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Let { A n } n be a sequence of square matrices such that size (A n) = d n → ∞ as n → ∞. We say that { A n } n has an asymptotic spectral distribution described by a Lebesgue measurable function f : D ⊂ R k → C if, for every continuous function F : C → C with bounded support, lim n → ∞ ⁡ 1 d n ∑ i = 1 d n F (λ i (A n)) = 1 μ k (D) ∫ D F (f (x)) d x , where μ k is the Lebesgue measure in R k and λ 1 (A n) , ... , λ d n (A n) are the eigenvalues of A n. In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A n arising from the discretization of differential equations, whose size d n diverges to ∞ along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit relation is satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. [ABSTRACT FROM AUTHOR]

AbstractLet the function β be strictly increasing and continuous on an interval I ⊂ ℝ. The β-difference operator is defined by Dβ f (t) = (f(β(t)) − f(t)/(β(t) – t), where t ≠ β(t), and Dβ f (t) = f′ t(t) when t = s0 is a fixed point of the function β. This quantum operator is a generalization of q-Jackson, Hahn, power and other quantum operators. As a convenience of the β-function: β(t) turns into the probability distribution function with the probability measure 1, and the sample space ℝ, in the case of its conditions are relaxed to be increasing and continuous from the right, that is, limt→∞ β(t) = 1 and limt→∞ β(t) = 0, and also by using the Lebesgue-Stieltjes measure of the interval [a, b] to be β(b) − β(a). In this paper, we investigate a β-Taylor’s formula associated with the operator Dβ when the function β has a unique fixed point s0 ∈ I, which may allow for more flexible and accurate approximations of functions. An estimation of its remainder is given. Additionally, the β-power series is defined. Furthermore, as application, the β-expansion form of some fundamental functions is introduced. Finally, we find the unique solution of the β-shifting problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a marginal screening method for variable screening in high-dimensional binary classification based on the Wasserstein distance accounting for the distributional difference. Many existing screening methods, such as the two-sample t-test and Kolmogorov test, have been developed under the parametric/nonparametric modeling assumptions to reduce the dimension of the predictors. However, such modeling specifications or nonparametric approaches are associated with the probability measure induced by the predictor in a Euclidean space. While many machine learning methods have successfully found the nonlinear decision boundary in the transformed space, called the reproducing kernel Hilbert space (RKHS), we consider the Wasserstein filter's capacity to detect the distributional difference between two probability measures induced by the nonlinear function of the predictor in the RKHS. Thereby, we can flexibly filter out the non-informative predictors associated with the binary classification, as well as escape the modeling assumptions required in a Euclidean space. We prove that the Wasserstein filter satisfies the sure screening property under some mild conditions. We also demonstrate the advantages of our proposed approach by comparing the finite sample performance of it with those of the existing choices through simulation studies, as well as through application to lung cancer data. [ABSTRACT FROM AUTHOR]

In this research article, we introduced the notion of q-probabilty distributions in quantum calculus. We characterized the concept of q-density by connecting it to a probability measure and investigated some of their outstanding properties. In this case, the Transfer theorem was extended in order to compute afterwards the q-moments, q-entropy, q-moment generating function, and q-quantiles. We are also interested in finding the centered q-Gaussian distribution N q (0 , σ 2) with variance σ 2 . We also proved that this q-distribution belongs to a class of classical discrete distributions. The centered q-Gaussian law N q (0 , σ 2) is also naturally related to the q-Gaussian distribution N q (μ , σ 2) with mean μ and standard deviation σ. We corroborated that the q-moments of these q-distributions are q-analogs of the moments of classical distributions. Numerical studies demonstrated that N q (0 , σ 2) interpolates between the classical Uniform and Gaussian distributions when q goes to 0 and 1, respectively. Subsequently, simulation studies for various q parameter values and samples sizes of the Gaussian q-distributions were conducted to demonstrate the effectiveness of the proposed model. Eventually, we provided some pertinent closing remarks and offered new perspectives for future works. [ABSTRACT FROM AUTHOR]

Let 퐺 be a real semisimple Lie group with finite centre and without compact factors, Q < G a parabolic subgroup and 푋 a homogeneous space of 퐺 admitting an equivariant projection on the flag variety G / Q with fibres given by copies of lattice quotients of a semisimple factor of 푄. Given a probability measure 휇, Zariski-dense in a copy of H = SL 2 ⁡ (R) in 퐺, we give a description of 휇-stationary probability measures on 푋 and prove corresponding equidistribution results. Contrary to the results of Benoist–Quint corresponding to the case G = Q , the type of stationary measures that 휇 admits depends strongly on the position of 퐻 relative to 푄. We describe possible cases and treat all but one of them, among others using ideas from the works of Eskin–Mirzakhani and Eskin–Lindenstrauss. [ABSTRACT FROM AUTHOR]

One of the fundamental results of ergodic optimization asserts that for any dynamical system on a compact metric space with the specification property and for a generic continuous function f every invariant probability measure that maximizes the space average of f must have zero entropy. We establish the analogical result in the context of constrained ergodic optimization, which is introduced by [Garibaldi and Lopes, Functions for relative maximization, Dyn. Syst. 22(4) (2007), pp. 511-528]. [ABSTRACT FROM AUTHOR]

