Approximation of Sojourn Times of Gaussian Processes

We investigate the tail asymptotic behavior of the sojourn time for a large class of centered Gaussian processes X, in both continuous- and discrete-time framework. All results obtained here are new for the discrete-time case. In the continuous-time case, we complement the investigations of Berman (Commun Pure Appl Math 38(5):519–528, 1985a and Probab Theory Relat Fields 20(1):113–124, 1987) for non-stationary X. A by-product of our investigation is a new representation of Pickands constant which is important for Monte-Carlo simulations and yields a sharp lower bound for Pickands constant.

     

Properization: constructing proper scoring rules via Bayes acts

Scoring rules serve to quantify predictive performance. A scoring rule is proper if truth telling is an optimal strategy in expectation. Subject to customary regularity conditions, every scoring rule can be made proper, by applying a special case of the Bayes act construction studied by Grünwald and Dawid (Ann Stat 32:1367–1433, 2004) and Dawid (Ann Inst Stat Math 59:77–93, 2007), to which we refer as properization. We discuss examples from the recent literature and apply the construction to create new types, and reinterpret existing forms, of proper scoring rules and consistent scoring functions. In an abstract setting, we formulate sufficient conditions under which Bayes acts exist and scoring rules can be made proper.

     

An Infinite Antichain of Planar Tanglegrams

Contrary to the expectation arising from the tanglegram Kuratowski theorem of Czabarka et al. (SIAM J. Discrete Math. 31(3), 1732–1750, 2017), we construct an infinite antichain of planar tanglegrams with respect to the induced subtanglegram partial order. R.E. Tarjan, R. Laver, D.A. Spielman and M. Bóna, and possibly others, showed that the partially ordered set of finite permutations ordered by deletion of entries contains an infinite antichain, i.e., there exists an infinite collection of permutations, such that none of them contains another as a pattern. Our construction adds a twist to the construction of Spielman and Bóna (Electr. J. Comb. 7, N2, 2000).

     

AM-Totally Boundedness on Normed Riesz Spaces

In this paper we introduce and study the class of AM-totally bounded operators. Also we introduce the notion of graph lattice norm to prove a domination result for this class of operators. In particular, we extend the main theorem of Aqzzouz et al. [8]. Further, we give new results concerning the power domination problem by a totally bounded operator.

     

Counting the Number of τ-Exceptional Sequences over Nakayama Algebras

The notion of a τ-exceptional sequence was introduced by Buan and Marsh in (2018) as a generalisation of an exceptional sequence for finite dimensional algebras. We calculate the number of complete τ-exceptional sequences over certain classes of Nakayama algebras. In some cases, we obtain closed formulas which also count other well known combinatorial objects, and exceptional sequences of path algebras of Dynkin quivers.

     

Weighted Counting of Inversions on Alternating Sign Matrices

We generalize the author’s formula (2011) on weighted counting of inversions on permutations to one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix which essentially follows from the earlier work of Lascoux-Schützenberger (1996).

     

Existence and Asymptotic Behavior of Solutions for the Unstirred Chemostat Model with Ratio-Dependent Function

A system of reaction-diffusion equations arising from the unstirred chemostat model with ratio-dependent function is considered. The asymptotic behavior of solutions is given and all positive steady-state solutions to this model lie on a single smooth solution curve. It turns out that the ratio-dependence effect will not affect the dynamics, compared with (Hsu and Waltman in SIAM J. Appl. Math. 53(4):1026–1044, 1993) and (Nie and Wu in Sci. China Math. 56(10):2035–2050, 2013).

     

Existence of Two Non-zero Weak Solutions for a Nonlinear Navier Boundary Value Problem Involving the $p$-Biharmonic

In this paper, we consider a class of non-cooperative elliptic systems of Kirchhoff type involving \(p\)-biharmonic operator and critical growth. With the help of the Limit index theory due to Li (Nonlinear Anal. TMA 30(7):4619–4627, 1997) and the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity.

     

New kernel estimators of the hazard ratio and their asymptotic properties

We propose a kernel estimator of a hazard ratio that is based on a modification of Ćwik and Mielniczuk (Commun Stat-Theory Methods 18(8):3057–3069, 1989)’s method. A naive nonparametric estimator is Watson and Leadbetter (Sankhyā: Indian J Stat Ser A 26(1):101–116, 1964)’s one, which is naturally given by the kernel density estimator and the empirical distribution estimator. We compare the asymptotic mean squared error (AMSE) of the hazard estimators, and then, it is shown that the asymptotic variance of the new estimator is usually smaller than that of the naive one. We also discuss bias reduction of the proposed estimator and derived some modified estimators. While the modified estimators do not lose nonnegativity, their AMSE is small both theoretically and numerically.

     

Cohen positive strongly p-summing and p-convex multilinear operators

The aim of this work is to give and study the notion of Cohen positive p-summing multilinear operators. We prove a natural analog of the Pietsch domination theorem for these classes and characterize their conjugates. As an application, we generalize a result due to Bu and Shi (J. Math. Anal. Appl. 401:174–181, 2013), and we compare this class with the class of multiple p-convex m-linear operators.

     

On the Convergence Complexity of Gibbs Samplers for a Family of Simple Bayesian Random Effects Models

The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, n, and/or the number of parameters, p, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed n and p are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (Stat Comput 6:269–275, 1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et al. Electron J Stat 13:1790–1812, 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as n → ∞. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (Stat Comput 6:269–275, 1996). Our results show that, under a strong growth condition, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases.

