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User shahrooz - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T04:55:42Z http://mathoverflow.net/feeds/user/19885 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130883/is-there-any-proof-that-you-feel-you-do-not-understand/130955#130955 Answer by Shahrooz for Is there any proof that you feel you do not "understand"? Shahrooz 2013-05-17T13:13:31Z 2013-05-17T13:13:31Z
"Edgar Allan Poe:" "Any cryptosystem that human invented are breakable." So, any proofs are understandable. However until now, the zodiac 340 letter (or 340 symbol code) is not cracked.

For me, these two problems were very outstanding:

$1)$ Suppose there are infinite number of points in a plane and the distances between each two point is an integer, then they are all on a single straight line. This problem proposed by Sylvester and the solution by Paul Erdős is very strange for me.

$2)$ For $n\geq 2r$, the chromatic number of Kneser graph $K_{(n:r)}$ is $n-2r+2$. When I studied the proof of this theorem again and again (and also I understood it), I found that I can not understand it much more.
 http://mathoverflow.net/questions/129495/quantum-algorithms-for-dummies/129559#129559 Answer by Shahrooz for Quantum algorithms for dummies Shahrooz 2013-05-03T18:03:07Z 2013-05-03T19:10:37Z
I think this book is good for beginning:

"An introduction to quantum computing algorithms" by Pittenger A.O.

Also, the course by Prof. $Vazirani$ on $edx$ "(www.edx.org)" is good way for learning the necessary material in this way.

Since, there are two main algorithms and other algorithms uses these basic algorithms, you can understand deeply these two quantum algorithms and modify your problem again to find an approach. These two basic and important algorithms are Grover's algorithm (for searching problem) and Shor's algorithm (for factorization). Also, Simons algorithm is good for studying. But, in general, you must learn Fourier sampling and superposition and the quantum gates and their works.
 http://mathoverflow.net/questions/128487/ramsey-numbers-and-graph-spectra/128504#128504 Answer by Shahrooz for Ramsey numbers and graph spectra Shahrooz 2013-04-23T16:31:07Z 2013-04-23T19:13:08Z
Actually, I think you are right and there are not such papers that exactly match with your subject. But, some probabilistic techniques used spectral theory to find some bounds for Ramsey number. One of this paper is famous:

"Asymptotically tight bounds for some multicolored Ramsey numbers" by "Noga Alon" and "Vojtech Rodl"

So, if you change your keywords for searching based on this paper and its references, you might be more successful.

Also, the below paper is good:

"Extremal and Probabilistic Combinatorics" by "N. Alon" and "M. Krivelevich".
 http://mathoverflow.net/questions/128176/will-quantum-computing-kill-cryptography/128189#128189 Answer by Shahrooz for Will quantum computing kill cryptography ? Shahrooz 2013-04-20T19:34:41Z 2013-04-20T19:34:41Z
There is a very good book that you can find your answer there completely. This book's name is:

"Post-Quantum Cryptography" by "Daniel J. Bernstein, Johannes Buchmann and Erik Dahmen".

As a part of this book, today we know that these cryptosystems can be broken by quantum computers:

$1)$ RSA public key encryption

$2)$ Diffie-Hellman key-exchange

$3)$ Elliptic curve cryptography

$4)$ Buchmann-Williams key-exchange

$5)$ Algebraically Homomorphic

and these cryptosystems (and also with some variants) are safe:

$1)$ McEliece public key encryption

$2)$ NTRU public key encryption

$3)$ Lattice-based public key encryption

Also, the good cryptosystems is not usable today because of the storage space problem and complexity. We have some limit on quantum computers that help us to design some good cryptosystems. There are some problems that if we have very large quantum computer and the best quantum algorithm,still we need exponential time for solving them. For example, searching among very large database to find special data, is very hard problem for quantum computer. We can prove that if we have $N$ cases that there is only one suitable case, the best quantum algorithm need $O(N^\frac{1}{2})$ to solve it. Also, it is proved that there is not better result. So, we can hope that we can find some efficient algorithms against the power of quantum computer and quantum algorithms.
 http://mathoverflow.net/questions/125817/mathematical-paper-that-just-links-two-different-fields-of-sciences Mathematical Paper That Just Links Two Different Fields of Sciences Shahrooz 2013-03-28T12:24:58Z 2013-03-29T09:27:42Z
I have a soft question that is interesting for me in some aspects. I appreciate your answers and comments about it.

Four years ago, one of my friends in MIT, in the biology lab, had working on neuroscience and specially he worked on Deja-Vu phenomenon. When he asked me about writing a program with Matlab for simulating this phenomenon with a network of cells that they want simulate the Sinc function, I found that there are many good theorems in graph theory that can be useful for his research. When I suggested him this idea, he found it very interesting.

My question are about this event in a little bit different way. Is it possible that we publish a paper in some mathematical journals that:

1) The only new thing in the paper is relation between a real phenomena and a field of mathematics that is well known. For example, we just model the controversy with bandwidth problem, and no more things just using the theorems that proved for bandwidth problem.

2) This paper does not have new theorems as like as theorems that are common in mathematical papers. This paper just use mathematical theorems in its direction.

Also, do we have some mathematical journals that publish such a papers? And if yes, is there some evidences for this type of publication?

Maybe someone think about the Hilbert Spaces and quantum mechanic. But, in my view, this is not the case. We use Hilbert spaces to model some aspects of quantum mechanics and we get some new results and theorems in quantum mechanic. If we want to think this relation, the paper only must be contain the modeling of quantum mechanic by Hilbert spaces and no more.

Briefly, suppose we found a connection between a real phenomenon and a field of mathematics that can be acceptable or a new view point for analysis the phenomenon. For example, if we found a relation between Darwin's evolutionary theory and a game on graph, is it possible that we can publish such a results as a paper in a mathematical journal? And what kind of mathematical journal is good for this work?

Sorry me for long question.
 http://mathoverflow.net/questions/94742/examples-of-interesting-false-proofs/125082#125082 Answer by Shahrooz for Examples of interesting false proofs Shahrooz 2013-03-20T15:38:34Z 2013-03-20T15:38:34Z
I think nobody point to these interesting false proof:

Let $i=\sqrt{-1}$ be the complex number.

$1)$ $1=\sqrt{-1\times-1}=\sqrt{-1}\times\sqrt{-1}=i\times i=-1$.

$2)$ We know that $x^\frac{2}{6}=x^\frac{1}{3}\Rightarrow (\sqrt{x^2})^\frac{1}{6}=(\sqrt{x})^\frac{1}{3}$. Now, let $x=-1$ and so we have: $$(\sqrt{(-1)^2})^\frac{1}{6}=(\sqrt{-1})^\frac{1}{3}\Rightarrow1=-1.$$
 http://mathoverflow.net/questions/121097/spectrum-of-a-laplacianized-matrix/121101#121101 Answer by Shahrooz for Spectrum of a Laplacianized matrix Shahrooz 2013-02-07T18:03:09Z 2013-02-07T18:03:09Z
It is just a point of view. But it is more long for writing it as a comment.

If $A$ be the adjacency matrix of graph $G$, then $R-A$ is its Laplacian matrix and there are some good bounds for the radius (or maximum eigenvalues of this matrix) of matrix. For example, if we denote the maximum eigenvalue of this matrix by $\mu$, we have: $$\mu\leq \sqrt(2)\times max(d(v)^2+\sum_{uv\in E(G)}{d(u)})^\frac{1}{2}$$ where $v$ changes in the vertices of the graph $G$. In most cases, we can extend such relations to the positive definite matrices. For more such bounds you can see the paper:

Bounds on the (Laplacian) spectral radius of graphs, written by $Lingsheng$ $Shi$ and published in Linear Algebra and Its Application.

But in general, as you want, I think there is not good bound.
 http://mathoverflow.net/questions/119364/minimum-sum-among-fixed-length-factors-of-a-number Minimum sum among fixed length factors of a number Shahrooz 2013-01-19T21:17:37Z 2013-01-21T06:17:37Z
I think this question is famous, but I searched a lot in the internet with many keywords and unfortunately I did not find any related problem. I will appreciate any helpful answers and any guidance for similar questions and problems.

Suppose the integers $n$ and $k$ are given and the set $S$ shows all possible factorization of the integer $n$ into $k$ factors. The factors of these factorization not necessarily primes and can be composite. Also, at most one factor can be the number $1$. For example, if $n=128$ and $k=3$, we have:

$S=\lbrace 2\times4\times16, 4\times4\times8, 2\times 2\times32, 1\times2\times64, 1\times4\times32,1\times8\times16\rbrace$.

For fixed numbers $n$ and $k$ (and also the set $S$), let $m\in S$ and $m=l_1\times l_2 \ldots\times l_k$. We define $S(m)=\sum_{i=1}^k{l_i}$.

What can we say about the minimum of $S(m)$, where $m$ changes in the set $S$? Do we have any formula for finding this minimum respect to the $n$ and $k$?

For the above example, the minimum is happen for $m=4\times4\times8$ and $S(m)=16$.

Is this a famous problem and are there any works related to this problem?

$Note:$ It can be seen that if the factors of $m\in S$ be as much possible as close to each other, then the sum of its factors converges to the minimum.
 http://mathoverflow.net/questions/118545/what-are-the-major-open-problems-in-design-theory-nowaday/118566#118566 Answer by Shahrooz for What are the major open problems in design theory nowaday? Shahrooz 2013-01-10T20:10:09Z 2013-01-10T20:22:06Z
These four below problems are interesting and still open conjecture in design theory and its related topics:

$1.$ There exist Ucycles for $k$-subsets of $[n]$, provided $k$ divides $Cr(n-1,k-1)$ and $n$ is sufficiently large.

$2.$ For $n≥6$ even, it is not possible to have a length $\frac{n^2}{2}$ cyclic covering word for the $(n−2)$-subsets of an $n$-set.

$3.$ For each $k≥3$, there exists a constant $c_k$ such that for all $v≥c_k$ and $λ≥1$, the $1$ block-intersection graph of any $BIBD(v,k,λ)$ is Hamiltonian.