The generalized mean-field orthoplicial model is a mean-field model on a space of continuous spins on R n that are constrained to a scaled (n - 1) -dimensional ℓ 1 -sphere, equivalently a scaled (n - 1) -dimensional orthoplex, and interact through a general interaction function. The finite-volume Gibbs states of this model correspond to singular probability measures. In this paper, we use probabilistic methods to rigorously classify the infinite-volume Gibbs states of this model, and we show that they are convex combinations of product states. The predominant methods utilize the theory of large deviations, relative entropy, and equivalence of ensembles, and the key technical tools utilize exact integral representations of certain partition functions and locally uniform estimates of expectations of certain local observables. [ABSTRACT FROM AUTHOR]

In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron's method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation. [ABSTRACT FROM AUTHOR]

In this paper, we analyze a certain family of holomorphic correspondences on ... x ... and prove their equidistribution properties. In particular, for any correspondence in this family we prove that the naturally associated multivalued map F is such that for any a ∈ ..., we have that (Fn)*(δa) converges to a probability measure μF for which F*(μF) = μF d where d is the degree of F. This result is used to show that the minimal Hutchinson invariant set, introduced by P. Alexandersson, P. Brändén, and B. Shapiro [An inverse problem in Pólya--Schur theory. I. Non-degenerate and degenerate operators, preprint, 2024], of a large class of operators and for sufficiently large n exists and is the support of the aforementioned measure. We prove that under a minor additional assumption, the minimal Hutchinson-invariant set is a Cantor set. [ABSTRACT FROM AUTHOR]

Predicting how nature's contributions to people (NCP) vary spatially remains a challenge. For NCP provided by mobile species, it is unclear how predictions need to account for the influence of multiple habitat types that act as sources, sinks and potential distractors of the NCP‐providing species. Existing approaches that do not account for these effects may inaccurately predict outcomes in real landscapes. To move beyond these limitations, we transfer quantitative inference approaches from movement ecology to explore how spatial habitat patterns determine the negative NCP of the invasive common brushtail possum Trichosurus vulpecula in New Zealand. We used a Bayesian model to investigate how the size of, and distance from, grassland and indigenous and exotic forest patches together contribute to relative possum density (measured by capture probability across a trapping network) in a heterogeneous 11,000‐ha landscape. We found that indigenous and exotic forest area were the most important factors in determining possum density. Although capture probability declined with increasing grassland area, the addition of grassland did not improve the relative model fit above one with indigenous forest as the only habitat. We expected differences in predicted possum density at habitat boundaries, for example, due to preferential foraging at edges. We found that indigenous and exotic forests contributed to capture probability interactively, such that capture probability at the between‐habitat edge was lower than expected, given the habitat area. We also found that models allowing for non‐linear habitat effects of exotic forests or grasslands, but not indigenous forests, were significantly better at predicting possum density than simpler models. Synthesis and applications. Our novel approach for spatial prediction can be applied to any of nature's contributions to people (NCP), and extended to identify trade‐offs and synergies among multiple NCP. For example, the negative NCP of possum density trades off with multiple known positive NCP from indigenous forests, including culturally significant non‐material NCP, and material NCP produced by exotic forests. We recommend that landscape management plans to maximise these positive NCP in future scenarios also consider how the risk of possum density may dampen net NCP provision. To minimise this negative NCP, our results support trap deployment in both indigenous and exotic forest. [ABSTRACT FROM AUTHOR]

In this article, we use the notion of generalized orthogonality for a sequence of polynomials introduced by Bryc, Fakhfakh, and Mlotkowski (2019) to extend the characterizations of the Feinsilver, Meixner, and Shanbhag based on orthogonal polynomials. These new versions subsume the real natural exponential families (NEFs) having polynomial variance function in the mean of arbitrary degree. We also relate generalized orthogonality to Sheffer systems. We show that the generalized orthogonality of Sheffer systems occurs if and only if the associated classical additive convolution semigroup of probability measures generates NEFs with polynomial variance function. In addition, we use the raising and lowering operators for quasi-monomial polynomials associated with NEFs to give a characterization of NEFs with polynomial variance function of arbitrary degree. [ABSTRACT FROM AUTHOR]

AbstractThis paper proposes a new association statistic for determining whether random variables are statistically independent. The proposed association statistic can also be used to examine the strength of both linear and non-linear dependency between variables. This statistic is derived by examining how the conditional probabilities of events differ from their corresponding marginal probabilities. The new statistic is defined in terms of the probability of acceptance commonly associated with the Accept-Reject algorithm and has a very simple formula. The simulated density of the probability of acceptance can be used to measure the degree of uncertainty of the estimated values of the new association statistic. The results from simulations as well as examples that employ real data indicate that this new association statistic is very powerful in detecting linear and non-linear associations between two random variables. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

We consider the (rough) stochastic differential equations driven by a linear multiplicative fractional Brownian motion with Hurst index H ∈ (1 2 , 1) (or a linear multiplicative geometric fractional Brownian rough path with Hurst index H ∈ (1 3 , 1 2 ]), we construct the pullback random attractors for these two type noises via the transformation method. Furthermore, we obtain the invariant probability measures, which are supported on the pullback random attractors. Finally, we obtain the limit behavior of the invariant probability measures. [ABSTRACT FROM AUTHOR]

