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We review what is known, unknown, and expected about the mathematical properties of Coulomb and Riesz gases. Those describe infinite configurations of points in R d interacting with the Riesz potential ±|x|−s (respectively, −log |x| for s = 0). Our presentation follows the standard point of view of statistical mechanics, but we also mention how these systems arise in other important situations (e.g., in random matrix theory). The main question addressed in this Review is how to properly define the associated infinite point process and characterize it using some (renormalized) equilibrium equation. This is largely open in the long range case s < d. For the convenience of the reader, we give the detail of what is known in the short range case s > d. Finally, we discuss phase transitions and mention what is expected on physical grounds. [ABSTRACT FROM AUTHOR]

In this article, we study a family of subgraphs of the Farey graph, denoted as ℱN for every N ∈ ℕ. We show that ℱN is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as ℱN -continued fractions for each N > 1. We establish a relation between ℱN-continued fractions and certain paths from infinity in the graph ℱN. Using this correspondence, we discuss the existence and uniqueness of ℱN-continued fraction expansions of real numbers. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Hyers-Ulam stability problem for the following function inequalities ... in β-homogeneous F-spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributed field estimation problem using a sensor network, and the main purpose is to design a local filter for each sensor node to estimate a spatially-distributed physical process using the measurements of the whole network. The finite element method is employed to discretize the infinite dimensional process, which is described by a partial differential equation, and an approximate finite dimensional linear system is established. Due to the sparsity on the spatial distribution of the source function, the ℓ 1 -regularized H ∞ filtering is introduced to solve the estimation problem, which attempts to provide better performance than the classical centralized Kalman filtering. Finally, a numerical example is provided to demonstrate the effectiveness and applicability of the proposed method. [ABSTRACT FROM AUTHOR]

Continued fraction and quasi-reciprocal continued fraction expansions of the generating function of Bernoulli numbers have been obtained. The convergence and uniform convergence of continued fraction expansions have been proved. Representations of the generating function of Bernoulli polynomials in the form of the product of three continued fractions, as well as the product of three quasi-reciprocal continued fractions, have been found. [ABSTRACT FROM AUTHOR]

We investigate the applications of slowly varying functions (in Karamata's sense) to the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment but regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine the known limit results. [ABSTRACT FROM AUTHOR]

We study transport processes on infinite metric graphs with non-constant velocities and matrix boundary conditions in the L ∞ L ∞ -setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions. [ABSTRACT FROM AUTHOR]

We construct an infinite volume spatial random permutation (X , σ) , where X ⊂ R d is locally finite and σ : X → X is a permutation, associated to the formal Hamiltonian H (X , σ) = ∑ x ∈ X ‖ x - σ (x) ‖ 2. The measures are parametrized by the point density ρ and the temperature α . Spatial random permutations are naturally related to boson systems through a representation originally due to Feynman (Phys Rev 91:1291–1301, 1953). Let ρ c = ρ c (α) be the critical density for Bose–Einstein condensation in Feynman's representation. Each finite cycle of σ induces a loop of points of X . For ρ ≤ ρ c we define (X , σ) as a Poisson process of finite unrooted loops of a random walk with Gaussian increments that we call Gaussian loop soup, analogous to the Brownian loop soup of Lawler and Werner (Probab Theory Related Fields 128(4):565–588, 2004). We also construct Gaussian random interlacements, a Poisson process of doubly infinite trajectories of random walks with Gaussian increments analogous to the Brownian random interlacements of Sznitman (Ann Math 2 171(3):2039–2087, 2010). For d ≥ 3 and ρ > ρ c we define (X , σ) as the superposition of independent realizations of the Gaussian loop soup at density ρ c and the Gaussian random interlacements at density ρ - ρ c . In either case we call (X , σ) a Gaussian random permutation at density ρ and temperature α . The resulting measure satisfies a Markov property and it is Gibbs for the Hamiltonian H. Its point marginal X has the same distribution as the boson point process introduced by Shirai-Takahashi (J Funct Anal 205(2):414–463, 2003) in the subcritical case, and by Tamura-Ito (J Funct Anal 243(1): 207–231, 2007) in the supercritical case. [ABSTRACT FROM AUTHOR]

