Search

In this article, we prove new results for the anticipating stochastic integral introduced by Ayed and Kuo. We present an existence criterion for the integral, a Fubini's theorem, a Leibniz integral rule, and an alternative definition for a class of integrands. [ABSTRACT FROM AUTHOR]

The cornerstone of osteopathic care lies in the osteopathic tenets—first of all, the idea of a self-regulating, dynamic unit made of body-mind-spirit. The clinical application of the osteopathic principles mainly relies on the structure-function models, but the practitioners’ community is still trying to reach a consensus on the fundamental theoretical framework. Mostly, the debate swings between the biomechanical-structural pole and the biopsychosocial pole. However, there is a compelling need for a robust conceptual framework in osteopathic care. It is necessary to draw up a more consistent interprofessional framework, emphasizing the distinctive focus of the osteopathic intervention in health care. In the present hypothesis paper, the different osteopathic care models are integrated into the 4-quadrant model of the Integral Theory. In light of the Integral Theory, osteopathic care can be construed to improve the individual mind-body function and spiritual behavior integrated with the environment. [ABSTRACT FROM AUTHOR]

n this paper we introduce the concepts of Wijsman ρ-statistical convergence of order α and Wijsman strongly ρ-convergence of order α. In addition, some inclusion theorems are also presented. [ABSTRACT FROM AUTHOR]

We present a position Langevin equation for overdamped particle motion on rough two-dimensional surfaces. A Brownian dynamics algorithm is suggested to evolve this equation numerically, allowing for the prediction of effective (projected) diffusion coefficients over corrugated surfaces. In the case of static surface roughness, we find that a simple area-scaling prediction for the projected diffusion coefficient leads to seemingly quantitative agreement with numerical results. To study the effect of dynamic surface evolution on the diffusive process, we consider particle diffusion over a thermally fluctuating elastic membrane. Surface fluctuation has the effect of increasing the effective diffusivity toward a limiting annealed-surface value discussed previously. We argue that protein motion over cell surfaces spans a variety of physical regimes, making it impossible to identify a single approximation scheme appropriate to all measurements of interest. [ABSTRACT FROM AUTHOR]

Abstract This paper addresses the problem of synchronization control for semi-Markov jump neural networks with mode-dependent stochastic parametric uncertainties covering a finite-time period. A more general semi-Markov jump neural network is developed by the mode-dependent stochastic parametric uncertainties technique, where both upper and lower bounds of parametric uncertainties are taken into consideration in determining the imprecise measurements. The time-varying semi-Markov chain information subject to uncertainty is established, and sufficient conditions are achieved combine with some integral method. Finally, two numerical examples are exhibited to verify the effectiveness of the produced scheme. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we prove some new fixed point theorems for self-mappings defined on a complete metric space with a w -distance. These results extend some previous fixed point theorems in this field to more general contractive conditions in the setting of w -distances for selfmappings which satisfy certain weaker Meir–Keeler conditions. [ABSTRACT FROM AUTHOR]

In this work, firstly, we established Hermite-Hadamard's inequalities for harmonically convex functions via Katugampola fractional integrals. Then we give some Hermite-Hadamard type inequalities of these classes functions. [ABSTRACT FROM AUTHOR]

Perturbation to Noether symmetry and adiabatic invariants are investigated for the generalized fractional Birkhoffian system with the combined Riemann-Liouville fractional derivative and the combined Caputo fractional derivative, respectively. Firstly, differential equations of motion for the generalized fractional Birkhoffian system are established. Secondly, Noether symmetry and conserved quantity are studied. Thirdly, perturbation to Noether symmetry and adiabatic invariants are presented for the generalized fractional Birkhoffian mechanics. And finally, several applications are discussed to illustrate the results and methods. [ABSTRACT FROM AUTHOR]

The existing content-based recommendation methods have two major limitations. First, due to the defects of the items and the user model matching algorithms, the recommendation results are very narrow. Second, scant attention is paid to the scenario, making the recommendation system not context-aware. It is essential to improve user satisfaction through high-quality recommendation. In this paper, two state-of-the-art methods are analyzed and extended to enhance recommendation performance. The first method is the context-aware recommender, which integrates context information into the recommendation process. The second method is the semantic analysis-based recommender, which incorporates domain semantics. Despite their compatibility, the challenge is to combine them in a way that will fully exploit their potential. An improved content-based model is proposed in this paper incorporating both semantics and context. Context-aware recommendation is performed to improve sensitivity to the context. Semantic relevance-based instance similarity is computed to address the problem of narrowness. The proposed recommendation system is evaluated using metrics (for instance, recall metric) and paralleled with the current methods grounded on the content. Results demonstrate the superiority of the proposed system in terms of accuracy. [ABSTRACT FROM AUTHOR]

