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According to Lidstone's interpolation theory, an entire function of a single variable of exponential type < π is determined by it derivatives of even order at 0 and 1. In a previous paper, we gave a survey of this classical univariate theory. Here, we generalize it to two variables. Multivariate Lidstone interpolation will be the topic of a forthcoming paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new subclass of starlike functions with respect to (2 ȷ , ℓ) -symmetric conjugate points using the principle of subordination. Several relationship with the well-known classes have been established. We have focussed on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of the results are presented here as corollaries, most of which are extensions of well-known results. [ABSTRACT FROM AUTHOR]

In this paper, we study exponential tractability of multivariate approximation problem for Hermite spaces of analytic functions over ℝ d in the worst case setting. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. The considered algorithms based on tensor products of Gauss–Hermite rules for multivariate integration are constructive and use the class Λ std of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we find a necessary and sufficient condition and inclusion relation for Poisson distribution series to be in the class (k , α , β) of analytic functions with negative coefficients defined by Sălăgean differential operator involving Stirling numbers. Also, we consider an integral operator related to Poisson distribution series. Several corollaries and consequences of the main results are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we study some new results such as coefficients bounds and Fekete–Szegö inequalities of the certain classes starlike and convex functions associated with shell-like defined using the concept of Ruscheweyh q -differential operator. Comparisons of new results with those that were obtained in earlier investigation are given as corollaries. [ABSTRACT FROM AUTHOR]

In this paper, the following composite analytic functions F (z) = f (Φ (z)) and H (z) = G (Φ (z) , ... , Φ (z)) are considered, where f : ℂ → ℂ , Φ : 𝔹 n → ℂ , G : ℂ m → ℂ. We established conditions which provide equivalence of boundedness of the l -index of the function f and boundedness of the L -index in joint variables of the function F , where l : ℂ → ℝ + is a continuous function, L (z) = (l (Φ (z)) | ∂ Φ (z) ∂ z 1 | , ... , l (Φ (z)) | ∂ Φ (z) ∂ z n |). For the function H with additional restrictions, the function L is constructed such that H has bounded L -index in joint variables in the case when the function G has bounded L -index in the direction 1 = (1 , ... , 1) , where L : ℂ n → ℝ + n is a positive continuous function. Our proofs are based on the application of analog of Hayman's theorem for these classes of functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to define the operator of q -derivative based upon the Borel distribution and by using this operator, we familiarize a new subclass of β -uniformly starlike functions β - (α , q , λ , δ). Further, we obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions f ∈ β - (α , q , λ , δ). We also determine the second Hankel inequality for functions belonging to this subclass. [ABSTRACT FROM AUTHOR]

Ma–Minda class (of starlike functions) consists of normalized analytic functions f defined on the unit disk for which the image of the function z f ′ (z) / f (z) is contained in some starlike region lying in the right-half plane. In this paper, we obtain the best possible bounds on some initial coefficients for the inverse functions of Ma–Minda starlike functions. Further, the bounds on the Fekete–Szegö functional and the second Hankel determinant are computed for such functions. In addition, some sharp radius estimates are also determined. [ABSTRACT FROM AUTHOR]

The area of q -calculus has attracted the serious attention of researchers. This great interest is due its application in various branches of mathematics and physics. The application of q -calculus was initiated by Jackson [Jackson, On q -definite integrals, Quart. J. Pure Appl. Math.41 (1910) 193–203; On q -functions and a certain difference operator, Trans. Roy. Soc. Edinburgh46 (1908) 253–281.], who was the first to develop q -integral and q -derivative in a systematic way. In this paper, we make use of the concept of q -calculus in the theory of univalent functions, to obtain the bounds for certain coefficient functional problems of q -starlike and q -convex functions. Further, we also obtain similar type of inequalities related to lemniscate of Bernoulli. The authors sincerely hope that this paper will revive this concept and encourage other researchers to work in this q -calculus in the near-future in the area of complex function theory. Also, we present a direct and shortened proof for the estimates of | a 3 − μ a 2 2 | found in [Mishra and Gochhayat, Fekete–Szego problem for k -uniformly convex functions and for a class defined by Owa–Srivastava operator, J. Math. Anal. Appl.347(2) (2008) 563–572] for f ∈ k − 𝒰 𝒞 𝒱 , 0 ≤ k < ∞. [ABSTRACT FROM AUTHOR]

