8,131 results
Search Results
202. Paradoxical Euler: Integrating by Differentiating
- Author
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Hieu D. Nguyen and Andrew Fabian
- Subjects
Integral calculus ,symbols.namesake ,Differential equation ,General Mathematics ,medicine ,Calculus ,Euler's formula ,symbols ,medicine.disease ,Calculus (medicine) ,Mathematics ,Exposition (narrative) - Abstract
Every student of calculus learns that one typically solves a differential equation by integrating it. However, as Euler showed in his 1758 paper (E236), Exposition de quelques paradoxes dans le calcul intégral (Explanation of certain paradoxes in integral calculus) [1], there are differential equations that can be solved by actually differentiating them again. This initially seems paradoxical or, as Euler describes it in the introduction of his paper:Here I intend to explain a paradox in integral calculus that will seem rather strange: this is that we sometimes encounter differential equations in which it would seem very difficult to find the integrals by the rules of integral calculus yet are still easily found. not by the method of integration. but rather in differentiating the proposed equation again; so in these cases, a repeated differentiation leads us to the sought integral.
- Published
- 2013
203. Frieze groups, cylinders, and quotient groups
- Author
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Alan F. Beardon
- Subjects
Algebra ,Combinatorics ,Frieze ,Group (mathematics) ,General Mathematics ,Frieze group ,Homogeneous space ,Order (group theory) ,Invariant (mathematics) ,Symmetry (geometry) ,Quotient group ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
This paper is about the classification of frieze groups. A frieze is a decorative strip of paper (or wood, stone,…) on which a pattern is produced by the periodic repetition of a picture along the strip. A symmetry of a frieze is an isometry of the plane that leaves the pattern unchanged, and a frieze group is the group of symmetries of some frieze. A popular exercise is to start with a given frieze and then try to identify its frieze group. However, in order to do this we need to know that there are only seven possible frieze groups, and what these groups are. Seven friezes, with different frieze groups, are illustrated below, in such a way that each pattern is invariant under the same translation, namely x → x + 1. Our task is to show that (up to a change in the motif, and a simple change of coordinates in the plane) these are the only frieze patterns.
- Published
- 2013
204. The Lax–Oleinik semi-group: a Hamiltonian point of view
- Author
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Patrick Bernard, Université Paris sciences et lettres (PSL), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), European Project: 307062,EC:FP7:ERC,ERC-2012-StG_20111012,SAW(2012), Université Paris Dauphine-PSL, École normale supérieure - Paris (ENS-PSL), and Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris)
- Subjects
Pure mathematics ,Kolmogorov–Arnold–Moser theorem ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Fixed point ,Invariant (physics) ,01 natural sciences ,Convexity ,Hamiltonian system ,010101 applied mathematics ,symbols.namesake ,Compact space ,symbols ,Configuration space ,0101 mathematics ,Hamiltonian (quantum mechanics) ,Mathematics - Abstract
International audience; The weak KAM theory was developed by Fathi in order to study the dynamics of convex Hamiltonian systems. It somehow makes a bridge between viscosity solutions of the Hamilton–Jacobi equation and Mather invariant sets of Hamiltonian systems, although this was fully understood only a posteriori. These theories converge under the hypothesis of convexity, and the richness of applications mostly comes from this remarkable convergence. In this paper, we provide an elementary exposition of some of the basic concepts of weak KAM theory. In a companion paper, Albert Fathi exposed the aspects of his theory which are more directly related to viscosity solutions. Here, on the contrary, we focus on dynamical applications, even if we also discuss some viscosity aspects to underline the connections with Fathi's lecture. The fundamental reference on weak KAM theory is the still unpublished book Weak KAM theorem in Lagrangian dynamics by Albert Fathi. Although we do not offer new results, our exposition is original in several aspects. We only work with the Hamiltonian and do not rely on the Lagrangian, even if some proofs are directly inspired by the classical Lagrangian proofs. This approach is made easier by the choice of a somewhat specific setting. We work on R d and make uniform hypotheses on the Hamiltonian. This allows us to replace some compactness arguments by explicit estimates. For the most interesting dynamical applications, however, the compactness of the configuration space remains a useful hypothesis and we retrieve it by considering periodic (in space) Hamiltonians. Our exposition is centred on the Cauchy problem for the Hamilton–Jacobi equation and the Lax–Oleinik evolution operators associated to it. Dynamical applications are reached by considering fixed points of these evolution operators, the weak KAM solutions. The evolution operators can also be used for their regularizing properties; this opens an alternative route to dynamical applications. 1. The method of characteristics, existence and uniqueness of regular solutions We consider a C 2 Hamiltonian H(t, q, p) : R × R d × R d * → R and study the associated Hamiltonian system ˙ q(t) = ∂ p H(t, q(t), p(t)), ˙ p(t) = −∂ q H(t, q(t), p(t)), (HS) * This paper is a late addition to the papers surveying active areas in partial differential equations , published in issue 141.2, which were based on a series of mini-courses held in the International Centre for Mathematical Sciences (ICMS) in Edinburgh during 2010. and Hamilton–Jacobi equation ∂ t u + H(t, q, ∂ q u(t, q)) = 0. (HJ) We denote by X H (x) = X H (q, p) the Hamiltonian vector field X H = J dH, where J is the matrix J = 0 I −I 0. The Hamiltonian system can be written in condensed terms ˙ x(t) = X H (t, x(t)). We shall always assume that the solutions extend to R. We denote by ϕ t τ = (Q t τ , P t τ): R d
- Published
- 2012
205. Maclaurin's inequalities: reflections on a STEP question
- Author
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Stephen T C Siklos
- Subjects
Inequality ,General Mathematics ,media_common.quotation_subject ,Mathematical economics ,Mathematics ,media_common - Abstract
Step is a public examination administered by the admissions tests division of Cambridge Assessment (which is the parent body of the OCR examination board). It used as a basis for conditional offers by Cambridge and Warwick.A grade one on a STEP paper is awarded to candidates who produce nearly complete solutions to four questions. The duration of the paper is three hours, so each question is designed to take a good candidate about 45 minutes to complete. Not surprisingly, then, the questions can be quite profound, often having depths far beyond what is required for the examination. One such question haunted me, on and off, for many years. It turns out that the underlying idea is elementary and far from new; but it is nevertheless very striking and not, I suspect, well known. A pleasing by-product is a relatively easy proof of the AM–GM (algebraic mean–geometric mean) inequality.
