Partially 2-colored permutations and the Boros–Moll polynomials

We find a combinatorial setting for the coefficients of the Boros–Moll polynomials P _m (a) in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of P _m (a). This approach enables us to give a combinatorial interpretation of the log-concavity of P _m (a) which was conjectured by Moll and confirmed by Kauers and Paule.     

The ℱ-Limit of a Sequence of Prime Ideals

In this article, we introduce new topologies on the prime spectrum of a commutative ring by using the ?-limit of a sequence of prime ideals where ? is a nonprincipal ultrafilter on the natural numbers ?. These topologies are strictly finer than the Zariski's Topology, countably compact and Hausdorff. We construct 2^𝔠 new topologies on Spec(?^?) which are pairwise non-homeomorphic.     

A Lower Bound for the Differences of Powers of Linear Operators

Let T be a bounded linear operator in a Banach space, with σ（T）={1}. In 1983, Esterle-Berkani＇ s conjecture was proposed for the decay of differences （I - T） T^n as follows： Eitheror lim inf （n→∞（n＋1）｜｜（I-T）T^n｜｜≥1/e or T = I. We prove this claim and discuss some of its consequences.     

RESEARCH ANNOUNCEMENTS The Lower Bound of the Number of Input Bits to Strong n×n S-boxes

ThesecurityoftheDES-likecrytosystemsdependsheavilyonthestrengthofthesub-stitutionboxes(S-boxes)used.ThedesignofnewS-boxesisthereforeanimportantconcerninthecreationofnewandmoresecurecryptosystems.ThefullsetofdesigucriteriafortheS-boxesoftheDEShasnever...     

A Fréchet-Optimal Strengthening of the Dawson-Sankoff Lower Bound

This paper proposes a lower bound for the probability that at least one out of arbitrary events occurs. The information used consists of the first- and second- degree Bonferroni summations in conjunction with and , where is the probability that exactly one event occurs and is the probability that all events occur. We prove that the proposed bound is a Fréchet optimal lower bound, which is a criterion difficult to achieve in general. The two additional non-negative terms used in the proposed bound make it at least as good as the Dawson–Sankoff lower bound, a Fréchet optimal degree two lower bound using the first- and second- degree Bonferroni summations only. A numerical example is presented to illustrate that in some cases, the improvement can be substantial. *Part of the work of this author was done as Visiting Scholar, 2004, at the University of Sydney.     

On the Lower Bound of the Minimum Area of Convex Lattice Polygons

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Rigidity and the Lower Bound Theorem for Doubly Cohen–Macaulay Complexes

We prove that for d≥3, the 1-skeleton of any (d?1)-dimensional doubly Cohen–Macaulay (abbreviated 2-CM) complex is generically d-rigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes (Barnette, D. Isr. J. Math. 10:121–125, 1971; Barnette, D. Pac. J. Math. 46:349–354, 1973) hold for every 2-CM complex of dimension ≥2 (see Kalai, G. Invent. Math. 88:125–151, 1987). Moreover, the initial part (g ₀,g ₁,g ₂) of the g-vector of a 2-CM complex (of dimension ≥3) is an M-sequence. It was conjectured by Björner and Swartz (J. Comb. Theory Ser. A 113:1305–1320, 2006) that the entire g-vector of a 2-CM complex is an M-sequence.     

On Asymptotic and Strict Monotonicity of a Sharper Lower Bound for Student’s t Percentiles

Let z _α and t _ν,α denote the upper 100α% points of a standard normal and a Student’s t _ν distributions respectively. It is well-known that for every fixed $0<\alpha <\frac{1}{2}$ and degree of freedom ν, one has t _ν,α ?>?z _α and that t _ν,α monotonically decreases to z _α as ν increases. Recently, Mukhopadhyay (Methodol Comput Appl Probab, 2009) found a new and explicit expression b _ν (?>?1) such that t _ν,α ?>?b _ν z _α for every fixed $0<\alpha <\frac{1}{2}$ and ν. He also showed that b _ν converges to 1 as ν increases. In this short note, we prove three key results: (i) $\log(b_{\nu+1}/b_{\nu})\sim -\frac{1}{4}\nu^{-2}$ for large enough ν, (ii) b _ν strictly decreases as ν increases, and (iii) $b_{\nu}\sim 1+\frac14\nu^{-1}+\frac1{32}\nu^{-2}$ for large enough ν.     

