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A new proof is given of a simple probabilistic lemma which implies the solution of the Dirichlet problem for simply connected domains in the plane. This proof uses the conformal invariance of planar Brownian motion in place of the previously existing measure-theoretic arguments. [ABSTRACT FROM AUTHOR]

In this paper, we establish some new results on the existence and nonexistence of radial large positive solutions for a modified Schrödinger system with a nonconvex diffusion term by a successive iteration technique and the dual approach. The necessary and sufficient condition for the existence of radial large positive solutions is established. Our results improve and extend many previous work in this field of research. [ABSTRACT FROM AUTHOR]

We introduce a new method to reformulate certain classes of problems involving non-local terms as local problems. This allows us to apply the techniques developed in [24,25] to prove upper and lower bounds for problems arising in Micromagnetics and in the variational study of the Method of Vanishing Viscosity for systems of conservation laws. [ABSTRACT FROM AUTHOR]

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results. [ABSTRACT FROM AUTHOR]

Abstract For the hypergeometric function of unit argument F 2 3 (1) we prove the existence and uniqueness of three-term contiguous relations with arbitrary integer shifts. We show that not only the original F 2 3 (1) function but also other five functions related to it satisfy one and the same three-term relation. This fact together with the uniqueness mentioned above provides three-term relations with a group symmetry of order 72. [ABSTRACT FROM AUTHOR]

In this paper, we study the hyperstability for the general linear equation in the setting of quasi-Banach spaces. We first extend the fixed point result of Brzdek et al. [5, Theorem 1] in metric spaces to b -metric spaces, in particular to quasi-Banach spaces. Then we use this result to generalize the main results on the hyperstability for the general linear equation in Banach spaces to quasi-Banach spaces. We also show that we can not omit the assumption of completeness in [5, Theorem 1] . As a consequence, we conclude that we need more explanations to replace a normed space by its completion in the proofs of some results in the literature. [ABSTRACT FROM AUTHOR]

Let U be a balanced open subset of a Hausdorff complex locally convex space E . In this paper we represent the predual G ( U ) of the space of holomorphic functions as the projective limit of the preduals of spaces of holomorphic functions that are bounded on certain subsets of U . As an application we prove that if E is a Banach space, then it has the approximation property if and only if G ( U ) has the approximation property. We also give a new proof of a result due to Aron and Schottenloher stating that if E is a complex Banach space with the approximation property then the space of holomorphic functions H ( U ) , with the compact-open topology, has the approximation property. [ABSTRACT FROM AUTHOR]

In this paper, we prove several congruences for Appell–Lerch sums which generalize some conjectures given by Chan. [ABSTRACT FROM AUTHOR]

The Banach spaces c e s ( p ) , 1 < p < ∞ , were intensively studied by G. Bennett and others. The largest solid Banach lattice in C N which contains ℓ p and which the Cesàro operator C : C N ⟶ C N maps into ℓ p is c e s ( p ) . For each 1 ≤ p < ∞ , the (positive) operator C also maps the Fréchet space ℓ p + = ⋂ q > p ℓ q into itself. It is shown that the largest solid Fréchet lattice in C N which contains ℓ p + and which C maps into ℓ p + is precisely c e s ( p + ) : = ⋂ q > p c e s ( q ) . Although the spaces ℓ p + are well understood, it seems that the spaces c e s ( p + ) have not been considered at all. A detailed study of the Fréchet spaces c e s ( p + ) , 1 ≤ p < ∞ , is undertaken. They are very different to the Fréchet spaces ℓ p + which generate them in the above sense. We prove that each c e s ( p + ) is a power series space of finite type and order one, and that all the spaces c e s ( p + ) , 1 ≤ p < ∞ , are isomorphic. [ABSTRACT FROM AUTHOR]

In the present paper we introduce a notion of ( ω , t ) -convexity as a natural generalization of the notion of usual t -convexity, t -strongly convexity, approximate t -convexity, delta t -convexity and many other. The main result of this paper establishes the necessary and sufficient conditions under which an ( ω , t ) -convex map can be supported at a given point by an ( ω , t ) -affine support function. Several applications of this support theorem are presented. For instance, new characterizations of inner product spaces among normed spaces involving the notion of ( ω , t ) -convexity are given. [ABSTRACT FROM AUTHOR]

