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A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

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.

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, 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.

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

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.

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 $U_{q}(N)$.

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.

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.

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.

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.

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.

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.

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

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).

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.

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.

One of the many definite integrals that Euler was the first to evaluate was(1)He did this, almost as an afterthought, at the end of his short, seven-page paper catalogued as E675 in [1] and with the matter-of-fact title, On the values of integrals from x = 0 to x = ∞. It is a beautiful Euler miniature which neatly illustrates the unexpected twists and turns in the history of mathematics. For Euler's derivation of (1) emerges as the by-product of a solution to a problem in differential geometry concerning the clothoid curve which he had first encountered nearly forty years earlier in his paper E65, [1]. As highlighted in the recent Gazette article [2], E675 is notable for Euler's use of a complex number substitution to evaluate a real-variable integral. He used this technique in about a dozen of the papers written in the last decade of his life. The rationale for this manoeuvre caused much debate among later mathematicians such as Laplace and Poisson and the technique was only put on a secure footing by the work of Cauchy from 1814 onwards on the foundations of complex function theory, [3, Chapter 1]. Euler's justification was essentially pragmatic (in agreement with numerical evidence) and by what Dunham in [4, p. 68] characterises as his informal credo, ‘Follow the formulas, and they will lead to the truth.’ Smithies, [3, p. 187], contextualises Euler's approach by noting that, at that time, ‘a function was usually thought of as being defined by an analytic expression; by the principle of the generality of analysis, which was widely and often tacitly accepted, such an expression was expected to be valid for all values, real or complex, of the independent variable’. In this article, we examine E675 closely. We have tweaked notation and condensed the working in places to reflect modern usage. At the end, we outline what is, with hindsight, needed to make Euler's arguments watertight: it is worth noting that all of his conclusions survive intact and that the intermediate functions of one and two variables that he introduces in E675 remain the key ingredients for much subsequent work on these integrals.

In this paper we demonstrate thatnon-commutative localizationsof arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category$\text{}\underline{C}$by a set$S$of morphisms in the heart$\text{}\underline{Hw}$of a weight structure$w$on it one obtains a triangulated category endowed with a weight structure$w^{\prime }$. The heart of$w^{\prime }$is a certain version of the Karoubi envelope of the non-commutative localization$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$(of$\text{}\underline{Hw}$by$S$). The functor$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of$S$invertible. For any additive category$\text{}\underline{A}$, taking$\text{}\underline{C}=K^{b}(\text{}\underline{A})$we obtain a very efficient tool for computing$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$coincides with the ‘abstract’ localization$\text{}\underline{A}[S^{-1}]$(as constructed by Gabriel and Zisman) if$S$contains all identity morphisms of$\text{}\underline{A}$and is closed with respect to direct sums. We apply our results to certain categories of birational motives$DM_{gm}^{o}(U)$(generalizing those defined by Kahn and Sujatha). We define$DM_{gm}^{o}(U)$for an arbitrary$U$as a certain localization of$K^{b}(Cor(U))$and obtain a weight structure for it. When$U$is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general$U$the result is completely new. The existence of the correspondingadjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over$U$.

A setisshyorHaar null(in the sense of Christensen) if there exists a Borel setand a Borel probability measureμonC[0, 1] such thatandfor allf∈C[0, 1]. The complement of a shy set is called aprevalentset. We say that a set isHaar ambivalentif it is neither shy nor prevalent.The main goal of the paper is to answer the following question: what can we say about the topological properties of the level sets of the prevalent/non-shy manyf∈C[0, 1]?The classical Bruckner–Garg theorem characterizes the level sets of the generic (in the sense of Baire category)f∈C[0, 1] from the topological point of view. We prove that the functionsf∈C[0, 1] for which the same characterization holds form a Haar ambivalent set.In an earlier paper, Balkaet al. proved that the functionsf∈C[0, 1] for which positively many level sets with respect to the Lebesgue measure λ are singletons form a non-shy set inC[0, 1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functionsf∈C[0, 1] for which positively many level sets with respect to the occupation measure λ ◦f–1are not perfect form a Haar ambivalent set inC[0, 1].We show that for the prevalentf∈C[0, 1] for the genericy∈f([0, 1]) the level setf–1(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functionsf∈C[0, 1] for which there exists a perfect setPf⊂ [0, 1] such thatfʹ(x) = ∞ for allx∈Pfis Haar ambivalent.

