On the convex hull of the multiform numerical range

Let A ₁,…,A_m be nxn hermitian matrices. Definine

W(A ₁,…,A_m )={(xA₁x ^?,…xA_mx ^?):x?C ⁿ ,xx ^?=1}. We will show that every point in the convex hull of W(A ₁,…,A_m ) can be represented as a convex combination of not more than k(m,n) points in W(A ₁,…,A_m ) where k(m,n)=min{n,[√m]+δ _n ² _m+1}.     

Polynomial sums over automorphs of a positive definite binary quadratic form

Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ $\text{Z}$ × $\text{Z}$ : Q(N) = m}. It is shown that if Σ_N∈A(m)P(N) = 0 for each m = 1, 2, 3,… then Σ_U∈GP(UX) ≡ 0.     

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.     

Almost sure central limit theorem for partial sums and maxima

Let X, X₁, X₂, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX² = 1. Let X_i and M_n = max{X_i, 1 ≤ i ≤ n }. Suppose there exists constants a_n > 0, b_n ∈ R and a nondegenrate distribution G (y) such that Then, we have almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Linear preservers of decomposable symmetric tensors

Let U be an n-dimensional vector space over an algebraically closed field F. Let U(^m) denote the mth symmetric power of U. For each positive integer k≤min{m,n}, let D_k denote the set of all nonzero decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) = k and E_k denote the set of all decomposable elements x₁ …x_m in U(^m) such that dim(x₁ …x_m ) ≤ k. In this paper we first show that E_k is an algebraic variety with D_k as a dense subset and determine the dimension of E_k . We next use these results to study the structure of linear mappings T on U^m such that T(D_k ) ? D_k or T(E_k ) ? E_k for some fixed k.     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

A new inequality for symmetric functions

Let x_i ≥ 0, y_i ≥ 0 for i = 1,…, n; and let a_j(x) be the elementary symmetric function of n variables given by a_j(x) = ∑_{1 ≤ ii < … <ij ≤ n}x_ii … x_ij. Define the partical ordering x <y if a_j(x) ≤ a_j(y), j = 1,… n. We show that $\text{x}\text{\$}\text{?}\text{y ? x}{}^{\mathrm{\alpha }}\text{\$}\text{?}\text{y}{}^{\mathrm{\alpha }}\text{, 0}\text{\$}\text{?}\text{α ≤ 1}$, where {x^α}_i = x^α_i. We also give a necessary and sufficient condition on a function f(t) such that x <y ? f(x) <f(y). Both results depend crucially on the following: If x <y there exists a piecewise differentiable path z(t), with z_i(t) ≥ 0, such that z(0) = x, z(1) = y, and z(s) <z(t) if 0 ≤ s ≤ t ≤ 1.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

Quasi‐symmetric designs with fixed difference of block intersection numbers

The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.

• (1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z³ then λ < x + 1 + z + z³.
• (2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
• (3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z².
• (4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
• (5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.

© 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007     

Banach spaces of operators that are complemented in their biduals

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T _i ) _i∈I in A(X, Y) satisfying lim _i 〈y*, T _i x y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.      

The Descent Monomials and a Basis for the Diagonally Symmetric Polynomials

Let R(X) = Q[x ₁, x ₂, ..., x _n] be the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and R*(X) denote the quotient of R(X) by the ideal generated by the elementary symmetric functions. Given a S _n, we let g In the late 1970s I. Gessel conjectured that these monomials, called the descent monomials, are a basis for R*(X). Actually, this result was known to Steinberg [10]. A. Garsia showed how it could be derived from the theory of Stanley-Reisner Rings [3]. Now let R(X, Y) denote the ring of polynomials in the variables X = {x ₁, x ₂, ..., x _n} and Y = {y ₁, y ₂, ..., y _n}. The diagonal action of S _n on polynomial P(X, Y) is defined as Let R (X, Y) be the subring of R(X, Y) which is invariant under the diagonal action. Let R *(X, Y) denote the quotient of R (X, Y) by the ideal generated by the elementary symmetric functions in X and the elementary symmetric functions in Y. Recently, A. Garsia in [4] and V. Reiner in [8] showed that a collection of polynomials closely related to the descent monomials are a basis for R *(X, Y). In this paper, the author gives elementary proofs of both theorems by constructing algorithms that show how to expand elements of R*(X) and R *(X, Y) in terms of their respective bases.     

