On the well-posedness of 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity

In this paper, we mainly study the well-posedness for the 3-D inhomogeneous incompressible Navier–Stokes equations with variable viscosity. With some smallness assumption on the BMO-norm of the initial density, we first get the local well-posedness of (1.1) in the critical Besov spaces. Moreover, if the viscosity coefficient is a constant, we can extend this local solution to be a global one. Our theorem implies that we have successfully extended the integrability index p of the initial velocity which has been obtained by Abidi, Gui and Zhang in [3], Burtea in [8] and Zhai and Yin in [32] to approach the ideal one i.e. $1<p<6$. The main novelty of this work is to apply the CRW theorem obtained by Coifman, Rochberg, Weiss in [11] to get a new a priori estimate for an elliptic equation with variable coefficients. The uniqueness of the solution also relies on a Lagrangian approach as in [16], [17], [18].     

Quasistatic crack growth in 2d-linearized elasticity

In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation ($GSBD$). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the $GSBD$ setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.     

The minimal measurement number for low-rank matrix recovery

The paper presents several results that address a fundamental question in low-rank matrix recovery: how many measurements are needed to recover low-rank matrices? We begin by investigating the complex matrices case and show that $4nr?4r^2$ generic measurements are both necessary and sufficient for the recovery of rank-r matrices in ${\mathbb{C}}^{n\times n}$. Thus, we confirm a conjecture which is raised by Eldar, Needell and Plan for the complex case. We next consider the real case and prove that the bound $4nr?4r^2$ is tight provided $n=2^k+r,k\in {\mathbb{Z}}_{+}$. Motivated by Vinzant's work [19], we construct 11 matrices in ${\mathbb{R}}^{4\times 4}$ by computer random search and prove they define injective measurements on rank-1 matrices in ${\mathbb{R}}^{4\times 4}$. This disproves the conjecture raised by Eldar, Needell and Plan for the real case. Finally, we use the results in this paper to investigate the phase retrieval by projection and show fewer than $2n?1$ orthogonal projections are possible for the recovery of $x\in {\mathbb{R}}^n$ from the norm of them, which gives a negative answer for a question raised in [1].     

Incomplete financial markets model with nominal assets: Variational approach

We deal with the analysis of the general equilibrium model with incomplete financial markets and nominal assets. We assume that there are 2 periods of time, say today and tomorrow. We define a consumption, portfolio holding, commodity and asset price vector as an equilibrium vector associated with a given economy if at those prices and economies households maximize utility under a budget constraints and markets clear. While the path breaking proofs of existence by Cass [6] and Werner [25] use a fixed point argument, we provide an independent existence proof in terms of variational inequalities (about the variational approach for the analysis of general equilibrium models see for example [9] and [10]). The analysis presented in this paper indicates that the variational inequality approach promises to be applicable in many specifications of the incomplete market model.     

High‐order algorithms for Riesz derivative and their applications (V)

In this article, based on the idea of combing symmetrical fractional centred difference operator with compact technique, a series of even‐order numerical differential formulas (named the fractional‐compact formulas) are established for the Riesz derivatives with order . Properties of coefficients in the derived formulas are studied in details. Then applying the constructed fourth‐order formula, a difference scheme is proposed to solve the Riesz spatial telegraph equation. By the energy method, the constructed numerical algorithm is proved to be stable and convergent with order , where τ and h are the temporal and spatial stepsizes, respectively. Finally, several numerical examples are presented to verify the theoretical results.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1754–1794, 2017     

Computing a Celis-Dennis-Tapia trust-region step for equality constrained optimization

We study an approach for minimizing a convex quadratic function subject to two quadratic constraints. This problem stems from computing a trust-region step for an SQP algorithm proposed by Celis, Dennis and Tapia (1985) for equality constrained optimization. Our approach is to reformulate the problem into a univariate nonlinear equation()=0 where the function() is continuous, at least piecewise differentiable and monotone. Well-established methods then can be readily applied. We also consider an extension of our approach to a class of non-convex quadratic functions and show that our approach is applicable to reduced Hessian SQP algorithms. Numerical results are presented indicating that our algorithm is reliable, robust and has the potential to be used as a building block to construct trust-region algorithms for small-sized problems in constrained optimization.This research was performed while the author was on a postdoctoral appointment in the Department of Mathematical Sciences, Rice University, Houston, TX, USA and was supported in part by AFOSR 85-0243 and DOE DEFG05-86ER 25017.     

Structure of Equations Solvable by the Inverse Scattering Transform for the Schrödinger Operator

We propose a new approach for deriving nonlinear evolution equations solvable by the inverse scattering transform. The starting point of this approach is consideration of the evolution equations for the scattering data generated by solutions of an arbitrary nonlinear evolution equation that rapidly decrease as x±. Using this approach, we find all nonlinear evolution equations whose integration reduces to investigation of the scattering-data evolution equations that are differential equations (in either ordinary or partial derivatives). In this case, the evolution equations for the scattering data themselves are linear and moreover solvable in the finite form.     

