SeparatingPH fromPP by relativization

We construct an oracleA such that . So the polynomial time hierarchy is separated from the polynomial time probabilistic complexity class in relativization. This research is supported in part by HTP863     

Symmetries and conservation laws of partial differential equations: Basic notions and results

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.     

Randomness and semigenericity

Let contain only the equality symbol and let be an arbitrary finite symmetric relational language containing . Suppose probabilities are defined on finite structures with `edge probability' . By , the almost sure theory of random -structures we mean the collection of -sentences which have limit probability 1. denotes the theory of the generic structures for (the collection of finite graphs with hereditarily nonnegative). . , the almost sure theory of random -structures, is the same as the theory of the -generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

     

Randomness and Nonlinear Evolution Equations

In this paper we survey some results on existence, and when possible also uniqueness, of solutions to certain evolution equations obtained by injecting randomness either on the set of initial data or as a perturbative term.     

Randomness friendly graphs

We consider the two problems from extremal graph theory: 1. Given integer N, real p ϵ (0, 1) and a graph G, what is the minimum number of copies of G a graph H with N vertices and pN²/2 edges can contain? 2. Given an integer N and a graph G, what is the minimum number of copies of G an N-vertex graph H and its complement H¯ can contain altogether? In each of the problems, we say that G is “randomness friendly” if the number of its copies is nearly minimal when H is the random graph. We investigate how the two classes of graphs are related: the graphs which are “randomness friendly” in Problem 1 and those of Problem 2. In the latter problem, we discover new families of graphs which are “randomness friendly.” © 1996 John Wiley & Sons, Inc.     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

Pairings and related symmetry notions

In this paper, we give a purely mathematical generalization of an information table. We call pairing on a given set \(\Omega \) a triple \(\mathfrak {P}=(U, F, \Lambda )\), where U and \(\Lambda \) are non-empty sets and \(F:U\times \Omega \rightarrow \Lambda \) is a map. We provide several examples of pairings: graphs, digraphs, metric spaces, group actions and vector spaces endowed with a bilinear form. Moreover, we reinterpret the usual notion of indiscernibility (with respect to a fixed attribute subset of an information table) in terms of local symmetry on U and, then, we study a global version of symmetry, that we called indistinguishability. In particular, we interpret the latter relation as the symmetrization of a pre-order \(\le _{\mathfrak {P}}\), that describes the symmetry transmission between subsets of \(\Omega \). Hence, we introduce a global average of symmetry transmission and studied it for some basic digraph families. Finally, we prove that the partial order of any finite lattice can be described in terms of the above pre-order.     

Universal forcing notions and ideals

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters. Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant no. 2002323). Also, we would like to thank the referee for valuable comments and suggestions concerning the exposition of the paper. This is publication 845 of the second author     

Randomness,chaos, and structure

We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. © 2009 Wiley Periodicals, Inc. Complexity 2009     

Generalized notions of amenability

New notions of amenability and contractability are introduced. Examples are given to show that for most of the new notions, the corresponding class of Banach algebras is larger than that for the classical amenable algebras introduced by Johnson. General theory is developed for these notions, and studied for several concrete classes of Banach algebras; special consideration is given to Banach algebras defined on locally compact groups.     

Curvature notions on graphs

We survey some geometric and analytic results under the assumptions of combinatorial curvature bounds for planar/semiplanar graphs and curvature dimension conditions for general graphs.     

Factorization,basic notions and applications

The idea to define an equivalence relation on the basic spaceW in terms of sample paths of some random processX _t (w) coinciding up to timet is the basis of the important concept of saturated filtration. Here we exploit the same idea to represent spaceW in the formW=W ×W, whereW is the space of all equivalence classes with respect to the above relation corresponding to a stopping time andW is an image ofW under some shifting operator. This representation allows us to work with spaceW ×W rather thanW and to investigate more precisely the properties of some random objects given onW.     

Renaissance notions of number and magnitude

In the 16th and 17th centuries the classical Greek notions of (discrete) number and (continuous) magnitude (preserved in medieval Latin translations of Euclid's Elements) underwent a major transformation that turned them into continuous but measurable magnitudes. This article studies the changes introduced in the classical notions of number and magnitude by three influential Renaissance editions of Euclid's Elements. Besides providing evidence of earlier discussions preparing notions and arguments eventually introduced in Simon Stevin's Arithmétique of 1585, these editions document the role abacus algebra and Renaissance views on the history of mathematics played in bridging the gulf between discrete numbers and continuous magnitudes.     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

Asymptotics of Multivariate Randomness Statistics

Kiefer considered the asymptotics of q-sample Cramer-Von Mises statistics for a fixed q and sample sizes tending to infinity. For univariate observations, McDonald proved the asymptotic normality of these statistics when q goes to infinity while the sample sizes stay fixed. Here we define a class of multivariate randomness statistics that generalizes the class considered by McDonald. We also prove the asymptotic normality of such statistics when the sample sizes stay fixed while q tends to infinity.     

Randomness on full shift spaces

We give various characterizations for algorithmically random configurations on full shift spaces, based on randomness tests. We show that all nonsurjective cellular automata destroy randomness and surjective cellular automata preserve randomness. Furthermore all one-dimensional cellular automata preserve nonrandomness. The last three assertions are also true if one replaces randomness by richness – a form of pseudorandomness, which is compatible with computability. The last assertion is true even for an arbitrary dimension.     

On independent varieties and some related notions

We investigate the relation of independence between varieties, as well as a generalisation of such which we call strict quasi-independence. Concerning the former notion, we specify a procedure for constructing an independent companion of a given solvable subvariety of a congruence modular variety; we show that joins of independent varieties inherit Mal’cev properties from the joinands; we investigate independence in 3- and 4-permutable varieties; we provide a more economical axiomatisation for the join of two independent varieties than the ones available in the literature. We also explore the latter notion, showing inter alia that joins of strictly quasi-independent varieties inherit the congruence extension property and the strong amalgamation property from the joinands, and conversely. An application section investigates independent varieties of Boolean algebras with operators (in particular, Akishev and Goldblatt’s bounded monadic algebras) and of groups. In particular, a complete characterisation of independent varieties of groups is given.