Let 𝒜 be the family of analytic and normalized functions f in the open unit disc | z | < 1. In this paper, we consider the following classes: ℛ (α , β) : = f ∈ 𝒜 : Re f ′ (z) + 1 + e i α 2 z f ″ (z) > β , | z | < 1 and ℒ α (b) : = f ∈ 𝒜 : f ′ (z) + 1 + e i α 2 z f ″ (z) − b < b , | z | < 1 , where − π < α ≤ π , 0 ≤ β < 1 and b > 1 / 2. We show that if f ∈ ℛ (α , β) , then Re { f ′ (z) } and Re { f (z) / z } are greater than β , and if f ∈ ℒ α (b) , then 0 < Re { f ′ (z) } < 2 b. Also, some another interesting properties of the class ℒ α (b) are investigated. Finally, the radius of univalence of 2nd section sum of f ∈ ℛ (α , β) is obtained. [ABSTRACT FROM AUTHOR]

A typical quandary in geometric functions theory is to study a functional composed of amalgamations of the coefficients of the pristine function. Conventionally, there is a parameter over which the extremal value of the functional is needed. The present paper deals with consequential functional of this type. By making use of Hohlov operator, a new subclass ℛ a , b c of analytic functions defined ins the open unit disk is introduced. For both real and complex parameter, the sharp bounds for the Fekete–Szegö problems are found. An attempt has also been taken to found the sharp upper bound to the second and third Hankel determinant for functions belonging to this class. All the extremal functions are express in term of Gauss hypergeometric function and convolution. Finally, the sufficient condition for functions to be in ℛ a , b c is derived. Relevant connections of the new results with well-known ones are pointed out. [ABSTRACT FROM AUTHOR]

In this paper, the analytic and fractional solutions of governing differential equations for helical flow of cylindrical nature have been presented. The series expansions and Laplace and Hankel transforms are applied to the governing equation of generalized Burger fluid flow for generating gamma functions. The analytical solutions of velocity fields and shear stresses are obtained through Caputo fractional approach. In order to justify the initial and boundary conditions, infinite series are invoked for expressing the analytical results of velocity fields and shear stresses in terms of H a , b + 1 1 , a (z) Fox-H function. At the end, few rheological parameters have been analyzed on four different types of models as shown in graphs. Finally, a comparative analysis of ordinary and fractional models has been focussed for angular and oscillating velocities of helical flow generated by circular cylinder. [ABSTRACT FROM AUTHOR]

In this paper, we study two new subclasses 𝒯 𝒮 m ∗ (α , λ) and 𝒯 𝒞 m (α , λ) of analytic functions which are defined by means of a differential operator. Some results connected to partial sums and neighborhoods and integral means related to these subclasses are obtained. [ABSTRACT FROM AUTHOR]

Let T ∈ B (ℋ) be a (p , k) -quasiposinormal operator on a complex Hilbert space ℋ. In this paper, we give basic properties for T and we show that a (p , k) -quasiposinormal operator T is polaroid. We also prove that all Weyl type theorems (generalized or not) hold and are equivalent for f (T) , where f is an analytic function defined on a neighborhood of σ (T). [ABSTRACT FROM AUTHOR]

In this paper, we define a sub-class of analytic bi-univalent functions associated with a certain differential operator . Bounds for the general Taylor-Maclaurin coefficients for the functions in this class are obtained. Estimates for the coefficient and the estimate for the functional for any real , are also found. Results for the specific values of the parameters , are also given mentioning some of the results obtained earlier. [ABSTRACT FROM AUTHOR]

