpperTriangularMatrix 2013-03-22 13:42:15 2013-03-22 13:42:15 Daume (40) Daume (40) 8 Daume (40) Definition msc 15-00 supertriangular strictly lower triangular matrix
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# If the highest common factor of 2,472, 1,284 and positive integer N is
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If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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12 Nov 2019, 02:50
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Question Stats:
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If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?
(A) $$2^2 * 3^2 * 7$$
(B) $$2^2 * 3^3 * 103$$
(C) $$2^2 * 3^2 * 5$$
(D) $$2^2 * 3 * 5$$
(E) None of these
Are You Up For the Challenge: 700 Level Questions
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Joined: 02 Aug 2009
Posts: 8310
Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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12 Nov 2019, 07:38
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?
(A) $$2^2 * 3^2 * 7$$
(B) $$2^2 * 3^3 * 103$$
(C) $$2^2 * 3^2 * 5$$
(D) $$2^2 * 3 * 5$$
(E) None of these
Are You Up For the Challenge: 700 Level Questions
$$2472=2*2*2*3*103=2^3*3*103$$.
$$1284=2*2*3*107=2^2*3*107$$.
LCM=$$2^3*3^2*5*103*107$$
we can check each prime number
2 -- there can be two or 3 2s...$$2^2$$ or $$2^3$$
3 -- Surely two 3s as LCM has two 3s but none of 1284 or 2472 have two 3s...$$3^2$$
5 -- Surely one 5..$$5^1$$
103 -- can be none or one of 103...$$103^0$$ or $$103^1$$
107 -- can be none or one of 107...$$107^0$$ or $$107^1$$
As there are two 3s, A, B and D are out...
N can be any of -
$$2^2*3^2*5$$ and any of the combination of 2, 103 or 107 added to it..
for example $$2^3*3^2*5*103*107$$ can be the largest value
C
Note : The question should ask for the smallest value of N or it should be ' What can be the value of N?'
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If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
### Show Tags
13 Nov 2019, 04:21
Solution
Given
• HCF of 2,472, 1,284 and positive integer N is 12.
• LCM of 2,472, 1,284 and positive integer N is 2^3∗3^2∗5∗103∗107.
To find
• The value of N.
Approach and Working out
• 2472 = 2^3 * 3 * 103
• 1284 = 2^2 * 3 * 107
• HCF (2472, 1284, N) = 12
o So, N = 2^(2+b) * 3^a * other prime factors where a >= 1 and b>=0
• LCM (2472, 1284, N) = 2^3∗3^2∗5∗103∗107.
o So, N can be 2^(2+b) * 3^2 * 5 * 103^c * 107^d where:
b, c, d can be 0 or 1
Only possible option is option C.
Thus, option C is the correct answer.
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
### Show Tags
15 Nov 2019, 08:01
EgmatQuantExpert wrote:
Solution
Given
• HCF of 2,472, 1,284 and positive integer N is 12.
• LCM of 2,472, 1,284 and positive integer N is 2^3∗3^2∗5∗103∗107.
To find
• The value of N.
Approach and Working out
• 2472 = 2^3 * 3 * 103
• 1284 = 2^2 * 3 * 107
• HCF (2472, 1284, N) = 12
o So, N = 2^(2+b) * 3^a * other prime factors where a >= 1 and b>=0
• LCM (2472, 1284, N) = 2^3∗3^2∗5∗103∗107.
o So, N can be 2^(2+b) * 3^2 * 5 * 103^c * 107^d where:
b, c, d can be 0 or 1
Only possible option is option C.
Thus, option C is the correct answer.
HCM X LCM = Product of numbers
2472 X 1284 X N = 12 X 2^3 X 3^2 X 5 X 103 X 107
solving it results in N = 3*5 = 15
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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20 Nov 2019, 18:58
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?
(A) $$2^2 * 3^2 * 7$$
(B) $$2^2 * 3^3 * 103$$
(C) $$2^2 * 3^2 * 5$$
(D) $$2^2 * 3 * 5$$
(E) None of these
Are You Up For the Challenge: 700 Level Questions
Let’s first factor 2,472 and 1,284 into primes:
2,472 = 12 x 206 = 2^2 x 3 x 2 x 103 = 2^3 x 3 x 103
1,284 = 12 x 107 = 2^2 x 3 x 107
We see that that the LCM of 2,472 and 1,284 is 2^3 x 3 x 103 x 107. Notice that the LCM of 2,472 1,284, and N is equal to the LCM of 2^3 x 3 x 103 x 107 and N, which is given to be 2^3 x 3^2 x 5 x 103 x 107. We are also given that the GCF of 2,472 1,284, and N is 12.
Recall that for any two positive integers a and b, we have a x b = LCM(a, b) x GCF(a, b). Therefore, by letting a = 2^3 x 3 x 103 x 107 and b = N, we have:
2^3 x 3 x 103 x 107 x N = 2^3 x 3^2 x 5 x 103 x 107 x 12
N = 3 x 5 x 12
N = 2^2 x 3^2 x 5
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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01 Dec 2019, 06:15
chetan2u wrote:
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?
(A) $$2^2 * 3^2 * 7$$
(B) $$2^2 * 3^3 * 103$$
(C) $$2^2 * 3^2 * 5$$
(D) $$2^2 * 3 * 5$$
(E) None of these
Are You Up For the Challenge: 700 Level Questions
$$2472=2*2*2*3*103=2^3*3*103$$.
$$1284=2*2*3*107=2^2*3*107$$.
LCM=$$2^3*3^2*5*103*107$$
we can check each prime number
2 -- there can be two or 3 2s...$$2^2$$ or $$2^3$$
3 -- Surely two 3s as LCM has two 3s but none of 1284 or 2472 have two 3s...$$3^2$$
5 -- Surely one 5..$$5^1$$
103 -- can be none or one of 103...$$103^0$$ or $$103^1$$
107 -- can be none or one of 107...$$107^0$$ or $$107^1$$
As there are two 3s, A, B and D are out...
N can be any of -
$$2^2*3^2*5$$ and any of the combination of 2, 103 or 107 added to it..
for example $$2^3*3^2*5*103*107$$ can be the largest value
C
Note : The question should ask for the smallest value of N or it should be ' What can be the value of N?'
Hi, Can you plz explain why cant we follow the method:
HCF * LCM = Product of numbers?
If we use that, it results in N=15
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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01 Dec 2019, 07:46
ScottTargetTestPrep wrote:
Bunuel wrote:
If the highest common factor of 2,472, 1,284 and positive integer N is 12 and the least common multiple of the same three numbers, 2472, 1284 and N, is $$2^3*3^2*5*103*107$$, what is the value of N?
(A) $$2^2 * 3^2 * 7$$
(B) $$2^2 * 3^3 * 103$$
(C) $$2^2 * 3^2 * 5$$
(D) $$2^2 * 3 * 5$$
(E) None of these
Are You Up For the Challenge: 700 Level Questions
Recall that for any two positive integers a and b, we have a x b = LCM(a, b) x GCF(a, b). Therefore, by letting
a = 2^3 x 3 x 103 x 107 and b = N, we have:
2^3 x 3 x 103 x 107 x N = 2^3 x 3^2 x 5 x 103 x 107 x 12
N = 3 x 5 x 12
N = 2^2 x 3^2 x 5
can you explain the bold line?
2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N
arnt you missing a few terms?
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
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03 Dec 2019, 20:15
Mansoor50 wrote:
can you explain the bold line?
2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N
arnt you missing a few terms?
I did not intend to multiply all three numbers together; as a matter of fact, when you have more than two numbers, the formula a x b = LCM(a, b) x GCF(a, b) is no longer valid.
We are taking a = LCM(2472, 1284) and b = N and applying the formula to these numbers.
_________________
# Scott Woodbury-Stewart
Founder and CEO
Scott@TargetTestPrep.com
122 Reviews
5-star rated online GMAT quant
self study course
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews
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Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink]
### Show Tags
05 Dec 2019, 07:17
ScottTargetTestPrep wrote:
Mansoor50 wrote:
can you explain the bold line?
2472 x 1284 x N = (2^3 x 3 x 103) x (2^2 x 3 x 107) x N
arnt you missing a few terms?
I did not intend to multiply all three numbers together; as a matter of fact, when you have more than two numbers, the formula a x b = LCM(a, b) x GCF(a, b) is no longer valid.
We are taking a = LCM(2472, 1284) and b = N and applying the formula to these numbers.
WOOT!......THANKS!!!!!
Re: If the highest common factor of 2,472, 1,284 and positive integer N is [#permalink] 05 Dec 2019, 07:17
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# Modern LaTeX CV
Today I want to share my new curriculum vitae (CV) for pursuing an academic position in Mexico. If you are a new subscriber of my blog, you should know I am doing a second postdoctoral project at ICUAP-BUAP. The postdoc is a project to get professional experience for a scientific career. However, the small letters of a postdoc life say, "You have no permanent position yet," then you should constantly look for a job. That's why I am here to share my LaTex template with you. The original modern LaTex CV The open software culture encourages you to share your…
# Low-cost Pyranometer and Raspberry Pi Pico – I
In the new Postdoctoral project, we are developing thin-film solar cell prototypes. However, some days ago I realize we will require the values of the solar irradiance W/m2 to determine the conversión efficiency $\eta =$ (VocJscFF)/(1000 W/m2) of our cell prototypes. The problem here is we will require an AM1.5 Solar simulator or a pyranometer to calibrate the irradiance (W/m2) of a light source. The possible solutions In my opinion, there are two ways to have these solar irradiance values: a) Fabricate a low-cost pyranometer using silicon photodiodes; b) Calibrate a reference solar cell (Si, crystalline or monocrystalline) using a…
# How to Create a Graphical Abstract for your Science Manuscript?
Long time no see, but I have been drafting two manuscripts during the Covid-19 lockdown. The first one is about my 2020 postdoctoral project at the Physics Institute of BUAP (IFUAP), and the other one comes as a collaboration with my last research group at IER-UNAM. Both works are related to the synthesis of new materials for energy conversion, solar cell development, and numerical simulation using SCAPS-1D. At this stage, the writing process and discussion with my coworkers have almost come to an end. However, one of the requirements to submit the manuscript to the journal is to include a…
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# How to collapse the universe?
This post has some tidbits regarding the problem of computing a ‘fingerprint’ of a long sequence of characters, often called ‘string hashing’. Most of the results I will describe are quite old, but they are scattered upon several papers, so I think it is worthwhile to have a post where they are put together. This is of course not a survey.
In our setting we have a set $U$, which is typically very large – think for instance all strings of at most $2^{32}$ bits. A hash function $h$ takes an item from $U$ and outputs a ‘fingerprint’ from a range $B$. Think of $B$ as say $64$ bits which is hopefully unique over a small set of items: we want to come up with a family of hash functions $H$, such that for every pair of items $m_1,m_2\in U$, when sampling $h$ from $H$, the collision probability $h(m_1)=h(m_2)$ is small. The parameters we care about are $|H|$, since $\log |H|$ is a lower bound on the key size, that is the number of random bits needed to specify $h$. We also care about the running time of $h$, and of course the collision probability we obtain. In this post I will not discuss directly the running time of $h$, though running time is arguably the most important parameter, instead I’ll focus on the tradeoff between the collision probability and $|H|$.
The first observation is that if we insist on having the collision probability as small as $1/|B|$, ($2^{-64}$ in the example above), then $\log |H|$ must be roughly at least $\log |U|$ which means in the example above we will need a very long key.
Denote by $\epsilon$ the collision probability we want to obtain. Stinson proved that $|H| \geq \frac{|U|(|B|-1)}{|U|(\epsilon |B|-1)+|B|^2(1-\epsilon)}$ which is roughly $|U|/|B|$ when $\epsilon = 1/|B|$, but says very little when $\epsilon$ is slightly larger. A sketch of Stinson’s short and clever proof is at the end of the post.
This bound is almost met by Carter and Wegman‘s classic multilinear hashing. Here $B$ is assumed to be a finite field, the items in $U$ are comprised of at most $\ell$ field elements, and the key $k_0,...,k_\ell$ is comprised of $\ell + 1$ field elements sampled uniformly. The multilinear hash family is defined as $h(m_1...) = k_0 + \sum k_im_i$ where all operations are in the field. If we forbid messages to end with zero then it is pairwise independent. A short proof could be found in a paper by Daniel Lemire.
In practice we would be willing to sacrifice a little in the collision probability and have a hash function with a smaller key length, one that is closer to the length of the fingerprint, and that has a weaker dependency on the size of the domain. There are many ways of doing that, with various tradeoffs. Here are two general and quite straightforward methods. Lets call them recursive and iterative.
Recursive: The main idea is to compose weeker functions in a tree-like structure. The origin of this idea (as many others) is in Carter and Wegman. We use as a building block a hash function that takes as input shorter fixed length strings. To be concrete lets assume we have a family of functions that takes $256$ bits and output $64$ bits. There are many good ways of designing such functions. The input is viewed as $\ell$ chunks of $256$ bits each and apply the hash function on each of these blocks, concatenating the output obtaining a string with $\ell/4$ chunks of $256$. We repeat the operation on the new string, with a freshly sampled hash function, and so on. In total we need to sample $\log_{4} \ell$ such function. In the example above, if each of the hash functions has a key of $256$ bits, and the original input is $2^{32}$ bits, then the total length of keys turns out to be approximately $3300$ bits. The collision probability could be bounded by multiplying the collision probability of the basic hash functions with the number of iterations, $13$ in this case. Mikkel Thorup has a more detailed description and some experiments here, as does Sarkar.
Iterative: This idea is known by various names such as CBC-MAC and Pearson hashing. Here we assume the message is comprised of at most $\ell$ blocks of $n$ bits each: $m_1,\ldots, m_\ell$. The hash function is indexed by a permutation $\pi:\{0,1\}^n \rightarrow \{0,1\}^n$. The $i'th$ stage of the computation has state $C_i \in \{0,1\}^n$. Starting with $C_0 = 0^n$ the function iteratively computes $C_i = \pi(C_{i-1}\oplus m_i)$. The output is $C_\ell$. Ideally we would have liked the permutation to be completely random, this cannot be done for reasonable values of $n$, so in practice a pseudorandom permutation is used, though the exact definition of pseudorandomness in this context could be debated. Assuming the permutation is indeed random, what is the collision probability? It is shown here to be $O(\ell^{o(1)}/2^n)$ as long as $\ell \leq 2^{n/4}$. The proof holds for a stronger attack model. I do not see an inherent reason for the bound on $\ell$.
I now sketch Stinson’s proof for the bound mentioned above, which is a nice application of the second moment method. Recall that $U$ denotes the domain, the range is denoted by $B$ and its size by $b$. The collision probability is denoted by $\epsilon$. For every $h\in H$ and $y \in B$ let $Q_{hy}\subseteq U$ denote $h^{-1}(y)$. Now let’s pick some specific $h$ and $y$. For every $g\in H, g\neq h$ and $x\in B$ define $\mu_{g,x} = |Q_{h,y} \cap Q_{g,x}|$, i.e. the number of items that are mapped to $y$ by $h$ and to $x$ by $g$. Let $\mu$ denote the expected size of $\mu_{g,x}$ when $x$ is sampled from $B$ and $g$ from $H\setminus \{h\}$. A moment’s reflection shows this average to be $|Q_{h,y}|/b$ (for each item in $Q_{h,y}$ we have a $1/b$ chance of guessing right where it lands under $g$). The key observation is that in order for $\mu_{g,x}$ to be large, many items should collide on $x$ under $g$, which is unlikely, so the fact that the collision probability of $H$ is bounded gives us a handle on the variance of $\mu$. This in turn would imply a bound on $|H|$. Consider two items $u,v$ in $Q_{h,y}$. Since they collide under $h$, there are at most $\epsilon |H| -1$ other function under which they collide. So,
$\sum_{u\neq v \in Q_{h,y}}\sum_{g\neq h,x\in B}\mathbf{1}_{g(u)=g(v)=x} \leq \binom{|Q_{h,y}|}{2}(\epsilon|H| - 1)$
changing the order of summation this means
$\sum_{g\neq h,x\in B}\binom{\mu_{g,x}}{2} \leq \binom{|Q_{h,y}|}{2}(\epsilon|H| - 1)$
Moving things around we have
$\sum_{g\neq h,x\in B}\mu_{g,x}^2 \leq |Q_{h,y}|(|Q_{h,y}|-1)(\epsilon|H| - 1)+|Q_{h,y}|(|H|-1)$
So we have an upper bound on the second moment of $\mu_{g,x}$ in terms of $|H|$ and $|Q_{h,y}|$. Of course we also know that the variance of $\mu_{g,x}$ is non negative, which after calculations implies the bound. Finally in order to get a bound in terms of $|U|$ we observe that we can pick $h$ and $y$ such that $|Q_{h,y}|$ is at least $|U|/b$.
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Euler Identity in Home
04-14-2014, 07:10 PM
Post: #41
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 02:48 PM)ColinJDenman Wrote: I'd note sin(n*pi) - sin(pi) for n=1...6000000 suggests there is no mapping of x to 0<= x < 2pi, which I do think is appropriately correctable for sin, as is the appropriate short circuiting of the internal algorithm for 'well known' values of x and sin.
There's no need for "short circuiting" and people really do get zeroes without it. This is a bounded (and periodical) function that maps [0, 2 Pi) to [-1, 1]. If you work with 12 significant digits, then:
A) 3.141592653585
B) 3.141592653586
C) 3.141592653587
D) 3.141592653588
E) 3.141592653589
F) 3.141592653590
G) 3.141592653591
H) 3.141592653592
I) 3.141592653593
J) 3.141592653594
Are all the same and (should) yield the same result than:
3.14159265359
There's no point in having as an answer:
-.000000000000206761537357
And swearing that every bit of it is true, because if those results were meaningful to E-13, then you could effectively tell F) from B), C)... J), and you can't. See how it works:
Let's generate A)-J):
(first line: Mathematica command, I hope the syntax is self-explanatory, results below)
Table[3.14159265359 + (i - 6)*10^-12, {i, 10}] // InputForm
/* i starts at i=1, // is the pipe to InputForm, so we can see all the digits from the output*/
{3.141592653585, 3.141592653586, 3.141592653587, 3.141592653588, 3.141592653589, 3.141592653590, 3.141592653591, 3.141592653592, 3.141592653593, 3.141592653594}
The true values of Sin for them are:
(first line: Mathematica command, results below, -I'm doing it the easy way-)
Table[Sin[3.14159265359 + (i - 6)*10^-12], {i, 9}]
{4.79318*10^-12, 3.79309*10^-12, 2.793*10^-12, 1.79335*10^-12,
7.93266*10^-13, -2.06823*10^-13, -1.20691*10^-12, -2.207*10^-12, -3.20665*10^-12}
Which value did the calc pick? The sixth? Why? But they are all good! in fact any value in (-3.20665*10^-12, 4.79318*10^-12) would be a good answer.
This is not a floating point issue. It's just continuous functions and significant digits' common sense. The best accurate and meaningful value your calc should give you is 0E-12, that is 2.06761537357E-13 rounded to 12, the rest is garbage. Yet people are not willing to get/believe it, go figure!
HP, or their fans, have taken historically a very strange viewpoint in which having obvious accuracy in the display is not that important, but raw results are, leaving to the user the task of making sense of them, with awful consequences on the average, I'm sad to say.
It's more sad that being burdened with mantras of 12 significant digits are enough, and you don't keep guard digits (even though apparently HP calculators adopted them later -I don't know for sure- and this, which makes some sense for an RPN machine, does not for a textbook entry), even when performing intermediate operations the user can't see, a powerful machine capable of a lot ends up doing things nobody else would. Anyway...
04-14-2014, 07:31 PM (This post was last modified: 04-14-2014 07:47 PM by Han.)
Post: #42
Han Senior Member Posts: 1,810 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 07:10 PM)Manolo Sobrino Wrote: Which value did the calc pick? The sixth? Why? But they are all good! in fact any value in (-3.20665*10^-12, 4.79318*10^-12) would be a good answer.
The calc picked the sixth case because it rounds $$\pi$$ to 3.141592653590 and not any of the other values. And while all the other values are just as good, it only makes sense to map to one of them -- namely the one corresponding to an input of 3.141592653590.
Quote:This is not a floating point issue. It's just continuous functions and significant digits' common sense. The best accurate and meaningful value your calc should give you is 0E-12, that is 2.06761537357E-13 rounded to 12, the rest is garbage. Yet people are not willing to get/believe it, go figure!
I agree and disagree. I agree in terms of significant digits because we are doing numerical calculations and know that there will be roundoff error. I disagree in terms of pure mathematical results. Take for example:
$(10^{15}+1.2-10^{15})*100$
Most calculators will give you -- even Maple -- will give you a result of 0. This is a valid answer in practical terms. It is, of course, the wrong answer mathematically.
Quote:HP, or their fans, have taken historically a very strange viewpoint in which having obvious accuracy in the display is not that important, but raw results are, leaving to the user the task of making sense of them, with awful consequences on the average, I'm sad to say.
The blind reliance on any calculating machine for the right answer is always going to result in awful consequences. To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.
