Some estimations for the least degree of identities of subspaces M1(m,k) (F) of the matrix superalgebra M(m,k)(F)

We estimate the least degree of identities of subspaces M ₁^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over the field F for arbitrary m and k. For subspaces M ₁^(m,1) (F) (m≥1) and M ₁^(2,2) (F) we obtain concrete minimal identities.     

The polynomial Daugavet property for atomless L 1(μ)-spaces

For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial ${P:L_1(\mu)\longrightarrow L_1(\mu)}For any atomless positive measure μ, the space L ₁(μ) has the polynomial Daugavet property, i.e., every weakly compact continuous polynomial P:L₁(m)? L₁(m){P:L_1(\mu)\longrightarrow L_1(\mu)} satisfies the Daugavet equation ||Id + P||=1 + ||P||{\|{\rm Id} + P\|=1 + \|P\|}. The same is true for the vector-valued spaces L ₁(μ, E), μ atomless, E arbitrary.     

Irreducibility criteria of Schur-type and Pólya-type

Let f(x)=(x-a₁)?(x-a_m){f(x)=(x-a_1)\cdots (x-a_m)}, where a ₁, . . . , a _m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether (f(x))^2k+1{(f(x))^{2^k}+1} is irreducible for every k ≥ 1. In 1919 Pólya proved that if P(x) ? \mathbbZ[x]{P(x)\in\mathbb{Z}[x]} is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2^−N N! where N=ém/2ù{N=\lceil m/2\rceil}, then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

A q-analogue of the FKG inequality and some applications

Let L be a finite distributive lattice and μ: L → ℝ⁺ a log-supermodular function. For functions k: L → ℝ⁺ let

Parameterizing above Guaranteed Values: MaxSat and MaxCut

In this paper we investigate the parameterized complexity of the problems MaxSat and MaxCut using the framework developed by Downey and Fellows. LetGbe an arbitrary graph havingnvertices andmedges, and letfbe an arbitrary CNF formula withmclauses onnvariables. We improve Cai and Chen'sO(2^2ckcm) time algorithm for determining if at leastkclauses of ac-CNF formulafcan be satisfied; our algorithm runs inO(|f| + k²φ^k) time for arbitrary formulae and inO(cm + ckφ^k) time forc-CNF formulae, where φ is the golden ratio . We also give an algorithm for finding a cut of size at leastk; our algorithm runs inO(m + n + k4^k) time. We then argue that the standard parameterization of these problems is unsuitable, because nontrivial situations arise only for large parameter values (k ≥ m/2), in which range the fixed-parameter tractable algorithms are infeasible. A more meaningful question in the parameterized setting is to ask whether m/2 + kclauses can be satisfied, or m/2 + kedges can be placed in a cut. We show that these problems remain fixed-parameter tractable even under this parameterization. Furthermore, for up to logarithmic values of the parameter, our algorithms for these versions also run in polynomial time.     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.     

Differential operators associated with zonal polynomials. II

Summary LetC _κ(S) be the zonal polynomial of the symmetricm×m matrixS=(s_ij), corresponding to the partition κ of the non-negative integerk. If ∂/∂S is them×m matrix of differential operators with (i, j)th entry ((1+δ_ij)∂/∂s_ij)/2, δ being Kronecker's delta, we show that C_k(∂/∂S)C_λ(S)=k!δ_λkC_k(I), where λ is a partition ofk. This is used to obtain new orthogonality relations for the zonal polynomials, and to derive expressions for the coefficients in the zonal polynomial expansion of homogenous symmetric polynomials.     