Starting from the stationary Schrödinger equation for a system of identical interacting particles, the three-dimensional differential force law (DFL) is derived in terms of the kinetic energy density tensor with components tαβ(x), the particle density n(x), and the potential. The most general vector field h(x) is given such that integrating the scalar product of h with the DFL over an arbitrary volume Ω yields theorems involving in their volume integrals the tensor components only in the form t≡∑3α=1tαα (if at all) t being the positive definite density of kinetic energy. The procedure results in four integral theorems: (i) balance equation of forces, (ii) balance equation of torques, (iii) the generalized virial theorem, and (iv) a new exact theorem which can be regarded as vector theorem on the first moment of the kinetic energy density. The new theorem is shown to imply validity of the other three, and therefore is more comprehensive than they. [ABSTRACT FROM AUTHOR]

We show that a recent non-Born–Oppenheimer calculation of dipole moments exhibits obscure points and is not consistent with the well known Hellmann–Feynman theorem. [ABSTRACT FROM AUTHOR]

This paper reviews the fractional vectorial differential operators proposed previously and introduces the fractional versions of the classic Green's, Stokes', and Ostrogradski-Gauss's integral theorems. The suitability of fractional derivatives for sciences and the Laplacian definition are also discussed. [ABSTRACT FROM AUTHOR]

Green's formula and Gauss's formula are two important formulas in vector calculus. In 2008, Tarasov [12] developed the fractional Green and Gauss formulas and also suggested two possible extensions of his fractional vector formulas (see V.E. Tarasov (2008) [12] ). The first possible extension is to prove his fractional integral theorems for a general form of domains and boundaries, such as elementary regions. The second one is to generalize the formulations of fractional integral theorems for α > 1 . The purpose of this article is to follow the above two interesting suggestions and present the higher-order fractional Green and Gauss formulas that are the extensions of fractional integral theorems obtained by Tarasov. In particular, the obtained formulas can be reduced to the classical Green and Gauss formulas when α = 1 and the fractional Green and Gauss formulas in [12] when 0 < α ≤ 1 , respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some problems related to the almost sure central limit theorem for the hybrids of empirical and partial sum processes. More precisely, under mild technical conditions, we study the almost sure (a.s.) convergence of where denote the hybrids of empirical and partial sum processes, is a bounded Lipschitz function, denotes a sequence of positive weights, and [ABSTRACT FROM AUTHOR]

We consider monogenic functions taking values in a topological vector space being an expansion of a certain infinite-dimensional commutative Banach algebra associated with the three-dimensional Laplace equation. We establish also integral theorems for monogenic functions taking values in the mentioned algebra and the mentioned topological vector space. [ABSTRACT FROM AUTHOR]

Differintegral theorems are applied to solve some ordinary differential equations and fractional differential equations. By using these theorems, we obtain different results in the fractional differintegral forms. In this paper, we aim to solve the radial Schrödinger equation under the potential V (r) = H/r² - K/r + Lrκ in κ = 0, -1, -2 cases. We also obtain the solutions in the hypergeometric forms. [ABSTRACT FROM AUTHOR]

We give a new proof of the best presently-known error term in the prime geodesic theorem for compact hyperbolic surfaces, without the assumption of excluding a set of finite logarithmic measure. Stronger implications of the Gallagher-Koyama approach are derived, yielding to a further reduction of the error term outside a set of finite logarithmic measure. [ABSTRACT FROM AUTHOR]

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and ε p),θ( G − ,ω ) , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained. [ABSTRACT FROM AUTHOR]

For G-monogenic mappings taking values in the algebra of complex quaternions, we generalize some analogues of classical integral theorems of the holomorphic function theory of complex variable (the surface and curvilinear Cauchy integral theorems). [ABSTRACT FROM AUTHOR]