Let 풦 + ⁢ ( σ ) = { ℙ ( ϑ , σ ) ⁢ ( d ⁢ ζ ) : ϑ ∈ ( 0 , ϑ + ⁢ ( σ ) ) } {{\mathcal{K}_{+}}(\sigma)=\{\mathbb{P}_{(\vartheta,\sigma)}(d\zeta):\vartheta% \in(0,\vartheta_{+}(\sigma))\}} be the Cauchy–Stieltjes Kernel (CSK) family generated by a probability measure σ which is non degenerate and has support bounded from above. Consider the concept of V a {V_{a}} -transformation of measures introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545] for a ∈ ℝ {a\in\mathbb{R}} . We prove that V a ⁢ ( ℙ ( ϑ , σ ) ) ∈ 풦 + ⁢ ( σ ) {V_{a}(\mathbb{P}_{(\vartheta,\sigma)})\in{\mathcal{K}_{+}}(\sigma)} for all a ∈ ℝ ∖ { 0 } {a\in\mathbb{R}\setminus\{0\}} if and only if the measure σ is of the free Gaussian (semicircle) type law up to affinity. [ABSTRACT FROM AUTHOR]

For the set I (X) of probability measures on a compact Hausdorff space X, we propose a new way to introduce the topology by using the open subsets of the space X. Then, among other things, we give a new proof that for a compact Hausdorff space X, the space I (X) is also a compact Hausdorff space. For a Tychonoff space X, we consider the topological space I τ (X) of τ -smooth idempotent probability measures on X and show that the space I τ (X) is Čech-complete if and only if the given space X is Čech-complete. [ABSTRACT FROM AUTHOR]

Let K + (μ i) = { Q s i μ i , s i ∈ (m 0 μ i , m + μ i) } , i = 1 , 2 , be two CSK families generated by the nondegenerate probability measures μ 1 and μ 2 with support bounded from above. Define the set of measures L = K + (μ 1) • K + (μ 2) = { Q s 1 μ 1 • Q s 2 μ 2 , s 1 ∈ (m 0 μ 1 , m + μ 1) and s 2 ∈ (m 0 μ 2 , m + μ 2) } , where Q s 1 μ 1 • Q s 2 μ 2 denotes the Fermi convolution of Q s 1 μ 1 and Q s 2 μ 2 . We prove that if L is still a CSK family (that is, L = K + (σ) for some nondegenerate probability measure () σ), then the probability measures σ , μ 1 and μ 2 are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution t replacing the Fermi convolution • in the family of measures L. [ABSTRACT FROM AUTHOR]

This paper seeks to establish the measure attractors in stochastic fractional lattice systems. First, the presence of these attractor measures is proven by the uniform estimates of the solution. Subsequently, the study also looks at the upper semicontinuous dependence of the measure attractors on the noise intensity as the latter goes to zero. The given asymptotic compactness for the family of probability measures occurring with the solution probability distributions is exhibited by a uniform prior estimation of the far-field solution values. [ABSTRACT FROM AUTHOR]

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process. [ABSTRACT FROM AUTHOR]

Let Q be a probability measure on a finite group G, and let H be a subgroup of G. We show that a necessary and sufficient condition for the random walk driven by Q on G to induce a Markov chain on the double coset space H\backslash G/H is that Q(gH) is constant as g ranges over any double coset of H in G. We obtain this result as a corollary of a more general theorem on the double cosets H \backslash G / K for K an arbitrary subgroup of G. As an application we study a variation on the r-top to random shuffle which we show induces an irreducible, recurrent, reversible and ergodic Markov chain on the double cosets of \mathrm {Sym}_r \times \mathrm {Sym}_{n-r} in \mathrm {Sym}_n. The transition matrix of the induced walk has remarkable spectral properties: we find its invariant distribution and its eigenvalues and hence determine its rate of convergence. [ABSTRACT FROM AUTHOR]

We give a self-contained proof of the following result: Finitely additive probability measures (also known as "states") of the free boolean algebra F ω over the free generating set { X 1 , X 2 , ... } having the invariance property under finite permutations of the X i , coincide with states lying in the closure of the set of convex combinations of product states of F ω in the vector space R F ω equipped with the product topology. De Finetti's celebrated exchangeability theorem can be easily recovered from our proof. [ABSTRACT FROM AUTHOR]

The arboreal gas is the probability measure on (unrooted spanning) forests of a graph in which each forest is weighted by a factor β > 0 per edge. It arises as the q → 0 limit of the q -state random cluster model with p = β q . We prove that in dimensions d ⩾ 3 the arboreal gas undergoes a percolation phase transition. This contrasts with the case of d = 2 where no percolation transition occurs. The starting point for our analysis is an exact relationship between the arboreal gas and a non-linear sigma model with target space the fermionic hyperbolic plane H 0 | 2 . This latter model can be thought of as the 0-state Potts model, with the arboreal gas being its random cluster representation. Unlike the standard Potts models, the H 0 | 2 model has continuous symmetries. By combining a renormalisation group analysis with Ward identities we prove that this symmetry is spontaneously broken at low temperatures. In terms of the arboreal gas, this symmetry breaking translates into the existence of infinite trees in the thermodynamic limit. Our analysis also establishes massless free field correlations at low temperatures and the existence of a macroscopic tree on finite tori. [ABSTRACT FROM AUTHOR]

