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In this paper, we consider the following question: how many degree d curves are there in P 3 (passing through the right number of generic lines and points), whose image lies inside a P 2 , having δ nodes and one singularity of codimension k? We obtain an explicit formula for this number when δ + k ≤ 4 (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in the paper by A. Zinger [30] and which is further pursued by S. Basu and the second author in [1] , [2] and [3] . Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by S. Kleiman and R. Piene (in [17] ) and by T. Laarakker (in [19] ). We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author (in [22] ), where they compute the characteristic number of rational planar curves in P 3 with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande in [23] , regarding the enumerativity of BPS numbers for P 3 .

The norm of a partition is defined as the product of its parts. This paper aims to conduct a thorough study of norms of prime partitions which are partitions with all prime parts. The analysis begins by defining r p ( i , n ) , called the primal norm counting function, which refers to number of times i appears as a norm to the prime partitions of n. The generating function for this function is deduced, using which a wealth of intriguing relations are proved like series, product, integral representations, and recursive relations, etc. The special cases of these results bear resemblance to other results known in classical partition theory. An analogue of the Goldbach conjecture in the theory of norms is presented, and it is stressed that an explicit formula for r p ( i , n ) must be extracted. Two approaches are discussed for this purpose, and the paper is concluded with a potential scope for future work.

Most recently, some new double sequence spaces B ( M u ) , B ( C ϑ ) where ϑ = { b , b p , r , f , f 0 } and B ( L q ) for 0 q ∞ have been introduced as the domain of four-dimensional generalized difference matrix B ( r , s , t , u ) in the double sequence spaces M u , C ϑ where ϑ = { b , b p , r , f , f 0 } and L q for 0 q ∞ , and some topological properties, dual spaces, some new four-dimensional matrix classes and matrix transformations related to these spaces have also been studied by Tug and Basar and Tug (see [1] , [2] , [3] , [4] ). In this paper, we introduce new strongly almost convergent double sequence spaces B [ C f ] and B [ C f 0 ] whose B ( r , s , t , u ) -transforms are in the spaces [ C f ] and [ C f 0 ] , respectively. The main body of this paper is designed by the investigation of the following hypothesis. Firstly, we examine some topological properties and inclusion relations including the new double sequence spaces B [ C f ] and B [ C f 0 ] . Also, we determine the α-dual, β ( b p ) -dual and γ-dual of the space B [ C f ] . Finally, we give the necessary and sufficient conditions on an infinite matrix transforming from [ C f ] over C f , and we also characterize the classes ( [ C f ] ; M u ) , ( B [ C f ] ; C f ) and ( B [ C f ] ; M u ) .

In this paper we study the geometric properties of a couple of mutually orthogonal foliations with complementary dimensions. We recall that from Novikov's theorem, there is no foliation of S 3 by closed curves with integrable normal bundle. Nevertheless, for S 2 k + 1 , k ≥ 2 , Novikov's theorem is not applicable. In this paper we show that on odd-dimensional unit spheres there is no umbilical foliation with integrable normal bundle and divergence free mean curvature vector.

It is well-known that dyadic martingale transforms are a good model for Calderon–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A 2 matrix weight, then the weighted L 2 -norm of a Calderon–Zygmund operator with cancellation has the same dependence on the A 2 characteristic of W as the weighted L 2 -norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon–Zygmund operators on the A 2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderon–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon–Zygmund operators.

In this paper we describe the Hopf algebra approach to the center problem for the differential equation d v d x = ∑ i = 1 ∞ a i ( x ) v i + 1 , x ∈ [ 0 , T ] , and study some combinatorial properties of the first return map of this equation. The paper summarizes and extends previously developed approaches to the center problem due to Devlin and the author.

Let G be a locally compact group, B ( G ) the Fourier–Stieltjes algebra of G and B 0 ( G ) = B ( G ) ∩ C 0 ( G ) . The space B 0 ( G ) is a closed ideal of B ( G ) . In this paper, we study the Banach algebra B 0 ( G ) under various aspects. The main emphasis is on regularity and the existence of various kinds of approximate identities, the question of when the quotient of B 0 ( G ) modulo the Fourier algebra A ( G ) is radical, the Bochner–Schoenberg–Eberlein property and a characterization of elements in B 0 ( G ) in terms of continuity of translation properties. The paper also contains a number of illustrating examples.