$4.$ For each $k≥3$, there exists a constant $c_k$ such that for all $v≥c_k$ and $λ≥1$, the ${1,2}$-block-intersection graph of any $BIBD(v,k,λ)$ is Hamiltonian.

These four conjectures are respectively from:

$1.$ Chung, F., Diaconis, P., Graham, R.: Universal cycles for combinatorial structures. Discrete Math.110, 43–59 (1992)

$2.$ Stevens, B., Buskell, P., Ecimovic, P., Ivanescu, C., Malik, A., Savu, A., Vassilev, T., Verrall, H., Yang, B., Zhao, Z.: Solution of an outstanding conjecture: the non-existence of universal cycles withk=n−2. Discrete Math.258, 193–204 (2002)

$3.$ Jesso, Andrew T.(3-NF); Pike, David A.(3-NF); Shalaby, Nabil(3-NF) Hamilton cycles in restricted block-intersection graphs. (English summary) Des. Codes Cryptogr.61(3), 345–353

$Also,$ I believe that Hadamard conjecture is a nice diamond in the conjectures land.
 http://mathoverflow.net/questions/117815/the-cliques-of-cospectral-graphs The cliques of cospectral graphs Shahrooz 2013-01-01T22:54:19Z 2013-01-02T03:42:29Z
There are some facts that can be found by the spectrum of adjacency matrix of graph.For example, the number of edges and vertices, is bipartite or not, is complete multipartite or not and so on. Can we say anything about the clique number of two cospectral graphs?

We can construct the graphs $G_1$ and $G_2$ that they are cospectral and for arbitrary $k\in N$, the difference of clique number of these two graphs be grater than $k$. But, as I know, these graphs are disconnected.

So, my question is in connected case. I mean, suppose $G_1$ and $G_2$ are cospectral and connected. What can we say about their clique numbers?

Thanks for any helpful answer.
 http://mathoverflow.net/questions/116634/difference-of-the-maximum-eigenvalue-of-a-graph-with-the-one-of-one-edge-deleted/116651#116651 Answer by Shahrooz for Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph Shahrooz 2012-12-17T21:49:10Z 2012-12-17T22:02:02Z
This conjecture is not true in general. For example, let $G$ be a graph that obtained from joining the end vertex of $P_3$, $P_3$ and $P_4$, where it is an star-like tree. This graph has largest eigenvalue equal to $2.02852$. Now join the end vertex of $P_4$ to the end vertex of $P_3$. Therefore we added an edge to the previous graph. The largest eigenvalue of this graph is $2.13578$. Then the difference of this values is $0.10726$. But this graph has $8$ vertices and $1/8=0.125$. It is a counter example.

Also, we can construct a family of star-like tree that does not have this property. Also, I think for any polynomial $f(n)$, we can construct a graph $G$ such that $\lambda(G)-\lambda(G-e)\geq \frac{1}{f(n)}$ is not true.
 http://mathoverflow.net/questions/115738/decomposition-of-matrix-to-its-sub-matrix-with-constant-rank Decomposition of Matrix to its sub-matrix with constant rank Shahrooz 2012-12-07T19:19:16Z 2012-12-09T07:16:06Z
When we study the structure of simple graphs with a lot of $1$ or $-1$ as its adjacency eigenvalues, the rank of its adjacency matrix is very important. The reason is, in these case, we can study the matrix $A+I$ and $A-I$, where $A$ is the adjacency matrix of graph $G$.

Let $A$ be a symmetric $n\times n$ matrix such that $A+I$ has rank $k$. It is obvious to see that, $A$ has $k$ eigenvalues $-1$. Since $A$ is symmetric, we can write

\begin{equation*} A+I= \begin{bmatrix} A_1& B \newline B^T& A_2 \end{bmatrix} , \end{equation*}

where $A_1$ is a $k\times k$ matrix, $B$ is $k\times (n-k)$ matrix, $B^T$ is the transpose of $B$ and $A_2$ is $(n-k)\times (n-k)$ matrix. It is easy to see that $A_1$ is nonsingular.

My question is:

1) In general, is it true that $A_2=B^TA_1^{-1}B$, and if it is true, how we can prove it? We can see that $rank(A+I)=rank(A_1)$. May be it can help for proof as a strategy.

2) Would you please give me some references for these types of relations in matrices?

For rank$(A_1)=2$ or rank$(A_1)=3$, Professor Doob and Professor Haemers showed that, the structure of $A$ are determined, equivalently, the graphs with $n$ vertices and $n-2$ or $n-3$ eigenvalues $-1$ are determined by their adjacency spectrum. The answer for $n-4$ eigenvalues equal to $-1$ is negative. My another question is:

For which integer $n-k$, where $4\leq k\leq n$, the answer is known? I mean, if the multiplicity of $-1$ for graph $G$ with $n$ vertices be $n-k$, what are the known results about cospectral mate?

Professor Doob and Haemers, gave and example for $n-4$. I found some examples in other cases. I think these examples are important. Am I thinking truly?

I will appreciate any helpful answer and guidance.
 http://mathoverflow.net/questions/111684/does-every-bipartite-graph-with-512-edges-have-an-induced-subgraph-with-256-edges/112162#112162 Answer by Shahrooz for Does every bipartite graph with 512 edges have an induced subgraph with 256 edges? Shahrooz 2012-11-12T10:55:50Z 2012-11-12T20:46:58Z
You are right Dear domotorp,

Suppose the second largest eigenvalue of bipartite graph $G$ is one, i.e, $\lambda_2=1$. In this case, $G$ belong to the finite type, (there are infinite number of such graphs), bipartite graphs. With some calculation, we can see that your conjecture is true for this type of graphs. Also, we can say more, if we use some other spectral techniques.

The Paper Reference:

Petrovic M., On graphs with exactly one eigenvalue less than -1, J. Combin. Theory Ser. B 52 (1991), 102-112.
 http://mathoverflow.net/questions/84898/cayley-graphs-and-its-subgraphs Cayley graphs and its subgraphs Shahrooz 2012-01-04T18:54:19Z 2012-10-18T19:55:26Z
I have two questions about Cayley graphs. Any answers will be appreciate.

1) Do we have any Cayley graph that has Petersen graph as its induced subgraph?

2) Suppose $Cay(G,S)$ be a Cayley graph that $G$ is a finite group. Can we characterize any induced subgraphs of $Cay(G,S)$?

Thanks for any answer and guidance.
 http://mathoverflow.net/questions/102739/battle-of-the-brains-cultural-mathematics Battle of the brains; cultural mathematics Shahrooz 2012-07-20T13:09:54Z 2012-07-23T17:17:31Z
Firstly, I apologize if my question is long.

Three years ago, I watched a video with the name Battle of the Brains. It was a wonderful video about challenging some famous peoples to solve some special problems. I learned many new things from this video. Unfortunately, I could not find any similar video, but I interested in the abilities of people to attack a problem.

So, I studied many references about the system of educations in different countries. I found many parameters as like as; teacher education and qualifications, academic standards, teacher effectiveness, lesson plans and modules, teacher characteristics, instructional materials, program effectiveness, program evaluation, culture, history, class activities, educational games, number systems, cognitive ability, foreign influence, and fundamental concepts, are so important in learning mathematics.

Recently, I found the book African Mathematics: From Bones to Computers by Mamokgethi Setati, Abdul Karim Bangura that is very interesting. I think there are some good references as like as this book. My requests are:

Would you please introduce me video files as like as Battle of the Brains?

Would you please introduce textbooks or sites that are about cultural mathematics as like as the book above?

I think there are some references about the research of American mathematicians in Chinese cultural mathematics. I will appreciate, if someone reference these documents.
 http://mathoverflow.net/questions/100265/not-especially-famous-long-open-problems-which-anyone-can-understand/100405#100405 Answer by Shahrooz for Not especially famous, long-open problems which anyone can understand Shahrooz 2012-06-22T21:50:45Z 2012-06-24T15:51:41Z
I think nobody pointed this problem, if it is repeated, please say me to delete it. This problem killed me for three weeks, when I was a young student in high school. So, I want to recall it again.

$Problem:$ Find all right triangles with rational sides, where the area of these triangles are integer?

I think it is still open problem and if somebody can solve it, I will give 100$as a small award.

After I searched, I found these two interesting sources. I hope it will be helpful.

1) N.Koblitz, Introduction to elliptic curves and modular forms, volume 97 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1993.

2) Washington, Lawrence C., Elliptic Curves : Number Theory and Cryptography, CRC Press Series On Discrete Mathematics and Its Applications
 http://mathoverflow.net/questions/85244/good-surface-bad-surface-surface-classification Good Surface,Bad Surface-Surface classification Shahrooz 2012-01-09T10:31:06Z 2012-04-05T16:44:20Z
Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.

We say a surface$S$(2-dimensional metric(compact) Riemannian surface) is good (denote by$GS$), if every$2n$,$n\geq1$, points on surface can be separate by some geodesic to two distinct subsets$V_1$and$V_2$, where$|V_1|=|V_2|=n$. Also, if a surface$S$is not good, we say it is bad an denote it by$BS$.

For example, it is not difficult to show that plane is a$GS$. Also, a sphere is$GS$.

1) Do we have some$BS$examples(class of examples)?

2) Can we characterize the$GS$and$BS$surfaces?

I can't find any$BS$examples and also I can't prove that they are$GS$.

For example, is Klein Bottle$GS$or$BS$?

Is there any related works and questions about this post?
 http://mathoverflow.net/questions/90700/where-is-number-theory-used-in-the-rest-of-mathematics/90750#90750 Answer by Shahrooz for Where is number theory used in the rest of mathematics? Shahrooz 2012-03-09T19:10:08Z 2012-03-09T21:52:08Z
What is your idea about a problem in number theory that says:

$\frac{p^q-1}{p-1}$never divides$\frac{q^p-1}{q-1}$if$p,q$are distinct primes.

This is a$\textbf{conjecture}$and the validity of this conjecture would simplify the proof of solvability of groups of odd order, (W. Fiet, J. G. Thompson,$\textit{Pacific J. Math.}$, 13, no.3 (1963), 775-1029), rendering unnecessary the detailed use of generators and relations.

