Matrices Doubly Stochastic by Blocks

1977 ◽  
Vol 29 (3) ◽  
pp. 559-577 ◽  
Author(s):  
Pal Fischer ◽  
John A. R. Holbrook
Keyword(s):  
Convex Function ◽  
Classical Result ◽  
Doubly Stochastic ◽  
Continuous Convex Function ◽  
Image Position

The present work stems from the following classical result, due to G. H. Hardy, J. E. Littlewood, G. Pólya [7], and R. Rado [10].THEOREM 1. Concerning a pair of n-tuples x, y ϵ Rn, the following four statementsare equivalent:(a) for every continuous, convex function f : R → R

Inequalities for permanents and permanental minors

1965 ◽  
Vol 61 (3) ◽  
pp. 741-746 ◽  
Author(s):  
R. A. Brualdi ◽  
M. Newman
Keyword(s):  
Convex Function ◽  
Convex Set ◽  
Identity Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Counter Example ◽  
Doubly Stochastic Matrices ◽  
Permanent Function ◽  
Image Position

Let Ωndenote the convex set of alln×ndoubly stochastic matrices: chat is, the set of alln×nmatrices with non-negative entries and row and column sums 1. IfA= (aij) is an arbitraryn×nmatrix, then thepermanentofAis the scalar valued function ofAdefined bywhere the subscriptsi1,i2, …,inrun over all permutations of 1, 2, …,n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that ifA∈ Ωnthen 0 <cn≤ per (A) ≤ 1, where the constantcndepends only onn. It is natural to inquire if per (A) is a convex function ofAforA∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I+ ½A) ≤ ½ + ½ per (A) for allA∈ Ωn. HereI=Inis the identity matrix of ordern.

A Generalization of an Inequality of Bhattacharya and Leonetti

1996 ◽  
Vol 39 (4) ◽  
pp. 438-447 ◽  
Author(s):  
Ritva Hurri-Syrjänen
Keyword(s):  
Convex Function ◽  
Sobolev Class ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Continuous Convex Function ◽  
John Domains ◽  
Image Position

AbstractWe show that bounded John domains and bounded starshaped domains with respect to a point satisfy the following inequalitywhere F: [0, ∞) → [0, ∞) is a continuous, convex function with F(0) = 0, and u is a function from an appropriate Sobolev class. Constants b and K do depend at most on D. If F(x) = xp, 1 ≤ p < ∞, this inequality reduces to the ordinary Poincaré inequality.

Some remarks on probability inequalities for sums of bounded convex random variables

1975 ◽  
Vol 12 (1) ◽  
pp. 155-158 ◽  
Author(s):  
M. Goldstein
Keyword(s):  
Convex Function ◽  
Exponential Convergence ◽  
Random Variables ◽  
Upper Bounds ◽  
Independent Random Variables ◽  
Probability Inequalities ◽  
Continuous Convex Function ◽  
Bounded Convex

Let X1, X2, · ··, Xn be independent random variables such that ai ≦ Xi ≦ bi, i = 1,2,…n. A class of upper bounds on the probability P(S−ES ≧ nδ) is derived where S = Σf(Xi), δ > 0 and f is a continuous convex function. Conditions for the exponential convergence of the bounds are discussed.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

scholarly journals Separability of the L1-space of a vector measure

1992 ◽  
Vol 34 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Werner J. Ricker
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Classical Result ◽  
Vector Measure ◽  
Symmetric Difference ◽  
Outer Measure ◽  
Additive Measure ◽  
Image Position

Let Σ be a σ-algebra of subsets of some set Ω and let μ:Σ→[0,∞] be a σ-additive measure. If Σ(μ) denotes the set of all elements of Σ with finite μ-measure (where sets equal μ-a.e. are identified in the usual way), then a metric d can be defined in Σ(μ) by the formulahere E ΔF = (E\F) ∪ (F\E) denotes the symmetric difference of E and F. The measure μ is called separable whenever the metric space (Σ(μ), d) is separable. It is a classical result that μ is separable if and only if the Banach space L1(μ), is separable [8, p.137]. To exhibit non-separable measures is not a problem; see [8, p. 70], for example. If Σ happens to be the σ-algebra of μ-measurable sets constructed (via outer-measure μ*) by extending μ defined originally on merely a semi-ring of sets Γ ⊆ Σ, then it is also classical that the countability of Γ guarantees the separability of μ and hence, also of L1(μ), [8, p. 69].

