On the existence of smooth self‐similar blowup profiles for the wave map equation

Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self‐similar blowup profile. More generally, we study the relation between

• the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and
• the existence of a smooth blowup profile for the (hyperbolic) wave map problem.

This has several applications to questions of regularity and uniqueness for the wave map equation. © 2008 Wiley Periodicals, Inc.     

Interface boundary value problems of Robin‐transmission type for the Stokes and Brinkman systems on n‐dimensional Lipschitz domains: applications

In this paper, we describe a layer potential analysis in order to show an existence result for an interface boundary value problem of Robin‐transmission type for the Stokes and Brinkman systems on Lipschitz domains in Euclidean setting, when the given boundary data belong to some L^p or Sobolev spaces associated to such domains. Applications related to an exterior three‐dimensional Stokes flow past two concentric porous spheres with stress jump conditions on the fluid‐porous interface are also considered. Copyright © 2013 John Wiley & Sons, Ltd.     

Existence of a non‐reflexive embedding with birational Gauss map for a projective variety

We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ^N to re‐embed into some projective space ℙ^M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ^N is identically zero; hence the projective varietyX re‐embedded in ℙ^M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ^N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self‐dual and self‐petrie‐dual regular maps

Regular maps are cellular decompositions of surfaces with the “highest level of symmetry”, not necessarily orientation‐preserving. Such maps can be identified with three‐generator presentations of groups G of the form G = 〈a, b, c|a² = b² = c² = (ab)^k = (bc)^m = (ca)² = … = 1〉; the positive integers k and m are the face length and the vertex degree of the map. A regular map (G;a, b, c) is self‐dual if the assignment b?b, c?a and a?c extends to an automorphism of G, and self‐Petrie‐dual if G admits an automorphism fixing b and c and interchanging a with ca. In this note we show that for infinitely many numbers k there exist finite, self‐dual and self‐Petrie‐dual regular maps of vertex degree and face length equal to k. We also prove that no such map with odd vertex degree is a normal Cayley map. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 69:152‐159, 2012     

First‐Order system least squares for generalized‐Newtonian coupled Stokes‐Darcy flow

The coupled problem for a generalized Newtonian Stokes flow in one domain and a generalized Newtonian Darcy flow in a porous medium is studied in this work. Both flows are treated as a first‐order system in a stress‐velocity formulation for the Stokes problem and a volumetric flux‐hydraulic potential formulation for the Darcy problem. The coupling along an interface is done using the well‐known Beavers–Joseph–Saffman interface condition. A least squares finite element method is used for the numerical approximation of the solution. It is shown that under some assumptions on the viscosity the error is bounded from above and below by the least squares functional. An adaptive refinement strategy is examined in several numerical examples where boundary singularities are present. Due to the nonlinearity of the problem a Gauss–Newton method is used to iteratively solve the problem. It is shown that the linear variational problems arising in the Gauss–Newton method are well posed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1150–1173, 2015     

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L² norm. Numerical experiments are presented which confirm our theoretical findings.     

A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

In this paper, we consider the propagation of waves in a closed full or half waveguide where the index of refraction is periodic along the axis of the waveguide. Motivated by the limiting absorption principle, proven in the Appendix by a functional analytic perturbation theorem, we formulate a radiation condition that assures uniqueness of a solution and allows the existence of propagating modes. Our approach is quite different to the known one as, eg, considered recently by Fliss and Joly and allows an extension to open wave guides. After application of the Floquet‐Bloch transform, we consider the Bloch variable α as a parameter in the resulting quasiperiodic boundary value problem and study the behaviour of the solution when α tends to an exceptional value by a singular perturbation result, which goes back to Colton and Kress.     

Clique‐coloring some classes of odd‐hole‐free graphs

We consider the problem of clique‐coloring, that is coloring the vertices of a given graph such that no maximal clique of size at least 2 is monocolored. Whereas we do not know any odd‐hole‐free graph that is not 3‐clique‐colorable, the existence of a constant C such that any perfect graph is C‐clique‐colorable is an open problem. In this paper we solve this problem for some subclasses of odd‐hole‐free graphs: those that are diamond‐free and those that are bull‐free. We also prove the NP‐completeness of 2‐clique‐coloring K₄‐free perfect graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 233–249, 2006     

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

We exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ?² into a smooth compact revolution surface of ?³ that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with ??² target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc.     

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.     

Computable analysis of a boundary‐value problem for the generalized KdV–Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV‐Burgers equation with initial‐boundary value problem. Here, the solution operator is a nonlinear map H^{3m ? 1}(R⁺) × H^m(0,T)→C([0,T];H^{3m ? 1}(R⁺)) from the initial‐boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer m ≥ 2, and computable real number T > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Two‐grid methods for semilinear interface problems

In this article, we consider two‐grid finite element methods for solving semilinear interface problems in d space dimensions, for d = 2 or d = 3. We consider semilinear problems with discontinuous diffusion coefficients, which includes problems containing subcritical, critical, and supercritical nonlinearities. We establish basic quasioptimal a priori error estimates for Galerkin approximations. We then design a two‐grid algorithm consisting of a coarse grid solver for the original nonlinear problem, and a fine grid solver for a linearized problem. We analyze the quality of approximations generated by the algorithm and show that the coarse grid may be taken to have much larger elements than the fine grid, and yet one can still obtain approximation quality that is asymptotically as good as solving the original nonlinear problem on the fine mesh. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Continuous k‐to‐1 functions between complete graphs whose orders are of a different parity

