2,187 results
Search Results
2. Using A4-sized paper to illustrate that is irrational
- Author
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Nick Lord
- Subjects
General Mathematics ,Irrational number ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Published
- 2017
3. A Note on a Paper by Wong and Heyde
- Author
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Mikhail Urusov and Aleksandar Mijatović
- Subjects
Statistics and Probability ,Statistics::Theory ,Pure mathematics ,60G44, 60G48, 60H10, 60J60 ,General Mathematics ,Applied probability ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,60G48 ,FOS: Mathematics ,60G44 ,0101 mathematics ,60J60 ,Mathematics ,Local martingales versus true martingales ,010102 general mathematics ,Probability (math.PR) ,stochastic exponential ,Exponential function ,Mathematik ,60H10 ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Quantitative Finance - General Finance ,General Finance (q-fin.GN) ,Mathematics - Probability ,Counterexample - Abstract
In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed., Comment: To appear in Journal of Applied Probability, 11 pages
- Published
- 2011
4. The geometry of folding paper dolls
- Author
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Brigitte Servatius
- Subjects
Grade school ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,people.profession ,Art ,Folding (DSP implementation) ,Sister ,01 natural sciences ,Visual arts ,Girl ,0101 mathematics ,Dressmaker ,people ,media_common - Abstract
When a parent sees a little girl sitting on the floor cutting paper dolls, many thoughts may come to mind: ‘She’s keeping out of trouble’ or ‘She’s making a mess’ or even ‘There go my tax returns’. The thought that should have come to my parent’s mind, however, was ‘One day she’ll be a mathematician’. My grandmother, who worked as a dressmaker, often allowed my sister and me to use her razor sharp scissors on the strips of leftover tracing paper. This paper is inspired by a notebook that I kept in grade school when I ‘studied’ paper dolls, and the figures are based on dolls found pressed between the pages.
- Published
- 1997
5. A note on Saleh's paper ‘Almost continuity implies closure continuity’
- Author
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Julian Dontchev and Takashi Noiri
- Subjects
Pure mathematics ,Continuous function ,General Mathematics ,010102 general mathematics ,General Topology (math.GN) ,Closure (topology) ,Topology (electrical circuits) ,01 natural sciences ,010101 applied mathematics ,54C08 ,FOS: Mathematics ,0101 mathematics ,Mathematics - General Topology ,Mathematics - Abstract
Recently, Saleh claimed to have solved `a long standing open question' in Topology; namely, he proved that every almost continuous function is closure continuous (= $\theta$-continuous). Unfortunately, this problem was settled long time ago and even a better result is known., Comment: 2 pages, to appear in "Glasgow Math. J."
- Published
- 1998
6. Epidemics with carriers: A note on a paper of Dietz
- Author
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F. Downton
- Subjects
Statistics and Probability ,Entire population ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,Short interval ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Demography ,Mathematics - Abstract
In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.
- Published
- 1967
7. 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
8. 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
9. 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
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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
10. 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
11. Higher horospherical limit sets for G-modules over CAT(0)-spaces
- Author
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Ross Geoghegan and Robert Bieri
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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
12. 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
13. 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
14. Polynomial-Based Robust Adaptive Impedance Control of Electrically Driven Robots
- Author
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Alireza Izadbakhsh and Saeed Khorashadizadeh
- Subjects
0209 industrial biotechnology ,Polynomial ,Degree (graph theory) ,Computer science ,General Mathematics ,Stability (learning theory) ,02 engineering and technology ,01 natural sciences ,Computer Science Applications ,020901 industrial engineering & automation ,Impedance control ,Control and Systems Engineering ,Control theory ,Universal approximation theorem ,0103 physical sciences ,Robot ,010301 acoustics ,Electrical impedance ,Software - Abstract
SUMMARYThis paper presents a robust adaptive impedance controller for electrically driven robots using polynomials of degree N as a universal approximator. According to the universal approximation theorem, polynomials of degree N can approximate uncertainties including un-modeled dynamics and external disturbances. This fact is completely discussed and proved in this paper. The polynomial coefficients are estimated based on the adaptive law calculated in the stability analysis. A performance evaluation has been carried out to verify satisfactory performance of the controller. Simulation results on a two degree of freedom manipulator have been presented to guarantee its successful implementation.