Some 20 years after the dawn of participative democracy, there is little noticeable or substantial change in the living conditions of the average South African. The country remains divided by race, class and economics. Poverty, inequality and racial enmity remain looming challenges to human flourishing and social transformation. Some have begun to ask whether forgiveness for the sins of colonialism and apartheid are possible. This article engages with the (im)possibility of forgiveness as it is presented in Matthew 18:15-35. In particular, it does so from the bilingual perspective of a public theological engagement with the text and its contemporary readers in South Africa. By reading the text from an integral All Quadrants All Levels (AQAL) approach this article extrapolates a textured understanding of forgiveness that 'possibilises' the (im)possiblity of forgiveness between racially and socially divided groups of readers. [ABSTRACT FROM AUTHOR]

Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$, and we show that this bound is best possible. [ABSTRACT FROM PUBLISHER]

According to the Gottesman-Knill theorem, quantum algorithms that utilize only the operations belonging to a certain restricted set are efficiently simulable classically. Since some of the operations in this set generate entangled states, it is commonly concluded that entanglement is insufficient to enable quantum computers to outperform classical computers. I argue in this article that this conclusion is misleading. First, the statement of the theorem (that the particular set of quantum operations in question can be simulated using a classical computer) is, on reflection, already evident when we consider Bell's and related inequalities in the context of a discussion of computational machines. This, in turn, helps us to understand that the appropriate conclusion to draw from the Gottesman-Knill theorem is not that entanglement is insufficient to enable a quantum performance advantage, but rather that if we limit ourselves to the operations referred to in the Gottesman-Knill theorem, we will not have used the resources provided by an entangled quantum system to their full potential. [ABSTRACT FROM AUTHOR]

Similarity analysis shows that Nu x varies as Gr x 1/4 for natural convection on an isothermal vertical surface but Nu x varies as Gr x 1/5 for isothermal horizontal surfaces. It is thus difficult to develop a rigorously-derived, closed-form solution for Nu x on a surface with arbitrary inclination. In the present study we have formulated, for the first time, a unified integral theory for laminar natural convection on an arbitrarily inclined surface, both for specified variation in surface temperature ( T w ( x ) = T ∞ + f 1 ( x )) and surface heat flux ( q w = f 2 ( x )), such that the Nusselt number matches with results obtained from the similarity analysis in the limiting cases of vertical and horizontal surfaces. The predictions of the present formulation also agree well with previous computational and experimental results at intermediate angles of inclination between the vertical and the horizontal. f 1 ( x ) or f 2 ( x ) can be any arbitrary function, including power law variation, and represents a differentially heated surface. Another important feature of the present integral theory is that the developed generalized equations can accommodate arbitrary orders of polynomials ( λ and χ ) representing the velocity and temperature profiles, and optimum values for λ and χ have been systematically determined for various boundary conditions (i.e. λ = 4, χ = 2 for isothermal case and λ = 3, χ = 2 for constant-heat-flux case). Because of the simplicity of the present theory, it is easy to generate results for combinations of Grashof number, Prandtl number and inclination angle not presented here. The different physical mechanisms for natural convection on vertical and horizontal surfaces (buoyancy versus indirect pressure difference) are explained with the help of the present analysis. It is shown that for moderate to high Prandtl number fluids, the natural convection mechanism for vertical surface is the dominating factor for a large range of inclination angles except for near horizontal configurations. The range of inclination angles for which the vertical solution predominates decreases as the Prandtl number decreases. For very low Prandtl number fluids at low Grashof number, the vertical mechanism applies only to nearly vertical surfaces. A physical explanation for such behaviour is discovered here, for the first time, in terms of the relative magnitudes of the buoyancy and indirect pressure difference. Compact scaling laws for significant data reduction are proposed and explained. New algebraic correlations have been developed that give Nusselt number as explicit functions of Grashof number, Prandtl number and inclination angle. A new methodology for the representation of the results brings out more powerfully the role of inclination angle in determining the heat transfer rate as well as the mechanism of natural convection. [ABSTRACT FROM AUTHOR]

In this article we prove the Leibniz series for π which states that π/4 = ∞Σ/n=0 (-1)n/2·n+1 The formalization follows K. Knopp [8], [1] and [6]. Leibniz's Series for Pi is item #26 from the "Formalizing 100 Theorems" list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/ . [ABSTRACT FROM AUTHOR]

