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Abstract We obtain a simple two-sided inequality for the ratio L ν (x) / L ν − 1 (x) in terms of the ratio I ν (x) / I ν − 1 (x) , where L ν (x) is the modified Struve function of the first kind and I ν (x) is the modified Bessel function of the first kind. This result allows one to use the extensive literature on bounds for I ν (x) / I ν − 1 (x) to immediately deduce bounds for L ν (x) / L ν − 1 (x). We note some consequences and obtain further bounds for L ν (x) / L ν − 1 (x) by adapting techniques used to bound the ratio I ν (x) / I ν − 1 (x). We apply these results to obtain new bounds for the condition numbers x L ν ′ (x) / L ν (x) , the ratio L ν (x) / L ν (y) and the modified Struve function L ν (x) itself. Amongst other results, we obtain two-sided inequalities for x L ν ′ (x) / L ν (x) and L ν (x) / L ν (y) that are given in terms of x I ν ′ (x) / I ν (x) and I ν (x) / I ν (y) , respectively, which again allows one to exploit the substantial literature on bounds for these quantities. The results obtained in this paper complement and improve existing bounds in the literature. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of Lelong numbers of m -subharmonic currents of bidimension ( p , p ) on an open subset of C n , when m + p ≥ n . In the special case of m -subharmonic function φ , we give a relationship between the Lelong numbers of d d c φ and the mean values of φ on spheres and balls. As an application we study the integrability exponent of φ . We express the integrability exponent of φ in terms of volume of sub-level sets of φ and we give a link between this exponent and its Lelong number. [ABSTRACT FROM AUTHOR]

For a C 2 function u and an elliptic operator L , we prove a quantitative estimate for the derivative du in terms of local bounds on u and Lu . An integral version of this estimate is then used to derive a condition for the zero-mean value property of Δ u . An extension to differential forms is also given. Our approach is probabilistic and could easily be adapted to other settings. [ABSTRACT FROM AUTHOR]

We consider a junction regulated by a traffic lights, with n incoming roads and only one outgoing road. On each road the Phase Transition traffic model, proposed in [6] , describes the evolution of car traffic. Such model is an extension of the classic Lighthill–Whitham–Richards one, obtained by assuming that different drivers may have different maximal speed. By sending to infinity the number of cycles of the traffic lights, we obtain a justification of the Riemann solver introduced in [9] and in particular of the rule for determining the maximal speed in the outgoing road. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of nodal solutions for the following fractional Choquard equation ( − △ ) α u + u = ∫ R N | u ( z ) | p | x − z | μ d z | u ( x ) | p − 2 u ( x ) , x ∈ R N , where 0 < μ < 2 α < N and 2 N − μ N − 1 < p < 2 N − μ N − 2 α with α ∈ ( 1 2 , 1 ) . In view of the nonlocality of the fractional Laplacian operator and the so-called Hartree term ∫ R N | u ( z ) | p | x − z | μ d z | u ( x ) | p − 2 u ( x ) , the corresponding variational functional has entirely different properties with respect to the Laplacian case. For any k ∈ N , we prove that the problem possesses at least a radially symmetrical solution which changes sign k times. [ABSTRACT FROM AUTHOR]

This paper is concerned with an evolution problem having an elliptic equation involving the 1-Laplacian operator and a dynamical boundary condition. We apply nonlinear semigroup theory to obtain existence and uniqueness results as well as a comparison principle. Our main theorem shows that the solution we found is actually a strong solution. We also compare solutions with different data. [ABSTRACT FROM AUTHOR]

In this article we prove that, as n → ∞ , ∑ j , k = 1 n − 1 1 3 − cos ⁡ 2 π j n − cos ⁡ 2 π k n − cos ⁡ 2 π ( j + k ) n ∼ n 2 log ⁡ n 3 π . We also obtain the secondary term of size ≍ n 2 to be followed by an error term of size O ( log ⁡ n ) . In this work, results and techniques from classical analysis involving roles by several special functions, from number theory, and from numerical analysis are needed. [ABSTRACT FROM AUTHOR]

