Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ ₁(w), . . . , Δ _r (w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δ _k (w)/Δ _k-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.     

On a function associated with the Bessel functions

Summary Defining the function Δn, 1,k;x(J) asΔn, 1,k;x(J)=J _n+1(x)−J _n(x)J _n+k+1(x) associated with the Bessel functionJ _n(x), we derive a series of products of Bessel functions for Δ_{n, f, k, x} (J). Whenk=1,k;x (J) becomes Turàn expression for Bessel functions. Some consequences have been pointed out.

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Riassunto Definita la Δ_{n, f, k, x} (J) come Δ_{n, f, k, x}, (J)=J _n+1(x)J _n+k(x)-J _n(_n+k+1)(x) associata alla funzioneJ _n(x) di Bessel, si ricava una serie di prodotti di funzioni di Bessel per Δ_{n, f, k, x}, (J). 3 Quandok=1, Δ_{n, f, k, x}, (J) diventa una espressione di Turàn per le funzioni di 2 Bessel, vengono inoltre indicate alcune altre conseguenze.

     

On some generalized difference paranormed sequence spaces associated with multiplier sequence defined by modulus function

In this article we introduce the paranormed sequence spaces(f,Λ,△m,p),c0(f,Λ,△m,p) and ■∞(f,Λ,△m,p),associated with the multiplier sequence Λ =(λk),defined by a modulus function f.We study their different properties like solidness,symmetricity,completeness etc.and prove some inclusion results.     

Upper bound theorems for homology manifolds

In this paper we prove the Upper Bound Conjecture (UBC) for some classes of (simplicial) homology manifolds: we show that the UBC holds for all odd-dimensional homology manifolds and for all 2k-dimensional homology manifolds Δ such that β _k (Δ)⩽Σ{β _i (Δ):i ≠k-2,k,k+2 and 1 ⩽i⩽2k-1}, where β _i (Δ) are reduced Betti numbers of Δ. (This condition is satisfied by 2k-dimensional homology manifolds with Euler characteristic χ≤2 whenk is even or χ≥2 whenk is odd, and for those having vanishing middle homology.) We prove an analog of the UBC for all other even-dimensional homology manifolds. Kuhnel conjectured that for every 2k-dimensional combinatorial manifold withn vertices, . We prove this conjecture for all 2k-dimensional homology manifolds withn vertices, wheren≥4k+3 orn≤3k+3. We also obtain upper bounds on the (weighted) sum of the Betti numbers of odd-dimensional homology manifolds.     

Equipartition of mass distributions by hyperplanes

We consider the problem of determining the smallest dimensiond=Δ(j, k) such that, for anyj mass distributions inR ^d , there arek hyperplanes so that each orthant contains a fraction 1/2 ^k of each of the masses. The case Δ(1,2)=2 is very well known. The casek=1 is answered by the ham-sandwich theorem with Δ(j, 1)=j. By using mass distributions on the moment curve the lower bound Δ(j, k)≥j(2 ^k −1)/k is obtained. We believe this is a tight bound. However, the only general upper bound that we know is Δ(j, k)≤j2 ^k−1. We are able to prove that Δ(j, k)=⌈j(^2k−1/k⌉ for a few pairs (j, k) ((j, 2) forj=3 andj=2 ⁿ withn≥0, and (2, 3)), and obtain some nontrivial bounds in other cases. As an intermediate result of independent interest we prove a Borsuk-Ulam-type theorem on a product of balls. The motivation for this work was to determine Δ(1, 4) (the only case forj=1 in which it is not known whether Δ(1,k)=k); unfortunately the approach fails to give an answer in this case (but we can show Δ(1, 4)≤5). This research was supported by the National Science Foundation under Grant CCR-9118874.     

Some new generalized difference sequence spaces defined by a sequence of moduli

The idea of difference sequence sets X( ) = {x = (x k ) : x ∈ X} with X = l ∞ , c and c 0 was introduced by Kizmaz [12]. In this paper, using a sequence of moduli we define some generalized difference sequence spaces and give some inclusion relations.     

