Convolution Formula Research Articles

Coherent deeply virtual Compton scattering off He4

Coherent Deeply virtual Compton scattering off the $^4$He nucleus is studied in impulse approximation. A convolution formula for the nuclear Generalized Parton Distribution (GPD) is derived in terms of the $^4$He non-diagonal spectral function and of the GPD of the struck nucleon. A model of the nuclear non diagonal spectral function, based on the momentum distribution corresponding to the AV18 nucleon-nucleon interaction, is used in the actual calculation. Typical impulse approximation results are reproduced, in proper limits, for the nuclear form factor and for nuclear parton distributions. The nuclear generalized parton distribution and the Compton form factor are evaluated using, as nucleonic ingredient, a well known generalized parton distribution model. An overall very good agreement is found with the data recently published by the EG6 experiment at the Jefferson Laboratory (JLab). More refined nuclear calculations are addressed and will be necessary for the expected improved accuracy of the next generation of experiments at JLab, with the 12 GeV electron beam and high luminosity.

Documenting and treating a dysregulated gut-immune-brain axis in schizophrenia

We study generalized Catalan matrices based on the Riordan array and Fuss–Catalan numbers. A unified combinatorial interpretation for the entries of the generalized Catalan matrices is presented by means of m-Dyck paths. Some properties and examples of the generalized Catalan matrices are given including a new convolution formula for the generalized Catalan numbers. Finally, we present applications of generalized Catalan matrices to the problems in counting the hill-free and lower peak-free m-Dyck paths.

Extreme values of the Riemann zeta function and its argument

We combine our version of the resonance method with certain convolution formulas for $\zeta(s)$ and $\log\, \zeta(s)$. This leads to a new $\Omega$ result for $|\zeta(1/2+it)|$: The maximum of $|\zeta(1/2+it)|$ on the interval $1 \le t \le T$ is at least $\exp\left((1+o(1)) \sqrt{\log T \log\log\log T/\log\log T}\right)$. We also obtain conditional results for $S(t):=1/\pi$ times the argument of $\zeta(1/2+it)$ and $S_1(t):=\int_0^t S(\tau)d\tau$. On the Riemann hypothesis, the maximum of $|S(t)|$ is at least $c \sqrt{\log T \log\log\log T/\log\log T}$ and the maximum of $S_1(t)$ is at least $c_1 \sqrt{\log T \log\log\log T/(\log\log T)^3}$ on the interval $T^{\beta} \le t \le T$ whenever $0\le \beta < 1$.

Context-Free Grammars for Several Polynomials Associated with Eulerian Polynomials

In this paper, we present grammatical descriptions of several polynomials associated with Eulerian polynomials, including q-Eulerian polynomials, alternating run polynomials and derangement polynomials. As applications, we get several convolution formulas involving these polynomials.

Hopf algebras and Tutte polynomials

By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the literature arise from this framework. These polynomials include the classical Tutte polynomial of graphs and matroids, Las Vergnas' Tutte polynomial of the morphism of matroids and his Tutte polynomial for embedded graphs, Bollobás and Riordan's ribbon graph polynomial, the Krushkal polynomial, and the Penrose polynomial.We show that our Tutte polynomials of Hopf algebras share common properties with the classical Tutte polynomial, including deletion-contraction definitions, universality properties, convolution formulas, and duality relations. New results for graph polynomials from the literature are then obtained as examples of the general results.Our results offer a framework for the study of the Tutte polynomial and its analogues in other settings, offering the means to determine the properties and connections between a wide class of polynomial invariants.

Universal Tutte characters via combinatorial coalgebras

This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.

Aumann Fuzzy Improper Integral and Its Application to Solve Fuzzy Integro-Differential Equations by Laplace Transform Method

We introduce the Aumann fuzzy improper integral to define the convolution product of a fuzzy mapping and a crisp function in this paper. The Laplace convolution formula is proved in this case and used to solve fuzzy integro-differential equations with kernel of convolution type. Then, we report and correct an error in the article by Salahshour et al. dealing with the same topic.

