On coercivity and irregularity for some nonlinear degenerate elliptic systems

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W ₀^1,p which are nowhere C ¹ on their supports. We also give examples of universally p-coercive variational integrals in W ₀^1,p for p ⩾ with L ^∞ coefficients for which uniqueminimizers under affine boundary conditions are nowhere C ¹.      

Recovery of the C ∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the ??boundary rigidity problem??. If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the Taylor series of the metric tensor (C ^?? jet) at the boundary (see Lassas et?al. (Math Ann 325:767?C793, 2003), Uhlmann et?al. (Adv Appl Math 31:379?C387, 2003)). In this paper we give a new reconstruction procedure for the C ^?? jet at non-concave points on the boundary using the localized boundary distance function. A closely related problem is the ??lens rigidity problem??, which asks whether the lens data determine metric structure uniquely. Lens data include information of boundary distance function, the lengths of all geodesics, and the locations and directions where geodesics hit the boundary. We give the first examples that show that lens data cannot uniquely determine the C ^?? jet. The example include two manifolds with the same boundary and the same lens data, but different C ^?? jets. With some additional careful work, we can find examples with different C ¹ jets, which means the boundaries in the two lens-equivalent manifolds have different second fundamental forms.     

New examples of curves with a one-dimensional family of pencils of minimal degree

We give a geometric construction of sub-linear systems on a K3 surface consisting of smooth curves C with infinitely many ${g^1_{{\rm gon}(C)}}$ ??s.     

On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Propagation des singularités des courants positifs fermés

Given a closed positive currentT on a bounded Runge open subset Ω ofC ⁿ , we study sufficient conditions for the existence of a global extension ofT toC ⁿ . WhenT has a sufficiently low density, we show that the extension is possible and that there is no propagation of singularities, i.e.T may be extended by a closed positiveC ^∞-form outside \(\bar \Omega \) . Conversely, using recent results ofH. Skoda andH. El Mir, we give examples of non extendable currents showing that the above sufficient conditions are optimal in bidegree (1, 1).     

<Emphasis Type="Italic">t</Emphasis>-Structures are Normal Torsion Theories

We characterize t-structures in stable ∞-categories as suitable quasicategorical factorization systems. More precisely we show that a t-structure ?? on a stable ∞-category C is equivalent to a normal torsion theory ?? on C, i.e. to a factorization system ?? = (??, ?) where both classes satisfy the 3-for-2 cancellation property, and a certain compatibility with pullbacks/pushouts.     

Optimal Parameters for 2-cyclic AOR

Assume A∈C^n×n is a 2-cyclic consistently ordered matrix and J is denoted as its associated block Jacobi iteration matrix. We consider the 2-cyclic AOR method for solving the consistent linear systems Ax=b. In the case that σ(J²) is either nonnegative or nonpositive, we give detailed discussion and derive definite expressions on optimal parameters and spectral radius by efficient method. Moreover, we give some numerical examples.     

The question on multipliers in L 1 and C

The connection between Belinsky?CDveyrin?CMalamud??s theorem on multipliers and some other known theorems on multipliers in the spaces L ¹ and C is clarified. It is shown that Belinsky?CDveyrin?CMalamud??s theorem can be obtained from them. The relevant examples of multipliers in the spaces L ¹ and C indicating the nonequivalence of the compared theorems are presented.     

Curves in Banach spaces which allow a C 1,BV parametrization or a parametrization with finite convexity

We give a complete characterization of those f: [0, 1] → X (where X is a Banach space) which allow an equivalent C ^1,BV parametrization (i.e., a C ¹ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for X = ? ⁿ . We present examples which show applicability of our characterizations. For example, we show that the C ^1,BV and C ² parametrization problems are equivalent for X = ? but are not equivalent for X = ? ⁿ .     

A simple proof for persistence of snap-back repellers

In this article, we show that if f has a snap-back repeller then any small C¹ perturbation of f has a snap-back repeller, and hence has Li-Yorke chaos and positive topological entropy, by simply using the implicit function theorem. We also give some examples.     

Smooth diffeomorphisms of the plane with stable periodic points in a neighborhood of a homoclinic point

We consider self-diffeomorphisms of the plane of the class C ^r (1 ?? r < ??) with a fixed hyperbolic point and a nontransversal point homoclinic to it. We present a method for constructing a set of diffeomorphisms for which the neighborhood of a homoclinic point contains countably many stable periodic points with characteristic exponents bounded away from zero.     

A Proof of the Welch and Niho Conjectures on Cross-Correlations of Binary m-Sequences

Binary m-sequences are widely applied in navigation, radar, and communication systems because of their nice autocorrelation and cross-correlation properties. In this paper, we consider the cross-correlation between a binary m-sequence of length 2^m−1 and a decimation of that sequence by an integer t. We will be interested in the number of values attained by such cross-correlations. As is well known, this number equals the number of nonzero weights in the dual of the binary cyclic code C_1,t of length 2^m−1 with defining zeros α and α^t, where α is a primitive element in GF(2^m). There are many pairs (m, t) for which C^⊥_1,t is known or conjectured to have only few nonzero weights. The three-weight examples include the following cases:We present a method of proving many of these known or conjectured results, including all of the above cases, in a unified way.     

On rigidity of critical circle maps

We give examples of analytic critical circle maps which are not C ^1+α rigid.     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

Exotic analytic structures and Eisenman intrinsic measures

Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onC ⁿ under which we understand smooth complex affine algebraic varietiers which are diffeomorphic toR ²ⁿ but not biholomorphic toC ⁿ . We also develop a new method of constructing these structures which enables us to produce exotic analytic structures onC ³ with a given number of hypersurfaces isomorphic toC ² and a family of these structures with a given number of moduli.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

Numerical existence and uniqueness proof for solutions of nonlinear hyperbolic equations

We consider a numerical method to verify the existence and uniqueness of the solutions of nonlinear hyperbolic problems with guaranteed error bounds. Using a C¹ finite element solution and an inequality constituting a bound on the norm of the inverse operator of the linearized operator, we numerically construct a set of functions which satisfy the hypothesis of Banach's fixed point theorem for a continuous map on L^p-space in a computer. We present detailed verification procedures and give some numerical examples.     

Random periodic solutions of random dynamical systems

In this paper, we give the definition of the random periodic solutions of random dynamical systems. We prove the existence of such periodic solutions for a C¹ perfect cocycle on a cylinder using a random invariant set, the Lyapunov exponents and the pullback of the cocycle.     

Non-Pointed Strongly Protomodular Theories

We give criterions for strong protomodularity and prove that the strong protomodularity of an algebraic theory is inherited by its models in a category with finite limits. We give examples of strongly protomodular theories with several constants: C ^*-algebras, rings, Heyting algebras and Boolean algebras.