Adelic Feynman amplitudes in lower orders of perturbation theory

We formulate convergency conditions for adelic Feynman amplitudes and prove that for spaces of sufficiently high dimension, there exists a nonempty domain in the space of powers of propagators in which the adelic amplitude is correctly defined. We investigate an analytic continuation w.r.t. the power of propagators in amplitudes of the ϕ⁴ theory in the third and fourth order of the perturbation theory. We demonstrate that these amplitudes cannot be continued to the whole complex plane (assuming the validity of the Riemann hypothesis about zeros of the zeta-function) and physically meaningful values of the propagator powers lie at the boundary of the analyticity domain. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 1, pp. 95–109, July, 2000.     

Asymptotic behavior of the distortion-rate function for Gaussian processes in Banach spaces

Let μ be a Gaussian measure on a separable Banach space. We prove a tight link between the logarithmic small ball probabilities of μ and certain moment generating functions. Based upon this link we provide a new lower bound for the distortion-rate function (DRF) against the small ball function. This allows us to use results of the theory of small ball probabilities to deduce lower bounds for the DRF. In particular, we obtain the correct weak asymptotics of the distortion rate function in many important cases (e.g. Brownian motion).     

Improved upper and lower bounds for the spectral radius of digraphs

Let G be a digraph with n vertices and m arcs without loops and multiarcs. The spectral radius ρ(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, sharp upper and lower bounds on ρ(G) are given. We show that some known bounds can be obtained from our bounds.     

Underestimating the cost of a soft constraint is dangerous: revisiting the edit-distance based soft regular constraint

Many real-life problems are over-constrained, so that no solution satisfying all their constraints exists. Soft constraints, with costs denoting how much the constraints are violated, are used to solve these problems. We use the edit-distance based SoftRegular constraint as an example to show that a propagation algorithm that sometimes underestimates the cost may guide the search to incorrect (non-optimal) solutions to an over-constrained problem. To compute correctly the cost for the edit-distance based SoftRegular constraint, we present a quadratic-time propagation algorithm based on dynamic programming and a proof of its correctness. We also give an improved propagation algorithm using an idea of computing the edit distance between two strings, which may also be applied to other constraints with propagators based on dynamic programming. The asymptotic time complexity of our improved propagator is always at least as good as the one of our quadratic-time propagator, but significantly better when the edit distance is small. Our propagators achieve domain consistency on the problem variables and bounds consistency on the cost variable. Our method can also be adapted for the violation measure of the edit-distance based Regular constraint for constraint-based local search.     

Boundary smoothness properties of Lipα analytic functions

LetU be an open set andb ∈ bdy(U). Let 0 < α< 1. Let A(U) denote the space of Lipα functions that are analytic onU, and a(U) the subspace lipα ∩ A(U). The space a(U ∪b), consisting of the functions that are analytic nearb, is dense in a(U). Letk be a natural number. We say that a(U) admits ak-th order continuous point derivation (cpd) atb if the functionalf → f^{(k) (b)} is continuous on a(U ∪b), with respect to the Lipα norm.     

Optimal designs for weighted approximation and integration of stochastic processes on [0,∞)

We study minimal errors and optimal designs for weighted L₂-approximation and weighted integration of Gaussian stochastic processes X defined on the half-line [0,∞). Under some regularity conditions, we obtain sharp bounds for the minimal errors for approximation and upper bounds for the minimal errors for integration. The upper bounds are proven constructively for approximation and non-constructively for integration. For integration of the r-fold integrated Brownian motion, the upper bound is proven constructively and we have a matching lower bound.     

How to minimize the cost of iterative methods in the presence of perturbations

We consider iterative methods for approximating solutions of nonlinear equations, where the iteration cannot be computed exactly, but is corrupted by additive perturbations. The cost of computing each iteration depends on the size of the perturbation. For a class of cost functions, we show that the total cost of producing an ε-approximation can be made proportional to the cost c(ε) of one single iterative step performed with the accuracy proportional to ε. We also demonstrate that for some cost functions the total cost is proportional to c(ε)². In both cases matching lower bounds are shown. The results find natural application to establishing the complexity of nonlinear boundary-value problems, where they yield an improvement over the known upper bounds, and remove the existing gap between the upper and lower bounds.     

On the distance distribution of trees

Let G be a tree and k a non-negative integer. We determine best possible upper and lower bounds on the number of pairs of vertices at distance exactly k in G in terms of order alone, and in terms of order and radius or diameter.     

Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators

We study the inheritance of properties of free backward propagators associated with transition probability functions by backward Feynman-Kac propagators corresponding to functions and time-dependent measures from non-autonomous Kato classes. The inheritance of the following properties is discussed: the strong continuity of backward propagators on the space , the -smoothing property of backward propagators, and various generalizations of the Feller property. We also prove that a propagator on a Banach space is strongly continuous if and only if it is separately strongly continuous and locally uniformly bounded.

     

A non-steady flow of liquid in a porous pipe with variable permeability

LetB denote the infinitesimal operator of a strongly continuous semigroup S(t), with resolvent R_λ, on Banach space L. We define related operators P and V so that λR_λf = Pf + λVf + o(λ), as λ → 0⁺. For α, η > 0 and possibly unbounded, linear operator A, we let U_{α, η}(t) represent a strongly continuous semigroup generated by αA + ηB. We show that under appropriate simultaneous convergence of α and η, U_{α, η}(t) converges strongly to a strongly continous semigroup U(t), having infinitesimal operator characterized through PA(VA)^rf where r =min{j ? 0, PA(VA)^j ≠ 0}. We apply the abstract perturbation theorem to a singular perturbation initial-value problem, of Tihonov-type, for a non-linear system of ordinary differential equations.     

Über die Führer von Gaußschen Summen als Größencharaktere

Let K be an absolute abelian number field. The conductor of the Hecke characters of K formed by Gaussian sums is investigated. In some cases we obtain formulas for this conductor while in others a bound is given. These enable us to obtain a general upper bound for the conductor in terms of the class field theoretic conductor of K. Our bound is an improvement on existing bounds in the literature.     

On the first factor of the class number of prime cyclotomic fields

Let H(l) be the first factor of the class number of the field $\text{Q}$(exp 2πi/l), l a prime. The best-known upper and lower bounds on H(l) are improved for small l. The methods would also improve the best-known bounds for large l. It is shown that H(l) is the absolute value of the determinant of an easily written down matrix whose only entries are 0 and 1. The upper bounds obtained on H(l) significantly improve the Hadamard bound on the determinant of this matrix. Results of Lehmer on the factors of H(l) are explained via class field theory.     

Homological properties of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over an algebraically closed field k. We will investigate homological properties of piecewise hereditary algebras Λ. In particular we give lower and upper bounds of the strong global dimension, show the behavior of the strong global dimension under one point extensions and tilting. Moreover we show that the “pieces” of modΛ have Auslander–Reiten sequences.     

Weakly continuous holomorphic functions on Banach spaces with a shrinking and unconditional basis

Let E and F be complex Banach spaces, and let U be an open ball in E. We show that if E has a shrinking and unconditional basis, then every holomorphic function that is weakly continuous on U-bounded sets is weakly uniformly continuous on U-bounded sets.     

Extremal traces on some group-invariant C1-algebras

Let $\text{U}$ be a UHF-algebra of Glimm type n^∞, and {α_g: g?G} a strongly continuous group of ¹-automorphisms of product type on $\text{U}$, for G compact. Let $\text{U}$^α be the C¹-subalgebra of fixed elements of $\text{U}$. We show that any extremal normalized trace on $\text{U}$^α arises as the restriction of a symmetric product state ? on $\text{U}$ of the form ? = ?_k?1 ω. As an example we classify the extremal traces on $\text{U}$^α for the case G = SU(n), α_g = ?_{k ? 1} Ad(g).     

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L²(X) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e−tL whose kernels pt(x,y) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted Lp-norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L² estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.     

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.     

Resolution Complexity of Random Constraint Satisfaction Problems: Another Half of the Story

Let C_n,cn^2,k,t be a random constraint satisfaction problem(CSP) of n binary variables, where c > 0 is a fixed constant and the cn constraints are selected uniformly and independently from all the possible k-ary constraints each of which contains exactly t tuples of the values as its restrictions. We establish upper bounds for the tightness threshold for C_n,cn^2,k,t to have an exponential resolution complexity. The upper bounds partly answers the open problems regarding the CSP resolution complexity with the tightness between the existing upper and lower bound [1].     

Demicompactness Results for Strongly Continuous Semigroups,Generators and Resolvents

Let \((U(t))_ {t\ge 0}\) be a strongly continuous semigroup of bounded linear operators on a Banach space X and B be a bounded operator on X. In this paper, we develop some aspects of the theory of semigroup for which U(t)B (respectively, BU(t), BU(t)B) is demicompact for some (respectively, every) \(t>0\). In addition, we study the demicompactness of similar, subspace and product semigroups. We also investigate the demicompactness of the resolvent. We close this paper by giving some conditions guaranteeing the demicompactness of a generator of a strongly continuous semigroup in a Hilbert space.