Let P 1 and P 2 be two points in C 2 such that the complex line joining them is tangent to the unit sphere ∂ B 2 . We prove that if a real analytic function f on ∂ B 2 extends holomorphically on every complex line through P 1 and P 2 , then f extends holomorphically to the whole ball B 2 . [ABSTRACT FROM AUTHOR]

We extend the study of the multifractal analysis of the class of equicontractive self-similar measures of finite type to the non-equicontractive setting. Although stronger than the weak separation condition, the finite type property includes examples of IFS that fail the open set condition. The important combinatorial properties of equicontractive self-similar measures of finite type are extended to the non-equicontractive setting and we prove that many of the results from the equicontractive case carry over to this new, more general, setting. In particular, previously it was shown that if an equicontractive self-similar measure of finite type was regular , then the calculations of local dimensions were relatively easy. We modify this definition of regular to define measures to be generalized regular . This new definition will include the non-equicontractive case and we obtain similar results. Examples are studied of non-equicontractive self-similar generalized regular measures, as well as equicontractive self-similar measures which are generalized regular in this new sense, but which are not regular. [ABSTRACT FROM AUTHOR]

We prove that the uniform probability measure μ on every ( n − k ) -dimensional projection of the n -dimensional unit cube verifies the variance conjecture with an absolute constant C Var μ | x | 2 ≤ C sup θ ∈ S n − 1 ⁡ E μ 〈 x , θ 〉 2 E μ | x | 2 , provided that 1 ≤ k ≤ n . We also prove that if 1 ≤ k ≤ n 2 3 ( log ⁡ n ) − 1 3 , the conjecture is true for the family of uniform probabilities on its projections on random ( n − k ) -dimensional subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we mainly extend the Khintchine's recurrence theorem, Furstenberg's multiple recurrence theorem, and multiple Birkhoff recurrence theorem from single-valued maps to set-valued maps. [ABSTRACT FROM AUTHOR]

In this paper we consider a nonlocal viscous Burgers' equation and study the well-posedness and asymptotic behavior of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers' equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories. [ABSTRACT FROM AUTHOR]

We prove that the Berkovich space (or multiplicative spectrum) of the algebra of bounded analytic functions on the open unit disk of an algebraically closed nonarchimedean field contains multiplicative seminorms that are not norms and whose kernel is not a maximal ideal. We also prove that in general these seminorms are not univocally determined by their kernels, and provide a method for obtaining families of different seminorms sharing the same kernel. The relation with the Berkovich space of the Tate algebra is also given. [ABSTRACT FROM AUTHOR]

The paper deals with a general optimal control problem for age-structured systems. A necessary optimality condition of Pontryagin type is obtained, where the novelty is in that mixed control-state constraints are present. The proof uses an abstract Lagrange multiplier theorem, and the main difficulty is to obtain regularity of the Lagrange multipliers in the particular problem at hand. [ABSTRACT FROM AUTHOR]

We prove that every surjective isometry between the unit spheres of two atomic JBW ⁎ -triples E and B admits a unit extension to a surjective real linear isometry from E into B . This result constitutes a new positive answer to Tingley's problem in the Jordan setting. [ABSTRACT FROM AUTHOR]

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the SOT-closed operator algebra acting on L 2 ( R ) that is generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on L p ( R ) , where 1 < p < ∞ . In the last section, it is also shown that the reflexive closures of the Fourier binests on L p ( R ) are all order isomorphic for 1 < p < ∞ . [ABSTRACT FROM AUTHOR]

We prove that every unconditionally summable sequence in a Hilbert space can be factorized as the product of a square summable scalar sequence and a Bessel sequence. Some consequences on the representation of unconditionally convergent multipliers are obtained, thus providing positive answers to a conjecture by Balazs and Stoeva in some particular cases. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of infinitely many bounded entire solutions, ground state entire solutions and entire solutions with prescribed limit at infinity to PDEs involving the ϕ -Laplace operator in the setting of Orlicz–Sobolev spaces. The main tools used are the sub-solution and super-solution technique in conjunction with the Schauder–Tychonoff fixed point theorem. [ABSTRACT FROM AUTHOR]

We propose and study a time-dependent Ginzburg–Landau–Chern–Simons model. The local existence of solutions to the model in a bounded domain Ω is proved by applying contraction mapping theorem. We use covariant energy estimate to extend a local solution to global one. [ABSTRACT FROM AUTHOR]

