Search

The aim of this paper is to develop an analytical approach to obtain the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a general Sturm-Liouville problem y ″ = q (t) y + λ m (t) y , where q is a nonnegative potential and another potential m admits to change sign. A typical example of such problems is the well-known Camassa-Holm equations with indefinite potentials, which corresponds to the case q (t) ≡ 1 4. It is shown that the solution of the minimization problems of the lowest positive periodic eigenvalues and Dirichlet eigenvalues will lead to more general distributions of potentials which have no densities with respect to the Lebesgue measure. As a result, it is very natural to choose the general setting of the measure differential equations d y • = y (t) d μ (t) + λ y d ν (t) , to understand the eigenvalues and their minimization, where μ and ν are two suitable measures. The variational characterization of lowest positive eigenvalues, together with a strong continuous dependence of eigenvalues on the potentials, will play crucial roles in our analysis. Different from the periodic case, we are able to obtain the optimal bounds for higher Dirichlet eigenvalues. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we show the existence of the first nontrivial family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in F. de la Hoz et al. (2016) [13]. [ABSTRACT FROM AUTHOR]

Abstract In this paper we obtain a very general Gauss–Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing (A , D u) when A is a bounded divergence measure vector field and u is a bounded function of bounded variation. [ABSTRACT FROM AUTHOR]

Abstract Let T f : [ 0 , 1 ] → [ 0 , 1 ] be an expanding Lorenz map, this means T f x : = f (x) (mod 1) where f : [ 0 , 1 ] → [ 0 , 2 ] is a strictly increasing map satisfying inf ⁡ f ′ > 1. Then T f has two pieces of monotonicity. In this paper, sufficient conditions when T f is topologically mixing are provided. For the special case f (x) = β x + α with β ≥ 2 3 a full characterization of parameters (β , α) leading to mixing is given. Furthermore relations between renormalizability and T f being locally eventually onto are considered, and some gaps in classical results on the dynamics of Lorenz maps are corrected. [ABSTRACT FROM AUTHOR]

Abstract The parameter space of n ordered points in projective d -space that lie on a rational normal curve admits a natural compactification by taking the Zariski closure in (P d) n. The resulting variety was used to study the birational geometry of the moduli space M ‾ 0 , n of n -tuples of points in P 1. In this paper we turn to a more classical question, first asked independently by both Speyer and Sturmfels: what are the defining equations? For conics, namely d = 2 , we find scheme-theoretic equations revealing a determinantal structure and use this to prove some geometric properties; moreover, determining which subsets of these equations suffice set-theoretically is equivalent to a well-studied combinatorial problem. For twisted cubics, d = 3 , we use the Gale transform to produce equations defining the union of two irreducible components, the compactified configuration space we want and the locus of degenerate point configurations, and we explain the challenges involved in eliminating this extra component. For d ≥ 4 we conjecture a similar situation and prove partial results in this direction. [ABSTRACT FROM AUTHOR]

We show new upper and lower bounds for the effective differential Nullstellensatz for differential fields of characteristic zero with several commuting derivations. Seidenberg was the first to address this problem in 1956, without giving a complete solution. In the case of one derivation, the first bound is due to Grigoriev in 1989. The first bounds in the general case appeared in 2009 in a paper by Golubitsky, Kondratieva, Szanto, and Ovchinnikov, with the upper bound expressed in terms of the Ackermann function. D'Alfonso, Jeronimo, and Solernó, using novel ideas, obtained in 2014 a new bound if restricted to the case of one derivation and constant coefficients. To obtain the bound in the present paper without this restriction, we extend this approach and use the new methods of Freitag and León Sánchez and of Pierce, which represent a model-theoretic approach to differential algebraic geometry. [ABSTRACT FROM AUTHOR]