In this paper, we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel–Rockafellar duality theory, we give an explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving the problem which relies on a tensor-kernel representation. This analysis overcomes the typical difficulties connected to learning in Banach spaces. We finally present a large class of tensor-kernels to which our theory fully applies: power series tensor kernels. This type of kernels describes Banach spaces of analytic functions and includes generalizations of the exponential and polynomial kernels as well as, in the complex case, generalizations of the Szegö and Bergman kernels. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated Fekete–Szegö problems and coefficient estimates for some subclasses of bi-univalent functions defined in terms of quasi-subordination. These inequalities result in the bounds of the third coefficient in Maclaurin's series expansion in the unit disc. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the q -Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a q -analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in q -analogs of Lidstone polynomials which are in fact q -Bernoulli polynomials as in the classical case. The definitions of these q -Bernoulli polynomials and numbers are introduced. We also introduce q -analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points z 1 and z 2 either in terms of the polynomials { 1 , ∏ k = 0 n − 1 (z − z 1 q k) (z − z 2 q k) , n ∈ ℕ } or in terms of the polynomials { 1 , ∏ k = 0 n − 1 (z 1 − z q k) (z 2 − z q k) , n ∈ ℕ }. As an application, we introduce a new series expansion for the basic hypergeometric series 2 ϕ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we introduce some interesting inequalities regarding the argument of certain p-valent functions. The subclass under investigation is defined by using certain new integral operator. Moreover, special cases and connections of the results presented here with those obtained in earlier works are pointed out. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to introduce a new kind of functions, Weakly Analytic Functions. The idea is to use this new kind of mathematical tools to describe many physical phenomena and to study the proprieties of some of the most important functions in mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a unified subclass of analytic functions by making use of the principle of subordination, involving generalized Ruscheweyh Derivative operator 𝕁 p λ , μ . The properties such as inclusion relationships, distortion theorems, coefficient inequalities and differential sandwich theorem for the above class have been discussed. [ABSTRACT FROM AUTHOR]

Let f : (ℝn, 0) → (ℝ, 0) be a nonconstant analytic function defined in a neighborhood of the origin 0 ∈ ℝn. The classical Łojasiewicz inequality states that there exist positive constants δ, c and l such that |f(x)| ≥ cd(x, f-1(0))l for |x| ≤ δ, where d(x, f-1(0)) denotes the distance from x to the set f-1(0). The Łojasiewicz exponent of f at the origin 0 ∈ ℝn, denoted by , is the infimum of the exponents l satisfying the Łojasiewicz inequality. In this paper, we establish a formula for computing the Łojasiewicz exponent of f in terms of the Newton polyhedron of f in the case where f is nonnegative and nondegenerate. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of quasi-convex functions by considering a subclass of normalized analytic functions in the unit disk for some convex function with fixed second coefficient of order , such that We obtain results on growth and distortion theorems and radius of convexity. [ABSTRACT FROM AUTHOR]

This is the first of two papers in which we study the modular invariance of pseudotraces of logarithmic intertwining operators. We construct and study genus-one correlation functions for logarithmic intertwining operators among generalized modules over a positive-energy and -cofinite vertex operator algebra . We consider grading-restricted generalized -modules which admit a right action of some associative algebra , and intertwining operators among such modules which commute with the action of (-intertwining operators). We obtain duality properties, i.e. suitable associativity and commutativity properties, for -intertwining operators. Using pseudotraces introduced by Miyamoto and studied by Arike, we define formal -traces of products of -intertwining operators, and obtain certain identities for these formal series. This allows us to show that the formal -traces satisfy a system of differential equations with regular singular points, and therefore are absolutely convergent in a suitable region and can be extended to yield multivalued analytic functions, called genus-one correlation functions. Furthermore, we show that the space of solutions of these differential equations is invariant under the action of the modular group. [ABSTRACT FROM AUTHOR]

We extend our previous analysis of a model for "dark energy" based on a Weyl scaling invariant dark energy action. We reexpress all prior results in terms of proper time, using the fluctuation amplitude Φ without approximation, and derive a compact formula for the squared effective Hubble parameter. This formula involves effective dark energy and matter densities that differ from their expressions in the standard Λ CDM cosmology. We also give new analytic results for the function Φ and discuss their implications. [ABSTRACT FROM AUTHOR]

