scholarly journals Cuntz-Krieger Algebras Associated with Hilbert $C^*$-Quad Modules of Commuting Matrices

2015 ◽  
Vol 117 (1) ◽  
pp. 126 ◽  
Author(s):  
Kengo Matsumoto
Keyword(s):  
Euclidean Algorithm ◽  
C Algebra ◽  
Commuting Matrices ◽  
Tiling Space ◽  
Positive Integers

Let $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ be the $C^*$-algebra associated with the Hilbert $C^*$-quad module arising from commuting matrices $A,B$ with entries in $\{0,1\}$. We will show that if the associated tiling space $X_{A,B}^\kappa$ is transitive, the $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{A,B}_{\kappa}}$ is simple and purely infinite. In particular, for two positive integers $N,M$, the $K$-groups of the simple purely infinite $C^*$-algebra $\mathscr{O}_{\mathscr{H}^{[N],[M]}_{\kappa}}$ are computed by using the Euclidean algorithm.

The Euclidean Algorithm as a Matrix Transformation

1972 ◽  
Vol 65 (3) ◽  
pp. 228-229
Author(s):  
Aziz Ibrahim ◽  
Edward Gucker
Keyword(s):  
Euclidean Algorithm ◽  
Matrix Transformation ◽  
Greatest Common Divisor ◽  
Repeated Application ◽  
Division Algorithm ◽  
Positive Integers

The algorithm of Euclid for finding the greatest common divisor of two positive integers is based on repeated application of the division algorithm.

scholarly journals A Note on the Computation of the Modular Inverse for Cryptography

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 116
Author(s):  
Michele Bufalo ◽  
Daniele Bufalo ◽  
Giuseppe Orlando
Keyword(s):  
Periodic Solution ◽  
Computational Cost ◽  
Euclidean Algorithm ◽  
Extended Euclidean Algorithm ◽  
Modulo Operation ◽  
Cryptographic Algorithms ◽  
Naive Algorithm ◽  
Positive Integers

In literature, there are a number of cryptographic algorithms (RSA, ElGamal, NTRU, etc.) that require multiple computations of modulo multiplicative inverses. In this paper, we describe the modulo operation and we recollect the main approaches to computing the modulus. Then, given a and n positive integers, we present the sequence (zj)j≥0, where zj=zj−1+aβj−n, a<n and GCD(a,n)=1. Regarding the above sequence, we show that it is bounded and admits a simple explicit, periodic solution. The main result is that the inverse of a modulo n is given by a−1=⌊im⌋+1 with m=n/a. The computational cost of such an index i is O(a), which is less than O(nlnn) of the Euler’s phi function. Furthermore, we suggest an algorithm for the computation of a−1 using plain multiplications instead of modular multiplications. The latter, still, has complexity O(a) versus complexity O(n) (naive algorithm) or complexity O(lnn) (extended Euclidean algorithm). Therefore, the above procedure is more convenient when a<<n (e.g., a<lnn).

Microcomputer-assisted Discoveries: Euclidean Algorithm and Continued Fractions

1983 ◽  
Vol 76 (7) ◽  
pp. 510-548
Author(s):  
Clark Kimberling
Keyword(s):  
Continued Fractions ◽  
Euclidean Algorithm ◽  
Greatest Common Divisor ◽  
Positive Integers

Students can use microcomputers to cut through algorithms and computations to gain mathematical insights. This approach is especially true for the Euclidean algorithm, so often used to find the greatest common divisor (GCD) of two positive integers. The Euclidean algorithm also yields continued fractions, at least far enough for students to find patterns and discover truths about numbers.

A Complete Classification of AI Algebras with the Ideal Property

2011 ◽  
Vol 63 (2) ◽  
pp. 381-412 ◽  
Author(s):  
Kui Ji ◽  
Chunlan Jiang
Keyword(s):  
Inductive Limit ◽  
Complete Classification ◽  
C Algebra ◽  
Ideal Property ◽  
Positive Integers ◽  
The Ideal

Abstract Let A be an AI algebra; that is, A is the C*-algebra inductive limit of a sequencewhere are [0, 1], kn, and [n, i] are positive integers. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. In this article, we give a complete classification of AI algebras with the ideal property.

Dynamical systems of type and their -algebras

2012 ◽  
Vol 33 (5) ◽  
pp. 1291-1325 ◽  
Author(s):  
PERE ARA ◽  
RUY EXEL ◽  
TAKESHI KATSURA
Keyword(s):  
Dynamical Systems ◽  
Disjoint Union ◽  
Crossed Product ◽  
Discrete Groups ◽  
Partial Action ◽  
Partial Isometries ◽  
Generating Set ◽  
C Algebra ◽  
Finite Dimensional ◽  
Positive Integers

AbstractGiven positive integers$n$and$m$, we consider dynamical systems in which (the disjoint union of)$n$copies of a topological space is homeomorphic to$m$copies of that same space. The universal such system is shown to arise naturally from the study of a C*-algebra denoted by${\cal O}_{m,n}$, which in turn is obtained as a quotient of the well-known Leavitt C*-algebra$L_{m,n}$, a process meant to transform the generating set of partial isometries of$L_{m,n}$into a tame set. Describing${\cal O}_{m,n}$as the crossed product of the universal$(m,n)$-dynamical system by a partial action of the free group$\mathbb {F}_{m+n}$, we show that${\cal O}_{m,n}$is not exact when$n$and$m$are both greater than or equal to 2, but the corresponding reduced crossed product, denoted by${\cal O}_{m,n}^r$, is shown to be exact and non-nuclear. Still under the assumption that$m,n\geq 2$, we prove that the partial action of$\mathbb {F}_{m+n}$is topologically free and that${\cal O}_{m,n}^r$satisfies property (SP) (small projections). We also show that${\cal O}_{m,n}^r$admits no finite-dimensional representations. The techniques developed to treat this system include several new results pertaining to the theory of Fell bundles over discrete groups.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

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