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.

     

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.

     

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).

     

A Level-1 Limit Order Book with Time Dependent Arrival Rates

We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (SIAM J Financial Math 4(1), 1–25 2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.

     

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.

     

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).

     

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.

     

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.

     

A coarsening algorithm on adaptive red-green-blue refined meshes

Adaptive meshing is a fundamental component of adaptive finite element methods. This includes refining and coarsening meshes locally. In this work, we are concerned with the red-green-blue refinement strategy in two dimensions and its counterpart-coarsening. In general, coarsening algorithms are mostly based on an explicitly given refinement history. In this work, we present a coarsening algorithm on adaptive red-green-blue meshes in two dimensions without explicitly knowing the refinement history. To this end, we examine the local structure of these meshes, find an easy-to-verify criterion to adaptively coarsen red-green-blue meshes, and prove that this criterion generates meshes with the desired properties. We present a MATLAB implementation built on the red-green-blue refinement routine of the ameshref-package (Funken and Schmidt 2018, 2019).

     

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).

     

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.

     

Uniform in Time $L^{\infty }$ -Estimates for Nonlinear Aggregation-Diffusion Equations

In this paper, we consider the general reaction–diffusion system proposed in Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017) as a generalization of the original Lengyel–Epstein model developed for the revolutionary Turing-type CIMA reaction. We establish sufficient conditions for the global existence of solutions. We also follow the footsteps of Lisena (Appl. Math. Comput. 249:67–75, 2014) and other similar studies to extend previous results regarding the local and global asymptotic stability of the system. In the local PDE sense, more relaxed conditions are achieved compared to Abdelmalek and Bendoukha (Nonlinear Anal., Real World Appl. 35:397–413, 2017). Also, new extended results are achieved for the global existence, which when applied to the Lengyel–Epstein system, provide weaker conditions than those of Lisena (Appl. Math. Comput. 249:67–75, 2014). Numerical examples are used to affirm the findings and benchmark them against previous results.

     

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.

     

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 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.

     

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.

     

Characterizing bounded orthogonally additive polynomials on vector lattices

We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in Kusraeva (Vladikavkaz Math J 16(4):49–53, 2014) actually characterize them. Secondly, by employing complexifications of the unique symmetric multilinear maps associated with orthogonally additive polynomials, we derive new characterizing formulas.

     

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.

     

Optimal investment with random endowments and transaction costs: duality theory and shadow prices

This paper studies the utility maximization on the terminal wealth with random endowments and proportional transaction costs. To deal with unbounded random payoffs from some illiquid claims, we propose to work with the acceptable portfolios defined via the consistent price system such that the liquidation value processes stay above some stochastic thresholds. In the market consisting of one riskless bond and one risky asset, we obtain a type of super-hedging result. Based on this characterization of the primal space, the existence and uniqueness of the optimal solution for the utility maximization problem are established using the duality approach. As an important application of the duality theorem, we provide some sufficient conditions for the existence of a shadow price process with random endowments in a generalized form similar to Czichowsky and Schachermayer (Ann Appl Probab 26(3):1888–1941, 2016) as well as in the usual sense using acceptable portfolios.