A New $$q$$-Congruence on Double Sums

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.

We obtain several estimates for trilinear form with double Kloosterman sums. In particular, these bounds show the existence of nontrivial cancellations between such sums.

Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of character sums and their estimation with respect to non homogeneous Beatty sequences 𝛽𝛼 = ⌊𝛼𝑛 + 𝛽 : 𝑛 = 1, 2, 3...⌋ where 𝛽 in real numbers and 𝛼 greater than zero isirrational. In order to estimate the cardinality, the discrepancy is used to measure the number of uniform distribution for Beatty sequences. Pigeonhole principle is discussed on the estimation of the fractional part of Beatty sequences involve. Meanwhile, Cauchy inequalities is applied to expand the double character sums. Then, the cardinality of double character sums is obtained by applying the extension properties of additive and multiplicative character sums. The resultobtained is depend on the existing of identity of additive and multiplicative character sums and the uniformly distribution modulo 1. The result of the estimation in this study over composite modules is more general compared to previous studies, which only cover prime modules.

Lindelöf’s hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann’s zeta function $\zeta (1/2+it)$ is of order $O(t^{\varepsilon })$ for any $\varepsilon>0$. It is well known that for large $t$, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum. However, since such estimates cannot yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf’s hypothesis. Here a completely different approach is introduced. In particular, a novel linear integral equation is derived for $|\zeta (\sigma +it)|^2, \ 0<\sigma <1$ whose asymptotic analysis yields asymptotic results for a certain Riemann zeta-type double exponential sum. This sum has the same structure as the sum describing the leading asymptotics of $|\zeta (\sigma +it)|^2$, namely it involves $m_1^{-\sigma -it}m_2^{-\sigma -it}$, but its summation limits are different than those of the sum corresponding to $|\zeta (\sigma +it)|^2$. The analysis of the above integral equation requires the asymptotic estimation of four different integrals denoted by $I_1,I_2,\tilde{I}_3,\tilde{I}_4$, as well as the derivation of an exact relation between certain double exponential sums. Here the latter relation is derived, and also the rigourous analysis of the first two integrals $I_1$ and $I_2$ is presented. For the remaining two integrals, formal results are only derived that suggest a possible roadmap for the derivation of rigourous asymptotic results of the above double exponential sum, as well as for other sums associated with $|\zeta (\sigma +it)|^2$. Additional developments suggested by the above novel approach are also discussed.

The paper brings an elementary approach to Lipschitz type estimates for functions of operators. The approach is based on the reduction to the case of operators on finite-dimensional inner product spaces. This allows us to avoid double and triple operator integrals and consider instead double and triple operator sums. Unlike in the case of double and triple operator integrals, double and triple operator sums can be defined for arbitrary functions which completely eliminates the problem of definitions of double operator integrals and various types of triple operator integrals.

Evanescent acoustical waves occur in a variety of situations such as when sound is incident on a fluid interface beyond the critical angle and when flexural waves on a plate are subsonic with respect to the surrounding fluid. The scattering by circular cylinders at normal incidence was calculated to give insight into the consequences on the scattering of the evanescence of the incident wave. To analyze the scattering, it is necessary to express the incident wave using a modified expansion involving cylindrical functions. For plane evanescent waves, the expansion becomes a double summation with products of modified and ordinary Bessel functions. The resulting modified series is found for the scattering by a fluid cylinder in an unbounded medium. The perfectly soft and rigid cases are also examined. Unlike the case of an ordinary incident wave, the counterpropagating partial waves of the same angular order have unequal magnitudes when the incident wave is evanescent. This is a consequence of the exponential dependence of the incident wave amplitude on the transverse coordinate. The associated exponential dependence of the scattering on the location of a scatterer was previously demonstrated [T. J. Matula and P. L. Marston, J. Acoust. Soc. Am. 93, 1192–1195 (1993)].

A hit-or-miss transform architecture and a canonical Boolean logic format and threshold decomposition algorithm are proposed to process binary and gray-tone images with morphological combined operations. The transformation of morphological combined image processing functions to hit-or-miss transforms is explicit: Binary functions are formulated by a summation of hit-or-miss transforms and gray-tone functions formulated by a double summation. A simple optical hit-or-miss transform processor is suggested on the basis of incoherent two-channel correlation with dual-rail spatial coding. The hit-or-miss transforms can be realized optically in one parallel processing step. Thus, binary and gray-tone image processing functions can be implemented by the repeated use of the optical processor with different structuring elements.

