A global Inverse Map Theorem and biLipschitz maps in the Heisenberg group

Abstract We prove a global Inverse Map Theorem for a map f from the Heisenberg group into itself, provided the Pansu differential of f is continuous, non singular and satisfies some growth conditions at infinity. An estimate for the Lipschitz constant (with respect to the Carnot–Carathéodory distance in ) of a continuously Pansu differentiable map is included. This gives a characterization of (continuously Pansu differentiable) globally biLipscitz deformations of in term of a pointwise estimate of their differential. Keywords: Inverse problem, Heisenberg group     

Beurling-Lax representations using classical Lie groups with many applications II: GL(n,©) and Wiener-Hopf factorization

The classical Beurling-Lax-invariant subspace theorem characterizes the full range simply invariant subspacesM of L _n ² as being of the formM=H _n ² where L _n×n is a phase function. Here L _n ² is the Hilbert space of measurable ⁿ-valued functions on the unit circle {e^it|0t2} which are square-integrable in norm, H _n ² is the subspace of functions in L _n ² with analytic continuation to the interior of the disk {zz|<1}, L _n×n ² is the space of measurable essentially bounded n×n matrix functions on the unit circle, and a phase function is one whose values (e^it) are unitary for a.e. t (i.e., (e^it) is in the Lie group U(n) a.e.). Halmos extended this to L ² . A subspace ML _n ² is said to beinvariant if e^it MM,simply invariant if in addition e^ikt M=(0), andfull range if 0} $$ " align="middle" border="0"> e^–iNt M is dense in L _n ² . In the Beurling-Lax representationM=H _n ² ,M uniquely determines up to a unitary constant factor on the right if one insists that (e^it)U(n). If one demands only that (e^it) GL(n,) (the group of invertible n×n complex matrices), however, there is considerably more freedom; in fact H _n ² =₁H _n ² where ₁ F and FL _n×n is outer with inverse F^–1L _n×n . More generally, we have H _n ² =[₁H _n ]^– whenever ₁=F and F is outer with F and F^–1 in L _n×n ² . (An FL _n×n ² will be said to beouter if FH _n is a dense subset of H _n ² .) In particular one can use this freedom to obtain representationsM=[H _n ]^– where the representor has values (e^it) in other matrix Lie groups. This program was carried out in accompanying work of the authors [B-H1-4] for the classical simple Lie groups U(m,n), O(p,q), O^*(2n), Sp(n,C), Sp(n,R), Sp(p,q), O(n,C), GL(n,R), U^*(2n), GL(n,R) and SL(n,C) and many applications were given. In this paper we give a natural theorem for GL(n,), by introducing the extra structure of preassigning the spaceM ^x=[H _n ]^– as well asM=[H _n ]^–. The theorems in [B-H1-4] can be derived by specializing our main result here for GL(n,) to the various subgroups which we listed.Both authors are partially supported by the National Science Foundation.     

On the reductibility of the general existence problem for closed convex surfaces to Miranda's equation

It is proved that the solution of the general existence problem for closed convex surfaces with prescribed local propertiesf(R ₁ R ₂,R ₁+R ₂,n)=(n) can be obtained as the solution of Miranda's equationR ₁ R ₂+(f)+cn=((n),(n)) with right-hand side depending on the unknown surface under the hypothesis that the latter satisfies the closure condition , where is the unit sphere andd is its element of area.Translated from Ukrainskií Geometricheskií Sbornik, Issue 29, 1986, pp. 103–112.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

The isotone mappings on Ockham algebras

In this paper we shall describe some algebraic concepts of Ockham algebras. We show that, ifL, M K _1,1 thenH(S(L),S(M)) S(H(L,M)).     

Galois Module Structure and the γ-Filtration

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the -filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic in the Grothendieck group of finitely generated Z[G]-modules, when X is a curve over Z and G has prime order.     

The Structural Nature of the Nerve Functor for n-Groupoids

Given a left exact category B, the construction of the nerve functor _n for n-groupoids in B is related to a certain property of the category S-S i m p l _{n – 1} B of the split (n – 1)-truncated simplicial objects in B, which allows us to define the split n-truncated simplicial objects in B completely internally to S-S i m p l _{n – 1} B and thus to construct intrisincally from it the category S-S i m p l _n B.     