Lower Bound for the Poles of Igusa’s p-adic Zeta Functions

Let K be a p-adic field, R the valuation ring of K, P the maximal ideal of R and q the cardinality of the residue field R/P. Let f be a polynomial over R in n >1 variables and let χ be a character of . Let M _i (u) be the number of solutions of f = u in (R/P ⁱ ) ⁿ for and. These numbers are related with Igusa’s p-adic zeta function Z _f,χ(s) of f. We explain the connection between the M _i (u) and the smallest real part of a pole of Z _f,χ(s). We also prove that M _i (u) is divisible by , where the corners indicate that we have to round up. This will imply our main result: Z _f,χ(s) has no poles with real part less than − n/2. We will also consider arbitrary K-analytic functions f.     

On a Lower Bound of the Uniform Distance in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive a lower bound of the uniform distance in the central limit theorem for real -mixing random variables under the finiteness of the eighth moments of summands. The main result of the present paper generalizes the corresponding author"s result obtained in 1997 for m-dependent random variables to the case of -mixing random variables.     

On a Lower Bound of L p Norms in the Central Limit Theorem for ϕ-Mixing Random Variables

We derive lower bounds of the L _p norms _np for all p, 1p, in the central limit theorem for -mixing random variables with finite sixth-order moments in a strictly stationary case and finite eighth-order moments in a not necessarily stationary one.     

The attractor of a parabolic–elliptic system

Following Coclite, Holden and Karlsen [G.M. Coclite, H. Holden and K.H. Karlsen, Well-posedness for a parabolic-elliptic system, Discrete Contin. Dyn. Syst. 13 (3) (2005) 659–682] and Tian and Fan [Lixin Tian, Jinling Fan, The attractor on viscosity Degasperis-Procesi equation, Nonlinear Analysis: Real World Applications, 2007], we study the dynamical behaviors of the parabolic–elliptic system

_ut+_(f(t,x,u))x+g(t,x,u)+_Px−ε_uxx=0$u_t+{\left(f(t,x,u)\right)}_x+g(t,x,u)+P_x-\epsilon u_{xx}=0$

An Unusual Application of Cramér-Rao Inequality to Prove the Attainable Lower Bound for a Ratio of Complicated Gamma Functions

Methodology and Computing in Applied Probability - A specific function f(r) involving a ratio of complicated gamma functions depending upon a real variable r(&gt;?0) is handled. Details...     

The U—property of Orlicz Sequence Spaces

LetXbeaBanachspace,X'bethedualofX,S(X)={xeX:IIxII=l},B(X)={x6X:llxIlS1}betheunitsphereandunitballofX,respectively.ForxeX,let7xdenotethesetofsupportingfunctionalsfofS(X.)atx.Inl978,LauKa-singintroducedthefollowingnotiontostudytheChebyshevsubsetofX[1j:DefinitionABanachspaceXissaidtohaveU-propertyifforanyE>O,thereexists3>osuchthatSomeoftheresultsofU-propertyin[ljaresummarizedinthefollowing.Theorem(I)IfXhasU-property,thenXisuniformlynonSquare,inparticular,Xissuper-reflexive(infactr…     

The non-gap sequence of a subcode of a generalized Reed–Solomon code

This paper addresses the question how often the square code of an arbitrary l-dimensional subcode of the code GRS _k (a, b) is exactly the code GRS_2k-1(a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first used and stated by Wieschebrink where he applied the Sidelnikov–Shestakov attack to break the Berger–Loidreau cryptosystem.     

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2~(1/m-1),2~(1/2)}, Such thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

The Brill–Noether rank of a tropical curve

We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill–Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill–Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill–Noether rank of a tropical curve with the dimension of the Brill–Noether locus of an algebraic curve.     

The convex hull of a Banach–Saks set

A subset A of a Banach space is called Banach–Saks when every sequence in A has a Cesàro convergent subsequence. Our interest here focuses on the following problem: is the convex hull of a Banach–Saks set again Banach–Saks? By means of a combinatorial argument, we show that in general the answer is negative. However, sufficient conditions are given in order to obtain a positive result.     

The Moore–Penrose inverse of a companion matrix

Necessary and sufficient conditions are given for the Moore–Penrose inverse of a companion matrix over an arbitrary ring to exist.     

The Krull–Gabriel Dimension of a Serial Ring

Abstract

We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair. Using this we calculate the Krull–Gabriel dimension of the module category over serial rings. For instance, we show that this dimension cannot be equal to 1.