In this paper, associated with convolutions, linear operators and are defined with the requirement of certain recurring relations. Some results based on analytic functions involving these operators are derived. Taking some specific forms of such operators, results involving the Dziok-Srivastava operator, the Jung-Kim-Srivastava integral operator, the Srivastava-Attiya linear operator and others, are also mentioned. [ABSTRACT FROM AUTHOR]

In this paper, a new Hamiltonian lattice hierarchy is analytically investigated, which can be reduced to some classic integrable lattice hierarchies, such as Ablowitz-Ladik hierarchy, Volterra hierarchy and multi-Hamiltonian lattice hierarchy, etc. By choosing the auxiliary problem , we present a Darboux transformation (DT) to the new discrete matrix spectral problem. As its applications, a series of analytical solutions are generated in a recursive manner. Finally, the graphical analysis of these analytical solutions are presented, respectively. The DT of other lattice hierarchies can be also constructed in this method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class 풞(ϕ, ψ, η) of analytic functions defined by means of subordination. Coefficient bounds, Fekete-Szegö problem and norm estimates of the pre-Schwarzian derivatives of functions belonging to the class 풞(ϕ, ψ, η) are investigated. A class of multiple close-to-convex functions is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form ψ(z; r) are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of ξ at each pole, as well as some well-known real analytic functions, that appear for instance in the completion of Appell-Lerch sums. [ABSTRACT FROM AUTHOR]

In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self-consistent, if the whole energy content is not uniquely defined in a specific volume. Here, we generalize the Einstein gravitational energy–momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator □ − 1 are taken into account. We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy–momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations. Furthermore, the energy–momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy–momentum complex, taking into account both gravitational and matter components, can be derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications. [ABSTRACT FROM AUTHOR]

We analyze the structure of the eigenvalue of the color-singlet Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation in N = 4 SYM in terms of the meromorphic functions obtained by the analytic continuation of harmonic sums from positive even integer values of the argument to the complex plane. The meromorphic functions we discuss have pole singularities at negative integers and take finite values at all other points. We derive the reflection identities for harmonic sums at weight five decomposing a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or non-negative values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible set of bilinear reflection identities at weight five which presents the main result of the paper. We show how the reflection identities can be used to restore the functional form of the next-to-leading eigenvalue of the color-singlet BFKL equation in N =4 SYM, i.e. we argue that it is possible to restore the full functional form on the entire complex plane provided one has information how the function looks like on just two lines on the complex plane. Finally we discuss how nonlinear reflection identities can be constructed from our result with the use of well known quasi-shuffle relations for harmonic sums. [ABSTRACT FROM AUTHOR]

We consider the class of λ -pseudo starlike functions f (z) = z + a 2 z 2 + a 3 z 3 + ⋯ such that z (f ′ (z)) λ / f (z) maps the open unit disk | z | < 1 onto a strip domain w with α − π 2 sin α < ℜ (w (z)) < α 2 sin α for some α , π / 2 ≤ α < π. We estimate | a 2 | , | a 3 | and solve the Fekete–Szegö problem for functions in this class. [ABSTRACT FROM AUTHOR]

Based on q -Hypergeometric Series, a linear operator H s r [ α 1 ; q ] : → is considered and involving this operator a class S q r , s ([ α 1 ] ; k , α) of analytic functions is defined by using q -derivatives. As a special case, a class H q (a , b ; c ; k , α) by involving q -analogue of Hohlov operator I a , b , c q is defined. Coefficient inequality for a function f to be in the class S q r , s ([ α 1 ] ; k , α) is obtained. Further, in terms of subordination, an equivalent condition for a function f to be in this class is given and using this equivalent class condition results on coefficient estimates including Fekete–Szegö inequality and a convolution result are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study certain properties involving the generalized Srivastava-Attiya operator. A set of subordination results are obtained and some special cases connected with the Hurwitz-Lerch zeta function and their relevances with known results are also pointed out. [ABSTRACT FROM AUTHOR]