- Published
- 2012
206. The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers
- Author
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Marcelo Laca and Siegfried Echterhoff
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,Mathematics::Operator Algebras ,Semigroup ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Algebraic number field ,Primary 46L05, 46L80, Secondary 20Mxx, 11R04 ,Space (mathematics) ,01 natural sciences ,Ring of integers ,Primitive ideal ,Prime (order theory) ,Crossed product ,0103 physical sciences ,FOS: Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Operator Algebras (math.OA) ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Mathematics - Abstract
The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.
- Published
- 2012
207. Ruelle operator with weakly contractive iterated function systems
- Author
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Yuan-Ling Ye
- Subjects
Sequence ,Pure mathematics ,Operator (computer programming) ,Iterated function system ,Dynamical systems theory ,Triple system ,Applied Mathematics ,General Mathematics ,Lipschitz continuity ,Mathematics - Abstract
The Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS $(X, \{w_j\}_{j=1}^m)$ and an associated family of positive continuous potential functions $\{p_j\}_{j=1}^m$, a triple system $(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ruelle operator is quasi-compact, and the iterations sequence of the Ruelle operator converges with a specific geometric rate, if the potential functions are Lipschitz continuous.
- Published
- 2012
208. Boundary behaviour of special cohomology classes arising from the Weil representation
- Author
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Jens Funke and John J. Millson
- Subjects
Pure mathematics ,Cover (topology) ,General Mathematics ,Cohomology of locally symmetric spaces ,Holomorphic function ,Orthogonal group ,Theta function ,Compactification (mathematics) ,Weil representation ,Cohomology ,Mathematics ,Singular homology ,Siegel modular form - Abstract
In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.
- Published
- 2012
209. SOME PROPERTIES OF A SEQUENCE ANALOGOUS TO EULER NUMBERS
- Author
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Zhi-Wei Sun
- Subjects
Combinatorics ,Sequence ,General Mathematics ,Floor and ceiling functions ,Congruence relation ,Mathematics - Abstract
Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.
- Published
- 2012
210. Cup-products for the polyhedral product functor
- Author
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Anthony Bahri, Martin Bendersky, S. Gitler, and Frederick R. Cohen
- Subjects
Combinatorics ,Functor ,Brown's representability theorem ,Derived functor ,General Mathematics ,Künneth theorem ,Tor functor ,Exact functor ,Topology ,Cohomology ,Cohomology ring ,Mathematics - Abstract
Davis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [6]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [4]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [4].Subsequent developments were given in work of Denham–Suciu [7] and Franz [9] which were followed by [1, 2]. Namely, given a family of based CW-pairs X, A) = {(Xi, Ai)}mi=1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z(K;(X,A)) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case (X,A) = (D2, S1) [1, 2]. A decomposition theorem was proven which splits the suspension of Z(K; (X, A)) into a bouquet of spaces determined by the full sub-complexes of K.This paper is a study of the cup-product structure for the cohomology ring of Z(K; (X, A)). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z(K; (X, A)) [1, 2]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results.Explicit computations are made for families of suspension pairs and for the cases where Xi is the cone on Ai. These results complement and extend those of Davis–Januszkiewicz [6], Buchstaber–Panov [3, 4], Panov [13], Baskakov–Buchstaber–Panov, [3], Franz, [8, 9], as well as Hochster [12]. Furthermore, under the conditions stated below (essentially the strong form of the Künneth theorem), these theorems also apply to any cohomology theory.
- Published
- 2012
211. Variational characterizations of weighted Hardy spaces and weighted spaces
- Author
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Yongming Wen, Huoxiong Wu, Weichao Guo, and Dongyong Yang
- Subjects
symbols.namesake ,Pure mathematics ,General Mathematics ,symbols ,Hardy space ,Mathematics - Abstract
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
- Published
- 2021
212. Dynamical profile of a class of rank-one attractors
- Author
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Qiudong Wang and Lai Sang Young
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Rank (linear algebra) ,Dynamical systems theory ,Differential equation ,Applied Mathematics ,General Mathematics ,Lyapunov exponent ,Nonlinear Sciences::Chaotic Dynamics ,symbols.namesake ,Attractor ,symbols ,Ergodic theory ,Large deviations theory ,Central limit theorem ,Mathematics - Abstract
This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.
- Published
- 2012
213. Convergence of Brownian motion with a scaled Dirac delta potential
- Author
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Martin Grothaus, Florian Conrad, Janna Lierl, and Olaf Wittich
- Subjects
symbols.namesake ,Simple (abstract algebra) ,General Mathematics ,Convergence (routing) ,Mathematical analysis ,symbols ,Dirac delta function ,Statistical physics ,Scaling ,Brownian motion ,Mathematics - Abstract
The method of deriving scaling limits using Dirichlet-form techniques has already been successfully applied to a number of infinite-dimensional problems. However, extracting the key tools from these papers is a rather difficult task for non-experts. This paper meets the need for a simple presentation of the method by applying it to a basic example, namely the convergence of Brownian motions with potentials given by n multiplied by the Dirac delta at 0 to Brownian motion with absorption at 0.