An other interesting application of number theory is in real world. Many years ago, cables used for communication. A lot of cables must be gathered near to each other for more efficiency. But for blocking the noises of each cable to the other cable, a special arrange of cables needed. For this arrangement and neighboring of cables, scientist used reminder theorem and number theory. I think first time, Bell company's scientists invented it.

Also, I think this relation between group theory, graph theory and number theory is very nice example: Suppose the order of group$G$,$|G|$, is$n=p_1^{k_1}p_2^{k_2}\ldots p_s^{k_s}$. We fix two prime numbers$p_i$and$p_j$of divisors of$n$and define a graph$\Gamma(G)$as fallow:

The vertices of$\Gamma(G)$are the elements of group$G$and two vertices$g_1$and$g_2$are adjacent if and only if$o(g_1g_2)=p_ip_j$, where$o$means the order of element$g_1g_2$as a group element. These graphs have very nice structures and well defined as$\textit{Prime Graph}$of group.
 http://mathoverflow.net/questions/40454/practical-applications-of-eigenvalues-and-eigenvectors/87416#87416 Answer by Shahrooz for practical applications of eigenvalues and eigenvectors Shahrooz 2012-02-03T10:14:40Z 2012-02-03T10:14:40Z
I think the book$SpectraofGraphs:TheoryandApplications$by Dragos M. Cvetkovic, Michael Doob, Horst Sachs and M. Cvetkovic is very good source for practical applications of eigenvalues and eigenvectors.

In communication theory, coding theory and cryptography, the minimum distance of codes is very important parameter in decoding and also is very important in coding based cryptography (for example McEliece cryptosystem). It is interesting that the second largest eigenvalue of related graph to a code, can determine a good lower-bond for minimum distance of code.
 http://mathoverflow.net/questions/4279/interesting-applications-of-the-pigeon-hole-principle/86704#86704 Answer by Shahrooz for Interesting applications of the Pigeon-hole Principle Shahrooz 2012-01-26T09:15:04Z 2012-01-26T09:15:04Z
I think the solutions of these questions are very interesting (by using pigeon-hole principle), first question is easy, but second question is more advanced:

1) For any integer$n$, There are infinite integer numbers with digits only$0$and$1$where they are divisible to$n$.

2) For any sequence$s=a_1a_2\cdots a_n$, there is at least one$k$, such that$2^k$begin with$s$.
 http://mathoverflow.net/questions/85124/homotopical-combinatorics Homotopical Combinatorics Shahrooz 2012-01-07T11:08:50Z 2012-01-13T22:19:22Z
I have a question about the situation of homotopical combinatorics. There are many topics about combinatorial homotopy. But, I can't find any topic about homotopical combinatorics. More precisely, are there any definitions for some combinatorial objects as like as Latin Squares, Designs and etc in homotopy theory?

Do we have any examples about the applications of homotopy theory in combinatorics and graph theory? I think there are some generalizations for the definitions of some combinatorial objects in the language of homotopy theory.

Is this true thinking or not? Please guide me, if you have some experiences.
 http://mathoverflow.net/questions/43057/looking-for-a-collection-of-entry-level-proofs/85156#85156 Answer by Shahrooz for Looking for a collection of entry level proofs Shahrooz 2012-01-07T21:47:21Z 2012-01-07T21:47:21Z
I think you have to determine some categories, as like as number theory, combinatorics, geometry and etc. But I think this book is so interesting:

"Ingenuity in Mathematics" by Ross Hansberger
 http://mathoverflow.net/questions/84692/graphical-representation-of-chromatic-polynomial Graphical representation of chromatic polynomial Shahrooz 2012-01-01T19:03:00Z 2012-01-06T07:20:47Z
Suppose$c(G,u)$is chromatic polynomial of connected simple graph$G$. We know that$|c(G,-1)|$, as Stanley proved, is the total number of directed graph on$G$, without any cycle. Also, we know some other graphical representations of the value of$c(G,u)$.

1) Do we have any graphical representation for$|c(G,2)|$?

2) Do we have any graphical representation for the multiplicity of$2$as a root of$c(G,u)$?

I found some graphical representation for these values, but I didn't prove them yet.

Thanks for any helpful answer and good references.
 http://mathoverflow.net/questions/50872/how-might-m-c-escher-have-designed-his-patterns/84766#84766 Answer by Shahrooz for How might M.C. Escher have designed his patterns? Shahrooz 2012-01-02T19:41:48Z 2012-01-02T20:41:42Z
When I was in high-school, my teacher suggested a very interesting method for generating some interesting Escher's like pictures. I think it is useful for you.

Get a mirror(or something like mirror) and construct some geometrical shape with this mirror, for example an cylinder, that the inner surface of this shape is mirror. Now, collect some isomorphic shape, for example shell, flower, or simply some triangles. If we put these objects inside the cylinder, we find very beautiful Escher-like shape on the mirror. With some simple method in photography, we can obtain the picture that is inside the cylinder. We can change the shape of mirror and create some elegant symmetry.

I didn't see this method anywhere. I hope it will be useful for you.

To memory of Escher.
 http://mathoverflow.net/questions/84310/generalizing-euclids-proof-of-the-infinity-of-primes/84765#84765 Answer by Shahrooz for Generalizing Euclid's proof of the infinity of primes Shahrooz 2012-01-02T19:20:40Z 2012-01-02T19:20:40Z
Dear Currie

The below problem is still open:

Do we have infinite prime number of the form$2p+1$, where$p$is prime?

One of interesting way to examine this problem is %Sandram$ table. For further information about Sandram table, you can see the book

" Ingenuity in Mathematics" by Ross Hansberger

Also, with some calculation and using the Jacobi symbol, you will find this problem is equivalent to this problem:

The polynomial $n^2+n-1$ generate infinite prime number, that it is open problem. You can find some further information about this question in the Richard Guy's book with name:

" Open problem in number theory".

So I think your problem is a kind of open problem.
 http://mathoverflow.net/questions/84615/learning-sources-about-ihara-coefficient learning sources about Ihara Coefficient Shahrooz 2011-12-30T22:42:05Z 2011-12-31T16:41:39Z
Do we have any good sources(lecture notes or books) for learning about $Ihara$ Coefficient?

Is there any relation between $Ihara$ Coefficient and the eigenvalues of graphs?

Thanks for any help.
 http://mathoverflow.net/questions/84286/estimation-of-ds-graph-growth Estimation of DS graph growth Shahrooz 2011-12-25T22:49:16Z 2011-12-26T01:47:53Z
We know that $DS$ graphs are such connected graphs that determinable by their adjacency spectrum.

Suppose $DS(n)$ and $G(n)$ show the number of $DS$ graphs and all graphs with $n$ vertices,respectively.

$1)$ Do we have any good approximation for $DS(n)$(even if $n$ be sufficiently large)?

$2)$ what is the behavior of $\alpha$, if we have:

 $lim (DS(n)/ (G(n)-DS(n))^\alpha=c\neq0)$

$n \rightarrow \infty$

Is there any new survey about DS graphs, after 2010?
 http://mathoverflow.net/questions/84091/non-isomorphic-graphs-with-the-same-numbers-of-closed-walks Non-isomorphic graphs with the same numbers of closed walks Shahrooz 2011-12-22T13:53:35Z 2011-12-23T15:41:34Z
Can somebody help me to construct two family of finite simple connected graph $G_i$ and $H_i$, $i=1, 2, \cdots,n$ ($n$ possibly large), such that:

$1)$ $G_i\ncong H_i$ for $i=1, 2, \cdots, n$

$2)$ $|V(G_i)|=|V(H_i)|, |E(G_i)|=|E(H_i)|$

$3)$ If $C_k(G)$ denotes the number of closed walk of length $k$ in graph $G$, we have:

$C_k(G_i)=C_k(H_i)$ for $i=1, 2, \cdots, n$

$4)$ Preferably, I need these graphs be $a)$minimal and $b)$highly irregular(or has one of these two conditions $(a)$ or $(b)$).

$Definition 1:$ A graph $G$ is Highly irregular, if every vertex $v$ of $G$ is adjacent only to vertices with distinct degrees.

$Definition 2:$ The sequence of graphs $G_i$,$i=1,2,\cdots,n$, is minimal, if the number of vertices of every $G_i$ is minimum.

For example, two trees $T_1$ and $T_2$ with degree sequences $4,4,1,1,1,1,1,1$ and $5,2,2,1,1,1,1,1$ respectively, are minimal, because they are minimum vertices co-spectral trees.

I will appreciate any help and guidance.
 http://mathoverflow.net/questions/83817/operation-on-isospectral-graphs Operation on Isospectral graphs Shahrooz 2011-12-18T19:15:08Z 2011-12-18T23:01:30Z
Suppose $G$ and $H$ are two isospectral connected graphs. Can we say anything about isospectrality of graphs that obtain by binary operation between $G$ and $H$? For example,in special case, is $G\otimesG$(Kronecker product) isospectral? which binary operations between $G$ and $H$, preserve the isospectrality?
 http://mathoverflow.net/questions/83627/counting-special-graphs Counting Special Graphs Shahrooz 2011-12-16T15:25:01Z 2011-12-17T20:09:02Z
Do we have any formula for counting the number of graphs with $n$ vertices, that has exactly $k$ vertices with degree $d$ and the other vertices have different and disjoint degrees? (Different and disjoint are the same, $d_1$ is different or disjoint rather than $d_2$, iff $d_1\neq d_2$.)