In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.

This paper considers robustness for linear evolution equations with non-instantaneous impulsive effects in Banach spaces. We present sufficient conditions to guarantee robustness of nonuniform exponential contractions and exponential dichotomies with both non-instantaneous linear impulsive actions of the associated equation and linear perturbations of the associated linear evolution equation. This paper establishes a new framework to investigate robustness for linear evolution equations with non-instantaneous impulsive conditions and also generalizes results from classical impulsive equations using several nontrivial techniques.

This paper presents an improvement of Palmer's linearization theorem in [10] . Palmer's linearization theorem extended the Hartman–Grobman theorem to the nonautonomous case. It requires two essential conditions: (i) the nonlinear term is bounded and Lipschitzian; (ii) the linear system as a whole possesses an exponential dichotomy. The main purpose of this paper is to weaken assumptions (i) and (ii). Also in this paper we prove that the topologically equivalent function H ( t , x ) in the linearization theorem is always Holder continuous (and has a Holder continuous inverse), so as a result we generalize and improve Palmer's linearization theorem.

The characterization of all Darboux polynomials and rational first integrals of the generalized Lorenz system, x ˙ = a ( y − x ) , y ˙ = b x + c y − x z , z ˙ = d z + x y , was published very recently in [K. Wu, X. Zhang, Bull. Sci. Math. 136 (2012) 291–308]. In this paper we improve that work in two aspects. On the one hand, we obtain the same results in a much more straightforward way. To do that, we show the equivalence under generic conditions, c ≠ 0 , between the Lorenz system and the generalized Lorenz system by means of a linear scaling in time and coordinates. Thus, from the well-known results on Darboux polynomials and its algebraic integrability for the Lorenz system, it is direct to obtain the corresponding results for the generalized Lorenz system. On the other hand, in the case c = 0 , we find a new Darboux polynomial of the generalized Lorenz system, not detected in the above paper, which is also a first integral. In this way, we complete the list provided by the authors of all Darboux polynomials and rational first integrals of the generalized Lorenz system.

This paper is devoted to the construction of a solution for the “Inhomogeneous skew Brownian motion” equation, which first appeared in a seminal paper by Sophie Weinryb, and recently, studied by Etore and Martinez. Our method is based on the use of the Balayage formula. At the end of this paper we study a limit theorem of solutions.

This paper is concerned with a class of reflected backward stochastic differential equations (RBSDEs in short) with two barriers. The first purpose of the paper is to establish existence and uniqueness results of adapted solutions for such RBSDEs. Most of existing results on adapted solutions for RBSDEs with two barriers are heavily based on either the Mokobodski condition or other restrictive regularity conditions. In this paper, the two barriers are modeled by stochastic differential equations with coefficients satisfying the local Lipschitz condition and the linear growth condition, which enables us to weaken the regularity conditions on the boundary processes. Existence is proved by a penalization scheme together with a comparison theorem under the Lipschitz condition on the coefficients of RBSDEs. As an application, it is proved that the initial value of an RBSDE with two barriers coincides with the value function of a certain Dynkin game under Knightian uncertainty.

This article extends results contained in Buzzi et al. (2006) [4] , Llibre et al. (2007, 2008) [12] , [13] concerning the dynamics of non-smooth systems. In those papers a piecewise C k discontinuous vector field Z on R n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F : U → R a polynomial function defined on the open subset U ⊂ R n . The set F − 1 ( 0 ) divides U into subdomains U 1 , U 2 , … , U k , with border F − 1 ( 0 ) . These subdomains provide a Whitney stratification on U . We consider Z i : U i → R n smooth vector fields and we get Z = ( Z 1 , … , Z k ) a discontinuous vector field with discontinuities in F − 1 ( 0 ) . Our approach combines several techniques such as e-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an e-regularization of Z (see Sotomayor and Teixeira, 1996 [18] ; Llibre and Teixeira, 1997 [15] ). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16] ), in systems with hysteresis (Seidman, 2006 [17] ) and in mechanical systems with impacts (di Bernardo et al., 2008 [5] ).

