On the existence of a minimum integer representation for?weighted voting systems

A basic problem in the theory of simple games and other fields is to study whether a simple game (Boolean function) is weighted (linearly separable). A second related problem consists in studying whether a weighted game has a minimum integer realization. In this paper we simultaneously analyze both problems by using linear programming. For less than 9 voters, we find that there are 154 weighted games without minimum integer realization, but all of them have minimum normalized realization. Isbell in 1958 was the first to find a weighted game without a minimum normalized realization, he needed to consider 12 voters to construct a game with such a property. The main result of this work proves the existence of weighted games with this property with less than 12 voters. This research was partially supported by Grant MTM 2006-06064 of “Ministerio de Ciencia y Tecnología y el Fondo Europeo de Desarrollo Regional” and SGRC 2005-00651 of “Generalitat de Catalunya”, and by the Spanish “Ministerio de Ciencia y Tecnología” programmes ALINEX (TIN2005-05446 and TIN2006-11345).     

A note on Ramanujan’s identities involving the hypergeometric function {_{2}}F_{1}(\frac{1}{6},\frac{5}{6};1;z)

We study a class of elliptic functions associated with the hypergeometric function ${_{2}}F_{1}(\frac{1}{6},\frac{5}{6};1;z)$ . From the perspective of the properties of conformal mappings and differential equations, we provide new insight into a set of identities of Ramanujan associated with the above hypergeometric function.     

Fibers of the Baum-Bott map for foliations of degree two on ?2

The Baum-Bott map associates to a foliation the Baum-Bott indexes of their singularities. In this paper we study the fibers of the Baum-Bott map in the space of foliations of degree two on the projective plane ℙ². In the main result we prove that its generic fiber contains exactly 240 orbits of the natural action of Aut(ℙ²) onthespace of foliations.     

On the equivalence of the centered Gaussian measure in L 2 with the correlation operator (−d 2/dx 2)−1 and the conditional Wiener measure

Let w and µ be respectively the conditional Wiener measure in C ₀([0, 1]) and the centered Gaussian measure in L ₂[0, 1] with the correlation operator (?d ²/dx ²)^?1. We prove the equivalence of these two measures in the following sense: for any Borel set A ? L ₂[0, 1] the set A ∩ C ₀([0, 1]) is a Borel subset of C ₀([0, 1]) and µ(A) = w(A∩C ₀([0, 1])).     

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index {{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}

Let B?=?(B ₁(t), . . . ,B _d (t)) be a d-dimensional fractional Brownian motion with Hurst index ???<?1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low H?lder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using ??standard?? tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, 2011), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d??emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, 2011).     

On some entire modular forms of weights\tfrac{7}{2} and\tfrac{9}{2} for the congruence group Γ0(4N)for the congruence group Γ0(4N)

Entire modular forms of weights $\tfrac{7}{2}$ and $\tfrac{9}{2}$ for the congruence group Γ₀(4N) are constructed, which will be useful for revealing the arithmetical sense of additional terms in formulas for the number of representations of positive integers by quadratic forms in 7 and 9 variables.     

On the collection of spectral multiplicities {2, 4, …, 2 n } for totally ergodic ℤ2-actions

The paper is devoted to the realization of collections of spectral multiplicities for ergodic ?²-actions. Sufficient conditions ensuring the realizability of multiplicities of the form {2, 4, …, 2 ⁿ } are given.     

On an estimate,uniform on the semiaxis t ≥ 0, for the rate of convergence of galerkin approximations for the equations of motion of Kelvin-Voight fluids

An estimate, uniform on the semiaxis t 0, is obtained for the rate of convergence of spectral Galerkin approximations for the solutions of an initial-boundary value problem for the Eqs. (1), (2) of motion of Kelvin-Voight fluids. Namely, under the condition that the solution v is asymptotically stable for t in the norm of the Dirichlet integral.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 182, pp. 123–130, 1990.     

A necessary and sufficient condition for the biorthogonality of {ρ 1,ρ 2}

In this paper, we present a necessary and sufficient condition for the biorthogonality of a class of special functionsρ ₁ andρ ₂. The functions are useful in the theory of biorthogonal wavelet.     

Gradient Estimates and Harnack Inequalities for Positive Solutions of $$\mathfrak{L}u=\frac{\partial u}{\partial t}$$ L u = ∂ u ∂ t on Self-shrinkers

In this paper, we investigate the positive solutions of Lu=∂u/∂t on self-shrinkers, then get some gradient estimates and Harnack inequalities for the positive solutions.     

Knaster’s Problem for (Z 2) k -Symmetric Subsets of the Sphere S^{2^{k}-1}

We prove a result about partitioning an absolutely continuous measure in ℝ ^d into 2d equal parts by a system of cones with common vertex, where d is an odd prime power. The proof is topological and based on the calculation of the equivariant Euler class of a certain vector bundle.     

