In ac2005, Prof. Hiss introduced how to calculate decomposition numbers by the algrithm MOC. I have implemented a part of MOC in GAP. I will show my calulation of decompositions numbers from ordinary irrecucible characters by GAP.
There are many interesting subalgebras in the affine vertex operatora algebras. For the study of these subalgebras, it is important to know the operator product expansions. In this talk we show the computation of the operator product expansions by a computer algebra system Risa/Asir for a certain $W$ algera which appear in an affine vertex operator algebra.
E. R. van Dam gave an example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $45$ on $\text{GF}(2^{12})$. The aim of this paper is to present a new example of primitive non-amorphous association schemes in which every nontrivial relation is a strongly regular graph, as a fusion scheme of the cyclotomic scheme of class $75$ on $\text{GF}(2^{20})$. We also propose an infinite family of parameters of association schemes containing both of these two examples.
We consider some strongly regular graphs related with sporadic simple groups and the sizes of their maximum cliques and cocliques. Further, we give new constructions of these graphs by using their maximum clique and coclique designs.
Let $R$ be a finite commutative ring. A code $C$ of length $n$ is an $R$-submodule of the collection of $n$-tuples over $R$. An important type of code is the self-dual code, composed of all $n$-tuples which are orthogonal to every element of the code. In this talk, I give a survey of work on the classification of self-dual codes over certain rings, including recent results on the rings of integers modulo $m$, based of joint work with K Nagata and H Wada.
It has become possible to compute the complete coset distributions of the second order Reed-Muller code of length 64. In the present talk we report the group theoretical techniques that reduce the computing time largely, and some methods to check the correctness of the computation.
Using the classification of extremal unimodular lattices of rank 28 due to Bacher and Venkov (2001), we classify ternary extremal self-dual codes of length 28. This is achieved by classifying 3-frames of each of these lattices up to isometry of the lattice. There are 38 such lattices, and four of them have no 3-frames. The total number of 3-frames in the other 34 lattices is 13399474, and up to isometry, it is 6931. Among these, we find 18 codes constructed by Huffman (1992), 16 codes constructed by Harada (1998), and a code constructed by Harada (2001).
A set $\{a_1,\dots,a_m\}$ of $m$ distinct positive integers is called a Diophantine $m$-tuple if $a_i a_j+1$ is a perfect square for all $i,j$ with $1 \leq i<j \leq m$. It is conjectured that any Diophantine quadruple $\{a,b,c,d\}$ $(a<b<c<d) $ is regular, i.e., satisfies $d=d:=a+b+c+2abc+2rst$, where $r=\sqrt{ab+1},\,s=\sqrt{ ac+1},\,t=\sqrt{bc+1}$. In this talk, we show that if $\{a,b,c,d,e\}$ $(a<b<c<d<e)$ i s a Diophantine quintuple, then $\{a,b,c,d\}$ is regular, i.e., $d=d_+$.
Asymptotic expansions including the Edgeworth expansions provide effective approximate alternatives for many useful statistics to their exact distributions without computability. Higher-order expansion involves the higher-order moments of the statistic concerned as well as algebraic computations, sometimes huge in size. Use of base transformation of symmetric polynomials is a well-known approach applicable to most statistics required in practice, but suffers from the combinatorial intermediate swell. Several improvements are implemented in the algorithms employed; including tailored sorting of terms to their generation order, reusing the existing intermediary results, earlier deleting of unnecessary higher-order terms and other application-specific computations in statistics.
Let d be a positive non-square integer which is congruent to 1 modulo 4 and t and u be positive integers such that (t+u*sqrt(d))/2 is the fundamental unit of the quadratic order of discriminant d. Searching d's satisfying (i) the continued fraction expansion of (1+sqrt(d))/2 is with a period of fixed length and (ii) u > d, we find that some of such d's can be parameterized.
In the area of public key cryptosystems, we apply computational problems to construct schemes and discuss their security in the mathematical model. In this talk, we first review the models of public key cryptosystems and their security, and then review some computational problems utilized in this area. Moreover, we survey the notion of provable security and review some fundamental schemes and ones with additional properties.
Let $k$ be an arbitrary field. We study a general method to solve the field isomorphism problem of generic polynomials over $k$ via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution of the field isomorphism of cubic generic polynomials for $S_3$ and $C_3$ over $k$. As an application of the cubic case, we also give several sextic generic polynomials over $k$.
NZMATH, a number theory oriented calculation system based on Python, have been continuously developed for fifth years since AC2003. We will summarise recent two years of the development with problems and issues recognized through the experience, and will present a medium term future vision of development for the coming two years.
In this talk, we will discuss a system for number theory called "NZMATH" which is being developed at Tokyo Metropolitan University. From the viewpoint of an NZMATH user, we will show how you can use this system and what you can do with it. Moreover, we will demonstrate how to install, execute commands and link NZMATH with other softwares.