This article formalizes the proof of Niven's theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr1fWiki (https://proofwiki.org/wiki/Niven's_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9]. [ABSTRACT FROM AUTHOR]

This paper descibes our hunt for the solver of Problem 157 in the Scottish Book, a problem originally posed by A. J. (Gus) Ward in 1937. We first make the observation that a theorem of Richard O'Malley from 1975 yields an immediate positive solution. A further look at O'Malley's references revealed a 1970 paper by Donald Ornstein that we now believe contains the first solution of SB 157. We isolate the common elements in the machinery used by both Ornstein and O'Malley and discuss several consequences. We also examine an example function given by Ornstein. There are some difficulties with this function but we provide a fix, and show moreover that functions of that kind are typical in the sense of the Baire category theorem. [ABSTRACT FROM AUTHOR]

A definition of classical and quantum chaos on the basis of the Liouville-Arnold theorem is proposed. According to this definition, a chaotic quantum system that has N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) that are determined by the symmetry of the Hamiltonian for the system being considered. Quantitative measures of quantum chaos are established. In the classical limit, they go over to the Lyapunov exponent or the classical stability parameter. The use of quantum-chaos parameters in nuclear physics is demonstrated. [ABSTRACT FROM AUTHOR]

Consensus problem of high-order integral multi-agent systems under switching directed topology is considered in this study. Depending on whether the agent’s full state is available or not, two distributed protocols are proposed to ensure that states of all agents can be convergent to a same stationary value. In the proposed protocols, the gain vector associated with the agent’s (estimated) state and the gain vector associated with the relative (estimated) states between agents are designed in a sophisticated way. By this particular design, the high-order integral multi-agent system can be transformed into a first-order integral multi-agent system. Also, the convergence of the transformed first-order integral agent’s state indicates the convergence of the original high-order integral agent’s state, if and only if all roots of the polynomial, whose coefficients are the entries of the gain vector associated with the relative (estimated) states between agents, are in the open left-half complex plane. Therefore, many analysis techniques in the first-order integral multi-agent system can be directly borrowed to solve the problems in the high-order integral multi-agent system. Due to this property, it is proved that to reach a consensus, the switching directed topology of multi-agent system is only required to be ‘uniformly jointly quasi-strongly connected’, which seems the mildest connectivity condition in the literature. In addition, the consensus problem of discrete-time high-order integral multi-agent systems is studied. The corresponding consensus protocol and performance analysis are presented. Finally, three simulation examples are provided to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

A finite element model is proposed to simulate the residual stresses (thermal and intrinsic stresses) of a system consisting of a DLC coating on a steel substrate with rough surfaces. The risk of failure of residually stressed coating–substrate systems is evaluated by using the cohesive zone model, extended finite element method, and J -integral theory. It is found that relatively large residual stresses as well as coating cracks are normally generated near the convex asperities of the interface, and the steel plasticity is concentrated around the concave asperity of the substrate surface near the edge. Moreover, driving forces and evolutions of failures in the coating–substrate system are presented. One can see that the interfacial failure is more sensitive to the shear traction delamination than to the normal one. For the issue of multi-crack, two closely spaced crack tips are vulnerable to coalesce during the propagation process. Additionally, in the light of mechanics, it is demonstrated that the interlayer Ti is effective in failure protection for the entire system. Numerical results have also been compared with other computational or experimental works, and can establish a theoretical basis for enhancing the durability of PVD coating–substrate systems. [ABSTRACT FROM AUTHOR]

Several classical problems are related to mixed polynomial-exponential equations. Such equations have been also considered recently by many authors. In the present paper, extending a theorem of the second and last authors, we completely solve the title equation in positive integers a, b, y, n with n ≥ for all values of x, z with 1 ≤ x, z ≤ 50 and x ≢ z (mod 2). It is interesting to note that apparently deep effective tools (e.g. Baker's method) alone are not sufficient to handle the problem completely. In our arguments we combine local arguments and Baker's method to prove our results. [ABSTRACT FROM AUTHOR]

We extend a result of the second author [27, Theorem 1.1] to dimensions d ≥ 3 which relates the size of Lp-norms of eigenfunctions for 2 < p < 2(d + 1)/(d - 1) to the amount of L²-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an "ε-removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] L² oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, the L²-norm of eigenfunctions eλ over unit-length tubes of width λ-1/2 goes to zero. Using our main estimate, we deduce that, in this case, the Lp-norms of eigenfunctions for the above range of exponents are relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions d ≥ 3 of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature. [ABSTRACT FROM AUTHOR]