Currently, there is no mature method for calculating the probability of human failure that occurs in multi-unit. In this study, the single-unit HRA is expanded and improved, and we have compiled a method flow suitable for human reliability analysis in multi-unit. This method considers the communication of commands between different organizations and analyzes the equipment shared by multiple units. This research demonstrates the feasibility of the THERP+HCR method to calculate the human-caused failure probability of multiple units. In this paper, the human failure probability of the fifth emergency diesel engine is investigated during a station blackout at a multi-unit site. The calculated failure probability of personnel deploying a fifth diesel-engine between two units is 0.1061, which is higher than the failure probability of a single unit, and the consequences of failure of a dual-unit invocation would be more severe. Due to the uncertainty of some conditions and parameters, the variation of the failure probability after sensitivity analysis ranges from–0.0846–0.2083. Involvement of more organizations during the accident increases the risk of the plant site. Among the most influential factors on the probability of failure are the presence of inspection clauses in the operating procedures and the level of tension of personnel. [ABSTRACT FROM AUTHOR]

Background and Objectives: This study aimed to examine the relationship between cardiometabolic risk factors and atrial fibrillation (AF) and the simultaneous presence of AF and metabolic syndrome (MetS) in the Pakistani population. Materials and Methods: A total of 690 subjects were enrolled (n = 230 patients with AF, n = 230 patients with AF and MetS, and n = 230 controls). The associations between cardiometabolic parameters and AF with and without MetS were analyzed by univariable and multivariable binary regression analyses. Results: Body mass index (BMI), fasting blood glucose (FBG), and triglycerides (TG) were independently positively correlated, but the glomerular filtration rate (GFR) and sodium were independently negatively correlated with AF. An increase in BMI, FBG, and TG levels by one unit measure increased the probability by 55.1%, 20.6%, and 1.3%, respectively, for the AF occurrence. A decrease in GFR and sodium levels increased the probability by 4.3% and 33.6%, respectively, for the AF occurrence. On the other hand, uric acid was independently negatively correlated, whereas sodium was independently positively correlated, with MetS and AF. A decrease in uric acid levels and an increase in sodium levels by 1 unit measure increased the probability for MetS and AF by 23.2% and 7.5%, respectively. Conclusions: Cost-effective and routinely measured parameters, i.e., BMI, FBG TG, GFR, and sodium levels, can be reliable indicators of AF, whereas serum uric acid and sodium levels are independently associated with AF and MetS in the Pakistani population. Timely recognition and the control of modifiable cardiometabolic risk factors are of great significance in the prevention of AF development. [ABSTRACT FROM AUTHOR]

We provide a definition of a ≺-asymptotic (we suggest the pronunciation "prec-asymptotic") pair in a topological action (X , G) of a countable amenable group G , where ≺ is an order on G of type Z. In the case where for some G -invariant Borel probability measure μ on X , the measure-preserving system (X , μ , G) factors, via a map φ , onto a multiorder (O ˜ , ν , G) , we also introduce the notion of a φ -asymptotic pair. Then we prove that if μ has positive measure-theoretic conditional entropy with respect to the multiorder factor, then the set of points which belong to φ -asymptotic pairs has positive measure μ. This result is a generalization of the Blanchard-Host-Ruette Theorem for classical topological dynamical systems (actions of Z). As a strengthening of our theorem, we show that for any system (X , G) of positive topological entropy, any multiorder (O ˜ , ν , G) and ν -almost every ≺ ∈ O ˜ , there exist ≺-asymptotic pairs in X. Finally, we characterize systems (X , G) of topological entropy zero as factors of topologically multiordered systems (in which case φ is defined μ -almost everywhere for every G -invariant measure μ) with no φ -asymptotic pairs. [ABSTRACT FROM AUTHOR]

Quantum Markov chains (QMCs) are positive maps on a trace-class space describing open quantum dynamics on graphs. Such objects have a statistical resemblance with classical random walks, while at the same time they allow for internal (quantum) degrees of freedom. In this work we study continuous-time QMCs on the integer line, half-line and finite segments, so that we are able to obtain exact probability calculations in terms of the associated matrix-valued orthogonal polynomials and measures. The methods employed here are applicable to a wide range of settings, but we will restrict ourselves to classes of examples for which the Lindblad generators are induced by a single positive map, and such that the Stieltjes transforms of the measures and their inverses can be calculated explicitly. [ABSTRACT FROM AUTHOR]

It was long assumed that the pseudorandom distribution of prime numbers was free of biases. Specifically, while the prime number theorem gives an asymptotic measure of the probability of finding a prime number and Dirichlet’s theorem on arithmetic progressions tells us about the distribution of primes across residue classes, there was no reason to believe that consecutive primes might “know” anything about each other — that they might, for example, tend to avoid ending in the same digit. Here, we show that the Iterated Function System method (IFS) can be a surprisingly useful tool for revealing such unintuitive results and for more generally studying structure in number theory. Our experimental findings from a study in 2013 include fractal patterns that reveal “repulsive” phenomena among primes in a wide range of classes having specific congruence properties. Some of the phenomena shown in our computations and interpretation relate to more recent work by Lemke Oliver and Soundararajan on biases between consecutive primes. Here, we explore and extend those results by demonstrating how IFS points to the precise manner in which such biases behave from a dynamic standpoint. We also show that, surprisingly, composite numbers can exhibit a notably similar bias. [ABSTRACT FROM AUTHOR]

