Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

Spin depolarization decay rates in α-symmetric stable fields on cubic lattices

We study the asymptotic, long-time behavior of the energy function where {X_s : 0 ≤ s < ∞} is the standard random walk on the d-dimensional lattice Z^d, 1 < α ≤ 2, and f:R⁺ → R⁺ is any nondecreasing concave function. In the special case f(x) = x, our setting represents a lattice model for the study of transverse magnetization of spins diffusing in a homogeneous, α-stable, i.i.d., random, longitudinal field {λV(x) : x ∈ Z^d} with common marginal distribution, the standard α-symmetric stable distribution; the parameter λ describes the intensity of the field. Using large-deviation techniques, we show that S_c(λ α f) = lim_t→∞ E(t; λ f) exists. Moreover, we obtain a variational formula for this decay rate S_c. Finally, we analyze the behavior S_c(λ α f) as λ → 0 when f(x) = x^β for all 1 ≥ β > 0. Consequently, several physical conjectures with respect to lattice models of transverse magnetization are resolved by setting β = 1 in our results. We show that S_c(λ, α, 1) ≈ λ^α for d ≥ 3, λ^agr;(ln 1/λ)^α−1 in d = 2, and in d = 1. © 1996 John Wiley & Sons, Inc.     

On uniformly continuous Nemytskij operators generated by set-valued functions

Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all Hölder functions ${\varphi : I \to C}Let I = [0, 1], let Y be a real normed linear space, C a convex cone in Y and Z a real Banach space. Denote by clb(Z) the set of all nonempty, convex, closed and bounded subsets of Z. If a superposition operator N generated by a set-valued function F : I × C → clb(Z) maps the set H _α (I, C) of all H?lder functions j: I ? C{\varphi : I \to C} into the set H _β (I, clb(Z)) of all H?lder set-valued functions f: I ? clb(Z){\phi : I \to clb(Z)} and is uniformly continuous, then