This paper deals with the geometric local theta correspondence at the Iwahori level for dual reductive pairs of type II over a non Archimedean field $F$ of characteristic $p\neq 2$ in the framework of the geometric Langlands program. First we construct and study the geometric version of the invariants of the Weil representation of the Iwahori-Hecke algebras. In the particular case of $(GL_1, GL_m)$ we give a complete geometric description of the corresponding category. The second part of the paper deals with geometric local Langlands functoriality at the Iwahori level in a general setting. Given two reductive connected groups $G$, $H$ over $F$ and a morphism $\check{G}\times \mathrm{SL}_2\to\check{H}$ of Langlands dual groups, we construct a bimodule over the affine extended Hecke algebras of $H$ and $G$ that should realize the geometric local Arthur-Langlands functoriality at the Iwahori level. Then, we propose a conjecture describing the geometric local theta correspondence at the Iwahori level constructed in the first part in terms of this bimodule and we prove our conjecture for pairs $(GL_1, GL_m)$., Comment: To appear in the Journal of the Institute of Mathematics of Jussieu

SUMMARYThis paper presents a new numerical approach using a Genetic algorithm (GA) to search for the singularity-free cylindrical space of a 4-RRR planar redundant parallel manipulator and investigates the effects of the joint position (namely the length ratios of two links) of each leg on the singularity-free cylindrical space. A previous method investigated the maximal singularity-free zone in a 3-dimensional (3-D) space within a given workspace. The method in this paper is improved by optimizing the maximal singularity-free zone in a 2-dimensional (2-D) plane while considering the whole workspace. This improvement can be helpful for reducing the searching time and for finding a larger singularity-free zone. Furthermore, the effect of the joint position of each leg on the maximal singularity-free zone is studied in this paper, which reveals a larger singularity-free zone than before. This result shows that changing the joint positions of one or two legs may be more practical than changing the joint positions of more legs. The approach in this paper can be used to analyze the maximal singularity-free zone of any other three-degree-of-freedom (3-DOF) planar parallel mechanisms and will be useful for the optimal design of redundant parallel manipulators.

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion, and demonstrate that Bismut–Zhang’s equivariant Cheeger–Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and torsion cohomology comparison for arithmetic locally symmetric spaces to an equivariant setting.

SUMMARYThe weightless planar 2R underactuated manipulators with passive last joint are considered in this paper for investigating a feasible method to stabilize the system, which is a second-order nonholonomic-constraint mechanical system with drifts. The characteristics including the controllability of the linear approximation model, the minimum phase property, the Small Time Local Controllability (STLC), the differential flatness, and the exactly nilpotentizable properties, are analyzed. Unfortunately, these negative characteristics indicate that the simplest underactuated mechanical system is difficult to design a stable closed-loop control system. In this paper, nilpotent approximation and iterative steering methods are utilized to solve the problem. A globally effective nilpotent approximation model is developed and the parameterized polynomial input is adopted to stabilize the system to its non-singularity equilibrium configuration. In accordance with this scheme, it is shown that designing a stable closed-loop control system for the underactuated mechanical system can be ascribed to solving a set of nonlinear algebraic equations. If the nonlinear algebraic equations are solvable, then the controller is asymptotically stable. Some numerical simulations demonstrate the effectiveness of the presented approach.

Tent spaces of vector-valued functions were recently studied by Hytönen, van Neerven and Portal with an eye on applications to $H^{\infty }$-functional calculi. This paper extends their results to the endpoint cases $p=1$ and $p=\infty $ along the lines of earlier work by Harboure, Torrea and Viviani in the scalar-valued case. The main result of the paper is an atomic decomposition in the case $p=1$, which relies on a new geometric argument for cones. A result on the duality of these spaces is also given.

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

The purpose of this paper is to give a complete description of the primitive ideal space of the C*-algebra [R] associated to the ring of integers R in a number field K in the recent paper [5]. As explained in [5], [R] can be realized as the Toeplitz C*-algebra of the affine semigroup R ⋊ R× over R and as a full corner of a crossed product C0() ⋊ K ⋊ K*, where is a certain adelic space. Therefore Prim([R]) is homeomorphic to the primitive ideal space of this crossed product. Using a recent result of Sierakowski together with the fact that every quasi-orbit for the action of K ⋊ K* on contains at least one point with trivial stabilizer we show that Prim([R]) is homeomorphic to the quasi-orbit space for the action of K ⋊ K* on , which in turn may be identified with the power set of the set of prime ideals of R equipped with the power-cofinite topology.

