Homotopy types for tamely approximable Sobolev maps between manifolds

We define a class ^p (M,N) of Sobolev maps from a manifoldM into a manifoldN, in such a way that each mapu ^p (M, N) has a well defined [p]-homotopy type, providedN satisfies a topological hypothesis. Using this, we prove the existence of minimizers in [p]-homotopy classes for some polyconvex variational problems.     

Speed of convergence as a function of given accuracy for random search methods

The essence of this article lies in a demonstration of the fact that for some random search methods (r.s.m.) of global optimization, the number of the objective function evaluations required to reach a given accuracy may have very slow (logarithmic) growth to infinity as the accuracy tends to zero. Several inequalities of this kind are derived for some typical Markovian monotone r.s.m. in metric spaces including thed-dimensional Euclidean space ^d and its compact subsets. In the compact case, one of the main results may be briefly outlined as a constructive theorem of existence: if is a first moment of approaching a good subset of-neighbourhood ofx ₀=arg maxf by some random search sequence (r.s.s.), then we may choose parameters of this r.s.s. in such a way that E c(f) In² . Certainly, some restrictions on metric space and functionf are required.     

Complete problems withL-samplable distributions

The average case complexity classes P, L-samplable and NL, L-samplable are defined. We show that Deterministic Bounded Halting is complete for P, L-samplable and that Graph Reachability is complete for NL-samplable, both problems with a universal logspace samplable distribution.     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

A note on closed ideals in rings of smooth functions

We prove that if finitely many smooth functions on a manifoldM generate a closed ideal inC (M), then they generate a closed ideal inC (M×N).     

A free convenient vector space for holomorphic spaces

In this paper we show that there exists a free convenient vector space for the case of holomorphic spaces and holomorphic maps. This means that for every spaceX with a holomorphic structure, there exists an appropriately complete locally convex vector space X and a holomorphic mapl _X:XX, such that for any vector space of the same kind the map (l _X )^*:L(X,E)(X,E) is a bijection. Analogously to the smooth case treated in [2, 5.1.1] the free convenient vector space X can be obtained as the Mackey closure of the linear subspace spanned by the image of the canonical mapX(X).In the second part of the paper we prove that in the case whereX is a Riemann surface, one hasX=(X,).     

New minimal geodesics in the group of symplectic diffeomorphisms

We compute the Hofer distance for a certain class of compactly supported symplectic diffeomorphisms of ²ⁿ. They are mainly characterized by the condition that they can be generated by a Hamiltonian flow _H ^t which possesses only constantT-periodic solutions for 0 <T 1. In addition, we show that on this class Hofer's and Viterbo's distances coincide.     

Subfactors with principal graph E 6 (1)

We characterize irreducible II₁ subfactorsN M with principal graphE ₆ ⁽¹⁾ as N=P Z ₃ P A ₄, whereA ₄ acts outerly on a factorP.     

Idempotents of Fourier multiplier algebra

We prove that if the indicator-function1 _E of a measurable setE is a Fourier multiplier in the spaceE ^p () for somep2 thenE is an open set (up to a set of measure zero).     

Arithmetic distance on compact symmetric spaces

Let M be a compact Riemannian symmetric space. Then M=G/K, where G is the identity component of the isometry group of M and K is the isotropy subgroup of G at a point. In 1965 Nagano studied and classified the geometric transformation groups of compact symmetric spaces. Roughly speaking they are larger groups L that act on M, (i) G/L; (ii) L is a Lie transformation group acting effectively on M; (iii) L preserves the symmetric structure of M; and (iv) L is simple.Using Helgason spheres, S(), the minimal totally geodesic spheres in a compact irreducible symmetric space, we define an arithmetic distance for compact irreducible symmetric spaces and prove: THEOREM. Let M=G ^p(K ⁿ ), K=, H, or R, or M=AI(n), of rank greater that 1 and dimension greater that 3, let L be the geometric transformation group of M. Let L={: MM: is a diffeomorphism and preserves arithmetic distance}. Then L=L     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

The circle problem on surfaces of variable negative curvature

In this note we study the problem of orbit counting for certain groups of isometries of simply connected surfaces with possibly variable negative curvature. We show that ifN(t) denotes the orbit counting function for a convex co-compact group of isometries then for some constantsC, h>0,N(t) Ce ^ht , ast +.     

Minimal submanifolds in a Riemannian manifold with pinched positive curvature

In this paper, we prove the following result: LetM be ann-dimensional compact curvature-invariant minimal subinanifoid immersed in a(> 2n–2/5n–2)- pinched Riemannian manifoldN. Denote by the second fundamental form ofM. If ¦¦²<3/9((5n–2)–2(n–1)), thenM is totally geodesic. This result generalizes the Simons pinching theorem.Supported by the JSPS postdoctoral fellowship and NSF of China.     

Decompositions of factor maps involving bi-closing maps

We describe all possible decompositions of a finite-to-one factor map : _A S, from an irreducible shift of finite type onto a sofic shift, into two maps =, such that the range of is a shift of finite type, and is bi-closing. We also give necessary and sufficient conditions for to be almost topologically conjugate overS to a bi-closing map.     

Compact groups on compact projective planes

LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE ₀ inE, cf. [31]. Here we consider the case E ₀. It is shown that the action of on the point spaceP is equivalent to the classical action ofE ₀. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO₂ Aut().Für H. Reiner Salzmann zum 65. Geburtstag     

The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps

We prove a global existence of solutions for the Landau-Lifshitz equation of the ferromagnetic spin chain from am-dimensional manifoldM into the unit sphereS ² of ³ and establish some new links between harmonic maps and the solutions of the Landau-Lifshitz equation.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.