A New Formulation of the Fractional Optimal Control Problems Involving Mittag–Leffler Nonsingular Kernel

The aim of this paper is to propose a new formulation of the fractional optimal control problems involving Mittag–Leffler nonsingular kernel. By using the Lagrange multiplier within the calculus of variations and by applying the fractional integration by parts, the necessary optimality conditions are derived in terms of a nonlinear two-point fractional boundary value problem. Based on the convolution formula and generalized discrete Gronwall’s inequality, the numerical scheme for solving this problem is developed and its convergence is proved. Numerical simulations and comparative results show that the suggested technique is efficient and provides satisfactory results.

Generalized Trigonometric Integral Transforms

Some generalizations of the Fourier and Hartley transforms on ℝ and ℝ n are studied. These transforms involve functional coefficients. The Parseval identities, convolution formulas, conditions of self-adjointness and unitary property for new transforms are obtained.

On a motivic invariant of the arc-analytic equivalence

To a Nash function germ, we associate a zeta function similar to the one introduced by J. Denef and F. Loeser. Our zeta function is a formal power series with coefficients in the Grothendieck ring $\mathcal{M}$ of $\mathcal{AS}$-sets up to $\mathbb{R}^*$-equivariant $\mathcal{AS}$-bijections over $\mathbb{R}^*$, an analog of the Grothendieck ring constructed by G. Guibert, F. Loeser and M. Merle. This zeta function generalizes the previous construction of G. Fichou but thanks to its richer structure it allows us to get a convolution formula and a Thom-Sebastiani type formula. We show that our zeta function is an invariant of the arc-analytic equivalence, a version of the blow-Nash equivalence of G. Fichou. The convolution formula allows us to obtain a partial classification of Brieskorn polynomials up to the arc-analytic equivalence by showing that the exponents are arc-analytic invariants.

Skew Motzkin paths

In this paper, we study the class S of skew Motzkin paths, i.e., of those lattice paths that are in the first quadrat, which begin at the origin, end on the x-axis, consist of up steps U = (1, 1), down steps D = (1,−1), horizontal steps H = (1, 0), and left steps L = (−1,−1), and such that up steps never overlap with left steps. Let S n be the set of all skew Motzkin paths of length n and let s n = |S n |. Firstly we derive a counting formula, a recurrence and a convolution formula for sequence {s n } n ≥0. Then we present several involutions on S n and consider the number of their fixed points. Finally we consider the enumeration of some statistics on S n .

Some identities related to the r-Whitney numbers

ABSTRACTStarting from the definitions of the r-Whitney numbers of both kinds, in this paper we give two convolution formulas for the r-Whitney numbers. The convolution formula for the r-Whitney numbers of the second kind is new and it helps us to find the LU factorization of a new type of matrices consisting of the r-Whitney numbers of the second kind . Meanwhile, the evaluation of the determinant of this type of matrices is given. Finally, the Dobinski and Spivey types of identities for the r-Whitney numbers of the second kind are also presented.

Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution

The present paper deals with the Jain–Durrmeyer operators based on inverse Pólya–Eggenberger distribution. First, we give the moments with the help of Vandermonde convolution formula and then study approximation properties of these operators which include uniform convergence and degree of approximation.

ENUMERATION OF A DUAL SET OF STIRLING PERMUTATIONS BY THEIR ALTERNATING RUNS

In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

Aberration in qualitative multilevel designs

Generalized Word Length Pattern (GWLP) is an important and widely-used tool for comparing fractional factorial designs. We consider qualitative factors, and we code their levels using the roots of the unity. We write the GWLP of a fraction F using the polynomial indicator function, whose coefficients encode many properties of the fraction. We show that the coefficient of a simple or interaction term can be written using the counts of its levels. This apparently simple remark leads to major consequence, including a convolution formula for the counts. We also show that the mean aberration of a term over the permutation of its levels provides a connection with the variance of the level counts. Moreover, using mean aberrations for symmetric sm designs with s prime, we derive a new formula for computing the GWLP of F. It is computationally easy, does not use complex numbers and also provides a clear way to interpret the GWLP. As case studies, we consider non-isomorphic orthogonal arrays that have the same GWLP. The different distributions of the mean aberrations suggest that they could be used as a further tool to discriminate between fractions.