In this paper, we prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant c depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as c . Moreover, the CIUPM is unique up to translation if the diameter of the support equals c . [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem − Δ p u + a ( x ) | u | p − 2 u = | u | p ⁎ − 2 u + μ | u | q − 2 u in R N , where 1 < p < N , max ⁡ { p , p ⁎ − 1 } < q < p ⁎ = N p N − p , μ > 0 , Δ p is the p -Laplacian operator, and the potential a ( x ) is positive and verifies suitable decay assumptions. We prove that the problem has infinitely many solutions by using truncation method. [ABSTRACT FROM AUTHOR]

We prove that a finite (state and action spaces) semi-Markov decision process with limiting ratio average (undiscounted) payoff has an optimal pure semi-stationary policy (i.e., a semi-Markov policy independent of decision epoch count). We conclude by showing (with the aid of an example) that the result cannot be strengthened further. A crude but finite step algorithm is given to compute such an optimal policy. [ABSTRACT FROM AUTHOR]

In this paper, we find a new auxiliary function and use the maximum principle to give a new proof for the gradient of a solution for semi-linear elliptic equations with capillarity-type problem. [ABSTRACT FROM AUTHOR]

We present a beautiful but relatively unknown theorem that every differentiable function f : P → R , with P ⊂ R being closed, admits differentiable extension F : R → R . We present an elementary proof of this result based on a construction sketched in a hard-to-access 1923 paper [4] of V. Jarník. Using this construction, we also obtain an elegant version of Whitney extension theorem characterizing when such an f admits continuously differentiable extension. [ABSTRACT FROM AUTHOR]

This paper concerns about the regularity criteria for the three-dimensional Navier–Stokes equations. By establishing a more subtle estimate of the crucial convective term in Navier–Stokes equations, we improve [4] and [10] a lot. [ABSTRACT FROM AUTHOR]

We prove that every representation of the Cuntz algebra O N on a separable Hilbert space H arises from a pure isometry V whose wandering space H ⊖ im V has dimension N . We identify the permutative representations in this construction. [ABSTRACT FROM AUTHOR]

In this note, we use recent zero duality results arising from Monotropic Programming problem for analyzing consistency of the convex feasibility problem in Hilbert spaces. We characterize consistency in terms of the lower semicontinuity of the infimal convolution of the associated support functions. [ABSTRACT FROM AUTHOR]

The viscous and inviscid aggregation equation with Newtonian potential models a number of different physical systems and has close analogs in 2D incompressible fluid mechanics. We consider a slight generalization of these equations in the whole space establishing well-posedness of the viscous and inviscid equations, spatial decay of the viscous solutions, and the convergence of viscous solutions to the inviscid solution as the viscosity goes to zero. [ABSTRACT FROM AUTHOR]

Let X be an abstract L -space and let Y be any Banach space. Motivated by a classical result of T. J. Abatzoglou that describes smooth points of the space of operators on a Hilbert space, we give a characterization of very smooth points in the space of operators from X to Y . We also show that a similar result can be proved when Y is an abstract M -space such that the set of extreme points of the dual unit ball is a weak ⁎ -compact set. [ABSTRACT FROM AUTHOR]

Let G = ( V , E ) be a connected finite graph. In this short paper, we reinvestigate the Kazdan–Warner equation Δ u = c − h e u with c < 0 on G , where h defined on V is a known function. Grigor'yan, Lin and Yang [3] showed that if the Kazdan–Warner equation has a solution, then h ‾ , the average value of h , is negative. Conversely, if h ‾ < 0 , then there exists a number c − ( h ) < 0 , such that the Kazdan–Warner equation is solvable for every 0 > c > c − ( h ) and it is not solvable for c < c − ( h ) . Moreover, if h ≤ 0 and h ≢ 0 , then c − ( h ) = − ∞ . Inspired by Chen and Li's work [1] , we ask naturally: Is the Kazdan–Warner equation solvable for c = c − ( h ) ? In this paper, we answer the question affirmatively. We show that if c − ( h ) = − ∞ , then h ≤ 0 and h ≢ 0 . Moreover, if c − ( h ) > − ∞ , then there exists at least one solution to the Kazdan–Warner equation with c = c − ( h ) . [ABSTRACT FROM AUTHOR]