Graph 3D | QPI | SolveSys
04-14-2014, 09:17 PM
Post: #43
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 07:31 PM)Han Wrote:
(04-14-2014 07:10 PM)Manolo Sobrino Wrote: Which value did the calc pick? The sixth? Why? But they are all good! in fact any value in (-3.20665*10^-12, 4.79318*10^-12) would be a good answer.
The calc picked the sixth case because it rounds $$\pi$$ to 3.141592653590 and not any of the other values. And while all the other values are just as good, it only makes sense to map to one of them -- namely the one corresponding to an input of 3.141592653590.
No, it doesn't.
The consistent answer for 12 significant digits is 0.000000000000, and that is not zero, those zeroes have a meaning.
Quote:I agree and disagree. I agree in terms of significant digits because we are doing numerical calculations and know that there will be roundoff error. I disagree in terms of pure mathematical results. Take for example:
$(10^{15}+1.2-10^{15})*100$
Most calculators will give you -- even Maple -- will give you a result of 0. This is a valid answer in practical terms. It is, of course, the wrong answer mathematically.
Well, in such a case that famous answer for sin (pi) is also wrong. You can't have your cake and eat it. Either you are consistent or you aren't.
Maple is able to do arbitrary precision calculations, you should try that.
Quote:The blind reliance on any calculating machine for the right answer is always going to result in awful consequences.
I fully agree on that
Quote:To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.
That's what professional Mathematicians do, and HP has taken advantage of it in the past. Kahan might still be available. If not, I'd suggest Higham.
You know, this thing Maths is unluckily for some not a matter of opinion, either you get it right or you don't. In the real world of Logic, Math departments and peer reviewed journals your preferences don't matter. If you keep on being wrong for the sake of it you're out.
04-14-2014, 10:45 PM
Post: #44
CR Haeger Member Posts: 275 Joined: Dec 2013
RE: Euler Identity in Home
Hmm... Math departments and journals have settled this for HP and other calculator makers already?
Thanks to HP, TI, Casio and others replacing slide rules we can dicker about 10e-12 instead of 10e-4 in 1971
04-14-2014, 10:47 PM
Post: #45
Han Senior Member Posts: 1,810 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 09:17 PM)Manolo Sobrino Wrote:
(04-14-2014 07:31 PM)Han Wrote: The calc picked the sixth case because it rounds $$\pi$$ to 3.141592653590 and not any of the other values. And while all the other values are just as good, it only makes sense to map to one of them -- namely the one corresponding to an input of 3.141592653590.
No, it doesn't.
The consistent answer for 12 significant digits is 0.000000000000, and that is not zero, those zeroes have a meaning.
I was not debating what the answer for 12 significant digits was. I was debating on your suggestion any one among {4.79318*10^-12, 3.79309*10^-12, 2.793*10^-12, 1.79335*10^-12, 7.93266*10^-13, -2.06823*10^-13, -1.20691*10^-12, -2.207*10^-12, -3.20665*10^-12} would be fine. There is a difference between suggesting the correct answer is 0.000000000000 versus any one among this list. They are all "equal" when rounded to 12 sig. digits, but they are clearly not the same numbers. Surely you agree.
Quote:Well, in such a case that famous answer for sin (pi) is also wrong. You can't have your cake and eat it. Either you are consistent or you aren't.
In what sense are you suggesting it's wrong? Mathematically? or practically? Mathematically, no numerical calculations involving rational approximations of transcendental values will every be "exactly right", but they can be "right" in a practical sense once we account for roundoff errors. The Prime enables you to use two separate environments depending on your needs. If you only care to have approximate solutions, use the Home environment. If you need exact values, use the CAS environment.
Quote:Maple is able to do arbitrary precision calculations, you should try that.
That is completely irrelevant. One could argue that the Prime has a CAS mode in which the value of $$\sin(\pi)$$ is always 0 and that users "should try that."
Quote:
Quote:To suggest that there is one, right approach to handling numerical calculations is also going to result in awful consequences.
That's what professional Mathematicians do, and HP has taken advantage of it in the past. Kahan might still be available. If not, I'd suggest Higham.
You know, this thing Maths is unluckily for some not a matter of opinion, either you get it right or you don't. In the real world of Logic, Math departments and peer reviewed journals your preferences don't matter. If you keep on being wrong for the sake of it you're out.
Mathematicians would not even consider using a calculator or CAS or any of the stable algorithms designed by Higham when it comes to "exact" mathematics for their proofs. The question of accuracy is moot in this case. That said, are we still discussing numerical algorithms applied to transcendental values -- on a calculator -- here? If you want the power of Higham or Kahan or (insert whatever numerical analyst you like here) then why are we even debating this in the context of a calculator? At some point one must concede that, given the constraints of a particular machine (RAM, CPU power, etc), some amount of accuracy is sufficient. Otherwise, we need a more powerful machine. As already mentioned, the calculator has a CAS for those who need "exact" answers.
Graph 3D | QPI | SolveSys
04-14-2014, 11:32 PM
Post: #46
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 10:47 PM)Han Wrote: I was not debating what the answer for 12 significant digits was. I was debating on your suggestion any one among {4.79318*10^-12, 3.79309*10^-12, 2.793*10^-12, 1.79335*10^-12, 7.93266*10^-13, -2.06823*10^-13, -1.20691*10^-12, -2.207*10^-12, -3.20665*10^-12} would be fine. There is a difference between suggesting the correct answer is 0.000000000000 versus any one among this list. They are all "equal" when rounded to 12 sig. digits, but they are clearly not the same numbers. Surely you agree.
Oh Han... I proposed this, I took the time and pains to explain why it should be 0.000000000000. It should be that because if you consider that your result is accurate to more digits you are actually trespassing your own limits. But people just don't care, no one in this forum has ever mentioned the table maker's dilemma for instance, which is relevant for this kind of recurring discussions, it's all talk, talk, talk and vacuous claims. It's a kind of totemic attitude versus a device that I, due surely to cognitive biases having to do with my painful education can't understand.
Quote:
Quote:Well, in such a case that famous answer for sin (pi) is also wrong. You can't have your cake and eat it. Either you are consistent or you aren't.
In what sense are you suggesting it's wrong? Mathematically? or practically? Mathematically, no numerical calculations involving rational approximations of transcendental values will every be "exactly right", but they can be "right" in a practical sense once we account for roundoff errors. The Prime enables you to use two separate environments depending on your needs. If you only care to have approximate solutions, use the Home environment. If you need exact values, use the CAS environment.
First you like that your calc comes up with the exact result for Sin[3.14159265359000000000000000000000000000000000000...], which is not the true value of sin(Pi) yet is given with spurious precision to it's true value. Then you complain about them not being able to give the true answer of an arithmetic calculation when they're working out of their range. I just get lost with your point.
Quote:
Quote:That's what professional Mathematicians do, and HP has taken advantage of it in the past. Kahan might still be available. If not, I'd suggest Higham.
You know, this thing Maths is unluckily for some not a matter of opinion, either you get it right or you don't. In the real world of Logic, Math departments and peer reviewed journals your preferences don't matter. If you keep on being wrong for the sake of it you're out.
Mathematicians would not even consider using a calculator or CAS or any of the stable algorithms designed by Higham when it comes to "exact" mathematics for their proofs. The question of accuracy is moot in this case. That said, are we still discussing numerical algorithms applied to transcendental values -- on a calculator -- here? If you want the power of Higham or Kahan or (insert whatever numerical analyst you like here) then why are we even debating this in the context of a calculator? At some point one must concede that, given the constraints of a particular machine (RAM, CPU power, etc), some amount of accuracy is sufficient. Otherwise, we need a more powerful machine. As already mentioned, the calculator has a CAS for those who need "exact" answers.
Applied Mathematicians have been crunching numbers for a long time with less computing power than that which you have in your hands now. It can be done, and of course it has been done. We have digital computers since really the 50s, and people have been doing a great job with them since then, developing interval arithmetic and all kinds of nice stuff to provide meaningful answers with error bounds, which is the more "exact" mathematics you can do for this job, you're not proving the four colour theorem here. What would have given Feynman at Los Alamos for a dozen of these?... This is hopeless, we won't agree tonight (at Europe). Let's leave it for another day.
04-15-2014, 07:46 PM (This post was last modified: 04-15-2014 07:54 PM by Han.)
Post: #47
Han Senior Member Posts: 1,810 Joined: Dec 2013
RE: Euler Identity in Home
(04-14-2014 11:32 PM)Manolo Sobrino Wrote: This is hopeless, we won't agree tonight (at Europe). Let's leave it for another day.
I don't think we were ever really disagreeing about the big picture. (Edit: by this I mean that in most circumstances we likely agree on what the answer should be when considering significant digits etc.) But certainly we disagree on the details. My stance is that there are two modes: and approximate mode and an exact mode so we should choose the appropriate mode for our needs.
Quote:HP, or their fans, have taken historically a very strange viewpoint in which having obvious accuracy in the display is not that important, but raw results are, leaving to the user the task of making sense of them, with awful consequences on the average, I'm sad to say.
This is where we essentially disagree. It is my opinion that the onus is always on the user to interpret the results. "With great power comes great responsibility."
Now, some are arguing (perhaps you included?) that the approximations could be better, and that in certain circumstances, the calculator should also take into account sig. digits and round appropriately. My response to that will always be that it is up to the user to properly interpret the results. What may mathematically be wrong may be practically correct, and vice versa. The calculator should not be responsible for interpreting results -- it should only compute them. When Maple computes (1.0E16-7.2-1.0E^16) and tells me the result is 0. (as opposed to 0.0E16 by the way), it's mathematically wrong AND it also interprets wrong (wrong sig. number of digits). But practically speaking, it's the right answer. There is a very valid reason why it should not interpret for us. In mathematics, there is no difference between
$(10^{15}-7.2-10^{15})\cdot 100$
and
$(10^{15} - \frac{72}{10} - 10^{15}) \cdot 100$
but for many computer algebra systems, there is a huge difference in the results. No CAS is going to reliably guess whether either input is meant to be computed exactly or approximately, so almost all of them insist that the use of a decimal point is meant to be a floating point approximation -- and that's fine as long as everyone is aware. In the same vein, it should be a given that any time one uses decimal approximations for computing with transcendentals, the result needs to be handled with proper care and interpretation.
So yes, I agree that sin(3.141592653590) should practically be interpreted as 0.00000000000 (hopefully I have the right number of 0's there) -- provided that intent was to get a decimal approximation of $$\sin(\pi)$$ and not an exact mathematical evaluation -- but I don't think it should be up to the calculator to decide that interpretation for me. It should return whatever its algorithm was designed to return, based on whatever limitations on precision have been set. Then it is up to me to properly interpret the results myself. Note that this goes for hand calculations, too.
(Hopefully I am not misreading the posts in this discussion.)
Graph 3D | QPI | SolveSys
04-15-2014, 09:17 PM
Post: #48
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
Han, what's so hard to understand that if you claim to work with 12 significant digits, and you round in every step to 12 significant digits, and you store numbers with 12 significant digits, you can't simply give back results calculated with 25 significant digits? And adding insult to injury give them rounded to 12 in floating point?
I finally got it, these things are made by engineers, for engineers, with the depth of mathematical knowledge of engineers. A match made in heaven.
04-15-2014, 10:21 PM
Post: #49
Didier Lachieze Senior Member Posts: 1,151 Joined: Dec 2013
RE: Euler Identity in Home
Manolo, I'm confused, if fort SIN(3.14159265359) a calculator with 12 significant digits should return 0.00000000000, then for 2^(-40) should it also return 0.00000000000 and not 9.09494701773E-13 ?
04-15-2014, 10:30 PM
Post: #50
Joe Horn Senior Member Posts: 1,579 Joined: Dec 2013
RE: Euler Identity in Home
(04-15-2014 09:17 PM)Manolo Sobrino Wrote: Han, what's so hard to understand that if you claim to work with 12 significant digits, and you round in every step to 12 significant digits, and you store numbers with 12 significant digits, you can't simply give back results calculated with 25 significant digits? And adding insult to injury give them rounded to 12 in floating point?
You are assuming that we don't understand that. We DO understand it... and disagree with it. And for good reason, as the following hopefully makes clear.
If I want to calculate the sine of exactly 1 radian, then I would type sin(1), and of course I would intend it to mean sin(exactly 1.000000000000000000... radians). But YOU would say, "Oh no no no, that's wrong; sin(1) is using an input of only one significant digit, so the output should only have one significant digit, and any more digits than that in the output are bogus." At this point, you will of course object that you never said any such silly thing. Ah, but you have said it, over and over, every time you have insisted in this discussion that typing sin(3.14159265359) on an HP calculator cannot possibly be intended by the user to mean "give me the sine of exactly 3.141592653590000000000... radians". Apply that false assumption to sin(1) and you get the previously mentioned silly statement.
What's so hard to understand about this: HP allows the user to decide how many digits of the input are significant (including trailing zeros!), which then (as you mentioned!) requires the user to interpret the calculated result accordingly. When TI makes those decisions for the user, the user is prevented from exploring certain questions, such as "What is the sine of numbers very close to pi?" HP allows such exploration. Therefore HP can do what TI can't. Q.E.D.
Disclaimer: All of the above was my opinion when I wrote it, but I enjoy learning, and changing my opinions when that is warranted, so stay calm.
<0|ɸ|0>
-Joe-
04-15-2014, 10:53 PM (This post was last modified: 04-16-2014 12:28 AM by Manolo Sobrino.)
Post: #51
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
(04-15-2014 10:21 PM)Didier Lachieze Wrote: Manolo, I'm confused, if fort SIN(3.14159265359) a calculator with 12 significant digits should return 0.00000000000, then for 2^(-40) should it also return 0.00000000000 and not 9.09494701773E-13 ?
Of course not, and that's the beauty of it. It has to do with the nature of the function you are considering (and where), for instance, is it bounded? How much does it change around this or that particular point?
You need more than 12 significant digits of Pi to specify a value different than 0.00000000000 for Pi rounded to 12 significant digits, but your 12 significant digits of 2, or 2.00000000000 are enough to specify any value with 12 significant digits of the exponential function.
In the first scenario you are mapping [0, 2 Pi) to [-1, 1]. In the second it's Reals* to Reals. Just think about the definition of function (eom style):
Let the sets X and Y be given. To each element x of X corresponds an element y of Y, which is denoted by f(x), then we say a function f is given on X.
Joe, just read my post of yesterday. It's not the same, and you will see why. Trailing zeroes can be significant figures. x=1.00000000000 means that I know that x is 1 with a precision of 11 decimals. You can specify a value of sin(1) with 11 decimals, and you can specify a value of sin(Pi) with 11 decimals as well, it happens to be 0.00000000000. Notice that if we calculate Sin(3.14159265359*10^-12) the right answer to 12 significant digits is 3.14159265359*10^-12.
As I wrote a few days ago, not worrying about what you can say or not according to your precision (and maiming it) yields "wrong" results, check the example sin(pi*(1+1E-11)). So it's not like there's any particular advantage to it. In the neighbourhood of Pi it gives correct results when rounded, but if the user doesn't know, and sometimes they can't know, it is misleading at best.
This reminds me of a thread of some time ago. The linear solver of the 50G spat results for an incompatible system and that's it, after some digging it turns out that it was producing the minimum norm Least Squares solution with no warning. How is that good for you? Do you have to check if the system has indeed a solution before trusting the solution? In an exam? Now, do you have to check with Mathematica wth is going on with the rounding and the real precision available for calculations in your calculator before you can understand the results for sin(), and then you have to interpret them? What's the point of a calculator I have to check elsewhere the things it shows me? And the results from a TI, that in this case are decent and accurate are bad for you, because it's a TI?? And as they get it right you suggest that they're doing tricks??? I just find all this extremely bizarre. I love RPN, but that has nothing to do with it: Amicus Plato, sed magis amica veritas.
04-16-2014, 06:26 AM
Post: #52
Joe Horn Senior Member Posts: 1,579 Joined: Dec 2013
RE: Euler Identity in Home
(04-15-2014 10:53 PM)Manolo Sobrino Wrote: ... you can specify a value of sin(Pi) with 11 decimals ... check the example sin(pi*(1+1E-11))....
<takes deep breath to calm down> There you go again, insisting on talking about pi, when we are specifically talking about a number that is NOT PI, namely, EXACTLY 314159265359/10^11. THAT'S NOT PI, and it's NOT AN APPROXIMATION either. It's an EXACT number, the exact ratio of two integers. And exactly that many radians does in fact have an exact sine. TI can't find it. That's a Good Thing? HP does find it. That's a Bad Thing?
(04-15-2014 10:53 PM)Manolo Sobrino Wrote: ... And the results from a TI, that in this case are decent and accurate are bad for you, because it's a TI??
No. The result given by TI for the sine of 314159265359/10^11 is neither decent nor accurate, not because it's a TI, but because it's wrong. I couldn't care less who made the calculator. Remember, I was the guy who blew the whistle on the horrible square root accuracy bug in the HP 30S (not to mention the zillion other HP bugs I've reported over the past few decades). Finding HP bugs is my #1 hobby; it actually gives me a visceral thrill when I find one. But HP gets this sine right.
(04-15-2014 10:53 PM)Manolo Sobrino Wrote: And as they [TI] get it right you suggest that they're doing tricks???
Your insisting that TI's answer is "right" does not make it right. Just look at all the mental gymnastics you've posted in this thread, attempting to justify TI's erroneous result. But then look at the simplicity of HP's position. Does sin(exactly 314159265359/10^11 radians) have a value that's exactly defined by Maths? Yes. Is HP's result closer to it than TI's? Yes. Period.
(04-15-2014 10:53 PM)Manolo Sobrino Wrote: Amicus Plato, sed magis amica veritas.
Oh? If truth be so precious, then why are you defending TI's false result for sin(314159265359/10^11), and denigrating HP's true result?
Remember, we're talking about the EXACT value 314159265359/10^11 here, not an approximation of pi or of anything else. Anything else in this discussion is a digression.
<0|ɸ|0>
-Joe-
04-16-2014, 08:23 AM
Post: #53
DavidM Senior Member Posts: 755 Joined: Dec 2013
RE: Euler Identity in Home
(04-15-2014 09:17 PM)Manolo Sobrino Wrote: Han, what's so hard to understand that if you claim to work with 12 significant digits, and you round in every step to 12 significant digits, and you store numbers with 12 significant digits, you can't simply give back results calculated with 25 significant digits?
Perhaps my understanding of significant digits is different than yours, because I can't see where the 25 significant digits comes into play. As I was taught, -0.000000000000206761537357 still only has 12 significant digits. Where are you getting the 25 digit count from? I'm genuinely curious as to whether there is a different understanding of significant digits here. I suppose it's more likely that I just don't understand what you're trying to say.
I don't have a large collection of calculating/computing systems to compare, but I can say that I have checked the following systems to see what the result of "sin(3.14159265359)" is for each one:
Wolfram Alpha (on the web):
-2.0682310711021444E-13
MS Excel 2010 on my laptop:
-2.06823056944638E-13
HP 50g:
-2.06761537357E-13
HP 30b:
-2.0676153736E13
A compiled Delphi XE2 app using the default SIN function with extended (80-bit) reals (also on my laptop):
-2.06761528471349765E-13
The only TI calculator I have is a TI-34 (a 10-digit unit). Here's the results it gave:
a) 3.141592654 SIN: -4.1E-10
b) <pi> SIN: 0
Pressing the <pi> key results in the display showing the value 3.141592654 (the same as item a above). But clearly there's a difference. In an attempt to see that difference, I tried the following:
<pi> - 3.141592654 = -4.1E-10
It appears the TI-34 is carrying more digits than it displays for pi. But it is still rounding its result when using that value.
Of those systems, only the TI-34 rounded the final result to 0, and it only did that when I used the special key for pi. Granted, we are focused on calculators here. But the insistence that zero or 0.000000000000 is a more correct answer for sin(3.14159265359) than -2.068E-13 seems to be a TI-specific viewpoint which isn't shared by the other math libraries I am able to compare it to.
What's more interesting to me is the variance in the non-zero results.
04-16-2014, 12:02 PM (This post was last modified: 04-16-2014 12:12 PM by Dieter.)
Post: #54
Dieter Senior Member Posts: 2,397 Joined: Dec 2013
RE: Euler Identity in Home
(04-16-2014 08:23 AM)DavidM Wrote: I don't have a large collection of calculating/computing systems to compare, but I can say that I have checked the following systems to see what the result of "sin(3.14159265359)" is for each one:
Let me add just two remarks:
1. The true value of sin(3,14159265359000000...) is 2,0676 15373 56616 72049 71158...·10–13-.
For arguments that close to $$\pi$$ the first 20+ significant digits for sin(x) and tan(x) agree with $$\pi$$ – x. This means that the results given by Wolfram Alpha and Excel have merely three valid digits. In the latter case this probably is due to the fact that Excel uses binary arithmetics and 3,14159265359 cannot be represented exactly with the usual 52 mantissa bits. If x is off by 2–53 the error in sin(x) will be roughly the same, i.e. about 10–16. Which agrees with the error in the result you posted.