A function between graphs is k‐to‐1 if each point in the codomain has precisely k pre‐images in the domain. Given two graphs, G and H, and an integer k≥1, and considering G and H as subsets of ?³, there may or may not be a k‐to‐1 continuous function (i.e. a k‐to‐1 map in the usual topological sense) from G onto H. In this paper we consider graphs G and H whose order is of a different parity and determine the even and odd values of k for which there exists a k‐to‐1 map from G onto H. We first consider k‐to‐1 maps from K_2r onto K_2s+1 and prove that for 1≤r≤s, (r, s)≠(1, 1), there is a continuous k‐to‐1 map for k even if and only if k≥2s and for k odd if and only if k≥?s?_o (where ?s?_o indicates the next odd integer greater than or equal to s). We then consider k‐to‐1 maps from K_2s+1 onto K_2s. We show that for 1≤r<s, such a map exists for even values of k if and only if k≥2s. We also prove that whatever the values of r and s are, no such k‐to‐1 map exists for odd values of k. To conclude, we give all triples (n, k, m) for which there is a k‐to‐1 map from K_n onto K_m in the case when n≤m. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 35–60, 2010.     

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

The homotopy limit problem for Karoubi?s Hermitian K-theory (Karoubi, 1980) [26] was posed by Thomason (1983) [44]. There is a canonical map from algebraic Hermitian K-theory to the Z/2-homotopy fixed points of algebraic K-theory. The problem asks, roughly, how close this map is to being an isomorphism, specifically after completion at 2. In this paper, we solve this problem completely for fields of characteristic 0 (Theorems 16, 20). We show that the 2-completed map is an isomorphism for fields F of characteristic 0 which satisfy cd₂(F[i])<∞, but not in general.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

Uncoupling evolutionary groundwater‐surface water flows using the Crank–Nicolson Leapfrog method

Consider an incompressible fluid in a region Ω_f flowing both ways across an interface into a porous media domain Ω_p saturated with the same fluid. The physical processes in each domain have been well studied and are described by the Stokes equations in the fluid region and the Darcy equations in the porous media region. Taking the interfacial conditions into account produces a system with an exactly skew symmetric coupling. Spatial discretization by finite element method and time discretization by Crank–Nicolson LeapFrog give a second‐order partitioned method requiring only one Stokes and one Darcy subphysics and subdomain solver per time step for the fully evolutionary Stokes‐Darcy problem. Analysis of this method leads to a time step condition sufficient for stability and convergence. Numerical tests verify predicted rates of convergence; however, stability tests reveal the problem of growth of numerical noise in unstable modes in some cases. In such instances, the addition of time filters adds stability. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Scoring Concept Maps: An Expert Map‐Based Scheme Weighted for Relationships

The use of student‐constructed concept maps in assessment is congruent with the changing emphases set forth by the National Science Education Standards. Authorities have expressed concern about concept map scoring systems and their associated validity and reliability. They favor methods that employ expert/criterion maps as referents and emphasize the use of accurate concept relationships in deriving scores, which have been found to correlate with performance on standardized tests. In this study, student constructed concept maps (n= 17) that emerged from post‐instructional interviews about chlorofluorocarbons were scored against a teacher‐expert map using a scheme weighted for relationships. Interrater reliability for the scoring scheme was high (r= .959). Students' map scores correlated highly with their scores on the California Achievement Test component total (r= .729) and moderately with their Pathfinder index (r= .474), the latter believed to be an excellent measure of structural knowledge. A revised map score, derived only from relationships containing one or more of the concepts employed in Pathfinder analysis, was a statistically significant (p= .031) predictor of the Pathfinder index. The findings of this study support the recommendations of others to use expert referents and emphasize concept relationships in assessing concept maps.     

Multi‐to One‐Dimensional Optimal Transport

We consider the Monge‐Kantorovich problem of transporting a probability density on to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level‐set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples. More specifically, fix probability densities f and g on open sets and with . Consider transporting f onto g so as to minimize the cost . We give a nondegeneracy condition on that ensures the set of x paired with [g‐a.e.] y ∈ Y lie in a codimension‐n submanifold of X. Specializing to the case m > n = 1, we discover a nestedness criterion relating s to (f,g) that allows us to construct a unique optimal solution in the form of a map . When and g and f are bounded, the Kantorovich dual potentials (u,υ) satisfy , and the normal velocity V of with respect to changes in y is given by . Positivity of V locally implies a Lipschitz bound on f; moreover, if intersects transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer the norms of and are controlled by these bounds, , and the smallness of . We give examples showing regularity extends from $X to part of , but not from Y to . We also show that when s remains nested for all (f,g), the problem in reduces to a supermodular problem in . © 2017 Wiley Periodicals, Inc.     

Nonlocal Tug‐of‐War and the Infinity Fractional Laplacian

Motivated by the “tug‐of‐war” game studied by Peres et al. in 2009, we consider a nonlocal version of the game that goes as follows: at every step two players pick, respectively, a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving a fixed amount ϵ > 0 (as is done in the classical case), it is an s‐stable Levy process that chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically derive a deterministic nonlocal integrodifferential equation that we call the “infinity fractional Laplacian.” We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double‐obstacle problem, both problems having a natural interpretation as tug‐of‐war games. © 2011 Wiley Periodicals, Inc.     

An application of Newton–Puiseux charts to the Jacobian problem

We study 2-dimensional Jacobian maps using so-called Newton–Puiseux charts. These are multi-valued coordinates near divisors of resolutions of indeterminacies at infinity of the Jacobian map in the source space as well as in the target space. The map expressed in these charts takes a very simple form, which allows us to detect a series of new analytical and topological properties. We prove that the Jacobian Conjecture holds true for maps (f,g) whose topological degree is ≤5, for maps with gcd(degf,degg)≤16 and for maps with. gcd(degf,degg) equal to 2 times a prime.