The knowledge, attitudes, and practices (KAP) framework is often used in public health to explore people's adaptive behaviors and to guide intervention programs. Still, psychometric and structure-wise refinements could provide new insights, more aligned with an epidemiological view of people's role in disease transmission and containment. Our study aims to (1) develop a psychometrically adequate KAP instrument for measuring COVID-19 pandemic responses, and (2) test a structural configuration of KAP that can aid in estimating behavior-related epidemiological parameters in a population: (a) the average number of people that an infected person comes into contact with per unit of time, (b) the probability of each of these exposures to result in infection. One relates to behavioral prevention practices through social distancing or isolation and was measured by the Contact Practices subscale. The other links to health practices, like mask-wearing or hand-washing, being measured by the Probability Practices subscale. A first multi-mediation model suggests that Knowledge has both direct and indirect effects on Probability Practices, but only indirect on Contact Practices, mediated by Attitudes. Two distinct content areas emerged for measuring Knowledge: Preventive Measures, and Causes and Consequences. The final path model demonstrates an excellent fit, indicating that knowledge regarding Preventive Measures has effects on Practices, both directly and indirectly through Attitudes. Disease containment depends on balancing Probability Practices and Contact Practices as distinct but equifinal behavioral strategies, both amenable to attitude-centered intervention. Consistent with other infectious disease research, knowledge about prevention practices far outweighs specialized medical information in promoting adaptive timely responses. [ABSTRACT FROM AUTHOR]

Accounting for uncertainty is essential for precautionary approaches to managing seabird bycatch in commercial fisheries. However, there is no existing mechanism to explicitly quantify the uncertainty of seabird‐vessel interactions (i.e. co‐occurrence in space and time). Here we develop a time geographic method to measure the probability of individual birds encountering (co‐occurring within 30 km) and attending (within 5 km) individual fishing vessels. The approach involves creating voxel‐based probabilistic space–time prisms (PSTPs) to model the movements of individual birds and vessels, with trajectory data from bird‐borne GPS devices and vessel Automatic Identification Systems (AIS). We intersected these PSTPs to quantify the probability of interaction between bird‐vessel pairs over time and space. We demonstrate the approach with a case study of interactions of Endangered Toroa (Antipodean Albatross; Diomedea antipodensis antipodensis) with pelagic longline vessels in part of the South Pacific high seas. We found 15 vessels within 150 km and 3 h of two birds, yet interaction occurred with only two of those vessels. We visualised the probability of encounter and attendance over time and space and determined that interactions lasted several hours each (up to 6.2–14.1 h attendance, 20.8–26.1 h encounter for one bird‐vessel pair). Our time geographic approach adds to existing tools to quantify seabird bycatch risk by providing an explicit measure of uncertainty of seabird‐vessel interactions. We provide a flexible methodological pathway and R scripts, the application of which would allow managers to estimate interaction probability for multiple marine species and fisheries, including those with lower‐resolution positional datasets. [ABSTRACT FROM AUTHOR]

Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function f:K→R$f\colon K \rightarrow \mathbb {R}$ defined on a convex domain K⊆Rd$K \subseteq \mathbb {R}^{d}$ and any random variable X$X$ taking values in K$K$, E[f(X)]⩾f(E[X])$\mathbb {E}[f(X)] \geqslant f(\mathbb {E}[X])$. In this paper, sharp upper and lower bounds on E[f(X)]$\mathbb {E}[f(X)]$, termed ‘graph convex hull bounds’, are derived for arbitrary functions f$f$ on arbitrary domains K$K$, thereby extensively generalizing Jensen's inequality. The derivation of these bounds necessitates the investigation of the convex hull of the graph of f$f$, which can be challenging for complex functions. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for any K$K$‐valued random variable X$X$. Therefore, these bounds are of particular interest in cases where f$f$ is relatively simple and X$X$ is complicated or unknown. Both finite‐ and infinite‐dimensional domains and codomains of f$f$ are covered as well as analogous bounds for conditional expectations and Markov operators. [ABSTRACT FROM AUTHOR]

Optimization problems on probability measures in ℝ d are considered where the cost functional involves multi-marginal optimal transport. In a model of N interacting particles, for example in Density Functional Theory, the interaction cost is repulsive and described by a two-point function c ⁢ (x , y) = ℓ ⁢ (| x - y |) where ℓ : ℝ + → [ 0 , ∞ ] is decreasing to zero at infinity. Due to a possible loss of mass at infinity, non-existence may occur and relaxing the initial problem over sub-probabilities becomes necessary. In this paper, we characterize the relaxed functional generalizing the results of [4] and present a duality method which allows to compute the Γ-limit as N → ∞ under very general assumptions on the cost ℓ ⁢ (r) . We show that this limit coincides with the convex hull of the so-called direct energy. Then we study the limit optimization problem when a continuous external potential is applied. Conditions are given with explicit examples under which minimizers are probabilities or have a mass < 1 . In a last part, we study the case of a small range interaction ℓ N ⁢ (r) = ℓ ⁢ (r / ε) ( ε ≪ 1 ) and we show how the duality approach can also be used to determine the limit energy as ε → 0 of a very large number N ε of particles. [ABSTRACT FROM AUTHOR]

