8,131 results
Search Results
102. 346. [V. a.] Squared Paper Solution of the Equation α cos θ+b sin θ=c
- Author
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A. C. Dixon
- Subjects
Combinatorics ,Pure mathematics ,General Mathematics ,Mathematics - Published
- 1911
103. Note on a paper by S. C. R. Dennis
- Author
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E. T. Goodwin
- Subjects
General Mathematics ,Mathematics - Published
- 1965
104. SELF-COMPLEMENTARY VERTEX-TRANSITIVE GRAPHS OF ORDER A PRODUCT OF TWO PRIMES
- Author
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Cai Heng Li and Guang Rao
- Subjects
Combinatorics ,Discrete mathematics ,Vertex (graph theory) ,Transitive relation ,General Mathematics ,Short paper ,Mathematics - Abstract
In this short paper, we characterise graphs of order $pq$ with $p, q$ prime which are self-complementary and vertex-transitive.
- Published
- 2013
105. WIENER INDEX AND TRACEABLE GRAPHS
- Author
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Lihui Yang
- Subjects
Combinatorics ,Discrete mathematics ,General Mathematics ,Short paper ,Wiener index ,Connectivity ,Mathematics - Abstract
In this short paper, we show that, with three exceptions, if the Wiener index of a connected graph of order $n$ is at most $(n+ 5)(n- 2)/ 2$, then it is traceable.
- Published
- 2012
106. On generation of the root lattice by roots
- Author
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Simon M. Goodwin
- Subjects
Pure mathematics ,Lattice (module) ,General Mathematics ,Short paper ,Root (chord) ,Basis (universal algebra) ,Root system ,Mathematics - Abstract
Let Φ be a root system and let Γ ⊆ Φ. In this short paper we prove that Γ contains a ${\mathbb Z}$-basis of the lattice that it generates.
- Published
- 2007
107. The Γ-opial property
- Author
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Małgorzata Michalska, Monika Budzyńska, and Tadeusz Kuczumow
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Set (abstract data type) ,Pure mathematics ,General Mathematics ,Bounded function ,Short paper ,Banach space ,Opial property ,Mathematics ,Zero-product property - Abstract
In this short paper we show that if (X, ∥ · ∥) is a Banach space, Γ a norming set for X and C is a nonempty, bounded and Γ sequentially compact subset of X, then in C the Γ-Opial condition for nets is equivalent to the Γ-Opial condition.
- Published
- 2006
108. THE BIOLOGICAL TREATMENT OF WASTEWATER: MATHEMATICAL MODELS
- Author
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Asma O. M. Alharbi
- Subjects
General Mathematics ,Biomass ,02 engineering and technology ,010501 environmental sciences ,Residence time (fluid dynamics) ,Pulp and paper industry ,01 natural sciences ,020303 mechanical engineering & transports ,Activated sludge ,0203 mechanical engineering ,Wastewater ,Volume (thermodynamics) ,Bioreactor ,Sewage treatment ,Water treatment ,0105 earth and related environmental sciences ,Mathematics - Abstract
The activated sludge process is one of the major aerobic processes used in the biological treatment of wastewater. A significant drawback of this process is the production of excess sludge, the disposal of which can account for 50-60% of the running costs of a plant. Thus there is a growing interest in methods that reduce the volume and mass of excess sludge produced as part of biological wastewater treatment processes. In practice a target value is often set for the sludge content inside the bioreactor. If the sludge content is higher than the target value, the process is stopped and the reactor is cleaned. This is undesirable as it increases running costs. In chapter 2 we investigate a simple model for the activated sludge process in which the influent contains a mixture of soluble and biodegradable particulate substrate. Within the bioreactor the biodegradable particulate substrate is hydrolyzed to form soluble substrate. The soluble organics are used for energy and growth by the biomass. Biomass decay produces soluble substrate in addition to inert material. We use steady-state analysis to investigate how the amount of sludge formed depends upon the residence time and the use of a settling unit. We show that when the steady-state sludge content is plotted as a function of the residence time that there are five generic response diagrams, depending upon the value of the effective recycle parameter. Four of them are desirable because the sludge content is below the target value if the residence time is higher than some critical value that is not ‘too large’ in practice. In chapter 3 we investigate how the volume and mass of excess sludge produced by the activated sludge process can be reduced by coupling the bioreactor used in the process to a sludge disintegration unit. In chapter 4 a seemingly minor modification is made to the model in chapter 2.