Let N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 and the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , respectively, where a , b , c , d are positive integers, n is a nonnegative integer and x , y , z , w are integers. Sun established many relations between N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) and posed 23 conjectures. Yao proved five of them by using ( p , k ) -parametrization of theta functions. In this paper, we confirm four conjectures of Sun by employing theta function identities. [ABSTRACT FROM AUTHOR]

We consider functions of the form F c → , a , T ( s ) = ∑ j = 0 M c j ( − 1 ) j L 2 j ξ ( a + 2 j ) ( s ) , with L = log ⁡ T 2 π and c j real constants satisfying a certain constraint. We show that as T → ∞ , the proportion of zeros of F c → , a , T ( s ) on the critical line Re ( s ) = 1 / 2 tends to 1. [ABSTRACT FROM AUTHOR]

Let S be a pure bounded rationally multicyclic subnormal operator on a separable complex Hilbert space H and let M z be the minimal normal extension on a separable complex Hilbert space K containing H . Let b p e ( S ) be the set of bounded point evaluations and let a b p e ( S ) be the set of analytic bounded point evaluations. We show a b p e ( S ) = b p e ( S ) ∩ I n t ( σ ( S ) ) . The result affirmatively answers a question asked by J. B. Conway concerning the equality of the interior of b p e ( S ) and a b p e ( S ) for a rationally multicyclic subnormal operator S . As a result, if λ 0 ∈ I n t ( σ ( S ) ) and d i m ( k e r ( S − λ 0 ) ⁎ ) = N , where N is the minimal number of cyclic vectors for S , then the range of S − λ 0 is closed, hence, λ 0 ∈ σ ( S ) ∖ σ e ( S ) . [ABSTRACT FROM AUTHOR]

In this paper we present quantitative upper estimates for the number of negative eigenvalues of a two dimensional Schrödinger operator with potential supported by an unbounded Lipschitz curve. The estimates are given in terms of weighted L 1 and L log ⁡ L type Orlicz norms of the potential. [ABSTRACT FROM AUTHOR]

In this paper, we study the number of limit cycles in the family d H − ε ω = 0 , where H = y 2 2 − ∫ 0 x g ( u ) d u , ω = y f ( x ) d x , with g ( x ) = x ( x 2 − 1 ) ( x 2 − 1 4 ) 2 , and f ( x ) an even polynomial of degree 10. We will consider mainly the bifurcation of limit cycles near the eye-figure loop and the center of d H = 0 . Our investigation focuses on the lower bound of the maximal number of limit cycles for these systems. In particular, we show that the perturbed system can have at least 8 limit cycles when deg ⁡ ( f ( x ) ) = 10 . [ABSTRACT FROM AUTHOR]

We show that certain terminating ϕ 5 6 series can be factorized into a product of two ϕ 2 3 series. As applications we prove a summation formula for a product of two q -Delannoy numbers along with some congruences for sums involving q -Delannoy numbers. This confirms three recent conjectures of the second author. [ABSTRACT FROM AUTHOR]

In this paper, we confirm five conjectures on the relations between N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) given by Sun by utilizing ( p , k ) -parametrization of theta functions, where N ( a , b , c , d ; n ) and t ( a , b , c , d ; n ) denote the number of representations of n as a x 2 + b y 2 + c z 2 + d w 2 and the number of representations of n as a x ( x + 1 ) 2 + b y ( y + 1 ) 2 + c z ( z + 1 ) 2 + d w ( w + 1 ) 2 , respectively, with a , b , c , d ∈ N + , n ∈ N , and x , y , z , w ∈ Z . [ABSTRACT FROM AUTHOR]

In terms of the derivative operator, integral operator and a Saalschütz-type identity, two families of summation formulae involving generalized harmonic numbers are established. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center x ˙ = − y + 16 3 x 2 − 4 3 y 2 , y ˙ = x + 8 3 x y by the averaging method of first order, when it is perturbed inside a class of discontinuous quadratic polynomial differential systems. The Chebyshev criterion is used to show that this maximum number is 5 and can be realizable. In some sense, the result and that in paper [6] also answer the questions left in the paper [9] . [ABSTRACT FROM AUTHOR]

In this work we study a class of planar Hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles. Employing Melnikov functions of any order, we show that the class of systems can have at most three limit cycles and the limit is reached. [ABSTRACT FROM AUTHOR]