- Published
- 2012
214. The Darboux problem involving the distributional Henstock–Kurzweil integral
- Author
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Guoju Ye, Ying Wang, Yueping Lu, and Wei Liu
- Subjects
Set (abstract data type) ,Pure mathematics ,Distribution (mathematics) ,Schauder fixed point theorem ,Henstock–Kurzweil integral ,General Mathematics ,Mathematical analysis ,Structure (category theory) ,Integral of inverse functions ,Darboux integral ,Mathematics - Abstract
In this paper, using the Schauder Fixed Point Theorem and the Vidossich Theorem, we study the existence of solutions and the structure of the set of solutions of the Darboux problem involving the distributional Henstock–Kurzweil integral. The two theorems presented in this paper are extensions of the previous results of Deblasi and Myjak and of Bugajewski and Szufla.
- Published
- 2012
215. Non-standard real-analytic realizations of some rotations of the circle – CORRIGENDUM
- Author
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Shilpak Banerjee
- Subjects
Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.
- Published
- 2016
216. ON SOLUBILITY OF GROUPS WITH FEW NORMALISERS
- Author
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Mohammad Zarrin
- Subjects
Discrete mathematics ,Algebra ,Conjecture ,Group (mathematics) ,General Mathematics ,Finite set ,Mathematics - Abstract
In this paper we prove that every group with at most 26 normalisers is soluble. This gives a positive answer to Conjecture 3.6 in the author’s paper [On groups with a finite number of normalisers’, Bull. Aust. Math. Soc.86 (2012), 416–423].
- Published
- 2014
217. Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings
- Author
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Annette Werner, Amaury Thuillier, and Bertrand Rémy
- Subjects
Linear algebraic group ,Pure mathematics ,Absolutely irreducible ,General Mathematics ,010102 general mathematics ,General linear group ,01 natural sciences ,010101 applied mathematics ,Analytic geometry ,Algebraic group ,Embedding ,Equivariant map ,Compactification (mathematics) ,0101 mathematics ,Mathematics - Abstract
In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.
- Published
- 2011
218. EXPLICIT REPRESENTATIONS OF THE INTEGRAL CONTAINING THE ERROR TERM IN THE DIVISOR PROBLEM II
- Author
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Jun Furuya and Yoshio Tanigawa
- Subjects
Differentiation under the integral sign ,Pure mathematics ,General Mathematics ,Gauss ,Natural number ,Divisor (algebraic geometry) ,Term (logic) ,Riemann zeta function ,Algebra ,symbols.namesake ,Divisor summatory function ,symbols ,Complex number ,Mathematics - Abstract
In our previous paper [2], we derived an explicit representation of the integral ∫1∞t−θΔ(t)logjtdt by differentiation under the integral sign. Here, j is a fixed natural number, θ is a complex number with 1 < θ ≤ 5/4 and Δ(x) denotes the error term in the Dirichlet divisor problem. In this paper, we shall reconsider the same formula by an alternative approach, which appeals to only the elementary integral formulas concerning the Riemann zeta- and periodic Bernoulli functions. We also study the corresponding formula in the case of the circle problem of Gauss.
- Published
- 2011
219. NOTE ON q-DEDEKIND-TYPE SUMS RELATED TO q-EULER POLYNOMIALS
- Author
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Taekyun Kim
- Subjects
Euler function ,Discrete mathematics ,Pure mathematics ,Euler's criterion ,General Mathematics ,Proof of the Euler product formula for the Riemann zeta function ,Prime (order theory) ,symbols.namesake ,symbols ,Order (group theory) ,Dedekind cut ,Euler number ,Mathematics ,Euler summation - Abstract
Recently, q-Dedekind-type sums related to q-zeta function and basic L-series are studied by Simsek in [13] (Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), 333–351) and Dedekind-type sums related to Euler numbers and polynomials are introduced in the previous paper [11] (T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contem. Math. 18 (2009), 249–260). It is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of the higher order Dedekind the type sums related to q-Euler polynomials and numbers by using an invariant p-adic q-integrals.
- Published
- 2011
220. The Hardy space H1 on non-homogeneous metric spaces
- Author
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Tuomas Hytönen, Dongyong Yang, and Dachun Yang
- Subjects
Mathematics::Functional Analysis ,Dual space ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,Banach space ,Duality (optimization) ,Context (language use) ,Hardy space ,Space (mathematics) ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Combinatorics ,Metric space ,symbols.namesake ,Mathematics - Classical Analysis and ODEs ,42B30 (Primary) 42B20, 42B35 (Secondary) ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
Let $({\mathcal X}, d, \mu)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrical doubling condition. In this paper, we introduce the atomic Hardy space $H^1(\mu)$ and prove that its dual space is the known space ${\rm RBMO}(\mu)$ in this context. Using this duality, we establish a criterion for the boundedness of linear operators from $H^1(\mu)$ to any Banach space. As an application of this criterion, we obtain the boundedness of Calder\'on--Zygmund operators from $H^1(\mu)$ to $L^1(\mu)$., Comment: This paper has been withdrawn by the authors, since it has already been published
- Published
- 2011
221. On the polynomial vector fields on
- Author
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Jaume Llibre and Yulin Zhao
- Subjects
Combinatorics ,Polynomial ,Polynomial vector fields ,Degree (graph theory) ,General Mathematics ,Homogeneous polynomial ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Vector field ,Mathematics - Abstract
Let X be a polynomial vector field of degree n on M, M = ℝm. The dynamics and the algebraic-geometric properties of the vector fields X have been studied intensively, mainly for the case when M = ℝm, and especially when n = 2. Several papers have been dedicated to the study of the homogeneous polynomial vector field of degree n on $\mathbb{S}^2$, mainly for the case where n = 2 and M = $\mathbb{S}^2$. But there are very few results on the non-homogeneous polynomial vector fields of degree n on $\mathbb{S}^2$. This paper attempts to rectify this slightly.