In this paper, we define I-convergence and I*-convergence of an infinite product by Cauchy conditions and prove the relation between these two notions. [ABSTRACT FROM AUTHOR]

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is wellknown that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is -autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics. [ABSTRACT FROM AUTHOR]

We introduce two groups of duplication processes that extend the well known Cayley–Dickson process. The first one allows to embed every 4 -dimensional (4D) real unital algebra into an 8D real unital algebra denoted by FD (). We also find the conditions on under which FD () is a division algebra. This covers the most classes of known 4 D real division algebras. The second process allows us to embed particular classes of 4 D RDAs into 8 D RDAs. Besides, both duplication processes give an infinite family of non-isomorphic 8 D real division algebras whose derivation algebras contain su (2). [ABSTRACT FROM AUTHOR]

• A fully discretization scheme by the implicit Euler method for stochastic population models. • The preservation of the total population with a "suitable" numerical boundary condition. • The proposed numerical basic reproduction number R h through embedded infinite stochastic Leslie operators. • The preservation and detection of the analytic stability through numerical solutions for small stepsize. In this paper, a fully discretization scheme based on the implicit Euler method (IM) is considered for stochastic age-structured population models. The preservation of the total population with a suitable numerical boundary condition according to the biological meanings are shown. An explicit formula of the numerical basic reproductive number R h is proposed by the technique that the numerical process is embedded into an l 1 (R) -valued integrable stochastic process with infinite stochastic Leslie operators. Furthermore, the convergence and connection between R h and the stability of numerical solution is analyzed. The preservation and detection of the analytic stability through the numerical solutions are discussed for small stepsize. Finally, some numerical experiments including an infection-age model for modified SARS epidemic illustrate the verification and efficiency of our analysis. [ABSTRACT FROM AUTHOR]

We propose a class of numerical schemes for mixed optimal stopping and control of processes with infinite activity jumps and where the objective is evaluated by a nonlinear expectation. Exploiting an approximation by switching systems, piecewise constant policy timestepping reduces the problem to nonlocal semi-linear equations with different control parameters, uncoupled over individual time steps, which we solve by fully implicit monotone approximations to the controlled diffusion and the nonlocal term, and specifically the Lax–Friedrichs scheme for the nonlinearity in the gradient. We establish a comparison principle for the switching system and demonstrate the convergence of the schemes, which subsequently gives a constructive proof for the existence of a solution to the switching system. Numerical experiments are presented for a recursive utility maximization problem to demonstrate the effectiveness of the new schemes. [ABSTRACT FROM AUTHOR]

Biological processes are inherently continuous, and the chance of phenotypic discovery is significantly restricted by discretising them. Using multi-parametric active regression we introduce the Regression Plane (RP), a user-friendly discovery tool enabling class-free phenotypic supervised machine learning, to describe and explore biological data in a continuous manner. First, we compare traditional classification with regression in a simulated experimental setup. Second, we use our framework to identify genes involved in regulating triglyceride levels in human cells. Subsequently, we analyse a time-lapse dataset on mitosis to demonstrate that the proposed methodology is capable of modelling complex processes at infinite resolution. Finally, we show that hemocyte differentiation in Drosophila melanogaster has continuous characteristics. High-content screening prompted the development of software enabling discrete phenotypic analysis of single cells. Here, the authors show that supervised continuous machine learning can drive novel discoveries in diverse imaging experiments and present the Regression Plane module of Advanced Cell Classifier. [ABSTRACT FROM AUTHOR]

Many studies have stressed that human activities may cause the extinction of single species. Anthropogenic activities, however, may affect not only the number of individuals of single species, but also their behavior. To investigate this issue, we propose a growth model in which agents may care not only for the species' survival but also for the typicality of their behavior. We assume that the environmental defensive expenditures can protect the species avoiding their extinction, but can induce the species to modify their behavior. Results emerging from the model suggest that if the social planner cares for typicality of species behavior, then an infinite growth process may no longer be optimal. Numerical simulations, moreover, show the possible existence of a trade-off between number and behavior of the species, leading the system to a high number of species' members that behave in an atypical way or to few members behaving very typically. [ABSTRACT FROM AUTHOR]