F(x,y)=A(x,y) \text+* B(x),       x ? I, y ? CF(x,y)=A(x,y) \stackrel{*}{\text{+}} B(x),\qquad x \in I, y \in C      6. A descent method for submodular function minimization      Satoru Fujishige  Satoru Iwata 《Mathematical Programming》2002,92(2):387-390 We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 V with ?,V∈? and f(?)=0 and for any vector x∈R V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|2) times, the descent method finds a sequence of subsets Z 1,Z 2,···,Z k of V such that f(Z 1)>f(Z 2)>···>f(Z k )=min{f(Y) | Y∈?}, where k is O(|V|2). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001      7. Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case      N. I. Kavallaris  A. A. Lacey  C. V. Nikolopoulos  D. E. Tzanetis 《Mathematical Methods in the Applied Sciences》2007,30(13):1507-1526 We estimate the blow‐up time for the reaction diffusion equation ut=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ*, where λ* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ* with ∥w(?, λ)∥∞→∞ as 0<λ→λ*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=es, for λ?λ*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.      8. Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting      《随机分析与应用》2013,31(3):491-509 Abstract Let X 1, X 2… and B 1, B 2… be mutually independent [0, 1]-valued random variables, with EB j  = β > 0 for all j. Let Y j  = B 1 … sB j?1 X j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y 1,…,Y n ):=sup{EY τ:τ is a stopping rule for Y 1,…,Y n } and E(max 1≤j≤n Y j ). It is shown that the set of ordered pairs {(x, y):x = V(Y 1,…,Y n ), y = E(max 1≤j≤n Y j ) for some sequence Y 1,…,Y n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ n, β(x)}, where Ψ n, β(x) = [(1 ? β)n + 2β]x ? β?(n?2) x 2 if x ≤ β n?1, and Ψ n, β(x) = min j≥1{(1 ? β)jx + β j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.      9. Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions      Abbas Salemi 《Mathematische Nachrichten》2000,217(1):141-146 Let T(λ, ε ) = λ2 + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ0 be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ2 + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ0, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ0 is analytic.      10. From regular boundary graphs to antipodal distance-regular graphs      M. A. Fiol  E. Garriga  J. L. A. Yebra 《Journal of Graph Theory》1998,27(3):123-140 Let Γ be a regular graph with n vertices, diameter D, and d + 1 different eigenvalues λ > λ1 > ··· > λd. In a previous paper, the authors showed that if P(λ) > n − 1, then D ≤ d − 1, where P is the polynomial of degree d − 1 which takes alternating values ± 1 at λ1, …, λd. The graphs satisfying P(λ) = n − 1, called boundary graphs, have shown to deserve some attention because of their rich structure. This paper is devoted to the study of this case and, as a main result, it is shown that those extremal (D = d) boundary graphs where each vertex have maximum eccentricity are, in fact, 2-antipodal distance-regular graphs. The study is carried out by using a new sequence of orthogonal polynomials, whose special properties are shown to be induced by their intrinsic symmetry. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 123–140, 1998      11. The linearity coefficient of metric projections onto a Chebyshev subspace      P. A. Borodin 《Mathematical Notes》2009,85(1-2):168-175 The linearity coefficient λ(Y) of a metric projection P Y onto a subspace Y in a Banach space X is determined. This coefficient turns out to be related to the Lipschitz norm of the operator P Y . It is proved that, for any Chebyshev subspace Y in the space C or L 1, either λ(Y) = 1 (which corresponds to the linearity of P Y ) or λ(Y) ≤ 1/2.      12. Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS      Thomas Aubriot 《代数通讯》2013,41(12):3919-3936 Pour toute algèbre enveloppante quantique Uq(𝔤) de Drinfeld–Jimbo et toute famille λ = (λij)1≤i ∈ k? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien Aλ de Uq(𝔤) et nous montrons que tout objet galoisien de Uq(𝔤) est homotope à un unique objet de la forme Aλ. For any Drinfeld–Jimbo quantum enveloping algebra Uq(𝔤) and for any family λ = (λij)1≤i ∈ k? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.      13. Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)      《Quaestiones Mathematicae》2013,36(4):539-545 The Padé table of 2 F 1(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z-, the denominator polynomial Q mn (z) in the [m/n] Padé approximant P mn (z)/Q mn (z) for 2 F 1(a, 1; c; z) and the remainder term Q mn (z)2 F 1(a, 1; c; z)-Pmn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of Pmn (z)/Qmn (z) lie on the cut (1,∞). We deduce that the sequence of approximants Pmn (z)/Qmn (z) converges to 2 F 1(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.      14. Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1      Badr Alharbi 《代数通讯》2013,41(5):1939-1966 Let ? = ??, ?1(𝔖 n ) be the Hecke algebra of the symmetric group 𝔖 n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d λν for all partitions of the form λ = (a, c, 1 b ) and ν 2 ? regular.      15. On the zeros of exponential polynomials      Cerino E Avellar  Jack K Hale 《Journal of Mathematical Analysis and Applications》1980,73(2):434-452 Suppose r = (r1, …, rM), rj ? 0, γkj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γk · r = ∑jγkjrj. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑j = 1Naje?λγj · r, where each aj is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑k = 1MAkx(t ? rk) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.      16. Bilinear isometries on subspaces of continuous functions      Juan J. Font  M. Sanchis 《Mathematische Nachrichten》2010,283(4):568-572 Let A and B be strongly separating linear subspaces of C0(X) and C0(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C0(Z) be a bilinear isometry. Then there exist a nonempty subset Z0 of Z, a surjective continuous mapping h: Z0 → ?A × ?B and a norm‐one continuous function a: Z0 → K such that T (f, g)(z) = a (z)f (πx (h (z))g (πy (h (z)) for all z ∈ Z0 and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)      17. Multiobjective programming and penalty functions      D. J. White 《Journal of Optimization Theory and Applications》1984,43(4):583-599 This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R m, with each component equal to unity;f:R n →R m, represents them objective functions {f i} defined onX \( \subseteq \) R n; λ ∈R 1, λ>0;P:R n →R 1 X \( \subseteq \) Z \( \subseteq \) R n,P(x)≦0, ∨x ∈R n,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ r )} for a sequence of positive {λ r } converging to 0 in relationship toE(X), whereE(Z, λ r ) is the efficient set ofZ with respect tog(·, λr) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.      18. On the Almost Sure Convergence of Floating-Point Mantissas and Benford's Law      Peter Schatte 《Mathematische Nachrichten》1988,135(1):79-83 Let Y1, Y2,… be a sequence of random variables and let Mn be the floating-point mantissa of Yn. Further let 11,x)(·) denote the indicator of the interval (1,x). If Yn/n→Z a.s., where Z ≠ 0 is a further random variable, then the sequence 1(1,x)(Mn) converges a.s. to log x in the sense of ?? ∞-means and logarithmic means, respectively. The speed of convergence in this relations is estimated. As a conclusion, a further argument for Benford's law is provided.      19. Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell      Nguyen Xuan Tan 《Mathematical Methods in the Applied Sciences》1993,16(1):13-33 The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.      20. An identity involving partitional generalized binomial coefficients      Christopher Bingham 《Journal of multivariate analysis》1974,4(2):210-223 Define coefficients (κλ) by Cλ(Ip + Z)/Cλ(Ip) = Σk=0l Σ?∈$\text{P}$k (?λ) Cκ(Z)/Cκ(Ip), where the Cλ's are zonal polynomials in p by p matrices. It is shown that C?(Z) etr(Z)/k! = Σl=k∞ Σλ∈$\text{P}$l (?λ) Cλ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (?λ), ? ∈ $\text{P}$k, k ≤ 3. Several identities involving the (?λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in sj = tr Zj of degree k ≤ 5.

