Convolution Product Research Articles

Existence and concentration of ground state solutions for a quasilinear Choquard equation with critical exponential growth in ℝ2

In this paper, we consider the following quasilinear Choquard equation − ϵ 2 Δ u + V ( x ) u − ϵ 2 Δ ( u 2 ) u = ϵ μ − 2 ( 1 | x | μ ∗ F ( u ) ) f ( u ) in R 2 , where ϵ > 0 is a parameter, 0 < μ < 2 , ∗ is the convolution product in R 2 , V ( x ) is a continuous real function in R 2 , F ( u ) is the primitive function of f ( u ) and f has critical exponential growth with respect to the Trudinger-Moser inequality. By employing a change of variables, the quasilinear equation can be reduced to a semilinear equation, whose associated functional is well defined in a nonstandard Orlicz space and exhibits a mountain pass geometry. Under suitable assumptions on V and f, we investigate the existence and concentration behavior of positive ground state solutions for the above problem by variational methods.

Some Estimates for Certain q-analogs of Gamma Integral Transform Operators

The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.

An easy-to-use analytical model for standing column wells operating with bleed

Accurate assessment of subsurface heat transport is vital for the design of standing column well systems. When bleed is utilized, the mathematical problem is governed by coupled heat transfer and groundwater flow within a radially convergent flow field, making the development of models challenging. While numerical models exist to simulate these transient processes, the absence of simple and accessible analytical solutions limits their broader application. This study addresses this gap by developing a novel easy-to-use analytical model to accurately represent the heat advection–diffusion problem in standing column wells operating with bleed. The proposed model combines the well-known infinite line source model with an innovative scaling function, inspired from the field of solute transport, through a simple convolution product. Notably, the developed model depends on only three dimensionless parameters: dimensionless advection time, Péclet number, and bleed ratio. Rigorous validation against two distinct sets of reference numerical solutions demonstrated the model’s efficiency, accuracy, and reliability across a broad spectrum of nine physical parameters. Key results include relative root mean square errors on the order of 10−3 across 200 reference solutions, confirming the model’s robustness. These findings highlight the model’s potential to significantly advance both research and practical applications in the design and optimization of standing column wells.

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

A real-time and energy-efficient SRAM with mixed-signal in-memory computing near CMOS sensors

In-memory computing (IMC) represents a promising approach to reducing latency and enhancing the energy efficiency of operations required for calculating convolution products of images. This study proposes a fully differential current-mode architecture for computing image convolutions across all four quadrants, intended for deep learning applications within CMOS imagers utilizing IMC near the CMOS sensor. This architecture processes analog signals provided by a CMOS sensor without the need for analog-to-digital conversion. Furthermore, it eliminates the necessity for data transfer between memory and analog operators as convolutions are computed within modified SRAM memory. The paper suggests modifying the structure of a CMOS SRAM cell by incorporating transistors capable of performing multiplications between binary (−1 or +1) weights and analog signals. Modified SRAM cells can be interconnected to sum the multiplication results obtained from individual cells. This approach facilitates connecting current inputs to different SRAM cells, offering highly scalable and parallelized calculations. For this study, a configurable module comprising nine modified SRAM cells with peripheral circuitry has been designed to calculate the convolution product on each pixel of an image using a 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$3 \ imes 3$$\\end{document} mask with binary values (−1 or 1). Subsequently, an IMC module has been designed to perform 16 convolution operations in parallel, with input currents shared among the 16 modules. This configuration enables the computation of 16 convolutions simultaneously, processing a column per cycle. A digital control circuit manages both the readout or memorization of digital weights, as well as the multiply and add operations in real-time. The architecture underwent testing by performing convolutions between binary masks of 3 × 3 values and images of 32 × 32 pixels to assess accuracy and scalability when two IMC modules are vertically integrated. Convolution weights are stored locally as 1-bit digital values. The circuit was synthesized in 180 nm CMOS technology, and simulation results indicate its capability to perform a complete convolution in 3.2 ms, achieving an efficiency of 11,522 1-b TOPS/W (1-b tera-operations per second per watt) with a similarity to ideal processing of 96%.

Convolution Theorem for (p,q)-Gamma Integral Transforms and Their Application to Some Special Functions

This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems.

Generalized Fractional Integral Operator in a Complex Domain

A new fractional integral operator is used to present a generalized class of analytic functions in a complex domain. The method of definition is based on a Hadamard product of analytic function, which is called convolution product. Then we formulate a convolution integral operator acting on the sub-class of normalized analytic functions. Consequently, we investigate the suggested convolution operator geometrically. Differential subordination inequalities, taking the starlike formula are given. Some consequences of well-known results are illustrated. Keywords: Analytic function, subordination and superordination, univalent function, open unit disk, fractional integral operator, convolution operator, fractional calculus.

The continuous wavelet transform for a Laguerre type operator on the half line

In this paper, we consider a Laguerre differential operator on half line by accomplishing harmonic analysis tools with respect to the operator. We study some definitions and properties of Laguerre continuous wavelet transform. We also explore generalized Laguerre Fourier transform and convolution product on half line associated with the Laguerre differential operator. Also a new continuous wavelet transform associated with Laguerre function is constructed and investigated.

On the distributional fractional derivative: From unidimensional to multidimensional

In this paper, we use the generalized notions of Riemann–Liouville (fractional calculus with respect to a regular function) to extend the definitions of fractional integration and derivative from the functional sense to the distributional sense. First, we give some properties of fractional integral and derivative for the functions infinitely differentiable with compact support. Then, we define the weak derivative, as well as the integral and derivative of a distribution with compact support, and the integral and derivative of a distribution using the convolution product. Then, we generalize those concepts from the unidimensional to the multidimensional case. Finally, we propose the definitions of some usual differential operators.

Wave packet transform and wavelet convolution product involving the index Whittaker transform

Wave packet transform and wavelet convolution product involving the index Whittaker transform

On Certain Analogues of Noor Integral Operators Associated with Fractional Integrals

In this paper, we employ a q-Noor integral operator to perform a q-analogue of certain fractional integral operator defined on an open unit disc. Then, we make use of the Hadamard convolution product to discuss several related results. Also, we derive a class of convex functions by utilizing the q-fractional integral operator and apply the inspired presented theory of the differential subordination, to geometrically explore the most popular differential subordination properties of the aforementioned operator. In addition, we discuss an exciting inclusion for the given convex class of functions. Over and above, we investigate the q-fractional integral operator and obtain some applications for the differential subordination.

Coagulation, non-associative algebras and binary trees

We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by: (i) grafting together the roots of all pairs of order-compatible trees at preceding orders, or (ii) attaching binary branches to all free branches of trees at the previous order. We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.

Inverse and Moore–Penrose inverse of conditional matrices via convolution

AbstractMoore–Penrose inverse emerges in statistics, neural networks, machine learning, applied physics, numerical analysis, tensor computations, solving systems of linear equations and in many other disciplines. Especially after the 2000s, the topic of Moore–Penrose inverse has started to attract great attention by researchers and has become a popular subject. In this paper, we investigate the Moore–Penrose inverse of the conditional matrices via convolution product formula. In order to use convolution formula effectively, we derive some useful identities by using some properties of the generalized conditional sequence. Moreover, we express the Moore–Penrose inverse of the conditional matrices in the form of block matrices. Finally, we not only present more general results compared to earlier works, but also provide many novel results using analytical techniques.

Topological amenability of semihypergroups

Abstract In this article, we introduce and explore the notion of topological amenability in the broad setting of (locally compact) semihypergroups. We acquire several stationary, ergodic and Banach algebraic characterizations of the same in terms of convergence of certain probability measures, total variation of convolution with probability measures and translation of certain functionals, as well as the F-algebraic properties of the associated measure algebra. We further investigate the interplay between restriction of convolution product and convolution of restrictions of measures on a sub-semihypergroup. Finally, we discuss and characterize topological amenability of sub-semihypergroups in terms of certain invariance properties attained on the corresponding measure algebra of the parent semihypergroup. This in turn provides us with an affirmative answer to an open question posed by J. Wong in 1980.