This article proposes frequentist multiple-equation least-squares averaging approaches for multistep forecasting with vector autoregressive (VAR) models. The proposed VAR forecast averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multistep forecast averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecast averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step-ahead forecast averaging, whereas for direct multistep forecast averaging, the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. To present our methodology, we investigate the finite-sample behavior of the proposed averaging procedures under model misspecification via simulation experiments. [ABSTRACT FROM AUTHOR]

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) that contain a stochastic process in their vector field functions. They have been used for many years in a wide range of applications, but have been a shadow existence to stochastic differential equations (SDEs) despite being able to model a wider and often physically more adequate range of disturbances. In this article, we study the safety verification problem over both finite time horizons and the infinite time horizon for RODEs incorporating Wiener processes. Concretely, we investigate the ${p}$ -safety problem, where we identify the set of initial states from which the probability to satisfy safety specifications is at least ${p}$. Based on identifying a set of sample paths whose probability measure is larger than ${p}$ , we propose a method of reducing stochastic reachability to adversary reachability of ODEs for solving the ${p}$ -safety problem over finite time horizons. This method permits an efficient lifting of reach-set computation methods for perturbed ODEs to RODEs. In this method, the ${p}$ -safety problem over finite time horizons is reduced to the problem of inner-approximating robust backward reachable sets for ODEs with time-varying perturbation inputs. We then extend the method to the ${p}$ -safety problem over the infinite time horizon. Finally, we demonstrate our method on several examples. [ABSTRACT FROM AUTHOR]

Since Wiener pointed out that the RSA can be broken if the private exponent d is relatively small compared to the modulus N , it has been a general belief that the Wiener attack works for d < N 1 4 . On the contrary, in [1], it was shown that the bound d < N 1 4 is not accurate as it has been thought of. Specifically, for the standard assumption of the two primes p and q that q < p < 2 q , the Wiener continued fraction technique is proven to work for d ≤ 1 18 4 N 1 4 . In this paper, we consider a general condition on the RSA primes, namely q < p < α q , and we give the corresponding bound for the Wiener attack to work, which is d ≤ α 4 2 (α + 1) N 1 4 . In a special case when α = 2 , this general bound agrees with the result of [1]. [ABSTRACT FROM AUTHOR]

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a nonnegative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight. [ABSTRACT FROM AUTHOR]

An explicit form of genus character L‐functions of quadratic orders is presented in full generality. As an application, we generalize a formula due to Hirzebruch and Zagier on the class number of imaginary quadratic fields expressed in term of the continued fraction expansion. [ABSTRACT FROM AUTHOR]

We study the coarse-grained distribution of a Hamiltonian system on the space partition determined by the initial measurement inaccuracies. Using methods of coding theory, introduced by Shannon and further researchers, Kolmogorov treated the stationary case for a discretized time, when the microscopic system is initially uniformly distributed. Following his work, we consider the non-stationary mesoscopic process induced by the Hamiltonian evolution from an inhomogeneous initial distribution. In general, this process has an infinite memory, but we show that its memory fades out with time: with any finite accuracy a, it can be approximated by a process with a memory limited to the n past events, n depending only on a. As a result, under suitable hypotheses, the mesoscopic process obeys an approximate Markov equation on groups of n successive states. More roughly, one obtains an ordinary Markov system by introducing a time coarse-graining on n successive elementary time steps. So, in a generic case, the system eventually tends to equilibrium for any initial mesoscopic distribution. [ABSTRACT FROM AUTHOR]

In some applications the considered multistage stochastic programs have a periodical behavior. We show that in such cases it is possible to drastically reduce the number of stages by introducing a periodical analogue of the so-called Bellman equations for discounted infinite horizon problems used in Markov decision processes and stochastic optimal control. Furthermore, we describe a variant of the stochastic dual dynamic programming algorithm, applied to the constructed periodical Bellman equations, and provide numerical experiments for the Brazilian interconnected power system problem. [ABSTRACT FROM AUTHOR]