Multiple summation inequalities and their application to stability analysis of discrete-time delay systems

Heat sources on rectangular plates with double-sided convective cooling are a common application of heat transfer in electronic packaging. However, it is considerably difficult to obtain analytical solutions for improving thermal performance because of the mixed boundary conditions on a plate’s upper surface. In this study, mixed boundary conditions were adjusted twice. Based on the method of separation of variables, a new analytical model was established and used to calculate the thermal performance of a random application case, including temperature distribution, thermal spreading resistance, and total thermal resistance. The results indicated that accurate analytical solutions can be obtained with few computing resources when more than 30 terms are used in each of single and double summations with a total mesh number of 196 in the heat source region. Subsequently, a series of thermal performances were calculated for random application cases and simulations were also conducted using the commercial software Icepak to offer a reference. The data comparisons revealed that the analytical model is able to calculate the temperature values at arbitrary point coordinates accurately over wide ranges of a plate’s thermal conductivity and heat transfer coefficient.

A general method of calculating the intensity of X-ray diffraction from small crystalline particles whose boundary is defined by a shape function is discussed. The intensity formula which is generally given by a double sum over the reciprocal-lattice points is simplified into the form of a single sum, using 'the random-shift treatment' which assumes that the position of the boundary relative to the crystal lattice varies at random from crystal to crystal. By the use of Fourier theorems, the intensity formulas are also converted into a single sum over the direct lattice. Although the electron distribution in the particle has been defined in various ways by the shape function, a more reasonable expression of the electron density appropriate to small crystals is introduced. The intensity formulas derived on the basis of the new form of the electron density are compared with other intensity formulas which have so far been proposed.

The article addresses the problem whether indefinite double sums involving a generic sequence can be simplified in terms of indefinite single sums. Depending on the structure of the double sum, the proposed summation machinery may provide such a simplification without exceptions. If it fails, it may suggest a more advanced simplification introducing in addition a single nested sum where the summand has to satisfy a particular constraint. More precisely, an explicitly given parameterized telescoping equation must hold. Restricting to the case that the arising unspecified sequences are specialized to the class of indefinite nested sums defined over hypergeometric, multi-basic or mixed hypergeometric products, it can be shown that this constraint is not only sufficient but also necessary.

An estimate for double sums of multiplicative characters with additive convolutions of the characteristic functions of sets having small additive doubling is obtained.

We develop a new algorithm for the computation of the rotationally averaged elastic molecular diffraction signal for the cases of perpendicular or parallel pump-probe geometries. The algorithm first collocates the charge density from an arbitrary ab initio wave function onto a Becke quadrature grid [A. Becke, J. Chem. Phys. 1988 , 88 , 2457 ], providing a high-fidelity multiresolution representation of the charge density. A double sum is then performed over the Becke grid points, and the interaction between points computed using the scattering kernels of Williamson and Zewail [J. C. Williamson and A. H. Zewail, J. Phys. Chem. 1994 , 98 , 2766 ]. These kernels analytically average over the molecular orientations with the cos2 γ selection factor appropriate for one-photon dipole absorption in a perpendicular pump-probe geometry. We show that the method is converged with small grids containing <500 points/atom. We implement the algorithm on a GPU for increased efficiency and emonstrate the algorithm for molecules with up to a few dozen atoms. We explore the accuracy of the independent atom model (IAM) by comparison with our new and more accurate method. We also investigate the possibility of detecting signatures of electronic transitions in polyatomic pump-probe diffraction experiments.

For any fixed integer a and real variables x , y with y x let N a ( x, y ) denote the number of primes p ≤ x for which p + a has at least one prime factor greater than y . As an elementary application of the following deep theorem of Bombieri on arithmetic progressions, Theorem (Bombieri, [1]). For any constant A > 0, there exists a positive constant B such that if with l = log x, then for x > 1 where Φ(x; m, a) denotes the number of primes less than x which are congruent to a mod m ; we shall prove the following theorem: Theorem 1. Let a be any fixed integer and let x > e. We then have where the double sum is taken over primes p and q .

The method of summation by diagonals is shown to be a practical method for the summation of double series which can give good results. Where problems arise an elementary analysis can often be used which enables the sum to be calculated to a high degree of accuracy. The method can be extended in an obvious manner to higher dimensional series.