k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

Cohomology and extension problems for semi <Emphasis Type="Italic">q</Emphasis>-coronae

We prove some extension theorems for analytic objects, in particular sections of a coherent sheaf, defined in semi q-coronae of a complex space. Semi q-coronae are domains whose boundary is the union of a Levi flat part, a q-pseudoconvex part and a q-pseudoconcave part. Such results are obtained mainly using cohomological techniques.     

On the Eigenvalue Accumulation of Sturm-Liouville Problems Depending Nonlinearly on the Spectral Parameter

A nonlinear spectral problem for a Sturm-Liouville equation-(p(x, λ)y'(x, λ))' + q(x, λ) y(x, λ) = 0 on a finite interval [a, b] with λ-dependent boundary conditions is considered. The spectral parameter λ is varying in an interval ∧ and p(x, λ), q(x, A) are real, continuous functions on [a, b] × ∧ Some criteria to the eigenvalue accumulation at the endpoints of A will be established. The results are applied to concrete problems arising in magnetohydrodynamics.     

Partial Regularity up to the Boundary for Weak Solutions of Elliptic Systems with Nonlinearity q Greater Than Two

Nonlinear elliptic systems with q-growth are considered. It is assumed that additional nonlinear terms of the systems have q-growth in the gradient, q < 2. For Dirichlet and Neumann boundary-value problems we study the regularity of weak bounded solutions in the vicinity of the boundary. In the case of small dimensions (n q + 2), the Hölder continuity or partial Hölder continuity up to the boundary is proved for the solutions considered. In the previous article, the author studied the same problem for q = 2. Bibliography: 12 titles.     

The Simultaneous Approximation Order by Hermite Interpolation in a Smooth Domain

Let D be a smooth domain in the complex plane. In D consider the simultaneous approximation to a function and its ith (0 ≤i≤q) derivatives by Hermite interpolation. The orders of uniform approximation and approximation in the mean, are obtained under some domain boundary conditions. Some known results are included as particular cases of the theorems of this paper. Received May 25, 2000, Revised November 3, 2000, Accepted December 7, 2000     

A coupled complex boundary method for an inverse conductivity problem with one measurement

We recently proposed in [Cheng, XL et al. A novel coupled complex boundary method for inverse source problems Inverse Problem 2014 30 055002] a coupled complex boundary method (CCBM) for inverse source problems. In this paper, we apply the CCBM to inverse conductivity problems (ICPs) with one measurement. In the ICP, the diffusion coefficient q is to be determined from both Dirichlet and Neumann boundary data. With the CCBM, q is sought such that the imaginary part of the solution of a forward Robin boundary value problem vanishes in the problem domain. This brings in advantages on robustness and computation in reconstruction. Based on the complex forward problem, the Tikhonov regularization is used for a stable reconstruction. Some theoretical analysis is given on the optimization models. Several numerical examples are provided to show the feasibility and usefulness of the CCBM for the ICP. It is illustrated that as long as all the subdomains share some portion of the boundary, our CCBM-based Tikhonov regularization method can reconstruct the diffusion parameters stably and effectively.     

Filling of a given boundary by p-gons and related problems

We consider here (p,s)-polycycles (3ps) i.e. plane graphs, such that all interior faces are p-gons, all interior vertices are s-valent and any vertex of the boundary (i.e. the exterior face) has valency within [2,s]. The boundary sequence of a (p,s)-polycycle P is the sequence b(P) enumerating, up to a cyclic shift or reversal, the consecutive valencies of vertices of the boundary. We show that the values p=3,4 are the only ones, such that the boundary sequence defines its (p,3)-filling (i.e. a (p,3)-polycycle with given boundary) uniquely.Also we give new results in the enumeration of maps M_n(p,q) (i.e. plane 3-valent maps with only p- and q-gonal faces, such that the q-gons are organized in an n-ring) and two of their generalizations.Both problems are similar (3-valent filling by p-gons of a boundary or of a ring of q-gons) and the same programs were used for both computations.     

On the mutually non isomorphic ℓp(ℓq) spaces

We extend a result of Pe?czyński showing that {?_p(?_q): 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of ?_p(?_q) are also given.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

Local solvability of the -equation with boundary regularity on weakly q-convex domains

We consider weakly q-convex domains with smooth boundary and show that the -equation is locally solvable with regularity up to the boundary for smooth forms of degree (p,s) for s≥q.     

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter-dependent boundary conditions

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h₁,h₂, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H₁,H₂, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.     