- Published
- 2016
109. 87.17 An unusual product for sinz and variations of Wallis's product
- Author
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Thomas J. Osler
- Subjects
General Mathematics ,Product (mathematics) ,Pulp and paper industry ,Mathematics - Published
- 2003
110. 77.8 The greatest product
- Author
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B.S. Beevers
- Subjects
General Mathematics ,Product (mathematics) ,Pulp and paper industry ,Mathematics - Published
- 1993
111. On the line-comitants of a space cubic
- Author
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R. W. Weitzenböck
- Subjects
Pure mathematics ,General Mathematics ,Short paper ,Cubic form ,Projective invariants ,Invariant (mathematics) ,Mathematics - Abstract
In a short paper recently published in these Proceedings, E. J. F. Primrose determines amongst other things a projective invariant N of a space cubic K and a linear complex L, N being of the third degree in the coefficients cik of the complex L. The purpose of the following remarks is to establish the connexion between this invariant N and the well-known projective comitants of the cubic K.
- Published
- 1951
112. A cardboard approach to factor groups
- Author
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Keith Austin
- Subjects
General Mathematics ,visual_art ,visual_art.visual_art_medium ,cardboard ,Pulp and paper industry ,Mathematics - Abstract
Example 1Take a cardboard equilateral triangle. It is well known that this has six symmetries. To display these symmetries we colour the vertices of the triangle on both sides of the card with three different colours.
- Published
- 1983
113. Chaotic behavior of the p-adic Potts–Bethe mapping II
- Author
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Otabek Khakimov and Farrukh Mukhamedov
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Chaotic ,Mathematics - Abstract
The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).
- Published
- 2021
114. Topologically mixing tiling of generated by a generalized substitution
- Author
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Tyler M. White
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Substitution (logic) ,Mixing (physics) ,Mathematics - Abstract
This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.
- Published
- 2021
115. Multiplicative constants and maximal measurable cocycles in bounded cohomology
- Author
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Marco Moraschini, Alessio Savini, Moraschini M., and Savini A.
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Multiplicative function ,Lattice ,Geometric Topology (math.GT) ,Cohomology ,Mathematics - Geometric Topology ,Maximal cocycle ,Mathematics::Quantum Algebra ,Bounded function ,FOS: Mathematics ,Bounded cohomology ,Boundary map ,Invariant (mathematics) ,Zimmer cocycle ,Mathematics - Abstract
Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices., Comment: 35 pages; Major corrections along the paper following the referee's suggestions. To appear in Ergod. Theory Dyn. Syst
- Published
- 2021
116. EXTREME POINT METHODS IN THE STUDY OF ISOMETRIES ON CERTAIN NONCOMMUTATIVE SPACES
- Author
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Jurie Conradie and Pierre de Jager
- Subjects
Pure mathematics ,General Mathematics ,Extreme point ,Noncommutative geometry ,Mathematics - Abstract
In this paper, we characterize surjective isometries on certain classes of noncommutative spaces associated with semi-finite von Neumann algebras: the Lorentz spaces $L^{w,1}$ , as well as the spaces $L^1+L^\infty$ and $L^1\cap L^\infty$ . The technique used in all three cases relies on characterizations of the extreme points of the unit balls of these spaces. Of particular interest is that the representations of isometries obtained in this paper are global representations.
- Published
- 2021
117. A group of Pythagorean triples using the inradius
- Author
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Howard Sporn
- Subjects
Combinatorics ,Coprime integers ,Group (mathematics) ,General Mathematics ,Pythagorean triple ,Right triangle ,Mathematics ,Incircle and excircles of a triangle - Abstract
Pythagorean triples are triples of integers (a, b, c) satisfying the equation a2 + b2 = c2. For the purpose of this paper, we will take a, b and c to be positive, unless otherwise stated. Then, of course, it follows that a triple represents the lengths of sides of a right triangle. Also, for the purpose of this paper, we will consider the triples (a, b, c) and (b, a, c) to be distinct, even though they represent the same right triangle. A primitive Pythagorean triple is one for which a, b and c are relatively prime.