Unit-distance graphs,graphs on the integer lattice and a Ramsey type result

Summary Let (R ², 1) denote the graph withR ² as the vertex set and two vertices adjacent if and only if their Euclidean distance is 1. The problem of determining the chromatic number(R ², 1) is still open; however,(R ², 1) is known to be between 4 and 7. By a theorem of de Bruijn and Erdös, it is enough to consider only finite subgraphs of (R ², 1). By a recent theorem of Chilakamarri, it is enough to consider certain graphs on the integer lattice. More precisely, forr > 0, let (Z ²,r, ) denote a graph with vertex setZ ² and two vertices adjacent if and only if their Euclidean distance is in the closed interval [r – ,r + ]. A simple graph is faithfully -recurring inZ ² if there exists a real numberd > 0 such that, for arbitrarily larger, G is isomorphic to a subgraph of (Z ²,r, ) in which every pair of vertices are at least distancedr apart. Chilakamarri has shown that, ifG is a finite simple graph, thenG is isomorphic to a subgraph of (R ², 1) if and only ifG is faithfully -recurring inZ ². In this paper we prove that(Z ²,r, ) 5 for integersr 1. We also prove a Ramsey type result which states that for any integerr > 1, and any coloring ofZ ² either there exists a monochromatic pair of vertices with their distance in the closed interval [r – ,r + ] or there exists a set of three vertices closest to each other with three distinct colors.     

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksA _ii irreducible. LetS be the set of indices such that the diagonal blockA is singular. We define the singular graph ofA to be the setS with partial order defined by > if there exists a chain of non-zero blocksA _i, A_ij, , A_l.Let ₁ be the set of maximal elements ofS, and define thep-th level _p ,p = 2, 3, , inductively as the set of maximal elements ofS \( _{1 p-1}). Denote by _p the number of elements in _p . The Weyr characteristic (associated with 0) ofA is defined to be (A) = ( ₁, ₂,, _h ), where ₁ + + _p = dim KerA ^p ,p = 1, 2, , and _h > 0, _h+1 = 0.Using a special type of basis, called anS-basis, for the generalized eigenspaceE(A) of 0 ofA, we associate a matrixD withA. We show that(A) = ( ₁, , _h) if and only if certain submatricesD _p,p+1 ,p = 1, , h – 1, ofD have full column rank. This condition is also necessary and sufficient forE(A) to have a basis consisting of non-negative vectors, which is a Jordan basis for –A. We also consider a given finite partially ordered setS, and we find a necessary and sufficient condition that allM-matricesA with singular graphS have(A) = ( ₁, , _h). This condition is satisfied ifS is a rooted forest.The work of the second-named author was partly supported by the National Science Foundation, under grant MPS-08618 A02.     

The uniqueness of the adjoint operation I

Let H be a complex, infinite-dimensional Hilbert space. Let B(H) denote the set of bounded linear operators on H. This paper contains a nonlinear characterization of the adjoint operation on B(H). The statement of this result is:THEOREM:Let h: B(H) B(H)be a function such that h(I)0.Then h(ST)=h(T)h(S)and h(S)S0for all elements Sand Tof B(H)if and only if h(S)=S^* for all S B(H).     

On the Signs of Fourier Coefficients of Cusp Forms

Let be a discrete subgroup of SL(2, ) with a fundamental region of finite hyperbolic volume. (Then, is a finitely generated Fuchsian group of the first kind.) Let 0}} {a(n)e^{2\pi i(n + {\kappa })z/{\lambda }} } ,{ }z \in \mathcal{H}.$$ " align="middle" border="0"> be a nontrivial cusp form, with multiplier system, with respect to . Responding to a question of Geoffrey Mason, the authors present simple proofs of the following two results, under natural restrictions upon . Theorem. If the coefficients a(n) are real for all n, then the sequence {a(n)} has infinitely many changes of sign. Theorem. Either the sequence {Re a(n)} has infinitely many sign changes or Re a(n) = 0 for all n. The same holds for the sequence {Im a(n)}.     

A study of semiiterative methods for nonsymmetric systems of linear equations

Summary Given a nonsingular linear systemA x=b, a splittingA=M–N leads to the one-step iteration (1)x _m =T X _m–1 +c withT:=M ^–1N andc:=M ^–1 b. We investigate semiiterative methods (SIM's) with respect to (1), under the assumption that the eigenvalues ofT are contained in some compact set of , with 1. There exist SIM's which are optimal with respect to , but, except for some special sets , such optimal methods are not explicitly known in general. Using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory, we describe here SIM's which are asymptotically optimal with respect to . It is shown that Euler methods, extensively studied by Niethammer-Varga [NV], are special SIM's. Various algorithms for SIM's are also derived here. A 1-1 correspondence between Euler methods and SIM's, generated by generalized Faber polynomials, is further established here. This correspondence gives that asymptotically optimal Euler methods are quite near the optimal SIM's.Dedicated to Professor Karl Zeller (Universität Tübingen) on the occasion of his sixtieth birthday (December 28, 1984)     

Abel maps of Gorenstein curves

Abstract  For a Gorenstein curve X and a nonsingular point P∈ X, we construct Abel maps and , where J_Xⁱ is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L⊂ X meeting in separating nodes. Finally we establish a connection with Seshadri’s moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1). Keywords Abel map, Torsion-free rank-1 sheaf, Compactified Jacobian, Gorenstein singularity Mathematics Subject Classification (2000) 14H40, 14H60     

Local Lipschitz continuity of the stop operator

On a closed convex set Z in ^N with sufficiently smooth (W ^2,) boundary, the stop operator is locally Lipschitz continuous from W ^1,1([0,T]^N) × Z into W ^1,1([0,T],^N). The smoothness of the boundary is essential: A counterexample shows that C ¹-smoothness is not sufficient.     