This paper primarily grows from the paper of Llibre and Zhang [J. Llibre, X. Zhang, Polynomial first integrals for quasi-homogeneous polynomial differential systems, Nonlinearity 15 (2002) 1269–1280] with the following essential generalizations: (i) we prove that the link established in the mentioned paper between the Kowalevskaya exponents and the degree of the polynomial first integrals holds not only for ( 1 , … , 1 ) -2 type systems but also for any ( s 1 , … , s n ) - d type systems. (ii) by using different methods, we obtain necessary and sufficient conditions for planar ( s 1 , s 2 ) - d systems to have rational first integrals, whereas in the mentioned paper, only ( s 1 , s 2 ) -2 type systems and only polynomial integrability are considered. As an application of the methods and the results, we present an illustrative and well studied example to show its non-existence of polynomial first integrals.

The first Brauer-Thrall conjecture asserts that algebras of bounded representation type have finite representation type. This conjecture was solved by Roiter in 1968. The induction scheme which he used in his proof prompted Gabriel to introduce an invariant which we propose to call Gabriel-Roiter,measure. This invariant is defined for any finite length module and it will be studied in detail in this paper. Whereas Roiter and Gabriel were dealing with algebras of bounded representation type only, it is the purpose of the present paper to demonstrate the relevance of the Gabriel-Roiter measure for algebras in general, in particular for those of infinite representation type. (c) 2005 Elsevier SAS. All rights reserved.

The center problem and bifurcation of limit cycles for degenerate singular points are far to be solved in general. In this paper, we study center conditions and bifurcation of limit cycles at the degenerate singular point in a class of quintic polynomial vector field with a small parameter and eight normal parameters. We deduce a recursion formula for singular point quantities at the degenerate singular points in this system and reach with relative ease an expression of the first five quantities at the degenerate singular point. The center conditions for the degenerate singular point of this system are derived. Consequently, we construct a quintic system, which can bifurcates 5 limit cycles in the neighborhood of the degenerate singular point. The positions of these limit cycles can be pointed out exactly without constructing Poincaré cycle fields. The technique employed in this work is essentially different from more usual ones. The recursion formula we present in this paper for the calculation of singular point quantities at degenerate singular point is linear and then avoids complex integrating operations.

1. Introduction and results The main purpose of this paper is to complete one step in our previous paper [6]. That is, we passed too rapidly and incorrectly in obtaining (4.37) and (4.38) of [6]. Let us preserve all notations in [6]. In particular, we consider the following stochastic differential equation (1) { dz(t) = cJ(s(t))dw(t) + b(x(t))dt z(0) = 20 where o : R” -+ R” x R” and b : R” + R” are smooth functions with bounded derivatives of all orders, w(t) is the r-dimensional standard Brownian motion, realized on the classical Wiener space (X, H, ,u) as the coordinate process. As in [6] we consider the Stroock-Varadhan approximation series of (1) (2)

Let f = h + g ‾ be a normalized and sense-preserving convex harmonic mapping in the unit disk D . In a recent paper, Ponnusamy and Sairam Kaliraj conjectured that there is a θ ∈ [ 0 , 2 π ) such that the function h + e i θ g is convex in D . In this article, we first disprove a more flexible conjecture: “Let f = h + g ‾ be a convex harmonic mapping in the disk D . Then there is a θ ∈ [ 0 , 2 π ) such that the function h + e i θ g is starlike in D ”. In addition, we present an example to show that there exists a harmonic automorphism f = h + g ‾ of a disk such that the function h + e i θ g is convex in only one direction for θ ≠ 0 , and that the analytic function h + g is not starlike therein. The article concludes with a new conjecture.

It seems that the index theory for non-compact spaces has found its ultimate formulation in the realm of coarse spaces and K-theory of related operator algebras. Relative and partitioned index theorems may be mentioned as two important and interesting examples of this program. In this paper we formulate a combination of these two theorems and establish a partitioned-relative index theorem.

