Lipschitz lower sigma-exponents of linear differential systems

For the lower sigma-exponent of the linear differential system ? = A(t)x, x ∈ R ⁿ , t ≥ 0, defined by the formula Δ_σ(A) ≡ inf_λ[Q]≤-σ λ ₁(A + Q), σ > 0, on the basis of the lower characteristic exponents λ ₁(A+Q) of perturbed linear systems with Lyapunov exponents λ[Q] ≤ ?σ < 0 of perturbations Q, we prove the following general form as a function of the parameter σ > 0. For any nondecreasing bounded function f(σ) of the parameter σ ∈ (0,+∞) that coincides with a constant on some infinite interval (σ ₀,+∞), σ ₀ ≥ 0, and satisfies the Lipschitz condition on the complementary interval (0, σ ₀], we prove the existence of a linear system with coefficient matrix A _f (t) bounded on the half-line [0,+∞) whose lower sigma-exponent Δ_σ(A _f ) coincides with the function f(σ) on the entire interval (0,+∞).     

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u^′=A(t)u+?H(t,u)+f(t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ? is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u^′=A(t)u+f(t) with general conditions on f. For small ? we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.     

Uniqueness properties in finite-continuous additive measurement

Let ? be a binary relation on A×X, and suppose that there are real valued functions f on A and g on X such that, for all ax, by ∈ A×X, ax ? by if and only if f (a)+g(x) ? f(b)+g(y). This paper establishes uniqueness properties for f and g when A is a finite set, X is a real interval with g increasing on X, and for any a, b and x there is a y for which f(a)+g(x)=f(b)+g(y). The resultant uniqueness properties occupy an intermediate position among uniqueness properties for other structural cases of two-factor additive measurement.It is shown that f is unique up to a positive affine transformation (αf+β₁ with α > 0), but that g is unique up to a similar positive affine transformation (αg+β₂) if and only if the ratio [f(a)?f(b)]/[f(a)?f(c)] is irrational for some a, b, c ∈ A. When the f ratios are rational for all cases where they are defined, there will be a half-open interval (x₀, x₁) in X such that the restriction of g on (x₀, x₁) can be any increasing function for which sup {g(x)?g(x₀): x₀ ? x < x₁} does not exceed a specified bound, and, when g is thus defines on (x₀, x₁), it will be uniquely determined on the rest of X. In general, g must be continuous only in the ‘irrational’ case.     

Brouwer's degree without properness

A degree deg(f,y) is defined for every continuous function , which possesses all the properties of Brouwer's degree provided that y is restricted to the complement of some closed set A(f) of “asymptotic” values. Sufficient conditions are given for A(f) to be nowhere dense. It is also shown that, in the opposite direction, A(f) having nonempty interior has a direct impact on the solutions of f(x)=y, which cannot be discovered by degree arguments.     

The extreme points of some convex sets in the theory of majorization

Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ω_{µ, w}⁺f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩ_{µ, w} ⁺f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given.     

Topologically inseparable functions II: Infinitary case

Given a set A and a function A: A → A, we study the set of all functions g: A → A that are continuous for all topologies for which f continuous. We prove that in a sense to be made precise in the text, for any essentially infinitary function f, any non-constant such g equals f ⁿ , for some n∈ ?. We also prove a similar result for the clone of n-ary functions from A ⁿ → A.     

A composition for matroids

The composition of a quotient matroid Q over a collection of component matroids f₁, …, f_n indexed on the cells of Q, is described. This composition, called quotient composition, may be viewed as an application of clutter composition to matroids, or as a generalization of matroid direct sum composition to the next higher connectivity. It may also be viewed as equivalent to compositions described by Minty in 1966, and Brylawski in 1971.Quotient composition is characterized, and the circuits and rank function of a composed matroid are given. Various other properties are described, along with a category for quotient composition.     

Strong Hypercontractivity and Relative Subharmonicity

A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, Q_D and Q_max, given by Q(f)=∫_M g(grad f(z), grad f(z)) dμ(z). Q_D is the form closure of Q on C^∞_c(M) and Q_max is the form closure of Q on C¹_b(M). The corresponding operators, A_D and A_max, generate semigroups having standard hypercontractivity properties in the scale of L^p spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e^−tA_D has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of L^p. These results are extended here to A_max. When (M, g) is not complete it is necessary that the elliptic differential operator A_max degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both A_D and A_max is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z^1/n and the weighted Bergman spaces in the unit disc are given as examples.     