For example, for $n=3, k=2, d=1$, we only have one graph($P_3$) with this property. Also, for $n=4, k=2, d=2$, we have the only graph with degree sequence $1, 2, 2, 3$.
 http://mathoverflow.net/questions/128930/solvable-groups-in-which-all-subgroups-are-supplemented Comment by Shahrooz Shahrooz 2013-04-28T19:20:29Z 2013-04-28T19:20:29Z Dear Prof. Abdollahi, after some attempts, I could not find suitable $A$. http://mathoverflow.net/questions/128930/solvable-groups-in-which-all-subgroups-are-supplemented Comment by Shahrooz Shahrooz 2013-04-27T20:31:14Z 2013-04-27T20:31:14Z Dear Prof. Abdollahi, I am thinking about $Q_8 \times G$, where $G$ is a suitable infinite abelian poly-cyclic group. Did you examined such groups? http://mathoverflow.net/questions/125817/mathematical-paper-that-just-links-two-different-fields-of-sciences/125824#125824 Comment by Shahrooz Shahrooz 2013-03-29T09:26:03Z 2013-03-29T09:26:03Z Dear Eremenko, your answer was interesting for me, since I did not know that Mandelbrot did not prove new theorem in fractal geometry. I think general relativity and Non-Euclidean geometry has same story. Einstein just found the relation between his theory and a suitable geometry. http://mathoverflow.net/questions/124764/does-there-exist-a-regular-graph-of-degree-4-diameter-3-32-vertices/124787#124787 Comment by Shahrooz Shahrooz 2013-03-18T18:55:36Z 2013-03-18T18:55:36Z Dear geoffreyexoo, thanks for your explanation. For me, the number of examples that you found is sufficient to accept my wrong imagination. But, these number of graphs among all 4-regular graphs with 32 vertices is very small. http://mathoverflow.net/questions/124764/does-there-exist-a-regular-graph-of-degree-4-diameter-3-32-vertices/124787#124787 Comment by Shahrooz Shahrooz 2013-03-17T20:55:50Z 2013-03-17T20:55:50Z Very nice, dear geoffreyexoo, what was your strategy for finding this graph? I found some 4-regular graphs with diameter 4. Also by some papers that BOLLOBAS and his coworkers wrote, I think there are a little number of such graph that you found one of them. http://mathoverflow.net/questions/124764/does-there-exist-a-regular-graph-of-degree-4-diameter-3-32-vertices Comment by Shahrooz Shahrooz 2013-03-17T12:09:10Z 2013-03-17T12:09:10Z By this address: mathworld.wolfram.com/QuarticGraph.html and a probabilistic program that I wrote in Maple (and with 20 min run, that I know it is very small time), I think there is not such a graph. It is interesting for me if such a graph exist. Also, I think Dear McKay with Naughty software, can help you more than me. Also, it is obvious that, such a graph (if it exist) is not strongly regular. http://mathoverflow.net/questions/124443/group-of-a-graph-vs-graph-of-a-group Comment by Shahrooz Shahrooz 2013-03-13T19:38:59Z 2013-03-13T19:38:59Z Dear Valette, I did not aware about these groups of graphs. Thank you very much because of the links. Your first paper is "666.02" Kb that we discussed this number with dear Meyerowitz. So strange!! http://mathoverflow.net/questions/124443/group-of-a-graph-vs-graph-of-a-group Comment by Shahrooz Shahrooz 2013-03-13T19:28:31Z 2013-03-13T19:28:31Z Dear Hopkins, this group is interesting. Thanks for your answer. Dear Meyerowitz, It is obvious that I did not used my imagination for asking this question. I used the Google and some famous paper databases, and after that I asked this question. But, I am agree with you that my imagination is a limit. About, 666, I know it belongs to our dear common friend "Satan". But it is very strange that, how you know my mobile number ends with this three digits. Finally, thanks for your good comments. http://mathoverflow.net/questions/121097/spectrum-of-a-laplacianized-matrix/121101#121101 Comment by Shahrooz Shahrooz 2013-02-08T10:24:08Z 2013-02-08T10:24:08Z In general no, but these references maybe good for further study: 1. "THE NORMALIZED LAPLACIAN MATRIX AND GENERAL RANDIC INDEX OF GRAPHS", a thesis by "Michael Scott Cavers" 2. "Generalizing some results to the normalized Laplacian" by "Steve Butler" http://mathoverflow.net/questions/121049/generalization-of-friendship-theoremn-vertices-any-m-vertices-have-exactly-one Comment by Shahrooz Shahrooz 2013-02-08T10:09:43Z 2013-02-08T10:09:43Z Oops, you are absolutely right. http://mathoverflow.net/questions/121049/generalization-of-friendship-theoremn-vertices-any-m-vertices-have-exactly-one/121112#121112 Comment by Shahrooz Shahrooz 2013-02-07T22:02:50Z 2013-02-07T22:02:50Z Dear Aaron, did you assume that any $m$ adjacent vertices have exactly one common vertex as a neighbor? In the question, this assumption does not exist. Am I thinking truly? http://mathoverflow.net/questions/121049/generalization-of-friendship-theoremn-vertices-any-m-vertices-have-exactly-one Comment by Shahrooz Shahrooz 2013-02-07T14:27:24Z 2013-02-07T14:27:24Z Let $A_m$ be the set of all graphs with $m$ vertices and that they have not dominating vertex. So, why the cone over any subset of $A_m$ is not an answer? http://mathoverflow.net/questions/121049/generalization-of-friendship-theoremn-vertices-any-m-vertices-have-exactly-one Comment by Shahrooz Shahrooz 2013-02-07T14:19:43Z 2013-02-07T14:19:43Z But for your problem, the graph has a dominating vertex and also the graph is $P_4$ free. http://mathoverflow.net/questions/121049/generalization-of-friendship-theoremn-vertices-any-m-vertices-have-exactly-one Comment by Shahrooz Shahrooz 2013-02-07T14:12:00Z 2013-02-07T14:12:00Z I think if you want to generalize the friendship theorem, you have to see one of important parameter in its definition. In the definition of friendship graph, any two adjacent vertices have exactly one common neighbor. By your definition, you deleted this condition and it is not the generalization of friendship graph. http://mathoverflow.net/questions/119364/minimum-sum-among-fixed-length-factors-of-a-number/119400#119400 Comment by Shahrooz Shahrooz 2013-01-21T15:59:56Z 2013-01-21T15:59:56Z Dear Kohl, thanks for your note.


text:
# Approaches to integrate $\int_0^1 \frac{x}{\sqrt{a+bx+cx^2}} dx$

I search for a good approach (or a general solution technique) to integrate

$$\int_0^1 \frac{x}{\sqrt{a+bx+cx^2}} dx$$

My knowledge on integration ends with rational functions.

-
Hint: complete the square. –  1015 Jan 29 '13 at 18:23
Hint the second: You will need the derivative of arcsin. –  Harald Hanche-Olsen Jan 29 '13 at 18:29
Thanks, that got me on the right track. –  shuhalo Jan 29 '13 at 18:38

We give an outline, but a complete outline. If we want to avoid complex numbers (and to some degree even if we don't) there are a number of cases, and these cases tend to have subcases. Each case can be handled without much difficulty. We do not end up with a universal formula. However, for each case you could follow the procedure described below and produce a formula for that case in terms of $a$, $b$, and $c$.

$1.$ Suppose $c=0$. The subcase $b=0$ is trivial. If $b\ne 0$ we can let $u=a+bx$.

If $c=ne 0$, the manipulation suggested at the beginning of the answer by juantheron is useful. After that manipulation, the top is replaced by a constant. From now on assume that manipulation has been done.

$2.$ Suppose $c\lt 0$. Complete the square in the expression $a+bx+c$. We get something like $k-(px+q)^2$, where in any interesting case $k$ is positive. The substitution $px+q=\sqrt{k}u$ transforms our definite integral into something of shape $\int\frac{a}{\sqrt{1-u^2}}\,du$, and we are in $\arcsin$ land.

$3.$ Suppose $c\gt 0$. Complete the square. Then our quadratic takes on shape $(px+q)^2 +k$ where $k$ is positive, negative, or $0$, three subcases.

The subcase $k=0$ is easy, just take the square root (carefully), we get $1$ over a linear expression, and we are in $\log$ land.

If $k$ is positive, let $px+q=\sqrt{k}\,u$. After some minor algebra, we want to find $\int \frac{du}{\sqrt{u^2+1}}$. This can be done in various ways, such as $u=\tan t$, or $u=\sinh t$. There are several other methods.

If $k$ is negative, let $px+q=\sqrt{-k}\,u$. soon we end up needing $\int \frac{du}{\sqrt{u^2-1}}$. Now again there are various tricks, such as $u=\sec t$, or $u=\cosh t$, other tricks.

-

$\displaystyle \int\frac{x}{\sqrt{a+bx+cx^2}}dx = \frac{1}{2c}\int\frac{\left(2cx+b\right)-b}{\sqrt{a+bx+cx^2}}dx$

$\displaystyle = \frac{1}{2c}\int\frac{2cx+b}{\sqrt{a+bx+cx^2}}dx-\frac{b}{2c}\int\frac{1}{\sqrt{a+bx+cx^2}}dx$

for First one put $a+bx+cx^2 = t^2$ and $\left(2cx+b \right)dx = 2tdt$

$\displaystyle = \frac{1}{c}\sqrt{a+bx+cx^2}-\frac{b}{2c}.\frac{1}{\sqrt{c}}\int\frac{1}{\sqrt{x^2+\frac{b}{c}.x+\frac{a}{c}}}dx$

$\displaystyle = \frac{1}{c}\sqrt{a+bx+cx^2}-\frac{b}{2c\sqrt{c}}\int\frac{1}{\sqrt{\left(x+\frac{b}{2c}\right)^2+\left(\frac{\sqrt{b^2-4ac}}{2c}\right)^2}}dx$

$\displaystyle = \frac{1}{c}\sqrt{a+bx+cx^2}-\frac{b}{2c\sqrt{c}}\ln \left|\left(x+\frac{b}{2c}\right)+\sqrt{x^2+\frac{b}{c}.x+\frac{a}{c}}\right|+\mathbb{C}$