For G = G(R), a split, simply connected, semisimple Lie group of rank n and K the maximal compact subgroup of G, we give a method for computing Iwasawa coordinates of K\G using the Chevalley generators and the Steinberg presentation. When K\G is a scalar coset for a supergravity theory in dimensions = 3, we determine the action of the integral form G(Z) on K\G. We give explicit results for the action of the discrete U-duality groups SL2(Z) and E7(Z) on the scalar cosets SO(2)\SL2(R) and [SU(8)/{±Id}] \ E7(+7)(R) for type IIB supergravity in ten dimensions and 11-dimensional supergravity reduced to D = 4 dimensions, respectively. For the former, we use this to determine the discrete U-duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum-generating symmetry group for fundamental BPS solitons of type IIB supergravity in D = 10 dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U-duality groups in general. [ABSTRACT FROM AUTHOR]

We obtain the integral limit theorems for the first passage time through an arbitrary remote boundary by a compound renewal process both for the cases of finite and infinite variance of the process. In the latter case, we assume that some distributions belong to the attraction domain of the stable law. [ABSTRACT FROM AUTHOR]

Nearly 70 % of sub-Saharan Africa's urban population live in informal settlements and populations are expected to double by 2030. Based on the constitutional right to adequate housing and growing public pressure and dissent, the South African government has begun a process of large-scale formalisation through the provision of housing and infrastructure to informal areas. A disjuncture has however occurred and conflicts have arisen between what are understood as modernist ideas of how cities should appear and function (formality) and an alternate, organic and flexible mode of thought (the informal). This conflict is seen by some authors as a 'clash of governmentalities' which goes deeper than a simple lack of dialogue, inadequate participation and/or a disinclination to see others' points of view. For a number of years, calls have been made for a way to organise these governmentalities and perspectives, understand what goes on at their interface and unpack the complexity that exists between them. Integral Theory and its AQAL (All Quadrants, All Levels) framework are fast becoming a sought-after arena of academic discourse. It deals specifically with complex interactions and perspectives and offers a methodology that draws together a number of already existing separate paradigms and perspectives into a unified, interrelated framework. This paper uses the AQAL framework and Integral Theory as a tool to map the differing governmentalities (rationalities, techniques and practices) that exist in the upgrading of informal settlements in Cape Town and understand their relationship and interactions. [ABSTRACT FROM AUTHOR]

Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the threesphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition. [ABSTRACT FROM AUTHOR]

External scattering from a finite phononic crystal (PC) is studied using the Helmholtz-Kirchhoff integral theorem integrated with a Bloch wave expansion (BWE). The BWE technique is used to describe the internal pressure field of a semi-infinite or layered PC subject to an incident monochromatic plane wave. Following the BWE solution, the Helmholtz-Kirchhoff integral is used to determine the external scattered field. For cubic PCs, the scattered results are compared to numerical treatments in both the frequency and time domain. The presented approach is expected to be valid when the PC size is larger than the acoustic wavelength. However, very good agreement in the spatial beam pattern is also documented for both large and small (with respect to the wavelength) PCs. The result of this work is a fully-analytical, efficient, and verified approach for accurately predicting external scattering from finite, three-dimensional PCs. [ABSTRACT FROM AUTHOR]

We are concerned with the following third-order boundary value problem with integral boundary condition:   u′′′(t)=f(t,u(t)),  t∈[0,1],  u′(0)=u(1)=0,  u′′(η)+∫αβ‍u(t)dt=0, where 1/2<α≤β≤1,  α+β≤4/3, and η∈(1/2,α]. Although the corresponding Green's function is sign-changing, we still obtain the existence of at least two positive and decreasing solutions under some suitable conditions on f by using the two-fixed-point theorem due to Avery and Henderson. An example is also included to illustrate the main results obtained. [ABSTRACT FROM AUTHOR]

The article presents a proof of calculus theorem for Lebesgue integral. Topics discussed include the Vitali convergence theorem related to convergence of integrals, continuous function, real numbers' compact interval, absolutely continuous being uniformly absolutely continuous and inability of Riemann integral to solve integral calculus problem.