KNOPPIX/Math is a bootable CD/DVD that contains a collection of mathematical software. It includes over 100 mathematical software, and anyone can use KNOPPIX/Math for research, education, and presentations, etc.
On the other hand, we have too many mathematical software in one DVD, it's like a deep forest. I'd like to guide some walking trails in this forest.
To treat various problems of Waring type ( W-problem), for example the linear diophantine problem of Frobenius or the problem G(3) = 4 ? , with a unified aspect, we introduce (a) R-sieve and (b) wirklich method. ( A suitable pair of ( sieve, wirklich ) works well for various problems. Some examples will be explained. ) Using these tools, we see that W-problems satisfy common properties such as Cell Principle, Net Structure, Quantitative Relations and (curious) Divisor Property. On the other hand, to overcome proper difficulties of each problem, we need new ideas such as Trunk-Star Principle, Equality Property etc.
We investigate linear three term recurrence relations of the form $z_n=T(n)z_{n-1}+U(n)z_{n-2}$ associated with two sequences of integers $(T(n))_{n\ge 0}$ and $(U(n))_{n\ge 0}$. If these sequences are ultimately periodic modulo $m$, then for some fixed integers $r$ and $i$ we have $A(n)z_{rn+i}+B(n)z_{r(n-1}+i}+C(n)z_{r(n-2)+i}=0$. In special, if $U(n)=1$ for all $n\ge 0$, this is reduced to the three term recurrence relation for leaping convergents of a Hurwitz continued fraction.
In this talk, we are concerned with the ``multiple zeta value (MZV)''. More than fifteen years ago, D. Zagier gave a conjecture on MZVs based on numerical computations on PARI. Since then there have been various derived conjectures and two kinds of efforts for attacking them: one is a mathematical proof and another one is a computational experiment to get more confidence to verify a conjecture. We have checked one of these conjectures up to weight $k=20$ with Risa/Asir function for non-commutative polynomials and special parallel programs of linear algebra designed for this purpose.
By Mordell's theorem, the group of an elliptic curve over the rationals is the product of a finite subgroup consisting of all torsion points and $r \ge 0$ copies of an infinite cyclic group. There are exactly 15 possible torsion groups, but little is known about which values of rank $r$ are possible. The conjecture is that rank can be arbitrary large, but it seems to be very hard to find examples with large rank. The current record is an example of elliptic curve over $\mathbb{Q}$ with rank $\geq 28$, found by Elkies in May 2006.
There is even a stronger conjecture that for any of 15 possible torsion groups $T$ we have $B(T)=\infty$, where $B(T)=\sup \{ {\rm rank}\,(E(\mathbb{Q})) \,:\, \mbox{torsion group of $E$ over $ \mathbb{Q}$ is $T$} \}$. It follows from results of Montgomery and Atkin \& Morain that $B(T)\geq 1$ for all admissible torsion groups $T$. We improved this result by showing that $B(T)\geq 3$ for all $T$. In this talk, we will describe some recent improvements on lower bounds for $B(T)$. The information about current records for all admissible torsion groups can be found on my web page {\tt http://web.math.hr/$\sim$duje/tors/tors.html}.
Construction of high-rank curves in families of elliptic curves appears naturally in several Diophantine problems. We will present results related to elliptic curves induced by Diophantine triples, i.e. curves of the form $y^2=(ax+1)(bx+1)(cx+1)$, where $a,b,c$ are non-zero rationals such that $ab+1$, $ac+1$ and $bc+1$ are perfect squares. We show that there are exactly four types of possible torsion groups, and construct curves with rank equal to $r$, for $r \le 9$. We will also consider arithmetic progressions consisting of integers which are solutions of a Pellian equation. In a recent joint paper with A.~Peth\H{o} and P.~Tadi\'{c}, we have constructed a seven-term arithmetic progression with the given property, and also several five-term arithmetic progressions which satisfy two different Pellian equations. These results are obtained by studying properties of a parametric family of elliptic curves.
It is well-known that the whole set of computable numbers forms an algebraically closed field. In contrast, it is not well-known that the set of whole polynomial time computable numbers also forms an algebraically closed field, shown by Sch\"onhage in 1980s. We introduce the result, and show numbers defined by a certain kind of dicision problems are transcendental by using the result. We also discuss about limitations of this method.
We study pattern sequences for patterns in <q,r>-numaration systems through their generating functions. Our result implies that any nontrivial linear combination over ${\Bbb C}$ of pattern sequences chosen from different <q,r>-numaration systems can not be a linear recurrence sequence.
Weber proved that the class number h_n of Q(2cos(2*pi/2^{n+2})) is odd for all n>0. For an odd prime number p, Washington's theorem says that p-part of h_n is bounded as n tends to infinity. Unfortunately Washington's bound is not explicit. We show any odd prime number p < 10^5 does not divide h_n for all n>0.
$Date: 2007/11/16 00:46:58 $+ 9:00:00 (JST)