Let (X, O) be a germ of a normal surface singularity, π: → X be the minimal resolution of singularities and let A = (ai,j) be the n × n symmetrical intersection matrix of the exceptional set of In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme , and defines a map from the set of irreducible components of to the set of exceptional components of the minimal resolution of singularities of (X,O). He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition •For any couple (Ei,Ej) of distinct exceptional components, we define Numerical Nash condition (NN(i,j)). We have that (NN(i,j)) implies In this paper we prove that (NN(i,j)) is always true for at least the half of couples (i,j).•The condition (NN(i,j)) is true for all couples (i,j) with i ≠ j, characterizes a certain class of negative definite matrices, that we call Nash matrices. If A is a Nash matrix then the Nash map N is bijective. In particular our results depend only on A and not on the topological type of the exceptional set.•We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.•We give infinitely many other classes of singularities where Nash Conjecture is true.The proofs are based on my old work [8] and in Plenat [10].

In this paper, we give a characterization of the simultaneous unitarizability of any finite set of SL(2, ℂ)-valued functions on and determine all possible ways of the unitarization. Such matrix functions can be regarded as images of the generators for the fundamental group of a surface in an -family, and the results of this paper have applications in the construction of constant mean curvature surfaces in space.

An algebraic group of type E7 over an algebraically closed field has an irreducible representation in a vector space of dimension 56 and is, in fact, the identity component of the automorphism group of a quartic form on the space. This paper describes the construction of the quartic form if the characteristic is ≠ 2, 3, taking into account a field of definition F. Certain F-forms of E7 appear in the automorphism groups of quartic forms over F, as well as forms of E6. Many of the results of the paper are known, but are perhaps not easily accessible in the literature.

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal $\a$ with rational exponent $t \ge 0$. We first prove a key lemma of this paper, which gives a characterization of the ideal $\tau(\a^t)$. As applications of this key lemma, we generalize the preceding results on the behavior of the test ideal $\tau(R)$. Moreover, we prove an analog of so-called Skoda's theorem, which is formulated algebraically via adjoint ideals by Lipman in his proof of the "modified Brian\c{c}on--Skoda theorem.", Comment: 11 pages, AMS-LaTeX; v.2: minor changes, to appear in Nagoya Math. J

We correct two technical errors in the original paper. The main result in the original paper remains valid without any changes.

Suppose that {X s , 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X s ), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

This paper is concerned with a finitely generated module M over a (commutative Noetherian) local ring R. In the case when R is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of M with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when R is universally catenary and such that all its formal fibres are Cohen–Macaulay.These formulae involve certain subsets of the spectrum of R called the pseudosupports of M; these pseudo-supports are closed in the Zariski topology when R is universally catenary and has the property that all its formal fibres are Cohen–Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

We set out some general criteria to prove the K-property, refining the assumptions used in an earlier paper for the flow case, and introducing the analogous discrete-time result. We also introduce one-sided $\lambda $ -decompositions, as well as multiple techniques for checking the pressure gap required to show the K-property. We apply our results to the family of Mañé diffeomorphisms and the Katok map. Our argument builds on the orbit decomposition theory of Climenhaga and Thompson.

A gap mapping is a discontinuous interval mapping with two strictly increasing branches that have a gap between their ranges. They are one-dimensional dynamical systems, which arise in the study of certain higher dimensional flows, for example the Lorenz flow and the Cherry flow. In this paper, we prove hyperbolicity of renormalization acting on $C^3$ dissipative gap mappings, and show that the topological conjugacy classes of infinitely renormalizable gap mappings are $C^1$ manifolds.

In this paper we prove a local exponential synchronization for Markovian random iterations of homeomorphisms of the circle $S^{1}$ , providing a new result on stochastic circle dynamics even for $C^1$ -diffeomorphisms. This result is obtained by combining an invariance principle for stationary random iterations of homeomorphisms of the circle with a Krylov–Bogolyubov-type result for homogeneous Markov chains.