We consider the Noetherian symmetries of second-order ODEs subjected to forces with nonzero curl. Both position and velocity dependent forces are considered. In the former case, the first integrals are shown to follow from the symmetries of the celebrated Emden-Fowler equation. [ABSTRACT FROM AUTHOR]

We study the escape rate for the Farey map, an infinite measure preserving system, with a hole including the indifferent fixed point. Due to the ergodic properties of the map, the standard theoretical approaches to this problem cannot be applied. It has been recently shown in [Knight & Munday, 2016] how to apply the standard analytical methods to a piecewise-linear version of the Farey map with holes depending on the associated partition, but their results cannot be obtained in the general case we consider here. To overcome these difficulties we propose here to study approximations of the hole by means of real analytic functions. We introduce a particular family of approximations and study numerically the behavior of the escape rate for approximated holes with vanishing measure. The results suggest that the scaling of the escape rate depends on the 'shape' of the approximation, and we show that this is a typical feature of systems with an indifferent fixed point, not an artifact of the particular family we consider. [ABSTRACT FROM AUTHOR]

We study the impact of nonlocal modifications of General Relativity on stellar structure. In particular, assuming an analytic distortion function and specific equations of state, we made use of remnant stars to put qualitative constraints on a parameter not directly restricted by solar system tests. Using current data sets available for white dwarfs and strange quark stars candidates, we find that the most stringent bounds come from the objects displaying the highest core densities, namely strange quark stars. Specifically, the constraints obtained from this class of stars are three to four orders of magnitude tighter than those obtained using white dwarfs. [ABSTRACT FROM AUTHOR]

The article focuses on the theory of weighted composition operators that appears in a natural way on different spaces of functions and play a pivotal role in the theory of isometries of Bergman spaces, Hardy spaces, Lpspaces, spaces of continuous functions and many other spaces of functions. Topics include examines many mathematicians start investigating different properties of the products of differentiation, multiplication and composition operators on different spaces of analytic functions.

The article focuses on using the theory of differential subordination as investigate differential inequalities satisfied by an analytic function w defined on the unit disc, that imply the boundedness of the function w. Topics include considered the results are applied to find sufficient conditions for the normalized analytic functions to be either starlike or close-to-convex.

We are interested in a WKB analysis of the logarithmic nonlinear Schrödinger equation with "Riemann-like" variables in an analytic framework in semiclassical regime. We show that the Cauchy problem is locally well posed uniformly in the semiclassical constant and that the semiclassical limit can be performed. In particular, our framework is not only compatible with the Gross–Pitaevskii equation with logarithmic nonlinearity, but also allows initial data (and solutions) which can converge to 0 at infinity. [ABSTRACT FROM AUTHOR]

Let f be an analytic function defined on the open unit disc , with f (0) = f ′ (0) − 1 = 0 , satisfying the subordination z f ′ (z) / f (z) ≺ L s (z) : = (1 + s z) 2 , where 0 < s ≤ 1 / 2 . The domain L s () is bounded by a Limaçon and the function f is called starlike function associated with Limaçon domain. For (1 − s) 2 < a < (1 + s) 2 , we find the smallest disc (a ; R a) and the largest disc (a ; r a) , centered at (a , 0) such that the domain L s () is contained in (a ; R a) and contains (a ; r a). By using this result, we find the radius of Limaçon starlikeness for the class of starlike functions of order α (0 ≤ α < 1) and the class of functions f satisfying Re (z f ′ (z) / f (z)) < β , 1 < β < 4 / 3. We give extension of our results for Janowski starlike functions. [ABSTRACT FROM AUTHOR]

Assume Ω ⊂ ℝ 2 is a planar domain, and u is a locally bounded distributional solution to the elliptic equation − Δ u = | x | 2 β h (x) f (u) in  Ω , where β > − 1 is a constant, h and f are real analytic functions defined on Ω and the real line ℝ , respectively. We establish asymptotic expansions of u (x) to arbitrary orders near 0 , which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction. [ABSTRACT FROM AUTHOR]