- Published
- 2020
15. 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
16. Triangles whose sides form an arithmetic progression
- Author
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Sadi Abu-Saymeh and Mowaffaq Hajja
- Subjects
Computer science ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Arithmetic progression ,010307 mathematical physics ,0101 mathematics ,Arithmetic ,Mathematical proof ,01 natural sciences ,ComputingMethodologies_COMPUTERGRAPHICS ,Arithmetic mean - Abstract
This article is motivated by, and is meant as a supplement to, the recent paper [1]. That paper proves three geometric characterisations of triangles whose sides are in arithmetic progression, or equivalently triangles in which one of the sides is the arithmetic mean of the other two. More precisely, it gives three geometric contexts in which such triangles appear. In this Article, we supply references for the results in [1] and we provide more proofs of these results. We also add more contexts in which such triangles appear, and we raise related issues for future work. We hope that this will be a source of problems for training for, and for including in, mathematical competitions.
- Published
- 2020
17. 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
18. 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
19. Codimension growth for weak polynomial identities, and non-integrality of the PI exponent
- Author
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David Levi da Silva Macedo and Plamen Koshlukov
- Subjects
010101 applied mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences - Abstract
Let K be a field of characteristic zero. In this paper, we study the polynomial identities of representations of Lie algebras, also called weak identities, or identities of pairs. These identities are determined by pairs of the form (A, L) where A is an associative enveloping algebra for the Lie algebra L. Then a weak identity of (A, L) (or an identity for the representation of L associated to A) is an associative polynomial which vanishes when evaluated on elements of L⊆ A. One of the most influential results in the area of PI algebras was the theory developed by Kemer. A crucial role in it was played by the construction of the Grassmann envelope of an associative algebra and the close relation of the identities of the algebra and its Grassmann envelope. Here we consider varieties of pairs. We prove that under some restrictions one can develop a theory similar to that of Kemer's in the study of identities of representations of Lie algebras. As a consequence, we establish that in the case when K is algebraically closed, if a variety of pairs does not contain pairs corresponding to representations of sl2(K), and if the variety is generated by a pair where the associative algebra is PI then it is soluble. As another consequence of the methods used to obtain the above result, and applying ideas from papers by Giambruno and Zaicev, we were able to construct a pair (A, L) such that its PI exponent (if it exists) cannot be an integer. We recall that the PI exponent exists and is an integer whenever A is an associative (a theorem by Giambruno and Zaicev), or a finite-dimensional Lie algebra (Zaicev). Gordienko also proved that the PI exponent exists and is an integer for finite-dimensional representations of Lie algebras.
- Published
- 2020
20. 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
21. 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 $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.
- Published
- 2020
22. Least energy solution for a scalar field equation with a singular nonlinearity
- Author
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Yeonho Kim, Sun-Ho Choi, Jaeyoung Byeon, and Sang-Hyuck Moon
- Subjects
010101 applied mathematics ,Physics ,Nonlinear system ,General Mathematics ,Quantum electrodynamics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Scalar field ,Energy (signal processing) - Abstract
We are concerned with a nonnegative solution to the scalar field equation $$\Delta u + f(u) = 0{\rm in }{\open R}^N,\quad \mathop {\lim }\limits_{|x|\to \infty } u(x) = 0.$$ A classical existence result by Berestycki and Lions [3] considers only the case when f is continuous. In this paper, we are interested in the existence of a solution when f is singular. For a singular nonlinearity f, Gazzola, Serrin and Tang [8] proved an existence result when $f \in L^1_{loc}(\mathbb {R}) \cap \mathrm {Lip}_{loc}(0,\infty )$ with $\int _0^u f(s)\,{\rm d}s < 0$ for small $u>0.$ Since they use a shooting argument for their proof, they require the property that $f \in \mathrm {Lip}_{loc}(0,\infty ).$ In this paper, using a purely variational method, we extend the previous existence results for $f \in L^1_{loc}(\mathbb {R}) \cap C(0,\infty )$. We show that a solution obtained through minimization has the least energy among all radially symmetric weak solutions. Moreover, we describe a general condition under which a least energy solution has compact support.