We study probabilistic iterated function systems (IFSs), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R d . If our strong open set condition is also satisfied, we show that both upper and lower bounds for the Hausdorff and packing dimensions of the invariant measure can be found. Both bounds take on the familiar form of ratio of entropy to the Lyapunov exponent. Proving these bounds in this setting requires methods which are quite different from the standard methods used for average-contracting IFSs. [ABSTRACT FROM AUTHOR]

In this article, we present multiple-term refinements of Young's inequality for real numbers and operators. In particular, given a natural number N , we find N positive terms refining Young's inequality. Detailed properties of this refinement are studied leading to a new interesting mean. [ABSTRACT FROM AUTHOR]

Let A and B be selfadjoint operators in a Krein space and assume that the resolvent difference of A and B is of rank one. In the case that A is nonnegative and I is an open interval such that σ ( A ) ∩ I consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of B in I . The general result is illustrated with eigenvalue estimates for singular indefinite Sturm–Liouville problems. [ABSTRACT FROM AUTHOR]

In this paper we investigate the asymptotic behaviour of singular numbers of the operator P h C P h , where C is Cauchy's operator and P h is the orthogonal projection from L 2 ( D ) onto harmonic function subspace. We prove that s n ( P h C P h ) = O ( 1 n ) , as n → + ∞ . Moreover, we find the upper and lower asymptotic estimates, π − 1 ⩽ lim n → + ∞ ⁡ n s n ( P h C P h ) ⩽ π − 1 ( 35 + 21 2 6 ) + 7 6 . [ABSTRACT FROM AUTHOR]

Let A t , k ( n ) denote the number of partition k -tuples of n where each partition is t -core. In this paper, we establish formulas of A t , k ( n ) for some values of t and k by employing the method of modular forms, which extends Wang's result for t = 3 and k = 2 , 3 . [ABSTRACT FROM AUTHOR]

We construct several families of tight wavelet frames in L 2 ( R 2 ) associated to the quincunx matrix. A couple of those families has five generators. Moreover, we construct a family of tight wavelet frames with three generators. Finally, we show families with only two generators. The generators have compact support, any given degree of regularity, and any fixed number of vanishing moments. Our construction is made in Fourier space and involves some refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler. In addition, we will use well known results on construction of tight wavelet frames with two generators on R with the dyadic dilation. The refinable functions we use are constructed from the Daubechies low pass filters and are compactly supported. The main difference between these families is that while the refinable functions associated to the five generators have many symmetries, the refinable functions used in the construction of the others families are merely even. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of the solution to some time harmonic wave problems when the wave number is taken as a small asymptotic parameter. Our basic strategy is to introduce suitable Lagrangian multipliers into the governing equations, and transforming them into saddle point problems. These saddle point problems are uniformly invertible with respect to the wave number k ∈ [ 0 , k 0 ] , with k 0 being an arbitrary but fixed positive number. The asymptotic expansion is then derived by the standard regular perturbation technique. [ABSTRACT FROM AUTHOR]

In this article, we obtain a regularized trace formula for magic Gribov operator H = λ ″ G + H μ , λ acting on Bargmann space where G = a ⁎ 3 a 3 and H μ , λ = μ a ⁎ a + i λ a ⁎ ( a + a ⁎ ) a Here a and a ⁎ are the standard Bose annihilation and creation operators and in Reggeon field theory, the real parameters λ ″ is the magic coupling of Pomeron, μ is Pomeron intercept, λ is the triple coupling of Pomeron and i 2 = − 1 . By applying some abstract results of Sadovnichii–Podol'skii (2002) [23] , we give the number of corrections sufficient for the existence of finite formula of the trace of concrete magic Gribov's operator. [ABSTRACT FROM AUTHOR]

The problem of constructing quasiperiodic solutions of the Kadomtsev–Petviashvili equation is revisited. Following the idea of Krichever and the technique developed by Gesztesy and Holden, we shall explicitly construct the Baker–Akhiezer function from the compatible solutions which satisfy infinite numbers of equations in the Broer–Kaup hierarchy. Analytic and asymptotic properties will also be studied by introducing a special meromorphic function ψ 1 ( P ) ψ ( P ⁎ ) , from which we derive theta function representations for the Baker–Akhiezer function and quasiperiodic solutions of the Kadomtsev–Petviashvili equation. [ABSTRACT FROM AUTHOR]