This paper is devoted to the study of existence of solutions to some integro-differential problems. We establish the existence of solutions by employing a singular perturbations method as a natural tool. The perturbed problems are classical, nonlocal, non-linear, hyperbolic and the limits of the subsequences of their solutions, in weighted Sobolev type spaces, are solutions of the main problem. The existence of a solution to the nonlocal hyperbolic problem can be ensured using the Schauder fixed point theorem.

The goal of this paper is to establish the relation between the notion of sets of finite perimeter and the theory of the Ornstein-Uhlenbeck semigroup in the setting of the Gauss space. We prove a characterization of sets of finite perimeter in terms of a short time behavior of the Ornstein-Uhlenbeck semigroup and we also give a new characterization of BV functions in terms of a near-diagonal energy in this space.

Assume E is an imaginary quadratic field and O is its ring of integers. For each positive integer m, let I m be the free Hermitian lattice of rank m over O having an orthonormal basis. For each positive integer n, let S O ( n ) be the set of all Hermitian lattices of rank n over O that can be represented by some I m . Denote by g O ( n ) the smallest positive integer g such that each Hermitian lattice in S O ( n ) can be represented by I g . In this paper, we shall provide an explicit upper bound for g O ( n ) for all imaginary quadratic fields E and all positive integers n.

This paper considers backward stochastic differential equations (BSDEs for short) on the infinite horizon under consistent nonlinear expectations which are dominated by consistent sublinear expectations. We derive the existence and uniqueness result by discretization and approximation method. Besides, we show the existence of the solution of Markovian ergodic BSDEs under G ˜ -expectations and state some applications.

In this paper we study pseudo-differential operators with operator valued symbols on the Similitude group, SIM ( 2 ) . We characterize the unique infinite-dimensional, unitary, irreducible representations of SIM ( 2 ) and using that the Fourier inversion formula is given. The operator valued Wigner transform and the Weyl transform are also studied here. Furthermore we also obtain a necessary and sufficient condition on the symbols for which the pseudo-differential operators SIM ( 2 ) are Hilbert-Schmidt and trace class operators. Finally, we also give the trace formula for these operators.

In this paper, we are mainly concerned with the following integral equations: (0.1) u ( x ) = C n ∫ R n ln ⁡ ( 1 | x − y | ) f ( y , u ( y ) ) d y + γ , x ∈ R n , where n ≥ 2 , γ ∈ R , u ∈ C ( R n ) and f ( x , u ) may change signs and satisfies some assumptions. By using the method of scaling spheres developed by Dai and Qin in [14] , we first derive nonexistence of positive solutions to the above IEs under some assumptions. Then, based on the equivalence between the above IEs and the following 2D PDEs: (0.2) − Δ u ( x ) = f ( x , u ) , x ∈ R 2 , we also obtain nonexistence of positive solutions to the 2D PDEs under some assumptions. One should note that there are no growth conditions on u and hence f ( x , u ) can grow exponentially (or even faster) on u.

Let X = P ( E 1 ) × C P ( E 2 ) , where C is any smooth irreducible curve and E 1 , E 2 are vector bundles over C . In this paper, we calculate the nef cone and pseudoeffective cone of X whenever E 1 and E 2 are semistable, and show that they coincide. We also calculate the nef cone and pseudoeffective cone of X in case both E 1 and E 2 have rank 2, and neither E 1 nor E 2 is semistable or at least one of them is semistable. As a result, in the case when both E 1 and E 2 are of rank 2 bundles on C , we get that nef cone and pseudoeffective cone of X coincide iff both E 1 and E 2 are semistable.