-
That's a huge leap from the 2nd to last line to the last line. Care to fill in some details? –  Ron Gordon Jan 29 '13 at 20:36

format: plain


text:
# Daniel J Karmgard

Ontology type: schema:Person

NAME

Daniel J

SURNAME

Karmgard

### Publications in SciGraph latest 50 shown

• 2019-11-25 Measurements of triple-differential cross sections for inclusive isolated-photon+jet events in pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2019-05-20 Combined measurements of Higgs boson couplings in proton–proton collisions at s=13Te in EUROPEAN PHYSICAL JOURNAL C
• 2019-01-10 Measurement of differential cross sections for inclusive isolated-photon and photon+jet production in proton-proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-11-28 Search for heavy resonances decaying into a vector boson and a Higgs boson in final states with charged leptons, neutrinos and b quarks at s=13 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-11-22 Measurement of differential cross sections for Z boson production in association with jets in proton-proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-11-15 Studies of Bs2∗(5840)0 and Bs1(5830)0 mesons including the observation of the Bs2∗(5840)0→B0KS0 decay in proton-proton collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-11-02 Measurement of the top quark mass with lepton+jets final states using pp collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-09-28 Search for new physics in dijet angular distributions using proton–proton collisions at s=13TeV and constraints on dark matter and other models in EUROPEAN PHYSICAL JOURNAL C
• 2018-09-03 Measurement of the Z/γ∗→ττ cross section in pp collisions at s=13TeV and validation of τ lepton analysis techniques in EUROPEAN PHYSICAL JOURNAL C
• 2018-09-03 Search for third-generation scalar leptoquarks decaying to a top quark and a τ lepton at s=13Te in EUROPEAN PHYSICAL JOURNAL C
• 2018-09-01 Measurement of the weak mixing angle using the forward–backward asymmetry of Drell–Yan events in pp collisions at 8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-08-31 Measurement of charged particle spectra in minimum-bias events from proton–proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-07-20 Electroweak production of two jets in association with a Z boson in proton–proton collisions at s= 13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-07-10 Azimuthal correlations for inclusive 2-jet, 3-jet, and 4-jet events in pp collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-07-09 Erratum to: Measurement of b hadron lifetimes in pp collisions at s=8Te in EUROPEAN PHYSICAL JOURNAL C
• 2018-06-22 Erratum to: Measurements of the pp→ZZ production cross section and the Z→4ℓ branching fraction, and constraints on anomalous triple gauge couplings at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-06-20 Measurement of prompt and nonprompt charmonium suppression in PbPb collisions at 5.02TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-06-07 Measurement of b hadron lifetimes in pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-06-05 Search for single production of vector-like quarks decaying to a b quark and a Higgs boson in JOURNAL OF HIGH ENERGY PHYSICS
• 2018-04-11 Search for new physics in events with a leptonically decaying Z boson and a large transverse momentum imbalance in proton–proton collisions at s = 13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-04-09 Measurement of associated Z + charm production in proton–proton collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-03-21 Study of dijet events with a large rapidity gap between the two leading jets in pp collisions at s=7TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-02-24 Measurements of the pp→ZZ production cross section and the Z→4ℓ branching fraction, and constraints on anomalous triple gauge couplings at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2018-02-19 Search for standard model production of four top quarks with same-sign and multilepton final states in proton–proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-12-08 Search for dark matter produced in association with heavy-flavor quark pairs in proton-proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-11-08 Measurements of the associated production of a Z boson and b jets in pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-11-07 Measurement of the triple-differential dijet cross section in proton-proton collisions at s=8TeV and constraints on parton distribution functions in EUROPEAN PHYSICAL JOURNAL C
• 2017-10-26 Search for new phenomena with the MT2 variable in the all-hadronic final state produced in proton–proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-09-22 Search for new phenomena with multiple charged leptons in proton–proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-09-22 Search for heavy resonances that decay into a vector boson and a Higgs boson in hadronic final states at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-09-01 Search for physics beyond the standard model in events with two leptons of same sign, missing transverse momentum, and jets in proton–proton collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-07-14 Measurement of the jet mass in highly boosted tt¯ events from pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-07-11 Measurement of double-differential cross sections for top quark pair production in pp collisions at s=8TeV and impact on parton distribution functions in EUROPEAN PHYSICAL JOURNAL C
• 2017-05-29 Measurement of the top quark mass using single top quark events in proton-proton collisions at s=8 TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-05-18 Searches for pair production of third-generation squarks in s=13TeV pp collisions in EUROPEAN PHYSICAL JOURNAL C
• 2017-05-08 A search for new phenomena in pp collisions at s=13TeV in final states with missing transverse momentum and at least one jet using the αT variable in EUROPEAN PHYSICAL JOURNAL C
• 2017-04-27 Measurement of prompt and nonprompt J/ψ production in pp and pPb collisions at sNN=5.02TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-04-21 Search for electroweak production of a vector-like quark decaying to a top quark and a Higgs boson using boosted topologies in fully hadronic final states in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-04-19 Suppression and azimuthal anisotropy of prompt and nonprompt J/ψ production in PbPb collisions at sNN=2.76TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-04-12 Measurement of the WZ production cross section in pp collisions at s=7 and 8TeV and search for anomalous triple gauge couplings at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-04-05 Measurements of differential production cross sections for a Z boson in association with jets in pp collisions at s=8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2017-03-20 Measurement of the tt¯ production cross section using events in the eμ final state in pp collisions at s=13TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-02-11 Measurement of the production cross section of a W boson in association with two b jets in pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2017-01-07 Measurements of the tt¯ production cross section in lepton+jets final states in pp collisions at 8 TeV and ratio of 8 to 7 TeV cross sections in EUROPEAN PHYSICAL JOURNAL C
• 2016-09-30 Measurement of dijet azimuthal decorrelation in pp collisions at s=8TeV in EUROPEAN PHYSICAL JOURNAL C
• 2016-09-12 Erratum to: Search for direct pair production of scalar top quarks in the single- and dilepton channels in proton-proton collisions at s=8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2016-09-09 Search for Higgs boson off-shell production in proton-proton collisions at 7 and 8 TeV and derivation of constraints on its total decay width in JOURNAL OF HIGH ENERGY PHYSICS
• 2016-09-06 Search for s channel single top quark production in pp collisions at s=7 and 8 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2016-08-24 Azimuthal decorrelation of jets widely separated in rapidity in pp collisions at s=7 TeV in JOURNAL OF HIGH ENERGY PHYSICS
• 2016-08-22 Measurement of the differential cross section and charge asymmetry for inclusive pp→W±+X production at s=8 TeV in EUROPEAN PHYSICAL JOURNAL C

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text:
# Why are most (or all?) polynomial time algorithms practical?

I read an interesting comment in a paper recently about how weirdly useful maths turns out to be. It mentions how polynomial time doesn't have to mean efficient in reality (e.g., $O(n^{999999999999999999999})$ is polynomial time, but not efficient). Yet, isn't it the case that all algorithms in polynomial time also happen to be realistic, like at most $O(n^4)$ or something? I guess my questions are:

1. Is this surprising?

2. Are there any examples of algorithms that are polynomial time but not practical?

The comment is wrong. It is very easy to give examples of polynomial time algorithms which aren't practical at all:

1. The ellipsoid algorithm for solving linear programs runs in polynomial time but is too slow to be practical. The simplex algorithm, whose worst-case running time is exponential, is preferred in practice.

2. The AKS primality testing algorithm runs in polynomial time but is too slow to be practical. Instead, randomized polynomial time algorithms are used. We sacrifice certainty for performance.

3. Fast matrix multiplication algorithms run asymptotically faster than high-school matrix multiplication (both are polynomial), but are too slow to be practical. The high-school algorithm is used in practice.

4. A similar issue occurs in fast integer multiplication algorithms. The algorithm with the best asymptotic complexity, Fürer's algorithm, is too slow to be used in practice. Instead, relatively simple algorithms are used even for quite large integers.

5. Big data algorithms need to run in linearithmic time (which is $O(n \log^{O(1)}n)$) to be practical.

These examples show that the identification of polynomial time with practical is not accurate, and might depend on the circumstances. Researchers in theoretical algorithms feel a need to justify their field of research, and so they ideologically believe in the sentiments expressed in the comment you mention. You shouldn't take such comments too literally.

In fact, many algorithms used in practice are heuristic and we don't have any estimates on their running time other than empirical results. Such algorithm don't fit into the theoretical framework at all, yet there are very useful in practice. Several (but not all) machine learning algorithms belong to this class just in terms of running time (not to mention in terms of performance), as do search algorithms such as A* and alpha-beta, and SAT solving algorithms.

Question (1) is a tricky one that I have never seen a good reason for. One suggestion may be that we're much more likely to comprehend and find the answers for simpler problems, the harder ones take special skills, so the chance the right person works on it and comes up with the answer is low. This is all very hand-waving though.

For (2) there's definitely examples, in fact, this question has already been asked and answered! Note also that there's a chain of links to a cstheory.se question and a math.se question in there that have other examples.

I have seen the following explanation regarding your question (1):

The powers of $n$ in the running time usually arise out of for loops or similar constructs. Each for loop in turn is needed, because we, as solvers, had an idea how to break up the problem or index over something useful. Since we usually use only a small number of ideas, the powers tend to be small. It's hard to imagine an algorithm with 20 nested for loops, where inside the very last for loop we actually do something that depends on all 20 indices.

This argument is appealing to me, but it is quite weak because it only deals with for loops. For instance, one can easily create any number of nested for loops with recursion, but then one can find good reasons against doing such a bizarre construction. In any case, I think this argument can be strengthened with an analysis of a lot of different cases, but I present just the high-level idea.


text:
## Arithmetic Mean

For a Continuous Distribution function, the arithmetic mean of the population, denoted , , , or , is given by

 (1)

where is the Expectation Value. For a Discrete Distribution,
 (2)

The population mean satisfies
 (3)

 (4)

and
 (5)

if and are Independent Statistics. The sample mean,'' which is the mean estimated from a statistical sample, is an Unbiased Estimator for the population mean.