Let A 3 ( n ) (resp. B 3 ( n ) ) denote the number of partition pairs (resp. triples) of n where each partition is 3-core. By applying Ramanujan's ψ 1 1 formula and Bailey's ψ 6 6 formula, we find the explicit formulas for A 3 ( n ) and B 3 ( n ) . Using these formulas, we confirm a conjecture of Xia and establish many arithmetic identities satisfied by A 3 ( n ) and B 3 ( n ) . [ABSTRACT FROM AUTHOR]

It is well known that the vertex number of the typical k -face of a stationary random hyperplane tessellation in R d has, under some mild conditions, an expectation equal to 2 k , independent of the underlying distribution. The variance of this vertex number can vary widely. Under Poisson assumptions, we give sharp bounds for this variance, showing, in particular, that its maximum is attained if the tessellation is isotropic (that is, its distribution is rotation invariant) with respect to a suitable scalar product on R d . The employed representation of the second moment of the vertex number is a special case of formulas providing the covariance matrix of the random vector ( ℓ 0 , … , ℓ k ) , where ℓ r is the total r -face content of the typical k -face of a stationary Poisson hyperplane mosaic. For k = d in the isotropic case, such formulas were first obtained by Miles in 1961. We give a more elementary proof and extend the formulas to general orientation distributions and to k -dimensional faces. [ABSTRACT FROM AUTHOR]

For an ergodic hyperbolic measure ω of a C 1 + r diffeomorphism f there is an ω -full measured set Λ ˜ such that for every invariant measure μ ∈ M inv ( Λ ˜ , f ) the exponential growth rate of the number of periodic points whose atomic measures approximate μ equals to the metric entropy h μ ( f ) . [ABSTRACT FROM AUTHOR]

The connective eccentricity index (CEI) of a graph G is defined as ξ ce ( G ) = ∑ v i ∈ V ( G ) d ( v i ) ε ( v i ) where ε ( v i ) and d ( v i ) are the eccentricity and the degree of vertex v i , respectively, in G . In this paper we obtain some lower and upper bounds on the connective eccentricity index for all trees of order n and with matching number β and characterize the corresponding extremal trees. And the maximal graphs of order n and with matching number β and n edges have been determined which maximize the connective eccentricity index. Also the extremal graphs with maximal connective eccentricity index are completely characterized among all connected graphs of order n and with matching number β . Moreover we establish some relations between connective eccentricity index and eccentric connectivity index, as another eccentricity-based invariant, of graphs. [ABSTRACT FROM AUTHOR]

We study the growth at the golden rotation number ω = ( 5 − 1 ) / 2 of the function sequence P n ( ω ) = ∏ r = 1 n | 2 sin ⁡ π r ω | . This sequence has been variously studied elsewhere as a skew product of sines, Birkhoff sum, q-Pochhammer symbol (on the unit circle), and restricted Euler function. In particular we study the Fibonacci decimation of the sequence P n , namely the sub-sequence Q n = | ∏ r = 1 F n 2 sin ⁡ π r ω | for Fibonacci numbers F n , and prove that this renormalisation subsequence converges to a constant. From this we show rigorously that the growth of P n ( ω ) is bounded by power laws. This provides the theoretical basis to explain recent experimental results reported by Knill and Tangerman (2011) [10] . [ABSTRACT FROM AUTHOR]

We give a general formula for the determinants of a class of Hankel matrices which arise in combinatorics theory. We revisit and extend existent results on Hankel determinants involving the sum of consecutive Catalan, Motzkin and Schröder numbers and we prove a conjecture of [10] about the recurrence relations satisfied by the Hankel transform of linear combinations of Catalans numbers. [ABSTRACT FROM AUTHOR]