- Published
- 2021
118. Fourier restriction in low fractal dimensions
- Author
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Bassam Shayya
- Subjects
Conjecture ,Measurable function ,Characteristic function (probability theory) ,General Mathematics ,Second fundamental form ,010102 general mathematics ,42B10, 42B20 (Primary), 28A75 (Secondary) ,0102 computer and information sciences ,Function (mathematics) ,Lebesgue integration ,01 natural sciences ,Measure (mathematics) ,Combinatorics ,symbols.namesake ,Hypersurface ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,symbols ,0101 mathematics ,Mathematics - Abstract
Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each radius, then the problem of determining the range of exponents $(p,q)$ for which the estimate $\| Ef \|_{L^q(X)} \leq C \| f \|_{L^p(S)}$ holds is equivalent to the restriction conjecture. In this paper, we study the estimate under the following assumption on the set $X$: there is a number $0 < \alpha \leq n$ such that $|X \cap B_R| \leq c \, R^\alpha$ for all balls $B_R$ in $\Bbb R^n$ of radius $R \geq 1$. On the left-hand side of this estimate, we are integrating the function $|Ef(x)|^q$ against the measure $\chi_X dx$. Our approach consists of replacing the characteristic function $\chi_X$ of $X$ by an appropriate weight function $H$, and studying the resulting estimate in three different regimes: small values of $\alpha$, intermediate values of $\alpha$, and large values of $\alpha$. In the first regime, we establish the estimate by using already available methods. In the second regime, we prove a weighted H\"{o}lder-type inequality that holds for general non-negative Lebesgue measurable functions on $\Bbb R^n$, and combine it with the result from the first regime. In the third regime, we borrow a recent fractal Fourier restriction theorem of Du and Zhang and combine it with the result from the second regime. In the opposite direction, the results of this paper improve on the Du-Zhang theorem in the range $0 < \alpha < n/2$., Comment: 31 pages. Minor revision
- Published
- 2021
119. CARLESON INTERPOLATING SEQUENCES FOR BANACH SPACES OF ANALYTIC FUNCTIONS
- Author
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Paweł Mleczko, David Norrbo, Michał Rzeczkowski, Mikael Lindström, and Mieczysław Mastyło
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics::Classical Analysis and ODEs ,Banach space ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics ,Analytic function - Abstract
This paper presents an approach, based on interpolation theory of operators, to the study of interpolating sequences for interpolation Banach spaces between Hardy spaces. It is shown that the famous Carleson result forH∞can be lifted to a large class of abstract Hardy spaces. A description is provided of the range of the Carleson operator defined on interpolation spaces between the classical Hardy spaces in terms of uniformly separated sequences. A key role in this description is played by some general interpolation results proved in the paper. As by-products, novel results are obtained which extend the Shapiro–Shields result on the characterisation of interpolation sequences for the classical Hardy spacesHp. Applications to Hardy–Lorentz, Hardy–Marcinkiewicz and Hardy–Orlicz spaces are presented.
- Published
- 2021
120. Local limit theorems in relatively hyperbolic groups I: rough estimates
- Author
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Matthieu Dussaule
- Subjects
Pure mathematics ,Series (mathematics) ,010201 computation theory & mathematics ,Spectral radius ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Limit (mathematics) ,0101 mathematics ,Random walk ,01 natural sciences ,Mathematics - Abstract
This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.
- Published
- 2021
121. Higher horospherical limit sets for G-modules over CAT(0)-spaces
- Author
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Ross Geoghegan and Robert Bieri
- Subjects
Pure mathematics ,Discrete group ,Euclidean space ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Space (mathematics) ,01 natural sciences ,Action (physics) ,Zeroth law of thermodynamics ,010201 computation theory & mathematics ,Tropical geometry ,Limit (mathematics) ,0101 mathematics ,Group theory ,Mathematics - Abstract
The Σ-invariants of Bieri–Neumann–Strebel and Bieri–Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Σ-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The “zeroth stage” of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the “nth stage” for any n.
- Published
- 2021
122. THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS’S CONJECTURE
- Author
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Alexander Bertoloni Meli
- Subjects
Conjecture ,Mathematics - Number Theory ,General Mathematics ,Prove it ,Type (model theory) ,Cohomology ,Combinatorics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Morphism ,Square-integrable function ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We study the $l$-adic cohomology of unramified Rapoport-Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm{GL_n}$ and to show local-global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms, $\mathrm{Mant}_{b, \mu}$, of Grothendieck groups of representations constructed from the cohomology of the above spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin, and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm{Mant}_{b, \mu}(\rho)$ for $\rho$ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm{Mant}_{b, \mu}(\rho)$ for all $\rho$ and prove it when $\rho$ is essentially square integrable. Our proof works for general $\rho$ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup., Comment: 50 pages, published version
- Published
- 2021
123. The factorisation property ofl∞(Xk)
- Author
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Paul F. X. Müller, Thomas Schlumprecht, Pavlos Motakis, and Richard Lechner
- Subjects
Pure mathematics ,Property (philosophy) ,Basis (linear algebra) ,General Mathematics ,010102 general mathematics ,Diagonal ,Banach space ,01 natural sciences ,Identity (music) ,Bounded operator ,Factorization ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the following problem: letXk, be a Banach space with a normalised basis (e(k, j))j, whose biorthogonals are denoted by${(e_{(k,j)}^*)_j}$, for$k\in\N$, let$Z=\ell^\infty(X_k:k\kin\N)$be theirl∞-sum, and let$T:Z\to Z$be a bounded linear operator with a large diagonal,i.e.,$$\begin{align*}\inf_{k,j} \big|e^*_{(k,j)}(T(e_{(k,j)})\big|>0.\end{align*}$$Under which condition does the identity onZfactor throughT? The purpose of this paper is to formulate general conditions for which the answer is positive.