The minimum increment of f-divergences given total variation distances

Let (P _i , Q _i ), i = 0, 1, be two pairs of probability measures defined on measurable spaces (Ω _i ,F _i ) respectively. Assume that the pair (P₁, Q₁) is more informative than (P₀,Q₀) for testing problems. This amounts to say that If (P₁,Q₁) ≥ If (P₀,Q₀), where If (·, ·) is an arbitrary f-divergence. We find a precise lower bound for the increment of f-divergences I_f(P₁,Q₁) ? I_f(P₀,Q₀) provided that the total variation distances ||Q₁ ? P₁|| and ||Q₀ ? P₀|| are given. This optimization problem can be reduced to the case where P₁ and Q₁ are defined on the space consisting of four points, and P₀ and Q₀ are obtained from P₁ and Q₁ respectively by merging two of these four points. The result includes the well-known lower and upper bounds for I_f(P,Q) given ||Q ? P||.     

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C^∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ?C × H for which $\text{D?}{}_{\mathrm{c}}\text{(x) = 0}$; here we write $\text{?}{}_{\mathrm{c}}\text{(x)}$ for $\text{?(c, x)}$. Say ? is of standard type if at all critical points of ?_c it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f?F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ? 5, then for most a?A the function f_o?F(B,H) is of standard type. Using this it is shown that those f?F(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thom's theorem for catastrophes in $\text{R}$ⁿ.     

Points of continuity and quasicontinuity

Let C(f), Q(f), E(f) and A(f) be the sets of all continuity, quasicontinuity, upper and lower quasicontinuity and cliquishness points of a real function f: X → ℝ, respectively. The triplets (C(f),Q(f),A(f)), (C(f),E(f),A(f) and (Q(f),E(f),A(f)are characterized for functions defined on Baire metric spaces without isolated points.     

On the pointwise matrix product and the mean value theorem

Bounds for the pointwise or Schur product of two matrices are derived with respect to the spectral norm 6·6. For real symmetric and positive semidefinite matrices Q=(q_ij) one of them gives a bound of 6|Q6, |Q|=(|q_ij|). Two of these bounds are applied to obtain a mean value theorem for g: t→f(A(t)) where A(t) is a matrix depending on a parameter t and f is a function on the spectrum of A(t).     

On meromorphic extendibility

Let D be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint real-analytic simple closed curves. Let f be an integrable function on bD. In the paper we show how to compute the candidates for poles of a meromorphic extension of f through D and thus reduce the question of meromorphic extendibility to the question of holomorphic extendibility. Let A(D) be the algebra of all continuous functions on which are holomorphic on D. We prove that a continuous function f on bD extends meromorphically through D if and only if there is an N∈N∪{0} such that the change of argument of Pf+Q along bD is bounded below by −2πN for all P,Q∈A(D) such that Pf+Q≠0 on bD. If this is the case then the meromorphic extension of f has at most N poles in D, counting multiplicity.     

Optimal solution of a Monge-Kantorovitch transportation problem

Suppose that P and Q are probabilities on a separable Banach space $\text{E}$. It is known that if (P, Q) satisfies certain regularity conditions and a random variable X has law P, then there exists a function $\text{f :}\text{E}\text{→}\text{E}$, such that the function f(X) has the law Q and the random pair (X, f(X)) is an optimal coupling for the Monge-Kantorovitch problem. In this paper we provide an approximation of the function f when the law Q is discrete. Thenwe extend this main result to any law Q. The proofs are based on a relationship between optimal couplings and nonlinear equations.     