In this paper, we establish approximate representations of solutions to stochastic Volterra integral equations (SVIEs, for short), by virtue of solutions to a family of stochastic differential equations. As an application, we present two algorithms for solving SVIEs: stochastic theta method and splitting method, and prove the global 1/2 order convergence rates in L 2 norm. [ABSTRACT FROM AUTHOR]

We prove that zeros of each basis element of the space of weakly holomorphic modular forms of weight k for the Fricke group Γ 0 + ( 2 ) of level 2 interlace, extending the result for SL 2 ( Z ) of Jenkins and Pratt [4] . [ABSTRACT FROM AUTHOR]

Let H and K be two Hilbert spaces. The algebra of all bounded linear operators acting on H is denoted by B ( H ) . The main purpose of this paper is to obtain a characterization of bijective maps Φ : B ( H ) → B ( K ) satisfying the following condition Δ λ ( Φ ( A ) Φ ( B ) ) = Φ ( Δ λ ( A B ) ) for all A , B ∈ B ( H ) , where Δ λ ( T ) stands for the λ -Aluthge transform of the operator T ∈ B ( H ) . More precisely, we prove that a bijective map Φ satisfies the above condition, if and only if, there exists a unitary operator U : H → K , such that Φ ( A ) = U A U ⁎ for all A ∈ B ( H ) . [ABSTRACT FROM AUTHOR]

This paper presents a systematic study for harmonic analysis of metaplectic wave-packets in L 2 ( R ) via group representation theory. The abstract notions of metaplectic wave-packet groups and metaplectic wave-packet representations will be presented and as the main result, we prove an admissibility condition on closed subgroups of the special linear group SL ( 2 , R ) , which guarantees square integrability of the associated metaplectic wave-packet representation on L 2 ( R ) . Finally, we present an analytic study of metaplectic wave-packet representations over admissible subgroups of SL ( 2 , R ) . [ABSTRACT FROM AUTHOR]

Let β > 1 be a real number. A basic interval of order n is a set of real numbers in ( 0 , 1 ] having the same first n digits in their β -expansion which contains x ∈ ( 0 , 1 ] , denote by I n ( x ) and write the length of I n ( x ) as | I n ( x ) | . In this paper, we prove that the extremely irregular set containing points x ∈ [ 0 , 1 ] whose upper limit of − log β ⁡ | I n ( x ) | n equals to 1 + λ ( β ) is residual for every λ ( β ) > 0 , where λ ( β ) is a constant depending on β . [ABSTRACT FROM AUTHOR]

We study [ φ t , X ] , the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup { φ t } t ≥ 0 of analytic self-maps of the unit disk, when X is BMOA, H ∞ or the disk algebra. In particular, we show that [ φ t , BMOA ] ≠ BMOA for all nontrivial semigroups. We also prove, for every semigroup { φ t } t ≥ 0 , that lim t → 0 + ⁡ φ t ( z ) = z not just pointwise, but in H ∞ norm. This provides a unified proof of known results about [ φ t , X ] when X ∈ { H p , A p , B 0 , VMOA } . [ABSTRACT FROM AUTHOR]

We prove that a solution of the Schrödinger-type equation i ∂ t u = H u , where H is a Jacobi operator with asymptotically constant coefficients, cannot decay too fast at two different times unless it is trivial. [ABSTRACT FROM AUTHOR]

We give a new proof of Zagier's theorem on evaluation of multiple zeta (star) values at ( { 2 } a , 3 , { 2 } b ) . The novelty of the proof consists in employing a hypergeometric identity of Andrews, which expresses a very-well-poised hypergeometric series in terms of a multiple series, and the residue theorem. [ABSTRACT FROM AUTHOR]

The objective of this work is to obtain Liouville type theorems for positive supersolutions of the elliptic system { − Δ u + | ∇ u | = λ v p , − Δ v + | ∇ v | = μ u s , in exterior domains of R N , where p , s > 0 and λ , μ > 0 are parameters. We prove that positive supersolutions do not exist in some ranges of the parameters which turn out to be optimal. [ABSTRACT FROM AUTHOR]

We generalize the case p = 2 of the Caccioppoli-type inequality from [5] to multidimensional domains. The argument also provides a simpler proof of the case p = 2 of the original inequality. [ABSTRACT FROM AUTHOR]