The difficulties students have in blending mathematics and physics are here analyzed, by focusing on the issue of a convergent series. We present an experimental and a theoretical analysis of some phenomena which can be investigated employing series, as the bouncing marble and Zeno's paradox of Achilles and the turtle. Measurements were carried out by students employing ICT instruments, such as the smartphone microphone, the smartphone camera or online motion sensors and results were the grounds for a deep discussion about the apparent paradox and the sources of students' misunderstanding. The activities were designed for students on introductory university courses or in advanced high-school classes and was implemented with 90 students of mathematics and physics who are interested in a curriculum addressed to the teaching of mathematics and physics at high school level. Results about their preconceptions before the sequence and some quotes of their metacognitive thinking after the activities are reported. [ABSTRACT FROM AUTHOR]

Many real phenomena exhibit non‐Gaussian behavior. The non‐Gaussianity is manifested by impulsive behavior of the real data that can be found in both one‐dimensional and multi‐dimensional cases. Especially the multi‐dimensional datasets with non‐Gaussian behavior pose substantial analysis challenges to scientists and statisticians. In this article, we analyze the bidimensional vector autoregressive (VAR) model based on general bidimensional α‐stable distribution. This time series can be applied in modeling bidimensional data with impulsive behavior. We focus on the description of the spatio‐temporal dependence for analyzed bidimensional time series which in the considered case cannot be expressed in the language of the classical cross‐covariance or cross‐correlation function. We propose a new cross measure based on the alternative measure of dependence adequate for infinite variance processes, namely cross‐covariation. This article is an extension of the authors' previous work where the cross‐codifference was considered as the spatio‐temporal measure of the components of VAR model based on sub‐Gaussian distribution. In this article, we demonstrate that cross‐codifference and cross‐covariation can give different information about the relationships between components of bidimensional VAR models. [ABSTRACT FROM AUTHOR]

We investigate the existence and uniqueness of strong solutions for state-dependent regime-switching diffusion processes in an infinite state space with singular coefficients. Non-explosion conditions are given by using the Zvonkin's transformation. The strong Feller property is proved by further assuming that the diffusion in each fixed environment generates a strong Feller semigroup, and our results can also be applied to irregular or degenerate situations. [ABSTRACT FROM AUTHOR]

We give a new class of multidimensional p-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that the multidimensional p-adic version of Lagrange's Theorem will hold. [ABSTRACT FROM AUTHOR]

A variant of the standard symmetric system of two parallel queues under the join-the-shortest-queue policy is introduced. Here, the shortest queue has service rate μ 1 , while the longest queue has rate μ 2 , where μ 1 + μ 2 = 1 . In the case of equality, both queues are served at rate 1 / 2. Each queue has capacity K, which may be finite or infinite, and the global Poisson arrival rate is ρ . Couplings show that, as μ 2 varies, the model is totally ordered, in terms of both total number N(t) of customers in the system and longest queue length. The two extreme cases μ 2 = 0 and μ 2 = 1 then provide simple stochastic bounds for N(t) for arbitrary μ 2 . The ordering partially extends to the enlarged model in which, whenever the shortest queue is empty, the idle server at that queue helps—or on the contrary disturbs—the other server. The previous bounds remain valid. In the extended setup, different martingales are next built from the infinite capacity process. Those lead to simple explicit formulations of the means and Laplace transforms of the hitting time of saturation, for the process with finite K started from any initial state. Asymptotics are then derived, as K gets large. Using one particular mean time, the stationary blocking probability is explicitly obtained, extending the result by Dester et al. regarding the standard symmetric model. Finally, the joint distribution of the time and state of the system at the first queue-length equality is expressed as a function of the inverse of a simple explicit matrix. [ABSTRACT FROM AUTHOR]

This study furthers the theory of conceptual blending as a useful tool for revealing the structure and process of student reasoning in relation to the Sierpinski triangle (ST). We use conceptual blending to investigate students' reasoning, revealing how students engage with the ST and coordinate their understandings of its area and perimeter. Our analysis of ten individual interviews with mathematics education masters' student documents diverse ways in which students reason about this situation through the constituent processes of blending: composition, completion, and elaboration. This reveals that students who share basic understandings of the area and perimeter of the ST recruit idiosyncratic ideas to engage with and resolve the paradox of a figure with infinite perimeter and zero area. [ABSTRACT FROM AUTHOR]