This paper deals with the long-time dynamics for a class of highly nonlinear fractional nonclassical diffusion equations with nonlinear colored noise and time delay defined on the whole space R n . The existence and uniqueness of tempered pullback random attractors of the equations are established in C ( [ − ρ , 0 ] , H α ( R n ) ) ( ρ > 0 and α ∈ ( 0 , 1 ) ) for polynomial growth drift and diffusion terms as well as Lipschitz time-delay terms. In a special case where the nonlinear diffusion term depends only on the space variable, the approximation of those random attractors is also investigated in C ( [ − ρ , 0 ] , H α ( R n ) ) when the correlation time of the colored approaches zero. The pullback asymptotical compactness of the solutions in C ( [ − ρ , 0 ] , H α ( R n ) ) is proved by virtue of the arguments of Arzela-Ascoli theorem, spectral decomposition as well as uniform tail-estimates developed by Wang (1999) [59] in order to surmount several difficulties caused by the lack of compact Sobolev embeddings on unbounded domains as well as the weakly dissipative distinguishing structures of the equations.

In this paper, the question of when the k-subdirect sum of two Dashnic-Zusmanovich matrices is also a Dashnic-Zusmanovich matrix is studied, and some sufficient conditions are given. Furthermore, those sufficient conditions only depend on the elements of the given matrices. Moreover, examples are presented to illustrate the corresponding results.

In this paper, we obtain some boundedness of the following general multilinear square functions $T$ with non-smooth kernels, which extend some known results significantly. $$ T(\vec{f})(x)=\big( \int_{0}^\infty \big|\int_{(\mathbb{R}^n)^m}K_v(x,y_1,\dots,y_m) \prod_{j=1}^mf_{j}(y_j)dy_1,\dots,dy_m\big|^2\frac{dv}{v}\big)^{\frac 12}. $$ The corresponding multilinear maximal square function $T^*$ was also introduced and weighted strong and weak type estimates for $T^*$ were given., 19 pages

In this paper, we establish a universal variational characterization of the non-martingale components associated with weakly differentiable Wiener functionals in the sense of Le\~ao, Ohashi and Simas. It is shown that any Dirichlet process (in particular semimartingales) is a differential form w.r.t Brownian motion driving noise. The drift components are characterized in terms of limits of integral functionals of horizontal-type perturbations and first-order variation driven by a two-parameter occupation time process. Applications to a class of path-dependent rough transformations of Brownian paths under finite $p$-variation ($p\ge 2$) regularity is also discussed. Under stronger regularity conditions in the sense of finite $(p,q)$-variation, the connection between weak differentiability and two-parameter local time integrals in the sense of Young is established., Comment: Revised version. To appear in Bulletin des Sciences Math\'ematiques. arXiv admin note: text overlap with arXiv:1707.04972, arXiv:1408.1423

R. Geremia and F. Sullivan proved that 2-uniform rotundity is equivalent to 2-uniform convexity. Pei-Kee Lin extended this result for any integer k ≥ 1 . This paper gives a different proof for the equivalence of k-uniform rotundity and k-uniform convexity for any integer k ≥ 1 .

In this paper we study transformed trigonometric series with Hausdorff averages of Fourier coefficients. We prove Hardy–Littlewood and Pitt's inequalities for such series. The corresponding results for the Hausdorff averages of the Fourier transforms are also obtained.

In this paper, we characterize the centers of the quasi-homogeneous planar polynomial differential systems of any degree n ∈ N . First, we prove that the system does not admit centers if n is even, then present the sufficient and necessary conditions that the system admits a center if n is odd. Furthermore, we provide a simple method to obtain such these systems with center for a given order.

In this paper, we study holomorphic pairs over a class of non-compact Gauduchon manifolds. We prove that the stability implies the existence of Hermitian–Yang–Mills metric, and the semi-stability implies the existence of approximate Hermitian–Yang–Mills structure. We generalize the result in Bradlow (1991) [7] to the non-compact and non-Kahler case. Our proof is a combination of heat flow method and continuity method.

In this paper we continue our investigation of Wold-type decomposition in Baer ⁎-rings. We obtain further information on building blocks of commuting pairs of isometries, by constructing Popovici decomposition of bi-isometries in Baer ⁎-rings. Then we introduce Suciu decomposition of an Abelian isometric semigroup. Moreover, we clarify the relationship between the Wold-Slocinski's decomposition of a given bi-isometry and Suciu decomposition of the semigroup induced by this bi-isometry.

