Compactness principle for periodic singular and fine structures

We consider the compactness principle in the variable space L ² related to a periodic Borel measure. It is assumed that the periodic Borel measure determines a periodic singular (i.e., irregular) or fine structure. We prove the compactness principle for the periodic singular and fine grids, box structures, and composite structures in the plane and in space.     

Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W _p ¹ on the “zero” cusp with a Hölder singularity at the vertex.     

Isomorphism and Embedding of Borel Systems on Full Sets

A Borel system consists of a measurable automorphism of a standard Borel space. We consider Borel embeddings and isomorphisms between such systems modulo null sets, i.e. sets which have measure zero for every invariant probability measure. For every t>0 we show that in this category, up to isomorphism, there exists a unique free Borel system (Y,S) which is strictly t-universal in the sense that all invariant measures on Y have entropy <t, and if (X,T) is another free system obeying the same entropy condition then X embeds into Y off a null set. One gets a strictly t-universal system from mixing shifts of finite type of entropy ≥t by removing the periodic points and “restricting” to the part of the system of entropy <t. As a consequence, after removing their periodic points the systems in the following classes are completely classified by entropy up to Borel isomorphism off null sets: mixing shifts of finite type, mixing positive-recurrent countable state Markov chains, mixing sofic shifts, beta shifts, synchronized subshifts, and axiom-A diffeomorphisms. In particular any two equal-entropy systems from these classes are entropy conjugate in the sense of Buzzi, answering a question of Boyle, Buzzi and Gomez.     

An extension of Prohorov's theorem for transition probabilities with applications to infinite-dimensional lower closure problems

A criterion for the relative weak compactness of sets of transition probabilitieson T×B(S) was given by Balder (1984 a),T being a space equipped with a fixed measure andB(S) denoting the σ-algebra of a standard Borel spaceS. This result, which generalizes Prohorov's classical tightness criterion, is extended here so as to cover the case where the spaceS is σ-standard Borel. As a consequence, the method applied in Balder (1984 a) now yields a new general infinite, dimensional lower closure result that can be used in the existence theory for optimal control.     

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting on a Borel space S and on an underlying ??-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.     

PMEA implies proposition P

The Product Measure Extension Axiom (PMEA) asserts that for every set A, Haar measure on 2^A can be extended to all subsets of 2^A. PMEA implies the normal Moore space conjecture. Proposition P is the statement that every point-finite analytic-additive family of subsets of a metrizable space is σ-discretely decomposible. Proposition P is useful in nonseparable Borel theory. We show in this paper that PMEA implies Proposition P.     

A mathematical programming problem with singular mixed pointwise-integral constraints

We investigate an infinite dimensional optimization problem which constraints are singular integral-pointwise ones. We give some partial results of existence for a solution in some particular cases. However, the lack of compactness, even in L¹ prevents to conclude in the general case. We give an existence result for a weak solution (as a measure) that we are able to describe. The regularity of such a solution is still an open problem.     

Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75-94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L²(m) (not only unitarily equivalent to it), then it has a cyclic vector in L^∞(m). We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L²(C,τ) is the norm-closure of the polynomials in z.     

On approximative properties for thin structures

In this paper, we study approximative properties connecting a periodic Borel measure µ^h with its weak-limit measure µ as h → 0. V. V. Zhikov showed that these properties are necessary for averaging problems with two small parameters arising in thin periodic structures with thickness tending to zero.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Compactness of embeddings of sobolev type on metric measure spaces

We establish conditions for the compactness of embeddings for some classes of functions on metric space with measure satisfying the duplication condition. These classes are defined in terms of the L ^p -summability of maximal functions measuring local smoothness.     

Entropy of Hadamard spaces

We consider a geodesically complete and proper Hadamard metric measure space X endowed with a Borel measure. Assuming that there exists a certain non-amenable group of isometry of X which acts freely, properly discontinuously and cocompactly on X and preserves the measure we show that the topological entropy of the geodesic flow on the orbit space is positive.     

板几何中一类具周期边界条件的奇异迁移算子的谱

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Singular one-dimensional transport equations on Lp spaces

We prove the well-posedness of the Cauchy problem governed by a linear mono-energetic singular transport equation (i.e., transport equation with unbounded collision frequency and unbounded collision operator) with specular reflecting and periodic boundary conditions on L_p spaces. The large time behaviour of its solution is also considered. We discuss the compactness properties of the second-order remainder term of the Dyson-Phillips expansion for a large class of singular collision operators. This allows us to evaluate the essential type of the transport semigroup from which the asymptotic behaviour of the solution is derived.     

On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

Riemann-measurable selections

LetX be a Polish space equipped with a σ-finite regular Borel measure μ. IfE is a metric space andF a set-valued function:X → 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.     

Product sigma-ideals

Quotients of the Borel sigma-algebra with respect to the sigma-ideals of countable, first category, and measure zero sets have been generalized to an essentially new multi-dimensional context. The question of isomorphisms between these quotients is examined. These techniques are used to obtain regularity theorems about a measurable space X from an assumed isomorphism between products Xⁿ and X^m.     

On Borel equivalence relations in generalized Baire space

We construct two Borel equivalence relations on the generalized Baire space κ ^κ , κ ^<κ ?=?κ >?ω, with the property that neither of them is Borel reducible to the other. A small modification of the construction shows that the straightforward generalization of the Glimm-Effros dichotomy fails.     

Pathwise random periodic solutions of stochastic differential equations

In this paper, we study the existence of random periodic solutions for semilinear stochastic differential equations. We identify these as the solutions of coupled forward-backward infinite horizon stochastic integral equations in general cases. We then use the argument of the relative compactness of Wiener-Sobolev spaces in C⁰([0,T],L²(Ω)) and generalized Schauder?s fixed point theorem to prove the existence of a solution of the coupled stochastic forward-backward infinite horizon integral equations. The condition on F is then further weakened by applying the coupling method of forward and backward Gronwall inequalities. The results are also valid for stationary solutions as a special case when the period τ can be an arbitrary number.     

Is Lebesgue measure the only σ-finite invariant Borel measure?

S. Saks and recently R.D. Mauldin asked if every translation invariant σ-finite Borel measure on Rd is a constant multiple of Lebesgue measure. The aim of this paper is to investigate the versions of this question, since surprisingly the answer is “yes and no,” depending on what we mean by Borel measure and by constant. According to a folklore result, if the measure is only defined for Borel sets, then the answer is affirmative. We show that if the measure is defined on a σ-algebra containing the Borel sets, then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant σ-finite Borel measure (in the wider sense) on Rd can be non-σ-finite when we restrict it to the Borel sets.