- Published
- 2020
124. On a new stochastic model for cascading failures
- Author
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Hyunju Lee
- Subjects
Statistics and Probability ,Stochastic modelling ,General Mathematics ,010102 general mathematics ,Residual ,01 natural sciences ,Stochastic ordering ,Cascading failure ,010104 statistics & probability ,Control theory ,Component (UML) ,Life test ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.
- Published
- 2020
125. ON THE OPTIMAL EXTENSION THEOREM AND A QUESTION OF OHSAWA
- Author
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Xiangyu Zhou, Zhi Li, and Sha Yao
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Extension (predicate logic) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we present a version of Guan-Zhou’s optimal $L^{2}$ extension theorem and its application. As a main application, we show that under a natural condition, the question posed by Ohsawa in his series paper VIII on the extension of $L^{2}$ holomorphic functions holds. We also give an explicit counterexample which shows that the question fails in general.
- Published
- 2020
126. On moderate deviations in Poisson approximation
- Author
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Qingwei Liu and Aihua Xia
- Subjects
Statistics and Probability ,Random graph ,Matching (graph theory) ,Distribution (number theory) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,Birthday problem ,Normal distribution ,010104 statistics & probability ,symbols.namesake ,FOS: Mathematics ,Rare events ,symbols ,Applied mathematics ,Moderate deviations ,0101 mathematics ,Statistics, Probability and Uncertainty ,Primary 60F05, secondary 60E15 ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures
- Published
- 2020
127. THE MINIMAL MODULAR FORM ON QUATERNIONIC
- Author
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Aaron Pollack
- Subjects
Algebra ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Modular form ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Suppose that $G$ is a simple reductive group over $\mathbf{Q}$, with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$-valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$, which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$. We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$, which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$.
- Published
- 2020
128. Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems
- Author
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Lior Fishman, Tushar Das, Mariusz Urbański, and David Simmons
- Subjects
Class (set theory) ,Pure mathematics ,Conjecture ,Mathematics - Number Theory ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Diophantine equation ,010102 general mathematics ,11J13, 11J83, 28A75, 37F35 ,Open set ,Dynamical Systems (math.DS) ,Rational function ,01 natural sciences ,Measure (mathematics) ,010101 applied mathematics ,Hausdorff dimension ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Abstract
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [{\it Invent. Math.} {\bf 138}(3) (1999), 451--494] resolving Sprind\v zuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. {\it Selecta Math.} {\bf 10} (2004), 479--523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson--Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In the first of this series of papers [{\it Selecta Math.} {\bf 24}(3) (2018), 2165--2206], we introduce and develop a systematic account of two classes of measures, which we call {\it quasi-decaying} and {\it weakly quasi-decaying}. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ``inherited exponent of irrationality'' version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson--Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying., Comment: Link to Part I: arXiv:1504.04778
- Published
- 2020
129. Bernoulliness of when is an irrational rotation: towards an explicit isomorphism
- Author
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Christophe Leuridan
- Subjects
Rational number ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diophantine approximation ,01 natural sciences ,Irrational rotation ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Bernoulli scheme ,Isomorphism ,0101 mathematics ,Real number ,Unit interval ,Mathematics - Abstract
Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.
- Published
- 2020
130. Martingale decomposition of an L2 space with nonlinear stochastic integrals
- Author
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Clarence Simard
- Subjects
Statistics and Probability ,Optimization problem ,General Mathematics ,010102 general mathematics ,Stochastic calculus ,01 natural sciences ,010104 statistics & probability ,Nonlinear system ,Integrator ,Bounded function ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Lp space ,Martingale (probability theory) ,Brownian motion ,Mathematics - Abstract
This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.
- Published
- 2019
131. Type classification of extreme quantized characters
- Author
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Ryosuke Sato
- Subjects
Pure mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Context (language use) ,01 natural sciences ,Representation theory ,Quantization (physics) ,symbols.namesake ,Character (mathematics) ,Operator algebra ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Quantum ,Mathematics ,Von Neumann architecture - Abstract
The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory forquantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.