- Published
- 2011
222. Strong renewal theorems and Lyapunov spectra forα-Farey andα-Lüroth systems
- Author
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Marc Kesseböhmer, Sara Munday, and Bernd O. Stratmann
- Subjects
Lyapunov function ,Pure mathematics ,Gauss map ,Computer Science::Information Retrieval ,Applied Mathematics ,General Mathematics ,symbols.namesake ,Number theory ,symbols ,Countable set ,Farey sequence ,Ergodic theory ,Partition (number theory) ,Mathematics ,Unit interval - Abstract
In this paper, we introduce and study theα-Farey map and its associated jump transformation, theα-Lüroth map, for an arbitrary countable partitionαof the unit interval with atoms which accumulate only at the origin. These maps represent linearized generalizations of the Farey map and the Gauss map from elementary number theory. First, a thorough analysis of some of their topological and ergodic theoretical properties is given, including establishing exactness for both types of these maps. The first main result then is to establish weak and strong renewal laws for what we have calledα-sum-level sets for theα-Lüroth map. Similar results have previously been obtained for the Farey map and the Gauss map by using infinite ergodic theory. In this respect, a side product of the paper is to allow for greater transparency of some of the core ideas of infinite ergodic theory. The second remaining result is to obtain a complete description of the Lyapunov spectra of theα-Farey map and theα-Lüroth map in terms of the thermodynamical formalism. We show how to derive these spectra and then give various examples which demonstrate the diversity of their behaviours in dependence on the chosen partitionα.
- Published
- 2011
223. A CHARACTERIZATION OF (−1, −1)-FREUDENTHAL–KANTOR TRIPLE SYSTEMS
- Author
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Susumu Okubo, Noriaki Kamiya, and Daniel Mondoc
- Subjects
Algebra ,Pure mathematics ,Triple product ,General Mathematics ,Product (mathematics) ,Bilinear interpolation ,Characterization (mathematics) ,Connection (algebraic framework) ,Mathematics - Abstract
In this paper, we discuss a connection between (−1, −1)-Freudenthal–Kantor triple systems, anti-structurable algebras, quasi anti-flexible algebras and give examples of such structures. The paper provides the correspondence and characterization of a bilinear product corresponding a triple product.
- Published
- 2011
224. On exponential limit laws for hitting times of rare sets for Harris chains and processes
- Author
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Peter W. Glynn
- Subjects
Harris recurrent Markov process ,Statistics and Probability ,Exponential distribution ,60G70 ,General Mathematics ,01 natural sciences ,Time reversibility ,Combinatorics ,Hitting time ,010104 statistics & probability ,60J05 ,60K05 ,60J25 ,60F05 ,Phase-type distribution ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Markov chain ,010102 general mathematics ,regenerative process ,Harris chain ,Harris recurrent Markov chain ,Markov property ,Statistics, Probability and Uncertainty - Abstract
This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.
- Published
- 2011
225. ON THE SOLVABILITY OF SYSTEMS OF SUM–PRODUCT EQUATIONS IN FINITE FIELDS
- Author
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Le Anh Vinh
- Subjects
Finite field ,General Mathematics ,Product (mathematics) ,Applied mathematics ,Mathematics - Abstract
In an earlier paper, for ‘large’ (but otherwise unspecified) subsets , , , of q, Sárközy showed the solvability of the equations a + b = cd with a ∈ , b ∈ , c ∈ , d ∈ . This equation has been studied recently by many other authors. In this paper, we study the solvability of systems of equations of this type using additive character sums.
- Published
- 2011
226. Fisher information and statistical inference for phase-type distributions
- Author
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Mogens Bladt, Bo Friis Nielsen, and Luz Judith R. Esparza
- Subjects
Statistics and Probability ,Fisher information ,General Mathematics ,Fisher kernel ,Fisher consistency ,Newton--Raphson ,symbols.namesake ,60J27 ,Observed information ,Scoring algorithm ,Expectation–maximization algorithm ,Statistics ,symbols ,Fiducial inference ,60J10 ,Applied mathematics ,62F25 ,60J75 ,Statistics, Probability and Uncertainty ,EM algorithm ,Likelihood function ,Phase-type distribution ,Mathematics - Abstract
This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.
- Published
- 2011
227. Self-adaptive Monte Carlo localization for mobile robots using range finders
- Author
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René Zapata, Lei Zhang, Pascal Lepinay, Robotique mobile pour l'exploration de l'environnement (EXPLORE), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), and Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)
- Subjects
0209 industrial biotechnology ,business.industry ,General Mathematics ,Monte Carlo method ,Monte Carlo localization ,Sample (statistics) ,Mobile robot ,02 engineering and technology ,Kidnapped robot problem ,[SPI.AUTO]Engineering Sciences [physics]/Automatic ,Computer Science Applications ,Set (abstract data type) ,020901 industrial engineering & automation ,Control and Systems Engineering ,0202 electrical engineering, electronic engineering, information engineering ,Robot ,020201 artificial intelligence & image processing ,Computer vision ,Artificial intelligence ,Focus (optics) ,business ,Algorithm ,Software ,Mathematics - Abstract
SUMMARYIn order to achieve the autonomy of mobile robots, effective localization is a necessary prerequisite. In this paper, we propose an improved Monte Carlo localization algorithm using self-adaptive samples (abbreviated as SAMCL). By employing a pre-caching technique to reduce the online computational burden, SAMCL is more efficient than the regular MCL. Further, we define the concept of similar energy region (SER), which is a set of poses (grid cells) having similar energy with the robot in the robot space. By distributing global samples in SER instead of distributing randomly in the map, SAMCL obtains a better performance in localization. Position tracking, global localization and the kidnapped robot problem are the three sub-problems of the localization problem. Most localization approaches focus on solving one of these sub-problems. However, SAMCL solves all the three sub-problems together, thanks to self-adaptive samples that can automatically separate themselves into a global sample set and a local sample set according to needs. The validity and the efficiency of the SAMCL algorithm are demonstrated by both simulations and experiments carried out with different intentions. Extensive experimental results and comparisons are also given in this paper.
