where the a,are r fixed nonzero elements of Z, and 0<«i(a) <Ui(a) (i = 2, ■ ■ • , r), then A=Z. In [3, Theorem 11; 5; 2; l] specialized forms of (1) (e.g. cn(o) £Z, an(-a) —aEZ) are shown to imply commutativity at least for semi-simple rings. It is natural therefore to seek an extension of Nakayama's result to semi-simple rings. Since a semi-simple ring is a subdirect sum of primitive rings and...

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for l...

We prove that the binary complexity of solving ordinary polynomial differential equations in terms of Puiseux series is single exponential in the number of terms in the series. Such a bound was given by Grigoriev [10] for Riccatti differential polynomials associated to ordinary linear differential operators. In this paper, we get the same bound for arbitrary differential polynomials. The algori...

Explicit expressions for restricted partition function W (s,d) and its quasiperiodic components Wj(s,d ) (called Sylvester waves) for a set of positive integers d = {d1, d2, . . . , dm} are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is establis...

A normal complex algebraic variety X is called a symplectic variety (cf. [Be]) if its regular locus Xreg admits a holomorphic symplectic 2-form ω such that it extends to a holomorphic 2-form on a resolution f : X̃ → X. Affine symplectic varieties are constructed in various ways such as nilpotent orbit closures of a semisimple complex Lie algebra (cf. [CM]), Slodowy slices to nilpotent orbits (cf...

Several open problems related to the behavior of the monoid of effective divisors and the nef cone for smooth projective surfaces over an algebraically closed field are discussed, motivating and putting into historical context concepts such as Mori dream spaces, Seshadri constants and the resurgence of homogeneous ideals in polynomial rings. Some recent work on these topics is discussed along w...

The independence equivalence class of a graph $G$ is the set graphs that have same polynomial as $G$. A whose contains only itself, up to isomorphism, unique. Beaton, Brown and Cameron (2019) showed paths with an odd number vertices are unique raised problem finding even vertices. completely solved in this paper.

We make progress on some questions related to polynomial approximations of AC0. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. 6th CCC 1991), that any AC0 circuit of size s and depth d has an ε-error probabilistic polynomial over the reals of degree (log(s/ε))O(d). We improve this upper bound to (log s)O(d) · log(1/ε), which is much better ...

the topological index of a graph g is a numeric quantity related to g which is invariant underautomorphisms of g. the vertex pi polynomial is defined as piv (g)  euv nu (e)  nv (e).then omega polynomial (g,x) for counting qoc strips in g is defined as (g,x) =cm(g,c)xc with m(g,c) being the number of strips of length c. in this paper, a new infiniteclass of fullerenes is constructed. the ...

Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, European J. Combin. 32 (2011) 954–974] introduced the subgraph component polynomial Q(G;x, y) of a graph G, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying gra...