We obtain some well-known expansions for the Lerch transcendent and the Hurwitz zeta function using elementary Fourier analytic methods. These Fourier series can be used to analytically continue the functions and prove the classical functional equations, which arise from the relations satisfied by the Fourier conjugate and flat Fourier series. In particular, the functional equation for the Riemann zeta function can be obtained in this way without contour integrals. The conjugate series for special values of the parameters yields analogous results for the Bernoulli and Apostol–Bernoulli polynomials. Finally, we give some consequences derived from the Fourier series. [ABSTRACT FROM AUTHOR]

Let η ≠ − 1 be a non-zero complex number, and let ϕ be a not necessarily linear bijection between two von Neumann algebras, one of which has no central abelian projections, satisfying ϕ ( I ) = I and preserving the Jordan triple η -⁎-product. It is showed that ϕ is a linear ⁎ -isomorphism if η is not real and ϕ is the sum of a linear ⁎ -isomorphism and a conjugate linear ⁎ -isomorphism if η is real. [ABSTRACT FROM AUTHOR]

Let R 0 2 = R 2 ∖ { ( 0 , 0 ) } , R ⁎ 2 = { ( x , y ) ∈ R 2 : x 2 ≠ y 2 } and f : R 0 2 → R , g : R ⁎ 2 → R . In this paper we consider the Ulam–Hyers stability of the functional equations f ( u x + v y , u y − v x ) = f ( x , y ) + f ( u , v ) , f ( u x − v y , u y + v x ) = f ( x , y ) + f ( u , v ) , g ( u x − v y , u y − v x ) = g ( x , y ) + g ( u , v ) , g ( u x + v y , u y + v x ) = g ( x , y ) + g ( u , v ) for all ( x , y , u , v ) ∈ Γ , where Γ ⊂ R 4 is of 4-dimensional Lebesgue measure zero. The above functional equations are modified versions of the equations in [9,11,14,18,24] which arise from number theory and are in connection with characterizations of determinant and permanent of two-by-two matrices. [ABSTRACT FROM AUTHOR]

In this paper we study a class of planar piecewise smooth quadratic integrable non-Hamiltonian systems, which have a center. By using the averaging method, we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n . Our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems. [ABSTRACT FROM AUTHOR]

A variety of different orthogonal wavelet bases has been found for L 2 ( R ) for the last three decades. Such bases are not only interesting by themselves, a number of theoretical and applied problems were solved using them. It appeared that similar non-trivial wavelet constructions also exist for functions defined on some other algebraic structures, such as local fields of positive characteristic including the Vilenkin/Cantor groups. We note that most of the bases involved consist of Bruhat–Schwartz functions, i.e. compactly supported and band-limited ones. There were several attempts to develop wavelet theory in a more general setting, e.g., for zero-dimensional groups. Analyzing these papers, one can see that actually nothing except for the Haar basis and its trivial modifications was constructed. In the present paper we give an explanation for these failures. Namely, we describe the situation for the field Q p of p -adic numbers and prove that any orthogonal wavelet basis for L 2 ( Q p ) that consists of band-limited (periodic) functions, is a modification of the Haar basis. [ABSTRACT FROM AUTHOR]

For a fixed simple curve in a nonautonomous flow, the fluxing index of a passively-advected Lagrangian particle is the total number of times it goes across the curve within a given time interval. Such indices naturally induce flux sets, equivalence classes of the particles at the initial time. This line of research mainly concerns donating regions, the explicit construction of the flux sets from a sufficiently continuous velocity field. In the author's previous paper (Zhang, 2013 [13] ) the flux sets with indices ±1 were constructed from characteristic curves of the flow field. This work generalizes the notion of donating regions and shows that flux sets are index-by-index equivalent to the generalized donating regions for any finite integer index, provided that the two backward streaklines seeded at the two endpoints of the simple curve neither intersect nor self-intersect. All Lagrangian particles marked by their initial positions are thus classified by their fluxing indices. [ABSTRACT FROM AUTHOR]

We consider a variation of the Lévy–Steinitz theorem where one rearranges only a small number of terms. In particular, we study the set of obtainable rearrangements of a multidimensional series by a permutation σ : ω → ω such that { n ∈ ω : σ ( n ) ≠ n } ∈ I for some proper ideal I ⊂ P ( ω ) . [ABSTRACT FROM AUTHOR]