For small samples, the mean is more efficient than the Median and approximately less (Kenney and Keeping 1962, p. 211). A general expression which often holds approximately is

 (6)

Given a set of samples , the arithmetic mean is

 (7)

Hoehn and Niven (1985) show that
 (8)

for any Positive constant . The arithmetic mean satisfies
 (9)

where is the Geometric Mean and is the Harmonic Mean (Hardy et al. 1952; Mitrinovic 1970; Beckenbach and Bellman 1983; Bullen et al. 1988; Mitrinovic et al. 1993; Alzer 1996). This can be shown as follows. For ,
 (10)

 (11)

 (12)

 (13)

 (14)

with equality Iff . To show the second part of the inequality,
 (15)

 (16)

 (17)

with equality Iff . Combining (14) and (17) then gives (9).

Given independent random Gaussian Distributed variates , each with population mean and Variance ,

 (18)

 (19)

so the sample mean is an Unbiased Estimator of population mean. However, the distribution of depends on the sample size. For large samples, is approximately Normal. For small samples, Student's t-Distribution should be used.

The Variance of the sample mean is independent of the distribution.

 (20)

From k-Statistic for a Gaussian Distribution, the Unbiased Estimator for the Variance is given by
 (21)

where
 (22)

so
 (23)

The Square Root of this,
 (24)

is called the Standard Error.
 (25)

so
 (26)

See also Arithmetic-Geometric Mean, Arithmetic-Harmonic Mean, Carleman's Inequality, Cumulant, Generalized Mean, Geometric Mean, Harmonic Mean, Harmonic-Geometric Mean, Kurtosis, Mean, Mean Deviation, Median (Statistics), Mode, Moment, Quadratic Mean, Root-Mean-Square, Sample Variance, Skewness, Standard Deviation, Trimean, Variance

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Alzer, H. A Proof of the Arithmetic Mean-Geometric Mean Inequality.'' Amer. Math. Monthly 103, 585, 1996.

Beckenbach, E. F. and Bellman, R. Inequalities. New York: Springer-Verlag, 1983.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 471, 1987.

Bullen, P. S.; Mitrinovic, D. S.; and Vasic, P. M. Means & Their Inequalities. Dordrecht, Netherlands: Reidel, 1988.

Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities. Cambridge, England: Cambridge University Press, 1952.

Hoehn, L. and Niven, I. Averages on the Move.'' Math. Mag. 58, 151-156, 1985.

Kenney, J. F. and Keeping, E. S. Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, 1962.

Mitrinovic, D. S.; Pecaric, J. E.; and Fink, A. M. Classical and New Inequalities in Analysis. Dordrecht, Netherlands: Kluwer, 1993.

Vasic, P. M. and Mitrinovic, D. S. Analytic Inequalities. New York: Springer-Verlag, 1970.


text:
# Deducing the closed form for pentagonal numbers

Consider the sequence:

$0,1,5,12,22,35,51,70,92,117,145,176,\ldots$

Find both a recurrence and a closed form for this sequence.

I've done some research and found out that the majority of these (except 0) are "pentagonal numbers." We haven't learned about these yet, but further research did allow me to find a formula:

$A_n = ((3n^2)-n)/2$

That is the closed form for this sequence, correct? Could someone possibly lead me as to how to get to that equation?

In terms of a recurrence, I'm somewhat lost.

• Whys is 0 a part of that sequence? – Daniel W. Farlow Apr 21 '16 at 3:52
• Not too sure. It was given in the problem. I realize that it is not a pentagonal number. – d3m0nxdrums Apr 21 '16 at 3:53

Let $Sa_n=a_{n+1}$, then we can compute from the table that $(S-1)^3a_n=0$. The solution for this Linear Recurrence Relation is of the form $$a_n=bn^2+cn+d\tag{1}$$ Using the known values for $n=0,1,2$, we get \begin{align} 0&=d\\ 1&=b+c+d\\ 5&=4b+2c+d \end{align}\tag{2} Solving $(2)$ and substituting into $(1)$ gives \begin{align} a_n &=\frac32n^2-\frac12n\\ &=\frac{(3n-1)n}2\tag{3} \end{align}
When you have a polynomial expressed as a sequence, you can find it by taking differences $$\begin {array} {l l l l l l} 0&1&5&12&22&35\\1&4&7&10&13\\3&3&3&3 \end {array}$$ where the first line is your sequence and subsequent entries are the difference between the one up and to the right and the one above. The fact that the second differences are constant says you have a second power polynomial. To get the leading term, you take the constant in the second line and divide it by $2!$, so your polynomial starts with $\frac 32n^2$. You can subtract that off and get a linear polynomial, which will be constant in the first line down.
You can use this to get a recurrence. The first line is $A_n-A_{n-1}$ and you can observe it is $1+3(n-1)$, so $A_n=A_{n-1}+3n-2$


text:
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A priority encoder accepts three input signals $\text{(A, B and C)}$ and produces a two-bit output $(X1, X0 )$ corresponding to the highest priority active input signal. Assume $A$ has the highest priority followed by $B$ and $C$ has the lowest priority. If none of the inputs are active the output should be $00$, design the priority encoder using $4:1$ multiplexers as the main components.
edited | 956 views
0
+1

This might help ...

0
be aware, in this above lecture, he did some mistake ..!
+1
yes @ankit, he did a mistake, 0's will be put only in 01 rows in second k-map.

$\text{MSB} - \text{Most Significant Bit}$
$\text{LSB} - \text{Least Significant Bit}$

$\begin{array}{ccccc} \rlap{\text{TRUTH TABLE}} \\ \rlap{\text{Inputs}: A,B,C}\\ \rlap{\text{Outputs}: \text{MSB},\text{LSB}}\\ A &B&C&\quad\text{MSB}&\text{LSB}\\ 0&0&0&0&0\\ 0&0&1&0&1\\ 0&1&X&1&0\\ 1&X&X&1&1\\\hline \end{array}$

$\text{MSB} = A+B$
$\text{LSB} =A + \bar B C$

It can be implemented using two $4 \times 1$ Multiplexers.

edited
0
Here why u select C as input of MUX?

shouldnot it be signal to see which MUX is active at a time and other deactive at that time?
+1
no we can't do like that, we need both MSB and LSB at any time thats why both the mux should be active all the time
0

https://en.wikipedia.org/wiki/Priority_encoder

why u took o/p 1 for 001 i/p??

0

अनुराग पाण्डेय how you derive expression (K-map??) if yes then how you have taken the inputs please comment.

0
Is this realisation possible if "If none of the inputs are active the output should be 00" - this statement is not present in question?

$\begin{array}{ccccc} A &B&C&\quad\text{MSB}&\text{LSB}\\ 0&0&0&X&X\\ 0&0&1&0&0\\ 0&1&X&0&1\\ 1&X&X&1&0\\\hline \end{array}$

$MSB=A$

$LSB=A \bar{B}$
0

2018 Given in question "If none of the inputs are active the output should be 00", so lowest priority should start with "01" and not "00"

0

Assume A has the highest priority followed by Band C has the lowest priority.

and wiki says

If two or more inputs are given at the same time, the input having the highest priority will take precedence.[1] An example of a single bit 4 to 2 encoder is shown, where highest-priority inputs are to the left and "x" indicates an irrelevant value - i.e. any input value there yields the same output since it is superseded by higher-priority input.

0

srestha how the expression for MSB and LSB is derived?

0
for final circuit it is just an assumption

1


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# OpenGL Opengl scaling using ortho projection not working correctly

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Hi there. I was trying to write a sprite renderer using opentk and C#. Everything was working fine till I've noticed zooming out using ortho projection changes sprites colors. I've created a small example of what I'm doing.

First of all as I use linear texture filtering I add half texel to top and left vertices texture coordinates, and subtract half texel from right and bottom vertices texture coordinates. As this ends to make mapped area of the texture one pixel smaller I subtract 0.5 from bottom and right vertices positions, and add 0.5 to left and top vertices positions.

float x = 0;
?float y = 64;
float w = 400;
?float h = 16;

VertexPositionTexColor[] vertices = new VertexPositionTexColor[] {
new VertexPositionTexColor(){Position = new Vector2(x + 0.5f, y + 0.5f),TexCoord = new Vector2(22.5f / bmp.Width,0.5f/bmp.Height) // added 0.5f to uv to map center of the texel. did the same thing for position
,Color =new Vector4(1f,1f,1f,1f)},
new VertexPositionTexColor(){Position = new Vector2(x + w - 0.5f, y + 0.5f),TexCoord = new Vector2(22.5f / bmp.Width,0.5f/bmp.Height) // subtracted u by 0.5f and added 0.5f to v to map center of the texel. did the same thing for position
,Color =new Vector4(1f,1f,1f,1f)},
new VertexPositionTexColor(){Position = new Vector2(x + w - 0.5f, y + h - 0.5f),TexCoord = new Vector2(22.5f / bmp.Width,15.5f/bmp.Height) // subtracted uv by 0.5f to map center of the texel. did the same thing for position
,Color =new Vector4(1f,1f,1f,1f)},
new VertexPositionTexColor(){Position = new Vector2(x + 0.5f, y + h - 0.5f),TexCoord = new Vector2(22.5f / bmp.Width,15.5f/bmp.Height) // subtracted v by 0.5f and added 0.5f to u to map center of the texel. did the same thing for position
,Color =new Vector4(1f,1f,1f,1f)},
};

The texture is setup as below and I'm using an atlas.

GL.TexParameter(TextureTarget.Texture2D, TextureParameterName.TextureMinFilter, (int)TextureMinFilter.Linear);
GL.TexParameter(TextureTarget.Texture2D, TextureParameterName.TextureMagFilter, (int)TextureMagFilter.Linear);
GL.TexParameter(TextureTarget.Texture2D, TextureParameterName.TextureWrapS, (int)TextureWrapMode.ClampToEdge);
GL.TexParameter(TextureTarget.Texture2D, TextureParameterName.TextureWrapT, (int)TextureWrapMode.ClampToEdge);

Is this approach correct?

Now I setup my viewport and projection matrix

float viewPortWorldX = 400;
float viewPortWorldY = 300;

.
.
.
.

