A priori estimates for fluid interface problems

We consider the regularity of an interface between two incompressible and inviscid fluid flows in the presence of surface tension. We obtain local‐in‐time estimates on the interface in H^{(3/2)k + 1} and the velocity fields in H^(3/2)k. These estimates are obtained using geometric considerations which show that the Kelvin‐Helmholtz instabilities are a consequence of a curvature calculation. © 2007 Wiley Periodicals, Inc.     

Fourier–Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ₃û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H¹(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H¹(Ωˆ) is proved to be 𝒪(h+N⁻¹) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.     

Asymptotic modelling of conductive thin sheets

We derive and analyse models which reduce conducting sheets of a small thickness ε in two dimensions to an interface and approximate their shielding behaviour by conditions on this interface. For this we consider a model problem with a conductivity scaled reciprocal to the thickness ε, which leads to a nontrivial limit solution for ε → 0. The functions of the expansion are defined hierarchically, i.e. order by order. Our analysis shows that for smooth sheets the models are well defined for any order and have optimal convergence meaning that the H ¹-modelling error for an expansion with N terms is bounded by O(ε ^N+1) in the exterior of the sheet and by O(ε ^N+1/2) in its interior. We explicitly specify the models of order zero, one and two. Numerical experiments for sheets with varying curvature validate the theoretical results.     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.     

Higher Order Asymptotic Theory for Discriminant Analysis in Exponential Families of Distributions

This paper deals with the problem of classifying a multivariate observation X into one of two populations Π₁: p(x; w⁽¹⁾) S and Π₂: p(x; w⁽²⁾) S, where S is an exponential family of distributions and w⁽¹⁾ and w⁽²⁾ are unknown parameters. Let ; be a class of appropriate estimators ( ⁽¹⁾, ⁽²⁾) of (w⁽¹⁾, w⁽²⁾ based on training samples. Then we develop the higher order asymptotic theory for a class of classification statistics D = [ | = log{p(X; ⁽¹⁾)/p(X; ⁽²⁾)}, ( ⁽¹⁾, ⁽²⁾) ;]. The associated probabilities of misclassification of both kinds M( ) are evaluated up to second order of the reciprocal of the sample sizes. A classification statistic is said to be second order asymptotically best in D if it minimizes M( ) up to second order. A sufficient condition for to be second order asymptotically best in D is given. Our results are very general and give us a unified view in discriminant analysis. As special results, the Anderson W, the Cochran and Bliss classification statistic, and the quadratic classification statistic are shown to be second order asymptotically best in D in each suitable classification problem. Also, discriminant analysis in a curved exponential family is discussed.     

A recovery‐based error estimator for finite volume methods of interface problems: Conforming linear elements

In this article, we present a recovery‐based a posteriori error estimator for finite volume methods which use the conforming linear trial functions to approximate elliptic interface problems. The reliability and efficiency bounds for the error estimator are established by recovering the flux from a weighted L² projection to H(div) conforming finite element spaces. Numerical experiments are given to support the conclusions. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

On Fast Computations with the Inverse to Harmonic Potential Operators via Domain Decomposition

In this paper a method for fast computations with the inverse to weakly singular, hypersingular and double layer potential boundary integral operators associated with the Laplacian on Lipschitz domains is proposed and analyzed. It is based on the representation formulae suggested for above-mentioned boundary operations in terms of the Poincare-Steklov interface mappings generated by the special decompositions of the interior and exterior domains. Computations with the discrete counterparts of these formulae can be efficiently performed by iterative substructuring algorithms provided some asymptotically optimal techniques for treatment of interface operators on subdomain boundaries. For both two- and three-dimensional cases the computation cost and memory needs are of the order O(N log^p N) and O(N log² N), respectively, with 1 ≤ p ≤ 3, where N is the number of degrees of freedom on the boundary under consideration (some kinds of polygons and polyhedra). The proposed algorithms are well suited for serial and parallel computations.     

Mixed finite element methods for generalized Forchheimer flow in porous media

Mixed finite element methods are analyzed for the approximation of the solution of the system of equations that describes the flow of a single‐phase fluid in a porous medium in ?^d, d ≤ 3, subject to Forchhheimer's law—a nonlinear form of Darcy's law. Existence and uniqueness of the approximation are proved, and optimal order error estimates in L^∞(J; L²(Ω)) and in L^∞(J; H(div; Ω)) are demonstrated for the pressure and momentum, respectively. Error estimates are also derived in L^∞(J; L^∞(Ω)) for the pressure. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Two modular equations for squares of the Rogers–Ramanujan functions with applications