- Published
- 2019
132. Approximate lumpability for Markovian agent-based models using local symmetries
- Author
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Wasiur R. KhudaBukhsh, Arnab Auddy, Heinz Koeppl, and Yann Disser
- Subjects
Statistics and Probability ,Random graph ,Markov chain ,General Mathematics ,Probability (math.PR) ,Lumpability ,Neighbourhood (graph theory) ,Markov process ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,60J28 ,010201 computation theory & mathematics ,Approximation error ,Local symmetry ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,State space ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Abstract
We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed., Comment: 28 pages, 4 figures
- Published
- 2019
133. Comparison results for M/G/1 queues with waiting and sojourn time deadlines
- Author
-
Yoshiaki Inoue
- Subjects
Statistics and Probability ,Waiting time ,Discrete mathematics ,021103 operations research ,Service time ,General Mathematics ,0211 other engineering and technologies ,Comparison results ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,M/G/1 queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.
- Published
- 2019
134. Combinatorial proofs of two theorems of Lutz and Stull
- Author
-
Tuomas Orponen
- Subjects
FOS: Computer and information sciences ,28A80 (primary), 28A78 (secondary) ,General Mathematics ,kombinatoriikka ,Combinatorial proof ,Computational Complexity (cs.CC) ,01 natural sciences ,Combinatorics ,Mathematics - Metric Geometry ,Hausdorff and packing measures ,0103 physical sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Algorithmic information theory ,Lemma (mathematics) ,Euclidean space ,Pigeonhole principle ,010102 general mathematics ,Orthographic projection ,Hausdorff space ,Metric Geometry (math.MG) ,Projection (relational algebra) ,Computer Science - Computational Complexity ,Mathematics - Classical Analysis and ODEs ,fraktaalit ,010307 mathematical physics ,mittateoria - Abstract
Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if $K \subset \mathbb{R}^{n}$ is any set with equal Hausdorff and packing dimensions, then $$ \dim_{\mathrm{H}} π_{e}(K) = \min\{\dim_{\mathrm{H}} K,1\} $$ for almost every $e \in S^{n - 1}$. Here $π_{e}$ stands for orthogonal projection to $\mathrm{span}(e)$. The primary purpose of this paper is to present proofs for Lutz and Stull's projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman's "potential theoretic" method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to slightly generalise Lutz and Stull's theorems: the versions in this paper apply to orthogonal projections to $m$-planes in $\mathbb{R}^{n}$, for all $0 < m < n$., 11 pages. v2: Incorporated referee suggestions
- Published
- 2021
135. Weak containment of measure-preserving group actions
- Author
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Alexander S. Kechris and Peter Burton
- Subjects
Containment (computer programming) ,Group action ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Calculus ,Measure (physics) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Weak equivalence ,Mathematics - Abstract
This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.
- Published
- 2019
136. Regularity results for the 2D critical Oldroyd-B model in the corotational case
- Author
-
Zhuan Ye
- Subjects
Logarithm ,Cauchy stress tensor ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dissipation ,Vorticity ,01 natural sciences ,010101 applied mathematics ,A priori and a posteriori ,Oldroyd-B model ,Gravitational singularity ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.
- Published
- 2019
137. LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS
- Author
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Saminathan Ponnusamy, Navneet Lal Sharma, and Karl-Joachim Wirths
- Subjects
010101 applied mathematics ,Combinatorics ,Logarithm ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Convex function ,01 natural sciences ,Upper and lower bounds ,Unit disk ,Mathematics ,Univalent function - Abstract
Let${\mathcal{S}}$be the family of analytic and univalent functions$f$in the unit disk$\mathbb{D}$with the normalization$f(0)=f^{\prime }(0)-1=0$, and let$\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$denote the logarithmic coefficients of$f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families${\mathcal{F}}(c)$and${\mathcal{G}}(c)$of functions$f\in {\mathcal{S}}$defined by$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)for some$c\in (0,3]$and$c\in (0,1]$, respectively. We obtain the sharp upper bound for$|\unicode[STIX]{x1D6FE}_{n}|$when$n=1,2,3$and$f$belongs to the classes${\mathcal{F}}(c)$and${\mathcal{G}}(c)$, respectively. The paper concludes with the following two conjectures:∙If$f\in {\mathcal{F}}(-1/2)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$for$n\geq 1$, and$$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$where$\text{Li}_{2}(x)$denotes the dilogarithm function.∙If$f\in {\mathcal{G}}(c)$, then$|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$for$n\geq 1$.