An application of L-functions of imaginary quadratic fields

Let F₁(x, y),…, F_2h+1(x, y) be the representatives of equivalent classes of positive definite binary quadratic forms of discriminant ?q (q is a prime such that q ≡ 3 mod 4) with integer coefficients, then the number of integer solutions of F_i(x, y) = n (i = 1,…, 2h + 1) can be calculated for each natural number n using L-functions of imaginary quadratic field $\text{Q}$((?q)^1/2).     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

On the existence of certain measures appearing in distance geometry

Abstract. We prove the following result: Let X be a compact connected Hausdorff space and f be a continuous function on X x X. There exists some regular Borel probability measure m\mu on X such that the value of¶¶ ò\limit _X f(x,y)dm(y)\int\limit _X f(x,y)d\mu (y) is independent of the choice of x in X if and only if the following assertion holds: For each positive integer n and for all (not necessarily distinct) x₁,x₂,...,x_n,y₁,y₂,...,y_n in X, there exists an x in X such that¶¶ ?_i=1ⁿ f(x_i,x)=?_i=1ⁿ f(y_i,x).\sum\limits _{i=1}^n f(x_i,x)=\sum\limits _{i=1}^n f(y_i,x).     

On The Postulation of The Multiples of The Nodes for Generic Nodal Curves on Generic Rational Surfaces

Fix integers x > 0, m₁ ≥ … ≥ ^m _x > 0 and P₁,…,P_x ∈ P² such that no 3 of them are collinear. Let C ? P² a “ general ” degree d plane curve with an ordinary point with multiplicity m _i at each P _i and y further singularities which are ordinary nodes. Fix any A ? Sing(C){P₁,…, P_x} and any integer m > 0. Here we study the postulation of the fat points m A ? ∪_Q∈AmQ.     

Stellar permutations of the two-element subsets of a finite set

Let X be a finite set and denote by X⁽²⁾ the set of 2-element subsets of X. A permutation ϕ of X⁽²⁾ is called stellar if, for each x in X, the image under ϕ of the star St(x) = {{x, y}: x ≠ y ∈ X} is a 2-regular graph spanning X − {x}. Several constructions of stellar permutations are given; in particular, there is a natural direct construction using self-orthogonal Latin squares, and a simple recursive construction using linear spaces having all line sizes at least four. Apart from some intrinsic interest, stellar permutations arise in the construction of certain designs. For example, applying such a map ϕ to each of the stars St(x) yields a double covering of the complete graph on X by near 2-factors. We also study stellar groups, that is groups {ϕ₁, …, ϕ_s} of permutations of X⁽²⁾ such that each ϕ_i is stellar (or the identity map). It is elementary to prove that s ≤ for any stellar group; when equality holds, one can construct an associated elation semibiplane. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 381–387, 1998     

Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ℕ², Ω an open bounded domain in ℝ^m , Y = [0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ℝ^m × ℝⁿ → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝ^m × ℝⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that

     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

On Partial Tilting Complexes

Let R be a commutative noetherian ring and A a noetherian R-algebra. Let P ^? ∈ 𝒦^b(𝒫 _A ) with Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i > 0. We will provide a sufficient condition for P ^? to be a direct summand of a silting complex. Also, in case Hom_𝒦(Mod-A)(P ^?, P ^?[i]) = 0 for i ≠ 0, we will provide a sufficient condition for P ^? to be a direct summand of a tilting complex.