Let |$(M^{n},g,f)$| be a Ricci shrinker such that |$\text{Ric}_{f}=\frac{1}{2}g$| and the measure induced by the weighted volume element |$(4\pi)^{-\frac{n}{2}}e^{-f}dv_{g}$| is a probability measure. Given a point |$p\in M$|⁠ , we consider two probability measures defined in the tangent space |$T_{p}M$|⁠ , namely the Gaussian measure |$\gamma $| and the measure |$\overline{\nu }$| induced by the exponential map of |$M$| to |$p$|⁠. In this paper, we prove a result that provides an upper estimate for the Wasserstein distance with respect to the Euclidean metric |$g_{0}$| between the measures |$\overline{\nu }$| and |$\gamma $|⁠ , and which also elucidates the rigidity implications resulting from this estimate. [ABSTRACT FROM AUTHOR]

Recent years witnessed the development of powerful generative models based on flows, diffusion, or autoregressive neural networks, achieving remarkable success in generating data from examples with applications in a broad range of areas. A theoretical analysis of the performance and understanding of the limitations of these methods remain, however, challenging. In this paper, we undertake a step in this direction by analyzing the efficiency of sampling by these methods on a class of problems with a known probability distribution and comparing it with the sampling performance of more traditional methods such as the Monte Carlo Markov chain and Langevin dynamics. We focus on a class of probability distribution widely studied in the statistical physics of disordered systems that relate to spin glasses, statistical inference, and constraint satisfaction problems. We leverage the fact that sampling via flow-based, diffusionbased, or autoregressive networks methods can be equivalently mapped to the analysis of a Bayes optimal denoising of a modified probability measure. Our findings demonstrate that these methods encounter difficulties in sampling stemming from the presence of a first-order phase transition along the algorithm's denoising path. Our conclusions go both ways: We identify regions of parameters where these methods are unable to sample efficiently, while that is possible using standard Monte Carlo or Langevin approaches. We also identify regions where the opposite happens: standard approaches are inefficient while the discussed generative methods work well. [ABSTRACT FROM AUTHOR]

In this paper, we shall discuss the extendability of probability and non-probability measures on Cayley trees to a σ -additive measure on Borel fields which has a fundamental role in the theory of Gibbs measures. [ABSTRACT FROM AUTHOR]

The problem of recommender system is very popular with myriad available solutions. Recommender systems recommend items to users and help them in narrowing their search from huge amount of options available to the user. In this work, a novel approach for the recommendation problem is proposed by incorporating techniques from the link prediction problem in social networks. The proposed approach models the typical user-item information as a bipartite network, and predicts future links using link prediction measures, in which link prediction would actually mean recommending an item to a user. The standard recommender system methods suffer from the problems of sparsity and scalability. Since link prediction measures involve computations pertaining to local neighborhoods in the network, this approach would lead to a scalable solution to recommendation. In this work, we present top k links that are predicted by link prediction measures as recommendations to the users. Our work initially applies different existing link prediction measures to the recommendation problem by making suitable adaptations. The prime contribution of this work is to propose a recommendation framework routed from link prediction problem in social networks, that effectively utilizes probabilistic measures of link prediction and embed temporal data accessible on existing links. The proposed approach is evaluated on one movie-rating dataset of MovieLens, two product-rating datasets of Epinions & Amazon and one hotel-rating dataset of TripAdvisor. Results show that the link prediction measures based on temporal probabilistic information prove to be more effective in improving the quality of recommendation. Especially, Temporal cooccurrence probability measure improves the area under ROC curve (AUROC) by 10% for MovieLens, 23% for Epinions, 17% for TripAdvisor, 9% for Amazon over standard item-based collaborative filtering method. Similar improved performance is observed in terms of area under Precision-Recall curve (AUPR) as well as Normalized Rank-Score. [ABSTRACT FROM AUTHOR]

In the paper, we prove a joint limit theorem in terms of the weak convergence of probability measures on C 2 defined by means of the Epstein ζ (s ; Q) and Hurwitz ζ (s , α) zeta-functions. The limit measure in the theorem is explicitly given. For this, some restrictions on the matrix Q and the parameter α are required. The theorem obtained extends and generalizes the Bohr-Jessen results characterising the asymptotic behaviour of the Riemann zeta-function. [ABSTRACT FROM AUTHOR]

Let M n = diag [ 3 p n , 3 q n ] with p n , q n ⩾ 1 for all n ⩾ 1 and let D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } . One can generate a Borel probability measure μ { M n } , D = δ M 1 − 1 D ∗ δ (M 2 M 1) − 1 D ∗ δ (M 3 M 2 M 1) − 1 D ∗ ⋯. Such measure μ { M n } , D is called a Moran Sierpinski-type measure. It is known Deng et al (Acta Math. Sin. submitted) that the associated Hilbert space L 2 (μ { M n } , D) has an exponential orthonormal basis. In this paper, we first characterize all the maximal exponential orthogonal sets for L 2 (μ { M n } , D) . For such a maximal orthogonal set, we then give some sufficient conditions to determine whether it is an orthonormal basis of L 2 (μ { M n } , D) or not. [ABSTRACT FROM AUTHOR]

In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980. [ABSTRACT FROM AUTHOR]