Approximation ofk-Monotone Functions

It is shown that an algebraic polynomial of degree k−1 which interpolates ak-monotone functionfatkpoints, sufficiently approximates it, even if the points of interpolation are close to each other. It is well known that this result is not true in general for non-k-monotone functions. As an application, we prove a (positive) result on simultaneous approximation of ak-monotone function and its derivatives inL_p, 0<p<1, metric, and also show that the rate of the best algebraic approximation ofk-monotone functions (with bounded (k−2)nd derivatives inL_p, 1<p<∞, iso(n^−k/p).     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Representation of polynomials as products of two values of a quadratic form

Let k be a field, char k ≠ 2, p(t) a monic irreducible polynomial over k, and k _p = k[t]/pk[t] the corresponding residue field. For a regular quadratic form φ over k we investigate the relationship between two conditions:

Parallel Itoh–Tsujii multiplicative inversion algorithm for a special class of trinomials

In this contribution, we derive a novel parallel formulation of the standard Itoh–Tsujii algorithm for multiplicative inverse computation over the field GF(2 ^m ). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best performance when using a special class of irreducible trinomials, namely, P(x) = x ^m  + x ^k  + 1, with m and k odd numbers and when implemented in hardware platforms. Under these conditions, our experimental results show that our parallel version of the Itoh–Tsujii algorithm yields a speedup of about 30% when compared with the standard version of it. Implemented in a Virtex 3200E FPGA device, our design is able to compute multiplicative inversion over GF(2¹⁹³) after 20 clock cycles in about 0.94 μS.      

On Four-Connecting a Triconnected Graph

We consider the problem of finding a smallest set of edges whose addition four-connects a triconnected graph. This is a fundamental graph-theoretic problem that has applications in designing reliable networks and improving statistical database security. We present an O(n · α(m, n) + m)-time algorithm for four-connecting an undirected graph G that is triconnected by adding the smallest number of edges, where n and m are the number of vertices and edges in G, respectively, and α(m, n) is the inverse Ackermann function. This is the first polynomial time algorithm to solve this problem exactly.In deriving our algorithm, we present a new lower bound for the number of edges needed to four-connect a triconnected graph. The form of this lower bound is different from the form of the lower bound known for biconnectivity augmentation and triconnectivity augmentation. Our new lower bound applies for arbitrary k and gives a tighter lower bound than the one known earlier for the number of edges needed to k-connect a (k − 1)-connected graph. For k = 4, we show that this lower bound is tight by giving an efficient algorithm to find a set of edges whose size equals the new lower bound and whose addition four-connects the input triconnected graph.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

Zagier's conjecture on L(E,2)

In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight two parts of the groups K ₂(E) and K ₁(E) for an elliptic curve E over an arbitrary field k. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on L(E,2) for modular elliptic curves over ℚ. Oblatum 3-VI-1996 & 16-V-1997     

Description of polygonal regions by polynomials of bounded degree

We show that every (possibly unbounded) convex polygon P in \mathbbR²{\mathbb{R}^2} with m edges can be represented by inequalities p ₁ ≥ 0, . . ., p _n ≥ 0, where the p _i ’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m, k) ￡ n(m, k) ￡ (1+e_m) s(m, k){s(m, k) \leq n(m, k) \leq (1+\varepsilon_m) s(m, k)} with s(m,k) ≔ max {m/k, log₂ m} and e_m ? 0{\varepsilon_m \rightarrow 0} as m ? ￥{m \rightarrow \infty}. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p ₁ > 0, . . ., p _n >  0 is also given. For k ≤ m/log₂ m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k.     

Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Computing the Top Betti Numbers of Semialgebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any ℓ>0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P ₁≤0,…,P _s ≤0, where each P _i ∈R[X ₁,…,X _k ] has degree≤2, and computes the top ℓ Betti numbers of S, b _k−1(S),…,b _k−ℓ (S), in polynomial time. The complexity of the algorithm, stated more precisely, is . For fixed ℓ, the complexity of the algorithm can be expressed as , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic sets in R ^k defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain, by letting ℓ=k, an algorithm for computing all the Betti numbers of S whose complexity is . An erratum to this article can be found at     

Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form f([`(x)]) = ?<sub>i = 1</sub><sup>k</sup> h<sub>i</sub> ([`(x)]) ·g_i ([`(x)])f(\bar x) = \sum\nolimits_{i = 1}^k {h_i (\bar x) \cdot g_i (\bar x)} , where each h _i is a polynomial that depends on only ρ linear functions, and each g _i is a product of linear functions (when h _i = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When max _i (deg(h _i · g _i )) = d we say that f is computable by a ΣΠΣ(k; d;ρ) circuit. We obtain the following results.