Sparse Resultant under Vanishing Coefficients

The main question of this paper is: What happens to the sparse (toric) resultant under vanishing coefficients? More precisely, let f ₁,...,f _n be sparse Laurent polynomials with supports A ₁,...,A _n A ₁ ? A ₁. Naturally a question arises: Is the sparse resultant of f ₁,f ₂,...,f _n with respect to the supports A ₁,A ₂,...,A _n in any way related to the sparse resultant of f ₁,f ₂,...,f _n with respect to the supports A ₁,A ₂,...,A _n ? The main contribution of this paper is to provide an answer. The answer is important for applications with perturbed data where very small coefficients arise as well as when one computes resultants with respect to some fixed supports, not necessarily the supports of the f _i's, in order to speed up computations. This work extends some work by Sturmfels on sparse resultant under vanishing coefficients. We also state a corollary on the sparse resultant under powering of variables which generalizes a theorem for Dixon resultant by Kapur and Saxena. We also state a lemma of independent interest generalizing Pedersen's and Sturmfels' Poisson-type product formula.     

A unified approach to the weighted Grtzsch and Nitsche problems for mappings of finite distortion

This note deals with the existence and uniqueness of a minimiser of the following Grtzsch-type problem inf f ∈F∫∫_(Q_1)φ(K(z,f))λ(x)dxdyunder some mild conditions,where F denotes the set of all homeomorphims f with finite linear distortion K(z,f)between two rectangles Q_1 and Q_2 taking vertices into vertices,φ is a positive,increasing and convex function,and λ is a positive weight function.A similar problem of Nitsche-type,which concerns the minimiser of some weighted functional for mappings between two annuli,is also discussed.As by-products,our discussion gives a unified approach to some known results in the literature concerning the weighted Grtzsch and Nitsche problems.     

Asymptotically stable sets and the stability of ω-limit sets

Let C be the collection of continuous self-maps of the unit interval I=[0,1] to itself. For f∈C and x∈I, let ω(x,f) be the ω-limit set of f generated by x, and following Block and Coppel, we take Q(x,f) to be the intersection of all the asymptotically stable sets of f containing ω(x,f). We show that Q(x,f) tells us quite a bit about the stability of ω(x,f) subject to perturbations of either x or f, or both. For example, a chain recurrent point y is contained in Q(x,f) if and only if there are arbitrarily small perturbations of f to a new function g that give us y as a point of ω(x,g). We also study the structure of the map Q taking (x,f)∈I×C to Q(x,f). We prove that Q is upper semicontinuous and a Baire 1 function, hence continuous on a residual subset of I×C. We also consider the map given by x?Q(x,f), and find that this map is continuous if and only if it is a constant map; that is, only when the set is a singleton.     

Univariate multiquadric approximation: Quasi-interpolation to scattered data

The univariate multiquadric function with centerx _j ∈R has the form {? _j (x)=[(x?x _j )²+c ²]^1/2, x∈R} wherec is a positive constant. We consider three approximations, namely, ? _A f, ?_? f, and ? _C f, to a function {f(x),x ₀≤x≤x _N } from the space that is spanned by the multiquadrics {? _j :j=0, 1, ...,N} and by linear polynomials, the centers {x _j :j=0, 1,...,N} being given distinct points of the interval [x ₀,x _N ]. The coefficients of ? _A f and ?_? f depend just on the function values {f(x _j ):j=0, 1,...,N}. while ? _A f, ? _C f also depends on the extreme derivativesf′(x ₀) andf′(x _N ). These approximations are defined by quasi-interpolation formulas that are shown to give good accuracy even if the distribution of the centers in [x ₀,x _N ] is very irregular. Whenf is smooth andc=O(h), whereh is the maximum distance between adjacent centers, we find that the error of each quasi-interpolant isO(h ²|logh|) away from the ends of the rangex ₀≤x≤x _N. Near the ends of the range, however, the accuracy of ? _A f and ?_? f is onlyO(h), because the polynomial terms of these approximations are zero and a constant, respectively. Thus, some of the known accuracy properties of quasiinterpolation when there is an infinite regular grid of centers {x _j =jh:j ∈F} given by Buhmann (1988), are preserved in the case of a finite rangex ₀≤x≤x _N , and there is no need for the centers {x _j :j=0, 1, ...,N} to be equally spaced.     

The expressive power of voting polynomials

We consider the problem of approximating a Boolean functionf∶{0,1} ⁿ →{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP ^A is properly contained in PSPACE ^A . We are also able to prove the old AC⁰ exponential-size lower bounds in a new way. This allows us to prove the new result that an AC⁰ circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.