     

Support Theorems and an Injectivity Result for Integral Moments of a Symmetric m-Tensor Field

In this work, we show an injectivity result and support theorems for integral moments of a symmetric m-tensor field on a simple, real analytic, Riemannian manifold. Integral moments of symmetric m-tensor fields were first introduced by Sharafutdinov. First we generalize a Helgason type support theorem proven by Krishnan and Stefanov (Inverse Probl Imaging 3(3):453–464, 2009). We use this extended result along with the first integral moments of a symmetric m-tensor field to prove the aforementioned results.

     

On maximal ideals of codimension one in m-convex algebras

In Studia Math. 58:291–298, 1976, W. Żelazko gives a characterization of complex commutative complete unital m-convex algebras in which all maximal ideals are of codimension one. In these algebras every element has a bounded spectrum.

Here we present a similar result concerning sufficient conditions so that all maximal ideals of a complex commutative unital m-convex algebra to be of codimension one. Moreover, these conditions are proved to be equivalent to each other. We also give an example of a complex commutative unital advertibly complete m-convex algebra such that all its maximal ideals are of codimension one and has an element with unbounded spectrum.

     

Characterizations of risk aversion in cumulative prospect theory

In this paper, we investigate the necessary and sufficient conditions for a decision maker to be monotone risk averse and left-monotone risk averse, respectively, in cumulative prospect theory (CPT). Our results show that the decision maker is more pessimistic than greedy if she is either monotone or left-monotone risk averse, which is similar to that of Chateauneuf et al. (Econ Theory 25(3):649–667, 2005) in the rank-dependent expected utility model. Detailed examples are presented to illustrate the main theorems. With this work, we make a progress in the characterizations of risk aversion in CPT, which is essential in understanding the features of CPT and its applications in finance and insurance.

     

Pollaczek contour integrals for the fixed-cycle traffic-light queue

The fixed-cycle traffic-light (FCTL) queue is the standard model for intersections with static signaling, where vehicles arrive, form a queue and depart during cycles controlled by a traffic light. Classical analysis of the FCTL queue based on transform methods requires a computationally challenging step of finding the complex-valued roots of some characteristic equation. Building on the recent work of Oblakova et al. (Exact expected delay and distribution for the fixed-cycle traffic-light model and similar systems in explicit form, 2016), we obtain a contour-integral expression, reminiscent of Pollaczek integrals for bulk-service queues, for the probability generating function of the steady-state FCTL queue. We also show that similar contour integrals arise for generalizations of the FCTL queue introduced in Oblakova et al. (2016) that relax some of the classical assumptions. Our results allow us to compute the queue-length distribution and all its moments using algorithms that rely on contour integrals and avoid root-finding procedures.

     

Brownian Loops Topology

In the seminal work of Symanzik (1969), Poisson ensembles of Brownian loops were implicitly used. Since the work of Lawler and Werner (Prob. Th. Rel. Fields 128:565–588 2004) on “loop soups”, these ensembles have also been the object of many investigations. The purpose of the present work is to determine the distributions related to their topological properties, using trace formula and zeta regularization. These results have been announced in Le Jan (2016) and Le Jan (2017).

     

New subspace minimization conjugate gradient methods based on regularization model for unconstrained optimization

In this paper, two new subspace minimization conjugate gradient methods based on p-regularization models are proposed, where a special scaled norm in p-regularization model is analyzed. Different choices of special scaled norm lead to different solutions to the p-regularized subproblem. Based on the analyses of the solutions in a two-dimensional subspace, we derive new directions satisfying the sufficient descent condition. With a modified nonmonotone line search, we establish the global convergence of the proposed methods under mild assumptions. R-linear convergence of the proposed methods is also analyzed. Numerical results show that, for the CUTEr library, the proposed methods are superior to four conjugate gradient methods, which were proposed by Hager and Zhang (SIAM J. Optim. 16(1):170–192, 2005), Dai and Kou (SIAM J. Optim. 23(1):296–320, 2013), Liu and Liu (J. Optim. Theory. Appl. 180(3):879–906, 2019) and Li et al. (Comput. Appl. Math. 38(1):2019), respectively.

     

Global Weak Solutions for the Cauchy Problem to One-Dimensional Heat-Conductive MHD Equations of Viscous Non-resistive Gas

We investigate the dynamics of the Vlasov-Poisson system in the presence of radiation damping. A propagation result for velocity moments of order \(k>3\) is established in (Kunze and Rendall in Ann. Henri Poincaré 2:857–886, 2001). In this paper, we prove existence of global solutions propagating velocity and velocity-spatial moments of order \(k>2\) and establish an explicit polynomially growing in time bound on the moments.

     

An observation concerning the vanishing topology of certain isolated determinantal singularities

We extend the results about the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities from a previous article by Frühbis-Krüger and Zach (On the vanishing topology of isolated Cohen–Macaulay codimension 2 singularities. arXiv:1501.01915, 2015). Due to the Hilbert–Burch theorem, these singularities have a canonical determinantal structure and a well behaved deformation theory, which, in particular, yields a unique Milnor fiber. Studying the case of possibly non-isolated singularities in the Tjurina transform, as introduced in Frühbis-Krüger and Zach (2015), we reveal that in dimension 3 and 2 there is always exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.