Kodaira Type Vanishing Theorem for the Hirokado Variety

The Hirokado variety is a Calabi–Yau threefold in characteristic 3 that is not liftable either to characteristic 0 or the ring W ₂ of the second Witt vectors. Although Deligne–Illusie–Raynaud type Kodaira vanishing cannot be applied, we show that H ¹(X, L ^?1) = 0, for an ample line bundle such that L ³ has a non-trivial global section, holds for this variety.     

On Pólya–Szegö reverse of Schwarz's inequality

We present some new results on the Cauchy–Schwarz inequality in inner product spaces, where four vectors are involved. This naturally extends Pólya–Szegö reverse of Schwarz's inequality onto complex inner product spaces. Applications to the famous Hadamard's inequality about determinants and the triangle inequality for norms are given.     

A combinatorial proof of the Kronecker–Weber Theorem in positive characteristic

In this paper we present a combinatorial proof of the Kronecker–Weber Theorem for global fields of positive characteristic. The main tools are the use of Witt vectors and their arithmetic developed by H.L. Schmid. The key result is to obtain, using counting arguments, how many p-cyclic extensions exist of fixed degree and bounded conductor where only one prime ramifies. We then compare this number with the number of subextensions of cyclotomic function fields of the same type and verify that these two numbers are the same.     

Well-posedness of the free boundary problem in compressible elastodynamics

We study the free boundary problem for the flow of a compressible isentropic inviscid elastic fluid. At the free boundary moving with the velocity of the fluid particles the columns of the deformation gradient are tangent to the boundary and the pressure vanishes outside the flow domain. We prove the local-in-time existence of a unique smooth solution of the free boundary problem provided that among three columns of the deformation gradient there are two which are non-collinear vectors at each point of the initial free boundary. If this non-collinearity condition fails, the local-in-time existence is proved under the classical Rayleigh–Taylor sign condition satisfied at the first moment. By constructing an Hadamard-type ill-posedness example for the frozen coefficients linearized problem we show that the simultaneous failure of the non-collinearity condition and the Rayleigh–Taylor sign condition leads to Rayleigh–Taylor instability.     

Fractional revival and association schemes

Fractional revival occurs between two vertices in a graph if a continuous-time quantum walk unitarily maps the characteristic vector of one vertex to a superposition of the characteristic vectors of the two vertices. This phenomenon is relevant in quantum information in particular for entanglement generation in spin networks. We study fractional revival in graphs whose adjacency matrices belong to the Bose–Mesner algebra of association schemes. A specific focus is a characterization of balanced fractional revival (which corresponds to maximal entanglement) in graphs that belong to the Hamming scheme. Our proofs exploit the intimate connections between algebraic combinatorics and orthogonal polynomials.     

An elementary,illustrative proof of the Rado–Horn theorem

The Rado–Horn theorem provides necessary and sufficient conditions for when a family of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subfamily of the vectors satisfies the so-called Rado–Horn inequality. In this paper we provide an elementary proof of the Rado–Horn theorem as well as results for the redundant case. Previous proofs give no information about how to actually partition the vectors; we use ideas present in our proof to find subfamilies of vectors which may be used to construct a kind of “optimal” partition.     

Time–frequency analysis of bivariate signals

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time–frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time–frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.     

Stability of the nonconstant stationary solution to the Poisson–Nernst–Planck–Navier–Stokes equations

We study the three-dimensional Cauchy problem of the Poisson–Nernst–Planck–Navier–Stokes equations. We first show that the corresponding stationary system has a unique semi-trivial solution under a general doping profile. Under initial small perturbations around such the semi-trivial stationary solution and small doping profile, we obtain the unique global-in-time solution to the non-stationary system. Moreover, we prove the asymptotic convergence of the solution toward the semi-trivial stationary solution as time tends to infinity.     

Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones,and convex domains

We find a new sharp trace Gagliardo–Nirenberg–Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell–Brascamp–Lieb inequality, coming from the Brunn–Minkowski theory.     

A blowup criterion for nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in 2-D

In this paper, we investigate nonhomogeneous incompressible Navier–Stokes–Landau–Lifshitz system in two-dimensional (2-D). This system consists of Navier–Stokes equations coupled with Landau–Lifshitz–Gilbert equation, an evolutionary equation for the magnetization vector. We establish a blowup criterion for the 2-D incompressible Navier–Stokes–Landau–Lifshitz system with finite positive initial density.     

Soliton interactions and Yang–Baxter maps for the complex coupled short-pulse equation

The complex coupled short-pulse equation (ccSPE) describes the propagation of ultrashort optical pulses in nonlinear birefringent fibers. The system admits a variety of vector soliton solutions: fundamental solitons, fundamental breathers, composite breathers (generic or nongeneric), as well as so-called self-symmetric composite solitons. In this work, we use the dressing method and the Darboux matrices corresponding to the various types of solitons to investigate soliton interactions in the focusing ccSPE. The study combines refactorization problems on generators of certain rational loop groups, and long-time asymptotics of these generators, as well as the main refactorization theorem for the dressing factors that leads to the Yang–Baxter property for the refactorization map and the vector soliton interactions. Among the results obtained in this paper, we derive explicit formulas for the polarization shift of fundamental solitons that are the analog of the well-known formulas for the interaction of vector solitons in the Manakov system. Our study also reveals that upon interacting with a fundamental breather, a fundamental soliton becomes a fundamental breather and, conversely, that the interaction of two fundamental breathers generically yields two fundamental breathers with a polarization shifts, but may also result into a fundamental soliton and a fundamental breather. Explicit formulas for the coefficients that characterize the fundamental breathers, as well as for their polarization vectors are obtained. The interactions of other types of solitons are also derived and discussed in detail and illustrated with plots. New Yang–Baxter maps are obtained in the process.     