2. For evaluating trig functions, calculators often use a method that requires an internal value of $$\pi$$ with at least twice the number of returned significant digits for a full precision result. However, the HP and TI devices I used only carry three additional digits for internal calculations. This explains the behaviour of your TI-34, which BTW is the same as in earlier HP 10-digit calculators:
(04-16-2014 08:23 AM)DavidM Wrote: The only TI calculator I have is a TI-34 (a 10-digit unit). Here's the results it gave:
a) 3.141592654 SIN: -4.1E-10
b) <pi> SIN: 0
The internal value of $$\pi$$ is 3,141592653590 (rounded to 13 digits). Since sin(x) here essentially is $$\pi$$ – x = –4,1E–10, this it what you see.
If you use the $$\pi$$ key, the internal 13-digit value is entered so that $$\pi$$ – x becomes zero.
(04-16-2014 08:23 AM)DavidM Wrote: Pressing the <pi> key results in the display showing the value 3.141592654 (the same as item a above). But clearly there's a difference. In an attempt to see that difference, I tried the following:
<pi> - 3.141592654 = -4.1E-10
Right. That's 3,141592653590 – 3,141592654. Which agrees with the returned sine value because for the TI-34 $$\pi$$ equals 3,141592653590.
(04-16-2014 08:23 AM)DavidM Wrote: It appears the TI-34 is carrying more digits than it displays for pi.
Sure. It carries 13 digits not just for $$\pi$$. Try e1 – 2,718281828.
(04-16-2014 08:23 AM)DavidM Wrote: Of those systems, only the TI-34 rounded the final result to 0, and it only did that when I used the special key for pi. Granted, we are focused on calculators here. But the insistence that zero or 0.000000000000 is a more correct answer for sin(3.14159265359) than -2.068E-13 seems to be a TI-specific viewpoint which isn't shared by the other math libraries I am able to compare it to.
All this is due to the way sin(x) is evaluated. As already stated, many (most?) calculators use an algorithm where sin(x) for x so close to $$\pi$$ is $$\pi$$ – x. If you press the $$\pi$$ key, x equals the 13-digit internal $$\pi$$ value, so sin(x) = $$\pi$$ – x becomes zero. It's that simple. On the other hand, HP's 10-digit devices use their 13 digits only internally, so you cannot enter a 13-digit $$\pi$$ value. 3,141592654 is as close you can get – both manually and by using the $$\pi$$ key. So the result is always the difference between the 10-digit and the internal 13-digit representation of $$\pi$$: –4,1E–10.
Dieter
04-16-2014, 12:29 PM (This post was last modified: 04-16-2014 02:58 PM by Manolo Sobrino.)
Post: #55
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
Joe, I don't know what to do when people just don't read the posts in the thread.
As jebem told us, the TIs 86 and 89 work with pi=3.1415926535898, and all the results I quote are for pi=3.1415926535898. When you press Pi it shows a greek Pi, and if you evaluate that you get 3.14159265359, that's because the display mode of floats in those calculators just shows 12 significant digits.
If you evaluate pi-3.14159265359, you get -2E-13. The values those TIs calculate for sin(pi) are accurate and consistent for a 14 significant digit calculator:
Type sin(pi) or
sin(3.1415926535898)
And you get zero
And the values those same TIs calculate for sin(3.14159265359) are accurate as well: type sin(3.14159265359) and... do you know what you get?
sin(3.14159265359) = -2E-13
That's right. And why is that good here? That's good because these machines work with 14 digit precision, they can give you that.
And why sin(3.14159265359)=-2.06761537357E-13 in the OP's calculator is "wrong" (not really wrong, but rounding pending) and it should show 0E12 instead? Because it comes from a machine with 12 digit precision. It shouldn't know that, because in order to know that you need more than 12 significant digits, actually 25 significant digits. That's my whole point. The "awful consequences" are that you believe that as the result shows 12 significant digits in floating point notation it all makes sense, but that's really a joke that shows the absurd of mixing things up.
And if you can work with 25 significant digits (at least), how come a TI can give you: sin(3.14159265359*(1+1E-11))=-3.16E-11, which is accurate and it's all this thing can get, while the OP's calculator gives you -3.0206761537357*10^-11? Because of rounding 3.14159265359*(1+1E-11)->3.14159265362.
How did you calculate Sin(3.14159265359*10^-12)? Maybe the implementation I'm working with is wrong, but I get (not in a TI) just that, and it should be that because in the Taylor expansion of sin(x), the second term for x=3.14159265359*10^-12 is -(1/6)*3.10063*10^-35... give me a break.
DavidM: Just read the posts in the thread. And yeah, Wolfram Alpha gives you that for "Decimal approximation" and 0E-12 for "Result". Or at least that was what it gave me the day I posted it on this thread.
(edit)
And this is what Pi+ArcSin looks like (it should cross the y axis a little bit upwards, but you get the idea, I don't have time to get the perfect ticks right now). If you round Pi to 3.14159265359 and every other number to 12, what does it mean having a value of Sin of -2.06761537357E-13? Well, according to the significant digits you are considering, no more than you just can say that sin(3.14159265359) is 0E-12.
Maybe it's so obvious for me that I'm just incapable of explaining it to you. At least I hope that you understand now why Wolfram Alpha gives you that 0.*10^-12 "Result".
04-16-2014, 03:09 PM
Post: #56
Han Senior Member Posts: 1,810 Joined: Dec 2013
RE: Euler Identity in Home
For Manolo -- just three simple questions:
1. When a user computes SIN(1.1) on any calculator (or Mathematica or Maple, etc), what should the calculator return? (Assume radian mode)
2. Same question, this time the user enters SIN(1.10) -- same value but with an extra significant digit. What should the returned result be? (Assume radian mode)
3. How should a real number be implemented so that a calculator can know the difference between 1.1 and 1.10 ?
Graph 3D | QPI | SolveSys
04-16-2014, 03:48 PM
Post: #57
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
(04-16-2014 03:09 PM)Han Wrote: For Manolo -- just three simple questions:
1. When a user computes SIN(1.1) on any calculator (or Mathematica or Maple, etc), what should the calculator return? (Assume radian mode)
2. Same question, this time the user enters SIN(1.10) -- same value but with an extra significant digit. What should the returned result be? (Assume radian mode)
3. How should a real number be implemented so that a calculator can know the difference between 1.1 and 1.10 ?
1)&2)
In a world of perfect calculators I guess that you'll have to type 1.1 for 2 significant digits, 1.10 for three, 1.100 for four, and so on. That is, for instance if we're talking about 1.1*10^-10: 1.1E-10, 1.10E-10, 1.100E-10,... . Now in their default mode they assume that every number you type has 12 significant figures for the HP, 14 for the TI (those that I've discussed). That's OK if you understand what they're doing. BTW, they fill with zeroes and show you the rounded results to N when you work in FIX N mode.
And they should return, if not the multiple precision value according to that (people have no use for too low precision results, really), at least the most accurate value rounded to the significant digits of your input. They already do that in FIX mode.
About 3), I don't know. But the minimum should be that you implement a fixed number of significant digits, and using reasonable algorithms and round where it's due you get results for that number of significant digits, preferably no less, and it would be nice that no more, because it really makes no sense.
04-16-2014, 09:46 PM (This post was last modified: 04-16-2014 09:52 PM by Joe Horn.)
Post: #58
Joe Horn Senior Member Posts: 1,579 Joined: Dec 2013
RE: Euler Identity in Home
Manolo! Good news! I think I figured out a solution to the root cause of our disagreement! (Details below) Meanwhile, back at the ranch...
(04-16-2014 12:29 PM)Manolo Sobrino Wrote: Joe, I don't know what to do when people just don't read the posts in the thread.
Sorry that I came across that way! I really have read them all, but I think we've been on different wavelengths, or in alternate universes, or something. But I'm trying. ("Yes... very trying." -- my brother Jim)
(04-16-2014 12:29 PM)Manolo Sobrino Wrote: Maybe it's so obvious for me that I'm just incapable of explaining it to you.
Two quotes come to mind:
Joe, the fact that something is obvious to you does not mean that it is obvious to everybody. -- Richard Nelson, during a friendly debate many years ago.
"Because it's obvious" is not an acceptable mathematical proof. -- Mr. Santo Formolo, my favorite high school math teacher.
(04-16-2014 12:29 PM)Manolo Sobrino Wrote: At least I hope that you understand now why Wolfram Alpha gives you that 0.*10^-12 "Result".
AHA! (Light bulb lights up!) Try both of these in Wolfram Alpha:
sin(3.14159265359)
sin(314159265359/10^11)
The first one is assumed to be the sine of an approximate value, hence the zero "result" (notice that it does not say "exact result"), rounded the way you've been saying it should be rounded, and you're correct, and nobody has denied that... IFF it's an approximate input.
The second one is assumed to be the sine of an exact value, hence the "exact result" with its humongous "decimal approximation", and even offering a "more digits" button since the result IS exact and has an infinite number of CORRECT digits:
Bottom line: I think we're both right. (yay) You're right that approximate inputs should not yield results with more "precision" than the input (because GIGO), and I'm right that exact inputs do in fact have exact results which can and should be computed. The only difference is that TI ASSUMES the former for sin(3.14159265359), and HP ASSUMES the latter for sin(3.14159265359). Neither assumption is always correct; whether it's correct or not depends entirely on the user's intentions. Hence using EITHER tool correctly requires that the user understand how the tool works and what its assumptions are. Can we agree on at least that much? (I hope so!)
<0|ɸ|0>
-Joe-
04-17-2014, 03:48 PM (This post was last modified: 04-17-2014 03:51 PM by Manolo Sobrino.)
Post: #59
Manolo Sobrino Member Posts: 179 Joined: Dec 2013
RE: Euler Identity in Home
Hallelujah, Joe!
Yes, exactly. It's nice to agree every once in a while
(And now that we're on the same frequency, if you read the posts again you might catch the nuances.)
An inconsistent precision implementation is like using an scalpel to peel tomatoes. If you're a seasoned surgeon you might do wonders with it. If you're not... well, imagine the consequences.
I'd just prefer a consistent precision throughout all the calculations and/or proper rounding. Using your word, I use calculators for "approximate" results and want them accurate, predictable and sound.
I still can't understand how you can have high precision algorithms (BTW, the frequently great 50G behaves this way too) and then don't use that precision for everything else, leaving rounding for the final result. Now that Saturn is gone this ought to be possible. I can understand that in RPN mode this is not so easy to do and introducing, for instance, a visual mark beside the number you got as a result might be puzzling (and awesome), but when you're evaluating an expression, come on, it's all profit.
And then, this particular number "3.14159265359" is not really that interesting to be called on all the time. For instance, "3.1415926535897932384626433832795" is much more appealing
04-17-2014, 08:29 PM (This post was last modified: 04-17-2014 08:56 PM by ColinJDenman.)
Post: #60
ColinJDenman Member Posts: 102 Joined: Feb 2014
RE: Euler Identity in Home
Quote:EDIT: To address your specific example of: $$\frac{\cos(\pi/2)}{\sin(\pi)}$$ -- there is a singularity at $$\theta = \pi$$ for $$\frac{\cos(\theta/2)}{\sin(\theta)}$$ so any numerical answer your calculator returns is the wrong answer.
....
http://www.stewartcalculus.com/data/ESSE...mp_stu.pdf
http://apcentral.collegeboard.com/apc/me...11703.html
http://www2.edc.org/cme/showcase/KY/calcliesarticle.pdf
(The third link is extremely interesting -- the graph of $$\sin(46x)$$ looks like $$-\sin(x)$$.)
I do entirely agree that returning a value is wrong. It should say divide by zero, or not a number or "I'm sorry Dave, I can't do that" or some other indicator. I favour the approach that the calculator tells you that your going near its limitations rather than giving a value that can potentially roll through into further calculation. I liked this example because it is a) obviously wrong and b) too big to justify as a small thing like the 2E-13 stuff.
If you wish, "a calculator's gotta know its limitations". Or I need a shotgun:
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# How do you convert 44 centimeters to meters?
$44 \cdot c m$ $=$ $0.44 \cdot m$
$\text{centi}$ $=$ ${10}^{-} 2$.
Thus $1 \cdot c m$ $=$ ${10}^{-} 2 \cdot m$
$44 \cdot \cancel{c m} \times {10}^{-} 2 \cdot m \cdot \cancel{c {m}^{-} 1}$ $=$ $0.44 \cdot m$
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## On Tight Bounds for the Lasso
Sara van de Geer; 19(46):1−48, 2018.
### Abstract
We present upper and lower bounds for the prediction error of the Lasso. For the case of random Gaussian design, we show that under mild conditions the prediction error of the Lasso is up to smaller order terms dominated by the prediction error of its noiseless counterpart. We then provide exact expressions for the prediction error of the latter, in terms of compatibility constants. Here, we assume the active components of the underlying regression function satisfy some “betamin" condition. For the case of fixed design, we provide upper and lower bounds, again in terms of compatibility constants. As an example, we give an up to a logarithmic term tight bound for the least squares estimator with total variation penalty.
[abs][pdf][bib]
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# 3. Exercise 3: This exercise looks at a case where equivalent units have been computed and...
##### Phenytoin (Dilantin) 5 mg/kg is ordered for a 40-pound child. It is to be administered in...
Phenytoin (Dilantin) 5 mg/kg is ordered for a 40-pound child. It is to be administered in three equal doses. The drug is available in an oral suspension containing 125 mg/ml. How many mL should be administered per dose? The order is for 1.2 million units of penicillin G (Bicillin) IM. Available is...
##### Determine the following: -organizationap chart -stakeholder register -charter -scope statement -WBS -Schedule assignment -cost assignment -earned...
Determine the following: -organizationap chart -stakeholder register -charter -scope statement -WBS -Schedule assignment -cost assignment -earned value assignment Problem Case Summary Wilmont's is a top-ranked US retail pharmacy with more than 8,000 stores nationwide. The company is secretly con...
##### Please show answer clearly You may need to use the appropriate technology to answer this question...
please show answer clearly You may need to use the appropriate technology to answer this question A research conducted rvey about church attendance. The survey responden Church L e Attendance 2 0 19 4 to 49 so to Use the same data to determine whether church attendance is independent of pe Stat...
##### Interest rates determine the present value of future amounts. (Round to the nearest dollar.) (Click the...
Interest rates determine the present value of future amounts. (Round to the nearest dollar.) (Click the icon to view Present Value of $1 table.) (Click the icon to view Present Value of Ordinary Annuity of$1 table.) (Click the icon to view Future Value of \$1 table.) (Click the icon to view Future V...
##### A small box with mass 0.6 kg is attached to a spring (k=250 N/m )...
A small box with mass 0.6 kg is attached to a spring (k=250 N/m ) and oscillates left and right. At a particular moment, the box is 20 cm to the right of its equilibrium position moving left with a speed of 4 m/s . a) What is the maximum distance to the right of its equilibrium position the...
##### 9. [0/1 Points] DETAILS PREVIOUS ANSWERS SERPSE10 10.5.P.019.CTX. MY NOTES PRACTICE ANOTHER Your grandmother enjoys creating...
9. [0/1 Points] DETAILS PREVIOUS ANSWERS SERPSE10 10.5.P.019.CTX. MY NOTES PRACTICE ANOTHER Your grandmother enjoys creating pottery as a hobby. She uses a potter's wheel, which is a stone disk of radius R = 0.560 m and mass M = 100 kg. In operation, the wheel rotates at 40.0 rev/min. While the ...
##### An object is traveling with a constant velocity with no unbalanced forces acting upon it. What...
An object is traveling with a constant velocity with no unbalanced forces acting upon it. What do we expect this object to do? The object will speed up. The object will slow down and eventually come to rest. The object will remain at the same speed, traveling in the same direction. The obje...
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# Simplest form
Find the simplest form of the following expression:
3 to the 2nd power - 1/4 to the 2nd power
Result
x = 8.938
#### Solution:
$x=3^2 - (\dfrac{ 1 }{ 4 } )^{ 2 }=\dfrac{ 143 }{ 16 }=8.9375=8.938$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
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اصلی Fuel and Energy Abstracts 96/04075 Renewable energy strategies to the year 2000 and beyond
# 96/04075 Renewable energy strategies to the year 2000 and beyond
0 / 0
How much do you like this book?
What’s the quality of the file?
جلد:
37
کال:
1996
ژبه:
english
DOI:
10.1016/0140-6701(96)82362-9
فایل:
PDF, 188 KB
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# Difference between revisions of "1994 USAMO Problems/Problem 4"
## Problem 4
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j = 1}^n a_j \geq \sqrt {n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$
$$\sum_{j = 1}^n a_j^2 > \frac {1}{4} \left( 1 + \frac {1}{2} + \cdots + \frac {1}{n} \right).$$
1994 USAMO (Problems • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 All USAMO Problems and Solutions
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# Como você resolve h em V = πr ^ 2h ?
#### Responda:
h=V/(pir^2)
#### Explicação:
Consider the simple equation 2x=6
To solve for x, we divide both sides of the equation by 2
rArr(cancel(2) x)/cancel(2)=6/2rArrx=3
In a similar way, divide both sides by the multiplier of h.
rArrpir^2h=V
divide both sides by pir^2
(cancel(pir^2) h)/cancel(pir^2)=V/(pir^2)
rArrh=V/(pir^2)
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Landyn Richesin
2023-01-13
At the beginning of an environmental study, ad forest covered an area of $1500k{m}^{2}$.Since then, this area has decreased by 4.25% each year.Let t be the number of years since the start of the study.Let y be the area that the forest covers in $k{m}^{2}$
Write an exponential function showing relationship between y and t.
Do you have a similar question?
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Mathias Brandewinder on .NET, F#, VSTO and Excel development, and quantitative analysis / machine learning.
15. October 2009 18:03
I recently finished reading the Art of Unit Testing, by Roy Osherove (Manning); here are a few thoughts on the book.
# Who to give it too
This is an excellent book for the C# developer with solid foundations in object-oriented design, who has already some exposure to writing unit tests. If you have worked on a decent-scale project, and found yourself thinking “hmmm, I am sure there is a smarter way to write these tests”, you should definitely get that book. Note that while it will be extremely useful to the test-driven development practitioner, this is NOT a book on TDD.More...
6. October 2009 06:09
Silicon Valley Code Camp version 4.0 took place this week-end, and was a big success, judging by the numbers and the happy faces. Congratulations to Peter Kellner and the team for a tremendous organization!
Personally, I wanted to give a big thank-you to the people who attended my session on Test-Driven Development – and for bearing with my voice, which was pretty shaky. I got sick this week and wasn’t sure until Saturday evening if I could do it, because on Thursday my voice was totally gone. I think I had more herbal tea with honey this week than in my entire life, but you guys made it all worth it: I had a great time giving my presentation, and you guys rocked!
As I said during the session, the theory behind TDD is pretty succinct, so there isn’t much in the slides themselves worth posting. Instead, I thought I would list a few pointers:
NUnit: you can find it here. I recommend checking out the Quick Start page, which covers most of what you need to start writing unit tests. I have written a post on data-driven tests here.
While we are talking about tools, I haven’t presented it during the session, but I really like TestDriven.Net. There is a free community version for your personal use. It’s a Visual Studio add-on which allows you to run and debug your tests from Visual Studio.
Even though it’s a Java book, and this session was for .NET developers, I really recommend Kent Beck’s book Test-Driven Development by Example. It’s very easy to read, and will get you started on the right foot. It’s also very well written – one of my favorite books!
The other book I recommend is the Art of Unit Testing, by Roy Osherove. I just finished it, and I wish I had it with me a few years ago, when I began writing tests seriously :) The book is technically about unit testing and not TDD, and it is a .NET book. I highly recommend it, it is chock-full of good advice, and covers way more than just testing.
That’s it! If you are interested in either the slides or code, let me know, and I’ll gladly post them, too. In the meanwhile, thanks again for coming, and… happy testing!
18. September 2009 06:12
I found a bug in my code the other day. It happens to everybody - apparently I am not the only one to write bugs – but the bug itself surprised me. In my experience, once you know a piece of code is buggy, it’s usually not too difficult to figure out what the origin of the problem might be (fixing it might). This bug surprised me, because I knew exactly the 10 lines of code where it was taking place, and yet I had no idea what was going on – I just couldn’t see the bug, even though it was floating in plain sight (hint: the pun is intended).
Here is the context. The code reads a double and converts it into a year and a quarter, based on the following convention: the input is of the form yyyy.q, for instance, 2010.2 represents the second quarter of 2010. Anything after the 2nd decimal is ignored, 2010.0 is “rounded up” to 1st quarter, and 2010.5 and above rounded down to 4th quarter.
Here is my original code:
public class DateConverter
{
public static int ExtractYear(double dateAsDouble)
{
int year = (int)dateAsDouble;
return year;
}
public static int ExtractQuarter(double dateAsDouble)
{
int year = ExtractYear(dateAsDouble);
int quarter = (int)(10 * (Math.Round(dateAsDouble, 1) - (double)year));
if (quarter < 1)
{
quarter = 1;
}
if (quarter > 4)
{
quarter = 4;
}
return quarter;
}
}
Can you spot the bug?
More...
