Strong Minimal Covers for Recursively Enumerable Degrees

We prove that there exists a nonzero recursively enumerable Turing degree possessing a strong minimal cover. Mathematics Subject Classification: 03D30.     

Arithmetical rank of squarefree monomial ideals of small arithmetic degree

In this paper, we prove that the arithmetical rank of a squarefree monomial ideal I is equal to the projective dimension of R/I in the following cases: (a) I is an almost complete intersection; (b) arithdeg I=reg I; (c) arithdeg I=indeg I+1. We also classify all almost complete intersection squarefree monomial ideals in terms of hypergraphs, and use this classification in the proof in case (c).     

Arithmetic duality for two-dimensional local rings with perfect residue field

We give a refinement of Saito's arithmetic duality for two-dimensional local rings by giving algebraic group structures for arithmetic cohomology groups.     

New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.     

Matlis Duals of Top Local Cohomology Modules and the Arithmetic Rank of an Ideal

Let I be an ideal of a local ring (R, 𝔪). Using local cohomology, we present new criteria (see 1.4, respectively 1.5) for the conditions ara (I) ≤ 1 respectively ara (I) ≤ 2, where ara (I) stands for the number of generators of I up to radical. Though this works equally well for the local and for the graded case, we show some subtle differences between the local and the graded situation in Section 2. Finally, in Section 3, we show that the Matlis dual of certain local cohomology modules, though not finite, is well behaved in some sense.     

Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

Abstract We prove that there are non-recursive r.e. sets A and C with A < _T C such that for every set . Both authors are supported by “863” and the National Science Foundation of China     

Spheres arising from multicomplexes

In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex Δ on the vertex set V with Δ≠V², the deleted join of Δ with its Alexander dual Δ^∨ is a combinatorial sphere. In this paper, we extend Bier?s construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.     

Sub-Ramsey Numbers for Arithmetic Progressions

Let the integers 1, . . . ,n be assigned colors. Szemerédi's theorem implies that if there is a dense color class then there is an arithmetic progression of length three in that color. We study the conditions on the color classes forcing totally multicolored arithmetic progressions of length 3. Let f(n) be the smallest integer k such that there is a coloring of {1, . . . ,n} without totally multicolored arithmetic progressions of length three and such that each color appears on at most k integers. We provide an exact value for f(n) when n is sufficiently large, and all extremal colorings. In particular, we show that f(n)=8n/17+O(1). This completely answers a question of Alon, Caro and Tuza.     

Analysis of Bounds for Multilinear Functions

We analyze four bounding schemes for multilinear functions and theoretically compare their tightness. We prove that one of the four schemes provides the convex envelope and that two schemes provide the concave envelope for the product of p variables over .     

算术半格的信息系统表示

本文给出了算术信息系统的概念,证明了算术信息系统是算术半格的表示。基于算术信息系统之间的逼近映射,我们得到了算术信息系统范畴和算术半格范畴之间的范畴等价。     

Estimation of Genus for Certain Arithmetic Groups

Abstract

We find the genera of arithmetic groups ?Γ₁(N), Φ? for N ≥ 1 and Φ the Fricke involution, from which we derive that the genus g(?Γ₁(N), Φ?) is zero if and only if 1 ≤ N ≤ 12 and N = 14, 15.     

论数值修约

数值修约的已有定义是从其现象来定义的。现在提出的定义是从数值修约的本质(用加减法使尾数化零)来定义的。因而按定义可直接推导得到种种结论。并可以此来定义各种数值修约规则。对一个数值连续修约的可行性作了解释。对修约间隔亦有论述。本文提出了五轮换规则的想法。关于两个数值修约规则的中文名称,主要是指出它们用常量(四、五、六)表示变量,因而产生逻辑上的错误。本文也讨论产生错误的原因,并同时指出其他错误。新名称的提出已不犯原有错误。     

Arithmetic structure in sparse difference sets

Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration {v₁,…,v?} of vectors in Zd, we show that if A⊂d[1,N] with |A|/Nd?CN−1/?, then there necessarily exists r≠0 such that {rv₁,…,rv?}⊆A−A.     

Arithmetic progressions on Pell equations

In this paper we consider arithmetic progressions on Pell equations, i.e. integral solutions (X,Y) whose X-coordinates or Y-coordinates are in arithmetic progression.     

Sharp Upper and Lower Bounds for Basket Options

Given a basket option on two or more assets in a one‐period static hedging setting, the paper considers the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. Sharp upper bounds are derived for the general n‐asset case and sharp lower bounds for the two‐asset case, both in closed forms, of the price of the basket option. In the case n = 2 examples are given of discrete distributions attaining the bounds. Hedge ratios are also derived for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves.     

Error Bounds for Polynomial Evaluation and Complex Arithmetic

Explicit and computable strict bounds of a posteriori type areconstructed for the evaluation of a polynomial and its derivativesby nested multiplication, using floating-point arithmetic. Thepolynomial may be real or complex, and may contain errors inits coefficients and argument. Some new error bounds for arithmetic operations with complexnumbers are included.     

Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one-dimensional groups, and analyze their spaces of solutions.     

An unexpected separation result in Linearly Bounded Arithmetic

The theories Sⁱ₁(α) and Tⁱ₁(α) are the analogues of Buss' relativized bounded arithmetic theories in the language where every term is bounded by a polynomial, and thus all definable functions grow linearly in length. For every i, a Σ^b_i+1(α)‐formula TOPⁱ(a), which expresses a form of the total ordering principle, is exhibited that is provable in Sⁱ⁺¹₁ (α), but unprovable in Tⁱ₁(α). This is in contrast with the classical situation, where Sⁱ⁺¹₂ is conservative over Tⁱ₂ w. r. t. Σ^b_i+1‐sentences. The independence results are proved by translations into propositional logic, and using lower bounds for corresponding propositional proof systems. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)