Some algebraic identities for the α-permanent

The permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers α and β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with some discussion and a conjecture for the computational complexity of the α-permanent, and provide some numerical illustrations.     

On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all α-limit sets is closed.     

Koszul duality in deformation quantization and Tamarkin's approach to Kontsevich formality

Let α be a quadratic Poisson bivector on a vector space V. Then one can also consider α as a quadratic Poisson bivector on the vector space V^∗[1]. Fixed a universal deformation quantization (prediction of some complex weights to all Kontsevich graphs [12]), we have deformation quantization of the both algebras S(V^∗) and Λ(V). These are graded quadratic algebras, and therefore Koszul algebras. We prove that for some universal deformation quantization, independent on α, these two algebras are Koszul dual. We characterize some deformation quantizations for which this theorem is true in the framework of the Tamarkin's theory [19].     

Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems

In this paper, we prove that a quasi-periodic linear differential equation in sl(2,?) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number and give some applications for Schrödinger operators. Our proof is a generalized KAM type iteration adapted to all irrational α.     

On zeros of the Alexander polynomial of an alternating knot

We prove that for any zero α of the Alexander polynomial of a two-bridge knot, −3<Re(α)<6. Furthermore, for a large class of two-bridge knots we prove −1<Re(α).     

Regularity for the nonlinear Signorini problem

We study the regularity of the solution to a fully nonlinear version of the thin obstacle problem. In particular we prove that the solution is C1,α for some small α>0. This extends a result of Luis Caffarelli of 1979. Our proof relies on new estimates up to the boundary for fully nonlinear equations with Neumann boundary data, developed recently by the authors.     

On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order α is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order α is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).     

A Bombieri-Vinogradov type exponential sum result with applications

We prove a Bombieri-Vinogradov type result for linear exponential sums over primes. Then we apply it to show that, for any irrational α and some θ>0, there are infinitely many primes p such that p+2 has at most two prime factors and ‖αp+β‖<p−θ.     

Existence for the α-patch model and the QG sharp front in Sobolev spaces

We consider a family of contour dynamics equations depending on a parameter α with 0<α?1. The vortex patch problem of the 2-D Euler equation is obtained taking α→0, and the case α=1 corresponds to a sharp front of the QG equation. We prove local-in-time existence for the family of equations in Sobolev spaces.     

Cauchy problems for fractional differential equations with Riemann–Liouville fractional derivatives

In this paper, we are concerned with Cauchy problems of fractional differential equations with Riemann–Liouville fractional derivatives in infinite-dimensional Banach spaces. We introduce the notion of fractional resolvent, obtain some its properties, and present a generation theorem for exponentially bounded fractional resolvents. Moreover, we prove that a homogeneous α-order Cauchy problem is well posed if and only if its coefficient operator is the generator of an α-order fractional resolvent, and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous α-order Cauchy problem.     

Dendrites with a countable closure of the set of end points

We investigate the problem of existence of universal elements in some families of dendrites with a countable closure of the set of end points. In particular, we prove that for each integer κ?3 and for each ordinal α?1 there exists a universal element in the family of all dendrites X such that ord(X)?κ and the α-derivative of the set clXE(X) contains at most one point.     

Existence d'une mesure de Sinai—Ruelle—Bowen pour des systèmes non uniformément hyperboliques

Let M be a smooth Riemannian manifold, and ƒ be a Csu1 + α-diffeomorphism of M onto itself Let Λ be a Pesin's set (see [5]). We prove that under some assumptions on Λ, and if the system (M, ƒ) is conservative for the Lebesgue measure, then there exists a SRB-measure, ƒ -invariant and σ-finite.     

Boundary Behavior of α-Harmonic Functions on the Complement of the Sphere and Hyperplane

We study α-harmonic functions on the complement of the sphere and on the complement of the hyperplane in Euclidean spaces of dimension bigger than one, for α?∈?(1,2). We describe the corresponding Hardy spaces and prove the Fatou theorem for α-harmonic functions. We also give explicit formulas for the Martin kernel of the complement of the sphere and for the harmonic measure, Green function and Martin kernel of the complement of the hyperplane for the symmetric α-stable Lévy processes. Some extensions for the relativistic α-stable processes are discussed.     

Inductive and projective limits of Banach spaces of measurable functions with order unities with respect to power parameter

We prove that a measurable function f is bounded and invertible if and only if there exist at least two equivalent norms by order unit spaces with order unities f^α and f^β with α > β > 0. We show that it is natural to understand the limit of ordered vector spaces with order unities f^α (α approaches to infinity) as a direct sum of one inductive and one projective limits. We also obtain some properties for the corresponding limit topologies.     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

Fixed Points of Set Functors: How Many Iterations are Needed?

The initial algebra for a set functor can be constructed iteratively via a well-known transfinite chain, which converges after a regular infinite cardinal number of steps or at most three steps. We extend this result to the analogous construction of relatively initial algebras. For the dual construction of the terminal coalgebra Worrell proved that if a set functor is α-accessible, then convergence takes at most α + α steps. But until now an example demonstrating that fewer steps may be insufficient was missing. We prove that the functor of all α-small filters is such an example. We further prove that for β ≤ α the functor of all α-small β-generated filters requires precisely α + β steps and that a certain modified power-set functor requires precisely α steps. We also present an example showing that whether a terminal coalgebra exists at all does not depend solely on the object mapping of the given set functor. (This contrasts with the fact that existence of an initial algebra is equivalent to existence of a mere fixed point.)     

Local Solution and Extension to the Calabi Flow

We consider the local solution of the Calabi flow for rough initial data. In particular, we prove that for any smooth metric, there is a C ^α neighborhood such that the Calabi flow has a short time solution for any C ^α metric in the neighborhood. We also prove that on a compact Kähler surface, if the evolving metrics of the Calabi flow are all L ^∞ equivalent, then the Calabi flow exists for all time and converges to an extremal metric subsequently.     

Subharmonic bifurcation from infinity

We are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods 2πn) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points ±αi with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods 2πn exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term.     

On Einstein Matsumoto metrics

We study a special class of Finsler metrics,namely,Matsumoto metrics F=α2α-β,whereαis a Riemannian metric andβis a 1-form on a manifold M.We prove that F is a(weak)Einstein metric if and only ifαis Ricci flat andβis a parallel 1-form with respect toα.In this case,F is Ricci flat and Berwaldian.As an application,we determine the local structure and prove the 3-dimensional rigidity theorem for a(weak)Einstein Matsumoto metric.