Lattice-theoretic characterizations of the group classes {\mathfrak{N}^{k-1}\mathfrak{A}} and {\mathfrak{N}^k} for k?�� 2

Let ${2\leq k\in \mathbb{N}}$ . Recently, Costantini and Zacher obtained a lattice-theoretic characterization of the classes ${\mathfrak{N}^k}$ of finite soluble groups with nilpotent length at most k. It is the aim of this paper to give a lattice-theoretic characterization of the classes ${\mathfrak{N}^{k-1}\mathfrak{A}}$ of finite groups with commutator subgroup in ${\mathfrak{N}^{k-1}}$ ; in addition, our method also yields a new characterization of the classes ${\mathfrak{N}^k}$ . The main idea of our approach is to use two well-known theorems of Gaschütz on the Frattini and Fitting subgroups of finite groups.     

On q-analogs of weight multiplicities for the Lie superalgebras \mathfrak{gl}(n,m) and \mathfrak{spo(}2n,M)

The paper is devoted to the generalization of Lusztig’s q-analog of weight multiplicities to the Lie superalgebras $\mathfrak{gl}(n,m)$ and $\mathfrak{spo(}2n,M).$ We define such q-analogs K _λ,μ (q) for the typical modules and for the irreducible covariant tensor $\mathfrak{gl}(n,m)$ -modules of highest weight λ. For $\mathfrak{gl}(n,m),$ the defined polynomials have nonnegative integer coefficients if the weight μ is dominant. For $\mathfrak{spo(}2n,M)$ , we show that the positivity property holds when μ is dominant and sufficiently far from a specific wall of the fundamental chamber. We also establish that the q-analog associated to an irreducible covariant tensor $\mathfrak{gl}(n,m)$ -module of highest weight λ and a dominant weight μ is the generating series of a simple statistic on the set of semistandard hook-tableaux of shape λ and weight μ. This statistic can be regarded as a super analog of the charge statistic defined by Lascoux and Schützenberger.     

The existence of (2 × c, λ) grid-block designs with {c\in \{3, 4, 5\}} and λ ≥ 1

Let X be a v-set and ${\mathcal{B}}$ a collection of r × c arrays with elements in X. Two elements of X are collinear if they are on the same grid line (row or column). A pair ${(X, \mathcal{B})}$ is called an (r × c, λ) grid-block design if every two distinct elements in X are collinear exactly λ times in the arrays of ${\mathcal{B}}$ . This design has absorbed much attention due to its use in DNA library screening. In this paper, we prove that the necessary conditions for the existence of (2 × c, λ) grid-block designs of order v with ${c\in \{3, 4, 5\}}$ and any integer λ ≥ 1 are also sufficient.     

Zolotarev's first problem — the best approximation by polynomials of degree ≤n − 2 tox n − nσx n−1 in [−1, 1]

Chebyshev determined $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n + a_1 x^{n - 1} + \cdots + a_n |$$ as 2^1?n , which is attained when the polynomial is 2^1?n T _n(x), whereT _n(x) = cos(n arc cosx). Zolotarev's First Problem is to determine $$\mathop {\min }\limits_{(a)} \mathop {\max }\limits_{ - 1 \le x \le 1} |x^n - n\sigma x^{n - 1} + a_2 x^{n - 2} + \cdots + a_n |$$ as a function ofn and the parameter σ and to find the extremal polynomials. He solved this in 1878. Another discussion was given by Achieser in 1928, and another by Erdös and Szegö in 1942. The case when 0≤|σ|≤ tan²(π/2n) is quite simple, but that for |σ|> tan²(π/2n) is quite different and very complicated. We give two new versions of the proof and discuss the change in character of the solution. Both make use of the Equal Ripple Theorem.     

On the primitive divisors of the recurrent sequence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1) with applications to group theory

Consider the sequence of algebraic integers un given by the starting values u0=0,u1=1 and the recurrence u_(n+1)=(4cos~2(2π/7)-1)u_n-u_(n-1).We prove that for any n ■{1,2,3,5,8,12,18,28,30}the n-th term of the sequence has a primitive divisor in Z[2 cos(2π/7)].As a consequence we deduce that for any sufficiently large n there exists a prime power q such that the groupcan be generated by a pair x,y with χ~2=y~3=(xy)~7=1 and the order of the commutator[x,y]is exactly n.The latter result answers in affirmative a question of Holt and Plesken.     

Bounds on the maximum numbers of clear two-factor interactions for 2~((n1+n2)-(k1+k2)) fractional factorial split-plot designs

Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures. There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors, which can form three types of two-factor interactions: WP2fi, WS2fi and SP2fi. This paper considers FFSP designs with resolutionⅢorⅣunder the clear effects criterion. It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs, and gives some methods for constructing the desired FFSP designs. It further examines the performance of the construction methods.     

Coincidence of the spaces\mathop {J_2^1 }\limits^ \circ (\Omega ) and for plane domains Ω, having exits at infinity

For a large class of plane domains Ω, having exits at infinity, one shows the coincidence of the spaces of solenoidal vector fields \(\mathop {J_2^1 }\limits^ \circ (\Omega )\) and , which play an important role in the investigation of initial-houndary-value problems for the Navier-Stokes equations.