Area density and regularity for soap film-like surfaces spanning graphs

For a given boundary Γ in Rⁿ consisting of arcs and vertices, with two or more arcs meeting at each vertex, we treat the problem of estimating the area density of a soap film-like surface Σ spanning Γ. Σ is assumed locally to minimize area, or more generally, to be strongly stationary for area with respect to Γ. We introduce a notion of total curvature (Γ) for such graphs, or nets, Γ. We show that 2π times the area density of Σ at any point is less than or equal to (Γ). For n=3, these density estimates imply, for example, that if (Γ)≤3.649π, then the only possible singularities of a piecewise smooth (M,0,δ)-minimizing set Σ are curves, along which three smooth sheets of Σ meet with equal angles of 120^°. Second author supported in part by NSF grant 00-71862.     

Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R ^N , N=2,3, surrounded by a thin layer Σ _ε , along a part Γ₂ of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ _ε with Reynolds’ number of order 1/ε in Σ _ε . Using Γ-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface Γ₂. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

On the Classification of Twisting Maps Between Kn and Km

We define the notion of admissible pair for an algebra A, consisting on a couple (Γ, R), where Γ is a quiver and R a unital, splitted and factorizable representation of Γ, and prove that the set of admissible pairs for A is in one to one correspondence with the points of the variety of twisting maps T_Aⁿ:=T(Kⁿ,A)\mathcal{T}_A^n:=\mathcal{T}(K^n,A). We describe all these representations in the case A = K ^m .     

A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces

Let Σ _g,1 be an oriented compact surface of genus g with 1 boundary component, and Γ _g,1 the mapping class group of Σ _g,1 . We define a bigraded series of cohomology classes m _i,j ∈H ^2i+j−2 (Γ _g,1 ;⋀ ^j H ₁(Σ _g,1 ;ℤ)), 2i+j−2≥1,i,j≥0. When j=0, the class m _i+1,0 is the i-th Morita- Mumford class [Mo][Mu]. It is proved that H ^r (Γ _g,1 ;⋀ ^s H ₁(Σ _g,1 ;ℚ)) is generated by m _i,j 's for the case r+s=2 and the case g≥5 and (r,s)=(1,3). Especially the Johnson homomorphism extended to the whole mapping class group by Morita [Mo3] has an implicit representation by the classes m _0,3 and m _0,2 m _1,1 over ℚ. Oblatum 28-IV-1995 & 8-II-1997     

On Σ-nilpotent ideals of topological PI-rings

We show that under certain conditions on the topology of a faithful module M over a topological PI-ring R, if M has at most countable dual topological Krull dimension, then the closure of the sum of all Σ-nilpotent ideals of the ring R is a Σ-nilpotent ideal too, and in the case of a bounded ring R its topological Baer radical is Σ-nilpotent. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 111–118, 2006.     

The Blow-up Locus of Heat Flows for Harmonic Maps

Abstract Let M and N be two compact Riemannian manifolds. Let u _k (x, t) be a sequence of strong stationary weak heat flows from M×R ⁺ to N with bounded energies. Assume that u _k→u weakly in H ^{1, 2}(M×R ⁺, N) and that Σ^t is the blow-up set for a fixed t > 0. In this paper we first prove Σ^t is an H ^m−2-rectifiable set for almost all t∈R ⁺. And then we prove two blow-up formulas for the blow-up set and the limiting map. From the formulas, we can see that if the limiting map u is also a strong stationary weak heat flow, Σ^t is a distance solution of the (m− 2)-dimensional mean curvature flow [1]. If a smooth heat flow blows-up at a finite time, we derive a tangent map or a weakly quasi-harmonic sphere and a blow-up set ∪_t<0Σ^t× {t}. We prove the blow-up map is stationary if and only if the blow-up locus is a Brakke motion. This work is supported by NSF grant     

On representation of the solution of the Neumann problem in a domain with peak by the harmonic simple layer potential

The problem of finding a solution of the Neumann problem for the Laplacian in the form of a simple layer potential Vρ with unknown density ρ is known to be reducible to a boundary integral equation of the second kind to be solved for density. The Neumann problem is examined in a bounded n-dimensional domain Ω⁺ (n > 2) with a cusp of an outward isolated peak either on its boundary or in its complement Ω⁻ = R ⁿ \Ω⁺. Let Γ be the common boundary of the domains Ω^±, Tr(Γ) be the space of traces on Γ of functions with finite Dirichlet integral over R ⁿ , and Tr(Γ)* be the dual space to Tr(Γ). We show that the solution of the Neumann problem for a domain Ω⁻ with a cusp of an inward peak may be represented as Vρ⁻, where ρ⁻ ∈ Tr(Γ)* is uniquely determined for all Ψ⁻ ∈ Tr(Γ)*. If Ω⁺ is a domain with an inward peak and if Ψ⁺ ∈ Tr(Γ)*, Ψ⁺ ⊥ 1, then the solution of the Neumann problem for Ω⁺ has the representation u ⁺ = Vρ⁺ for some ρ⁺ ∈ Tr(Γ)* which is unique up to an additive constant ρ₀, ρ₀ = V ⁻¹(1). These results do not hold for domains with outward peak.     

