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For all integers n ≥ 1 and k ≥ 2 , the Hadamard product v 1 ★ ⋯ ★ v k of k elements of K n + 1 (with K being the complex numbers or real numbers) is the element v ∈ K n + 1 which is the coordinate-wise product of v 1 , ... , v k (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map h : P n (K) k ⤏ P n (K) . When K = C , k = 2 and X i (C) ⊂ P n (C) , i = 1 , 2 are irreducible, we prove four theorems for the case dim X 2 (C) = 1 , three of them with X 2 (C) as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over R. Under mild assumptions, we prove that by knowing X 1 (R) ★ ⋯ ★ X k (R) , we know X 1 (C) ★ ⋯ ★ X k (C) . The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen P n (R) , very often provides real ghosts, i.e., images that do not come from a point of X 1 (R) × ⋯ × X k (R) . We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen. [ABSTRACT FROM AUTHOR]

Sălăgean differential operator D κ plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator D κ in connection with Stirling numbers are relatively new. In this paper, the differential operator D κ involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series Υ θ s (ϰ) written with the Pascal distribution are discussed for the subclass T κ (ϵ , ♭) . Also, we provide a sufficient condition for the inclusion relation I θ s R ϖ (E , D ) ⊂ T κ (ϵ , ♭) . Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of n = 2. Moreover, we give ratio inequalities of the operator power means by the Hadamard product. [ABSTRACT FROM AUTHOR]

Although neural networks have revolutionized various fields, their deployment in mobile environments confronts significant challenges due to limitations in battery and cooling capacity, especially for internet of things devices [1]. A lightweight neural network is urgently needed to address these issues. In this study, we explore the use of the Hadamard product and assess the usefulness of the method to enhance the neural network performance in mobile environments. The method has less computational complexity compared with other matrix multiplication methods [2]. Hadamard product methodologies are applied to the input features to amplify useful data and diminish noise. Our research involved the use of 48 electromyography signals sourced from the calves of three individuals with the signal time frame being iterated from 10 to 990 with a step size of 10. Findings indicate that the utilization of Hadamard products significantly improves the model performance relative to the increase in model size. [ABSTRACT FROM AUTHOR]

We introduce the Hadamard decomposition problem in the context of data analysis. The problem is to represent exactly or approximately a given matrix as the Hadamard (or element-wise) product of two or more low-rank matrices. The motivation for this problem comes from situations where the input matrix has a multiplicative structure. The Hadamard decomposition has potential for giving more succint but equally accurate representations of matrices when compared with the gold-standard of singular value decomposition (svd). Namely, the Hadamard product of two rank- h matrices can have rank as high as h 2 . We study the computational properties of the Hadamard decomposition problem and give gradient-based algorithms for solving it approximately. We also introduce a mixed model that combines svd and Hadamard decomposition. We present extensive empirical results comparing the approximation accuracy of the Hadamard decomposition with that of the svd using the same number of basis vectors. The results demonstrate that the Hadamard decomposition is competitive with the svd and, for some datasets, it yields a clearly higher approximation accuracy, indicating the presence of multiplicative structure in the data. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a linear operator and define a certain class Ep(a, c, λ, γ: h) of analytic and multivalent functions in the open unit disk associated with differential subordination. Also, we discuss some geometric properties for this class. [ABSTRACT FROM AUTHOR]

In order to examine various geometric features, we expanded the supertrigonometric function (STF) and superhyperbolic function (SHF) into the open unit disk. A convolution differential operator of the STF provides the formulas. The suggested operator works with both integral and double differential inequalities. As a result, we present a collection of findings that includes recent works. The idea of subordination and superordination serves as a guide for our method, and we then developed the primary conclusion as a double side’s theorem. [ABSTRACT FROM AUTHOR]

I n this work we introduce new subclasses of meromorphic functions in punctured unit disk and analyzes numerous connections and various features of these subclasses by making use of the linear operator that is connected to Gaussian hypergeometric functions. [ABSTRACT FROM AUTHOR]

Studies of Sălăgean differential operator Dκ in connection with Stirling numbers are relatively new. In this paper, the differential operator Dκ involving Stirling numbers is considered. New necessary and sufficient conditions involving Stirling numbers for the series ϒθs(ς) written by the Pascal distribution are discussed for the subclass Tκ(ϵ,α). Also, we provide sufficient condition for the inclusion relation Iθs(Rϖ(E,D))⊂Tκ(ϵ,α). Further, we consider the properties of integral operator related to Pascal distribution series. New special cases as consequences of the main results are also obtained.