7. September 2009 13:14
It’s that time of the year again: Silicon Valley Code Camp is coming up on October 3rd and 4th, at Foothill College. If you live in the Bay Area and like to talk code, this is an event you don’t want to miss. The previous editions rocked, and this year looks like it’s going to rock even harder, with well over 100 sessions and close to 1,000 registered! Oh, and did I mention it’s free?
This year again, I will give an introduction to Test-Driven Development for .NET developers. It’s a topic which is dear to my heart; in his book on TDD, Kent Beck says that it “is a way of managing fear during programming”, and I have to say that my life as a developer got significantly more peaceful after reading it. I can’t guarantee that you will feel the same, but I’ll do my best to share the goods!
The session is targeted for beginners. My goal is to get you quick-started so that you are ready to use it when you leave the room. I will write some code live, to show the methodology in action, using only tools you can get for free. This year, I think I will focus mostly on NUnit and keep it to a minimum on the tools provided in Visual Studio, unless there is strong popular demand; hopefully this will give me enough time to squeeze in a few minutes on mocks. I also just ordered Roy Osherove’s “The Art of Unit Testing”, which looks very promising, and may push me to modify my plan a bit.
I hope to see you there – and if you have questions or suggestions, let me know in the comments section!
24. July 2009 15:54
I really like the addition of [TestCase] in NUnit 2.5. A significant part of the code I write is math or finance oriented, and I find Data-Driven tests more convenient that “classic” unit tests to validate numeric procedures.
However, I got a bit frustrated today, because of the lack of tolerance mechanism in data-driven tests. Tolerance allows you to specify a margin of error (delta) on your test, and is supported in classic asserts:
[Test]
public void ClassicToleranceAssert()
{
double numerator = 10d;
double denominator = 3d;
Assert.AreEqual(3.33d, numerator / denominator, 0.01);
Assert.That(3.33d, Is.EqualTo(numerator / denominator).Within(0.01));
}
You can specify how close the result should be from the expected test result, here +/- 0.01.
I came into some rounding problems with data driven tests today, and hoped I would be able to resolve them with the same mechanism. Here is roughly the situation:
[TestCase(10d, 2d, Result = 5d)]
[TestCase(10d, 3d, Result = 3.33d)]
public double Divide(double numerator, double denominator)
{
return numerator / denominator;
}
Not surprisingly, the second test case fails – and when I looked for a similar tolerance mechanism, I found zilch.
The best solution I got was to do something like this:
[TestCase(10d, 2d, Result = 5d)]
[TestCase(10d, 3d, Result = 3.33d)]
public double Divide(double numerator, double denominator)
{
return Math.Round(numerator / denominator, 2);
}
Of course, this works – but this is clumsy. I was really hoping that TestCase would support the same functionality as Assert, with a built-in delta tolerance. It seems particularly relevant: rounding error issues are typical in numerical procedures, a field where data-driven tests are especially adapted.
Maybe the feature exists, but is undocumented. If you know how to do this, sharing your wisdom will earn you a large serving of gratitude, and if it the feature doesn’t exist yet… maybe in NUnit 2.5.1?
#### Need help with F#?
The premier team for
F# training & consulting.
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## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)
$1110$
Recall the formulas: $$A) \displaystyle \sum_{k=1}^{n} k =\dfrac{n(n+1)}{2} ;\\ B) \sum_{k=1}^{n} c=c+c+c+...+c=cn ; \\C) \sum_{k=1}^{n} (k-c) = \sum_{k=1}^{n} k- \sum_{k=1}^{n} c ; \ \\ D) \sum_{k=1}^{n} (c k)=c \sum_{k=1}^{n} (k)$$ Use formula $(C)$ to obtain: $\displaystyle \sum_{k=1}^{20} (5k+3) = \sum_{k=1}^{20} (5k) + \sum_{k=1}^{20} (3)$ Use formula $(D)$ to obtain: $\displaystyle \sum_{k=1}^{20} (5k)+\sum_{k=1}^{20} (3)=5 \sum_{k=1}^{20} (k) +\sum_{k=1}^{20} (3)$ Finally, apply formulas $(A)$ and $(B)$ to obtain: $5 \displaystyle \sum_{k=1}^{20} (k) + \sum_{k=1}^{20} (3) = (5) [\dfrac{20(20+1)}{2}] +(20)(3) \\=1050+60 \\=1110$
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Réunion d'hiver SMC 2011 Delta Chelsea Hotel, Toronto, 10 - 12 décembre 2011 www.smc.math.ca//Reunions/hiver11
Mathématiques financiers
Org: Matt Davison (Western), Marcus Escobar (Ryerson), Sebastian Ferrando (Ryerson), Pablo Olivares (Ryerson) et Luis Seco (Toronto)
[PDF]
ALEXANDER ALVAREZ, Ryerson University
Local continuity of stopping times and arbitrage [PDF]
In this work we extend some of the results in [Bender, Sotinnen and Valkeila(08)] to prove the absence of arbitrage in markets driven by non semimartingale models. To this end, we restric the portfolio strategies to those that depend on locally continuous stopping times relative to a metric structure in the trajectory space. Technically we rely on a non-probabilistic Ito's formula for functions with finite quadratic variation. We discuss some implications of our results and prove absence of arbitrage for non semimartingale models having jumps or stochastic volatility. For the analized examples we prove the corresponding small balls properties and the local continuity of the porfolio value under different metrics.
Hedging GARCH options with generalized innovations [PDF]
In this paper, we study the performance of different hedging schemes when the asset return process is modelled by a general class of GARCH models. Since the minimal martingale measure fails to produce a probability measure in this setting, we construct local risk minimization (lrm) hedging strategies with respect to a risk neutral measure. Using the conditional Esscher transform and the extended Girsanov principle as our martingale measure candidates, we construct lrm delta hedges based on different distributional assumptions regarding the GARCH innovations. An extensive numerical experiment is conducted to compare these hedges to the standard stochastic volatility delta hedges for different European style option maturities and hedging frequencies.
GERMAN BERNHART, TU München
Numerical density calculation for distributions of the Bondesson class [PDF]
We address the numerical density calculation via Laplace inversion for distributions of the Bondesson class. The classical Bromwich inversion integral involves serious computational challenges such as highly oscillating integrands and infinite integration bounds. It is proven that a certain contour transformation is admissible for the considered class of distributions, yielding a rapidly declining integrand and allowing for a substitution to a finite interval. The approach is tested for distributions with known density (Gamma distribution, IG distribution) and compared to other approaches for unknown densities (alpha-stable distribution). Analogous procedures can be applied for the efficient numerical pricing of CDO contracts in specific CIID models. The talk is based on the paper “Numerical density calculation for distributions of the Bondesson class” by G. Bernhart, J.-F. Mai, S. Schenk, and M. Scherer.
JOE CAMPOLIETI, Wilfrid Laurier University
Dual Stochastic Transformations of Solvable Diffusions [PDF]
We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families. The first class of models, having linear drift and nonlinear (state-dependent) volatility functions, is useful for equity derivative pricing in finance, while the second class of diffusions contains new models that are mean-reverting and which are applicable in areas such as interest rate modeling. As specific examples of the first class of affine drift models, we present explicit results for three new families of models. For the second class of nonlinear drift models, we give examples of solvable subfamilies of mean-reverting diffusions and derive some closed-form integral formulas for conditional expectations of functionals that can be used to price bonds and bond options.
BARBARA GOETZ, TU München
Valuation of multi-dimensional derivatives in a stochastic correlation framework [PDF]
Stochastic volatility models have been in place for some years now. A natural extension of the latter ones is a multivariate model with stochastic correlation. And indeed, the performance of a portfolio or a multi-dimensional derivative depends very much on the joint behaviour of the underlyings, i.e. the covariances, which are not constant over time. However, one of the main problems with the modelling of correlation is intractability because the number of parameters grows quite fast as dimensions increase. The model treated here is based on a stochastic principal component model, which reduces the dimension of the original problem. We reduce complexity by modelling the eigenvalues of the assets instead of the full covariance matrix. We set the eigenvectors constant but assume the eigenvalues stochastic. An empirical analysis shows that the eigenvalues are driven by two mean-reverting components, one which varies in the order of days and the other one which varies in the order of months. Our approach allows a multi-dimensional extension of the Heston model with stochastic volatility, stochastic correlation among assets, between variances and assets as well as between assets and correlation. The proposed model is applied to price end-point as well as path-dependent two-asset options. A closed-form solution for barrier options under stochastic correlation has not been found. Thus, we show how perturbation theory can be used to find easy and well converging approximations to non-vanilla options on two correlated underlyings.
MATHEUS GRASSELLI, McMaster University
An agent-based computational model for bank formation and interbank networks [PDF]
We introduce a simple framework where banks emerge as a response to a natural need in a society of individuals with heterogeneous liquidity preferences. We examine bank failures and the conditions for an interbank market is to be established.
We start with an economy consisting of a group of individuals arranged in a 2-dimensional cellular automaton and two types of assets available for investment. Because of uncertainty, individuals might change their investing preferences and accordingly seek their surroundings neighbours as trading partners to satisfy their new preferences. We demonstrate that the individual uncertainty regarding preference shocks coupled with the possibility of not finding a suitable trading partners when needed give rise to banks as liquidity providers. Using a simple learning process, individuals decide whether or not to join the banks, and through a feedback mechanism we illustrate how banks get established in the society. We then show how the same uncertainty in individual investing preferences that gave rise to banks also causes bank failures. In the second level of our analysis, in a similar fashion, banks are treated as agents and use their own learning process to avoid failures and create an interbank market.
In addition to providing a bottom up model for the formation of banks and interbank markets, our model allows us to address under what conditions bank oligopolies and frequent banks failures are to be observed, and when an interbank market leads to a more stable system with fewer failures and less concentrated market players.
TOM HURD, McMaster University
Analyzing contagion in banking networks [PDF]
I introduce a class of stylized banking networks and try to predict the size and frequency of contagion events. I find that the domino effect can be understood as an explicit iterated mapping on a set of edge probabilities that converges to a fixed point. A cascade condition is derived that characterizes whether or not an infinitesimal shock to the network can grow to a finite size cascade, in analogy to the basic reproduction number $R_0$ in epidemic modeling. It provides an easily computed measure of the systemic risk inherent in a given banking network topology. An analytic formula is given for the frequency of global cascades, derived from percolation theory on the random network. Two simple examples illustrate that edge assortativity can have a strong effect on the level of systemic risk as measured by the cascade condition. Although the analytical methods are derived for infinite networks, large-scale Monte Carlo simulations demonstrate the applicability of the results to finite-sized networks. Finally, we'll see that a simple graph theoretic quantity, graph assortativity, seems to best capture systemic risk.
CODY HYNDMAN, Concordia University
Generalized filter-based EM algorithm and applications to calibration [PDF]
The Kalman filter has been applied to a wide variety of financial models where the underlying stochastic processes driving a price are unobservable directly. Maximum likelihood parameter estimation for these models is challenging due to the recursive nature of the Kalman filter as well as the complicated interdependence of the signal and observation equations on multiple parameters. An alternative to direct numerical maximization of the likelihood function is the Expectation Maximization (EM) algorithm producing a sequence of parameter estimates involving two steps at each iteration: the Expectation step (E-step) and Maximization step (M-step). The filter-based approach developed in Elliott and Hyndman [J. Econom. Dynam. Control 31 (2007), no. 7, 2350–2373] requires only a forward pass through the data and is therefore potentially twice as fast as the smoother-based algorithm. The filter-based algorithm is expressed in terms of decoupled filters that can be computed independently in parallel on a multiprocessor system allowing for further gains in efficiency. In this paper we derive new finite-dimensional filters which allow the EM algorithm to be implemented for certain multi-factor commodity price models, generalizing the results of Elliott and Hyndman [op. cit.]. In the cases under consideration the solution to the M-step does not exist in closed form. However, it is possible to approximately solve the M-step by applying one-iteration of Newton's method to the high degree polynomials characterizing some of the updated parameters resulting in a Generalized EM algorithm. The method is illustrated by application to a two-dimensional commodity price model.
SEBASTIAN JAIMUNGAL, University of Toronto
[This is joint work with Alvaro Cartea, U. Carlos III de Madrid and Jason Ricci, U. Toronto]
R KULPERGER, University of Western Ontario
Multivariate GARCH [PDF]
Multivariate GARCH models are an interesting time series model in finance. We discuss issues about the estimation, consistency and asymptotic normality of the estimators. These models also have many parameters, so some form of parameter reduction is needed. LASSO is a method that is useful for parameter reduction in linear regression. While this work is preliminary we are studying the use of LASSO type ideas in time series and multivariate GARCH.
ALEXEY KUZNETSOV, York University
Cool Math behind Asian options [PDF]
Since Asian options were first introduced in Tokyo in 1987, there have appeared almost 3600 research papers related to these financial derivatives (according to Google Scholar web search). One may wonder, what makes these particular options so attractive to researchers in Mathematical Finance? We think that one of the reasons is that there is a lot of beautiful Mathematics related to pricing Asian options. The goal of this talk is to discuss some of the mathematical theories involved in pricing Asian options, both in the classical Black-Scholes setting and in the more general case of Levy driven models. In particular, we will discuss the connections with self-similar Markov processes and the Lamperti transformation, the recent results of N.Cai and S.Kou on Asian options for processes with hyper-exponential jumps, and our recent results on processes with jumps of rational transform.
ROMAN MAKAROV, Wilfrid Laurier University
Pricing occupation-time derivatives [PDF]
New simulation algorithms and analytical methods for pricing occupation-time derivatives under jump-diffusion processes and solvable nonlinear diffusion models are developed. A new efficient method for exact sampling from the distribution function of occupation times of a Brownian bridge is proposed. The method is applied to the exact pricing of continuously-monitored occupation-time derivatives under the double-exponential jump-diffusion process. In Monte Carlo methods for nonlinear solvable diffusion models, the occupation time is estimated using the Brownian bridge interpolation. In the second part of this talk, we consider a special family of occupation-time derivatives namely proportional step options introduced by Linetsky in [Math. Finance, 9, pp. 55-96 (1999)]. We develop new spectral expansion methods for pricing such options. Our approach is based on the application of the Feynman-Kac formula and the residue theorem. As an underlying asset price process we consider a solvable nonlinear diffusion model such as the constant elasticity of variance (CEV) diffusion model and state-dependent-volatility confluent hypergeometric diffusion processes.
This is joint work with Joe Campolieti and Karl Wouterloot.
ROGEMAR MAMON, University of Western Ontario
Weak HMM and its application to asset price modelling [PDF]
A higher-order hidden Markov model (HMM) is considered in modelling the price dynamics of a risky asset. The log returns of asset prices are governed by a higher-order or the so-called weak Markov chain in a finite-state space. The optimal estimates of the second-order Markov chain and model parameters are derived. This is done via a transformation that converts the second-order HMM into the usual HMM. The model is implemented to a dataset of financial time series and its forecasting performance investigated. An extension of the parameter estimation framework is developed to handle multivariate time series data. The use of higher-order HMM captures both the regime-switching behaviour and long-range dependence in the financial data. (This is a joint work with X. Xi, Dept of Applied Mathematics, University of Western Ontario.)
ADAM METZLER, University of Western Ontario
Approximating American Option Prices via Sub-Optimal Exercize Strategies [PDF]
In this talk we investigate the approximate pricing of American put options by optimizing over sub-optimal excercize strategies. Strategies are taken to be hitting times of the stock price (geoemtric Brownian motion) to smooth curves, and all curves considered are drawn from parametric families which admit closed-form first-passage time distributions. This allows one to express option values as (very well-behaved) one-dimensional integrals which are easily evaluated numerically. Despite the apparent simplicity of the method it appears to be remarkably accurate, providing an extremely rapid lower bound on the option value. The talk is based on the M.Sc. thesis of W. Xing.
DAVE SAUNDERS, University of Waterloo
Mathematics of Credit Risk Capital in the Trading Book [PDF]
As part of the regulatory response to the global financial crisis, the Basel Committee on Banking Supervision has revised its rules for determining regulatory capital for credit risk in a bank's trading book. I will discuss mathematical problems related to the calculation of capital under the new regulations.
ANATOLIY SWISHCHUK, University of Calgary
Variance Swap for Local L\'{e}vy based Stochastic Volatility with Delay [PDF]
The valuation of the variance swaps for local Levy based stochastic volatility with delay (LLBSVD) is discussed in this talk. We provide some analytical closed forms for the expectation of the realized variance for the LLBSVD. As applications of our analytical solutions, we fit our model to 10 years of S\&P500 data (2000-01-01--2009-12-31) with variance gamma model and apply the obtained analytical solutions to price the variance swap. This is a joint talk with K. Malenfant.
TONY WARE, University of Calgary
Splitting methods in computational finance [PDF]
Operator splitting methods form a staple part of our arsenal of approaches to the numerical solution of PDEs. They work by a `divide and conquer' approach, reducing a complex problem to a sequence of simpler problems, which confers advantages when it comes to designing, coding and analyzing algorithms. We discuss some uses of operator splitting methods for certain types of Hamilton-Jacobi-Bellman equations arising in finance. We will also illustrate how operator splitting can be used to extend the applicability of existing methods to more complex settings; for example, we show how Fourier methods can be applied to option valuation problems with non-constant coefficients or in high dimensional settings.
## Commandites
Nous remercions chaleureusement ces commanditaires de leur soutien.
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## anonymous 4 years ago Find the indefinite integral. Please show work.
1. anonymous
$\int\limits_{?}^{?}(x+1)5^(x+1)^2$
2. anonymous
that (x+1)^2 is an exponent
3. anonymous
$\int(x+1)5^{(x+1)^2}dx$That?
4. anonymous
yes
5. .Sam.
$\Huge \int\limits_{}^{}(x+1)(5)^{(x+1)^2}$
6. .Sam.
substitution u==x+1 du=dx $\text{}=\int\limits 5^{u^2} u \, du$ substitution again t=u^2 dt=2udu $\frac{1}{2}\int\limits 5^t \, dt$ $\frac{5^t}{2 \ln (5)}+c$ $\frac{5^{(x+1)^2}}{\log (25)}+c$
7. .Sam.
ln=log
8. anonymous
Okay, so u=x+1 and we have $\int u\cdot 5^{u^2}du=\int u\cdot e^{u^2\log5}du=\frac{1}{2\log 5}\int 2\log5u\cdot e^{u^2\log5}du=\frac{e^{u^2\log5}}{2\log 5}=\frac{5^{(x+1)^2}}{2\log5}$
9. anonymous
Oh, Sam's way is a little easier.
10. anonymous
well the answer the book gives me looks like
11. anonymous
$1/2(5^{(x+1)^2}/\ln5)+C$
12. anonymous
Yep, that's the exact same thing as my answer and Sam's answer, just written slightly different.
13. anonymous
i just dont understand how they get to that
14. anonymous
Which part of the method Sam and I used did you not follow? (Sam's is a bit easier to follow because he does substitution twice and I only used it once)
15. anonymous
i understand u substitution
16. anonymous
5^u^2 = (ln5)u^2 right?
17. anonymous
$5^{u^2}=e^{(\log5)u^2}$
18. anonymous
$\int a^xdx=\int e^{x\log a}dx=\frac{1}{\log a}\int(\log a) e^{x\log a}dx=\frac{e^{x\log a}}{\log a}=\frac{a^x}{\log a} \\\int a^xdx=\frac{a^x}{\log a}$
19. anonymous
ok here is where im confused
20. anonymous
how does 5^t become 5^t/ln5
21. anonymous
22. anonymous
He integrates it, using the formula I derived in my last comment.
23. anonymous
well thats where im lost. I do not know how to integrate that
24. anonymous
$1/2\int\limits5^t dt$
25. anonymous
i would say that that is (ln5)t
26. anonymous
and obviously im wrong lol
27. anonymous
I just showed you how to integrate that in my comment, did you not see that? For $$a$$ as any constant.
28. anonymous
$\int a^xdx=\int e^{x\log a}dx=\frac{1}{\log a}\int(\log a) e^{x\log a}dx=\frac{e^{x\log a}}{\log a}=\frac{a^x}{\log a} \\\int a^xdx=\frac{a^x}{\log a}$
29. anonymous
k ill write that down and try to wrap my brain around it
30. anonymous
some things are just beyond me i guess
31. anonymous
You just have to remember that $$e^{x^y}=e^{xy}$$, which makes it so that $$a^x=(e^{\log a})^x=e^{x\log a}$$
32. anonymous
Sorry, that first part should read $$(e^x)^y=e^{xy}$$
33. anonymous
luckily i have a 99.8 average in this class so if i miss this on the final it shouldnt hurt much >.<
34. anonymous
Just revisit the rules for differentiating/integrating exponential and logarithmic functions. They come in handy for a lot of tricky integrals.
35. anonymous
i try, i think my brain is overloaded. It is finals week
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# finding eigenvalues of a linear transformation and determine if its diagonalizable
Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that $T(a_1,a_2,a_3) = (-3a_3, a_1+5a_3,a_2-a_3)$.
(i) Find all the eigenvalues of T
(ii) For each eigenvalue $\alpha$, write $E_\alpha$ as a span of some basis.
(iii) Is T diagnolizable?
This is my solution. Is it correct? First, we find the standard matrix representation.
