On conservative spatial discretizations of the barotropic quasi-gasdynamic system of equations with a potential body force

A multidimensional barotropic quasi-gasdynamic system of equations in the form of mass and momentum conservation laws with a general gas equation of state p = p(ρ) with p′(ρ) > 0 and a potential body force is considered. For this system, two new symmetric spatial discretizations on nonuniform rectangular grids are constructed (in which the density and velocity are defined on the basic grid, while the components of the regularized mass flux and the viscous stress tensor are defined on staggered grids). These discretizations involve nonstandard approximations for ?p(ρ), div(ρu), and ρ. As a result, a discrete total mass conservation law and a discrete energy inequality guaranteeing that the total energy does not grow with time can be derived. Importantly, these discretizations have the additional property of being well-balanced for equilibrium solutions. Another conservative discretization is discussed in which all mass flux components and viscous stresses are defined on the same grid. For the simpler barotropic quasi-hydrodynamic system of equations, the corresponding simplifications of the constructed discretizations have similar properties.     

Efficient discrete implementations for a dynamic problem of linear elasticity

For a dynamic three-dimensional linear elasticity problem in velocities-stresses, we construct efficient difference schemes on the basis of various additive decompositions of the original spatial operator. They include a difference scheme whose efficient implementation at the “predictor” stage has the property of complete conservativeness. Another class of efficient difference schemes is related to the representation of the operator as a product of triangular operators, that is, an operator analog of the LU-decomposition. The parallelism degree of these difference schemes is the same as of explicit schemes.     

A study of diquark and meson condensation in the Nambu–Jona-Lasinio model and Fermi momentum

Using a three- and four-dimensional Pauli–Villars regularization scheme, we investigate quark–antiquark and diquark condensation in the framework of the Nambu–Jona-Lasinio model. Using the particle Fermi momentum as a cutoff parameter, we study the energy gap width and coherence length for the meson condensate 〈\(q\bar q\)〉. We also study the energy gap width and critical coherence length (the distance over which there would be no diquark condensation) for the diquark 〈qq〉 and the dependence on the Fermi momentum. We obtain an estimate of the Fermi momentum value for meson and diquark condensates with an energy gap width of the order of 100 MeV.     

Sequential and parallel domain decomposition methods for a singularly perturbed parabolic convection-diffusion equation

A boundary value problem for a singularly perturbed parabolic convection-diffusion equation is considered in a rectangular domain in x and t; the perturbation parameter ? multiplying the highest derivative takes arbitrary values in the half-open interval (0,1]. For the boundary value problem, we construct a scheme based on the method of lines in x passing through N ₀+1 points of the mesh with respect to t. To solve the problem on a set of intervals, we apply a domain decomposition method (on overlapping subdomains with the overlap width δ), which is a modification of the Schwarz method. For the continual schemes of the decomposition method, we study how sequential and parallel computations, the order of priority in which the subproblems are sequentially solved on the subdomains, and the value of the parameter ? (as well as the values of N ₀, δ) influence the convergence rate of the decomposition scheme (as N ₀ → ∞), and also computational costs for solving the scheme and time required for its solution (unless a prescribed tolerance is achieved). For convection-diffusion equations, in contrast to reaction-diffusion ones, the sequential scheme turns out to be more efficient than the parallel scheme.     

Unconditional and optimal <Emphasis Type="Bold">H</Emphasis><Superscript>2</Superscript>-error estimates of two linear and conservative finite difference schemes for the Klein-Gordon-Schrödinger equation in high dimensions

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrödinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H ²-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h ² + τ ²) with mesh-size h and time step τ in the discrete H ²-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.     

Computer difference scheme for a singularly perturbed elliptic convection–diffusion equation in the presence of perturbations

