Search results for "Mathematica"
showing 10 items of 7971 documents
Comparing the relative volume with a revolution manifold as a model
1993
Given a pair (P, M), whereM is ann-dimensional connected compact Riemannian manifold andP is a connected compact hypersurface ofM, the relative volume of (P, M) is the quotient volume(P)/volume(M). In this paper we give a comparison theorem for the relative volume of such a pair, with some bounds on the Ricci curvature ofM and the mean curvature ofP, with respect to that of a model pair\(\left( {\mathcal{P},\mathcal{M}} \right)\) where ℳ is a revolution manifold and\(\mathcal{P}\) a “parallel” of ℳ.
Asymptotics for thenth-degree Laguerre polynomial evaluated atn
1992
We investigate the asymptotic behaviour of ? n (n),n?? where ? n (x) denotes the Laguerre polynomial of degreen. Our results give a partial answer to the conjecture ?? n (n)>1 forn>6, made in 1984 by van Iseghem. We also show the connection between this conjecture and the continued fraction approximants of $$6\sqrt {{3 \mathord{\left/ {\vphantom {3 \pi }} \right. \kern-\nulldelimiterspace} \pi }} $$ .
Brauer’s Height Zero Conjecture for principal blocks
2021
Abstract We prove the other half of Brauer’s Height Zero Conjecture in the case of principal blocks.
Optimal Locations and Inner Products
1997
Abstract In a normed space X , we consider objective functions which depend on the distances between a variable point and the points of certain finite sets A . A point where such a function attains its minimum on X is generically called an optimal location. In this paper we obtain characterizations of inner product spaces with properties connecting optimal locations and the convex hull of A or barycenters of points of A with well chosen weights. We thus generalize several classical results about characterization of inner product spaces.
A simple proof of the polylog counting ability of first-order logic
2007
The counting ability of weak formalisms (e.g., determining the number of 1's in a string of length N ) is of interest as a measure of their expressive power, and also resorts to complexity-theoretic motivations: the more we can count the closer we get to real computing power. The question was investigated in several papers in complexity theory and in weak arithmetic around 1985. In each case, the considered formalism (AC 0 -circuits, first-order logic, Δ 0 ) was shown to be able to count up to a polylogarithmic number. An essential part of the proofs is the construction of a 1-1 mapping from a small subset of {0, ..., N - 1} into a small initial segment. In each case the expressibility of …
On the decision problem for the guarded fragment with transitivity
2002
The guarded fragment with transitive guards, [GF+TG], is an extension of GF in which certain relations are required to be transitive, transitive predicate letters appear only in guards of the quantifiers and the equality symbol may appear everywhere. We prove that the decision problem for [GF+TG] is decidable. This answers the question posed in (Ganzinger et al., 1999). Moreover, we show that the problem is 2EXPTIME-complete. This result is optimal since the satisfiability problem for GF is 2EXPTIME-complete (Gradel, 1999). We also show that the satisfiability problem for two-variable [GF+TG] is NEXPTIME-hard in contrast to GF with bounded number of variables for which the satisfiability pr…
On the Finite Satisfiability Problem for the Guarded Fragment with Transitivity
2005
We study the finite satisfiability problem for the guarded fragment with transitivity. We prove that in case of one transitive predicate the problem is decidable and its complexity is the same as the general satisfiability problem, i.e. 2Exptime-complete. We also show that finite models for sentences of GF with more transitive predicate letters used only in guards have essentially different properties than infinite ones.
Mappings of finite distortion: discreteness and openness for quasi-light mappings
2005
Abstract Let f ∈ W 1 , n ( Ω , R n ) be a continuous mapping so that the components of the preimage of each y ∈ R n are compact. We show that f is open and discrete if | D f ( x ) | n ⩽ K ( x ) J f ( x ) a.e. where K ( x ) ⩾ 1 and K n − 1 / Φ ( log ( e + K ) ) ∈ L 1 ( Ω ) for a function Φ that satisfies ∫ 1 ∞ 1 / Φ ( t ) d t = ∞ and some technical conditions. This divergence condition on Φ is shown to be sharp.
Old and New on the Quasihyperbolic Metric
1998
Let D be a proper subdomain of \( {\mathbb{R}^d}\). Following Gehring and Palka [GP] we define the quasihyperbolic distance between a pair x 1, x 2 of points in D as the infimum of \( {\smallint _\gamma }\frac{{ds}}{{D\left( {x,\partial D} \right)}}\) over all rectifiable curves γ joining x 1, x 2 in D. We denote the quasihyperbolic distance between x 1, x 2 by k D (x 1, x 2). As pointed out by Gehring and Osgood [GO], x 1 and x 2 can be joined by a quasihyperbolic geodesic; also see [Mr]. The quasihyperbolic metric is comparable to the usual hyperbolic metric in a simply connected plane domain by the Koebe distortion theorem. For a multiply connected plane domain D these two metrics are co…
Periodic Solutions of the Second Order Quadratic Rational Difference Equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}} $$ x n + 1 = α ( 1 + x n ) x n…
2016
The aim of this article is to investigate the periodic nature of solutions of a rational difference equation $$x_{n+1}=\frac{\alpha }{(1+x_n)x_{n-1}}. {(*)} $$ We explore Open Problem 3.3 given in Amleh et al. (Int J Differ Equ 3(1):1–35, 2008, [2]) that requires to determine all periodic solutions of the equation (*). We conclude that for the equation (*) there are no periodic solution with prime period 3 and 4. Period 7 is first period for which exists nonnegative parameter \(\alpha \) and nonnegative initial conditions.