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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

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."

In [5] Doeblin considered some classes of finite non-homogeneous Markov chains and gave without proofs several results concerning their asymptotic behaviour. In the present paper we first attempt to make Doeblin's results precise and try to reconstruct his arguments. Subsequently we investigate more general situations, where a state space decomposition is provided by the sets occurring in the representation of the atomic sets of the tail or-field. We show that Doeblin's notion of an associated chain, as well as considerations regarding the tail ar-field structure of the chain, can be used to solve such cases. FINITE MARKOV CHAIN; FINAL CLASS; CYCLICALLY MOVING SUBCLASS; TAIL ar-FIELD; ATOMIC SET; RECURRENCE; WHIRLPOOL

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.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

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

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.

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.

In this paper we extend to non-classical set theories the standard strategy of proving independence using Boolean-valued models. This extension is provided by means of a new technique that, combining algebras (by taking their product), is able to provide product-algebra-valued models of set theories. In this paper we also provide applications of this new technique by showing that: (1) we can import the classical independence results to non-classical set theory (as an example we prove the independence of$\mathsf {CH}$); and (2) we can provide new independence results. We end by discussing the role of non-classical algebra-valued models for the debate between universists and multiversists and by arguing that non-classical models should be included as legitimate members of the multiverse.

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.

This paper establishes a new version of integration by parts formula of Markov chains for sensitivity computation, under much lower restrictions than the existing researches. Our approach is more fundamental and applicable without using Girsanov theorem or Malliavin calculus as did by past papers. Numerically, we apply this formula to compute sensitivity regarding the transition rate matrix and compare with a recent research by an IPA (infinitesimal perturbation analysis) method and other approaches.

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.

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.

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.

This paper explores versions of the Yoneda Lemma in settings founded upon FM sets. In particular, we explore the lemma for three base categories: the category of nominal sets and equivariant functions; the category of nominal sets and all finitely supported functions, introduced in this paper; and the category of FM sets and finitely supported functions. We make this exploration in ordinary, enriched and internal settings. We also show that the finite support of Yoneda natural transformations is a theorem for free.

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

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 $\operatorname{GSp}_{2n}$.

In this paper, we analyze Boolean formulas in conjunctive normal form (CNF) from the perspective of read-once resolution (ROR) refutation schemes. A read-once (resolution) refutation is one in which each clause is used at most once. Derived clauses can be used as many times as they are deduced. However, clauses in the original formula can only be used as part of one derivation. It is well known that ROR is not complete; that is, there exist unsatisfiable formulas for which no ROR exists. Likewise, the problem of checking if a 3CNF formula has a read-once refutation is NP-complete. This paper is concerned with a variant of satisfiability called not-all-equal satisfiability (NAE-satisfiability). A CNF formula is NAE-satisfiable if it has a satisfying assignment in which at least one literal in each clause is set to false. It is well known that the problem of checking NAE-satisfiability is NP-complete. Clearly, the class of CNF formulas which are NAE-satisfiable is a proper subset of satisfiable CNF formulas. It follows that traditional resolution cannot always find a proof of NAE-unsatisfiability. Thus, traditional resolution is not a sound procedure for checking NAE-satisfiability. In this paper, we introduce a variant of resolution called NAE-resolution which is a sound and complete procedure for checking NAE-satisfiability in CNF formulas. The focus of this paper is on a variant of NAE-resolution called read-once NAE-resolution in which each clause (input or derived) can be part of at most one NAE-resolution step. Our principal result is that read-once NAE-resolution is a sound and complete procedure for 2CNF formulas. Furthermore, we provide an algorithm to determine the smallest such NAE-resolution in polynomial time. This is in stark contrast to the corresponding problem concerning 2CNF formulas and ROR refutations. We also show that the problem of checking whether a 3CNF formula has a read-once NAE-resolution is NP-complete.

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

In this paper we investigate the computational complexity of deciding if a given finite algebraic structure satisfies a fixed (strong) Maltsev condition $\Sigma$. Our goal in this paper is to show that $\Sigma$-testing can be accomplished in polynomial time when the algebras tested are idempotent and the Maltsev condition $\Sigma$ can be described using paths. Examples of such path conditions are having a Maltsev term, having a majority operation, and having a chain of J\'onsson (or Gumm) terms of fixed length., Comment: 33 pages, 19 figures

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

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.

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.

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 $U_{q}(N)$.

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V (in the sense that it correctly computes successors of singular cardinals greater than δ) or HOD is “far” from V (in the sense that all regular cardinals greater than or equal to δ are measurable in HOD). The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that (assuming the existence of an extendible cardinal) that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails.

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

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.

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.

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$n\geq 1$.

In this paper, we extend the strong laws of large numbers and entropy ergodic theorem for partial sums for tree-indexed nonhomogeneous Markov chains fields to delayed versions of nonhomogeneous Markov chains fields indexed by a homogeneous tree. At first we study a generalized strong limit theorem for nonhomogeneous Markov chains indexed by a homogeneous tree. Then we prove the generalized strong laws of large numbers and the generalized asymptotic equipartition property for delayed sums of finite nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we can get the similar results of some current literatures. In this paper, the problem settings may not allow to use Doob's martingale convergence theorem, and we overcome this difficulty by using Borel–Cantelli Lemma so that our proof technique also has some new elements compared with the reference Yang and Ye (2007).

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.

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.

This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at $p$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

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.

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.

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.

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.

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

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.

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.

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

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

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.

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).

We consider the question "Is every nonzero generic degree a density-1-bounding generic degree?" By previous results \cite{I2} either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees. We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide class of assumptions is sufficient to prove density-1-bounding. We also consider a historic difficulty in constructing a potential counterexample: By previous results \cite{I1} any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal. Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them. Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp \cite{HJKS}, this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees., Submitted. Revised. Our thanks to the reviewer and editor

In this paper we study a question related to the classical Erd\H{o}s-Ko-Rado theorem, which states that any family of $k$-element subsets of the set $[n] = \{1,\ldots,n\}$ in which any two sets intersect, has cardinality at most ${n-1\choose k-1}$. We say that two non-empty families are $\mathcal A, \mathcal B\subset {[n]\choose k}$ are {\it $s$-cross-intersecting}, if for any $A\in\mathcal A,B\in \mathcal B$ we have $|A\cap B|\ge s$. In this paper we determine the maximum of $|\mathcal A|+|\mathcal B|$ for all $n$. This generalizes a result of Hilton and Milner, who determined the maximum of $|\mathcal A|+|\mathcal B|$ for nonempty $1$-cross-intersecting families., Comment: This article was previously a portion of arXiv:1603.00938v1, which has been split

In this paper, some results on the complete moment convergence for arrays of rowwise negatively associated (NA, for short) random variables are established. The results obtained in this paper correct the corresponding one obtained in Ko [13] and also improve and generalize the corresponding ones of Kuczmaszewska [14] and Ko [13]. As an application of the main results, we present a result on complete consistency for the estimator in a non-parametric regression model based on NA errors. Finally, we provide a numerical simulation to verify the validity of our result.

Twisted commutative algebras (tca's) have played an important role in the nascent field of representation stability. Let A_d be the complex tca freely generated by d indeterminates of degree 1. In a previous paper, we determined the structure of the category of A_1-modules (which is equivalent to the category of FI-modules). In this paper, we establish analogous results for the category of A_d-modules, for any d. Modules over A_d are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra., Comment: 51 pages; v2: many revisions

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.