- Published
- 2019
138. The continued search for sums of powers
- Author
-
Nigel Derby
- Subjects
Discrete mathematics ,Matrix (mathematics) ,Sums of powers ,General Mathematics ,Natural number ,Pascal (programming language) ,Mathematical puzzle ,computer ,Mathematics ,computer.programming_language - Abstract
In a previous paper [1], I described a method for computing sums of powers via a matrix composed from Pascal's triangle. (The motivation for this came from a mathematical puzzle posed by Paul Maslanka in the Guardian.) Since then I have read or viewed a number of papers on sums of powers. Some were nearly four hundred years old and the most recent one was published this year. One rather obscure work I stumbled across was ‘On the sums of powers of natural numbers’ by the Japanese mathematician Mr T. Kariya, published in Tokyo in 1907. If any readers know more of Mr Kariya I would be glad to hear about him.
- Published
- 2019
139. Generalising a triangle inequality
- Author
-
Zoltan Retkes
- Subjects
Discrete mathematics ,Altitude (triangle) ,Triangle inequality ,Position (vector) ,General Mathematics ,Order (group theory) ,Point (geometry) ,Circumscribed circle ,Notation ,Incircle and excircles of a triangle ,Mathematics - Abstract
The main goal of this paper is to give a deeper understanding of the geometrical inequality proposed by Martin Lukarevski in [1]. In order to formulate our results we shall introduce and use the following notation throughout this paper. Let A1A2A3 be a triangle a1, a2, a3, the lengths of the sides opposite to A1, A2, A3 respectively, P an arbitrary inner point of it xi, the distance of P from the side of length ai. Let r, R be the inradius and circumradius of the triangle hi, the altitude belonging to side ai, Δ the area and finally let α be a real parameter. We adopt also the use of Σui to refer the sum taken over the suffices i = 1, 2, 3. Now we are in the position to reformulate the original problem into a more general form namely: find bounds for Σxαi in terms of r and R. The main results of our investigation are summarised in the following theorem.
- Published
- 2018
140. FOUR IDENTITIES FOR THIRD ORDER MOCK THETA FUNCTIONS
- Author
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Amita Malik, George E. Andrews, Bruce C. Berndt, Sun Kim, and Song Heng Chan
- Subjects
Lemma (mathematics) ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Rank (computer programming) ,Mathematical proof ,01 natural sciences ,Ramanujan's sum ,Ramanujan theta function ,Combinatorics ,symbols.namesake ,Third order ,Section (category theory) ,0103 physical sciences ,symbols ,0101 mathematics ,Mathematics - Abstract
In 2005, using a famous lemma of Atkin and Swinnerton-Dyer (Some properties of partitions, Proc. Lond. Math. Soc. (3)4(1954), 84–106), Yesilyurt (Four identities related to third order mock theta functions in Ramanujan’s lost notebook, Adv. Math. 190(2005), 278–299) proved four identities for third order mock theta functions found on pages 2 and 17 in Ramanujan’s lost notebook. The primary purpose of this paper is to offer new proofs in the spirit of what Ramanujan might have given in the hope that a better understanding of the identities might be gained. Third order mock theta functions are intimately connected with ranks of partitions. We prove new dissections for two rank generating functions, which are keys to our proof of the fourth, and the most difficult, of Ramanujan’s identities. In the last section of this paper, we establish new relations for ranks arising from our dissections of rank generating functions.
- Published
- 2018
141. ON THE BILINEAR SQUARE FOURIER MULTIPLIER OPERATORS ASSOCIATED WITH FUNCTION
- Author
-
Zhengyang Li and Qingying Xue
- Subjects
Multiplier (Fourier analysis) ,symbols.namesake ,Fourier transform ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,Applied mathematics ,Bilinear interpolation ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper will be devoted to study a class of bilinear square-function Fourier multiplier operator associated with a symbol $m$ defined by $$\begin{eqnarray}\displaystyle & & \displaystyle \mathfrak{T}_{\unicode[STIX]{x1D706},m}(f_{1},f_{2})(x)\nonumber\\ \displaystyle & & \displaystyle \quad =\Big(\iint _{\mathbb{R}_{+}^{n+1}}\Big(\frac{t}{|x-z|+t}\Big)^{n\unicode[STIX]{x1D706}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg|\int _{(\mathbb{R}^{n})^{2}}e^{2\unicode[STIX]{x1D70B}ix\cdot (\unicode[STIX]{x1D709}_{1}+\unicode[STIX]{x1D709}_{2})}m(t\unicode[STIX]{x1D709}_{1},t\unicode[STIX]{x1D709}_{2})\hat{f}_{1}(\unicode[STIX]{x1D709}_{1})\hat{f}_{2}(\unicode[STIX]{x1D709}_{2})\,d\unicode[STIX]{x1D709}_{1}\,d\unicode[STIX]{x1D709}_{2}\bigg|^{2}\frac{dz\,dt}{t^{n+1}}\Big)^{1/2}.\nonumber\end{eqnarray}$$ A basic fact about $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ is that it is closely associated with the multilinear Littlewood–Paley $g_{\unicode[STIX]{x1D706}}^{\ast }$ function. In this paper we first investigate the boundedness of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ on products of weighted Lebesgue spaces. Then, the weighted endpoint $L\log L$ type estimate and strong estimate for the commutators of $\mathfrak{T}_{\unicode[STIX]{x1D706},m}$ will be demonstrated.