The new computational methodology due to Yaroslav Sergeyev (see [22-24]) makes it possible to evaluate numerically the terminal features of complete, sequential decision-making processes. By standard numerical methods, these processes have indeterminate features or seem to support paradoxical conclusions. We show that they are better regarded as a class of problems for which the numerical methods based on Sergeyev's approach provide a uniform technique of resolution. [ABSTRACT FROM AUTHOR]

The viability of a switched system on a bounded polyhedron set, which is expressed by some linear inequalities, is investigated. Based on nonsmooth analysis and the properties of the tangent cone, a necessary and sufficient condition for viability is proposed. It is shown that the viability of a system is equivalent to the consistency of some systems of linear inequalities. Specifically, a viability condition for a switched system on a bounded polyhedron is presented. According to this condition, determining the viability of a bounded polyhedron can be transformed into verifying certain conditions at vertices of each facet. The method of determining viability, which transforms verifying the condition from infinite points to finite ones, can be implemented easily in practice. An algorithm to determine the viability for the switched system is constructed by using convex analysis. In addition, the approach can be extended to the switched system in which a control input is present. Finally, an example is listed to illustrate the effectiveness of the results. [ABSTRACT FROM AUTHOR]

We consider the problem of arranging a set of aircraft in a maintenance hangar operated by an independent aircraft service provider. The overall safety margins of the parking layout need to be maximized within the limited available space, measured by the weighted sum of the individual discrete safety margins of each aircraft. A mixed-integer linear programming model is developed, and the positions of the aircraft are determined by the position-controlling binary variables associated with a set of revised No-Fit Polygons (NFPs). Due to the nonconvex irregular shape of aircraft, the model involves a great number of binary variables associated with the revised NFP. The default branch-and-bound algorithm is inefficient in solving such a model as the infeasibility information of the precedent visited solution cannot be directly utilized by the default method to update the bounds. A heuristic algorithm is developed to provide practical solutions, and the intermediate infeasible solutions identified during searching are utilized to develop valid and approximate inequalities, tightening the optimality gap. The computational results demonstrate that the addition of inequalities improves the computational efficiency in solving a wide range of instances and in tightening the optimality gap while the stopping criterion is met. [ABSTRACT FROM AUTHOR]

We study the log-Minkowski inequality for centered convex bodies when the cone-volume body is a simplex or a parallelepiped. [ABSTRACT FROM AUTHOR]

By using the Riemann-Liouville fractional integral operator, we establish new weighted results of Chebyshev inequality type. Other integral inequalities of fractional order are also proved. Some classical results can be deduced as special cases. [ABSTRACT FROM AUTHOR]

In this article, we present an infinite number of refinements of the Heinz inequality for real numbers and operators. Making use of them, infinitely many refinements of the classical Pólya inequality and their operator versions are deduced. [ABSTRACT FROM AUTHOR]

We present a polynomial approximation method for realizing modified Landau-Kolmogorov type inequalities with small order derivatives. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]

Intrinsic inequalities involving Turán-type inequalities for someq-special functions are established. A special interest is granted toq-Dunkl kernel. The results presented here would provide extensions of those given in the classical case. [ABSTRACT FROM PUBLISHER]

We prove an inequality between the $$L^{\infty }$$ -norm of the contact Hamiltonian of a positive loop of contactomorphims and the minimal Reeb period. This implies that there are no small positive loops on hypertight or Liouville fillable contact manifolds. Non-existence of small positive loops for overtwisted 3-manifolds was proved by Casals et al. (J Symplectic Geom 14:1013-1031, 2016). As corollaries of the inequality we deduce various results. E.g. we prove that certain periodic Reeb flows are the unique minimisers of the $$L^{\infty }$$ -norm. Moreover, we establish $$L^\infty $$ -type contact systolic inequalities in the presence of a positive loop. [ABSTRACT FROM AUTHOR]

