Asymptotic model for free surface flow of an electrically conducting fluid in a high-frequency magnetic field

We study the behaviour of a layer of an electrically conducting inviscid incompressible fluid in a high-frequency alternating magnetic field. We derive nonlinear asymptotic equations governing the evolution of the fluid layer in the high-frequency limit. As a test for the model, we consider the linearised stability problem for an infinite planar free surface of a layer of finite depth.     

Exchange Relation Planar Algebras

An inclusion of II ₁ factors NM of finite index has as an invariant, a double sequence of finite-dimensional algebras known as the standard invariant. Planar algebras were introduced by V. Jones as a geometric tool for computing standard invariants of existing subfactors as well as generating standard invariants for new subfactors. In this paper we define a class of planar algebras, termed exchange relation planar algebras, that provides a general framework for understanding several classes of known subfactor inclusions: the Fuss–Catalan algebras (i.e. those coming from the presence of intermediate subfactors) and all depth 2 subfactors. In addition, we present a new class of planar algebras (and thus a new class of subfactors) coming from automorphism subgroups of finite groups.     

Guionnet–Jones–Shlyakhtenko subfactors associated to finite-dimensional Kac algebras

We analyse the Guionnet–Jones–Shlyakhtenko construction for the planar algebra associated to a finite-dimensional Kac algebra and identify the factors that arise as finite interpolated free group factors.     

Some remarks on the Balog–Wooley decomposition theorem and quantities <Emphasis Type="Italic">D</Emphasis><Superscript>+</Superscript>, <Emphasis Type="Italic">D</Emphasis><Superscript>×</Superscript>

A finite element with new properties of approximation of higher derivatives is constructed, and a method for the construction of a finite element space in the planar case is proposed. The method is based on Yu.N. Subbotin’s earlier results as well as on the results obtained in this paper. The constructed piecewise polynomial function possesses the continuity property and new approximation properties.     

On the depth of a planar graph

In this paper, we have defined the concept of the depth of a planar graph. We show that, if G is a simple finite planar graph with p vertices and q edges and q > 3(p ? 1) ? p/2^s?1, then the depth of G is at least equal to s.     

The planar algebra of group-type subfactors

If G is a countable, discrete group generated by two finite subgroups H and K and P is a II₁ factor with an outer G-action, one can construct the group-type subfactor PH⊂P?K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones' planar operad is determined explicitly.     

Generators for finite depth subfactor planar algebras

We show that a subfactor planar algebra of finite depth k is generated by a single s-box, for \(s \leq \min \{k+4,2k\}\).     

Self-similar planar curves related to modified Korteweg-de Vries equation

We exhibit a time reversible geometric flow of planar curves which can develop singularities in finite time within the uniform topology. The example is based on the construction of selfsimilar solutions of modified Korteweg-de Vries equation of a given (small) mean.     

CASCADIC MULTIGRID METHODS FOR MORTAR WILSON FINITE ELEMENT METHODS ON PLANAR LINEAR ELASTICITY

Cascadic multigrid technique for mortar Wilson finite element method of homogeneous boundary value planar linear elasticity is described and analyzed. First the mortar Wilson finite element method for planar linear elasticity will be analyzed, and the error estimate under L2 and H1 norm is optimal. Then a cascadic multigrid method for the mortar finite element discrete problem is described. Suitable grid transfer operator and smoother are developed which lead to an optimal cascadic multigrid method. Finally, the computational results are presented.     

Planar functions over fields of characteristic two

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over \(\mathbb {Z}_{4}\) . We then specialise to planar monomial functions f(x)=cx ^t and present constructions and partial results towards their classification. In particular, we show that t=1 is the only odd exponent for which f(x)=cx ^t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry.     

Construction of asymmetric connectors of depth two

An (n,N)-connector of depth d is an acyclic digraph with n inputs and N outputs in which for any injective mapping of input vertices into output vertices there exist n vertex disjoint paths of length at most d joining each input to its corresponding output. In this paper we consider the problem of construction of sparse depth two connectors with n?N. We use posets of star products and their matching properties to construct such connectors. In particular, this gives a simple explicit construction for connectors of size O(Nlogn/loglogn). Thus our earlier idea to use other posets than the family of subsets of a finite set was successful.     

Zimmermann type cancellation in the free Faà di Bruno algebra

Haiman and Schmitt showed that one can use the antipode SF of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode SH of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for SF are labeled by trees. By contrast, the non-commuting monomials of SH are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.     

How not to characterize planar-emulable graphs

We investigate the question of which graphs have planar emulators (a locally-surjective homomorphism from some finite planar graph)—a problem raised already in Fellows? thesis (1985) and conceptually related to the better known planar cover conjecture by Negami (1986). For over two decades, the planar emulator problem lived poorly in a shadow of Negami?s conjecture—which is still open—as the two were considered equivalent. But, at the end of 2008, a surprising construction by Rieck and Yamashita falsified the natural “planar emulator conjecture”, and thus opened a whole new research field. We present further results and constructions which show how far the planar-emulability concept is from planar-coverability, and that the traditional idea of likening it to projective embeddability is actually very out-of-place. We also present several positive partial characterizations of planar-emulable graphs.     

Complete embedded minimal surfaces of finite total curvature with planar ends of smallest possible order

Using a method by Traizet (J Differ Geom 60:103–153, 2002), which reduces the construction of minimal surfaces via the Weierstraß Theorem and the implicit function theorem to solving algebraic equations in several complex variables, we will show the existence of complete embedded minimal surfaces of finite total curvature with planar ends of least possible order.     

Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited

We show the David–Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.     

Realizing finite edge‐transitive orientable maps

J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) ( 5 ) established that the automorphism group of an edge‐transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge‐transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3,11 (mod 12), there exist finite, orientable, edge‐transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge‐transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge‐transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge‐transitive map, namely, as ?_n × ?₂ where n ≡ 2 (mod 4). © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 1–34, 2001     

On Planar Toeplitz Graphs

We describe several classes of finite, planar Toeplitz graphs and present results on their chromatic number. We then turn to counting maximal independent sets in these graphs and determine recurrence equations and generating functions for some special cases.     

Calculation of the Stress–Strain State of Plates Containing Cracks

A procedure for numerical modeling of the behavior of various planar samples of construction materials containing static and moving cracks under dynamic loading is formulated on the basis of the model of a linearly elastic solid and the application of a continuum mechanics approach. Calculations are carried out by a modified finite element method using a Lagrangian differencing scheme. The results of test calculations are given for comparison with data from a physical experiment. The comparison favorably supports the reliability of the results.