A grid approximation of a boundary value problem for a singularly perturbed elliptic convection–diffusion equation with a perturbation parameter ε, ε ∈ (0,1], multiplying the highest order derivatives is considered on a rectangle. The stability of a standard difference scheme based on monotone approximations of the problem on a uniform grid is analyzed, and the behavior of discrete solutions in the presence of perturbations is examined. With an increase in the number of grid nodes, this scheme does not converge -uniformly in the maximum norm, but only conditional convergence takes place. When the solution of the difference scheme converges, which occurs if N ₁ ^-1 N ₂ ^-1 ? ε, where N ₁ and N ₂ are the numbers of grid intervals in x and y, respectively, the scheme is not -uniformly well-conditioned or ε-uniformly stable to data perturbations in the grid problem and to computer perturbations. For the standard difference scheme in the presence of data perturbations in the grid problem and/or computer perturbations, conditions imposed on the “parameters” of the difference scheme and of the computer (namely, on ε, N ₁,N ₂, admissible data perturbations in the grid problem, and admissible computer perturbations) are obtained that ensure the convergence of the perturbed solutions as N ₁,N ₂ → ∞, ε ∈ (0,1]. The difference schemes constructed in the presence of the indicated perturbations that converges as N ₁,N ₂ → ∞ for fixed ε, ε ∈ (0,1, is called a computer difference scheme. Schemes converging ε-uniformly and conditionally converging computer schemes are referred to as reliable schemes. Conditions on the data perturbations in the standard difference scheme and on computer perturbations are also obtained under which the convergence rate of the solution to the computer difference scheme has the same order as the solution of the standard difference scheme in the absence of perturbations. Due to this property of its solutions, the computer difference scheme can be effectively used in practical computations.     

Improved difference scheme of the solution decomposition method for a singularly perturbed reaction-diffusion equation

A Dirichlet problem is considered for a singularly perturbed ordinary differential reaction-diffusion equation. For this problem, a new approach is developed in order to construct difference schemes that converge uniformly with respect to the perturbation parameter ?, ? ∈ (0, 1]. The approach is based on the decomposition of a discrete solution into regular and singular components, which are solutions of discrete subproblems on uniform grids. Using the asymptotic construction technique, a difference scheme of the solution decomposition method is constructed that converges ?-uniformly in the maximum norm at the rate O (N ^?2 ln² N), where N + 1 is the number of nodes in the grid used; for fixed values of the parameter ?, the scheme converges at the rate O(N ^?2). Using the Richardson technique, an improved scheme of the solution decomposition method is constructed, which converges ?-uniformly in the maximum norm at the rate O(N ^?4 ln⁴ N).     

Epimorphisms of <Emphasis Type="Italic">S</Emphasis>-quantales

In this paper we study certain actions of a pomonoid S on a complete lattice, which we call S-quantales. Our aim is to characterize epimorphisms in the category of S-quantales. For this purpose we show that this category is a monadic construct and has the amalgamation property.     

The local lifting problem for <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D₄-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D₄- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D₄-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D₄ is a local Oort group for the prime 2.     

Boundary value problem with normal derivatives for a higher-order elliptic equation on the plane

For an elliptic operator of order 2l with constant (and only leading) real coefficients, we consider a boundary value problem in which the normal derivatives of order (k _j ?1), j = 1,..., l, where 1 ≤ k ₁ < ··· < k _l, are specified. It becomes the Dirichlet problem for kj = j and the Neumann problem for k _j = j + 1. We obtain a sufficient condition for the Fredholm property of which problem and derive an index formula.     

Asymptotic expansions of periodic solutions of ordinary differential equations with large high-frequency terms

We consider the problem on the periodic solutions of a system of ordinary differential equations of arbitrary order n containing terms oscillating at a frequency ω ? 1 with coefficients of the order of ω ^n/2. For this problem, we construct the averaged (limit) problem and justify the averaging method as well as another efficient algorithm for constructing the complete asymptotics of the solution.     

Approximation by Linear Fractional Transformations of Simple Partial Fractions and Their Differences

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f ? ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ?⁺. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ?⁺. For odd functions we obtain such estimates on the whole axis ?.     

On weak* convergent sequences in duals of symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that the pre-dual of a σ-finite von Neumann algebra has property (k) in the sense of Figiel, Johnson and Pelczyński [12]. This resolves in the affirmative an open question raised in [12]. It is shown further that a weakly sequentially complete symmetric space E of τ-measurable operators affiliated with a semifinite σ-finite von Neumann algebra has property (k).     

Degeneration of the Hilbert pairing in formal groups over local fields

For an arbitrary local field K (a finite extension of the field Q_p) and an arbitrary formal group law F over K, we consider an analog c_F of the classical Hilbert pairing. A theorem by S.V. Vostokov and I.B. Fesenko says that if the pairing c_F has a certain fundamental symbol property for all Lubin–Tate formal groups, then c_F = 0. We generalize the theorem of Vostokov–Fesenko to a wider class of formal groups. Our first result concerns formal groups that are defined over the ring O_K of integers of K and have a fixed ring O₀ of endomorphisms, where O₀ is a subring of O_K. We prove that if the symbol c_F has the above-mentioned symbol property, then c_F = 0. Our second result strengthens the first one in the case of Honda formal groups. The paper consists of three sections. After a short introduction in Section 1, we recall basic definitions and facts concerning formal group laws in Section 2. In Section 3, we state and prove two main results of the paper (Theorems 1 and 2). Refs. 8.     

Plane Partitions and Their Pedestal Polynomials

For a linear extension P of a partially ordered set S, we consider a generating multivariate polynomial of certain reverse partitions on S, called P-pedestals. We establish a remarkable property of this polynomial: it does not depend on the choice of P. For S a Young diagram, we show that this polynomial generalizes the hook polynomial.     

Nonsingularity,positive definiteness,and positive invertibility under fixed-point data rounding

For a real square matrix A and an integer d ? 0, let A _(d) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A _(d) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists a real number α(d), computed solely from A _(d) (not from A), such that the following alternative holdsif d > α(d), then nonsingularity (positive definiteness, positive invertibility) of A _(d) implies the same property for A if d < α(d) and A _(d) is nonsingular (positive definite, positive invertible), then there exists a matrix A′ with A′_(d) = A _(d) which does not have the respective property.For nonsingularity and positive definiteness the formula for α(d) is the same and involves computation of the NP-hard norm ‖ · ‖_∞,1; for positive invertibility α(d) is given by an easily computable formula.     

C*-algebras have a quantitative version of Pełczyński’s property (V)

A Banach space X has Pe?czyński’s property (V) if for every Banach space Y every unconditionally converging operator T: X → Y is weakly compact. H.Pfitzner proved that C*-algebras have Pe?czyński’s property (V). In the preprint (Kruli?ová, (2015)) the author explores possible quantifications of the property (V) and shows that C(K) spaces for a compact Hausdorff space K enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we prove that in dual Banach spaces a quantitative version of the property (V) implies a quantitative version of the Grothendieck property.     

Noether-Type Discrete Conserved Quantities Arising from a Finite Element Approximation of a Variational Problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.     

Grid approximation of singularly perturbed parabolic equations with piecewise continuous initial-boundary conditions

The Dirichlet problem is considered for a singularly perturbed parabolic reaction-diffusion equation with piecewise continuous initial-boundary conditions in a rectangular domain. The highest derivative in the equation is multiplied by a parameter ? ², ? ε (0, 1]. For small values of the parameter ?, in a neighborhood of the lateral part of the boundary and in a neighborhood of the characteristic of the limit equation passing through the point of discontinuity of the initial function, there arise a boundary layer and an interior layer (of characteristic width ?), respectively, which have bounded smoothness for fixed values of the parameter ?. Using the method of additive splitting of singularities (generated by discontinuities of the boundary function and its low-order derivatives), as well as the method of condensing grids (piecewise uniform grids condensing in a neighborhood of boundary layers), we construct and investigate special difference schemes that converge ?-uniformly with the second order of accuracy in x and the first order of accuracy in t, up to logarithmic factors.     

Chosen ciphertext secure keyed-homomorphic public-key cryptosystems

In homomorphic encryption schemes, anyone can perform homomorphic operations, and therefore, it is difficult to manage when, where and by whom they are performed. In addition, the property that anyone can “freely” perform the operation inevitably means that ciphertexts are malleable, and it is well-known that adaptive chosen ciphertext (CCA) security and the homomorphic property can never be achieved simultaneously. In this paper, we show that CCA security and the homomorphic property can be simultaneously handled in situations that the user(s) who can perform homomorphic operations on encrypted data should be controlled/limited, and propose a new concept of homomorphic public-key encryption, which we call keyed-homomorphic public-key encryption (KH-PKE). By introducing a secret key for homomorphic operations, we can control who is allowed to perform the homomorphic operation. To construct KH-PKE schemes, we introduce a new concept, transitional universal property, and present a practical KH-PKE scheme with multiplicative homomorphic operations from the decisional Diffie-Hellman (DDH) assumption. For \(\ell \)-bit security, our DDH-based KH-PKE scheme yields only \(\ell \)-bit longer ciphertext size than that of the Cramer–Shoup PKE scheme. Finally, we consider an identity-based analogue of KH-PKE, called keyed-homomorphic identity-based encryption and give its concrete construction from the Gentry IBE scheme.