Let $H$ be a Hilbert space. In this paper we show among others that, if the selfadjoint operators $A$ and $B$ satisfy the condition $0$ $

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions Τ J (α i) . Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f ∈ Τ J (α i) . In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. [ABSTRACT FROM AUTHOR]

This work presents a novel investigation that utilizes the integral operator I p , λ n in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. [ABSTRACT FROM AUTHOR]

In this paper, we generalize and study the concept of Hadamard product of ideals of projective varieties to the case of monomial ideals. We have a research direction similar to the one of the join of monomial ideals contained in a paper of Sturmfels and Sullivant. In the second part of the paper, we give a brief tutorial on the Hadamard.m2 package of Macaulay2. [ABSTRACT FROM AUTHOR]

A new subclass T X q [ λ , A , B ] of analytic functions is introduced by making use of the q-derivative operator associated with the Pascal distribution. Certain properties of analytic functions in the subclass T X q [ λ , A , B ] are derived. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Let V3(z, f) and σ(1) 3 (z, f) be the cubic polynomials representing, respectively, the 3rd de la Vall'ee Poussin mean and the 3rd Cesaro mean of order 1 of a power series f(z). If K denotes the usual class of convex univalent functions in the open unit disk centered at the origin, we show that, in general, V3(z, f) ⊀ σ(1) 3 (z, f), for all f ∈ K. Making use of polylogarithms, we identify a transformation, Λ: K → K, such that V3(z, Λ(f)) ≺ σ(1) 3 (z, Λ(f)) for all f ∈ K. Here '≺' stands for subordination between two analytic functions. [ABSTRACT FROM AUTHOR]

Deep edge detection is challenging, especially with the existing methods, like HED (holistic edge detection). These methods combine multiple feature side outputs (SOs) to create the final edge map, but they neglect diverse edge importance within one output. This creates a problem: to include desired edges, unwanted noise must also be accepted. As a result, the output often has increased noise or thick edges, ignoring important boundaries. To address this, we propose a new approach called the normalized Hadamard-product (NHP) operation-based deep network for edge detection. By multiplying the side outputs from the backbone network, the Hadamard-product operation encourages agreement among features across different scales while suppressing disagreed weak signals. This method produces additional Mutually Agreed Salient Edge (MASE) maps to enrich the hierarchical level of side outputs without adding complexity. Our experiments demonstrate that the NHP operation significantly improves performance, e.g., an ODS score reaching 0.818 on BSDS500, outperforming human performance (0.803), achieving state-of-the-art results in deep edge detection. [ABSTRACT FROM AUTHOR]

The COVID-19 epidemic has surely disrupted daily life and resulted in unexpected shifts that have resulted in serious psychological reactions and mental health crises. In the early days of COVID-19, little was known about it, which understandably generates worry and distress, especially for those of us who suffer from anxiety disorders. In the early phases of the pandemic, there were large volume of calls to emergency lines, thus it was critical to divert non-emergency inquiries regarding the disease, particularly those looking for information, and to offer help for emergency calls about "breathing difficulty" and "person collapsing". It became vital to distinguish between emergency and non-emergency calls. This study examines the call content of calls landed in the emergency response support system during the pandemic and help in categorising the calls as "emergency" or "non-emergency" based on call content related to pandemic. The proposed model is a Hadamard product-based pretrained ensemble model of RoBERTa and XLNet using the ERSS dataset. The performance of the HPRXF proposed model is evaluated using the Hadamard product of the final probability vectors of the RoBERTa and XLNet models. The HPRXF Model outperforms existing models with an accuracy: 81.16% and an F1-score: 80.66%. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space. In this paper we show among others that, if the selfadjoint operators A and B satisfy the condition 0 < m ≤ A, B ≤ M, for some constants m, M, then... for all v ∈ [0;1]: We also have the inequalities for Hadamard product... for all v ∈ [0,1]. [ABSTRACT FROM AUTHOR]

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator Hölder type inequalities. Among other results, it is shown that if A = (At)t∈Ω and B=(Bt)t∈Ω are continuous fields of positive invertible operators in a unital C*-algebra A such that ∫Ω At dμ(t) ≤ IA and ∫Ω Bt dμ(t) ≤ IA, and if ωf is an arbitrary operator mean with the representing function f, then ...for all 0 < p ≤ 1, which is an extension of the operator Bellman inequality. [ABSTRACT FROM AUTHOR]

The generalized $M$-series is a hybrid function of generalized Mittag-Leffler function and generalized hypergeometric function. The principal aim of this paper is to investigate certain properties resembling those of the Mittag-Leffler and Hypergeometric functions including various differential and integral formulas associated with generalized $M$-series. Certain corollaries involving the generalized hypergeometric function are also discussed. Further, in view of Hadamard product of two analytic functions, we have represented our main findings in Hadamard product of two known functions.

Medical image segmentation has an important application value in the modern medical field, it can help doctors accurately locate and analyze the tissue structure, lesion areas, and organ boundaries in the image, which provides key information support for clinical diagnosis and treatment, but there are still a large number of problems in the accuracy of the segmentation, so in this paper, we propose a medical image segmentation network combining the Hadamard product and dual-scale attention gate (DAU-Net). First, the Hadamard product is introduced in the structure of the fifth layer of the codec for element-by-element multiplication, which can generate feature representations with more representational capabilities. Second, in the jump connection module, we propose a dual scale attention gating (DSAG), which can highlight more valuable features and achieve more efficient jump connections. Finally, in the decoder feature structure, the final segmentation result is obtained by aggregating the feature information provided by each part, and decoding is achieved by up-sampling operation. Through experiments on two public datasets, Luna and Isic2017, DAU-Net is able to extract feature information more efficiently using different modules and has better segmentation results compared to classical segmentation models such as U-Net and U-Net++, and also verifies the effectiveness of the model.

Let $H$ be a Hilbert space. In this paper we show among others that, if $f,$ $g$ are synchronous and continuous on $I$ and $A,$ $B$ are selfadjoint with spectra ${Sp}\left( A\right) ,$ ${Sp}\left( B\right) \subset I,$ then% \begin{equation*} \left( f\left( A\right) g\left( A\right) \right) \otimes 1+1\otimes \left( f\left( B\right) g\left( B\right) \right) \geq f\left( A\right) \otimes g\left( B\right) +g\left( A\right) \otimes f\left( B\right) \end{equation*}% and the inequality for Hadamard product% \begin{equation*} \left( f\left( A\right) g\left( A\right) +f\left( B\right) g\left( B\right) \right) \circ 1\geq f\left( A\right) \circ g\left( B\right) +f\left( B\right) \circ g\left( A\right) . \end{equation*}% Let either $p,q\in \left( 0,\infty \right) $ or $p,q\in \left( -\infty ,0\right) $. If $A,$ $B>0,$ then \begin{equation*} A^{p+q}\otimes 1+1\otimes B^{p+q}\geq A^{p}\otimes B^{q}+A^{q}\otimes B^{p}, \end{equation*}% and% \begin{equation*} \left( A^{p+q}+B^{p+q}\right) \circ 1\geq A^{p}\circ B^{q}+A^{q}\circ B^{p}. \end{equation*}

For all integers n≥1 and k≥2, the Hadamard product v1★⋯★vk of k elements of Kn+1 (with K being the complex numbers or real numbers) is the element v∈Kn+1 which is the coordinate-wise product of v1,…,vk (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map h:Pn(K)k⤏Pn(K). When K=C, k=2 and Xi(C)⊂Pn(C), i=1,2 are irreducible, we prove four theorems for the case dimX2(C)=1, three of them with X2(C) as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over R. Under mild assumptions, we prove that by knowing X1(R)★⋯★Xk(R), we know X1(C)★⋯★Xk(C). The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen Pn(R), very often provides real ghosts, i.e., images that do not come from a point of X1(R)×⋯×Xk(R). We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen.

Sălăgean differential operator Dκ plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator Dκ in connection with Stirling numbers are relatively new. In this paper, the differential operator Dκ involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series Υθs(ϰ) written with the Pascal distribution are discussed for the subclass Tκ(ϵ,♭). Also, we provide a sufficient condition for the inclusion relation IθsRϖ(E,D)⊂Tκ(ϵ,♭). Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained.