We prove that the relation of bisimilarity between countable labelled transition systems (LTS) is Σ-11complete (hence not Borel), by reducing the set of non-well orders over the natural numbers continuously to it. This has an impact on the theory of probabilistic and non-deterministic processes over uncountable spaces, since logical characterizations of bisimilarity (as, for instance, those based on the unique structure theorem for analytic spaces) require a countable logic whose formulas have measurable semantics. Our reduction shows that such a logic does not exist in the case of image-infinite processes. [ABSTRACT FROM AUTHOR]

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A brief exposition on some tools for proving the elliptic Harnack inequality is presented. [ABSTRACT FROM AUTHOR]

We use the inverse mean curvature flow with a free boundary perpendicular to the sphere to prove a geometric inequality involving the Willmore energy for convex hypersurfaces of dimension n ≥ 3 with boundary on the sphere. [ABSTRACT FROM AUTHOR]

We define a Bishop-type inequality on metric measure spaces with Riemannian curvature-dimension condition. The main result in this short article is that any RCD spaces with the Bishop-type inequalities possess only one regular set in not only the measure theoretical sense but also the set theoretical one. As a corollary, the Hausdorff dimension of such RCD* (K,N) spaces is exactly N. We also prove that every tangent cone at any point on such RCD spaces is a metric cone. [ABSTRACT FROM AUTHOR]

Some Hermite-Hadamard type inequalities for generalized k-fractional integrals (which are also named $(k,s)$ -Riemann-Liouville fractional integrals) are obtained for a fractional integral, and an important identity is established. Also, by using the obtained identity, we get a Hermite-Hadamard type inequality. [ABSTRACT FROM AUTHOR]

The notion of different kind of algebraic Casorati curvatures are introduced. Some results expressing basic Casorati inequalities for algebraic Casorati curvatures are presented. Equality cases are also discussed. As a simple application, basic Casorati inequalities for different δ-Casorati curvatures for Riemannian submanifolds are presented. Further applying these results, Casorati inequalities for Riemannian submanifolds of real space forms are obtained. Finally, some problems are presented for further studies. [ABSTRACT FROM AUTHOR]

In this short note, we present some trace inequalities for matrix means. Our results are generalizations of the ones shown by Bhatia, Lim, and Yamazaki. [ABSTRACT FROM AUTHOR]

A k-club is a subset of vertices of a graph that induces a subgraph of diameter at most k, where k is a positive integer. By definition, 1-clubs are cliques and the model is a distance-based relaxation of the clique definition for larger values of k. The k-club model is particularly interesting to study from a polyhedral perspective as the property is not hereditary on induced subgraphs when k is larger than one. This article introduces a new family of facet-defining inequalities for the 2-club polytope that unifies all previously known facets through a less restrictive combinatorial property, namely, independent (distance) 2-domination. The complexity of separation over this new family of inequalities is shown to be NP-hard. An exact formulation of this separation problem and a greedy separation heuristic are also proposed. The polytope described by the new inequalities (and nonnegativity) is then investigated and shown to be integral for acyclic graphs. An additional family of facets is also demonstrated for cycles of length indivisible by three. The effectiveness of these new facets as cutting planes and the difficulty of solving the separation problem in practice are then investigated via computational experiments on a test bed of benchmark instances. [ABSTRACT FROM AUTHOR]

Confidence intervals are commonly used to describe parameter uncertainty. In nonstandard problems, however, their frequentist coverage property does not guarantee that they do so in a reasonable fashion. For instance, confidence intervals may be empty or extremely short with positive probability, even if they are based on inverting powerful tests. We apply a betting framework and a notion of bet-proofness to formalize the 'reasonableness' of confidence intervals as descriptions of parameter uncertainty, and use it for two purposes. First, we quantify the violations of bet-proofness for previously suggested confidence intervals in nonstandard problems. Second, we derive alternative confidence sets that are bet-proof by construction. We apply our framework to several nonstandard problems involving weak instruments, near unit roots, and moment inequalities. We find that previously suggested confidence intervals are not bet-proof, and numerically determine alternative bet-proof confidence sets. [ABSTRACT FROM AUTHOR]

We show the positivity or negativity of truncated correlation functions in the quantum XY model with spin 1/2 (at any temperature) and spin 1 (in the ground state). These Griffiths-Ginibre inequalities of the second kind generalise an earlier result of Gallavotti. [ABSTRACT FROM AUTHOR]