- Published
- 2011
228. Dimension of the generalized 4-corner set and its projections
- Author
-
Balázs Bárány
- Subjects
Discrete mathematics ,Set (abstract data type) ,Iterated function system ,Dimension (vector space) ,Applied Mathematics ,General Mathematics ,Computation ,Hausdorff dimension ,Hausdorff space ,Dimension theory ,Calculus ,Fixed point ,Mathematics - Abstract
In the last two decades, considerable attention has been paid to the dimension theory of self-affine sets. In the case of generalized 4-corner sets (see Figure 1), the iterated function systems obtained as the projections of self-affine systems have maps of common fixed points. In this paper, we extend our result [B. Bárány. On the Hausdorff dimension of a family of self-similar sets with complicated overlaps. Fund. Math. 206 (2009), 49–59], which introduced a new method of computation of the box and Hausdorff dimensions of self-similar families where some of the maps have common fixed points. The extended version of our method presented in this paper makes it possible to determine the box dimension of the generalized 4-corner set for Lebesgue-typical contracting parameters.
- Published
- 2011
229. Differentiating potential functions of SRB measures on hyperbolic attractors
- Author
-
Miaohua Jiang
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Derivative ,Chain rule ,Measure (mathematics) ,Manifold ,Volume form ,symbols.namesake ,Attractor ,Jacobian matrix and determinant ,symbols ,Differentiable function ,Mathematics - Abstract
The derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the differentiation process of the unstable Jacobian and provide a complete derivation of its derivative formula. Our approach is to extend the volume form provided by the SRB measure on local unstable manifolds to a system of Hölder continuous local Riemannian metrics on the manifold so that under this system of local metrics, the unstable Jacobian becomes differentiable with respect to the base point and its derivative with respect to the map can be obtained by the chain rule.
- Published
- 2011
230. A NOTE ON EDGE-CONNECTIVITY OF THE CARTESIAN PRODUCT OF GRAPHS
- Author
-
Lakoa. Fitina, Terence M. Mills, and Christopher T. Lenard
- Subjects
Combinatorics ,Discrete mathematics ,symbols.namesake ,Cartesian product of graphs ,General Mathematics ,symbols ,Graph theory ,Edge (geometry) ,Cartesian product ,Mathematics - Abstract
The main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.
- Published
- 2011
231. Two-state trajectory tracking control of a spherical robot using neurodynamics
- Author
-
Qiang Zhan, Yao Cai, and Caixia Yan
- Subjects
Lyapunov function ,Lyapunov stability ,Nonholonomic system ,General Mathematics ,Control engineering ,Computer Science Applications ,Computer Science::Robotics ,Controllability ,symbols.namesake ,Control and Systems Engineering ,Control theory ,symbols ,Trajectory ,Robot ,Spherical robot ,Software ,Mathematics - Abstract
SUMMARYSpherical robot is a special kind of nonholonomic system that cannot be converted to chained form, which means most of the well-known control methodologies are not suitable for this system. For the trajectory tracking of BHQ-1, a spherical robot designed by our lab, a two-state trajectory tracking controller is proposed in this paper. First, the kinematic model of the robot is built using screw theory and exponential method and the controllability is proved based on the differential geometric control theory. Then to solve the two-state trajectory tracking problem of BHQ-1, a shunting model of neurodynamics and Lyapunov's direct method are combined to design a two-state trajectory tracking controller, of which the Lyapunov stability is validated. Finally, typical simulation examples, such as tracking linear, circular, and sinusoidal trajectories, are introduced to verify the effectiveness of the proposed controller. In this paper the proposed method can also be applied to the control of other spherical robots.
- Published
- 2011
232. Invariant rigid geometric structures and expanding maps
- Author
-
Yong Fang
- Subjects
Chaotic dynamical systems ,Pure mathematics ,Closed manifold ,Rigidity (electromagnetism) ,Homogeneous ,Applied Mathematics ,General Mathematics ,Invariant (mathematics) ,Algorithm ,Mathematics - Abstract
In the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.
- Published
- 2011
233. Euler's parallel oblique-angled diameters
- Author
-
Thomas J. Osler
- Subjects
symbols.namesake ,Conic section ,General Mathematics ,Euler's formula ,symbols ,Oblique case ,Geometry ,Mathematics - Abstract
In the paper [1], Euler was examining properties of the conic sections that could be shared by more general curves. Most of the paper is concerned with ‘oblique-angle diameters’, a concept that seems to have been familiar to his readers in the eighteenth century, but has been ignored today. In this paper we will explain this concept and, led by Euler, develop some of its consequences.
- Published
- 2011
234. Eigencircles and associated surfaces
- Author
-
Michael Englefield and Graham Farr
- Subjects
Pure mathematics ,Matrix (mathematics) ,General Mathematics ,Linear algebra ,Connection (mathematics) ,Mathematics - Abstract
Linear algebra has many fruitful connections with geometry. This article develops one such connection: the relationship between a 2 × 2 matrix and an associated circle which we call the eigencircle.This connection was first investigated in a previous paper of ours [1], but the present paper is self-contained, and in fact introduces eigencircles in a different way. Here we discuss some surfaces containing the eigencircle which also have a number of interesting properties and connections with the associated matrix.