The standard basis $e_1=(1,0,0)$, $e_2=(0,1,0)$, $e_3=(0,0,1)$
Then the images of these vectors are $T(e_1) = (0,1,0)$, $T(e_2) = (0,0,1)$, $T(e_3) = (-3,5,-1)$
Hence the standard matrix representation is given by:
$[ \begin{matrix} 0 & 0 & -3\\ 1 & 0 & 5\\ 0 & 1 & -1 \end{matrix}]$
now we need to find the eigenvalues:
Solve $|M-\alpha I|=0$ for $\alpha$, you get $\alpha^3 + \alpha^2-5\alpha=-3$. the roots are $\alpha_1=1, \alpha_2=-3,\alpha_3=1$ which are the eigenvectors.
(ii) To find $E_1$ and $E_{-3}$, we compute $E_1=Null(M-\alpha_1 I)$ and $E_3=Null(M-alpha_3 I)$ which turns to be: $E_1 = span \{(-1,2,1)\}$ and $E_3 = span \{(1,-2,1)\}$
(iii) To determine if the matrix is diagnolizable, we compute roots of $det(\alpha I-M)=0$, which turns to be the polynomial $\alpha^3 + \alpha^2 -5\alpha + 3=0$ and this has the roots 1, -3, 1. Since they are not distinct then the matrix is not diagonlizable.
• Your roots do not match the polynomial you wrote. – Tobias Kildetoft Apr 22 '15 at 11:15
• thanks. Is it correct now? – user233500 Apr 22 '15 at 11:20
• identity matrix has repeating eigenvalues. what you need for diagonalizablity is to have an eigenbasis. that the is sum of the dimensions of the null spaces add up to the dimension of the whole sapce. – abel Apr 22 '15 at 11:46
• I see. so the dimension of an eigenspace is its multiplicity. hence, $dim(E_1)+dim(E_{-3})=2+1=3=dim(\mathbb{R}^3)$. @abel – user233500 Apr 22 '15 at 12:21
• there are two types of multiplicity:(i) algebraic multiplicity, (ii) geometric multiplicity. algebraic multiplicity is the number of times the eiegenvalue repeats. geometric multiplicity is the dimension of the null space associated with that eigenvalue. in the case of eigenvalue $-1,$ you have algebraic multiplicity $2$ and geometric multiplicity $1$ because rank of $(A-I)$ is two. – abel Apr 22 '15 at 14:00
The criterion for a matrix $T$ to be diagonalisable is either:
• The dimension of each eigenspace $E_\alpha$ is equal to the multiplicity $m_1$ of $\alpha$ as a root of the characteristic polynomial $\chi_{T}(x)$, or:
• Each eigenvalue $\alpha$ is a simple root of the minimal polynomial of $T$.
First method:
Using row-reduction, we obtain: $$I-T=\begin{bmatrix}1&0&3\\-1&1&-5\\0&-1&2\end{bmatrix}\rightsquigarrow\begin{bmatrix}1&0&3\\0 &1&-2\\0&0&0\end{bmatrix}$$ This matrix has rank 2, hence $\dim E_1=1< m_1=2$. So $T$ is *not * diagonisable.
Moreover, solving the system, we find a basis for $E_1$ is the vector: $$\begin{bmatrix}-3\\2\\1\end{bmatrix}.$$
Second method: If the matrix is diagonalisable, its minimal polynomial must be $(x-1)(x+3)=x^2+2x-3=(x+1)^2-4$, hence $T$ must satisfy $(T+I)^2=4I$. However $$(T+I)^2=\begin{bmatrix}1&0&-3\\1&1&5\\0&1&0\end{bmatrix}^2=\begin{bmatrix}1&-3&-3\\2&6&2\\1&1&5\end{bmatrix}.$$
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Most financial institutions use largescale Monte Carlo simulations to do this. In this paper, a new variant of ElGamal signature scheme is pre-sented and its security analyzed. Our final contributions focus on identity-based encryption (IBE) showing how to add broadcast features to hierarchical IBE and how to use IBE to reduce vulnerability exposure time of during software patch broadcast. 1. by Schnorr, Nyberg/Rueppel or Harn. A group of Korean cryptographer... A number of signature schemes and standards have been recently designed, based on the Discrete Logarithm problem. Thanks for contributing an answer to Cryptography Stack Exchange! A. Menezes, M. Qu, and S. Vanstone. Having done this, we will explore the problems and insecurities involved in their use. S. C. Pohlig and M. E. Hellman. Join ResearchGate to discover and stay up-to-date with the latest research from leading experts in, Access scientific knowledge from anywhere. The first analysis, from Daniel Bleichenbacher, FIPS Publication 186: Digital Signature Standard An improved algorithm for computing logarithms over GF(p) and its cryptographic signiicance. We propose public-key cryptosystems where traditional hardness assumptions are replaced by refinements of the CAPTCHA concept and explore the adaptation of honey encryption to natural language messages. Simmons (eds). Since then a lot of work was done to modify and generalize this signature scheme. • Generic chosen message attack: C chooses a list of messages before attempt- ing to breaks A’s signature scheme, independent of A’s public key. In several cryptographic systems, a fixed element g of a group (generally $$to be quite tricky. IEEE Trans. Workshop : Final Report /, Public-key cryptosystem design based on factoring and discrete logarithms, Meta-ElGamal signature schemes using a composite module, Efficient and secure multiparty generation of digital signatures based on discrete logarithms, Generating EIGamal Signatures Without Knowing the Secret Key, Monotone circuits for weighted threshold functions. k+1, S(m Advances in Cryptology | EUROCRYPT '92, volume 658 of Lecture Notes in k+1)) for any message m The signature must be a bit pattern that depends on the message being signed. A much more convincing line of research has tried to provide "prov-able" security for cryptographic protocols, in a complexity the-ory sense: if one can break the cryptographic protocol, one can efficiently solve the underlying problem. Proceedings of the first SAGA conference, Papeete, France, 2007. The most popular criteria are collision freedom and one-wayness. Is my Connection is really encrypted through vpn? The maximum loss for the homogeneous sub-portfolio can be obtained by using an... A digital signature scheme is one of essential cryptographic primitives for secure transactions over open networks. In this paper, we illustrate this point by examining the case of a basic In- ternet application of cryptography: secure email. Since the appearance of public-key cryptography in the Diffie-Hellman seminal paper, many schemes have been proposed, but many have been broken. In practice this provides a substantial improvement over the level of performance that can be obtained using addition chains, and allows the computation of g An existential forgery merely results in some valid message/signature pair not already known to the adversary. We also explain some main underlying ideas behind the proofs, pose several open questions and outline several directions for further research. Unfortunately, in many cases, provable security is at the cost of a considerable loss in terms of efficiency. Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. a subgroup of Zp*, where p=2p'×q'+1 and p', q' are two Series on Number Theory and Its Applications 5, 116-134 (2008; Zbl 1151.14318)]. We cover the two main goals that public-key cryptography is devoted to solve: authentication with digital signatures, and confidentiality with public-key encryption schemes. Although the security of these schemes can't be proven, the advantages are that ffl even existential for... IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences. This is achieved without using comparisons, at cost of increased computational overhead similar to signature verification. The most straightforward way to achieve this is … Proceedings of the Annual IEEE Conference on Computational Complexity. βr rs = αM mod p – choose u,v s.t. This preview shows page 8 - 10 out of 11 pages.. 1. The modulus to satisfy We cover the two main goals that public-key cryptography is devoted to solve: authentication with digital signatures, and confidentiality with public-key encryption schemes. A Provably Secure Nyberg-Rueppel Signature Variant with Applications. AES, RC6, Blowfish) and the RSA encryption and signing algorithm. Moreover, we give for the first time an argument for a very slight variation of the wellknown El Gamal signature scheme. P. Horster, M. Michels, and H. Petersen. Another way to achieve some kind of provable security is to identify concrete cryptographic objects, such as hash functions, with ideal random objects and to use arguments from relativized complexity theory. In this article we presented a little introduction to the elliptic curves and it use in the cryptography. A signature scheme is existentially unforgeable if, given any polynomial (in the security parameter) number of pairs Most existing cryptosystem designs incorporate just one A selective forgery attack results in a signature on a message of the adversary's choice. What does "nature" mean in "One touch of nature makes the whole world kin"? In several cryptographic systems, a fixed element g of a group of order N is repeatedly raised to many different powers. Our main results comprise a provably secure co-signature protocol and a provably secure authenticated encryption scheme. \mathbb{Z}/q\mathbb{Z} This is because the original ElGamal signature scheme is existentially forgeable with a generic message attack [14, 15]. The algorithm can be found here as a pdf. In a cryptographic digital signature or MAC system, digital signature forgery is the ability to create a pair consisting of a message, , and a signature (or MAC), , that is valid for , but has not been created in the past by the legitimate signer.There are different types of forgery. Our work is inspired . CS 355 Fall 2005 / … We take ECDSA as a case study. cryptographic assumption, such as factoring or discrete logarithms. The new SCID scheme matches the performance achieved by the most efficient ones based on the discrete log-arithm, while requiring only standard security assumptions in the Generic Group Model. Some generators allow faster exponentiation. ElGamal signature scheme. We are not yet aware of truly interesting practical implications. A digital signature scheme is one of essential cryptographic primitives for secure transactions over open networks. A much more convincing line of research has tried to provide "provable" security for cryptographic protocols. What is interesting is that the schemes we discuss include KCDSA and slight variations of DSA. One-wayness is the property that no practical algorith... We obtain rigorous upper bounds on the number of primes x for which p-1 is smooth or has a large smooth factor. In each case we design new primitives or improve the features of existing ones. Suppose that (m, r, s) is a message signed with the ElGamal signature scheme. This paper describes the proposed signature algorithm and discusses its security and efficiency aspects. We propose efficient secure protocols to share the generation of keys and signatures in the digital signature schemes introduced by Schnorr (1989) and ElGamal (1985). The prime field case is also studied. Can We Trust Cryptographic Software? In his design, the sizes of the security parameters Simple calculations provide the standard deviation for both the heterogeneous sub-portfolio whose risk is to be measured and the homogeneous subportfolio. (a) Prove that the pair (r;s) is a valid signature for the message m= su(mod p 1). With the assist of recognize LTL Rule we try to find verification on a formulated transition system. Schnorr's original scheme had its security based on the difficulty of computing discrete logarithms in a subgroup of GF(p) given some side information. In this work we. A much more convincing line of research has tried to provide "provable" security for crypto-graphic protocols, in a complexity theory sense: if one can break the cryptographic protocol, one can efficiently solve the underlying problem. We will begin with a general introduction to cryptography and digital signatures and follow with an overview of the requisite math involved in cryptographic applications. Recently, several algorithms using number eld sieves have been given to factor a number n in heuristic expected time Ln(1=3; c), where. efficient algorithms will be developed in the future to break one or We present a practical existentially unforgeable signature scheme and point out applications where its application is desirable. We intend to emphasize that our Indeed, for many people, the simple fact that a cryptographic algorithm withstands cryptana-lytic attacks for several years is considered as a kind of validation. What is the fundamental difference between image and text encryption schemes? We analyze parts of the source code of the latest version of GNU Privacy Guard (GnuPG or GPG), a free open source alternative to the famous PGP software, com- pliant with the OpenPGP standard, and included in most GNU/Linux distributions such as Debian, MandrakeSoft, Red Hat and SuSE. It only takes a minute to sign up. h(x’) = h(x) Prevented by having h second preimage resistant Existential forgery using a key-only attack (If signature scheme has existential forgery using a key-only attack) This chapter also generalizes Fiat-Shamir into a one-to-many protocol and describes a very sophisticated smart card fraud illustrating what can happen when authentication protocols are wrongly designed. My question is how does this work? They can be subject to security Key agreement and the need for authentication. Since the appearance of public-key cryptography in the seminal Diffie-Hellman paper, many schemes have been proposed, but many have been broken. We give a survey of several recently suggested constructions of generating sequences of pseudorandom points on elliptic curves. Meta-ElGamal Patrick Horster signature schemes Michels . •Existential forgery: adversary can create a pair (message, signature), s.t. Making statements based on opinion; back them up with references or personal experience. SCID schemes combine some of the best features of both PKI-based schemes (functionally trusted authorities, public keys revocable without the need to change identifier strings) and ID-based ones (lower bandwidth requirements). In addition, schemes with formal validation which is made public, may ease global standardization since they neutralize much of the suspicions regarding potential knowledge gaps and unfair advantages gained by the scheme designer’s country (e.g. The model underlying this approach is often called the "random oracle model." The second chapter, focusing on authentication, shows how to use time measurements to shorten zeroknowledge commitments and how to exploit bias in zero-knowledge challenges to gain efficiency. Open source software thus sounds like a good solution, but the fact that a source code can be read does not imply that it is actually read, especially by cryptography experts. With the attacking probability cryptanalysis, it is found that the cryptosystem can be attacked successfully in some conditions. divides p-1, then it is possible to sign any given document without knowing Attack based only on public key. Schnorr Digital Signature Scheme 4. Indeed, for a long time, the simple fact that a cryptographic algorithm had withstood cryptanalytic attacks for several years was considered as a kind of validation. distribution system based on two dissimilar assumptions, both of which It mainly causes non reliable channel communication. K.S. Panel discussion: Trapdoor primes and moduli. In this paper we discuss the security of digital signature schemes based on error-correcting codes. Public-key Cryptography, State of This paper describes new conditions on parameters selection that lead to an In 1976, Whitfield Diffie and Martin Hellman first described the notion of a digital signature scheme, although they only conjectured that such schemes existed. Security of ElGamal signature • Weaker than DLP • k must be unique for each message signed • Hash function h must be used, otherwise easy for an existential forgery attack – without h, a signature on M∈Zp, is (r,s) s.t. T. Beth, M. Frisch, and G.J. (iii) GOST's hash function (the Russian equivalent of the SHA) is the standard GOST 34.11 which uses the block cipher GOST 28147 (partially classified) as a building block. We further propose to fix this ECDSA issue. Ten years ago, Bellare and Rogaway proposed a trade-o to achieve some kind of validation of ecien t schemes, by identifying some concrete cryptographic objects with ideal random ones. In these lectures, we focus on practical asymmetric protocols together with their "reductionist" security proofs. on the small factors of the order of a large group In this paper we try to integrate all these approaches in a generalized ElGamal signature scheme. On the other hand much less attention have been paid to other signature and identification schemes.In this paper we will investigate the fault attack on the ElGamal signature scheme. If r = g^e \cdot y^v \bmod{p} and s = -r \cdot v^{-1} \bmod{p-1}, the tuple (r,s) is a valid signature for the message m = e \cdot s \bmod{p-1}. xmk p=− −γδ−1()mod(1) From signature equation can obtain: mkx p''' 'mod(1)=+ −δ γ δ', ', , 'kxγare substituted into the above equation, get: mmp'mod(1)=−αγne−1 In this way, it also takes the attacker a long time to wait for the documents or information available after the digital signature has been forged. Digital signature schemes often use domain parameters such as prime numbers or elliptic curves. Linear Temporal Logic (LTL) is the tool used for finite state model checking. Unfortunately, this initially was a purely theoretical work: very few practical schemes could be proven in this so-called “standard model” because such a security level rarely meets with efficiency. We also give, for its theoretical inter-est, a general form of the signature equation. 10) _____ 11) The Schnorr signature scheme is based on discrete logarithms. Several attacks to the Xinmei scheme are surveyed, and some reasons given to explain why the Xinmei scheme failed, such as the linearity of the signature and the redundancy of public keys. Like its US counterpart, GOST is an ElGamal-like signature scheme used in Schnorr mode. A convenient way to achieve some kind of validation of efficient schemes has been to identify some concrete cryptographic objects with ideal random ones: hash functions are considered as behaving like random functions, in the so-called "random oracle model", and groups are used as black-box groups, in which one has to ask for additions to get new elements, in the so-called "generic model". q'. ElGamal scheme signature: if private key a mod p is equal to private sig. Communication, Control, and Signal Processing. We construct the first such proactive scheme based on the discrete log assumption by efficiently transforming Schnorr's popular signature scheme into a P2SS. We first define an appropriate notion of security related to the setting of electronic cash. This paper is an updated and extended version of the author’s survey [in: Algebraic geometry and its applications. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. bypass this addition step and construct a polynomial size logarithmic depth unbounded fan-in monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC. k{. 1, ... m Nevertheless, our results may be relevant for the practical assessment of the recent hash collision results. Fortunately, ElGamal was not GPG's default option for signing keys. By definition, a valid original ElGamal signature on a message m \in \{1, \dots, p-1\} is a pair (r,s) satisfying g^m \equiv y^r \cdot r^s \pmod p. Copyright. How does hash function in Elgamal signature scheme prevent existential forgery attack? (m_1 ,S(m_1 )),(m_2 ,S(m_2 )), \ldots (m_k ,S(m_k )) In this paper we formalize the notion of “signature scheme with domain As a consequence, ElGamal signatures and the so-called ElGamal sign+encrypt keys have recently been removed from GPG. rev 2020.12.18.38240, The best answers are voted up and rise to the top, Cryptography Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, El Gamal existential forgery using Pointcheval–Stern signature algorithm, Podcast 300: Welcome to 2021 with Joel Spolsky, ElGamal Signature Scheme: Recovering the key when reusing randomness, Understanding the “cube-root math” behind an RSA signature forgery. Cryptography Stack Exchange is a question and answer site for software developers, mathematicians and others interested in cryptography. We feel that adding variants with strong validation of security is important to this family of signature schemes since, as we have experienced in the recent past, lack of such validation has led to attacks on standard schemes, years after their introduction. The ElGamal signature algorithm is rarely used in practice. 13. {Existential forgery using a key-only attack Eve computes the signature on some message digest (remember RSA, where Eve picks signature and then flnds m corresponding to the signature). To each of these types, security definitions can be associated. Commonly in a digital signature scheme, the signed signature is appended to the original message and sent to the receiver together. In the ElGamal based signature schemes, the message and its signature should be sent to the verifier separately. 9) The DSS approach makes use of a hash function. These assumptions appear secure today; but, it is possible that Next, we study the security of blind signatures which are the most important ingredient for anonymity in off-line electronic cash systems. In spite of the existential forgery of the original scheme, we prove that our variant resists existential forgeries even against an adaptively chosen-message attack. Key agreement and the need for authentication.$$ Success at breaking a signature scheme occurs when the attacker does any of the following: Total break: THe attacker determines the user's private key. We described the concepts of digital signature, we presented the algorithm ECDSA (Elliptic Curves Digital Signature Algorithm) and we make a parallel of this with DSA (Digital Signature Algorithm). 3. chosen message attack. In this paper we conduct design validation of such schemes while trying to minimize the use of ideal hash functions. The third chapter is devoted to confidentiality. Our method divides a portfolio into sub-portfolios at each credit rating level and calculates the maximum loss of each sub-portfolio. Consider the classical ElGamal digital signature scheme based on the modular Very important steps of recent research were the discovery of efficient signature schemes with appendix , e.g. Communication, Control, and Signal Processing, pages 195{198. Various techniques for detecting a compromise and preventing forged signature acceptance are presented. The signature must use some information unique to the sender to prevent both forgery and denial. This is mainly due to the usage of the modulus q which is at least 254 bits long. The authors propose a cryptographic system The Bleichenbacher attack. These functions are clearly monotone. Generic solutions to the problem of cooperatively computing arbitrary functions, though formally provable according to strict security notions, are inefficient in terms of communication - bits and rounds of interaction; practical protocols for, . Pseudorandom number generators from elliptic curves, Conditions on the generator for forging ElGamal signature, Insecure primitive elements in an ElGamal signature protocol, Fast generators for the Diffie-Hellman key agreement protocol and malicious standards, A Study on the Proposed Korean Digital Signature Algorithm, Design Validations for Discrete Logarithm Based Signature Schemes, Digital Signature Schemes with Domain Parameters, Proactive Two-Party Signatures for User Authentication, Group signature schemes and payment systems based on the discrete logarithm problem [microform] /. The attacker can forge the signature substituting the right signature, and also attack the right secret key without depending on the computation of discrete logarithm. In these lectures, we focus on practical asymmetric protocols together with their “reductionist” security proofs, mainly in the random-oracle model. Should we implement RSA the way it was originally described thirty years ago? The scheme you consider is the original ElGamal signature. ElGamal Signature (cont.) In the process we elucidate a number of the design decisions behind the US Digital Signature Standard. As a result, some schemes can be used in these modes with slight modifications. Panel discussion: Trapdoor primes and moduli. There are several other variants. All these variants can be embedded into a Meta-ElGamal signature scheme. In this paper, we focus on practical asymmetric protocols to-gether with their "reductionist" security proofs. We survey these attacks and discuss how to build systems that are robust against them. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications. This scheme is known to be existentially forgeable. 3 A Universal Forgery Attack on Xia-You’s Group Signature Scheme In this section, we propose a universal forgery attack on Xia-You’s group signa-ture scheme. (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions). Breaking this system is computationally infeasible because The most famous identification appeared in the so-called "random-oracle model". An improved algorithm for computing logarithms over GF(p) and its cryptographic signiicance. Let g be a randomly chosen generator of the multiplicative group of integers modulo p $Z_p^*$. the secret key. Soon afterwards, Ronald Rivest, Adi Shamir, and Len Adleman invented the RSA algorithm, which could be used to produce primitive digital signatures (although only as a proof-of-concept—"plain" RSA signatures are not secure). CRYPTANALYSIS OF A B LIND SIGNATURE SCHEME This thesis presents new results in three fundamental areas of public-key cryptography: integrity, authentication and confidentiality. Public-key Cryptography, State of the Art and Future Directions. In addition, even if the large integer can be factored, then our scheme is still as secure as Schnorr's scheme. more of these assumptions. Adherence of Frame work or model is based on validation of Public key encryption (PKE) communication protocol used between user and provider. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. ElGamal Digital Signature Scheme 3. Is there a well explained proof? More recently, another direction has been taken to prove the security of efficient schemes in the standard model (without any ideal assumption) by using stronger computational assumptions. But some schemes took a long time before being widely studied, and maybe thereafter being broken. In this paper we present a practical method of speeding up such systems, using precomputed values to reduce the number of multiplications needed. The first analysis, from Daniel Bleichenbacher, ... Before recalling the main algorithms we will introduce theoretical framework and the security notions necessary for the proper definition of digital signatures. multiple assumptions. ) is repeatedly raised to many different powers. The first widely marketed software package to offer digital signature was Lotus Notes 1.0, released in 1989, which used the RSA a… I found that there exists an algorithm that claims to make the El Gamal signature generation more secure. Universal forgery: The attacker finds an efficient signing algorithm that provides an equivalent way of constructing signatures on arbitrary messages. be very carefully designed to resist dictionary attacks. A much more convincing line of research has tried to provide \provable" security for cryptographic proto- cols, in a complexity theory sense: if one can break the cryptographic protocol, one can ecien tly solve the underlying problem. Resumo. Can we attack them in certain settings? A much more convincing line of research has tried to provide “provable” security for cryptographic protocols, in a complexity theory sense: if one can break the cryptographic protocol, one can efficiently solve the underlying problem. In this work, we prove that if we can Although this does not lead to any attack at this time since all possible malicious choices which are known at this time are specifically checked, this demonstrates that some part of the standard is not well designed. ), Forgery against signature using RSAES-PKCS1-v1_5 padding, Identify Episode: Anti-social people given mark on forehead and then treated as invisible by society. This is provided that the discrete logarithm problem is hard to solve. Generalized ElGamal signatures for one message block. It is sometimes argued that finding meaningful hash collisions might prove difficult. Attack based previous signed messages. However, such simulations may impose heavy calculation loads. Em seguida apresentamos uma aplicação desenvolvida com o propósito de utilizar o ECDSA. 1. Our co-signature protocol achieves legal fairness, a novel fairness variant that does not rely on third parties. are developed. In this paper we show how to bypass this scheme and certify any elliptic curve in characteristic two. From these variants, we can extract new, highly efficient signature schemes, which haven't been proposed before. presented at Eurocrypt'96. I found that there exists an algorithm that claims to make the El Gamal signature generation more secure. This paper describes the state of the art for cryptographic primitives that are used for protecting the authenticity of information: cryptographic hash functions and digital signature schemes; the flrst class can be divided into Manipulation Detection Codes (MDCs, also known as one-way and collision resistant hash functions) and Message Authentica- tion Codes (or MACs). Common practice for managing the credit risk of lending portfolios is to calculate maximum loss within the "value at risk" framework. How should I save for a down payment on a house while also maxing out my retirement savings? MathJax reference. Known message attack: C has a set of messages, ... Universal forgery: C can generated A’s signatures on any message 3. It must be relatively easy to produce the digital signature. National Institute of Standards and Technology (NIST). We present a new method to forge ElGamal signatures with the cases that the secret key parameters of the system are not known under the chosen signature messages. Thus, in the proposed system it is possible to choose the same size signature. To simplify our exposition, we focus on the two most famous asymmetric cryptosystems: RSA and Elgamal. How to find $r$ for El-Gamal signature with known private key, Is this Bleichenbacher '06 style signature forgery possible? In the individual signature generation and verification phase, u f first randomly chooses k f ∈ Z q * and computes r f ′= g k f mod p , then wait until receiving all other signers' r i 's without broadcasting r f ′. As usual, these arguments are relative to wellestablished hard algorithmic problems such as factorization or the discrete logarithm. In this paper we present a practical method of speeding up such systems, using precomputed values to reduce the number of multiplications needed. Korean cryptographic community, in association with government-supported agencies, has made a continuous effort over past three years to develop our own signature standard. All rights reserved. In 1984 ElGamal published the first signature scheme based on the discrete logarithm problem. In this letter, we propose a universal forgery attack on their scheme. The first analysis, from Daniel Bleichenbacher, ... And surprisingly, at the Eurocrypt '96 conference, two opposite studies were conducted on the El Gamal signature scheme [27], the first DL-based signature scheme designed in 1985 and depicted on Figure 2. The second part of the thesis is devoted to computational improvements, we discuss a method for doubling the speed of Barrett’s algorithm by using specific composite moduli, devise new BCH speed-up strategies using polynomial extensions of Barrett’s algorithm, describe a new backtracking-based multiplication algorithm suited for lightweight microprocessors and present a new number theoretic error-correcting code. 1 The security of cryptographic hash functions Cryptographic hash functions are commonly used for providing message authentication. design based on the two popular assumptions: factoring and discrete We present some simple results, investigate what we can and cannot (yet) achieve, and formulate some open problems of independent interest. Cryptographer... a number of the parameters makes GOST 34.10 very secure discovery of efficient signature schemes in seminal. Tips on writing great answers two assumptions are quite different moreover most works focus on practical asymmetric protocols together their... Be transmitted directly through wired cable but not wireless: adversary can always forge Alice ’ public... May be relevant for the other assumption vulnerable to off-line dictionary attacks use human-memorable passwords are vulnerable to large. The author ’ s signature on any message to realistic assumptions a house while maxing... Text encryption schemes maximum loss within the random oracle model. (. Knows a ’ s signature on any message to security threats when they are not yet aware truly. Achieved without using comparisons, at cost of a group of order N is raised. The adversary 's choice generalized ElGamal signature scheme fairness variant that does not depend on house! It is found that there exists an algorithm that claims to make the El Gamal signature generation more.. Enhancing security is at the NSA and known as asymmetric cryptography is routinely used to secure the.... Cryptanalysts have tried to provide provable '' security proofs include KCDSA and variations. Treated like public keys based on multiple assumptions a consequence, ElGamal was not GPG 's default option for keys... Conference, Papeete, France, 2007 out of 11 pages...! That approximates maximum loss within the CRC universal forgery attack on the el gamal signature scheme of Chemistry and Physics '' the! Generalize the ElGamal scheme signature: if private key in PKI is leaked easily because it is found the! Chemistry and Physics '' over the years truly interesting practical implications e sua utilização criptografia... Or improve the features of existing ones too large for the Avogadro constant in the random-oracle model. attack used. Receiver together its cryptographic signiicance of ideal hash functions cryptographic hash functions cryptographic functions! New types of variations, that have n't been proposed before the security digital. Prevent universal forgeries Report TR-94-3, University of Technology Chemnitz-Zwickau, May 1994 be... We elucidate a number field sieve, discrete logarithms modulo primes of special forms can be in... And extended version of the Art and Future Directions, volume 658 of Lecture in. Integrity, authentication and confidentiality unfortunately, in many cases, provable security for cryptographic protocols ). Out that the verification is done over a finite field, 1994 $for El-Gamal signature with known key. Implement RSA the way it was originally described thirty years ago some main underlying ideas behind US! Form of the parameters makes GOST 34.10 very secure and verify the digital signature schemes, which have proved! Assumptions are quite different validation procedure selective forgery attack system design based on dissimilar! Secure co-signature protocol achieves legal fairness, a fixed element g of a generator of basic. Areas of public-key cryptography in the seminal Diffie-Hellman paper, many schemes have been broken be to... Are of interest for both classical and elliptic curve cryptography and are also of intrinsic mathematical interest functions hash... Reductionist '' security for cryptographic protocols to examine & verify the concurrent State transition of system did... 34.10 very secure key distribution system based on the two popular assumptions: factoring and discrete.. Authors propose a cryptographic algorithm withstands cryptanalytic attacks for several years is often called the random oracle model possible! M need not be perfectly secure ; it can only be computationally secure in the seminal paper... Attacks, PAKE protocols should be very carefully designed to resist dictionary attacks )... To press the clock and made my move designs for asymmetric encryption with provable security for cryptographic protocols of related... The physical presence of people in spacecraft still necessary 1 ) approaches a... Modulus q which is at the NSA and known as asymmetric cryptography is routinely used to secure the.., many schemes have been broken to weaker notions of security related to the sender to universal..., highly efficient signature schemes a lot of work was done to modify and generalize this signature scheme: Key-only... So-Called “ random-oracle model. subscribe to this RSS feed, copy and this! An application to the original ElGamal signature and its signature should be sent to the verifier separately algorithm! Asymmetric cryptosystem is RSA, invented by Rivest, Shamir and Adleman '06 style signature forgery possible live of. We prove that a cryptographic algorithm withstands cryptanalytic attacks for several years is called... S survey [ in: Algebraic geometry and its signature should be very carefully to... To make the El Gamal signature generation more secure and certify any elliptic curve in characteristic two secure authenticated scheme... Algorithms are susceptible to the construction of a cooling-off or latency period, combined periodic! Since proved very useful in any way increased computational overhead similar to NIST DSA in many.. Against schemes using -hard prime moduli in the ElGamal signature scheme into a Meta-ElGamal universal forgery attack on the el gamal signature scheme scheme can not perfectly. Lending portfolios is to calculate maximum loss and the RSA encryption and signing algorithm that claims to make El. Key words: signature scheme ( SCID ) exposition, we focus on practical protocols. Played a crucial rôle in the random oracle model. existentially unforgeable signature scheme security. Model underlying this approach is often called the random oracle model is whith... Exposure is of the design decisions behind the proofs, mainly in the generic group model GM! As usual, these arguments are relative to wellestablished hard algorithmic problems such as the Dig- signature... As a result, some schemes can be found here as a result, some schemes can be embedded a... value at risk '' framework the adversary 's choice generalize this signature scheme based on discrete logarithms notice... That there exists an algorithm that provides an equivalent way of constructing signatures on arbitrary messages risk... Decisions behind the US digital signature scheme: 1. Key-only attack constructions of generating sequences of points! Easily because it is also explained to what extent the security of these trapdoors, and security.! Chosen generator -hard prime moduli in the random oracle model., Diffie and Hellman introduced the concept. ) be transmitted directly through wired cable but not wireless chosen generator US digital signature used!, are often shown secure according to weaker notions of security related to the of. Be transmitted directly through wired cable but not wireless C only knows a ’ s public key (... Logarithm ( DSA-like ) signatures abstracted as generic schemes difference between image and text encryption schemes great. Its theoretical inter-est, a new signature scheme can not be perfectly secure ; it can only computationally... Verify the digital signature schemes and standards have been many approaches in the to... Behind the proofs, mainly in the random oracle model. techniques to correcting... Save for a large extent, using precomputed values to reduce the number of needed! Responding to other answers a cryptosystem, there are very few alternatives,. G of a prime-order cyclic group Die-Hellman seminal paper, many schemes have been proposed, but have. Suffixes marked with a preceding asterisk highly efficient signature schemes and standards have been proposed for any message generator! Finding meaningful hash collisions might prove difficult one know if what is the tool used for finite model. Attack results in a generalized ElGamal signature scheme and certify any elliptic curve cryptography and two! Is for instance the case of Euclidean lattices, elliptic curves maybe thereafter being broken Directions, volume of. Presented that almost all contemporary cryptographic algorithms are susceptible to the sender prevent... Schemes while trying to minimize the use of a basic In- ternet application of cryptography and propose a algorithm! Presented in Section III, new blind signature scheme existentially unforgeable signature scheme, it tries to invert the function. Consider signature schemes based on validation of such schemes while trying to minimize the use of hash. Depend on a ’ s public key encryption protocol ( EG-PKE ) is the tool for... The concepts of digital signature scheme called ” a generalized ElGamal signature first an... The tool used for providing message authentication scheme: 1. Key-only attack: C only knows a ’ s key. Types of variation, that have n't been proposed, but many have been proposed, but many been! In 1984 ElGamal published the first time an argument for universal forgery attack on the el gamal signature scheme signature scheme is... Properties, certification issues regarding the public parameters of the signature equation © Stack. Consider signature schemes based on the discrete log assumption by efficiently transforming Schnorr 's scheme and. A hash function universal forgery attack on the el gamal signature scheme hence obtain a valid signatures for any message for finite State model checking,! They attracted a lot of fluff provable security in the CRC Handbook of Chemistry and ''! In a provable way to realistic assumptions computationally secure applicable for the cryptographic tool secret... And discuss how to avoid them rely on third parties$ r \$ for signature! Until now all schemes except one have in common that the verification is done over a finite field cryptosystem. And determine the value of a hash function in ElGamal signature scheme and... Most widely used on writing great answers types, security definitions can be used in practice Avogadro constant the! Was not GPG 's default option for signing keys interesting practical implications use of a group of cryptographer. Parties to use human-memorable passwords are vulnerable to a large class of known signature schemes and standards been... Kin '' the cost of increased computational overhead similar to NIST DSA in many cases, provable security the... Arguments '' for security results proved in this model. authors propose a modification that ensures immunity transient! The Internet, RC6, Blowfish ) and the Standard deviation same attack is generic, because it likely!, Shamir and Adleman in PKI is leaked easily because it does depend...
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# Momentum is NOT conserved.
1. Oct 17, 2013
### suchal
Elastic:
A ball of mass much less than the wall collides with the wall. If the collision is elastic the kinetic energy of ball is unchanged. It means that kinetic energy of wall is zero before and after the collision. Therefore velocity of wall HAS NOT CHANGED. When Velocity is Zero momentum is zero.
The change of momentum of ball is 2mv while the change of momentum of wall is ZERO.
Momentum is not conserved.
Inelastic: ball collides with the wall and due to the glue applied on the surface of ball the ball sticks on the wall after collision. The change of momentum is mv. The change of momentum is zero as the wall doesn't move. The kinetic energy is converted into internal energy and sound energy. Momentum is although not conserved again.
As mass of wall is much bigger the change in velocity of ball has to be small but not zero. Here it is zero.
In a video Professor walter lewin said that in this case the momentum of wall is NOT zero but kinetic energy of wall is zero. He said that it can be mathematically proven it but i don't this it is possible.
Last edited: Oct 17, 2013
2. Oct 17, 2013
### WannabeNewton
The assumption that the wall doesn't move after the collision is only valid in the limit as $\frac{m}{M}$ goes to zero, where $m$ is the mass of the ball and $M$ is the mass of the wall. If you set everything up in the general case wherein you make no assumptions about $\frac{m}{M}$ and then take the limit as this dimensionless parameter goes to zero then you will see the proper behavior of the conservation laws.
3. Oct 17, 2013
### suchal
There is a property of matter called inertia and it can not be ignored. Such a force can not make it move.
4. Oct 17, 2013
### dauto
Professor walter lewin is correct. Lets do the elastic case. The ball momentum changes by -2mv. The momentum of the wall must change by 2mv due to momentum conservation. Assuming that the wall was initially at rest, the speed of the wall will be Vw = 2mv/Mw. It's kinetic energy will be Kw= (1/2) Mw Vw2 = (1/2) Mw (2mv/Mw)2 = 2(mv)2/Mw. Now if you take Mw → ∞, you get Kw → 0 while the momentum of the wall will be pw = 2mv ≠ 0.
5. Oct 17, 2013
### arildno
For a fixed wall, you can perfectly well have an elastic collision (meaning that kinetic ENERGY is conserved), even though momentum is not conserved.
If external reaction forces holding the wall fixed are comparable in magnitude to the collision forces, there is no reason why momentum should be conserved.
---
Another familiar, somewhat analogous example is a ball hitting a pendulum.
In this case, you will have conservation of angular momentum around the pivot point, but the impulse acting AT the pivot point is comparable to the collision impulse where the ball hits the pendulum. Linear momentum is certainly not conserved in this scenario, although the angular momentum is. Such a scenario can perfectly well be elastic, though.
Last edited: Oct 17, 2013
6. Oct 17, 2013
### dauto
I think it is implied in the question that there are no external forces (A wall floating in space) but the Original poster might want to clarify that point.
7. Oct 17, 2013
### arildno
I ought to have said that for a wall floating in space, momentum might certainly well be conserved, and the reason why we may regard the velocity of the wall to be zero, is that it is many orders of magnitude less than the velocity of the ball. (The resulting momenta, though, will be of the same order, indeed perfectly equal but oppositely directed)
8. Oct 17, 2013
### atyy
This is very unintuitive. It also says the velocity of the wall Vw = 2mv/Mw goes to zero, then it would seem the momentum of the wall is finite although its velocity is zero?
9. Oct 17, 2013
### WannabeNewton
The momentum is $p_w = M_w v_w$. We are taking the limit as $v_w$ goes to zero and $M_w$ goes to infinity so the momentum remains finite. This is why I said in post #2 to write things out generally and then take limits, which is what dauto later did anyways.
10. Oct 17, 2013
### atyy
I see. So for the kinetic energy, Mv2 goes to zero as M goes to inifnity and v goes to zero, because v2 goes to zero much faster than M gets large?
11. Oct 17, 2013
### WannabeNewton
Yep.
12. Oct 17, 2013
### atyy
And strictly speaking, do we have to say we take the double limit?
13. Oct 17, 2013
### dauto
That's right. The energy is proportional to the square of the speed, so it is an "infinitesimal" of higher order.
14. Oct 17, 2013
### A.T.
Infinities are. When you keep M constant, and let m -> 0, the effect on m/M is the same, but you don't get the mismatch between momentum and energy limits.
15. Oct 17, 2013
### kaplan
Momentum is conserved if the laws of physics are invariant under spatial translations (i.e. if they are the same everywhere in space). If you have an immovable wall somewhere, the laws of physics are NOT invariant under spatial translations, because there is a law that says that when you arrive at the wall something happens (you elastically rebound, or stick to it, or whatever).
So why would you expect momentum to be conserved when you treat the wall as immovable? It's not - you're right. (Of course in the real world there's no such thing as an immovable wall, but you're not asking about the real world.)
16. Oct 17, 2013
### UltrafastPED
This ball-wall problem is worked in detail here:
http://www.lhup.edu/~dsimanek/ideas/bounce.htm
Momentum is conserved in the absence of external forces.
Both cases are discussed here:
Last edited by a moderator: May 6, 2017
17. Oct 17, 2013
### suchal
When talking about conservation of momentum it is already clear that NO EXTERNAL FORCES ACT ON OBJECTS so those who raised this point should rethink about the concept of conservations of momenta.
Secondly, we say momentum of ball is 2mv. 2mv=Mw.Vw, Mw->infinity, Vw->0 but their product still is 2mv.
Atyy hope u got the answer.
Thanks to everyone for helping me out.
18. Oct 17, 2013
### atyy
Yes, I got the answer - thanks to everyone from me too. And suchal, thanks for asking the question - I'd never heard this crazy thing before!
19. Oct 17, 2013
### suchal
The next problem is that when we talk about momentum, the concept of inertia doesn't make sense. a little force will have some change on the momentum of the body. I think inertia is related to momentum. An object with large mass will have larger momentum so it will require more force to change it's velocity.
All forces i talk about are internal forces and there are no external forces acting on the closed system.
20. Oct 17, 2013
### suchal
It is good to see how helpful you people are. And please don't be shy to ask question no matter how much knowledge you have, you won't be insulted. These discussions clear the concepts and answer the question which otherwise disturb us.
21. Oct 17, 2013
### Staff: Mentor
No.
If you don't understand something, your first, second, and third approach should be "I don't get the concept" instead of "it has to be wrong, it does not make sense".
Sure
There is a relation, indeed.
No, it will require more force (or more time) to change its velocity by the same amount. Tiny velocity changes require tiny forces and short timescales, and there is no "minimal force" required to do anything (unless you have friction).
22. Oct 17, 2013
### suchal
F=ma, a=F/m, if m increase a decrease if F is same.
So for the same force, if the object with larger mass will have smaller acceleration.
When i say "doesn't make sense" it means i am talking about myself that the concept doesn't make sense to my weak mind. I never said it is wrong.
23. Oct 17, 2013
### Staff: Mentor
Right.
Okay.
24. Oct 17, 2013
### suchal
That is exactly what i said in simple words.
If the object has more mass a greater force will be required to accelerate it by the same amount.
25. Oct 17, 2013
Staff Emeritus
The very title of this thread, "Momentum is NOT conserved" is an example.
Posting wrong statements in the hope someone will correct them is a very inefficient way to learn. It also tends to annoy those who are trying to help you. That's why PF strongly discourages this. It's much better to ask questions.
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Difference Patterson map
Definition
An application of Patterson methods for solution of crystal structures, typically proteins with heavy-atom derivatives, where the Patterson function is calculated using structure-factor coefficients based on the difference between the heavy-atom derivative and the native molecule.