Solutions périodiques généralisées à énergie fixée pour un système hamiltonien singulier du premier ordre

We prove the existence of a generalized periodic solution (collision) of the first order singular Hamiltonian system (p,q) = (−∂H/∂q, ∂H/∂p) satisfying H(p(t), q(t)) = h where H(p, q) has a singularity at q = 0 like 1/3¦p¦^A − 1/¦q¦⁰ with 1 ≤ α < β and β ≥ 2.     

Chromatic uniqueness of certain complete tripartite graphs

Let P(G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P(H, λ) = P(G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniqueness of certain complete t-partite graphs. Ars Comb., 92, 353–376 (2009)] show that K(p − k, p − i, p) for i = 0, 1 are chromatically unique if p ≥ k + 2 ≥ 4. In this paper, we show that if 2 ≤ i ≤ 4, the complete tripartite graph K(p − k, p − i, p) is chromatically unique for integers k ≥ i and p ≥ k ²/4 + i + 1.     

On fixed points of conformal pseudogroups

We show in this paper Theorem 2 that if (H, H ₁) is a pseudogroup generated by a finite numberH ₁ of germs of conformal diffeomorphisms of defined on a sufficiently small discD, which is not linearizable and such that the linear group (L,H ₁)={g(0)/g(H,H ₁)}* is dense in *, then the set of fixed points of the pseudogroup (H, H ₁) is dense inD. This implies the abundance of distinct homotopy classes of loops in leaves of foliations defined in ² by generic polynomial vector fields as well as for germs of holomorphic vector fields in ² beginning with generic jets, both of degree at least 2. These homotopy classes may be realized arbitrarily close to the line at infinity or to 0, respectively. This shows the genericity of polynomial vector fields with infinite Petrovsky-Landis genus ([5]).The idea of the proof is very simple. Ifg is a non-linear conformal diffeomorphism with multiplier =g'(0), then the map obtained by the composition ofg and the linear map with multiplier ^–1 will have at 0 a fixed point of multiplicity at least 2. Since we may approximate ^–1 by elementsh in the pseudogroup and the multiplicity of fixed points satisfy a law of conservation of number, we obtain thath o g has fixed points close to 0. These fixed points appear as a by product of the relative nonlinearity of the generators of the pseudogroup, since linearizable pseudogroups have 0 as an isolated fixed point. The fixed points obtained are not conjugate since they have distinct multipliers.The main technical tool is the angular derivative introduced in [8]. It allows one to split the search for fixed points into two parts: One is to obtain a contraction and the other is to return arbitrarily close to the starting point without modifying the property of contraction. This is carried out since the angular derivative is multiplicative for compositions and is identically 1 for linear maps.Supported by CONACYT-CNRS and CONACYT 3398-E9307.     

Characteristic Functions for Multicontractions and Automorphisms of the Unit Ball

A multicontraction on a Hilbert space is an n-tuple of operators T = (T₁,..., T_n) acting on , such that . We obtain some results related to the characteristic function of a commuting multicontraction, most notably discussing its behaviour with respect to the action of the analytic automorphisms of the unit ball.     

On maximal ranges of polynomial spaces in the unit disk

Let be a domain in C, 0, and let _n ⁰ () be the set of polynomials of degreen such thatP(0)=0 andP(D), whereD denotes the unit disk. The maximal range _n is then defined to be the union of all setsP(D),P _n ⁰ (). We derive necessary and, in the case of ft convex, sufficient conditions for extremal polynomials, namely those boundaries whose ranges meet _n . As an application we solve explicitly the cases where is a half-plane or a strip-domain. This also implies a number of new inequalities, for instance, for polynomials with positive real part inD. All essential extremal polynomials found so far in the convex cases are univalent inD. This leads to the formulation of a problem. It should be mentioned that the general theory developed in this paper also works for other than polynomial spaces.Communicated by J. Milne Anderson.     

Absolute Continuity of the Spectrum of the Periodic Maxwell Operator in a Layer

The Maxwell operator in a layer is studied. It is assumed that the electric permittivity (x) and the magnetic permeability (x)are periodic along the layer. On the boundary of the layer, conditions of ideal conductivity are imposed. Under wide assumptions on (x) and (x), it is shown that the spectrum of the Maxwell operator is absolutely continuous. Bibliography: 10 titles.