Two modular identities of Gordon, McIntosh, and Robins are shown to be connected to the Rogers–Ramanujan continued fraction R(q), and in particular to Ramanujan’s parameter k:=R(q)R ²(q ²). Using this connection, we give new modular relations for R(q), and offer new and uniform proofs of several results of Ramanujan. In particular, we give a new proof of a famous and fundamental modular identity satisfied by the Rogers–Ramanujan continued fraction. We furthermore show that many analogous results hold for Ramanujan’s parameters μ:=R(q)R(q ⁴) and ν:=R ²(q ^1/2)R(q)/R(q ²). New proofs are offered for modular relations connecting R(q) to R(−q), R(q ²), and R(q ⁴), and new relations connecting R(q) at these arguments are offered. Eleven identities for the Rogers–Ramanujan functions are proved, including four new identities.      

Topology of Sobolev mappings III

We establish a necessary and sufficient topological condition for maps that are in W^1,p(M, N) to be connected by continuous paths in W^1,p(M, N) to maps in W^1,q(M, N), q > q ≥ 1. We also obtain a necessary and sufficient topological condition under which W^1,q(M, N) is strongly dense in W^1,p(M, N). Several results concerning sequential weak density of smooth (or W^1,q(M, N) maps in W^1,p(M, N) are also proven. © 2003 Wiley Periodicals, Inc.     

Reed-Muller codes and Barnes-Wall lattices: Generalized multilevel constructions and representation over GF(2<Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>)

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 ^q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 ^q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 ^l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall Λ(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 ^q ) and set partitioning based on partition chains of length-2 ^l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.      

On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

Approximation of the bifurcation function for elliptic boundary value problems

The bifurcation function for an elliptic boundary value problem is a vector field B(ω) on R ^d whose zeros are in a one‐to‐one correspondence with the solutions of the boundary value problem. Finite element approximations of the boundary value problem are shown to give rise to an approximate bifurcation function B^h(ω), which is also a vector field on R ^d. Estimates of the difference B(ω) − B^h(ω) are derived, and methods for computing B^h(ω) are discussed. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 194–213, 2000     

Recursive constructions of -polynomials over

This paper presents procedures for constructing irreducible polynomials over GF(2^s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F₀(x)GF(2^s) of degree n, polynomials F_k(x)GF(2^s) of degrees n2^k are constructed by iteratively applying the transformation x→x+x^-1, and their roots are shown to form a normal basis of GF(2^sn2k) over GF(2^s). In addition, the sequences are shown to be trace compatible, i.e., the trace map T_{GF(2^sn2k+1)/GF(2^sn2k)} fromGF(2^sn2k+1) onto GF(2^sn2k) maps the roots of F_k+1(x) onto those of F_k(x).     

Reducible Families of Curves with Ordinary Multiple Points on Surfaces in IP3 C

For a Coxeter system (G, S) the multi-parametric alternating subalgebra H ⁺(G) of the Hecke algebra and the alternating subgroup ?⁺(G) of the braid group are defined. Two presentations for H ⁺(G) and ?⁺(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.     

Regenerative embedding of Markov sets

Summary.  Given a closed Markov (i.e. regenerative) set in [0,∞), we characterize the laws of the Markov sets which are regeneratively embedded into the latter. Typically, let Φ⁽¹⁾ and Φ⁽²⁾ be two Laplace exponents corresponding to two regenerative laws, and M ⁽²⁾ a Markov set with exponent Φ⁽²⁾. There exists a Markov set M ⁽¹⁾ with exponent Φ⁽¹⁾ which is regeneratively embedded into M ⁽²⁾ if and only if Φ⁽¹⁾/Φ⁽²⁾ is a completely monotone function. Several examples and applications are discussed. Received: 12 April 1996 / In revised form: 12 March 1997     

Relationships between characteristics in periodic Poisson queues

We study single server periodic queues in the day equilibrium conditions. The following characteristics of interest are considered at time of dayt: V^p(t)-the work load, L^p(t)-the number of customers and u^p(t)-the departure rate. We give relationships between E[V^p(t)], E[L^p(t)] and u^p(t). We also prove that E[V^p(t)] < and E[L^p(t)] < provided the second moment of the service time is finite.     

On Dual Wavelet Tight Frames

A characterization of multivariate dual wavelet tight frames for any general dilation matrix is presented in this paper. As an application, Lawton's result on wavelet tight frames inL²( ) is generalized to then-dimensional case. Two ways of constructing certain dual wavelet tight frames inL²( ⁿ) are suggested. Finally, examples of smooth wavelet tight frames inL²( ) andH²( ) are provided. In particular, an example is given to demonstrate that there is a function ψ whose Fourier transform is positive, compactly supported, and infinitely differentiable which generates a non-MRA wavelet tight frame inH²( ).