Matrix4 ortho = Matrix4.CreateOrthographicOffCenter(0, viewPortWorldX,
viewPortWorldY, 0, -1, 1);

GL.Viewport(0, 0, 400, 300);
GL.UniformMatrix4(GL.GetUniformLocation(program, "projection"), false,
ref ortho);

The above code works fine because my ortho projection and viewport are the same size. Except one thing. The line is one pixel shorter than 400 while I've set w to 400. It is not surprising because I've subtracted 0.5 pixel from the right side.

The strange things start when I change ortho projection configuration.

float viewPortWorldX = 800;
float viewPortWorldY = 600;

.
.
.
.

Matrix4 ortho = Matrix4.CreateOrthographicOffCenter(0, viewPortWorldX,
viewPortWorldY, 0, -1, 1);

GL.Viewport(0, 0, 400, 300);
GL.UniformMatrix4(GL.GetUniformLocation(program, "projection"), false,
ref ortho);

The line color has changed! the middle of it was white but now it is blue!

if I change position of the line from 64 to 65 or any other odd number the result will be correct!

I've attached the visual studio 2013 project to this post:

[attachment=33274:OpenTKTest.zip]

Thanks in advance :)

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The color change may happen due to the linear filtering, because by using linear filtering, you're averaging between the 4 closest texels, and, since most of the texture is blueish, the resulting color will also be blueish.

The odd line position thing, may also be somehow related to this (though I can't properly explain it).

Have you tried setting the minification filtering to nearest instead of linear, to see if the results improve? You could keep the magnification filter linear, so that, if you zoom in, the result will be linearly filtered.

Anyway, I'm not sure this is the case, but I hope it helps.

Edited by __SKYe

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The color change may happen due to the linear filtering, because by using linear filtering, you're averaging between the 4 closest texels, and, since most of the texture is blueish, the resulting color will also be blueish.

The odd line position thing, may also be somehow related to this (though I can't properly explain it).

Have you tried setting the minification filtering to nearest instead of linear, to see if the results improve? You could keep the magnification filter linear, so that, if you zoom in, the result will be linearly filtered.

Anyway, I'm not sure this is the case, but I hope it helps.

Yes this solves the problem. But makes another problem in some viewport sizes the line is thicker. It seems that there is no real solution to this problem. Even if I solve this problem there is another problem. When zooming in or out at some viewport sizes the alignment of objects breaks. some objects may overlap. I tested this situation in gamemaker 8 and it has the alignment problem.

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Yes this solves the problem. But makes another problem in some viewport sizes the line is thicker. It seems that there is no real solution to this problem

Yeah, but you can't really help that, if you view such a small graphic zoomed out, stuff like this will happen.

Have you tried mipmaps? You might get a better result with them.

Even if I solve this problem there is another problem. When zooming in or out at some viewport sizes the alignment of objects breaks. some objects may overlap. I tested this situation in gamemaker 8 and it has the alignment problem.

This, I can't really help, except you could enforce sizes for the viewport zooming, instead of allowing arbitrary zooming amounts.

What I mean is, allow zooming, but in 50% increments (or something along those lines), so that you can ensure it doesn't break the alignment.

Obviously, depending on what you're trying to do, this may not be possible (or wanted), but it's a thought.

Anyway, good luck.

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Thanks. I just wanted to know if there is an out of the box way to solve this. I think using some fixed viewport sizes is a good way. I already force the aspect ration to be the same in all view sizes. I should choose some viewport sizes and use them.

I should take a look at mipmaps too.

Thank you

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text:
The Philosophy of Proving Trig Identities

1. Feb 6, 2009

romolo

I was wondering if anyone could help me out about trig identities. I'm a HS trig teacher and I'm going "by the books" and instructing them to manipulate the left and right sides of the identity independently of each other. They are not to treat it like an equation, e.g. no moving terms from one side to the other by way of adding etc, no squaring or rooting each side, no cross multiplying, etc. The rationale, I guess, is that we don't yet know whether it is a valid equality so the equals sign should have a little question mark above it until the two sides are identical.

I've always found that argument a little soft on logic. If I'm comparing two expressions and would find it convenient to add 15, say, to both of them, surely I have not changed their relationship either from equal to not equal or vice versa. Likewise, if I were to double each expression, I don't see any problem there either.

There might be an issue with multiplying variable expressions (as opposed to constants) on both sides, in that variable expressions may sometime equal zero or be undefined. Also, squaring both sides may introduce extraneous solutions and square rooting both sides might throw us into imaginary numbers. So maybe I'm comfortable banning these actions. (However, I would like to see an example of a "false" equation that, when using these inappropriate methods, results in the erroneous conclusion that the two sides are equal.)

Maybe this is all too theortical. Here's a student's answer to a verification problem that got me thinking of posting this.
The problem:
Verify cos^2(x) - sin^2(x) = 2cos^2(x) - 1
The work:
Step 1 -> add 1 to both sides -> 1 + cos^2(x) - sin^2(x) = 2cos^2(x)
Step 2 -> subtract cos^2(x) -> 1 - sin^2(x) = cos^2(x)
Step 3 -> Pythagorean Identity -> cos^2(x) = cos^2(x)
(I suppose step 3 could have been to add sin^2(x) to both sides, too.)

So here's my question. Do I allow my students to solve identities this way? Notice that in the example above, there was no "inappropriate" steps except adding and subtracting, so the concerns mentioned above are moot. I can think of no reason (apart from them going against my instruction) of disallowing it.

2. Feb 6, 2009

slider142

If they are experienced enough, you can let them use the same algebra that is valid for manipulating inequalities, keeping in mind that the specific inequality is unknown (or whether it is actually an equality, for which the same algebra is valid).
However, a less questionable method of presenting the same proof would have been to pretend it's an equality as the student did until you arrived at a real equality, and then turn it around and present the last line as the first line of the proof and the first line as the last line. this would allow him to catch any errors in reasoning that may have inadvertently been made.
It will also prepare them for epsilon-delta proofs of calculus, which usually get started on a scratch pad working backwards from epsilon to delta, then presenting the forwards approach cleanly from "let delta be ..." covering your tracks in the sand.

3. Feb 7, 2009

Ben Niehoff

My opinion is that to disallow perfectly valid mathematical steps will serve only to confuse the students and make them think that mathematics is haphazard and arbitrary.

4. Feb 7, 2009

Citan Uzuki

The problem here isn't whether they manipulate one side of the equation or both. The problem is one of logical validity. When a student is asked to verify some trigonometric identity, they are being asked to prove that it is true. The algebraic manipulations that they produce in response to such a question are a logical argument that the identity in question really is true, albeit one presented in algebraic shorthand rather than English words.

Now, when a student presents a response like the one in the example you gave, the argument he is implicitly making is:

Leaving out the algebraic manipulations and just focusing on the basic structure of the argument, the student's argument is:

Or in symbolic form:

I'm sure you see the problem here. The structure of the argument is NOT valid. It is the formal fallacy of affirmation of the consequent. That this argument is invalid may be seen by replacing A and B with other statements. For instance, we might let A be "I live in Massachusetts" and B be "I live in the United States of America." It is undoubtedly true that A implies B, since Massachusetts is in the U.S., and I do in fact live in the U.S., but the conclusion, that I live in Massachusetts, is not true. I live in Utah.

In general, if a student starts with the identity they are trying to prove is true, and algebraically manipulates both sides to produce an obviously true equation, they are giving an argument like the one produced by your student, which relies on a formal fallacy. That is why such a method of "verifying" trig identities cannot be allowed. However, that does not mean that the only valid method of proving an identity is to start with one side of the equation and transform it to the other. For instance, consider the following argument:

This argument has the structure:

Which IS perfectly valid, even though it involves manipulating two sides of an equation. The important thing is that we are arguing from premises we know to be true to the conclusion that we want to prove, and not from the conclusion we want to prove to premises we know to be true.

Now, this doesn't mean that starting with the conclusion and manipulating both sides is useless. As slider142 notes, in many cases an invalid proof which proceeds from conclusion to premises can be turned into a valid proof simply by reversing all the steps (which is pretty much what I did with your student's argument). And in many cases, it is actually easier to find a path from the conclusion to the premises and reverse it than it is to find a path from premises to conclusion in a more direct fashion. As such, trying to work backwards can be a useful method for discovering proofs. But it needs to be emphasized that:

• Working backwards is only a method for finding proofs, and cannot constitute a proof itself. Only the work from the premises to the conclusion counts as a proof.
• Working backwards is only useful if the steps can actually be reversed at the end. If a non-invertible step such as squaring both sides is used, this cannot be reversed, and assuming it can may lead you to present invalid proofs or worse, to incorrectly believe that some false equation is true (e.g. someone who forgets that squaring both sides is not invertible might conclude that x-y = y-x, because they both have the same square).
• Working backwards is NOT the only way to construct a proof. You should spend at least as much time trying to work forwards as backwards.

Now, as for what you should allow them to do as a teacher -- I would recommend that in general, you should allow them to prove these identities using any valid argument they like. There's no reason why they should be restricted to just manipulating one side of the equation at a time. But invalid arguments, such as the one presented by your student, should not be accepted. I would also recommend that you spend some time talking about exactly what constitutes a valid proof, and why proofs such as the one your student presented are invalid. Also spend some time discussing the circumstances in which such invalid proofs can be reversed to yield valid proofs, and how it relates to the problem of extraneous solutions in algebra.

(Briefly, the relation is this: The process of solving an equation in algebra amounts to manipulating an equation such as F(x) = G(x) (where F and G are expressions involving x) into the form x = C, where C is some expression not involving x. This amounts to producing a proof that "if F(x) = G(x), then x=C" -- i.e. that C is the only possible solution to the equation. However, such a proof is no guarantee that "if x=C, then F(x) = G(x)" -- i.e. that C actually IS a solution to the equation. Now, if all of the operations used to produce the proof that "if F(x) = G(x), then x=C" are invertible, then that proof can be reversed to show that "if x=C, then F(x) = G(x)" and that reversed proof guarantees that C really is a solution of the original equation. However, if non-invertible operations were involved in producing the original proof, then the proof cannot be reversed, and so you have no guarantee that C is actually a solution of the equation, so you really do have to check. Thus, solving an equation may produce extraneous solutions only if the proof cannot be reversed to show the solution x=C implies the original equation, which occurs only if non-invertible operations were used in deriving the solution.)