A reduced-order extrapolating space–time continuous finite element method for the 2D Sobolev equation

This paper mainly concerns with the order reduction to the coefficient vectors of the classical space–time continuous finite element (STCFE) solutions for a two-dimensional Sobolev equation. The classical STCFE model is first constructed for the governing equation, and the theoretical results of the existence, stability, and convergence are provided for the STCFE solutions. We then employ a proper orthogonal decomposition to develop a reduced-order extrapolating STCFE (ROESTCFE) vector model with the lower dimension, and demonstrate the existence, stability, and convergence for the ROESTCFE solutions by the matrix means, resulting in the very concise and flexible theoretical analysis. Lastly, we examine the effectiveness of the developed ROESTCFE model by several numerical tests. It is shown that the ROESTCFE method is computationally very cheap in actual applications.     

Standard orthogonal vectors in semilinear spaces and their applications

This paper investigates the standard orthogonal vectors in semilinear spaces of n-dimensional vectors over commutative zerosumfree semirings. First, we discuss some characterizations of standard orthogonal vectors. Then as applications, we obtain some necessary and sufficient conditions that a set of vectors is a basis of a semilinear subspace which is generated by standard orthogonal vectors, prove that a set of linearly independent nonstandard orthogonal vectors cannot be orthogonalized if it has at least two nonzero vectors, and show that the analog of the Kronecker–Capelli theorem is valid for systems of equations.     

A formula for the associated Buchsbaum–Rim multiplicities of a direct sum of cyclic modules II

The associated Buchsbaum–Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert–Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum–Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive associated Buchsbaum–Rim multiplicity in terms of the ordinary Buchsbaum–Rim and Hilbert–Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.     

Several logarithmic Caffarelli–Kohn–Nirenberg inequalities and applications

In this paper, based on the Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space and the weighted Hölder inequality, we establish the logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities, and give applications for the weighted ultracontractivity of positive strong solutions to a kind of evolution equations. We also prove corresponding logarithmic Caffarelli–Kohn–Nirenberg inequalities and parameter type logarithmic Caffarelli–Kohn–Nirenberg inequalities on the Heisenberg group and related to generalized Baouendi–Grushin vector fields. Some applications are provided.     

A new family of optimal FHS sets with composite lengths

In frequency-hopping spread spectrum systems, frequency-hopping sequences (FHSs) are widely used to alleviate the interferences caused by the hits of frequencies. In this paper, based on the Zeng–Cai–Tang–Yang cyclotomy and the Chinese Remainder Theorem, a family of FHS sets with composite lengths is constructed and the Hamming correlations of the new FHS sets are derived by some basic properties of cyclotomic numbers. The results show that the proposed FHS sets are optimal with respect to the Peng–Fan bound. Furthermore, it generalizes some earlier cyclotomic constructions of optimal FHS sets and produces optimal FHS sets with new parameters which are not covered in the literature.     

Identifiability issues of age–period and age–period–cohort models of the Lee–Carter type

The predominant way of modelling mortality rates is the Lee–Carter model and its many extensions. The Lee–Carter model and its many extensions use a latent process to forecast. These models are estimated using a two-step procedure that causes an inconsistent view on the latent variable. This paper considers identifiability issues of these models from a perspective that acknowledges the latent variable as a stochastic process from the beginning. We call this perspective the plug-in age–period or plug-in age–period–cohort model. Defining a parameter vector that includes the underlying parameters of this process rather than its realizations, we investigate whether the expected values and covariances of the plug-in Lee–Carter models are identifiable. It will be seen, for example, that even if in both steps of the estimation procedure we have identifiability in a certain sense it does not necessarily carry over to the plug-in models.     

Random attractor for the stochastic Cahn–Hilliard–Navier–Stokes system with small additive noise

The objective of this paper is to study the asymptotic behavior of solutions, in terms of the upper semi-continuous property of random attractor, of the Cahn–Hilliard–Navier–Stokes system with small additive noise. We prove the existence of a random attractor for the Cahn–Hilliard–Navier–Stokes system with small additive noise. Furthermore, we consider the stability of global attractor and prove the random attractor of the Cahn–Hilliard–Navier–Stokes system with small additive noise will convergent to the global attractor of the unperturbed Cahn–Hilliard–Navier–Stokes system when the parameter of the perturbation ε tends to zero.     

A view on multiple recurrence

In this note we present a proof of multiple recurrence for ergodic systems (and thereby of Szemerédi’s theorem) being a mixture of three known proofs. It is based on a conditional version of the Jacobs–de Leeuw–Glicksberg decomposition and properties of the Gowers–Host–Kra uniformity seminorms.     

A note on optimal investment–consumption–insurance in a Lévy market

In Shen and Wei (2014) an optimal investment, consumption and life insurance purchase problem for a wage earner with Brownian information has been investigated. This paper discusses the same problem but extend their results to a geometric Itô–Lévy jump process. Our modelling framework is very general as it allows random parameters which are unbounded and involves some jumps. It also covers parameters which are both Markovian and non-Markovian functionals. Unlike in Shen and Wei (2014) who considered a diffusion framework, ours solves the problem using a novel approach, which combines the Hamilton–Jacobi–Bellman (HJB) and a backward stochastic differential equation (BSDE) in a Lévy market setup. We illustrate our results by two examples.