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

From regular boundary graphs to antipodal distance-regular graphs

Let Γ be a regular graph with n vertices, diameter D, and d + 1 different eigenvalues λ > λ₁ > ··· > λ_d. In a previous paper, the authors showed that if P(λ) > n − 1, then D ≤ d − 1, where P is the polynomial of degree d − 1 which takes alternating values ± 1 at λ₁, …, λ_d. The graphs satisfying P(λ) = n − 1, called boundary graphs, have shown to deserve some attention because of their rich structure. This paper is devoted to the study of this case and, as a main result, it is shown that those extremal (D = d) boundary graphs where each vertex have maximum eccentricity are, in fact, 2-antipodal distance-regular graphs. The study is carried out by using a new sequence of orthogonal polynomials, whose special properties are shown to be induced by their intrinsic symmetry. © 1998 John Wiley & Sons, Inc. J Graph Theory 27: 123–140, 1998     

The linearity coefficient of metric projections onto a Chebyshev subspace

The linearity coefficient λ(Y) of a metric projection P _Y onto a subspace Y in a Banach space X is determined. This coefficient turns out to be related to the Lipschitz norm of the operator P _Y . It is proved that, for any Chebyshev subspace Y in the space C or L ₁, either λ(Y) = 1 (which corresponds to the linearity of P _Y ) or λ(Y) ≤ 1/2.     

Classification des Objets Galoisiens de U q (𝔤) à Homotopie PrèS

Pour toute algèbre enveloppante quantique U_q(𝔤) de Drinfeld–Jimbo et toute famille λ = (λ_ij)1≤i ∈ k^? d'éléments inversibles du corps de base, nous construisons explicitement par générateurs et relations un objet galoisien A_λ de U_q(𝔤) et nous montrons que tout objet galoisien de U_q(𝔤) est homotope à un unique objet de la forme A_λ.