- Published
- 2010
235. On maximal pattern complexity of some automatic words
- Author
-
Pavel V. Salimov and Teturo Kamae
- Subjects
Combinatorics ,Discrete mathematics ,Linear function (calculus) ,Applied Mathematics ,General Mathematics ,Bounded function ,Substitution (logic) ,Value (computer science) ,Function (mathematics) ,Fixed point ,Constant (mathematics) ,Word (group theory) ,Mathematics - Abstract
The pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.
- Published
- 2010
236. Lower bounds for Clifford indices in rank three
- Author
-
Herbert Lange and Peter E. Newstead
- Subjects
Projective curve ,Naturwissenschaftliche Fakultät -ohne weitere Spezifikation ,Rank (linear algebra) ,Degree (graph theory) ,Plane curve ,General Mathematics ,Vector bundle ,Clifford bundle ,Algebra ,Combinatorics ,Mathematics::Algebraic Geometry ,Genus (mathematics) ,ddc:510 ,Mathematics - Abstract
Clifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.
- Published
- 2010
237. Instability of one global transonic shock wave for the steady supersonic Euler flow past a sharp cone
- Author
-
Gang Xu and Huicheng Yin
- Subjects
Shock wave ,Shock (fluid dynamics) ,76N15 ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Mechanics ,35L70 ,01 natural sciences ,35L67 ,35L65 ,Shock diamond ,Oblique shock ,Supersonic speed ,Bow shock (aerodynamics) ,0101 mathematics ,Transonic ,Ludwieg tube ,Astrophysics::Galaxy Astrophysics ,Mathematics - Abstract
In this paper, we are concerned with the instability problem of one global transonic conic shock wave for the supersonic Euler flow past an infinitely long conic body whose vertex angle is less than some critical value. This is motivated by the following descriptions in the bookSupersonic Flow and Shock Wavesby Courant and Friedrichs: if there is a supersonic steady flow which comes from minus infinity, and the flow hits a sharp cone along its axis direction, then it follows from the Rankine-Hugoniot conditions, the physical entropy condition, and the apple curve method that there will appear a weak shock or a strong shock attached at the vertex of the cone, which corresponds to the supersonic shock or the transonic shock, respectively. A long-standing open problem is that only the weak shock could occur, and the strong shock is unstable. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand this. In particular, under some suitable assumptions, because of the essential influence of the rotation of Euler flow, we show that a global transonic conic shock solution is unstable as long as the related sharp circular cone is perturbed.
- Published
- 2010
238. ON THE RELATIVE LUSTERNIK–SCHNIRELMANN CATEGORY WITH RESPECT TO A REAL COHOMOLOGY CLASS
- Author
-
Dirk Schütz and Tieqiang Li
- Subjects
Discrete mathematics ,Pure mathematics ,Closed category ,General Mathematics ,Homotopy ,Lusternik–Schnirelmann category ,Homoclinic orbit ,Invariant (mathematics) ,Upper and lower bounds ,Cohomology ,Mathematics - Abstract
In this paper, we study a homotopy invariant cat(X, B, [ω]) on a pair (X, B) of finite CW complexes with respect to the cohomology class of a continuous closed 1-form ω. This is a generalisation of a Lusternik–Schnirelmann-category-type cat(X, [ω]), developed by Farber in [3, 4], studying the topology of a closed 1-form. This paper establishes the connection with the original notion cat(X, [ω]) and obtains analogous results on critical points and homoclinic cycles. We also provide a similar ‘cuplength’ lower bound for cat(X, B, [ω]).
- Published
- 2010
239. AN INVERSE THEOREM FOR THE GOWERSU4-NORM
- Author
-
Terence Tao, Tamar Ziegler, and Ben Green
- Subjects
Conjecture ,Mathematics - Number Theory ,General Mathematics ,010102 general mathematics ,Linear system ,Inverse ,Dynamical Systems (math.DS) ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,010201 computation theory & mathematics ,Norm (mathematics) ,Bounded function ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| = ��then there is a bounded complexity 3-step nilsequence F(g(n)��) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5, 49 pages, to appear in Glasgow J. Math. Fixed a problem with the file (the paper appeared in duplicate)
- Published
- 2010
240. Corrigendum: amenability of ultrapowers of Banach algebras
- Author
-
Matthew Daws
- Subjects
Rest (physics) ,Pure mathematics ,Tensor product ,Approximation property ,General Mathematics ,Diagonal ,Algebra over a field ,Characterization (mathematics) ,Ultraproduct ,Topology ,Banach *-algebra ,Mathematics - Abstract
Some of the results of § 5 of the cited paper are incorrect: in particular, the characterization of when an algebra is ultra-amenable, in terms of a diagonal like construction, is not proved; and Theorem 5.7 is stated wrongly. The rest of the paper is unaffected. We shall show in this corrigendum that Theorem 5.7 can be corrected and that the other results of § 5 are true if the algebra in question has a certain approximation property.