Discussion
Patterson methods for determining diffraction phases depend on the symmetries of interatomic vectors that show up as peaks in a three-dimensional map of the Patterson function. For small molecules containing a heavy atom, the heavy-atom positions can be determined directly from the Patterson function calculated using measured structure-factor amplitudes. For proteins, there are too few heavy atoms for this approach to be successful. However, if an isomorphous derivative crystal is available (i.e. one whose symmetry and dimensions and contents, with the exception of heavy-atom addition, are minimally changed), a Patterson map of derivative (FPH) minus native (FP) structure factors will be dominated by the vectors between the heavy atoms, and thus allow a solution of the coordinates of the heavy atoms.
A true difference Patterson function, representing the difference between the Patterson of the derivative minus the Patterson of the native protein, should be calculated using as coefficients $|F^2_{PH} - F^2_P|$.
In practice, protein crystallographers normally calculate a modulus difference-squared synthesis, also known as an isomorphous difference Patterson, using coefficients ( | FPH | − | FP | )2.
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## Causality: relaxing before exploring
This is a Perspective on "Causal hierarchy of multipartite Bell nonlocality" by Rafael Chaves, Daniel Cavalcanti, and Leandro Aolita, published in Quantum 1, 23 (2017). Quantum Views 1, 3 (2017).https://doi.org/10.22331/qv-2017-08-04-3By …
## Why the Quantum?
This is an Editorial on "Classification of all alternatives to the Born rule in terms of informational properties" by Thomas D. Galley and Lluis Masanes, published in Quantum 1, 15 …
## What kind of content is shared by Quantum?
Quantum is active on social media to foster engagement and spread news relevant for the quantum science community. As Quantum grows, it becomes important to have transparent guidelines about the …
## Quantum meets SciPost at YRM 2017
SciPost‘s Jean-Sébastien Caux and Quantum’s Christian Gogolin were both invited to speak about “the future of scientific publishing” at a dedicated session on that topic during YRM 2017 in Tarragona. …
## Quantum obtains an ISSN
Quantum has been assigned an International Standard Serial Number (ISSN) by the ISSN International Centre. The ISSN is similar to the more widely known ISBN, which is used to uniquely …
## On experimentally relevant quantum speedups
This is a Perspective on "Achieving quantum supremacy with sparse and noisy commuting quantum computations" by Michael J. Bremner, Ashley Montanaro, and Dan J. Shepherd, published in Quantum 1, 8 …
## Better local hidden variable models for two-qubit Werner states and an upper bound on the Grothendieck constant $K_G(3)$
Flavien Hirsch, Marco Túlio Quintino, Tamás Vértesi, Miguel Navascués, and Nicolas Brunner,
Quantum 1, 3 (2017). https://doi.org/10.22331/q-2017-04-25-3
We consider the problem of reproducing the correlations obtained by arbitrary local projective measurements on the two-qubit Werner state $\rho = v |\psi_- \rangle \langle\psi_- | + (1- v ) \frac{1}{4}$…
## Collaboration with Fermat’s library
Before our launch issue finally arrives we have further exciting news to share: Quantum is cooperating with Fermat’s Library, a project enabling collaborative online annotation of scientific papers. Papers published …
## 40 submissions
Quantum has received over 40 submissions since the launch in mid November! We would like to take this as an opportunity to thank all authors who have shown their support …
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# Spring MVC: difference between <mvc:annotation-driven/> and <context:annotation-config/> tags?
I am migrating from Spring 2.5 to Spring 3.
They have introduced <mvc:annotation-driven /> which does some black magic. This is expected to be declared in servlet configuration file only.
In Spring 2.5 I have just used <context:annotation-config /> and <context:component-scan base="..."/> tags declared both in application-context.xml and dispatcher servlet configuration XML with appropriate base packages to scan.
So I wonder what is the difference between mvc:annotation-driven and context:annotation-config tags in servlet config and what can I eliminate in Spring 3 config files?
<context:annotation-config> declares support for general annotations such as @Required, @Autowired, @PostConstruct, and so on.
<mvc:annotation-driven /> declares explicit support for annotation-driven MVC controllers (i.e. @RequestMapping, @Controller, although support for those is the default behaviour), as well as adding support for declrative validation via @Valid and message body marshalling with @RequestBody/ResponseBody.
For details: http://spring.io/blog/2009/12/21/mvc-simplifications-in-spring-3-0/
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# Tag Info
## Hot answers tagged topological-vector-spaces
1
Yes. Your argument is valid. In a nutshell, your argument is this: Take any $f \in X^*$. Since $X$ is finite dimensional, $f$ is bounded. So, $f \in X'$. So, $X^* \subseteq X'$, as desired. When $X$ is infinite dimensional, we can always construct an unbounded linear function $f$ on $X$. Finding an explicit construction of such an $X$ is tricky. ...
1
Hint: A basis of neighbourhoods of $x_0\in X$ for the weak topology is obtained by varying $\epsilon$, $k$, and the $f_i$'s in $E^\ast$ in the following expression: $$V(f_1,\cdots,f_k,\epsilon)=\{x \in E:\forall i=1,\cdots,k:|f_i(x-x_0)|<\epsilon\}$$ (see Proposition 3.4 of Brezis' "Functional Analysis, Sobolev Spaces and Partial Differential Equations")...
1
Such a topological vector space is called "locally bounded". Note that any subset of a bounded set is bounded, so actually it suffices for there to exist just a single bounded neighborhood of $0$.
1
Use the following lemma in topology: Let $\mathcal{B}$ and $\mathcal{B}'$ be bases for the topologies $\tau$ and $\tau'$, respectively, on a set $X$. Then $\tau=\tau'$ if and only if the following this true For each $x\in X$ and each basis element $B\in\mathcal{B}$, such that $x\in B$, there is a basis element $B'\in\mathcal{B}'$ such that \$x\in ...
Only top voted, non community-wiki answers of a minimum length are eligible
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# Wavelength and linear momentum
1. Nov 11, 2014
### mss90
1. The problem statement, all variables and given/known data
I had to find wavelenght and linear momenta of fotons with energies of 3eV, 50 KeV and 1.0 MeV
Are these correct?
2. Relevant equations
E=hc/λóλ=hc/E and p= h/ λ
3. The attempt at a solution
a. 3eV Hence λ=(6.63E-34*3E8)/3=6.63E-26m
p = 6.63E-34/6.63E-26 = 1E-8
b. 50 KeV = 50000 eVλ=(6.63E-34*3E8)/ 50000 =3.978E-30m
p = 6.63E-34/3.978E-30 = 1.66E-4
c. 1.0 MeV = 1 000000 eVλ=(6.63E-34*3E8)/ 1 000000 = 1.989E-31m
p = 6.63E-34/1.989E-31 = 0.0033
Last edited: Nov 11, 2014
2. Nov 11, 2014
### collinsmark
Hello mss90,
Welcome to PF! :)
Don't forget to first convert the energy (given in units of eV, keV, and MeV) to units of Joules first, before plugging in the numbers.
3. Nov 12, 2014
### mss90
Alright, can you confirm that this is correct:
3eV * 1.60E-19 = 4.8E-19 J Hence λ=(6.63E-34*3E8)/4.8E-19=4.17E-7m = 0.417µm
p = 6.63E-34/4.17E-7= 1.59E-27
4. Nov 12, 2014
### collinsmark
That looks about right, although there might be some rounding errors going on somewhere.
By the way, when calculating the photons' momentum magnitude, you can simply use the $p = \frac{E}{c}$ formula (after converting the energy into units of Joules, simply divide that by the speed of light, 3 × 108 m/s, and you have the magnitude of the photon's momentum. That way you don't need to depend on the λ intermediate step as part of the answer).
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# Sec squared formula
### Expansion form
$\sec^2{\theta} \,=\, 1+\tan^2{\theta}$
### Simplified form
$1+\tan^2{\theta} \,=\, \sec^2{\theta}$
## How to use
The secant squared identity is used as a formula in two cases in trigonometry.
1. The square of secant function is expanded as sum of one and tangent squared function.
2. The sum of one and tan squared function is simplified as secant squared function.
#### Proof
The secant squared formula is actually derived from the Pythagorean identity of secant and tan functions.
If theta is angle of a right triangle, then the subtraction of squares of tan function from sec function equals to one.
$\sec^2{\theta}-\tan^2{\theta} \,=\, 1$
$\therefore \,\,\,\,\,\, \sec^2{\theta} \,=\, 1+\tan^2{\theta}$
Therefore, it is proved that secant squared theta is equal to the summation of one and tan squared theta.
##### Alternative form
The secant squared identity is also often written in terms of different angles.
For example, if $x$ is used to represent angle of right triangle, then the sec squared formula is written as $\sec^2{x} \,=\, 1+\tan^2{x}$
Hence, the angle of right angled triangle can be denoted by any symbol, the sec squared formula must be written in terms of the corresponding symbol.
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legitimate. In logic, negation, also called the logical complement, is an operation that takes a proposition $${\displaystyle P}$$ to another proposition "not $${\displaystyle P}$$", written $${\displaystyle \neg P}$$, $${\displaystyle {\mathord {\sim }}P}$$ or $${\displaystyle {\overline {P}}}$$. train of thought. 10 Answers. These statements consist of those that have no quantifiers and are not Sufficient & Necessary statements. In logic, a set of symbols is commonly used to express logical representation. coherence. lucid. reasoning, sense. train of thought. e.g. This is usually referred to as "negating" a statement. “Logic.” Merriam-Webster.com Thesaurus, Merriam-Webster, https://www.merriam-webster.com/thesaurus/logic. "Betazed." The symbol to indicate negation is : ~ Original Statement Negation of Statement; Today is Monday. intelligent. Logical opposites are simply the negation of x (e.g. Emotions are not the opposite of logic, they aren’t even on the same spectrum: 1.) If both the operands are non-zero, then the condition becomes true. The Opposite of Logic. What is the definition of logical? Logic is a process of induction/deduction derived from information, it is not a sensation. Human beings are intuitive. (A || B) is true.! Staring into the tiny flames of the candles that lit his quarters, Spock didn't turn his head to acknowledge the man leaning in the open door frame. var ran1 = 1 + Math.floor (Math.random () * 100); var ran2 = 1 + Math.floor (Math.random () * 100); if (ran1 <= 63 && ran2 <= 18) { // code } else { // code } javascript logical-operators. 3. groundless, illogical, invalid, irrational, nonrational, nonsensical, nonvalid, unfounded, ... always have opposite truth values. noun. The point of this entry is to trace its history from thevantage point of the early twenty-first century, along with closelyrelated doctrines bearing on empty terms. LOGICAL OPPOSITION 3. 1. logical thinking. 2 is opposite of gold and represents those that prefer excitement over groundless, illogical, invalid, irrational, nonrational, nonsensical, nonvalid, unfounded, They can't be passed a number or a string directly; instead a comparison or test must be made. Logic is good. Antonyms for logical. phr. In classical logic, negation is normally identified with the truth function that takes truth to falsity (and vice versa). Synonyms. You can't get a SINGLE word which means the opposite of 'Logic'. Logic and Mathematical Statements Worked Examples. Thus, with the existence of just one single blueberry, we have negated the principle. Staring into the tiny flames of the candles that lit his quarters, Spock didn't turn his head to acknowledge the man leaning in the open door frame. But I'm not sure how to express the neither A nor B bit in JavaScript. How are logistics and logic related? 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. It refers to the relationship existing between two proposition having the same subject and the same predicate but differ in quantity or in quality or in quality or both in quantity or quality. Logically, I suppose that the opposite of A and B is both A or B and neither A nor B. Polar opposites deal with a spectrum. If we let A be the statement "You are rich" and B be the statement "You are happy", then the negation of "A or B" becomes "Not A and Not B." The opposite or reverse. sanity. by Kristen Elizabeth "Have you ever heard of a planet called Betazoid?" Another word for logic: science of reasoning, deduction, dialectics, argumentation, ratiocination | Collins English Thesaurus Imagination is not the opposite of logic. It is used to reverse the logical … not x), so the logical opposite of anything is not that thing. All trees are plants, but the converse , that all plants are trees, is not true. ¬ A ∨ ¬ B. 'All Intensive Purposes' or 'All Intents and Purposes'? Negation is thus a unary (single-argument) logical connective. Geometry: Logic Statements quizzes about important details and events in every section of the book. [1] 2) Imagination - T he faculty or action of forming new ideas, or images or concepts of external objects not present to the senses. "Class M, … argumentation. e.g. The ! Now I know that the logical opposite looks to negate the necessary condition, and therefore I could just ignore the the contrapositive's logical opposite. Logical opposite-original statement: A not B Logical opposite- contrapositive: not B A The logical opposite of the contrapositive looks like a mistaken reversal! (logic) Of a proposition or theorem of the form: given that "If A is true, then B is true", then "If B is true, then A is true."'' n. Logic is, well, “logical.” But human beings aren’t necessarily logical. An easy way to remember this idea is to look at the logical opposite of "all" as "not all." How to use logic in a sentence. Antonyms for logical. Imagination is not the opposite of logic. Now I know that the logical opposite looks to negate the necessary condition, and therefore I could just ignore the the contrapositive's logical opposite. An easy way to remember this idea is to look at the logical opposite of "all" as "not all." n. rationalism. operator is equivalent to Not.These functions work with logical values. Antonyms for logical. Truth tables get a little more complicated when conjunctions and … We gain insight from a number of areas – sight, sound, taste, touch, smell, and process information in ways that aren’t always logical. It is interpreted intuitively as being true when $${\displaystyle P}$$ is false, and false when $${\displaystyle P}$$ is true. philosophy. Nglish: Translation of logic for Spanish Speakers, Britannica English: Translation of logic for Arabic Speakers, Britannica.com: Encyclopedia article about logic. Synonyms for logical in Free Thesaurus. We gain insight from a number of areas – sight, sound, taste, touch, smell, and process information in ways that aren’t always logical. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition $${\displaystyle P}$$ is the proposition whose proofs are the refutations of $${\displaystyle P}$$. Thus, the logical opposite of our statement: "All berries are red," becomes "Not all berries are red." The ! LOGICAL OPPOSITION 3. Contradictory Opposition 3. by Kristen Elizabeth "Have you ever heard of a planet called Betazoid?" Does the phrase "gentle kingdom ringed in spears" ring any bells? If any of the two operands is non-zero, then the condition becomes true. || Called Logical OR Operator. The || operator is equivalent to Or.The Not function returns true if its argument is false; it returns false if its argument is true. the logical opposite of both love and hate is indifference. Contradictory Opposition 3. In other words, the opposite is to be not rich and not happy. convincing. The opposite or reverse. adj. Thinking that is coherent and logical. Synonyms for logic. Emotions are not the opposite of logic, they aren’t even on the same spectrum: 1.) A true opposite would be the "Light logic" where instead of earning the right to exist through war and combat, instead protecting those too weak to fight for themselves. False represents 0, and true represents 1. Princeton's WordNet (1.50 / 2 votes) Rate these antonyms: logical (adj) capable of or reflecting the capability for correct and valid reasoning "a logical mind" Antonyms: [2] ===== Imagination is not the opposite of logic and is quite arguably the very part of needed to pr A true opposite would be the "Light logic" where instead of earning the right to exist through war and combat, instead protecting those too weak to fight for themselves. At any given moment, every terminal is in one of the two binary conditions false (high) or true (low). 10 Answers. 37 synonyms for logical: rational, clear, reasoned, reasonable, sound, relevant, consistent, valid, coherent, pertinent, well … So yes, you are correct in your conjecture: If it is not the case that both A and B are true, this means that either A is not true, or B is not true, (or neither A nor B is true). logical-diagram | definition: a graphical representation of a program using formal logic | synonyms: logic diagram| antonyms: one-dimensional language Antonym.com is the web's best resource for English synonyms, antonyms, and definitions. In logic, a conjunction is a compound sentence formed by the word and to join two simple sentences. (A && B) is false. Today is not Monday. Emotions are feelings, it is a sensation; 2.) The And function returns true if all of its arguments are true. reasonable. That was fun. connection. Emotions are feelings, it is a sensation; 2.) List of Opposites in the English language in alphabetical order - A - F. Here you will find a table of words and their opposites. Logic is a process of induction/deduction derived from information, it is not a sensation. (A && B) is false. Polar opposites deal with a spectrum. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Opposite of of or according to the rules of logic or formal argument Opposite of characterized by or capable of clear, sound reasoning (of an action, decision, etc.) Which word describes a musical performance marked by the absence of instrumental accompaniment. ¬ A ∨ ¬ B. cogent. The Opposite of Logic. Test Your Knowledge - and learn some interesting things along the way. Opposites formed by prefixes (dis-, ex-, im-, in-, irr-, un- etc.) But acording to some senses of the word 'Logic' we may hit upon a … But note that in logic and mathematics, the use of "or" is always taken to be the inclusive ( ∨) sense of the word "or", where ( P or Q) ≡ P ∨ Q means exactly P or Q or both P and Q, … What is the meaning of logical? 28 synonyms for logic: science of reasoning, deduction, dialectics, argumentation, ratiocination, syllogistic reasoning, connection, rationale, coherence.... What are synonyms for logic? Logical opposites are simply the negation of x (e.g. Opposite of the … So, the logical opposite of our statement would be: “Not all the Dalmatian puppies from this litter have spots.” And lastly, we shall cover “regular” statements. Deductive reasoning is supported by deductive logic, for example: From general propositions: All ravens are black birds. Thus, the logical opposite of our statement: "All berries are red," becomes "Not all berries are red." [1] 2) Imagination - T he faculty or action of forming new ideas, or images or concepts of external objects not present to the senses. Logic is good. absurdity, brainlessness, insanity, irrationality, Following table shows all the logical operators supported by C language. n. rationalism. Indicates the opposite, usually employing the word not. consistent. which is equivalent, by one of DeMorgan's Laws, to. 3. Antonyms for logical ˈlɒdʒ ɪ kəl This page is about all possible antonyms and opposite words for the term logical. What made you want to look up logic? mentation abstract thought deductive reasoning prevision analytic thinking argument synthesis argumentation thinking regress synthetic thinking conjecture prediction ratiocination deduction intellection anticipation reasoning line thought process reasoning backward thought line of reasoning illation analysis logical argument … operator is equivalent to Not.These functions work with logical values. are not listed here. boolean-logic | definition: a system of symbolic logic devised by George Boole; used in computers | synonyms: mathematical logic, formal logic, symbolic logic, Boolean algebra Synonym.com is the web's best resource for English synonyms, antonyms, and definitions. A logic gate is a building block of a digital circuit.Most logic gates have two inputs and one output and are based on Boolean algebra. Negation Sometimes in mathematics it's important to determine what the opposite of a given mathematical statement is. The && operator is equivalent to And.The Or function returns true if any of its arguments are true. The doctrine of the square of opposition originated with Aristotle inthe fourth century BC and has occurred in logic texts ever since.Although severely criticized in recent decades, it is still regularlyreferred to. ... always have opposite truth values. The following table lists many common symbols, together with their name, pronunciation, and the related field of mathematics. Antonyms for logic. represents those that prefer competence and logic. phr. Can you spell these 10 commonly misspelled words? Logic definition is - a science that deals with the principles and criteria of validity of inference and demonstration : the science of the formal principles of reasoning. logicality. Opposite of the quality of being justifiable by reason, Opposite of the thought processes leading to accepted theories or solutions, “The belief in a higher being is largely rooted in deep, Opposite of the logic or reason for a decision or way of thinking, Opposite of the art of investigating or discussing the truth of opinions, Opposite of the quality of being clear, logical, and convincing, Opposite of the inference of particular instances by reference to a general law or principle, Opposite of the quality of being logical or rational, Opposite of the quality of being well organized and systematic in thought or action, Opposite of an actual event, situation, or fact, “The allegation of criminality is a complete, Opposite of the power of the mind to think, understand, and form judgments logically. Synonyms for logical in Free Thesaurus. The doctrine of the square of opposition originated with Aristotle inthe fourth century BC and has occurred in logic texts ever since.Although severely criticized in recent decades, it is still regularlyreferred to. Today’s article is up for you, the reading audience, to ratify. illogic, incoherence. But I'm not sure how to express the neither A nor B bit in JavaScript. Deductive reasoning is the process of reasoning from the general to the specific. Four Kinds of Logical Opposition 1. What are synonyms for logical? So yes, you are correct in your conjecture: If it is not the case that both A and B are true, this means that either A is not true, or B is not true, (or neither A nor B is true). Search all of SparkNotes Search. That was not fun. deduction. the thought processes that have been established as leading to valid solutions to problems. The point of this entry is to trace its history from thevantage point of the early twenty-first century, along with closelyrelated doctrines bearing on empty terms. The && operator is equivalent to And.The Or function returns true if any of its arguments are true. || Called Logical OR Operator. But note that in logic and mathematics, the use of "or" is always taken to be the inclusive ( ∨) sense of the word "or", where ( P or Q) ≡ P ∨ Q means exactly P or Q or both P and Q, so the added … Logically, I suppose that the opposite of A and B is both A or B and neither A nor B. equivalently: ''given that "All Xs are Ys", then "All Ys are Xs" . Geometry: Logic Statements quizzes about important details and events in every section of the book. I accept. logicality. But acording to some senses of the word 'Logic' we may hit upon a … Truth tables get a little more complicated when conjunctions and disjunctions of statements are included. [2] ===== Imagination is not the opposite of logic and is quite arguably the very part of needed to pr Conjunction. Assume variable A holds 1 and variable B holds 0, then − && Called Logical AND operator. , n. judgment. the logical opposite of both love and hate is indifference. The || operator is equivalent to Or.The Not function returns true if its argument is false; it returns false if its argument is true. Deductive and In ductive Logic . Contrary Opposition 2. Or if we rewrite it in terms of the original statement we get "You are not rich and not happy." It may be applied as an operation on notions, propositions, truth values, or semantic values more generally. All trees are plants, but the converse , that all plants are trees, is not true. var ran1 = 1 + Math.floor (Math.random () * 100); var ran2 = 1 + Math.floor (Math.random () * 100); if (ran1 <= 63 && ran2 <= 18) { // code } else { // code } javascript logical-operators. Search all of SparkNotes Search. rationale. , n. judgment. E.g. Thus, with the existence of just one single blueberry, we have negated the principle. [Definition] 1) Logic - A reasoning conducted or assessed according to strict principles of validity. If both the operands are non-zero, then the condition becomes true. Learn a new word every day. In logic, negation, also called the logical complement, is an operation that takes a proposition to another proposition "not ", written ¬, ∼ or ¯. It is interpreted intuitively as being true when P {\displaystyle P} is false, and false when P {\displaystyle P} is true. Logical opposite-original statement: A not B Logical opposite- contrapositive: not B A The logical opposite of the contrapositive looks like a mistaken reversal! It’s all based on logic I learned from the 90’s sitcom, Seinfeld. At any given moment, every terminal is in one of the two binary conditions false (high) or true (low). Sub- Contrary Opposition 4. Blue is opposite of green and represents those that prefer relationships over other preferences. Sub- Contrary Opposition 4. reasonable. Whatever you are feeling like today, be it hungover, hungry, hangry, angry, happy, 1+1 is always 2 Called Logical AND operator. Synonyms (Other Words) for Logical & Antonyms (Opposite Meaning) for Logical. not x), so the logical opposite of anything is not that thing. Contexts . The And function returns true if all of its arguments are true. Gold represents those that prefer organization. For every action, there is an opposite and equal reaction. But the polar opposite of love is hate. I accept. Contrary Opposition 2. Another word for logic: science of reasoning, deduction, dialectics, argumentation, ratiocination | Collins English Thesaurus sense. Does the phrase "gentle kingdom ringed in spears" ring any bells? . Synonyms for logical. Opposite of the thought processes leading to accepted theories or solutions. n. Synonyms. But the polar opposite of love is hate. Four Kinds of Logical Opposition 1. Delivered to your inbox! (logic) Of a proposition or theorem of the form: given that "If A is true, then B is true", then "If B is true, then A is true."'' This is shown in the truth table. adj. Opposite of expected or sensible under the circumstances Opposite of based on, or displaying, common sense The square of opposition is a group of theses embodied in a diagram.The … Thinking that is coherent and logical. You can't get a SINGLE word which means the opposite of 'Logic'. noun. How do you use logical in a sentence? This is shown in the truth table. A logic gate is a building block of a digital circuit.Most logic gates have two inputs and one output and are based on Boolean algebra. which is equivalent, by one of DeMorgan's Laws, to. Bomb logic is an alternative to sword logic. Logic is, well, “logical.” But human beings aren’t necessarily logical. Accessed 8 Dec. 2020. 1. logical thinking. E.g. equivalently: ''given that "All Xs are Ys", then "All Ys are Xs" . Human beings are intuitive. 'Nip it in the butt' or 'Nip it in the bud'? [Definition] 1) Logic - A reasoning conducted or assessed according to strict principles of validity. Synonyms for logic in Free Thesaurus. Bomb logic is an alternative to sword logic. "Betazed." (mass noun) Opposite of the power of the mind to think, understand, and form judgments logically. coherent. Orange is opposite of gold and represents those that prefer excitement over other preferences. mentation abstract thought deductive reasoning prevision analytic thinking argument synthesis argumentation thinking regress synthetic thinking conjecture prediction ratiocination deduction intellection anticipation reasoning line thought process reasoning backward thought line of reasoning illation analysis logical argument … According to strict principles of validity statement negation of statement ; Today Monday... 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The Size Variance Relationship of Business Firm Growth Rates
Author
Listed:
• Massimo Riccaboni
• Fabio Pammolli
• Sergey V. Buldyrev
• Linda Ponta
• H. Eugene Stanley
Abstract
The relationship between the size and the variance of firm growth rates is known to follow an approximate power-law behavior $\sigma(S) \sim S^{-\beta(S)}$ where $S$ is the firm size and $\beta(S)\approx 0.2$ is an exponent weakly dependent on $S$. Here we show how a model of proportional growth which treats firms as classes composed of various number of units of variable size, can explain this size-variance dependence. In general, the model predicts that $\beta(S)$ must exhibit a crossover from $\beta(0)=0$ to $\beta(\infty)=1/2$. For a realistic set of parameters, $\beta(S)$ is approximately constant and can vary in the range from 0.14 to 0.2 depending on the average number of units in the firm. We test the model with a unique industry specific database in which firm sales are given in terms of the sum of the sales of all their products. We find that the model is consistent with the empirically observed size-variance relationship.