Strongly almost convergent generalized difference sequences associated with multiplier sequences

Let Λ = (λ _k ) be a sequence of non-zero complex numbers. In this paper we introduce the strongly almost convergent generalized difference sequence spaces associated with multiplier sequences i.e. w ₀[A,Δ ^m ,Λ,p], w ₁[A,Λ ^m ,Λ,p], w _∞[A,Δ ^m ,Λ,p] and study their different properties. We also introduce Δ _Λ ^m -statistically convergent sequences and give some inclusion relations between w ₁[Δ ^m ,λ,p] convergence and Δ _Λ ^m -statistical convergence. Communicated by Pavel Kostyrko     

NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach

In this paper we investigate the problem of testing the coherence of an assessment of conditional probability following a purely logical setting. In particular we will prove that the coherence of an assessment of conditional probability χ can be characterized by means of the logical consistency of a suitable theory T _χ defined on the modal-fuzzy logic FP _k (RŁ_Δ) built up over the many-valued logic RŁ_Δ. Such modal-fuzzy logic was previously introduced in Flaminio (Lecture Notes in Computer Science, vol. 3571, 2005) in order to treat conditional probability by means of a list of simple probabilities following the well known (smart) ideas exposed by Halpern (Proceedings of the eighth conference on theoretical aspects of rationality and knowledge, pp 17–30, 2001) and by Coletti and Scozzafava (Trends Logic 15, 2002). Roughly speaking, such logic is obtained by adding to the language of RŁ_Δ a list of k modalities for “probably” and axioms reflecting the properties of simple probability measures. Moreover we prove that the satisfiability problem for modal formulas of FP _k (RŁ_Δ) is NP-complete. Finally, as main result of this paper, we prove FP _k (RŁ_Δ) in order to prove that the problem of establishing the coherence of rational assessments of conditional probability is NP-complete.      

Computation of betti numbers of monomial ideals associated with cyclic polytopes

We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ringk[Δ(P)]=A/I _Δ(P) of the boundary complex Δ(P) of an odd-dimensional cyclic polytopePover a fieldk. A corollary to the formula is that the Betti number sequence ofk[Δ(P)] is unimodal and does not depend on the base fieldk.     

Functional equations for Hecke-Maaass series

The Dirichlet (Hecke-Maass) series associated with the eigenfuctionsf andg of the invariant differential operator Δ_k=−y²(∂²/∂x²)+iky∂/∂x of weightk are investigated. It is proved that any relation of the form (f/kM)=g for thek-action of the groupSL ₂ SL ₂(ℝ) is equivalent to a pair of functional equations relating the Hecke-Maass series forf andg and involving only traditional gamma factors. This work was supported by the Russian Foundation for Basic Research (grant No. 96-01-10439). Institute of Applied Mathematics, Far East Division of Russian Academy of Sciences. Translated from Funktional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 23–32, April–June, 2000. Translated by V. M. Volosov     

Distributions of Points in <Emphasis Type="Italic">d</Emphasis> Dimensions and Large <Emphasis Type="Italic">k</Emphasis>-Point Simplices

We consider a variant of Heilbronn’s triangle problem by investigating for a fixed dimension d≥2 and for integers k≥2 with k≤d distributions of n points in the d-dimensional unit cube [0,1] ^d , such that the minimum volume of the simplices, which are determined by (k+1) of these n points is as large as possible. Denoting by Δ _k,d (n), the supremum of this minimum volume over all distributions of n points in [0,1] ^d , we show that c _k,d ⋅(log n)^1/(d−k+1)/n ^k/(d−k+1)≤Δ _k,d (n)≤c _k,d ′/n ^k/d for fixed 2≤k≤d, and, moreover, for odd integers k≥1, we show the upper bound Δ _k,d (n)≤c _k,d ″/n ^{k/d+(k−1)/(2d(d−1))}, where c _k,d ,c _k,d ′,c _k,d ″>0 are constants. A preliminary version of this paper appeared in COCOON ’05.     