- Published
- 2018
142. THE LATTICE OF VARIETIES OF STRICT LEFT RESTRICTION SEMIGROUPS
- Author
-
Peter R. Jones
- Subjects
010101 applied mathematics ,Pure mathematics ,Unary operation ,General Mathematics ,Lattice (order) ,010102 general mathematics ,0101 mathematics ,Identity element ,01 natural sciences ,Mathematics - Abstract
Left restriction semigroups are the unary semigroups that abstractly characterize semigroups of partial maps on a set, where the unary operation associates to a map the identity element on its domain. This paper is the sequel to two recent papers by the author, melding the results of the first, on membership in the variety $\mathbf{B}$ of left restriction semigroups generated by Brandt semigroups and monoids, with the connection established in the second between subvarieties of the variety $\mathbf{B}_{R}$ of two-sided restriction semigroups similarly generated and varieties of categories, in the sense of Tilson. We show that the respective lattices ${\mathcal{L}}(\mathbf{B})$ and ${\mathcal{L}}(\mathbf{B}_{R})$ of subvarieties are almost isomorphic, in a very specific sense. With the exception of the members of the interval $[\mathbf{D},\mathbf{D}\vee \mathbf{M}]$, every subvariety of $\mathbf{B}$ is induced from a member of $\mathbf{B}_{R}$ and vice versa. Here $\mathbf{D}$ is generated by the three-element left restriction semigroup $D$ and $\mathbf{M}$ is the variety of monoids. The analogues hold for pseudovarieties.
- Published
- 2018
143. Positive periodic solutions for singular fourth-order differential equations with a deviating argument
- Author
-
Fanchao Kong and Zaitao Liang
- Subjects
010101 applied mathematics ,Fourth order ,Singularity ,Differential equation ,Argument ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Continuation theorem ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.
- Published
- 2018
144. La conjecture de Manin pour une famille de variétés en dimension supérieure
- Author
-
Kevin Destagnol
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Inspired by a method of La Bretèche relying on some unique factorisation, we generalise work of Blomer, Brüdern and Salberger to prove Manin's conjecture in its strong form conjectured by Peyre for some infinite family of varieties of higher dimension. The varieties under consideration in this paper correspond to the singular projective varieties defined by the following equation$$ x_1 y_2y_3\cdots y_n+x_2y_1y_3 \cdots y_n+ \cdots+x_n y_1 y_2 \cdots y_{n-1}=0 $$in ℙℚ2n−1for alln⩾ 3. This paper comes with an Appendix by Per Salberger constructing a crepant resolution of the above varieties.
- Published
- 2018
145. Affine focal sets of codimension-2 submanifolds contained in hypersurfaces
- Author
-
Marcelo José Saia, Marcos Craizer, and Luis F. Sánchez
- Subjects
Pure mathematics ,General Mathematics ,020207 software engineering ,02 engineering and technology ,Codimension ,GEOMETRIA DIFERENCIAL CLÁSSICA ,01 natural sciences ,Darboux vector ,0104 chemical sciences ,010404 medicinal & biomolecular chemistry ,Hypersurface ,Hyperplane ,Affine focal set ,0202 electrical engineering, electronic engineering, information engineering ,Tangent space ,Affine sphere ,Affine transformation ,Mathematics - Abstract
In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.