- Published
- 2020
23. Design, Modeling, and Control of a New Manipulating Climbing Robot Through Infrastructures Using Adaptive Force Control Method
- Author
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H. Tourajizadeh and Vahid Boomeri
- Subjects
0209 industrial biotechnology ,Computer science ,General Mathematics ,Truss ,02 engineering and technology ,Robot end effector ,01 natural sciences ,Computer Science Applications ,Inverse dynamics ,law.invention ,020901 industrial engineering & automation ,Control and Systems Engineering ,Control theory ,law ,Climbing ,0103 physical sciences ,Robot ,Feedback linearization ,Motion planning ,010301 acoustics ,Software - Abstract
SUMMARYIn this paper, design, modeling, and control of a grip-based climbing robot are performed, which consists of a triangular chassis and three actuating legs. This robot can climb through any trusses, pipeline, and scaffolds structures and can perform any inspectional and operational tasks in the high height which decreases the falling danger of operation and increases the safety of the workers. The proposed robot can be substituted for the workers to decrease the risk of death danger and increase the safety of the operation. Since these kinds of infrastructures are truss shaped, the traditional wheel-based climbing robots are not able to travel through these structures. Therefore, in this paper, a grip-based climbing robot is designed to accomplish the climbing process through the trusses and infrastructures in order to perform inspecting and manipulating tasks. Hence, a proper mechanism for the mentioned robot is designed and its related kinematic and kinetic models are developed. Robot modeling is investigated for two different modes including climbing and manipulating phases. Considering the redundancy of the proposed robot and the parallel mechanism employed in it, the active joints are selected in a proper way and its path planning is performed to accomplish the required missions. Concerning the climbing mode, the required computed torque method (CTM) is calculated by the inverse dynamics of the robot. However, for the manipulation mode, after path planning, two controlling strategies are employed, including feedback linearization (FBL) and adaptive force control, and their results are compared as well. It is shown that the latter case is preferable since the external forces implemented on the end effector tool is not exactly predetermined and thus, the controller should adapt the robot with the exerted force pattern of the manipulator. The modeling correctness is investigated by performing some analytic and comparative simulation scenarios in the MATLAB and comparing the results with the MSC-ADAMS ones, for both climbing and manipulating phases. The efficiency of the designed controller is also proved by implementing an unknown force pattern on the manipulator to check its efficiency toward estimating the mentioned implemented forces and compensating the errors. It is shown that the designed robot can successfully climb through a truss and perform its operating task by the aid of the employed adaptive controller in an accurate way.
- Published
- 2020
24. 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
25. Robust Control of Semi-passive Biped Dynamic Locomotion based on a Discrete Control Lyapunov Function
- Author
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Ming Liu, Kang An, Jing Yang, Qijun Chen, and Chengju Liu
- Subjects
0209 industrial biotechnology ,Computer science ,General Mathematics ,Terrain ,02 engineering and technology ,Swing ,Impulse (physics) ,01 natural sciences ,Computer Science Applications ,020901 industrial engineering & automation ,Control and Systems Engineering ,Control theory ,Control system ,0103 physical sciences ,Robot ,Robust control ,010301 acoustics ,Walking gait ,Software ,Control-Lyapunov function - Abstract
SUMMARYThis paper focuses on robust control of a simplest passive model, which is established on a DCLF (discrete control Lyapunov function) -based control system, and presents gait transition method based on the study of purely passive walker. Firstly, the DCLF is introduced to stabilize walking process between steps exponentially by modulating the length of next step. Next, the swing leg trajectory from mid-stance position to foot-strike can be planned. Then the control law is calculated to resist external disturbance. Besides, an impulse is added just before foot-strike to realize a periodic walking pattern on flat or uphill ground. With walking terrain varying, the robot can transit to an adaptive walking gait in a few steps. With different push or pull disturbances acting on hip joint and the robot gait transiting on a continuously slope-changing downhill, the effectiveness of the presented DCLF-based method is verified using simulation experiments. The ability to walk on a changing environment is also presented by simulation results. The insights of this paper can help to develop a robust control method and adaptive walking of dynamic passive locomotion robots.
- Published
- 2019
26. Type classification of extreme quantized characters
- Author
-
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
27. Approximate lumpability for Markovian agent-based models using local symmetries
- Author
-
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
28. 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
29. Weak containment of measure-preserving group actions
- Author
-
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
30. 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
31. LOGARITHMIC COEFFICIENTS PROBLEMS IN FAMILIES RELATED TO STARLIKE AND CONVEX FUNCTIONS
- Author
-
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
32. FAT-Based Robust Adaptive Control of Electrically Driven Robots in Interaction with Environment
- Author
-
Saeed Khorashadizadeh, Payam Kheirkhahan, and Alireza Izadbakhsh
- Subjects
Lyapunov function ,0209 industrial biotechnology ,Control and Optimization ,Adaptive control ,Computer science ,General Mathematics ,Mechanical Engineering ,Orthogonal functions ,02 engineering and technology ,01 natural sciences ,Computer Science Applications ,Computer Science::Robotics ,symbols.namesake ,020901 industrial engineering & automation ,Function approximation ,Impedance control ,Control and Systems Engineering ,Control theory ,Approximation error ,Modeling and Simulation ,0103 physical sciences ,symbols ,010301 acoustics ,Fourier series ,Software - Abstract
SummaryThis paper presents a robust adaptive impedance controller for robot manipulators using function approximation techniques (FAT). Recently, FAT-based robust impedance controllers have been presented using Fourier series expansion for uncertainty estimation. In fact, sinusoidal functions can approximate nonlinear functions with arbitrary small approximation error based on the orthogonal functions theorem. The novelty of this paper in comparison with previous related works is that the number of required regressor matrices in this paper has been reduced. This superiority becomes more dominant when the manipulator degrees of freedom (DOFs) are increased. First, the desired signals for motor currents are calculated, and then the desired voltages are obtained. In the proposed approach, only a simple model of the actuator and manipulator dynamics is used in the controller design and all the rest dynamics are treated as external disturbance. The external disturbances can then be approximated by Fourier series expansion. The adaptation laws for Fourier series coefficients are derived from a Lyapunov-based stability analysis. Simulation results on a 2-DOF planar robot manipulator including the actuator dynamics indicate the efficiency of proposed method.
- Published
- 2018
33. 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
34. 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
35. 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
36. 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
37. 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
38. 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
39. 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
40. 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
41. 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
42. 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
43. Four notions of conjugacy for abstract semigroups
- Author
-
João Araújo, Michael Kinyon, António Malheiro, and Janusz Konieczny
- Subjects
Pure mathematics ,Endomorphism ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Group Theory (math.GR) ,0102 computer and information sciences ,01 natural sciences ,Representation theory ,Automaton ,Conjugacy class ,Areas of mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Special classes of semigroups ,0101 mathematics ,Mathematics - Group Theory ,Group theory ,Mathematics - Abstract
The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations or transformation semigroups). Here we study them in maximum generality. The paper ends with a large list of open problems., Comment: The paper is now more focused on abstract semigroups and a fourth notion of conjugacy was introduced for its importance in representation theory and finite semigroups
- Published
- 2017
44. Purely exponential growth of cusp-uniform actions
- Author
-
Wenyuan Yang
- Subjects
Cusp (singularity) ,Pure mathematics ,Lemma (mathematics) ,Mathematics::Dynamical Systems ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,Group Theory (math.GR) ,Dynamical Systems (math.DS) ,01 natural sciences ,Mathematics - Metric Geometry ,Exponential growth ,0103 physical sciences ,Shadow ,FOS: Mathematics ,Primary 20F65, 20F67 ,Countable set ,010307 mathematical physics ,Preprint ,Mathematics - Dynamical Systems ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal'bo-Otal-Peign\'e. For geometrically finite Cartan-Hadamard manifolds with pinched negative curvature this condition ensures the finiteness of Bowen-Margulis-Sullivan measures. In this case, our result recovers a theorem of Roblin (in a weaker form). Our main tool is the Patterson-Sullivan measures on the Gromov boundary of $X$, and a variant of the Sullivan shadow lemma called partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the condition of Dal'bo-Otal-Peign\'e. These results are further used in the paper \cite{YANG7}., Comment: Version 2: 34 pages, 2 figures. Sections 4 and 5 was rewritten following suggestions of the referee. Paper accepted by Ergodic Theory and Dynamical Systems
- Published
- 2017
45. Properties of bisect-diagonal quadrilaterals
- Author
-
Martin Josefsson
- Subjects
Quadrilateral ,General Mathematics ,Orthodiagonal quadrilateral ,010102 general mathematics ,Diagonal ,Regular polygon ,Class (philosophy) ,Computer Science::Computational Geometry ,01 natural sciences ,Connection (mathematics) ,Section (fiber bundle) ,Combinatorics ,0101 mathematics ,Equidiagonal quadrilateral ,Mathematics - Abstract
The general class of quadrilaterals where one diagonal is bisected by the other diagonal has appeared very rarely in the geometrical literature, but they have been named several times in connection with quadrilateral classifications. Günter Graumann strangely gave these objects two different names in [1, pp. 192, 194]: sloping-kite and sliding-kite. A. Ramachandran called them slant kites in [2, p. 54] and Michael de Villiers called them bisecting quadrilaterals in [3, pp. 19, 206]. The latter is a pretty good name, although a bit confusing: what exactly is bisected?We have found no papers and only two books where any theorems on such quadrilaterals are studied. In each of the books, one necessary and sufficient condition for such quadrilaterals is proved (see Theorem 1 and 2 in the next section). The purpose of this paper is to investigate basic properties of convex bisecting quadrilaterals, but we have chosen to give them a slightly different name. Let us first remind the reader that a quadrilateral whose diagonals have equal lengths is called an equidiagonal quadrilateral and one whose diagonals are perpendicular is called an orthodiagonal quadrilateral.
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- 2017
46. ANALYSIS OF CONTACT CAUCHY–RIEMANN MAPS II: CANONICAL NEIGHBORHOODS AND EXPONENTIAL CONVERGENCE FOR THE MORSE–BOTT CASE
- Author
-
Rui Wang and Yong-Geun Oh
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Cauchy–Riemann equations ,Homology (mathematics) ,01 natural sciences ,Moduli space ,symbols.namesake ,Symplectization ,0103 physical sciences ,symbols ,A priori and a posteriori ,Field theory (psychology) ,010307 mathematical physics ,0101 mathematics ,Exponential decay ,Symplectic geometry ,Mathematics - Abstract
This is a sequel to the papers Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817; Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3). In Oh and Wang (Real and Complex Submanifolds, Springer Proceedings in Mathematics and Statistics 106 (2014), 43–63, eds. by Y.-J. Suh and et al. for ICM-2014 satellite conference, Daejeon, Korea, August 2014; arXiv:1212.4817), the authors introduced a canonical affine connection on $M$ associated to the contact triad $(M,\unicode[STIX]{x1D706},J)$. In Oh and Wang (Analysis of contact Cauchy–Riemann maps I: a priori$C^{k}$estimates and asymptotic convergence, submitted, preprint, 2012, arXiv:1212.5186v3), they used the connection to establish a priori$W^{k,p}$-coercive estimates for maps $w:\dot{\unicode[STIX]{x1D6F4}}\rightarrow M$ satisfying $\overline{\unicode[STIX]{x2202}}^{\unicode[STIX]{x1D70B}}w=0$, $d(w^{\ast }\unicode[STIX]{x1D706}\circ j)=0$without involving symplectization. We call such a pair $(w,j)$ a contact instanton. In this paper, we first prove a canonical neighborhood theorem of the locus $Q$ foliated by closed Reeb orbits of a Morse–Bott contact form. Then using a general framework of the three-interval method, we establish exponential decay estimates for contact instantons $(w,j)$ of the triad $(M,\unicode[STIX]{x1D706},J)$, with $\unicode[STIX]{x1D706}$ a Morse–Bott contact form and $J$ a CR-almost complex structure adapted to $Q$, under the condition that the asymptotic charge of $(w,j)$ at the associated puncture vanishes.We also apply the three-interval method to the symplectization case and provide an alternative approach via tensorial calculations to exponential decay estimates in the Morse–Bott case for the pseudoholomorphic curves on the symplectization of contact manifolds. This was previously established by Bourgeois (A Morse–Bott approach to contact homology, Ph.D. dissertation, Stanford University, 2002) (resp. by Bao (On J-holomorphic curves in almost complex manifolds with asymptotically cylindrical ends, Pacific J. Math. 278(2) (2015), 291–324)), by using special coordinates, for the cylindrical (resp. for the asymptotically cylindrical) ends. The exponential decay result for the Morse–Bott case is an essential ingredient in the setup of the moduli space of pseudoholomorphic curves which plays a central role in contact homology and symplectic field theory (SFT).