Permanents of Random Doubly Stochastic Matrices

1974 ◽  
Vol 26 (3) ◽  
pp. 600-607 ◽  
Author(s):  
R. C. Griffiths
Keyword(s):  
Symmetric Group ◽  
Stochastic Matrix ◽  
Stochastic Matrices ◽  
Doubly Stochastic Matrix ◽  
Doubly Stochastic ◽  
Survey Article ◽  
Doubly Stochastic Matrices ◽  
Image Position

The permanent of an n × n matrix A = (aij) is defined aswhere Sn is the symmetric group of order n. For a survey article on permanents the reader is referred to [2]. An unresolved conjecture due to van der Waerden states that if A is an n × n doubly stochastic matrix; then per (A) ≧ n!/nn, with equality if and only if A = Jn = (1/n).

Traceless Maps as the Singular Minimizers in the Multi-dimensional Calculus of Variations

2017 ◽  
Vol 60 (3) ◽  
pp. 631-640
Author(s):  
M. S. Shahrokhi-Dehkordi
Keyword(s):  
Convex Function ◽  
Energy Functional ◽  
Uniformly Convex ◽  
Lagrange Equations ◽  
Regularity Of Minimizers ◽  
Uniformly Convex Function ◽  
Bounded Lipschitz Domain ◽  
Second Derivatives ◽  
Image Position ◽  
Singular Minimizers

AbstractLet Ω ⊂ ℝn be a bounded Lipschitz domain and consider the energy functionalover the space of W1,2(Ω, ℝm) where the integrand is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler–Lagrange equations. In particular, we introduce a class of singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional ℱ[ ·, Ω] using a method based on null Lagrangians.

An algebraic characterization of power set in countable standard models of ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 167-170
Author(s):  
George Metakides ◽  
J. M. Plotkin
Keyword(s):  
Standard Model ◽  
Boolean Algebra ◽  
Classical Result ◽  
Algebraic Characterization ◽  
Categorical Theory ◽  
Power Set ◽  
Standard Models ◽  
Image Position ◽  
Atomic Boolean Algebra

The following is a classical result:Theorem 1.1. A complete atomic Boolean algebra is isomorphic to a power set algebra [2, p. 70].One of the consequences of [3] is: If M is a countable standard model of ZF and is a countable (in M) model of a complete ℵ0-categorical theory T, then there is a countable standard model N of ZF and a Λ ∈ N such that the Boolean algebra of definable (in T with parameters from ) subsets of is isomorphic to the power set algebra of Λ in N. In particular if and T the theory of equality with additional axioms asserting the existence of at least n distinct elements for each n < ω, then there is an N and Λ ∈ N with 〈PN(Λ), ⊆〉 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of ω.From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.Definition 1.1. A countable Boolean algebra 〈B, ≤〉 is a pseudo-power set if there is a countable standard model of ZF, N and a set Λ ∈ N such thatIt is clear that a pseudo-power set is atomic.

On orthogonality of Riesz products

1974 ◽  
Vol 76 (1) ◽  
pp. 173-181 ◽  
Author(s):  
Gavin Brown ◽  
William Moran
Keyword(s):  
Haar Measure ◽  
Classical Result ◽  
Weak Limit ◽  
Absolutely Continuous ◽  
Riesz Product ◽  
Image Position ◽  
Riesz Products ◽  
Positive Integers

A typical Riesz product on the circle is the weak* limitwhere – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

On a conjecture of Watson

1983 ◽  
Vol 94 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Madhu Raka
Keyword(s):  
Quadratic Form ◽  
Classical Result ◽  
Preceding Paper ◽  
Real Numbers ◽  
Linear Forms ◽  
Indefinite Quadratic Form ◽  
Indefinite Quadratic ◽  
Image Position

Let Qn be a real indefinite quadratic form in n variables x1, x2,…, xn, of determinant D ≠ 0 and of type (r, s), 0 < r < n, n = r + s. Let σ denote the signature of Qn so that σ = r − s. It is known (see e.g. Blaney(4)) that, given any real numbers c1 c2, …, cn, there exists a constant C depending upon n and σ only such that the inequalityhas a solution in integers x1, x2, …, xn. Let Cr, s denote the infimum of all such constants. Clearly Cr, s = Cs, r, so we need consider non-negative signatures only. For n = 2, C1, 1 = ¼ follows from a classical result of Minkowski on the product of two linear forms. When n = 3, Davenport (5) proved that C2, 1 = 27/100. For all n and σ = 0, Birch (3) proved that Cr, r = ¼. In 1962, Watson(18) determined the values of Cr, s for all n ≥ 21 and for all signatures σ. He proved thatWatson also conjectured that (1·2) holds for all n ≥ 4. Dumir(6) proved Watson's conjecture for n = 4. For n = 5, it was proved by Hans-Gill and Madhu Raka(7, 8). The author (12) has proved the conjecture for σ = 1 and all n. In the preceding paper (13) we proved that C5, 1 = 1. In this paper we prove Watson's conjecture for σ = 2, 3 and 4.

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