5. Feb 13, 2009

m00npirate

I was taught to prove trig identities by only manipulating one side, which at the time I felt was silly for the reasons you all have mentioned, but I think it does serve a purpose. When doing calc you often have integrals of trig functions that do not immediately appear "nice," but through manipulation you can turn them into something more manageable. Proving these equalities using only one side of the equation (at least for me) helped me to be able to see and make these substitutions more easily.


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AC¯¯¯¯¯$\overline{AC}$ is a line segment with point B$B$ existing somewhere between points A$A$ and C$C$. If the measure of AC¯¯¯¯¯$\overline{AC}$ is 209, and the measure of AB¯¯¯¯¯$\overline{AB}$ is 163, then what is the measure of BC¯¯¯¯¯$\overline{BC}$?
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## Basic College Mathematics (10th Edition)

$513$ dollars
We have $513.499$ rounding to the nearest dollar we have $513$


text:
# Tag Info

12

This is not a trivial question. Here's a relevant excerpt (an appetizer, really) from Hull's book (7th Edition, P. 75): It is natural to assume that the rates on Treasury bills and Treasury bonds are the correct benchmark risk-free rates for derivative traders working for financial institutions. In fact, these derivative traders usually use LIBOR rates ...

10

I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality. You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution). I think there are no deeper truths to be found here.

9

Interest rates in general are far from independent and identically distributed. A high interest rate observation is quite likely to be followed by another high observation, and the volatility is likely to be higher as well. Interest rates are also mean reverting, as in most real-world situations (at least for developed markets) interest rates rarely rise ...

7

As with most derivatives that have early exercise, you are going to want to price this using a grid scheme. I have priced callable loans with floors using the Generalized Vasicek model at my old hedge fund, and it is fairly easy to handle. As a matter of fact my students are doing that very problem as homework this week, and my reference implementation ...

7

To answer a question with a question - are you assuming proportional or constant dividends? :) The general consensus of the market is that dividends are somewhere between proportional (fixed yield) and constant (fixed dollar). The carry embedded into the forward prices at different strikes reflects that consensus, in fact you can establish the ...

6

There are certainly (short-rate) models which assume bounded interest rates. I suppose I should clarify - the design of the model prohibits negative interest rates. Further, some models asymptotically reach some target, or mean rate which is considered mean reversion, the most famous perhaps the Vasicek. Short rate models where rates cannot go negative: ...

6

Recall that an interest rate swap has two legs, one fixed and one floating, each paid by one party to the transaction. Now, assume you go to a big bank like JPM, and want to borrow $100MM at fixed rate. JPM will have to fund that position, which because it is a big bank it will do at floating interest rates. But maybe JPM is worried about the effect such ... 6 I like to present to you a slightly different approach: Historically, only one single yield curve was derived from different instruments, such as OIS, deposit rates, or swap rates. However, market practice nowadays is to derive multiple swap curves, optimally one for each rate tenor. This idea goes against the idea of one fully-consistent zero coupon curve, ... 5 Take a look at historical short-term risk-free rate proxies such as Fed Funds, LIBOR, short Treasuries, and you will find plenty of periods where rates have been significantly above or below inflation (as measured by any CPI series) in the same period. In fact, controlling this difference, known as the real interest rate, is the primary tool of modern ... 5 The very easiest change you can make is to switch to quasirandom sampling. I favor the Niederreiter sequence, for which you can find implementations in most languages around the web. You can also get a (sometimes tremendous) speed boost by running using a control variate. Even a swap would probably reduce your variance somewhat. I don't recall the CIR ... 5 Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the ... 5 As you said,$\mu$is the expected return that is the expected value (mathematical expectation) of the random variable "stock return" under the objective probability measure. Assuming that returns are stationary*, the obvious way to estimate it is to compute a large number$N$of returns$R_i$, then to average them. You also want to annualize this average ... 5 The following paper, Interpolation Schemes in the Displaced-Diffusion LIBOR Market Model and the Efficient Pricing and Greeks for Callable Range Accruals, addresses this issue: We introduce a new arbitrage-free interpolation scheme for the displaced-diffusion LIBOR market model. Using this new extension, and the Piterbarg interpolation scheme, we study ... 5 The Macaulay duration is a measure of how sensitive a bond's price is to changes in interest rates. Duration is related to, but differs from, the slope of the plot of bond price against yield-to-maturity. The slope of the price-yield curve is$-\frac{D}{1+r}P,$where$D$is Macaulay duration,$P$is bond price, and$r$is yield. Here's how the definition ... 5 You should use the full yield curve, discounting cash flows at specific dates using the appropriate zero-coupon interest rate. As to which yield curve, that is often a matter of convention. Generally one uses the LIBOR/swaps curve for all but the most liquid products (in which case you use the treasury curve). The curve is constructed from LIBOR/Eurodollar ... 5 First step is to decide what instruments you want to include in your process for estimating the spot curve. You want to look at the following instruments for inclusion - treasury coupon strips, on-the-run treasury issues, and some off-the-run treasury issues (those not trading at liquidity discounts), or all treasury coupon securities and bills. You want ... 5 This is not an arbitrage because the transaction costs of the basket of goods is too high. Ever try to sell an item on eBay? I doubt you'll get 2-3% more for it next year, even new in box. Some of the items in the basket are current consumption goods. Good luck selling those fresh fruits and vegetables next year for 2-3% more than you paid. Others are ... 5 Dividends are the key. For simplicity, let's include a single dividend at the time of expiration, and assume that the options are European and expire ex. (There is really no reason not to assume that an option on a market index is European. EDIT: not quite true; that's discussed here.)$S+P = e^{-rt}K+C + e^{-rt}D$This is a certain fixed dividend, but ... 5 Indeed,$D(t)$is the discount factor used to compute the present value of a cash flow at time$t\$: $$PV = D(t) \cdot CF_t$$ It is more convenient to write it that way when you assume stochastic interest rate because you don't have to write the integral all the time.

5

There's no class at this time to add two curves as you want, but it won't be much difficult to write it. The closest you'll get in the library is the ZeroSpreadedTermStructure class, that shows the general idea: it inherits from YieldTermStructure (by way of ZeroYieldStructure) takes a YieldTermStructure and a spread (constant, in this case) and override ...

4

The risk implied by Euribor or EONIA (or their swaps) is for lending to another prime rated bank. These rate indexes represent where contributor banks are offering funds to each other in the interbank market. Contributing banks are mostly rated P-1 (Moody’s) or A-1 (S&P). You wouldn’t use these rates for govt discount curves because the risk doesn’t ...

4

Overnight funding is made through an auction, a fixing as you name it and it is achieved successfully (usually, i.e. when Lehman Brothers doesn't go bankrupt) because of the huge amounts on BOTH sides of this auction (i.e. liquidity). If you do an auction every minute, you will have more volatility as prices will potentially vary every minute and there will ...

4

If you are trying to arbitrage the put-call parity, then use your collateral interest rate for the options side, and your cost of funds on the stock side of the equation. Yes, that's right, 2 different interest rates. Also, don't forget to incorporate bid-ask spreads. If you are trying to turn a put into a call for your own book, you don't actually need ...

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I think you should put YOUR attainable interest rate. Because it is your view of how much the forward is worth. So, on the offer the rate at which you are indifferent is computed with the interest rate at which you borrow. And if you go short, the interest at which you can put your have your money fructified. No what rate can your money be fructified at ? ...

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If you are looking for good explanation with example than you may want to consider reading Jim Harper aka Bionic Turtle and his article on forward rates and spot rates. You can find excel spreadsheet for given example there as well.

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CMS adjustments in single curve context can be roughly explained if you consider a CMS swaplet by the fact that there is a single payment at the CMS rate at a single date and not on the whole strip of the underlying CMS tenor schedule. So if you are trying to hedge a CMS swaplet with the corresponding swap of CMS tenor length (with correct naïve nominal ...

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Forward interest rates are negative whenever the yield curve is negatively sloped. The US term structure was inverted most recently around 2007. Hard to find bank deposits that have negative yields (find countries experiencing deflation and you may find it), however, treasury bills during recent times of financial stress have yielded a negative rate. The ...

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A swap does not require a model because its price can be derived from the yield curve without any assumptions about how the yield curve may move in the future. The PFE however is an indication of by how much the swap's mark-to-market may move between now and a moment in the future. It is of course influenced by how volatile rates are. The more volatile ...

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There are two flaws in the argument. The simpler one is that expectations give information about probability distributions (premise D). I think this is what John was referring to in his comment. The fact that the expectation of a forward rate in period X is Y% tells us nothing about the implied probability distribution in that period, and certainly doesn't ...

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Taking the case of companies other than the bank, when you have a large amount of cash, you won't stock it in your backyard as there would be insurance and logistics costs that would cost you more than the negative government yield. I believe the main reason why people are willing to accept the negative yield is essentially for counterparty risk ...

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text:
# Does IND-CPA to INC-CCA effects on the error probability?

If we have a key encapsulation mechanism (KEM) that is IND-CPA secure, with some error probability $$P_e$$ (probability that the shared keys are not equal).

The goal is to use the Fujisaki-Okamoto transformation to obtain an IND-CCA security.

Does this transformation effect the error probability?


text:
# Strategy-proof Cardinal Decision Schemes

Working paper
Author/s:
Bhaskar Dutta, Hans Peters and Arunava Sen
Issue number:
722
Publisher:
University of Warwick
Year:
2005
We construct an example to show that the random dictatorship result no longer holds when individual utility functions satisfy this additional restriction. We then analyze the consequences of gradually reducing the grid size. That is, we consider an arbitrary sequence of strategy-proof and unanimous decision schemes deﬁned on a sequence of decreasing grid sizes approaching zero. We obtain a ‘limit’ random dictatorship result in the sense that the sequence of such decision schemes must converge to a random dictatorship for all proﬁles for which the limit exists.
Developed by Paolo Gittoi