In this paper we analyze the convergence of the following type series T N f ( x ) = ∑ j = N 1 N 2 v j ( e − a j + 1 Δ 2 f ( x ) − e − a j Δ 2 f ( x ) ) , x ∈ R n , where { e − t Δ 2 } t > 0 is the heat semigroup of the biharmonic operator Δ 2 with Δ being the classical laplacian, N = ( N 1 , N 2 ) ∈ Z 2 with N 1 N 2 , { v j } j ∈ Z is a bounded real sequences and { a j } j ∈ Z is a ρ-lacunary sequence of positive numbers, that is, 1 ρ ≤ a j + 1 / a j , for all j ∈ Z . Our analysis will consist in the boundedness, in L p ( R n ) and in B M O ( R n ) , of the operators T N and its maximal operator T ⁎ f ( x ) = sup N ⁡ | T N f ( x ) | . The proofs of these results need the language of semigroups in an essential way.

V tomto článku se zabýváme diferenční rovnicí druhého řádu s asymetrickými nelinearitami na množině celých čísel a zkoumáme rozložení nulových bodů spojitých rozšíření kladných půlvln. Vzdálenost mezi dvěma po sobě jdoucími nulovými body dvou různých kladných půlvln závisí nejenom na parametrech úlohy, ale také na pozici jednoho z těchto nulových bodů vzhledem k celočíselné mřížce. Pro vyjádření této vzdálenosti poskytujeme explicitní předpis, který nám dovoluje získat nový a velmi jednoduchý implicitní popis všech netriviálních Fučíkových křivek pro diskrétní Dirichletův operátor. Dále pro pevně zvolené parametry úlohy ukazujeme, že tato vzdálenost je omezená a nabývá svých globálních extrémů, které jsou popsány pomocí Čebyševových polynomů druhého druhu. Nakonec pro každou netriviální Fučíkovu křivku poskytujeme vhodné odhady pomocí dvou křivek s jednoduchým popisem, který je srovnatelný s popisem první netriviální Fučíkovy větve. In this paper, we deal with the second order difference equation with asymmetric nonlinearities on the integer lattice and we investigate the distribution of zeros for continuous extensions of positive semi-waves. The distance between two consecutive zeros of two different positive semi-waves depends not only on the parameters of the problem but also on the position of one of these zeros with respect to the integer lattice. We provide an explicit formula for this distance, which allows us to obtain a new simple implicit description of all non-trivial Fučík curves for the discrete Dirichlet operator. Moreover, for fixed parameters of the problem, we show that this distance is bounded and attains its global extrema that are explicitly described in terms of Chebyshev polynomials of the second kind. Finally, for each non-trivial Fučík curve, we provide suitable bounds by two curves with a simple description similar to the description of the first non-trivial Fučík curve.

This paper establishes the mapping properties of the singular integral operators, the Littlewood-Paley functions and the maximal Bochner-Riesz means on the Hardy local Morrey spaces with variable exponents by using extrapolation theory.

In this paper, we consider the uniqueness of nontrivial positive radial solutions to the following Schrodinger system with linear and nonlinear couplings: { − △ u 1 + V 1 ( x ) u 1 = μ 1 u 1 3 + β u 1 u 2 2 + κ u 2 in R N , − △ u 2 + V 2 ( x ) u 2 = μ 2 u 2 3 + β u 1 2 u 2 + κ u 1 in R N , u 1 ( x ) , u 2 ( x ) > 0 in R N ( N ≤ 3 ) , u 1 ( x ) , u 2 ( x ) → 0 as | x | → ∞ . By using the a priori estimate technique, we obtain interesting uniqueness results for two cases: (1) sufficiently small β , κ > 0 , (2) large β > 0 .

In this paper, we work on the scalar curvature of complete non-compact gradient Ricci-Bourguignon solitons and prove several results. In the first part, we prove that, with some conditions, any complete non-compact gradient Ricci-Bourguignon soliton has nonnegative scalar curvature. In the second part, we show that the quadratic decay at infinity of the Ricci curvature of complete non-compact gradient Ricci-Bourguignon soliton has constant scalar curvature.

In this paper, we study the existence of large orbits of odd-order subgroups of a finite solvable linear group G, and present some applications on bounding the odd-order parts of the chief series of an arbitrary linear group G. We also find applications to Gluck's conjecture as well as a related question of Navarro.