Sur le problème de Plateau pour un quadrilatère variable qui peut acquérir un point double

Résumé Soit Γ un quadrilatère variable, dont deux c?tés opposés A₂A₃, A₁A₄ sont dans les plans x₃=c, x₃=−c. Quand c tend vers0, Γ tend vers un quadrilatèreΓ ⁰ présentant un point double, A⁰. Le travail étudie la représentation conforme sur le demi-plan R(ix)<0 (ou sur le cercle - unité) de la surface minimale ∑ passant par Γ. Il montre (§ I) que si les affixes x de A₁, A₂, A₄ sont 0, 1, ∞, l'affixe de A₃ sera ɛ⁻², où ɛ tend vers 0 avec c. Il étudie (§ II) l'allure pour ɛ tendant vers 0 des intégrales canoniques de l'équation linéaire du problème. La forme de la relation entre ɛ et c est indiquée au n^o 19; on montre que dans la région de striction ɛ |x| reste borné et que la surface ∑ y est assimilable à une surface minimale simple: la surface de vis à filet carré. La représentation conforme de l'une des deux régions de ∑ qui tendent à se séparer l'une de l'autre tend à envahir tout le demi-plan (ou tout le cercle-unité). Les représentations conformes de ∑ pour c>0 et c<0 ne sont pas analytiquement distinctes (n⁰ 20). A titre d'exemple, on étudie (n⁰ 21) le cas où ∑ possède un axe de symétrie. A M. Enrico Bompiani pour son Jubilé scientifique     

A sum formula for the Casson-Walker invariant

We study the following question: How does the Casson-Walker invariant λ of a rational homology 3-sphere obtained by gluing two pieces along a surface depend on the two pieces? Our partial answer may be stated as follows. For a compact oriented 3-manifold A with boundary ∂A, the kernel L _A of the map from H ₁(∂A;Q) to H ₁(A;Q) induced by the inclusion is called the Lagrangian of A. Let Σ be a closed oriented surface, and let A, A′, B and B′ be four rational homology handlebodies such that ∂A, ∂A′, −∂B and −∂B′ are identified via orientation-preserving homeomorphisms with Σ. Assume that L _A = L _{A ′} and L _B = L _{B ′} inside H ₁(Σ;Q) and also assume that L _A and L _B are transverse. Then we express

On inhomogeneous diophantine approximation and Hausdorff dimension

Let Γ = Z A + Z ⁿ  ⊂ R ⁿ be a dense subgroup of rank n + 1 and let [^(w)] \hat{w} (A) denote the exponent of uniform simultaneous rational approximation to the generating point A. For any real number v ≥  [^(w)] \hat{w} (A), the Hausdorff dimension of the set B _v of points in R ⁿ that are v-approximable with respect to Γ is shown to be equal to 1/v.     

Unique solvability of the Cauchy problem for a system of Maxwell equations with initial data on a timelike surface

In this paper, a uniqueness theorem on the solution of the Cauchy problem for a system of Maxwell equations is proved in the case where the coefficients ε and μ are analytic functions of coordinates and the initial data are given on an “immovable” surface Σ=Γ×[0≤t≤2T], where Γ is an analytic surface in R³. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 30–37. Translated by N.S. Zabavnikova.     

On infinite real trace rational languages of maximum topological complexity

We consider the set ℝ^ω(Γ, D) of infinite real traces, over a dependence alphabet (Γ,D) with no isolated letter, equipped with the topology induced by the prefix metric. We prove that all rational languages of infinite real traces are analytic sets. We also reprove that there exist some rational languages of infinite real traces that are analytic but non-Borel sets; in fact, these sets are even Σ ₁ ¹ -complete, hence have maximum possible topological complexity. For this purpose, we give an example of a Σ ₁ ¹ -complete language that is fundamentally different from the known example of a Σ ₁ ¹ -complete infinitary rational relation given by Finkel (2003). Bibliography: 35 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 205–223.     

Generalizations of coatomic modules

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re _jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R _R(_RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ _i=1 ⁿ M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Smooth and path connected Banach submanifold Σ r of B(E,F) and a dimension formula in B(ℝ n ,ℝ m )

Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ _r the set of all operators of finite rank r in B(E,F), and Σ _r ^# the number of path connected components of Σ _r . It is known that Σ _r is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ _r . In this paper,the equality Σ _r ^# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ _r is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T _A Σ _r = {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ _r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of Σ _r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = ℝ ⁿ and F = ℝ ^m , then Σ _r is a smooth and path connected submanifold of B(ℝ ⁿ , ℝ ^m ) and its dimension is dimΣ _r = (m+n)r−r ² for each r, 0 <- r < min {n,m}. Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).     

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

Bipartite Distance-regular Graphs, Part I

Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra T=T(x) is the subalgebra of Mat _X(C) generated by A, E ^* ₀, E ^* ₁,…,E ^* _D, where A denotes the adjacency matrix for Γ and E ^* _i denotes the projection onto the i ^TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE ^* _i W≤1 for 0≤i≤Di. The endpoint of W is min{i|E ^* _i W≠0}. We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W. We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ are determined by the diameter D of Γ and the set of ordered pairs

Maximal sets in α-recursion theory

Let α be an admissible ordinal, and leta ^* be the Σ₁-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α^* if α^*−M is undounded and for any α-r.e. setA, eitherA∩(α^*-M) or (⇌^*-A)∩(α^*-M) is unbounded in α^*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α^*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ ^L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks.