Convolution Formula Research Articles

Principal angles between random subspaces and polynomials in two free projections

We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure.

Convolution and deconvolutional treatment on sample transparency aberration in Bragg–Brentano geometry

Exact and approximate mathematical models for the effects of sample transparency on the powder diffraction intensity data are examined. Application of the formula based on the first-order approximation about the deviation angle is justified for realistic measurement and computing systems. The effects of sample transparency are expressed by double convolution formulas applying two different scale transforms, including three parameters, goniometer radius R, penetration depth μ−1, and thickness of the sample t. The deconvolutional treatment automatically recovers the lost intensity and corrects the peak shift and asymmetric deformation of peak profile caused by the sample transparency.

Generalization of Duhamel's integral to multi-degree-of-freedom systems

This article presents a generalization to multi-degree-of-freedom (MDOF) systems of the classical convolution formula for single-degree-of-freedom (SDOF) systems that is widely known as Duhamel's integral. Remarkably, the extended convolution formula for a MDOF system is ultimately cast directly in terms of the system stiffness, mass and damping matrices in a manner that fully parallels the SDOF case, and without resorting to real or complex modal superposition. Still, although the proof based on the use of the Laplace transform might suggest that the obtained formulae are applicable to any type of viscous damping matrix, a more careful evaluation demonstrates that they contain a rather subtle constraint that implies that damping is of the classical type. The free vibration elicited by non-vanishing initial conditions is also considered. As a result, we ultimately obtain the general solution for MDOF systems with arbitrary dynamic loads and initial conditions. For undamped systems, the equivalence of the classical modal analysis with the direct approach proposed herein is demonstrated analytically. Special closed-form solutions, singular solutions and numerical examples are also investigated to a limited extent so as to further illustrate the application of the proposed formulae in the context of practical problems.

Topology of Augmented Bergman Complexes

The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex.
 It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.

New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials

New formulas including convolution, connection and radicals formulas of k-Fibonacci and k-Lucas polynomials

Another Proof of the Chu–Vandermonde Formula

Summary The Chu-Vandermonde convolution formula is a classic result on binomial coefficients with many known proofs. We provide a short and novel proof by using the method of finite differences.

Some identities involving the bi-periodic Fibonacci and Lucas polynomials

<abstract><p>In this paper, by using generating functions for the Chebyshev polynomials, we have obtained the convolution formulas involving the bi-periodic Fibonacci and Lucas polynomials.</p></abstract>

Analysis of Gegenbauer kernel filtration on the hypersphere

In this study, we aim to construct explicit forms of convolution formulae for Gegenbauer kernel filtration on the surface of the unit hypersphere. Using the properties of Gegenbauer polynomials, we reformulated Gegenbauer filtration as the limit of a sequence of finite linear combinations of hyperspherical Legendre harmonics and gave proof for the completeness of the associated series. We also proved the existence of a fundamental solution of the spherical Laplace-Beltrami operator on the hypersphere using the filtration kernel. An application of the filtration on a one-dimensional Cauchy wave problem was also demonstrated.

Compact BZT-ME-PML for modeling FDTD subsurface sensing and imaging

Depending on the matrix exponential (ME) method, a compact and highly-accurate implementation of the perfectly matched layer (PML) with the complex-frequency-shifted (CFS) scheme based upon the bilinear Z-transform (BZT) technique is developed for modeling the finite-difference time-domain (FDTD) problems on the very low frequency (VLF) subsurface sensing and the very high frequency (VHF) imaging. By using ME method, the proposal becomes a compact first-order differential matrix form so that it can completely circumvent convolution operations, formula rearrangements, or variable replacements during mathematical derivations. Furthermore, the BZT-ME-PML can not only flexibly terminate arbitrary media because of adding the DB constitutive relation, but efficiently attenuate evanescent waves and decrease late-time reflections due to incorporating the CFS scheme. Besides, the BZT-ME-PML can maintain higher absorption performances when it comes closer to truncate the FDTD domain with targets so that smaller computational regions can be adopted for modeling 3D open-domain subsurface sensing and imaging cases, resulting in palpable improvements in memory requirement and CPU time. The numerical simulations on the subsurface sensing and imaging have been carried out to illustrate the validity and accuracy of the approach.

Properties of Fundamental Solutions, Theorems on Integral Representations of Solutions and Correct Solvability of the Cauchy Problem for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Space Variables of Degeneration

The property of normality and the convolution formula are established for the classical fundamental solution of the Cauchy problem for a homogeneous ultraparabolic Kolmogorov-type equation with two groups of space variables of degeneration. Theorems on the integral representations of solutions and correct solvability of the Cauchy problem are presented for the classes of functions exponentially increasing as |x| → ∞ .

Third‐order differential subordinations for multivalent functions in the theory of source‐sink dynamics

We propose third‐order differential subordination results associated with new admissible classes of multivalent functions defined in the open unit disk on the complex plane. Besides, we investigate the geometric properties of multivalent functions associated with a novel convolution operator. This is done by taking suitable linear combinations of the classical Gaussian hypergeometric function and its derivatives up to third‐order and applying the Hadamard product (or convolution) formula for power series. The complex velocity potential and the stream function of two‐dimensional potential flow problems over a circular cylinder using both sources with sink and two sources are treated by the methods developed in the present paper. We further determine the fluid flow produced by a single source and construct a univalent function so that the image of source/sink is also source/sink for a given complex potential. Finally, some plot simulations are provided to illustrate the different results of this work.