- Published
- 2010
241. Kinematic calibration of the 3-DOF parallel module of a 5-axis hybrid milling machine
- Author
-
Jinsong Wang, Fugui Xie, Liping Wang, and Xin-Jun Liu
- Subjects
Basis (linear algebra) ,Robot calibration ,Calibration (statistics) ,General Mathematics ,Kinematics ,Linear interpolation ,Computer Science Applications ,Nonlinear system ,Matrix (mathematics) ,Control and Systems Engineering ,Control theory ,Linear combination ,Software ,Mathematics - Abstract
SUMMARYThis paper investigates the kinematic calibration of a 3-DOF parallel mechanism based on the minimal linear combinations of error parameters. The error mapping function between the geometric errors and the output errors is formulated and the identification matrix is generated and simplified. In order to identify the combinations of error parameters, four theorems to analyze the columns of the simplified identification matrix are introduced. Then, an anti-disturbance index is presented to evaluate the identification performance. On the basis of this index, measurement strategy is developed and optimal measuring configurations are given. After external calibration, linear interpolation compensation is applied to improve the terminal accuracy further. Results of experiment show that the method used in this paper is effective and efficient, and the errors are convergent within two iterations generally. This method can be extended to other parallel mechanisms with weakly nonlinear kinematics.
- Published
- 2010
242. THE ESSENTIAL SPECTRUM OF A PERTURBED OPERATOR ARISING IN TWO-DIMENSIONAL MAGNETOHYDRODYNAMICS
- Author
-
M. Faierman and Reinhard Mennicken
- Subjects
Algebra ,Pure mathematics ,General Mathematics ,Operator (physics) ,Essential spectrum ,Magnetohydrodynamics ,Mathematics - Abstract
Descloux and Geymonat considered a model problem in two-dimensional magnetohydrodynamics and conjectured that the essential spectrum has an explicitly given band structure. This conjecture was recently proved by Faierman, Mennicken, and Möller by reducing the problem to that for a 2×2 block operator matrix. In a subsequent paper Faierman and Mennicken investigated the essential spectrum for the problem arising from a particular type of perturbation of precisely one of the operator entries in the matrix representation cited above of the original problem considered by Descloux and Geymonat. In this paper we extend the results of that work by investigating the essential spectrum for the problem arising from particular types of perturbations of all but one of the aforementioned operators. It remains an open question whether one can perturb the exceptional operator in such a way as to leave the essential spectrum unchanged.
- Published
- 2010
243. A SUFFICIENT CONDITION FOR A GRAPH TO BE A FRACTIONAL (f, n)-CRITICAL GRAPH
- Author
-
Sizhong Zhou
- Subjects
Factor-critical graph ,Combinatorics ,Discrete mathematics ,Graph bandwidth ,Edge-transitive graph ,Graph power ,General Mathematics ,Voltage graph ,Quartic graph ,Cubic graph ,Distance-regular graph ,Mathematics - Abstract
Let a, b and n be non-negative integers such that 1 ≤ a ≤ b, and let G be a graph of order p with $\(p\geq\frac{(a+b-1)(a+b-2)+bn-2}{a}\)$ and f be an integer-valued function defined on V(G) such that a ≤ f(x) ≤ b for all x ∈ V(G). Let h: E(G) → [0, 1] be a function. If ∑e∋xh(e) = f(x) holds for any x ∈ V(G), then we call G[Fh] a fractional f-factor of G with indicator function h, where Fh = {e ∈ E(G): h(e) > 0}. A graph G is called a fractional (f, n)-critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. In this paper, it is proved that G is a fractional (f, n)-critical graph if $\(|N_G(X)|>\frac{(b-1)p+|X|+bn-1}{a+b-1}\)$ for every non-empty independent subset X of V(G), and $\(\delta(G)>\frac{(b-1)p+a+b+bn-2}{a+b-1}\)$. Furthermore, it is shown that the result in this paper is best possible in some sense.
- Published
- 2010
244. Generalized Increasing Convex and Directionally Convex Orders
- Author
-
Mhamed Mesfioui and Michel Denuit
- Subjects
Statistics and Probability ,Convex analysis ,Convex hull ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Proper convex function ,Convex set ,Subderivative ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Convex optimization ,Convex polytope ,Convex combination ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.
- Published
- 2010
245. On travelling wavefronts of Nicholson's blowflies equation with diffusion
- Author
-
Ming Mei and Chi-Kun Lin
- Subjects
Wavefront ,General Mathematics ,Mathematical analysis ,Reaction–diffusion system ,Perturbation (astronomy) ,Wave speed ,Mathematics - Abstract
This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.
- Published
- 2010
246. Orientation workspace analysis of a special class of the Stewart–Gough parallel manipulators
- Author
-
Weixi Ji, Yi Cao, Hui Zhou, and Zhen Huang
- Subjects
Inverse kinematics ,Discretization ,General Mathematics ,Parallel manipulator ,Context (language use) ,Kinematics ,Workspace ,Orientation (graph theory) ,Computer Science Applications ,Computer Science::Robotics ,Control and Systems Engineering ,Control theory ,Robot ,Software ,Mathematics - Abstract
SUMMARYThe workspace of a robotic manipulator is a very important issue and design criteria in the context of optimum design of robots, especially for parallel manipulators. Though, considerable research has been paid to the investigations of the three-dimensional (3D) constant orientation workspace or position workspace of parallel manipulators, very few works exist on the topic of the 3D orientation workspace, especially the nonsingular orientation workspace and practical orientation workspace. This paper addresses the orientation workspace analysis of a special class of the Stewart–Gough parallel manipulators in which the moving and base platforms are two similar semisymmetrical hexagons. Based on the half-angle transformation, a polynomial expression of 13 degree that represents the orientation singularity locus of this special class of the Stewart–Gough parallel manipulators at a fixed position is derived and graphical representations of the orientation singularity locus of this special class of the Stewart–Gough manipulators are illustrated with examples to demonstrate the result. Exploiting this half-angle transformation and the inverse kinematics solution of this special class of the Stewart–Gough parallel manipulators, a discretization method is proposed for computing the orientation workspace of this special class of the Stewart–Gough parallel manipulators taking limitations of active and passive joints and the link interference all into consideration. Based on this algorithm, this paper also presents a new discretization method for computing the nonsingular orientation workspace of this class of the manipulators, which not only can satisfy all the kinematics demand of this class of the manipulators but also can guarantee the manipulator is nonsingular in the whole orientation workspace, and the practical orientation workspace of this class of the manipulators, which not only can guarantee the manipulator is nonsingular and will never encounter any kinematic interference but also can satisfy the demand of the orientation workspace with a regular shape in practical application, respectively. Examples of a 6/6-SPS Stewart–Gough parallel manipulator of this special class are given to demonstrate these theoretical results.