A non-parametric Bayesian approach to decompounding from high frequency data

Given a sample from a discretely observed compound Poisson process, we consider non-parametric estimation of the density \(f_0\) of its jump sizes, as well as of its intensity \(\lambda _0.\) We take a Bayesian approach to the problem and specify the prior on \(f_0\) as the Dirichlet location mixture of normal densities. An independent prior for \(\lambda _0\) is assumed to be compactly supported and to possess a positive density with respect to the Lebesgue measure. We show that under suitable assumptions the posterior contracts around the pair \((\lambda _0,\,f_0)\) at essentially (up to a logarithmic factor) the \(\sqrt{n\Delta }\)-rate, where n is the number of observations and \(\Delta \) is the mesh size at which the process is sampled. The emphasis is on high frequency data, \(\Delta \rightarrow 0,\) but the obtained results are also valid for fixed \(\Delta .\) In either case we assume that \(n\Delta \rightarrow \infty .\) Our main result implies existence of Bayesian point estimates converging (in the frequentist sense, in probability) to \((\lambda _0,\,f_0)\) at the same rate. We also discuss a practical implementation of our approach. The computational problem is dealt with by inclusion of auxiliary variables and we develop a Markov chain Monte Carlo algorithm that samples from the joint distribution of the unknown parameters in the mixture density and the introduced auxiliary variables. Numerical examples illustrate the feasibility of this approach.     

Unimodality via Kronecker products

We present new proofs and generalizations of unimodality of the \(q\) -binomial coefficients  \(\left( {\begin{array}{c}n\\ k\end{array}}\right) _q\) as polynomials in  \(q\) . We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker coefficients of representations of  \(S_n\) . Other applications of this approach include strict unimodality of the diagonal \(q\) -binomial coefficients and unimodality of certain partition statistics.     

Computing steady-state solutions for a free boundary problem modeling tumor growth by Stokes equation

We consider a free boundary problem modeling tumor growth where the model equations include a diffusion equation for the nutrient concentration and the Stokes equation for the proliferation of tumor cells. For any positive radius R$R$, it is known that there exists a unique radially symmetric stationary solution. The proliferation rate μ$\mu$ and the cell-to-cell adhesiveness γ$\gamma$ are two parameters for characterizing “aggressiveness” of the tumor. We compute symmetry-breaking bifurcation branches of solutions by studying a polynomial discretization of the system. By tracking the discretized system, we numerically verified a sequence of μ/γ$\mu /\gamma$ symmetry breaking bifurcation branches. Furthermore, we study the stability of both radially symmetric and radially asymmetric stationary solutions.     

On-shell Extension of Distributions

We consider distributions on \({\mathbb{R}^{n}{\setminus}\{0\}}\) which satisfy a given set of partial differential equations and provide criteria for the existence of extensions to \({\mathbb{R}^n}\) that satisfy the same set of equations on \({\mathbb{R}^n}\) . We use the results to construct distributions satisfying specific renormalisation conditions in the Epstein and Glaser approach to perturbative quantum field theory. Contrary to other approaches, we provide a unified approach to treat Lorentz covariance, invariance under global gauge group and almost homogeneity, as well as discrete symmetries. We show that all such symmetries can be recovered by applying a linear map defined for all degrees of divergence. Using similar techniques, we find a relation between on-shell and off-shell time-ordered products involving higher derivatives of the fields.     

Approximations for Time-Dependent Distributions in Markovian Fluid Models

In this paper we analyse Markov-modulated fluid processes over finite time intervals. We study the joint distribution of the level at time \(\theta < \infty \) and of the maximum level over [0, θ], as well as the joint distribution of the level at time θ and the minimum level over [0, θ]. We approximate θ by a random variable T with Erlang distribution and so use an approach different from the usual Laplace transform to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration.     

3D Oriented Projective Geometry Through Versors of $${\mathbb{R}^{3,3}}$$

It is possible to set up a correspondence between 3D space and \({\mathbb{R}^{3,3}}\), interpretable as the space of oriented lines (and screws), such that special projective collineations of the 3D space become represented as rotors in the geometric algebra of \({\mathbb{R}^{3,3}}\). We show explicitly how various primitive projective transformations (translations, rotations, scalings, perspectivities, Lorentz transformations) are represented, in geometrically meaningful parameterizations of the rotors by their bivectors. Odd versors of this representation represent projective correlations, so (oriented) reflections can only be represented in a non-versor manner. Specifically, we show how a new and useful ‘oriented reflection’ can be defined directly on lines. We compare the resulting framework to the unoriented \({\mathbb{R}^{3,3}}\) approach of Klawitter (Adv Appl Clifford Algebra, 24:713–736, 2014), and the \({\mathbb{R}^{4,4}}\) rotor-based approach by Goldman et al. (Adv Appl Clifford Algebra, 25(1):113–149, 2015) in terms of expressiveness and efficiency.     

Proximal Hypertopologies Revisited

It is the aim of this paper to prove that for an arbitrary metric space (X,d) and a set of nonempty closed subsets of X which contains all singletons and which is closed under enlargements, we can construct a canonical approach distance on the hyperspace CL(X), having the -proximal topology (resp. the Hausdorff metric) as its topological (resp. p-metric) coreflection. We investigate some properties like, e.g., compactness and completeness of the introduced approach structures. In this way we obtain results which generalize their classical counterparts for proximal hit-and-miss hypertopologies. We also give a characterization of the completion of the introduced approach spaces.     

Perturbation Theory and the Analytic Approach in the Context of the Inclusive τ-Lepton Decay

We perform a comparative analysis of different forms of pertubative expansions in spacelike and timelike regions. In the context of the inclusive -lepton decay, we compare the results obtained using the standard perturbation theory and the Shirkov–Solovtsov analytic approach, which modifies the perturbative expansions such that the new approximations reflect basic principles of the theory, such as renormalization invariance, spectrality, and causality. We show the advantages and self-consistency of the analytic approach in describing the -lepton decay.