On the series ∑ k=1 ∞ ( k 3k )−1 k −n x k

In this paper we investigate the series ∑ _k=1 ^∞ ( _k ^3k )⁻¹ k ⁻ⁿ x ^k . Obtaining some integral representations of them, we evaluated the sum of them explicitly forn = 0, 1, 2.     

Centers of F-purity

In this paper, we study a positive characteristic analogue of the centers of log canonicity of a pair (R, Δ). We call these analogues centers of F-purity. We prove positive characteristic analogues of subadjunction-like results, prove new stronger subadjunction-like results, and in some cases, lift these new results to characteristic zero. Using a generalization of centers of F-purity which we call uniformly F-compatible ideals, we give a characterization of the test ideal (which unifies several previous characterizations). Finally, in the case that Δ = 0, we show that uniformly F-compatible ideals coincide with the annihilators of the F(E_R(k)){\mathcal{F}(E_R(k))} -submodules of E _R (k) as defined by Lyubeznik and Smith.     

List total colorings of planar graphs without triangles at small distance

Suppose that G is a planar graph with maximum degree Δ. In this paper it is proved that G is total-(Δ + 2)-choosable if (1) Δ ≥ 7 and G has no adjacent triangles (i.e., no two triangles are incident with a common edge); or (2) Δ ≥ 6 and G has no intersecting triangles (i.e., no two triangles are incident with a common vertex); or (3) Δ ≥ 5, G has no adjacent triangles and G has no k-cycles for some integer k ∈ {5, 6}.     

A generalization of Cohen- Eisenstein series and Shimura liftings and some congruences between Cusp forms and Eisenstein series

In this paper we introduce some modular forms of half-integral weight on congruence group Г_o(4N) withN an odd positive integer which can be viewed as a natural generalization of Cohen-Eisenstein series. Using these series, we can prove that the restriction of Shimura lifting on Eisenstein spaceE _k+1/2 ⁺ (4N,χl) gives an isomorphism fromE _k+1/2 ⁺ (4N,χl) toE _2k(N). We consider some congruence relationships between modular forms in use of Shimura lifting.     

On the general additive divisor problem

We obtain a new upper bound for the sum Σ _h≤H Δ _k (N, h) when 1 ≤ H ≤ N, k ∈ ℕ, k ≥ 3, where Δ _k (N, h) is the (expected) error term in the asymptotic formula for Σ _N<n≤2N d _k (n)d _k (n + h), and d _k (n) is the divisor function generated by ζ(s) ^k . When k = 3, the result improves, for H ≥ N ^1/2, the bound given in a recent work of Baier, Browning, Marasingha and Zhao, who dealt with the case k = 3.     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

A new method for smoothing surfaces and computing hermite interpolants

Let Ω be a rectangular bounded domain of a plane equipped with a rectangular partition Δ. Assume a piecewise bivariate function that is differentiable up to order (k,l) except at the knots of Δ, where it is less differentiable. In this paper, we introduce a new method for smoothing the above function at the knots. More precisely, we describe algorithms allowing one to transform it into another function that will be differentiable up to order (k,l) in the whole domain Ω. Then, as an application of this method, we give a recursive computation of tensor product Hermite spline interpolants. To illustrate our results, some numerical examples are presented. AMS subject classification (2000)  41A05, 41A15, 65D05, 65D07, 65D10     

Edge choosability of planar graphs without short cycles

In this paper we prove that if G is a planar graph with △= 5 and without 4-cycles or 6-cycles, then G is edge-6-choosable. This consequence together with known results show that, for each fixed k ∈{3,4,5,6}, a k-cycle-free planar graph G is edge-(△ 1)-choosable, where △ denotes the maximum degree of G.     

One class of singular complex-valued random variables of the Jessen-Wintner type

We study the structure of the distribution of a complex-valued random variable ξ = Σa _k ξ _k , where ξ _k are independent complex-valued random variables with discrete distribution and a _k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1653–1660, December, 1997.