M-P invertible matrices and unitary groups over Fq2

The Moor-Penrose generalized inverses (M-P inverses for short) of matrices over a finite field F_q ² which is a generalization of the Moor-Penrose generalized inverses over the complex field, are studied in the present paper. Some necessary and sufficient conditions for anm xn matrixA over F_q ² having an M-P inverse are obtained, which make clear the set ofm xn matrices over F_q ² having M-P inverses and reduce the problem of constructing and enumerating the M-P invertible matrices to that of constructing and enumerating the non-isotropic subspaces with respect to the unitary group. Based on this reduction, both the construction problem and the enumeration problem are solved by borrowing the results in geometry of unitary groups over finite fields.     

New inductive constructions of complete caps in PG(N,q), q even

Some new families of small complete caps in PG(N, q), q even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this article provide an improvement on the currently known upper bounds on the size of the smallest complete cap in PG(N, q), N≥4, for all q≥2³. In particular, substantial improvements are obtained for infinite values of q square, including q=2^2Cm, C≥5, m≥3; for q=2^Cm, C≥5, m≥9, with C, m odd; and for all q≤2¹⁸. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 177–201, 2010     

Types of superregular matrices and the number of n‐arcs and complete n‐arcs in PG (r,q)

Based on the classification of superregular matrices, the numbers of non‐equivalent n‐arcs and complete n‐arcs in PG(r, q) are determined (i) for 4 ≤ q ≤ 19, 2 ≤ r ≤ q ? 2 and arbitrary n, (ii) for 23 ≤ q ≤ 32, r = 2 and n ≥ q ? 8<$>. The equivalence classes over both PGL (k, q) and PΓL(k, q) are considered throughout the examinations and computations. For the classification, an n‐arc is represented by the systematic generator matrix of the corresponding MDS code, without the identity matrix part of it. A rectangular matrix like this is superregular, i.e., it has only non‐singular square submatrices. Four types of superregular matrices are studied and the non‐equivalent superregular matrices of different types are stored in databases. Some particular results on t(r, q) and m′(r, q)—the smallest and the second largest size for complete arcs in PG(r, q)—are also reported, stating that m′(2, 31) = 22, m′(2, 32) = 24, t(3, 23) = 10, and m′(3, 23) = 16. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 363–390, 2006     

Mixing Times of Critical Two‐Dimensional Potts Models

We study dynamical aspects of the q‐state Potts model on an n × n box at its critical β_c(q). Heat‐bath Glauber dynamics and cluster dynamics such as Swendsen–Wang (that circumvent low‐temperature bottlenecks) are all expected to undergo “critical slowdowns” in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is O(1), should at criticality be polynomial in n for 1 < q ≤ 4, and exponential in n for q > 4 in accordance with the predicted discontinuous phase transition. This was confirmed for q = 2 (the Ising model) by the second author and Sly, and for sufficiently large q by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for q=3, the inverse gap of Glauber dynamics is n^O(1); for q = 4, it is at most n^{O(log n)}; and for every q > 4 in the phase‐coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen‐Wang dynamics are exponential in n. For free or monochromatic boundary conditions and large q, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen‐Wang dynamics is exp(n^o(1)). © 2017 Wiley Periodicals, Inc.     

On Boundary Regularity of the Navier–Stokes Equations

Abstract

We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L ^{p, q} with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.     

Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions

We characterize the L^q-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

A Metric of Constant Curvature on Polycycles

We prove the following main theorem of the theory of (r, q)-polycycles. Suppose a nonseparable plane graph satisfies the following two conditions:(1) each internal face is an r-gon, where r ≥ 3;(2) the degree of each internal vertex is q, where q ≥ 3, and the degree of each boundary vertex is at most q and at least 2.Then it also possesses the following third property:(3) the vertices, the edges, and the internal faces form a cell complex.Simple examples show that conditions (1) and (2) are independent even provided condition (3) is satisfied. These are the defining conditions for an (r, q)-polycycle.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 223–233.Original Russian Text Copyright © 2005 by M. Deza, M. I. Shtogrin.     

The stationary and non-stationary stokes system in exterior domains with non-zero divergence and non-zero boundary values

In an exterior domain Ω??ⁿ, n ? 2, we consider the generalized Stokes resolvent problem in L^q-space where the divergence g = div u and inhomogeneous boundary values u = ψ with zero flux ∫_?Ωψ·N do = 0 may be prescribed. A crucial step in our approach is to find and to analyse the right space for the divergence g. We prove existence, uniqueness and a priori estimates of the solution and get new results for the divergence problem. Further, we consider the non-stationary Stokes system with non-homogeneous divergence and boundary values and prove estimates of the solution in L⁵(0, T;L^q(Ω)) for 1 < s, q < ∞.