- Published
- 2017
47. Existence of travelling pulses in a neural model
- Author
-
S. P. Hastings
- Subjects
Physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences - Abstract
In 1992 Ermentrout and McLeod published in this journal a landmark study of travelling wavefronts for a differential–integral equation model of a neural network. Since then a number of authors have extended the model by adding an additional equation for a ‘recovery variable’, thus allowing the possibility of travelling-pulse-type solutions. In a recent paper, Faye gave perhaps the first rigorous proof of the existence (and stability) of a travelling-pulse solution for a model of this type, treating a simplified version of equations originally developed by Kilpatrick and Bressloff. The excitatory weight function J used in this work allowed the system to be reduced to a set of four coupled ordinary differential equations (ODEs), and a specific firing-rate function S, with parameters, was considered. The method of geometric singular perturbation was employed, together with blow-ups. In this paper we extend Faye's results on existence by dropping one of his key hypotheses, proving the existence of pulses at least two different speeds, and, in a sense, allowing a wider range of the small parameter in the problem. The proofs are classical and self-contained aside from standard ODE material.
- Published
- 2017
48. SOME NORMALITY CRITERIA AND A COUNTEREXAMPLE TO THE CONVERSE OF BLOCH’S PRINCIPLE
- Author
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Kuldeep Singh Charak and S.D. Sharma
- Subjects
Pure mathematics ,Distribution (number theory) ,Mathematics::Complex Variables ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Holomorphic function ,01 natural sciences ,010101 applied mathematics ,Converse ,0101 mathematics ,Differential (infinitesimal) ,Value (mathematics) ,Normality ,Mathematics ,Meromorphic function ,Counterexample ,media_common - Abstract
In this paper we continue our earlier investigations on normal families of meromorphic functions\cite{CS2}. Here, we prove some value distribution results which lead to some normality criteria for a family of meromorphic functions involving the sharing of a holomorphic function by more general differential polynomials generated by members of the family and get some recently known results extended and improved. In particular, the main result of this paper leads to a counterexample to the converse of Bloch's principle.
- Published
- 2016
49. SEMIPERMUTABILITY IN GENERALISED SOLUBLE GROUPS
- Author
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James C. Beidleman, Adolfo Ballester-Bolinches, and R. Ialenti
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Finitely-generated abelian group ,0101 mathematics ,01 natural sciences ,Quotient ,Mathematics - Abstract
Some classes of finitely generated hyperabelian groups defined in terms of semipermutability and S-semipermutability are studied in the paper. The classification of finitely generated hyperabelian groups all of whose finite quotients are PST-groups recently obtained by Robinson is behind our results. An alternative proof of such a classification is also included in the paper.
- Published
- 2016
50. NILPOTENCY IN UNCOUNTABLE GROUPS
- Author
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Marco Trombetti, Francesco de Giovanni, De Giovanni, Francesco, and Trombetti, Marco
- Subjects
Pure mathematics ,nilpotent group ,uncountable group ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematics (all) ,Uncountable set ,010307 mathematical physics ,0101 mathematics ,soluble group ,01 natural sciences ,Mathematics - Abstract
The main purpose of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ in which all proper subgroups of cardinality $\aleph$ are nilpotent. It is proved that such a group $G$ is nilpotent, provided that $G$ has no infinite simple homomorphic images and either $\aleph$ has cofinality strictly larger than $\aleph _{0}$ or the generalized continuum hypothesis is assumed to hold. Furthermore, groups whose proper subgroups of large cardinality are soluble are studied in the last part of the paper.
- Published
- 2016
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