Global Regularity of 3D Nonhomogeneous Incompressible Micropolar Fluids

This paper is concerned with the global well-posedness of strong and classical solutions for the 3D nonhomogeneous incompressible micropolar equations with vacuum. We prove that the problem (1.1)–(1.5) has a unique global strong/classical solution \((\rho,u,w)\), provided \(\mu_{1}\) is sufficiently large, or \(\|\rho_{0}\|_{L^{\infty}}\) or \(\|\rho_{0}^{1/2}u_{0}\| ^{2}_{L^{2}}+\|\rho_{0}^{1/2}w_{0}\|^{2}_{L^{2}}\) is small enough.

     

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.

     

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.

     

Empirical Interpolation Decomposition

The Kantorovich metric provides a way of measuring the distance between two Borel probability measures on a metric space. This metric has a broad range of applications from bioinformatics to image processing, and is commonly linked to the optimal transport problem in computer science (Deng and Du in Electron. Notes Theor. Comput. Sci. 253: 73–82, 2009; Villani in Optimal Transport: Old and New, Grundlehren der mathematischen Wissenschaften, vol. 338, 2009). Noteworthy to this paper will be the role of the Kantorovich metric in the study of iterated function systems, which are families of contractive mappings on a complete metric space. When the underlying metric space is compact, it is well known that the space of Borel probability measures on this metric space, equipped with the Kantorovich metric, constitutes a compact, and thus complete metric space. In previous work, we generalized the Kantorovich metric to operator-valued measures for a compact underlying metric space, and applied this generalized metric to the setting of iterated function systems (Davison in Acta Appl. Math., 2014, https://doi.org/10.1007/s10440-014-9976-y; Generalizing the Kantorovich Metric to Projection-Valued Measures: With an Application to Iterated Function Systems, 2015; Acta Appl. Math., 2018, https://doi.org/10.1007/s10440-018-0161-6). We note that the work of P. Jorgensen, K. Shuman, and K. Kornelson provided the framework for our application to this setting (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005; Jorgenson et al. in J. Math. Phys. 48(8):083511, 2007; Jorgensen in Operator Theory, Operator Algebras, and Applications, Contemp. Math., vol. 414, pp. 13–26, 2006). The situation when the underlying metric space is complete, but not necessarily compact, has been studied by A. Kravchenko (Sib. Math. J. 47(1), 68–76, 2006). In this paper, we extend the results of Kravchenko to the generalized Kantorovich metric on operator-valued measures.

     

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).

     

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.

     

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).

     

Markov Semi-groups Associated with the Complex Unimodular Group $$Sl(2,{\mathbb {C}})$$ S l ( 2 , C )

In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group \(SL(2,{\mathbb {C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-\,1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb {R}})\) and to the Landau operator in the complex plane.

     

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.

     

Exponential Decay of Correlations for a Real-Valued Dynamical System Generated by a $k$ Dimensional System

As a first step towards modelling real time-series, we study a class of real-variable, bounded processes \(\{X_{n}, n\in \mathbb{N}\}\) defined by a deterministic \(k\)-term recurrence relation \(X_{n+k} = \varphi (X _{n}, \ldots , X_{n+k-1})\). These processes are noise-free. We immerse such a dynamical system into \(\mathbb{R}^{k}\) in a slightly distorted way, which allows us to apply the multidimensional techniques introduced by Saussol (Isr. J. Math. 116:223–248, 2000) for deterministic transformations. The hypotheses we need are, most of them, purely analytic and consist in estimates satisfied by the function \(\varphi \) and by products of its first-order partial derivatives. They ensure that the induced transformation \(T\) is dilating. Under these conditions, \(T\) admits a greatest absolutely continuous invariant measure (ACIM). This implies the existence of an invariant density for \(X_{n}\), satisfying integral compatibility conditions. Moreover, if \(T\) is mixing, one obtains the exponential decay of correlations.

     

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely

$$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$

with \(0 < a_{{+}} < a_{{-}}\). We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

     

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 Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar’s refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an Appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

     

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.

     

More good news on the HKM test for multivariate reflected symmetry about an unknown centre

We revisit the problem of testing for multivariate reflected symmetry about an unspecified point. Although this testing problem is invariant with respect to full-rank affine transformations, among the few hitherto proposed tests only a class of tests studied in Henze et al. (J Multivar Anal 87:275–297, 2003) that depends on a positive parameter a respects this property. We identify a measure of deviation \(\varDelta _a\) (say) from symmetry associated with the test statistic \(T_{n,a}\) (say), and we obtain the limit normal distribution of \(T_{n,a}\) as \(n \rightarrow \infty \) under a fixed alternative to symmetry. Since a consistent estimator of the variance of this limit normal distribution is available, we obtain an asymptotic confidence interval for \(\varDelta _a\). The test, when applied to a classical data set, strongly rejects the hypothesis of reflected symmetry, although other tests even do not object against the much stronger hypothesis of elliptical symmetry.

     

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.

     

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.