A 21:1 scaled sodar, operating at 40 kHz, has been built and tested in the laboratory. Sodars, which use sound scattered by turbulence to profile the lowest few hundred meters of the atmosphere, need good acoustic shielding to diminish annoyance and to reduce unwanted reflections from nearby objects. Design of the acoustic shielding is generally inhibited by the difficulty of testing on full-scale systems and uncertainty as to accuracy of models. In contrast, the scale model approach described allows for 'bench testing' of many designs under controlled conditions, and efficient comparison with models. Measured beam patterns from the scale model were compared with those from a numerical model based on the Kirchhoff integral theorem. Satisfactory agreement has allowed using the numerical model to optimize the acoustic shield design, both for the gross acoustic baffle geometry and for the geometry of rim modulations known as thnadners. Optimization was performed in the specific case of a scaled model of a commercial phased array sodar. [ABSTRACT FROM AUTHOR]

A development of generalized differentiation with respect to time is given within the structure of the functional approach to generalized function theory. The analysis is in parallel with techniques used in an earlier publication by F. Farassat and the author. The results are used to derive a generalized three dimensional Leibnitz' theorem that leads to a generalized Reynolds' transport theorem. This is then applied to derive the governing equations of continuum mechanics. It is shown that the process produces both the governing field equations and their associated discontinuity conditions in a single operation. [ABSTRACT FROM AUTHOR]

We deal with a new type of statistical convergence for double sequences, called Ψ-A-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces. [ABSTRACT FROM AUTHOR]

In this paper we generalize the classical Proth's theorem and the Miller-Rabin test for integers of the form N = Kpn +1. For these families, we present variations on the classical Pocklington’s results and, in particular, a primality test whose computational complexity is Ố (log2 N) and, what is more important, that requires only one modular exponentiation modulo N similar to that of Fermat's test. [ABSTRACT FROM AUTHOR]

The present paper investigates a class of disk functions via Laplace integral representation, where the disk polynomials appear as special cases. Recurrence relations involving the first-order derivative for them will be obtained. The connection of these functions with complex spherical harmonics also will be studied. Moreover, we exhibit an inductive method to construct bases of complex spherical harmonics via our disk functions. [ABSTRACT FROM AUTHOR]

In the present paper we introduce sufficient conditions for subordination and superordination involving the operator DRm;nλ and also we obtain sandwich-type results. [ABSTRACT FROM AUTHOR]

Expansion formulae associated with the interaction of oblique surface gravity waves with a floating flexible plate in the presence of a submerged horizontal flexible structure are derived using Green’s integral theorem in water of finite and infinite water depths. The associated Green’s functions are derived using the fundamental solution associated with the reduced wave equation. The integral forms of the Green’s functions and the velocity potentials are advantageous over the eigenfunction expansion method in situation when the roots of the dispersion relation coalesce. As an application of the expansion formulae, diffraction of oblique waves by a finite floating elastic plate in the presence of a submerged horizontal flexible membrane is investigated in water of finite depth. The accuracy of the numerical computation is demonstrated by analysing the convergence of the complex amplitude of the reflected waves and the energy relation. Effect of the submerged membrane on the diffraction of surface waves is studied by analysing the reflection and transmission coefficients for various parametric values. Further, the derivation of long wave equation under shallow water approximation is derived in a direct manner in the appendix. The concept and methodology can be easily extended to deal with acoustic wave interaction with flexible structures and related problems of mathematical physics and engineering. [ABSTRACT FROM PUBLISHER]

We show that almost all natural numbers n not divisible by 4, and not congruent to 7 modulo 8, are represented as the sum of three squares, one of which is the square of an integer no larger than . This answers a conjecture of Linnik for almost all natural numbers, and sharpens a conclusion of Bourgain, Rudnick, and Sarnak concerning nearest neighbor distances between normalized integral points on the sphere. [ABSTRACT FROM AUTHOR]

The article examines how the Banach-Tarski Paradox and the Sierpin'ski sets can be given concrete interpretations in the hyperbolic plane. It notes that a point in the plane can be selected for any discrete group of isometries of hyperbolic plane. The work of authors R. Dougherty and M. Foreman demonstrates that the Banach-Tarski paradox is possible using pieces having the property of Baire sets, the union between of a Borel set and a meager set.