Binomial Series-Confluent Hypergeometric Distribution and Its Applications on Subclasses of Multivalent Functions

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature.

Some notes on the algebraic structure of linear recurrent sequences

AbstractSeveral operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an R-algebra, given any commutative ring R with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these R-algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.

Initial Coefficients and Fekete-Szegő Inequalities for Functions Related to van der Pol Numbers (VPN)

ABSTRACT The purpose of this paper is to find coefficient estimates for the class of functions ℳ N ( γ , ϑ , λ ) consisting of analytic functions f normalized by f(0) = f′(0) – 1 = 0 in the open unit disk D subordinated to a function generated using the van der Pol numbers, and to derive certain coefficient estimates for a 2, a 3, and the Fekete-Szegő functional upper bound for f ∈ ℳ N ( γ , ϑ , λ ) . Similar results were obtained for the logarithmic coefficients of these functions. Further application of our results to certain functions defined by convolution products with a normalized analytic functions is given, and in particular, we obtain Fekete-Szegő inequalities for certain subclasses of functions defined through the Poisson distribution series.

A non-associative incidence near-ring with a generalized Möbius function

There is a convolution product on 3-variable partial flag functions of a locally finite poset that produces a generalized Möbius function. Under the product this generalized Möbius function is a one sided inverse of the zeta function and satisfies many generalizations of classical results. In particular we prove analogues of Phillip Hall’s Theorem on the Möbius function as an alternating sum of chain counts, Weisner’s Theorem, and Rota’s Crosscut Theorem. A key ingredient to these results is that this function is an overlapping product of classical Möbius functions. Using this generalized Möbius function we define analogues of the characteristic polynomial and Möbius polynomials for ranked lattices. We compute these polynomials for certain families of matroids and prove that this generalized Möbius polynomial has − 1 as root if the matroid is modular. Using results from Ardila and Sanchez we prove that this generalized characteristic polynomial is a matroid valuation.

Constant-Beamwidth LCMV Beamformer with Rectangular Arrays

This paper presents a novel approach utilizing uniform rectangular arrays to design a constant-beamwidth (CB) linearly constrained minimum variance (LCMV) beamformer, which also improves white noise gain and directivity. By employing a generalization of the convolutional Kronecker product beamforming technique, we decompose a physical array into virtual subarrays, each tailored to achieve a specific desired feature, and we subsequently synthesize the original array’s beamformer. Through simulations, we demonstrate that the proposed approach successfully achieves the desired beamforming characteristics while maintaining favorable levels of white noise gain and directivity. A comparative analysis against existing methods from the literature reveals that the proposed method performs better than the existing methods.

Performance of a commercially available artificial intelligence software for the detection of confirmed pulmonary nodules and masses in canine thoracic radiography.

Advancements in the field of artificial intelligence (AI) are modest in veterinary medicine relative to their substantial growth in human medicine. However, interest in this field is increasing, and commercially available veterinary AI products are already on the market. In this retrospective, diagnostic accuracy study, the accuracy of a commercially available convolutional neural network AI product (Vetology AI®) is assessed on 56 thoracic radiographic studies of pulmonary nodules and masses, as well as 32 control cases. Positive cases were confirmed to have pulmonary pathology consistent with a nodule/mass either by CT, cytology, or histopathology. The AI software detected pulmonary nodules/masses in 31 of 56 confirmed cases and correctly classified 30 of 32 control cases. The AI model accuracy is 69.3%, balanced accuracy 74.6%, F1-score 0.7, sensitivity 55.4%, and specificity 93.75%. Building on these results, both the current clinical relevance of AI and how veterinarians can be expected to use available commercial products are discussed.