Path integrals for quadratic lagrangians on <Emphasis Type="Italic">p</Emphasis>-adic and adelic spaces

Feynman’s path integrals in ordinary, p-adic and adelic quantum mechanics are considered. The corresponding probability amplitudes K(x″, t″; x′, t′) for two-dimensional systems with quadratic Lagrangians are evaluated analytically and obtained expressions are generalized to any finite-dimensional spaces. These general formulas are presented in the form which is invariant under interchange of the number fields ℝ ↔ ℚ _p and ℚ ↔ ℚ _p , p ≠ p′. According to this invariance we have that adelic path integral is a fundamental object in mathematical physics of quantum phenomena.     

Absolute upper semicontinuity

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ _t′ ^t″ g (s) ds. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.     

An extension of Ezeilo's result

Summary In a recent paper[1] Ezeilo considered the nonlinear third order differential equation x‴ + ω(x′)x″ + ω(x)x′ + ϑ(x, x′, x″)=p(t). He proved the ultimate boundedness of the solutions on rather general conditions for the nonlinear terms ϕ, ϕ, ϑ. These conditions (in a little weaker form) are also sufficient in order to prove the existence of forced oscillations in the case when the excitation is ω-periodic. For this purpose the Lerag-Schauder principle in a form suggested by G. Güssefeldt[2] is applicable. Dedicated to ProfessorKarl Klotter on his 70th birthday Entrata in Redazione il 21 ottobre 1971.     

Lower bound on the profile of degree pairs in cross-intersecting set systems

We raise the following problem. For natural numbers m, n ≥ 2, determine pairs d′, d″ (both depending on m and n only) with the property that in every pair of set systems A, B with |A| ≤ m, |B| ≤ n, and A ∩ B ≠ 0 for all A ∈ A, B ∈ B, there exists an element contained in at least d′ |A| members of A and d″ |B| members of B. Generalizing a previous result of Kyureghyan, we prove that all the extremal pairs of (d′, d″) lie on or above the line (n − 1) x + (m − 1) y = 1. Constructions show that the pair (1 + ɛ / 2n − 2, 1 + ɛ / 2m − 2) is infeasible in general, for all m, n ≥ 2 and all ɛ > 0. Moreover, for m = 2, the pair (d′, d″) = (1 / n, 1 / 2) is feasible if and only if 2 ≤ n ≤ 4. The problem originates from Razborov and Vereshchagin’s work on decision tree complexity. Research supported in part by the Hungarian Research Fund under grant OTKA T-032969.     

A theorem on level lines of continuous functions

A property of a continuous functionf(x), x ∈ E ₂, similar to the classical intermediate value property is established. Namely, let a Jordan compactJ ⊂ E ₂ be the domain of definition off. Then, for each parametrizationx(t), 0≦t≦T,x(0)=x(T), of the boundary Fr(J) ofJ there exists a unique real λ and a unique connected component Φ of the level set {x ∈ J: f(x)=λ} with the following property: any connected subset Ω ofJ containing “opposite” points of Fr(J) (i.e. pointsx(t′) andx(t″) such thatt″−t′=T/2) has a common element with Φ.     

A local variational relation and applications

In [BGH] the authors show that for a given topological dynamical system (X,T) and an open coveru there is an invariant measure μ such that infh _μ(T,ℙ)≥h _top(T,U) where infimum is taken over all partitions finer thanu. We prove in this paper that if μ is an invariant measure andh _μ(T,ℙ) > 0 for each ℙ finer thanu, then infh _μ(T,ℙ > 0 andh _top(T,U) > 0. The results are applied to study the topological analogue of the Kolmogorov system in ergodic theory, namely uniform positive entropy (u.p.e.) of ordern (n≥2) or u.p.e. of all orders. We show that for eachn≥2 the set of all topological entropyn-tuples is the union of the set of entropyn-tuples for an invariant measure over all invariant measures. Characterizations of positive entropy, u.p.e. of ordern and u.p.e. of all orders are obtained. We could answer several open questions concerning the nature of u.p.e. and c.p.e.. Particularly, we show that u.p.e. of ordern does not imply u.p.e. of ordern+1 for eachn≥2. Applying the methods and results obtained in the paper, we show that u.p.e. (of order 2) system is weakly disjoint from all transitive systems, and the product of u.p.e. of ordern (resp. of all orders) systems is again u.p.e. of ordern (resp. of all orders). Project supported by one hundred talents plan and 973 plan.     