Let An ∈ M2 (ℤ) be integral matrices such that the infinite convolution of Dirac measures with equal weights μ { A n , n ≥ 1 } := δ A 1 − 1 D ∗ δ A 1 − 1 A 2 − 2 D ∗ ⋯ is a probability measure with compact support, where D = { (0 , 0) t , (1 , 0) t , (0 , 1) t } is the Sierpinski digit. We prove that there exists a set Λ ⊂ ℝ2 such that the family {e2πi〈λ,x〉: λ ∈ Λ} is an orthonormal basis of L 2 (μ { A n , n ≥ 1 }) if and only if 1 3 (1 , − 1) A n ∈ Z 2 for n ≥ 2 under some metric conditions on An. [ABSTRACT FROM AUTHOR]

In this paper, we first derive Milstein schemes for an interacting particle system associated with point delay McKean–Vlasov stochastic differential equations, possibly with a drift term exhibiting super-linear growth in the state component. We prove strong convergence of order one and moment stability, making use of techniques from variational calculus on the space of probability measures with finite second-order moments. Then, we introduce an antithetic multilevel Milstein scheme, which leads to optimal complexity estimators for expected functionals of solutions to delay McKean–Vlasov equations without the need to simulate Lévy areas. [ABSTRACT FROM AUTHOR]

Large-scale socioeconomic studies of the impacts of floods are difficult and costly for countries such as Canada and the United States due to the large number of rivers and size of watersheds. Such studies are however very important for analyzing spatial patterns and temporal trends to inform large-scale flood risk management decisions and policies. In this paper, we present different flood occurrence and impact models based upon statistical and machine learning methods of over 31 000 watersheds spread across Canada and the US. The models can be quickly calibrated and thereby easily run predictions over thousands of scenarios in a matter of minutes. As applications of the models, we present the geographical distribution of the modelled average annual number of people displaced due to flooding in Canada and the US, as well as various scenario analyses. We find for example that an increase of 10 % in average precipitation yields an increase in the displaced population of 18 % in Canada and 14 % in the US. The model can therefore be used by a broad range of end users ranging from climate scientists to economists who seek to translate climate and socioeconomic scenarios into flood probabilities and impacts measured in terms of the displaced population. [ABSTRACT FROM AUTHOR]

The hexachordal theorem is an intriguing combinatorial property of the sets in Z=12Z, discovered and popularized by the musicologist Milton Babbitt (1916-2011). It has been given several explanations and partial generalizations. Here we enhance how this phenomenon can be understood by giving both a geometrical and a probabilistic perspective. [ABSTRACT FROM AUTHOR]

Let g$g$ be a random matrix distributed according to uniform probability measure on the finite general linear group GLn(Fq)$\mathrm{GL}_n(\mathbb {F}_q)$. We show that Tr(gk)$\mathrm{Tr}(g^k)$ equidistributes on Fq$\mathbb {F}_q$ as n→∞$n \rightarrow \infty$ as long as logk=o(n2)$\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of Tr(g1),...,Tr(gk)$\mathrm{Tr}(g^1),\ldots, \mathrm{Tr}(g^k)$ equidistribute as long as logk=o(n)$\log k =o(n)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for k⩽cqn$k \leqslant c_q n$, where cq$c_q$ depends on q$q$, due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of Tr(gk)$\mathrm{Tr}(g^k)$, we end up showing that certain explicit character sums modulo Tk+1$T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree n$n$ in Fq[T]$\mathbb {F}_q[T]$ as long as logk=o(n2)$\log k = o(n^2)$. This goes far beyond the classical range logk=o(n)$\log k =o(n)$ due to Montgomery and Vaughan. To study these sums, we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums. [ABSTRACT FROM AUTHOR]

In this paper, we show that a sufficient and necessary condition for that the Hoeffding concentration inequality of Banach space-valued (B -valued, for simplicity) random variables holds is that the Banach space in question admits the Rademacher type p for some 1 ≤ p ≤ 2 ; which is equivalent to that the Bernstein concentration inequality holds for such B -valued random variables. This is done by applying a refined McDiarmid inequality (stated and proved in this paper) via the entropy method. This result is also applied to the density property of induced subgraphs of random Cayley graphs generated by a finite abelian group. [ABSTRACT FROM AUTHOR]

We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient flow structure allows to give a new proof for the convergence of Kac's random walk to the homogeneous Boltzmann equation, exploiting the stability of gradient flows. [ABSTRACT FROM AUTHOR]