Suggested Citation
• Massimo Riccaboni & Fabio Pammolli & Sergey V. Buldyrev & Linda Ponta & H. Eugene Stanley, 2009. "The Size Variance Relationship of Business Firm Growth Rates," Papers 0904.1404, arXiv.org, revised Apr 2009.
• Handle: RePEc:arx:papers:0904.1404
as
File URL: http://arxiv.org/pdf/0904.1404
---><---
Citations
Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as
Cited by:
1. Massimo Riccaboni & Stefano Schiavo, 2009. "The Structure and Growth of Weighted Networks," Papers 0908.0348, arXiv.org, revised Dec 2009.
2. Fabio Pammolli & Massimo Riccaboni & Nicola Carmine Salerno, 2012. "I Farmaci Oncologici in Italia: innovazione e sostenibilità economica," Working Papers CERM 02-2012, Competitività, Regole, Mercati (CERM).
3. Xie, Wen-Jie & Gu, Gao-Feng & Zhou, Wei-Xing, 2010. "On the growth of primary industry and population of China’s counties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(18), pages 3876-3882.
4. Massimo Riccaboni & Stefano Schiavo, 2009. "The Structure and Growth of International Trade," Documents de Travail de l'OFCE 2009-24, Observatoire Francais des Conjonctures Economiques (OFCE).
5. Stanley, H.E. & Buldyrev, S.V. & Franzese, G. & Havlin, S. & Mallamace, F. & Kumar, P. & Plerou, V. & Preis, T., 2010. "Correlated randomness and switching phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(15), pages 2880-2893.
6. Misako Takayasu & Hayafumi Watanabe & Hideki Takayasu, 2013. "Generalised central limit theorems for growth rate distribution of complex systems," Papers 1301.2728, arXiv.org, revised Jan 2014.
7. Anindya S. Chakrabarti, 2013. "Bimodality in the firm size distributions: a kinetic exchange model approach," Papers 1302.3818, arXiv.org, revised May 2013.
8. Jerker Denrell & Christina Fang & Chengwei Liu, 2015. "Perspective—Chance Explanations in the Management Sciences," Organization Science, INFORMS, vol. 26(3), pages 923-940, June.
NEP fields
This paper has been announced in the following NEP Reports:
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Index of /macros/latex2e/contrib/carlisle
Name Last modified Size Description
Parent Directory 07-Dec-2019 05:17 -
dotlessj.sty 02-Apr-2018 03:37 3k
ltxtable.pdf 30-May-2018 04:02 69k
ltxtable.tex 15-Mar-2015 06:37 6k
mylatex.ltx 15-Mar-2015 06:37 8k
plain.sty 02-Apr-2018 03:37 8k
remreset.sty 02-Apr-2018 03:37 1k
scalefnt.sty 02-Apr-2018 03:37 1k
slashed.sty 02-Apr-2018 03:37 5k
CTAN:macros/latex/contrib/supported/carlisle
This directory contains various packages that are not related to
each other except by the fact that they have the same author.
All the files in this directory, and the LaTeX packages
that can be extracted from the source files in ths directory
may be redistributed and/or modified under the terms
of the LaTeX Project Public License Distributed from CTAN
archives in directory macros/latex/base/lppl.txt; either
version 1 of the License, or (at your option) any later version.
David Carlisle https://github.com/davidcarlisle/dpctex
2018-04-01
Currently the directory contains the following.
mylatex.ltx
This file provides a method of making a special format tailored to
one document, with all the class and packages, and other preamble
material pre-loaded. This can save quite a lot of time on some
systems. See the comments in the file.
dotlessj.sty
If you are using a font set without a dotless j (\j and \jmath) then
this package will fake one. It requires the LaTeX color package.
It does not require any explicit PostScript support.
plain.sty
Typeset plain TeX markup in LaTeX documents.
Within \begin{plain} ... \end{plain} most plain TeX constructs work
including commands such as \over which might have been disabled by
other LaTeX packages such as amsmath.
scalefnt.sty
\scalefont{2} selects the current font in twice the current size.
\scalefont{.75} reduces the current font size by three quarters.
slashed.sty
Commands for the Feynman slashed character' notation.
ltxtable.tex
ltxtable.pdf
This generates and documents the package ltxtable.sty. A merger of
longtable and tabularx packages from the Standard LaTeX tools
collection. This produces multipage tables in which the column
widths are automatically calculated to achieve a specified total
table width.
remreset.sty
\@removefromreset: a companion to the standard \@addtoreset command
allows counters to be removed from the reset list of a controlling
counter. For example, a class file based on book class may say
\@removefromreset{footnote}{chapter}
so that footnotes are no longer reset every chapter (the book class
default).
This file is obsolete with LaTeX releases from 2018 onwards as the
command is defined in the format.
Packages that were formally in this collection.
nopageno.sty
This file has now moved: see macros/latex/contrib/nopageno
blkarray.sty
This file has now moved: see macros/latex/contrib/blkarray
typehtml.dtx
typehtml.ins
These files have moved: see macros/latex/contrib/typehtml
comma.sty
These files have moved: see macros/latex/contrib/comma
colortbl.dtx
colortbl.ins
These files have moved: see macros/latex/contrib/colortbl
textcase.dtx
textcase.ins
These files have moved: see macros/latex/contrib/textcase
fix2col.dtx
fix2col.ins
These files have moved: see macros/latex/contrib/fix2col
pspicture.ins
pspicture.dtx
These files have now moved: see macros/latex/contrib/pspicture
tabulary.dtx
tabulary.ins
These files have now moved: see macros/latex/contrib/tabulary
`
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Sequence of Interpolating Values
Hero
New member
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?
Last edited:
Sudharaka
Well-known member
MHB Math Helper
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?
Hi Hero,
I am not very clear about your question. Do you have to construct interpolating polynomials for each, $$\frac{10}{n}$$ where $$n=1,2,\cdots,10$$ separately?
Kind Regards,
Sudharaka.
chisigma
Well-known member
Construct a sequence of interpolating values $$y_n$$ to $$f(1 +\sqrt{10})$$, where $$f(x)= \frac{1}{1+x^2 }$$ for $$−5≤x≤5$$, as follows: For each $$n = 1,2,…,10$$, let $$h =\frac{10}{n}$$ and $$y_n= P_n (1+\sqrt{10})$$, where $$P_n(x)$$ is the interpolating polynomial for $$f(x)$$ at nodes $$x_0^n,x_1^n,…,x_n^n=−5+jh$$, for each $$j=0,1,2,…,n$$. Does the sequence $${y_n }$$ appear to converge to $$f(1+\sqrt{10} )$$?
That is a classical 'example' of 'divergence' of a interpolating polynomial with equidistant points that was 'discovered' by the German Mathematician C.D.T. Runge. See...
Runge's phenomenon - Wikipedia, the free encyclopedia
The 'error function' is given by...
$\displaystyle f(x)-p(x)= \frac{f^{(n+1)}(\xi)}{(n+1)!}\ \prod_{k=1}^{n+1} (x-x_{k})$ (1)
... where $\xi$ is in the definition interval of f(*). In general is the behavior of the error at the edges of interval that causes divergence...
Kind regards
$\chi$ $\sigma$
chisigma
Well-known member
That is a classical 'example' of 'divergence' of a interpolating polynomial with equidistant points that was 'discovered' by the German Mathematician C.D.T. Runge. See...
Runge's phenomenon - Wikipedia, the free encyclopedia
The 'error function' is given by...
$\displaystyle f(x)-p(x)= \frac{f^{(n+1)}(\xi)}{(n+1)!}\ \prod_{k=1}^{n+1} (x-x_{k})$ (1)
... where $\xi$ is in the definition interval of f(*). In general is the behavior of the error at the edges of interval that causes divergence...
Kind regards
$\chi$ $\sigma$
A good method to avoid the ‘Runde’s phenomenon’is to avoid to use equidistant point and to interpolate in the so called ‘Chebysheff-Gauss-Lobatto’ points given by …
$\displaystyle x_{k}= - \cos \frac{k\ \pi}{n}\,\ k=0,1,…,n$ (1)
For the details see…
Chebyshev Interpolation
... where a very interesting 'animation' at the end of section 5.1 shows the better performance of the CGL points approach respect to the 'spontaneous' equidistant points approach...
Kind regards
$\chi$ $\sigma$
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# SSAT Upper Level Math : How to multiply fractions
## Example Questions
1 3 Next →
### Example Question #221 : Number Concepts And Operations
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Since this fraction is in its simplest form, that is the final answer.
### Example Question #21 : How To Multiply Fractions
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Since this fraction is in its simplest form, that is the final answer.
### Example Question #171 : Rational Numbers
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Then simplify the fraction accordingly:
### Example Question #231 : Number Concepts And Operations
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Then simplify the fraction accordingly:
But we are not done simplifying yet:
### Example Question #23 : How To Multiply Fractions
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Then simplify the fraction accordingly:
But we are not done simplifying yet:
### Example Question #24 : How To Multiply Fractions
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Then simplify the fraction accordingly:
### Example Question #41 : Operations With Fractions
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Since this fraction is in its simplest form, that is the final answer.
### Example Question #1301 : Ssat Upper Level Quantitative (Math)
Multiply these fractions:
Explanation:
To multiply the fractions, simply multiply the numerators together and the denominators together.
Since this fraction is in its simplest form, that is the final answer.
### Example Question #51 : Operations With Fractions
Multiple these fractions:
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# I (2,0) tensor is not a tensor product of two vectors?
Tags:
1. Jul 28, 2016
### voila
Hi. I'm trying to understand tensors and I've come across this problem:
"Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors".
Well, prior to that sentence, I would have thought it could... Why not?
2. Jul 28, 2016
### wrobel
let a (2,0) tensor be a product of two vectors. Take a coordinate system such that one of the vectors has the form (1,0,...0)
3. Jul 28, 2016
### Orodruin
Staff Emeritus
A (2,0) tensor is a linear combination of such tensor products. You must show that not all sych linear combinations are tensor products of two vectors.
4. Jul 28, 2016
### voila
May I state clearly that this is not a problem I must solve for class, this was just an example written somewhere which suggested we did it (thus why I didn't provide an attempt at solving it, just asking why it is that way). I still can't see why.
5. Jul 28, 2016
### voila
Oh, I think I'm getting it. Thinking about the matrix representation, that's just like stating that there are such matrices that can't be written as the tensor product of two vectors?
6. Jul 29, 2016
### Staff: Mentor
Yes. You can write all matrices as a sum of (2,0) tensors, but a single tensor $a \otimes b$ will always result in a matrix of rank 1.
7. Aug 2, 2016
### MisterX
The tensor notation such as $(2,0)$ only applies when the tensor is made up of a number of copies of a particular vector space and its dual vector space.
$\mathbf{a} \otimes \mathbf{b}$ with $\mathbf{a},\mathbf{b}\in V$ is a $(2,0)$ tensor which has total rank 2. The main point here was all $(2,0)$ tensors cannot be expressed $\mathbf{a} \otimes \mathbf{b}$ .
8. Aug 2, 2016
### Staff: Mentor
Just a remark.
Rank in this context is a bit of an ill-fated notation, since it has nothing to do with the rank of linear transformations which are also part of the context. Degree is (IMO) a far better word for it.
9. Aug 2, 2016
### Lucas SV
Here is the sketch of a proof. Let $V$ be a finite dimensional real vector space with dimension greater than 2 (the statement is simply not true for $\dim V=1$, since any real number $a$ can be written as $a=1\cdot a$). Let $(e_j)$ be a orthonormal basis and $T=e_1 \otimes e_2 - e_2 \otimes e_1$. Suppose $T=a \otimes b$, for some $a, b \in V$. By equating components, get a contradiction. So there exists $(2,0)$ tensors which cannot be written as the direct product of vectors.
Note: The counterexample above was inspired by the singlet state in Quantum Mechanics.
10. Aug 3, 2016
### voila
Thank you all for your answers. I reckon it's a rather simple question, but I was just beginning to study tensors and couldn't get my mind around it.
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# How do you solve y^2+y=20?
Dec 21, 2017
${y}^{2} + y = 20$
${y}^{2} + y - 20 = 0$
What $2$ numbers add to $+ 1$ and multiply to $- 20$?
$+ 5$ and $- 4$.
Therefore,
$\left(y + 5\right) \left(y - 4\right) = 20$
Therefore, $y$ can be color(red)(-5 or, +4
Dec 21, 2017
$y = - 5 , 4$
#### Explanation:
${y}^{2} + y = 20$.
Subtracting $20$ From Both Sides,
${y}^{2} + y - 20 = 0$.
Factoring,
$\left(y + 5\right) \left(y - 4\right) = 0$.
Since One Of The Answers Have To Be 0,
$y + 5 = 0$, $y = - 5$
$y - 4 = 0$, $y = 4$
The Answers Are: $- 5$ and $4$.
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SEARCH HOME
Math Central Quandaries & Queries
Question from Michael: Find the Quotient: ((a-5)/a)/(4/a)
Hi Michael,
If $a = 0$ the expression
$\frac{\frac{a-5}{a}}{\frac{4}{a}}$
is undefined. If $a \ne 0$ I would multiply the numerator and denominator by $a.$
Penny
Michael wrote back:
That answer did not help whatsoever.
Did you multiply the numerator $\large \frac{a-5}{a}$ by $a?$ What did you get?
Penny
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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DESeq2 independent filtering using filter=rv
1
0
Entering edit mode
@laurafancello-6772
Last seen 7.0 years ago
Belgium
Hi,
I'm using DESeq2 for DE analysis.
I found in the "Beginner's guide to the "DESeq2" that it's possible to perform both default independent filtering (which uses mean of normalized counts as a filter statistics) and independent filtering with other filters, such as filter=rv. Could you please explain me this last filter? Is it a filter about variance? are we speaking about gene variance within the same group?
I applied both default independent filtering and independent filtering with filter=rv on my dataset and whereas in the first case I had no improvement in the number of statistically significant DE genes, in the second case I had more DE genes.
Could it be that this difference be due to a too low number of samples in my dataset, therefore a high variance within groups. therefore more powerful filtering using variance?
Thanks a lot for your help!
Best
Laura
deseq2 INDEPENDENT FILTERING • 2.1k views
0
Entering edit mode
@mikelove
Last seen 38 minutes ago
United States
This line you refer to is in the main DESeq2 vignette:
rv <- rowVars(counts(dds,normalized=TRUE))
resFiltByVar <- results(dds, filter=rv)
Here, rv is the row variance of the normalized counts, and this is over all samples for each gene. So this is removing from the multiple test correction step those genes with small variance across all samples. It's hard to say why you get more experiment-wide power with the variance than the mean.
DESeq2 imports functionality from the genefilter package for independent filtering. You can read more about independent filtering from their paper:
http://www.ncbi.nlm.nih.gov/pmc/articles/pmid/20460310/
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# Left Eigenvectors vs. Right Eigenvectors
Suppose we have a matrix $A$ and a symmetric invertible matrix $D$ such that $DA$ is symmetric. The right eigenvectors of $A$ are $v_1,\cdots,v_n$ with eigenvalues $\lambda_1,\cdots, \lambda_n$. Can we use this information to derive (or estimate) left eigenvectors/eigenvalues of $A$?
I'm going to assume that $D$ is symmetric.
Let $x$ be an eigenvector of $A$ corresponding to eigenvalue $\lambda$. Let $y=Dx$. Then
$$A'y = (A'D)(D^{-1} y) = DAx =\lambda Dx = \lambda y.$$
So then $y$ is an eigenvector of $A'$.
• Thank you; am I right that this proof works even if $D$ is not a diagonal matrix? – Hoda Mar 5 '14 at 3:16
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Professor David Randall
6 Nov 2014
Poncie Rutsch
Professor David Randall gave the audience a historical perspective on international collaboration in Marine Science with his talk about the Stazione Zoologica.
Free for anyone to re-use, but must be credited to OIST.
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