Some arithmetic convolution identities

Let n be a positive integer. Let \(\delta _3(n)\) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum $$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$ satisfies the relation $$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$ In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σit(n)σ-it(N - n) (t is real) over 0 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.

Closed formulas for computing higher-order derivatives of functions involving exponential functions

For integers k ≥ 1 and n ≥ 0, the functions 1/(1−λeαt)k and the derivatives (1/(1−λeαt))(n) can be expressed each other by linear combinations. Based on this viewpoint, we find several new closed formulas for higher-order derivatives of trigonometric and hyperbolic functions, derive a higher-order convolution formula for the tangent numbers, and generalize a recurrence relation for the tangent numbers.

On the effect of intrafraction motion in a single fraction step-shoot IMRT.

The authors studied the respiratory motion effect in a single step-shoot intensity modulated radiotherapy (IMRT) to assess the basic properties of the uncertainty in the delivered dose due to the unknown starting phase of the motion. Using computer simulations, the motion-averaged dose for open beams with various field sizes was calculated for two one-dimensional trajectories with different motion amplitudes at 20 equally spaced starting phases. The properties of the standard deviation (SD) of delivered dose were analyzed. The dependence of SD on the field size, motion amplitude, and delivery time was investigated and experimentally validated. To study effect of number of small monitor unit (MU) segments on the dose uncertainty, the authors generated 1000 pairs of multisegment beams such that each pair consists of two beams with the same total MU and different segment MU. The SD at the central axis point was compared for each pair. The authors proved that the direct time-dependent dose accumulation can be calculated using a convolution formula for a single fraction step-shoot IMRT treatment. Single segment simulation showed that the maximum dose uncertainty occurred symmetrically at the beam penumbra for a sinusoidal motion. For other sinusoidal motions (sin(2n) n > 1), the maximum dose uncertainty occurred at asymmetrical locations and may be beyond the penumbra region. The SD of relative dose periodically varied with delivery time with decreasing envelope for both motion trajectories. The SD of absolute dose was a periodic function of the delivery time for a given field size and motion amplitude and was proved to be true for any periodic motion. The SD reduced to zero when the delivery time was an integer multiple of the motion period. Analytical function σA=3βsin(2)π/Tδt-4/3sin(4)π/Tδt+2/3sin(6)π/Tδt was found to fit the delivery time dependence of the SD for motions studied in this paper and was verified with experimental data and an irregular motion. The dose uncertainty increased with motion amplitude and decreased slowly with field size. Simulations for 1000 beam pairs with multiple segments demonstrated that the probability that more small MU segments did not introduce larger dose uncertainty at central axis point for three cutoff small MUs (2.5/5/10) was 55.8%/51.9%/43.4% and 54.6%/54.4%/45%, respectively, for a sin and a sin(4) motion in a conventional treatment. These probabilities became 53.6%/50.9%/47.2% and 51.0%/50.2%/47.8%, respectively, for a hypofractionated treatment. The periodic dependence of the dose uncertainty on the delivery time can be modeled with an analytic function, and the functional form is independent of motion trajectories in this paper. The relation of dose uncertainty between different dose schemes can be obtained using this function. The dose uncertainty at central axis point for a beam with more small MU segments may not be greater than a beam with less small MU segments. By varying the dose rate of each segment such that the delivery time is close to the integer multiples of the motion period, the interplay effect can be reduced.

An unabelian version of the Voronov higher bracket construction

Abstract In this paper, we show how to extend Voronov's construction of L ∞-structures by using the splitting of a graded Lie algebra into the direct vector space sum of two subalgebras, of which one is abelian, to just a graded Lie algebra inclusion without an algebraic complement. The construction uses certain Verma modules of the universal enveloping algebra of the graded Lie algebra, and it is fairly explicit in terms of convolution formulas. Voronov's result [J. Pure Appl. Algebra 202 (2005), 133–153] and Bandiera's result on nonabelian complements [http://arxiv.org/abs/1304.4097] occur as particular cases.