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

Abstract. In this Paper, the existence and uniqueness of solutions for boundary valueproblem     

Periodic solutions of second order non-Lagrangian systems

Using results in bifurcation theory, we show the existence of periodic solutions of a large class of non-Lagrangian systems of the formu″+A ₁ v′+B ₁u+F₁ (w, w′, w″)=0v″+A ₂ u′+B ₂v+F₂ (w, w′, w″)=0 wherew=(u, v).     

Ensembles minimaux sans mesure d'entropie maximale

Forh=1 we construct a compact dynamical system which is minimal and has the topological entropyh, but which does not carry any ergodic measure with entropyh. The method works for anyh(0<h ). It is an extension of the method used for the construction in [1], section 27.     

On the existence of almost periodic solutions of some dissipative second order differential equations

Summary The search for almost periodic solutions of any dissipative equation of the form(1.1), in which p(t) is an almost periodic function, has come to be closely linked up with a number of standard ? convergence ? restrictions on f, g′, g″ and k (see, for example,[2] and[3]). The object of the present paper is to show that as far as the existence, alone, of an almost periodic solution of(1.1) is concerned these ? convergence ? restrictions on f, g′, g″ and k are quite unnecessary. The first result (Theorem 1) shows in fact that the conditions(1.2) alone are quite sufficient for the existence of an almost periodic solution of(1.1); and Theorem 2 extends this result (though under stronger conditions on f and g) to the case in which the forcing function depends on x and x as well. Partially supported by N.S.F. research project G-57 at The University of Michigan.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

Bernoulli convolutions and an intermediate value theorem for entropies of<Emphasis Type="Italic">K</Emphasis>-partitions

We establish a strong regularity property for the distributions of the random sums Σ±λ ⁿ , known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed ℓ, the conditional distribution of (w _n+1...,w _n+ℓ) given the sum Σ _n=0 ^∞ w _n λ ⁿ , tends to the uniform distribution on {±1}^ℓ asn → ∞. More precise results, where ℓ grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems. The authors were partially supported by NSF grants DMS-9729992 (E. L.), DMS-9803597 (Y. P.) and DMS-0070538 (W. S.).     

A criterion of normality based on a single holomorphic function

Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any f ∈ F. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f ^(k)(z)| ≤ c, where c is a constant. Then F is normal on D.     

Certaines formes linéaires pathologiques sur un espace de Banach dual

We construct a Banach spaceE such thatE′ isw ^*-separable, andf∈E″/E, which isw ^*-continuous on every set ofE′ which is thew ^*-closure of a countablebounded set ofE′.      

Upward closure and cohesive degrees

It is shown that the degrees of unsolvability of sets having almost any sort of immunity or cohesiveness property studied in recursion theory are closed upwards. From this it follows that every degreea witha′≧0″ contains a cohesive set. This research was supported by National Science Foundation Grant GP 29223.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Conditions for convergence of trust region algorithms for nonsmooth optimization

This paper discusses some properties of trust region algorithms for nonsmooth optimization. The problem is expressed as the minimization of a functionh(f(x), whereh(·) is convex andf is a continuously differentiable mapping from ℝ″ to ℝ‴. Bounds for the second order derivative approximation matrices are discussed. It is shown that Powel’s [7, 8] results hold for nonsmooth optimization.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

Non-real zeros of linear differential polynomials

Let f be a real entire function with finitely many non-real zeros, not of the form f = Ph with P a polynomial and h in the Laguerre-Pólya class. Lower bounds are given for the number of non-real zeros of f″ + ω f, where ω is a positive real constant.