In this paper we study the maximum number N of limit cycles that can exhibit a planar piecewise linear differential system formed by two pieces separated by a straight line. More precisely, we prove that this maximum number satisfies 2 ≤ N ≤ 4 if one of the two linear differential systems has its equilibrium point on the straight line of discontinuity. [ABSTRACT FROM AUTHOR]

We prove a characterization showing when the ideal pointwise convergence does not imply the ideal equal (aka quasi-normal) convergence. The characterization is expressed in terms of a cardinal coefficient related to the bounding number b . We also prove a characterization showing when the ideal equal limit is unique. [ABSTRACT FROM AUTHOR]

For any real number β>1, we denote by Tβ the β-transformation on the unit interval [0,1] given by Tβ(x)=βx-βx, where denotes the integer part of We consider the size of the set of β>1 where the orbit of 1 under Tβ ultimately has a positive distance from a given point in [0,1]. For any x0∈[0,1] and any (β0,β1)(1,∞), we obtain the set of β∈(β0,β1) such that x0 is not an accumulation point of the orbit of 1 under Tβ with full Hausdorff dimension. Specifically,dimH{β∈(β0,β1):liminfn→∞|Tβn1-x0|>0}=1. This is a generalization of the result of Schmeling for the case where x0=0 and (β0,β1)=(1,∞). [ABSTRACT FROM AUTHOR]

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi-ordered linear space and the perturbation contains a family of set-valued maps satisfying certain property. From this general set-valued Ekeland variational principle, we deduce a number of particular versions of set-valued Ekeland variational principle, which include many known Ekeland variational principles, their improvements and some new results. [ABSTRACT FROM AUTHOR]

Abstract: Let K be a quadratic number field, and let denote the Dedekind zeta-function attached to K. Using the mixed second moments of derivatives of , we prove the existence of gaps between consecutive zeros of on the critical line which are at least times the average spacing. [Copyright &y& Elsevier]

Abstract: An n-dilation of a contraction T acting on a Hilbert space is a unitary dilation acting on . We show that if both defect numbers of T are equal to n, then the closure of the numerical range of T is the intersection of the closures of the numerical ranges of its n-dilations. We also obtain detailed information about the geometrical properties of the numerical range of T in case . [Copyright &y& Elsevier]

Abstract: In this note we construct Mabuchi functional and Aubin–Yau functionals on any compact complex surfaces, and establish a number of properties. Our construction coincides with the original one in the Kähler case. This method has been generalized to construct Mabuchi and Aubin–Yau functional over complex manifolds with complex dimension larger than two, see [5,6]. We will consider in [7] their applications to geometry. [Copyright &y& Elsevier]

Abstract: In this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers , and , when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity . Comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems, this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones. [Copyright &y& Elsevier]

Abstract: Motivated by a recent series of papers by K. Li and co-workers [14–18], we consider the problem of the existence and multiplicity of solutions for the Neumann or periodic BVPs associated to a class of scalar equations of the form . The class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems, one considered near zero and the other near infinity. We show how a rotation number approach, together with the Poincaré–Birkhoff theorem and a recent variant of it, allows to obtain multiplicity results in terms of the gap between the Morse indexes of the referred linear systems at zero and at infinity. These systems may also be resonant. When the gaps are sufficiently large, our multiplicity results improve the ones obtained by variational methods in the quoted papers. Also, our approach allows a description of the solutions obtained in terms of their nodal properties. [Copyright &y& Elsevier]

Abstract: We consider Ribaucour transformations for flat surfaces in the hyperbolic 3-space, . We show that such transformations produce complete, embedded ends of horosphere type and curves of singularities which generically are cuspidal edges. Moreover, we prove that these ends and curves of singularities do not intersect. We apply Ribaucour transformations to rotational flat surfaces in providing new families of explicitly given flat surfaces which are determined by several parameters. For special choices of the parameters, we get surfaces that are periodic in one variable and surfaces with any even number or an infinite number of embedded ends of horosphere type. [Copyright &y& Elsevier]

Abstract: Continuous Sheffer families have been recently introduced by the authors. These are continuous versions of the Sheffer sequences arising in the umbral calculus. We show here that quite a number of classical special functions are examples of such families. [Copyright &y& Elsevier]