For any Drinfeld–Jimbo quantum enveloping algebra U_q(𝔤) and for any family λ = (λ_ij)_{1≤i ∈ k^? of invertible elements of the base field, we explicitly construct a Galois object Aλ of Uq(𝔤) by generators and relations and we prove that any Galois object of Uq(𝔤) is homotopic to a unique object of type Aλ.}     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

On the zeros of exponential polynomials

Suppose r = (r₁, …, r_M), r_j ? 0, γ_kj ? 0 integers, k = 1, 2, …, N, j = 1, 2, …, M, γ_k · r = ∑_jγ_kjr_j. The purpose of this paper is to study the behavior of the zeros of the function h(λ, a, r) = 1 + ∑_{j = 1}^Na_je^{?λγ_j · r}, where each a_j is a nonzero real number. More specifically, if $\text{Z}\text{?}\text{(a, r) =}\text{closure}\text{{}\text{Re}\text{λ: h(λ, a, r) = 0}}$, we study the dependence of $\text{Z}\text{?}\text{(a, r)}\text{on}\text{a, r}$. This set is continuous in a but generally not in r. However, it is continuous in r if the components of r are rationally independent. Specific criterion to determine when $\text{0 ?}\text{Z}\text{?}\text{(a, r)}$ are given. Several examples illustrate the complicated nature of $\text{Z}\text{?}\text{(a, r)}$. The results have immediate implication to the theory of stability for difference equations x(t) ? ∑_{k = 1}^MA_kx(t ? r_k) = 0, where x is an n-vector, since the characteristic equation has the form given by h(λ, a, r). The results give information about the preservation of stability with respect to variations in the delays. The results also are fundamental for a discussion of the dependence of solutions of neutral differential difference equations on the delays. These implications will appear elsewhere.     

Bilinear isometries on subspaces of continuous functions

Let A and B be strongly separating linear subspaces of C₀(X) and C₀(Y), respectively, and assume that ?A ≠ ?? (?A stands for the set of generalized peak points for A) and ?B ≠ ??. Let T: A × B → C₀(Z) be a bilinear isometry. Then there exist a nonempty subset Z₀ of Z, a surjective continuous mapping h: Z₀ → ?A × ?B and a norm‐one continuous function a: Z₀ → K such that T (f, g)(z) = a (z)f (π_x (h (z))g (π_y (h (z)) for all z ∈ Z₀ and every pair (f, g) ∈ A × B. These results can be applied, for example, to non‐unital function algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

On the Almost Sure Convergence of Floating-Point Mantissas and Benford's Law

Let Y₁, Y₂,… be a sequence of random variables and let M_n be the floating-point mantissa of Y_n. Further let 1_1,x)(·) denote the indicator of the interval (1,x). If Y_n/n→Z a.s., where Z ≠ 0 is a further random variable, then the sequence 1_(1,x)(M_n) converges a.s. to log x in the sense of ?? ∞-means and logarithmic means, respectively. The speed of convergence in this relations is estimated. As a conclusion, a further argument for Benford's law is provided.     

Local bifurcation from characteristic values with finite multiplicity and its application to axisymmetric buckled states of a thin spherical shell

The purpose of this paper is to study bifurcation points of the equation T(v) = L(λ,v) + M(λ,v), (λ,v) ? Λ × D in Banach spaces, where for any fixed λ ? Λ, T, L(λ,·) are linear mappings and M(λ,·) is a nonlinear mapping of higher order, M(λ,0) = 0 for all λ ? Λ. We assume that λ is a characteristic value of the pair (T, L) such that the mapping T – L(λ ,·) is Fredholm with nullity p and index s, p > s ? 0. We shall find some sufficient conditions to show that (λ ,0) is a bifurcation point of the above equation. The results obtained will be used to consider bifurcation points of the axisymmetric buckling of a thin spherical shell subjected to a uniform compressive force consisting of a pair of coupled non-linear ordinary differential equations of second order.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.