Some sufficient conditions on certain constants which are involved in some first and second-order differential subordinations associated with certain functions with positive real part like modified Sigmoid function, exponential function and Janowski function are obtained so that the analytic function p normalized by the condition p (0) = 1 , is subordinate to Janowski function. The admissibility conditions for Janowski function are used as a tool in the proof of the results. As application, several sufficient conditions are also computed for Janowski starlikeness. [ABSTRACT FROM AUTHOR]

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from ℂ 1 + n with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space ℂ ℙ n and the complex hyperbolic disc n . We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of "polynomial" functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on ℂ ℙ n and n . More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the ∗ -involution of pointwise complex conjugation. [ABSTRACT FROM AUTHOR]

We study results related to a conjecture formulated by Strohmer and Beaver about optimal Gaussian Gabor frame set-ups. Our attention will be restricted to the case of Gabor systems with standard Gaussian window and rectangular lattices of density 2. Although this case has been fully treated by Faulhuber and Steinerberger, the results in this work are new and quite curious. Indeed, the optimality of the square lattice for the Tolimieri and Orr bound already implies the optimality of the square lattice for the sharp lower frame bound. Our main tools include determinants of Laplace–Beltrami operators on tori as well as special functions from analytic number theory, in particular Eisenstein series, zeta functions, theta functions and Kronecker's limit formula. We note that our results also carry over to energy minimization problems over lattices and a heat distribution problem over flat tori. [ABSTRACT FROM AUTHOR]

Suppose that L (∂ t , ∂ x) is a homogeneous constant coefficient strongly hyperbolic partial differential operator on ℝ t , x 1 + d and that H is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of H , the characteristic variety of L is the graph of a real analytic function τ (ξ) with rank τ ξ ξ identically equal to zero or the maximal possible value d − 1. Suppose that the source function f is compactly supported in t ≥ 0 and piecewise smooth with singularities only on H. Then the solution of L u = f with u = 0 for t < 0 is uniformly bounded on ℝ 1 + d . Typically when rank τ ξ ξ ≠ 0 on the conormal variety, the sup norm of the jump in the gradient of u across H grows linearly with t. [ABSTRACT FROM AUTHOR]

Using the Dziok–Srivastava operator, we describe the class of p-valent analytical functions ϰ p q , s (k , α 1 , α , β , μ , λ) and we obtain certain interesting properties of this class. [ABSTRACT FROM AUTHOR]

The axiomatic Wightman formulation for nonderivative conformal field theory is adopted to derive conformal bootstrap equation for the four-point function. The equivalence between PCT theorem and weak local commutativity, due to Jost plays a very crucial role in axiomatic field theory. The theorem is suitably adopted for conformal field theory to derive the desired equations in CFT. We demonstrate that the two Wightman functions are analytic continuation of each other. [ABSTRACT FROM AUTHOR]

Starting from divisibility problem for Fibonacci numbers, we introduce Fibonacci divisors, related hierarchy of Golden derivatives in powers of the Golden Ratio and develop corresponding quantum calculus. By this calculus, the infinite hierarchy of Golden quantum oscillators with integer spectrum determined by Fibonacci divisors, the hierarchy of Golden coherent states and related Fock–Bargman representations in space of complex analytic functions are derived. It is shown that Fibonacci divisors with even and odd k describe Golden deformed bosonic and fermionic quantum oscillators, correspondingly. By the set of translation operators we find the hierarchy of Golden binomials and related Golden analytic functions, conjugate to Fibonacci number F k . In the limit k → 0 , Golden analytic functions reduce to classical holomorphic functions and quantum calculus of Fibonacci divisors to the usual one. Several applications of the calculus to quantum deformation of bosonic and fermionic oscillator algebras, R -matrices, geometry of hydrodynamic images and quantum computations are discussed. [ABSTRACT FROM AUTHOR]

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore–Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. [ABSTRACT FROM AUTHOR]