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Following Ramanujan’s work on modular equations and approximations of π, there are formulas for 1/π of the form [PLEASE CHECK FORMULA IN THE PDF] for d = 2, 3, 4, 6, where λd are singular values that correspond to elliptic curves with complex multiplication, and α, δ are explicit algebraic numbers. In this paper we prove a ρ-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.

Let C1 and C2 be algebraic plane curves in ℂ 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1( ℂ 2 \ C1 U C2)) ≅ π1 ( ℂ 2 \ C1) × π1 ( ℂ 2 \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in ℂ 2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.

In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.

Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3/2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).

In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case.

In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation.

Motivated by the dynamics of a trimaran, an investigation of the dynamic behaviour of a double forcing parametrically excited system is carried out. Initially, we provide an outline of the stability regions, both numerically and analytically, for the undamped linear, extended version of the Mathieu equation. This paper then examines the anticipated form of response of our proposed nonlinear damped double forcing system, where periodic and quasiperiodic routes to chaos are graphically demonstrated and compared with the case of the single vertically-driven pendulum.

In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results.

Let \(\mathbb{N}_{0}\) be the set of all non-negative integers and \(\mathcal{P}(\mathbb{N}_{0})\) be its power set. Then, an integer additive set-indexer (IASI) of a given graph \(G\) is defined as an injective function \(f:V(G)\to \mathcal{P}(\mathbb{N}_{0})\) such that the induced edge-function \(f^+:E(G) \to\mathcal{P}(\mathbb{N}_{0})\) defined by \(f^+ (uv) = f(u)+ f(v)\) is also injective, where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). An IASI \(f\) of \(G\) is said to be a strong IASI of \(G\) if \(|f^+(uv)|=|f(u)|\,|f(v)|\) for all \(uv\in E(G)\). The nourishing number of a graph \(G\) is the minimum order of the maximal complete subgraph of \(G\) so that \(G\) admits a strong IASI. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers and determine their corresponding nourishing numbers.

In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.

We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD.

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.

This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin.

The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).

The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C.

Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An integer additive set-labeling (IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.

Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly.

In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former.

The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-structures.

This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.

We consider two principal bundles of embeddings with total space E m b ( M , N ) , with structure groups D i f f ( M ) and D i f f + ( M ) , where D i f f + ( M ) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B ( M , N ) = E m b ( M , N ) / D i f f ( M ) and B + ( M , N ) = E m b ( M , N ) / D i f f + ( M ) from the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. It is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in the mathematical litterature e.g. by H. Omori and by T. Ratiu. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B ( M , N ) . A case of particular interest is M = S 1 , where connected components of B + ( S 1 , N ) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + ( S 1 , N ) is a subgroup of some group G L r e s , following the classical notations of (Pressley, A., 1988). These constructions suggest some approaches in the spirit of one of our previous works on Chern-Weil theory that could lead to knot invariants through a theory of Chern-Weil forms.

In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques.

Burg’s entropy plays an important role in this age of information euphoria, particularly in understanding the emergent behavior of a complex system such as statistical mechanics. For discrete or continuous variable, maximization of Burg’s Entropy subject to its only natural and mean constraint always provide us a positive density function though the Entropy is always negative. On the other hand, Burg’s modified entropy is a better measure than the standard Burg’s entropy measure since this is always positive and there is no computational problem for small probabilistic values. Moreover, the maximum value of Burg’s modified entropy increases with the number of possible outcomes. In this paper, a premium has been put on the fact that if Burg’s modified entropy is used instead of conventional Burg’s entropy in a maximum entropy probability density (MEPD) function, the result yields a better approximation of the probability distribution. An important lemma in basic algebra and a suitable example with tables and graphs in statistical mechanics have been given to illustrate the whole idea appropriately.

In this paper we study some properties of conformal maps between equidimensional manifolds, we construct new example of non-harmonic biharmonic maps and we characterize the biharmonicity of some maps on the warped product manifolds.