We prove that all the nilpotent centers of planar analytic differential systems are limit of centers with purely imaginary eigenvalues, and consequently the Poincaré–Liapunov method to detect centers with purely imaginary eigenvalues can be used to detect nilpotent centers. [ABSTRACT FROM AUTHOR]

We study the hyperbolization problem of two hyperspaces over a metric space and prove that they can be hyperbolized by asymptotically PT − 1 metrics. [ABSTRACT FROM AUTHOR]

We prove the existence of bounded solutions of a new class of retarded functional differential equations with non-instantaneous impulses and delay on an unbounded domain. An application example is also included. [ABSTRACT FROM AUTHOR]

The bifurcation and stability structures of equilibria of a parabolic problem motivated by a population genetics model are completely described. In a previous work the authors considered weights having nonzero average and drew the bifurcation and stability diagrams of equilibria. Herein that study is completed by considering weights with null average over the boundary of the domain and other approach is employed to overcome a degeneration which occurs in such case. For each value of the parameter we prove existence of an unique nonconstant equilibrium solution. It belongs to a globally parametrized bifurcation curve constructed using a Lyapunov–Schmidt type reduction combined with the Morse lemma. That solution is showed to be a global minimizer of the corresponding energy functional which attracts all nontrivial semi-orbits. The behavior of its trace, when the parameter is large, is established and complete diagrams containing all bifurcation and stability information are also provided. [ABSTRACT FROM AUTHOR]

This paper studies the problem of determining the optimal cut-off for pairs trading rules. We consider two cointegrated assets whose spread is modelled by a general mean-reverting process, and the optimal pair trading rule is formulated as an optimal switching problem between three regimes: flat position (no holding stocks), one long and the other short and one short and the other long. A fixed commission cost is charged with each transaction. We use a viscosity solutions approach to prove the existence and the explicit characterisation of cut-off points via the resolution of quasi-algebraic equations. We illustrate our results by numerical simulations. [ABSTRACT FROM AUTHOR]

The minimal surfaces in Finsler geometry with respect to the Busemann–Hausdorff measure and the Holmes–Thompson measure are called BH-minimal and HT-minimal surfaces, respectively. Let ( p 1 , p 2 , p 3 , p 4 ) be the coordinates of R 4 and ( S 3 , F ˜ ) be a Randers sphere of flag curvature K = 1 with the navigation data ( h ˜ , W ˜ ) , where h ˜ is the standard sphere metric and W ˜ = ε ( 0 , 0 , − p 4 , p 3 ) , 0 < ε < 1 , is a Killing vector field. In this paper, we study the rotationally invariant minimal surface in ( S 3 , F ˜ ) generated by rotating the curve ( x ( s ) , y ( s ) , z ( s ) , 0 ) in the upper half sphere of S 2 around the p 1 p 2 -plane, s ∈ R . We first show that such a rotational BH-minimal surface in ( S 3 , F ˜ ) is either a great 2-sphere or the catenoid in ( S 3 , h ˜ ) . Then we give a classification of the rotational HT-minimal surfaces, where we use the angle data to analyze the solutions of the system of ODE that characterizes the HT-minimality and prove that, such a rotational HT-minimal surface must be a great 2-sphere, an HT-minimal torus, or a rotational surface of unduloid type. As a special case, we obtain a distinguished embedded compact HT-minimal torus depending on ε . The completeness of these surfaces is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the periodic higher-order KdV-type equation { ∂ t u + ( − 1 ) j + 1 ∂ x 2 j + 1 u + 1 2 ∂ x ( u 2 ) = 0 , ( t , x ) ∈ R × T , u ( 0 , x ) = u 0 ( x ) , u 0 ∈ H s ( T ) , is globally well-posed in H s for s ≥ − j 2 , j ≥ 3 . The proof is based on “I-method” introduced by Colliander et al. [4] . We also prove the nonsqueezing property of the periodic higher-order KdV-type equation. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and an approximation argument for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we apply the nonsqueezing theorem. By using the approximation argument, we extend this result to the infinite dimensional system. This argument was introduced by Kuksin [14] and made concretely by Bourgain [2] for the 1D cubic NLS flow, and Colliander et al. [5] for the KdV flow. One of our observations is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than that the KdV equation has. Hence, unlike the work of Colliander et al. [5] , we can get the nonsqueezing property for the solution flow without the Miura transform. [ABSTRACT FROM AUTHOR]