It is proved that every holomorphically convex complex space endowed with an action of a compact Lie group K can be realized as an open K-stable subspace of a holomorphically convex space endowed with a holomorphic action of the complexified group K . Similar results are obtained for holomorphic if-bundles over such spaces. Let G be a real Lie group which acts by holomorphic transformations on a (reduced) complex space X. Suppose that the Lie algebra of the complexification G c of G (see [Ho, p. 204]) is the complexification of the Lie algebra of G. This holds for example in the case where G is simply connected. Then, by integrating the holomorphic vector fields given by the G-action, the complexification G c acts locally and holomorphically on X (see [K]). Adapting the terminology of Palais (see [P]), we say that a complex space X* which contains X as an open subset is a globalization of the complex G-space X whenever the local G-action on X extends to a global holomorphic action on X* and G c X = X*. The following results are proved in this paper. THEOREM 1. Let G be a compact Lie group and X a holomorphically convex complex G-space X. Then there exists a globalization X* of X satisfying the following conditions (i) X* is holomorphically convex (ii) Every G-equivariant holomorphic map ψ from X into a complex space Received August 7, 1995. Γhe first author would like to thank the Department of Mathematics of Brandeis University for its hospitality while a part of this paper was written. This project was partially supported by Sonderforschungbereich 237 "Unordnung und groβe Fluktuationeon" of the Deutsche Forschungsgemeinschaft. 2 The second author is supported by an Instituto Nazionale di Alt a Matematica grant which is gratefully acknowledged.

In this paper we obtain some improved $L^p$ -Hardy and $L^p$ -Rellich inequalities on bounded domains of Riemannian manifolds. For Cartan–Hadamard manifolds we prove the inequalities with sharp constants and with weights being hyperbolic functions of the Riemannian distance.

We study the behavior of representation varieties of quivers with relations under the operation of node splitting. We show how splitting a node gives a correspondence between certain closed subvarieties of representation varieties for different algebras, which preserves properties like normality or having rational singularities. Furthermore, we describe how the defining equations of such closed subvarieties change under the correspondence. By working in the "relative setting" (splitting one node at a time), we demonstrate that there are many non-hereditary algebras whose irreducible components of representation varieties are all normal with rational singularities. We also obtain explicit generators of the prime defining ideals of these irreducible components. This class contains all radical square zero algebras, but also many others, as illustrated by examples throughout the paper. We also show the above is true when replacing irreducible components by orbit closures, for a more restrictive class of algebras. Lastly, we provide applications to decompositions of moduli spaces of semistable representations of certain algebras., 26 pages. Updated references, typos, metadata. Actually final version

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

A Birkhoff billiard is a system describing the inertial motion of a point mass inside a strictly convex planar domain, with elastic reflections at the boundary. The study of the associated dynamics is profoundly intertwined with the geometric properties of the domain: while it is evident how the shape determines the dynamics, a more subtle and difficult question is the extent to which the knowledge of the dynamics allows one to reconstruct the shape of the domain. This translates into many intriguing inverse problems and unanswered rigidity questions, which have been the focus of very active research in recent decades. In this paper we describe some of these questions, along with their connection to other problems in analysis and geometry, with particular emphasis on recent results obtained by the authors and their collaborators.

We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

Let G be a locally compact group and let ${\mathcal {SUB}(G)}$ be the hyperspace of closed subgroups of G endowed with the Chabauty topology. The main purpose of this paper is to characterise the connectedness of the Chabauty space ${\mathcal {SUB}(G)}$ . More precisely, we show that if G is a connected pronilpotent group, then ${\mathcal {SUB}(G)}$ is connected if and only if G contains a closed subgroup topologically isomorphic to ${{\mathbb R}}$ .

The aim of this paper is to study complete (noncompact) m-quasi-Einstein manifolds with λ=0 satisfying a fourth-order vanishing condition on the Weyl tensor and zero radial Weyl curvature. In this case, we are able to prove that an m-quasi-Einstein manifold (m>1) with λ=0 on a simply connected n-dimensional manifold(Mn, g), (n ≥ 4), of nonnegative Ricci curvature and zero radial Weyl curvature must be a warped product with (n–1)–dimensional Einstein fiber, provided that M has fourth-order divergence-free Weyl tensor (i.e. div4W =0).

In this paper, we study the structure of the rational cohomology groups of the IA-automorphism group$\mathrm {IA}_3$of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of$\mathrm {IA}_3$, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of$\mathrm {IA}_3$has finite index in that of the Andreadakis–Johnson filtration of$\mathrm {IA}_3$.

We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Brou\'e, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence., Comment: 10 pages