- Published
- 2010
247. ON POSITIVITY OF SEVERAL COMPONENTS OF SOLUTION VECTOR FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
- Author
-
Ravi P. Agarwal and Alexander Domoshnitsky
- Subjects
Stochastic partial differential equation ,Examples of differential equations ,Pure mathematics ,Matrix differential equation ,Linear differential equation ,Computer Science::Information Retrieval ,General Mathematics ,Linear form ,Ordinary differential equation ,Mathematical analysis ,First-order partial differential equation ,System of linear equations ,Mathematics - Abstract
In the classical theorems about lower and upper vector functions for systems of linear differential equations very heavy restrictions on the signs of coefficients are assumed. These restrictions in many cases become necessary if we wish to compare all the components of a solution vector. The formulas of the integral representation of the general solution explain that these theorems claim actually the positivity of all elements of the Green's matrix. In this paper we define a principle of partial monotonicity (comparison of only several components of the solution vector), which assumes only the positivity of elements in a corresponding row of the Green's matrix. The main theorem of the paper claims the equivalence of positivity of all elements in the nth row of the Green's matrices of the initial and two other problems, non-oscillation of the nth component of the solution vector and a corresponding assertion about differential inequality of the de La Vallee Poussin type. Necessary and sufficient conditions of the partial monotonicity are obtained. It is demonstrated that our sufficient tests of positivity of the elements in the nth row of the Cauchy matrix are exact in corresponding cases. The main idea in our approach is a construction of an equation for the nth component of the solution vector. In this sense we can say that an analog of the classical Gauss method for solving systems of functional differential equations is proposed in the paper.
- Published
- 2009
248. The Early Stage Behaviour of a Stochastic SIR Epidemic with Term-Time Forcing
- Author
-
Mathias Lindholm and Tom Britton
- Subjects
Statistics and Probability ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,Context (language use) ,Intersection graph ,01 natural sciences ,Giant component ,010104 statistics & probability ,Probability theory ,Statistics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Epidemic model ,education ,Branching process ,Mathematics - Abstract
This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R. The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied. Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied. The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.
- Published
- 2009
249. A Jacobian-based algorithm for planning the motion of an underactuated rigid body undergoing forward and reverse rotations
- Author
-
Sung k. Koh
- Subjects
Sequence ,Inverse kinematics ,Underactuation ,General Mathematics ,Rigid body ,Computer Science Applications ,Computer Science::Robotics ,symbols.namesake ,Control and Systems Engineering ,Control theory ,Orientation (geometry) ,Jacobian matrix and determinant ,symbols ,Motion planning ,Configuration space ,Algorithm ,Software ,Mathematics - Abstract
SUMMARYA Jacobian-based algorithm that is useful for planning the motion of a floating rigid body operated using two input torques is addressed in this paper. The rigid body undergoes a four-rotation fully reversed (FR) sequence of rotations which consists of two initial rotations about the axes of a coordinate frame attached to the body and two subsequent rotations that undo the preceding rotations. Although a Jacobian-based algorithm has been useful in exploring the inverse kinematics of conventional robot manipulators, it is not apparent how a correct FR sequence for a desired orientation could be found because the Jacobian of FR sequences is singular as well as being a null matrix at the identity. To discover the FR sequences that can synthesize the desired orientation circumventing these difficulties, the Jacobian algorithm is reformulated and implemented from arbitrary orientations where the Jacobian is not singular. Due to the insufficient degrees-of-freedom of four-rotation FR sequences required to achieve all possible orientations, the rigid body cannot achieve certain orientations in the configuration space. To best approximate these infeasible orientations, the Jacobian-based algorithm is implemented in the sense of least squares. As some orientations can never be attained by a single four-rotation FR sequence, two different four-rotation FR sequences are exploited alternately to ensure the convergence of the proposed algorithm. Assuming the orientation is supposed to be manipulated using three input torques, the switching Jacobian algorithm proposed in this paper has significant practical importance in planning paths for aerospace and underwater vehicles which are maneuvered using only two input torques due to the failure of one of the torque-generation mechanisms.
- Published
- 2009
250. Supplement to 'The group E6(q) and graphs with a locally linear group of automorphisms' by V. I. Trofimov and R. M. Weiss
- Author
-
V. I. Trofimov
- Subjects
Normal subgroup ,Vertex (graph theory) ,Combinatorics ,General Mathematics ,Mineralogy ,Permutation group ,Automorphism ,Linear subspace ,Prime power ,Graph ,Vector space ,Mathematics - Abstract
Let q be a prime power and let G be a group acting faithfully and vertex transitively on a graph such that for each vertex x, the stabilizer Gx is finite and contains a normal subgroup inducing on the set of neighbours of x a permutation group isomorphic to the linear group L5(q) acting on the 2-dimensional subspaces of a 5-dimensional vector space over Fq. In a companion paper, it is shown, except in some special situations where q = 2, that the kernel of the action of a vertex stabilizer Gx on the ball of radius 3 around x is trivial. In this paper we show that these special situations cannot occur.
- Published
- 2009
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