We give systematic ways of defining monotone quantum relative entropies and (multi-variate) quantum Rényi divergences starting from a set of monotone quantum relative entropies. Interestingly, despite its central importance in information theory, only two additive and monotone quantum extensions of the classical relative entropy have been known so far, the Umegaki and the Belavkin-Staszewski relative entropies, which are the minimal and the maximal ones, respectively, with these properties. Using the Kubo-Ando weighted geometric means, we give a general procedure to construct monotone and additive quantum relative entropies from a given one with the same properties; in particular, when starting from the Umegaki relative entropy, this gives a new one-parameter family of monotone (even under positive trace-preserving (PTP) maps) and additive quantum relative entropies interpolating between the Umegaki and the Belavkin-Staszewski ones on full-rank states. In a different direction, we use a generalization of a classical variational formula to define multi-variate quantum Rényi quantities corresponding to any finite set of quantum relative entropies (D q x ) x ∈ X and real weights (P (x)) x ∈ X summing to 1, as Q P b , q ((ϱ x) x ∈ X) : = sup τ ≥ 0 ⁡ { Tr τ − ∑ x P (x) D q x (τ ‖ ϱ x) }. We analyze in detail the properties of the resulting quantity inherited from the generating set of quantum relative entropies; in particular, we show that monotone quantum relative entropies define monotone Rényi quantities whenever P is a probability measure. With the proper normalization, the negative logarithm of the above quantity gives a quantum extension of the classical Rényi α -divergence in the 2-variable case (X = { 0 , 1 } , P (0) = α). We show that if both D q 0 and D q 1 are lower semi-continuous, monotone, and additive quantum relative entropies, and at least one of them is strictly larger than the Umegaki relative entropy then the resulting barycentric Rényi divergences are strictly between the log-Euclidean and the maximal Rényi divergences, and hence they are different from any previously studied quantum Rényi divergence. [ABSTRACT FROM AUTHOR]

The dynamics of stabilised concentrated emulsions presents a rich phenomenology including chaotic emulsification, non-Newtonian rheology and ageing dynamics at rest. Macroscopic rheology results from the complex droplet microdynamics and, in turn, droplet dynamics is influenced by macroscopic flows via the competing action of hydrodynamic and interfacial stresses, giving rise to a complex tangle of elastoplastic effects, diffusion, breakups and coalescence events. This tight multiscale coupling, together with the daunting challenge of experimentally investigating droplets under flow, has hindered the understanding of concentrated emulsions dynamics. We present results from three-dimensional numerical simulations of emulsions that resolve the shape and dynamics of individual droplets, along with the macroscopic flows. We investigate droplet dispersion statistics, measuring probability density functions (p.d.f.s) of droplet displacements and velocities, changing the concentration, in the stirred and ageing regimes. We provide the first measurements, in concentrated emulsions, of the relative droplet–droplet separations p.d.f. and of the droplet acceleration p.d.f., which becomes strongly non-Gaussian as the volume fraction is increased above the jamming point. Cooperative effects, arising when droplets are in contact, are argued to be responsible of the anomalous superdiffusive behaviour of the mean square displacement and of the pair separation at long times, in both the stirred and in the ageing regimes. This superdiffusive behaviour is reflected in a non-Gaussian pair separation p.d.f., whose analytical form is investigated, in the ageing regime, by means of theoretical arguments. This work paves the way to developing a connection between Lagrangian dynamics and rheology in concentrated emulsions. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the study of sequential interval reliability, a measure recently introduced in the literature. This measure represents the probability of the system working during a sequence of nonoverlapping time intervals. In the cited work, the authors proposed a recurrent-type formula for computing this indicator in the transient case and investigated the asymptotic behavior as all the time intervals go to infinity. The purpose of the present work is to further explore the asymptotic behavior when only some of the time intervals are allowed to go to infinity while the remaining ones are not. In this way, we provide a unique indicator that is able to describe the process evolution in the transient and asymptotic cases as well. It is important to mention that this is not a straightforward result since, in order to achieve it, we need to develop several mathematical ingredients that generalize the classical renewal and Markov renewal frameworks. A numerical example illustrates our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel two‐step modeling method to generalize cubic Hermite interpolators on the space of probability measures P+(I)$$ {\mathcal{P}}_{&#x0002B;}(I) $$. First, we develop new approaches to capture the Riemannian geometric structure of P+(I)$$ {\mathcal{P}}_{&#x0002B;}(I) $$ when equipped with Fisher–Rao metric. Furthermore, we develop and detail all numerical tools on P+(I)$$ {\mathcal{P}}_{&#x0002B;}(I) $$, namely, Levi–Civita connection, minimal geodesics, parallel transport, exponential map, and logarithm map. Then, we demonstrate that preliminary analysis results yield significant benefits in constructing an optimal cubic Hermite spline on P+(I)$$ {\mathcal{P}}_{&#x0002B;}(I) $$ as a nonlinear Riemannian manifold, precisely where conventional numerical methods fail. [ABSTRACT FROM AUTHOR]

In this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space H1((0,∞)d,γα)$H^1((0,\infty)^d,\gamma _\alpha)$ to L1((0,∞)d,γα)$L^1((0,\infty)^d,\gamma _\alpha)$ and also from L∞((0,∞)d,γα)$L^\infty ((0,\infty)^d,\gamma _\alpha)$ to the space of functions of bounded mean oscillation BMO((0,∞)d,γα)$\textup {BMO}((0,\infty)^d,\gamma _\alpha)$, with respect to the probability measure dγα(x)=∏j=1d2Γ(αj+1)xj2αj+1e−xj2dxj$d\gamma _\alpha (x)=\prod _{j=1}^d\frac{2}{\Gamma (\alpha _j+1)} x_j^{2\alpha _j+1} \text{e}^{-x_j^2} dx_j$ on (0,∞)d$(0,\infty)^d$ when α=(α1,⋯,αd)$\alpha =(\alpha _1, \dots,\alpha _d)$ is a multi‐index in −12,∞d$\left(-\frac{1}{2},\infty \right)^d$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting. [ABSTRACT FROM AUTHOR]