On the Extension of PM-Ring

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The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

On rings asymptotically close to associative rings

The subject of this work is an extension of A. R. Kemer’s results to a rather broad class of rings close to associative rings, over a field of characteristic 0 (in particular, this class includes the varieties generated by finite-dimensional alternative and Jordan rings). We prove the finite-basedness of systems of identities (the Specht property), the representability of finitely-generated relatively free algebras, and the rationality of their Hilbert series. For this purpose, we extend the Razymslov-Zubrilin theory to Kemer polynomials. For a rather broad class of varieties, we prove Shirshov’s theorem on height.     

Scalable and Systolic Architecture for Computing Double Exponentiation Over GF(2 m )

Double-exponentiation is a crucial arithmetic operation for many cryptographic protocols. Several efficient double-exponentiation algorithms based on systolic architecture have been proposed. However, systolic architectures require large circuit space, thus increasing the cost of the protocol. This would be a drawback when designing circuits in systems requiring low cost and low power consumption. However, some cost savings can be attained by compromising speed, as in portable devices and many embedded systems. This study proposes a scalable and systolic AB ² and a scalable and systolic A × B, which are the core circuit modules of double-exponentiation. A scalable and systolic double-exponentiation can thus be obtained based on the proposed scalable AB ² and A × B architecture. Embedded system engineers may specify a target double-exponentiation with appropriate scaling systolic circuits. The proposed circuit has lower circuit space/cost and low time/propagation than other circuits.     

On Browkin''''s conjecture about the elements of order five in K_2(Q)

Based on the work of Lenstra, a succinct proof of Browkin's conjecture about the elements of order five in K2(Q) is given.     

The strength of sharply bounded induction

We prove that the sharply bounded arithmetic T⁰₂ in a language containing the function symbol ?x /2^y ? (often denoted by MSP) is equivalent to PV₁. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on the decidability of exponential terms

In this paper we prove, modulo Schanuel's Conjecture, that there are algorithms which decide if two exponential polynomials in π are equal in ? and if two exponential polynomials in π and i coincide in ?. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A tropical nullstellensatz

We suggest a version of Nullstellensatz over the tropical semiring, the real numbers equipped with operations of maximum and summation.

     

一类多项式系统极限环的研究

讨论了系统dxdt=(-y y2-dx)(1-y)a-x2(1-y)a-1dydt=x[(1-y)a (ax)a]极限环的存在性和唯一性,其中α为正数.