A convolution formula for the local time of an Itô diffusion reflecting at 0 and a generalized Stroock–Williams equation

A new probabilistic insight into the structure of local time is presented. A convolution formula for the local time at 0 of Itô diffusions reflecting at 0 is obtained. A simple integro-differential equation for the cumulative distribution function of the local time is given. Finally, a probabilistic representation of a generalized Stroock–Williams equation is presented.

On the Generalized Laplace Transform

In this paper we introduce a generalized Laplace transform in order to work with a very general fractional derivative, and we obtain the properties of this new transform. We also include the corresponding convolution and inverse formula. In particular, the definition of convolution for this generalized Laplace transform improves previous results. Additionally, we deal with the generalized harmonic oscillator equation, showing that this transform and its properties allow one to solve fractional differential equations.

A remark on the distribution of products of independent normal random variables

We present a proof of the explicit formula of the probability density function of the product of normally distributed independent random variables using the multiplicative convolution formula for Meijer G functions.

The generalized convolution formula and its application are discussed

There are two ways to find the distribution of two-dimensional random variable function: distribution function method and convolution formula method, the distribution function method is too large; calculation is easy to make mistakes, the convolution formula method has limitations must be independent, Now let’s introduce the generalized convolution formula, The generalized convolution formula does not require independent constraints and can simplify the operation.

Experimental Study of Fractional-Order RC Circuit Model Using the Caputo and Caputo-Fabrizio Derivatives

This study employs the Caputo-Fabrizio fractional derivative to determine the model of fractional-order RC circuits with arbitrary voltage input which can be widely used in a variety of electrical systems. Analog circuit implementation of fractional-order RC circuits defined by Caputo-Fabrizio fractional derivative is presented and verified by comparing with the model proposed in this work. For the purpose of judging whether the fractional-order model defined by the Caputo-Fabrizio derivative is practical, the comparison experiments are carried out. By using Laplace transform, the analytical solutions of fractional-order RC circuits based on the Caputo-Fabrizio derivatives with constant and periodic voltage sources are deduced. Fractional-order model of RC circuits with arbitrary input are also calculated using the convolution formula. The correctness of the derivation of the model using the Caputo-Fabrizio derivative is verified. Through discussing the impedance model of capacitor in frequency domain, the analog realization of fractional capacitor based on the Caputo-Fabrizio derivative is derived. The fractional-orders of the RC circuits models defined by the Caputo and Caputo-Fabrizio fractional derivatives are fitted respectively through repeated charging and discharging experiment data. The fractional-order models based on the Caputo and Caputo-Fabrizio derivatives, and the integer-order model are all compared with the experiment data.

Moments of $L$-functions and the Liouville–Green method

We show that for at least 20% of primitive forms of level 1 and weight 4k \to \infty, k \in \mathbb N , the associated L -function at the central point is no less than (log k)^{–2} . The key ingredients of our proof are the Kuznetsov convolution formula and the Liouville–Green method.

Strange quark helicity in the proton from chiral effective theory

We compute the helicity-dependent strange quark distribution in the proton in the framework of chiral effective theory. Starting from the most general chiral SU(3) Lagrangian that respects Lorentz and gauge invariance, we derive the complete set of hadronic splitting functions at the one meson loop level, including the octet and decuplet rainbow, tadpole, Kroll-Ruderman and octet-decuplet transition configurations. By matching hadronic and quark level operators, we obtain generalized convolution formulas for the quark distributions in the proton in terms of hadronic splitting functions and quark distributions in the hadronic configurations, and from these derive model-independent relations for the leading nonanalytic behavior of their moments. Within the limits of parameters of the Pauli-Villars regulators derived from inclusive hyperon production, we find that the polarized strange quark distribution is rather small and mostly negative.

Incoherent deeply virtual Compton scattering off He4

Very recently, for the first time, the two channels of nuclear deeply virtual Compton scattering (DVCS), the coherent and incoherent ones, have been separated by the CLAS collaboration at JLab, using a $^4$He target. The incoherent channel, which can provide a tomographic view of the bound proton and shed light on its elusive parton structure, is thoroughly analyzed here in Impulse Approximation (IA). A convolution formula for the cross sections in terms of those for the bound proton is derived. Novel scattering amplitudes for a bound moving nucleon have been obtained and used. A state-of-the-art nuclear spectral function, based on the AV18 potential, exact in the two-body part, with the recoiling system in its ground state, and modelled in the remaining contribution, with the recoiling system in an excited state, has been used. Different parametrizations of the generalized parton distributions of the struck proton have been tested. A good overall agreement with the data for the beam spin asymmetry (BSA) is obtained. It is found that the predicted conventional nuclear effects are relevant in DVCS and in the competing Bethe-Heitler mechanism, but they cancel each other to a large extent in their ratio, to which the measured asymmetry is proportional. Besides, the calculated ratio of the BSA of the bound proton to that of the free one does not describe that estimated by the experimental collaboration. This points to possible interesting effects beyond the IA analysis presented here. It is therefore clearly demonstrated that the comparison of the results of a conventional realistic approach, as the one presented here, with future precise data, has the potential to expose quark and gluon effects in nuclei. Interesting perspectives for the next measurements at high luminosity facilities, such as JLab at 12 GeV and the future EIC, are addressed.

Finite free convolutions via Weingarten calculus

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain quadrature property, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.