We obtain estimates for the $L^{p}$-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of $\mathbb{R}^{2}$, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we will obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions $T\times W$ of the time-frequency plane using the Fourier transform to intertwine time and frequency, we will localize the representation of the signals in terms of the short-time Fourier transform in sets $\Delta $ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for $f\in L^{2}(\mathbb{R})$ from its STFT values on arbitrary discs. A discussion on optimality follows, along the lines of Donoho-Stark paper on uncertainty principles and signal recovery. We also consider the case of discrete Gabor systems, vector-valued STFT transforms and rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.

By Beurling–Lax–Halmos theorem the kernel of a block Hankel operator is described by an associated inner matrix. When the inner matrix is square, there is an explicit relation between the symbol function of the block Hankel operator and the inner matrix [16] . When the inner matrix is not square, little is known for the connection of the symbol function and the inner matrix. Recently, an insightful index of a matrix-valued function [18] illuminates a numerical relation between the index of the symbol and the size of the nonsquare inner matrix. In this paper, we find the explicit relation between the symbols of block Hankel operators and their associated nonsquare inner matrices. The basic result states that the symbol can be factorized into the product of a structured matrix with a certain index, otherwise free, and the adjoint of a square matrix which is a completion of the nonsquare inner matrix.

In this paper, moment estimates of Holder norm type are obtained for the solution to stochastic differential equation driven by fractional Brownian motion whose drift is measurable and has linear growth. As application, we prove the existence of a density for this kind of stochastic equation which extends the validity of a method introduced by N. Fournier and J. Printems (2010) and developed by M. Romito (2018).

Given a nonautonomous and nonlinear differential equation (0.1) x ′ = A ( t ) x + f ( t , x ) t ≥ 0 , on an arbitrary Banach space X, we formulate very general conditions for the associated linear equation x ′ = A ( t ) x and for the nonlinear term f : [ 0 , + ∞ ) × X → X under which the above system satisfies an appropriate version of the shadowing property. More precisely, we require that x ′ = A ( t ) x admits a very general type of dichotomy, which includes the classical hyperbolic behavior as a very particular case. In addition, we require that f is Lipschitz in the second variable with a sufficiently small Lipschitz constant. Our general framework enables us to treat various settings in which no shadowing result has been previously obtained. Moreover, we are able to recover and refine several known results. We also show how our main results can be applied to the study of the shadowing property for higher order differential equations. Finally, we conclude the paper by presenting discrete time versions of our results.

Quite recently, Bor (2021) [22] has proved two main theorems dealing with absolute Riesz summability factors of infinite series and Fourier series by using an almost increasing sequence. In this paper, we have generalized these theorems to the | A , p n | k summability method.

In this paper we study formulas for the coordinates of the Weierstrass coefficients a = ( a , A 1 , A 2 , … ) and b = ( b , B 1 , B 2 , … ) of the canonical lifting. More precisely we show that there are formulas for A 1 and B 1 that are given by modular functions, universal (meaning, single formulas that work in every possible case), and with no factor of Δ in their denominators. Two possible constructions are given. Moreover, a sufficient condition is given for the existence of A 2 and B 2 with these properties. Finally, we prove that the Hasse invariant, given as a polynomial on the Weierstrass coefficients of an elliptic curve of characteristic p ≥ 5 , has no repeated factors.

In this paper, we propose a theory for linear time fractional PDEs on L 2 ( R d ) , with two time parameters. The order of the time derivatives under consideration is less than 1. We study well-posedness, regularizing effects and dissipative properties. Regarding regularizing effects, we describe quite precisely the equations that have this effect or not. We highlight that, in purely fractional settings, the regularizing effect acts always only up to finite order; unlike to the standard case. Also, we investigate the properties of the three time variables solution operator generated by these PDEs.

Our first aim in this paper is to deal with asymptotic behavior of Poisson integrals in a cylinder. Next Carleman's formula for harmonic functions in it is also proved. As an application of them, we finally give the integral representation of harmonic functions in a cylinder.