- Published
- 2018
146. Kinematic and dynamic performance investigations of asymmetric (U-shape fixed base) planar parallel manipulators
- Author
-
Jayant Kumar Mohanta, Santhakumar Mohan, and Yogesh Singh
- Subjects
0209 industrial biotechnology ,Kinematic isotropy ,General Mathematics ,02 engineering and technology ,Workspace ,Kinematics ,Topology ,01 natural sciences ,Energy requirement ,Lower energy ,Computer Science Applications ,020901 industrial engineering & automation ,Planar ,Prismatic joint ,Control and Systems Engineering ,0103 physical sciences ,010301 acoustics ,Software ,Mathematics ,Fixed base - Abstract
SUMMARYIn this paper, a new family of 3-degree-of-freedom planar parallel manipulators (PPMs), namely U-shape fixed base PPMs starting with an active prismatic joint on each leg, is proposed. In order to identify the best manipulators of this family, comparative kinematic and dynamic performance studies are performed. The kinematic performances are quantified through the local performance index, namely the kinematic isotropy. From the kinematic isotropy analysis results, it is observed thatPPR-PRP-PRP,PRP-PRP-PRP andPRR-PRP-PRP configurations have better kinematic design aspects compared to other configurations of this family of U-shape fixed base parallel configurations. Further, from the workspace analysis, it is observed that thePPR-PRP-PRP configuration has a higher value of workspace to the total area required ratio compared to other configurations. This paper also presents a comparative dynamic performance analysis of these top-three U-shape fixed base configurations in terms of dynamic driving performance measures, and energy requirements for three different (fixed base size of the manipulators) aspect ratios under two different loading conditions. From the analyses results, it is perceived that thePRP-PRP-PRP configuration is requiring lower energy and dynamic driving performances than others. These analyses are done with the help of multi-body dynamic software, namely MSC ADAMS, and the results are validated through the help of real-time experiments conducted on in-house fabricated prototypes of these three PPMs. In specific, the energy consumption is measured and compared in this study. Experimental results demonstrated that thePRP-PRP-PRP manipulator displays a considerably better performance in terms of minimum energy requirement.
- Published
- 2018
147. Flows of measures generated by vector fields
- Author
-
Emanuele Paolini and Eugene Stepanov
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Lipschitz continuity ,01 natural sciences ,Measure (mathematics) ,Integral curve ,Flow (mathematics) ,Ordinary differential equation ,0103 physical sciences ,Vector field ,010307 mathematical physics ,0101 mathematics ,Borel measure ,Smooth structure ,Mathematics - Abstract
The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.
- Published
- 2018
148. Further characterisations of tangential quadrilaterals
- Author
-
Martin Josefsson
- Subjects
Background information ,Quadrilateral ,General Mathematics ,010102 general mathematics ,Vertex angle ,Regular polygon ,Tangent ,01 natural sciences ,Incircle and excircles of a triangle ,Combinatorics ,Point (geometry) ,0101 mathematics ,Inscribed figure ,Mathematics - Abstract
Tangential quadrilaterals are defined to be quadrilaterals in which a circle can be inscribed that is tangent to all four sides. It is well known and easy to prove that a convex quadrilateral is tangential if, and only if, the angle bisectors of all four vertex angles are concurrent at a point, which is the centre of the inscribed circle (incircle). The most well-known and in problem solving useful characterisation of tangential quadrilaterals is Pitot's theorem, which states that a convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d satisfy the relation a + c = b + d [1, pp. 64-67]. If you want to have more background information about characterisations of tangential quadrilaterals, then we recommend you to check out the lovely papers [2, 3, 4], as well as our previous contributions on the subject [5, 6, 7]. These six papers together include about 30 characterisations that are either proved or reviewed there with references.
- Published
- 2017
149. A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY
- Author
-
Zhihua Wang, Yinhuo Zhang, Libin Li, WANG, Zhihua, Li, Libin, and ZHANG, Yinhuo
- Subjects
Algebra ,Ring (mathematics) ,Pure mathematics ,General Mathematics ,Tensor (intrinsic definition) ,finite tensor category ,green ring ,Casimir number, Jacobson radical, Frobenius algebra ,010102 general mathematics ,0103 physical sciences ,Foundation (engineering) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper deals with the Green ring $\mathcal{G}(\mathcal{C})$ of a finite tensor category $\mathcal{C}$ with finitely many isomorphism classes of indecomposable objects over an algebraically closed field. The first part of this paper deals with the question of when the Green ring $\mathcal{G}(\mathcal{C})$, or the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over a field K, is Jacobson semisimple (namely, has zero Jacobson radical). It turns out that $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero in K. For the Green ring $\mathcal{G}(\mathcal{C})$ itself, $\mathcal{G}(\mathcal{C})$ is Jacobson semisimple if and only if the Casimir number of $\mathcal{C}$ is not zero. The second part of this paper focuses on the case where $\mathcal{C}=\text{Rep}(\mathbb {k}G)$ for a cyclic group G of order p over a field $\mathbb {k}$ of characteristic p. In this case, the Casimir number of $\mathcal{C}$ is computable and is shown to be 2p2. This leads to a complete description of the Jacobson radical of the Green algebra $\mathcal{G}(\mathcal{C})\otimes_{\mathbb {Z}}$K over any field K.
- Published
- 2017
150. Local rigidity of higher rank non-abelian action on torus
- Author
-
Zhenqi Jenny Wang
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Rigidity (electromagnetism) ,Applied Mathematics ,General Mathematics ,Torus ,Abelian group ,Mathematics - Abstract
In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.
- Published
- 2017
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