A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N ≤ 2 n , the defining functions φ ( z , z ¯ , u ) of all real-analytic hypersurfaces M = { v = φ ( z , z ¯ , u ) } ⊂ C n + 1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C N + 1 satisfy an universal algebraic partial differential equation D ( φ ) = 0 , where the algebraic-differential operator D = D ( n , N ) depends on n ≥ 1 , n < N ≤ 2 n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n , N as above there exists μ = μ ( n , N ) such that a Zariski generic real-analytic hypersurface M ⊂ C n + 1 of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C N + 1 . We also provide an explicit upper bound for μ in terms of n , N . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree. [ABSTRACT FROM AUTHOR]

An integer partition of n is a decreasing sequence of positive integers that add up to [ n ] . Back in 1979 Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In 1982 Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability 0. In 1997 we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of convergence rates. The main result in this paper is that each of the probabilities is e − 0.11 log ⁡ n / log ⁡ log ⁡ n , at most. A key element of the argument is a local limit theorem, with convergence rate, for the joint distribution of the [ n 1 / 4 − ε ] tallest columns and the [ n 1 / 4 − ε ] longest rows of the Young diagram representing the random partition. [ABSTRACT FROM AUTHOR]

In this paper we describe a machinery for homological calculations of representations of FI G , and use it to develop a local cohomology theory over any commutative Noetherian ring. As an application, we show that the depth introduced by the second author in [16] coincides with a more classical invariant from commutative algebra, and obtain upper bounds of a few important invariants of FI G -modules in terms of torsion degrees of their local cohomology groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of the Hermitian–Yang–Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder–Narasimhan–Seshadri filtration, this answers a question by Bando and Siu. [ABSTRACT FROM AUTHOR]

We devise a new geometric approach to study the propagation of disturbance – compactly supported data – in reaction–diffusion equations. The method builds a bridge between the propagation of disturbance and of almost planar solutions. It applies to very general reaction–diffusion equations. The main consequences we derive in this paper are: a new proof of the classical Freidlin–Gärtner formula for the asymptotic speed of spreading for periodic Fisher–KPP equations; extension of the formula to the monostable, combustion and bistable cases; existence of the asymptotic speed of spreading for equations with almost periodic temporal dependence; derivation of multi-tiered propagation for multistable equations. [ABSTRACT FROM AUTHOR]

Abstract: This paper derives an explicit formula for Branson''s Q-curvature in even-dimensional conformal geometry. The ingredients in the formula come from the Poincaré metric in one higher dimension; hence the formula is called holographic. When specialized to the conformally flat case, the holographic formula expresses the Q-curvature as a multiple of the Pfaffian and the divergence of a natural 1-form. The paper also outlines the relation between holographic formulae for Q-curvature and a new theory of conformally covariant families of differential operators due to the second author. [Copyright &y& Elsevier]

The principal aim of this paper is to establish an iteration method on the space of resurgent functions. We discuss endless continuability of iterated convolution products of resurgent functions associated with rooted trees and derive their estimates. We show the resurgence of formal series solutions of nonlinear differential and difference equations by applying the estimates. [ABSTRACT FROM AUTHOR]

Abstract: Given a point in the real Grassmannian, it is well-known that one can construct a soliton solution to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables , , and are considered on a large scale and the time is fixed. In this paper we use several decompositions of the Grassmannian in order to gain an understanding of the contour plots of the corresponding soliton solutions. First we use the positroid stratification of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at and . Next we use the Deodhar decomposition of the Grassmannian–a refinement of the positroid stratification–to study contour plots at . More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams, and then use these Go-diagrams to characterize the contour plots of solitons solutions when . Finally we use these results to show that a soliton solution is regular for all times if and only if comes from the totally non-negative part of the Grassmannian. [Copyright &y& Elsevier]

Abstract: In this paper we extend some notions, previously defined for log-concave functions, to the larger domain of the so-called -concave functions. We begin with a detailed discussion of support functions—first for log-concave functions, and then for general -concave functions. We continue by defining mean width, and proving some basic results such as an Urysohn type inequality. Finally, we demonstrate how such geometric results can imply Poincaré type inequalities. [Copyright &y& Elsevier]

The recent work of George Andrews and Mircea Merca on the truncated version of Euler's pentagonal number theorem has opened up a new study on truncated theta series. Since then several papers on the topic have followed. The main purpose of this paper is to generalize the study to Hecke-Rogers type double series, which are associated with some interesting partition functions. Our proofs heavily rely on a formula from the work of Zhi-Guo Liu on the q -partial differential equations and q -series. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we classify the centers localized at the origin of coordinates, and their isochronicity for the polynomial differential systems in of degree d that in complex notation can be written as where j is either 0 or 1. If then is an odd integer and n is an even integer satisfying . If then is an integer and n is an integer with converse parity with d and satisfying where denotes the integer part function. Furthermore and . Note that if and , we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in K.E. Malkin (1964) , N.I. Vulpe and K.S. Sibirskii (1988) , J. Llibre and C. Valls (2009) , and if , and , we are also obtaining as a particular case the quadratic polynomial differential systems studied in N.N. Bautin (1952) , H. Zoladek (1994) . So the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system Under a general twist condition for the Hamiltonian function in terms of the difference of the Conley–Zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions. Compared with the existing work in the literature, our results do not require the Hamiltonian function to have linearization at infinity. Our results allow interactions at infinity between the Hamiltonian and the linear spectra. The general twist condition raised here seems to resemble more the spirit of Poincaré''s last geometric theorem. [Copyright &y& Elsevier]

Abstract: In this paper the ω-limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms. The analogous problem is solved in full detail for analytic flows on the sphere and the projective plane. We also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces. [Copyright &y& Elsevier]

Abstract: In this paper, we give a systematical study of the local structures and fractal indices of the limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers. For a given Pisot number in the interval , we construct a finite family of non-negative matrices (maybe non-square), such that the corresponding fractal indices can be re-expressed as some limits in terms of products of these non-negative matrices. We are especially interested in the case that the associated Pisot number is a simple Pisot number, i.e., the unique positive root of the polynomial (). In this case, the corresponding products of matrices can be decomposed into the products of scalars, based on which the precise formulas of fractal indices, as well as the multifractal formalism, are obtained. [Copyright &y& Elsevier]

The formation of thinning filaments is commonly observed previously to the break-up of a very viscous jet. This paper shows that a fluid under capillary forces cannot break-up through the uniform collapse of a filament. [Copyright &y& Elsevier]

In this article, we construct a general series for 1/π. We indicate that Ramanujan''s 1/π-series are all special cases of this general series and we end the paper with a new class of 1/π-series. Our work is motivated by series recently discovered by Takeshi Sato. [Copyright &y& Elsevier]

In this paper, we will prove an explicit dimension formula for the hyperfunction solutions to a class of holonomic D-modules. This dimension formula can be considered as a higher dimensional analogue of a beautiful theorem on ordinary differential equations due to Kashiwara (Master''s Thesis, University of Tokyo, 1970) and Komatsu (J. Fac. Sci. Univ. Tokyo Math. Sect. IA 18 (1971) 379). In the course of the proof, we will make use of a recent innovation of Schmid–Vilonen (Invent. Math. 124 (1996) 451) in the representation theory (in the theory of index theorems for constructible sheaves). [Copyright &y& Elsevier]

Given a continuous vector field λ (t , ⋅) of Sobolev class H 3 2 on the unit circle S 1 , the flow maps η = g (t , ⋅) of the differential equation { d η d t = λ (t , η) η (0 , ζ) = ζ are known to be quasisymmetric homeomorphisms. Very recently, Gay-Balmaz-Ratiu [15] conjectured that the flow curve g (t , ⋅) is in the Weil-Petersson class WP (S 1) and is continuously differentiable with respect to the Hilbert manifold structure of WP (S 1) introduced by Takhtajan-Teo [40]. The first assertion had already been demonstrated in our previous paper [36]. In this sequel to [36] , we will continue to deal with the Weil-Petersson class WP (S 1) and completely solve this conjecture in the affirmative. [ABSTRACT FROM AUTHOR]