In this article, we investigate the problem of computing the $ * $ ∗ -product of domains in $ \mathbb {C}^N $ C N . Assuming that $ 0\in G\subset \mathbb {C}^N $ 0 ∈ G ⊂ C N is an arbitrary Runge domain and $ 0\in D\subset \mathbb {C}^N $ 0 ∈ D ⊂ C N is a bounded, smooth and linearly convex domain (or a non-decreasing union of such ones), we establish a geometric relation between $ D*G $ D ∗ G and another domain in $ \mathbb {C}^N $ C N which is 'extremal' (in an appropriate sense) with respect to a special coefficient multiplier dependent only on the dimension N. Next, for N = 2, we derive a characterization of the latter domain expressed in terms of planar geometry. These two results, when combined together, give a formula which allows to calculate $ D*G $ D ∗ G for two-dimensional domains D and G satisfying the outlined assumptions. [ABSTRACT FROM AUTHOR]

New possibilities of Gramian computation, by means of canonical transformations into diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the Gramian matrices can be represented as products of the Hadamard matrices of multipliers and the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these multiplier matrices are invariant under various canonical transformations of linear continuous systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI systems are developed. New algorithms for the computation of controllability Gramians in the form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of state in the controllable and observable canonical forms. The application of transformations to the canonical forms of controllability and observability allowed us to simplify the formulas of the spectral decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems are obtained, including spectral expansions of the finite and infinite cross - Gramians of continuous MIMO LTI systems. Recommendations on the use of the obtained results are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a class of bi-univalent functions, denoted by F(n, α, β), that depends on the Ruscheweyh operator and defined by the use of Gegenbauer Polynomials. For functions in this class, we derive the estimations for the initial Taylor-Maclaurin coefficients |a2| and |a3|. Moreover, we obtain the classical Fekete-Szegö inequality of functions belonging to this class. [ABSTRACT FROM AUTHOR]

The spectral radius of nonnegative tensor which is one of the significant research topics in tensor analysis has been extensively concerned by scholars. In this paper, we introduce the Hadamard product of tensors as a useful tool, and then present some inequalities with simple and elegant forms characterizing the upper bound on spectral radius for the Hadamard product of nonnegative tensors. Furthermore, the theoretical comparisons between the newly proposed upper bounds on spectral radius for the Hadamard product of nonnegative tensors are investigated and established. To further illustrate the effectiveness of the main results, some numerical examples are considered. [ABSTRACT FROM AUTHOR]

In this paper, we introduced a class of analytic functions defined by (H-R) fractional derivative defined in the unit disk, we obtained coefficient bounded, so we obtained some theorems of this class. 2000Mathematics Subject classification: 30C45. [ABSTRACT FROM AUTHOR]

In this paper, using operator Mowafy et. al. Which modified the operator of Al-Aboudi and Al-Amoudi we defined a new operator from which one can obtain many other new operators using the principle of Hadamard product (or convolution) by taking different values of its parameters. This operator has, three recurrence relations. Using one of its these recurrence relation we defined four classes related to univalent starlike and convex functions of order α, close-to-convex and quasi-convex functions of order η and type α (0≤α,η<1), which in turn generalize many other classes for different values of parameter. For these classes and by using Miller and Mocanu lemma we can obtain many inclusion results. Also some inclusion resulls for Libera integral operator are obtained. We have proved our rersults here by using the first recurrence relation. Using similar argument by applying the second and third recurrence relation one can get new results. [ABSTRACT FROM AUTHOR]

In this article, we determinate some applications of first order differential subordination and superordination results involving Hadamard product for λ-pseudo-starlike and λ-pseudo-convex functions with respect to symmetrical points defined in the open unit disk U. These results are applied to obtain sandwich results. [ABSTRACT FROM AUTHOR]

In this paper, we work with the umbrella matrix Lie group and its Lie algebra. We show that the Lie subalgebra of the PC matrix is the Lie algebra of the umbrella matrix. Furthermore, we obtain an isomorphism between the Lie subgroup of PC matrices and the umbrella matrix Lie group. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to find a necessary and sufficient condition for Miller-Ross-type Poisson distribution series to be in the class 𝒲δ(α, γ, β) of analytic functions with negative coefficients .Also, we investigate several inclusion properties of the classes 𝒮∗, 𝒦 and ℝτ(A, B) associated of the operator 𝕀ν,cm defined by this distribution. Further, we consider an integral operator related to Miller-Ross-type Poisson distribution series. Several corollaries and consequences of the main results are also considered.

In this search, we investigate a relation between generalized distribution series and particular subclasses of univalent functions. Further, we obtain the sufficient conditions for generalized distribution series 𝒩ψ(τ,z) and ℳ*ψ(η,τ,z) belongs to ℒθ(A,B;γ). Also, we investigate some mapping properties for this class. Finally, we obtain some corollaries and consequences of the main results.

Geometric function theory (GFT) is one of the richest research disciplines in complex analysis. This discipline also deals with the extended differential inequality theory, known as the differential subordination theory. Based on these theories, this study focuses on analyzing intriguing aspects of the geometric subclass of meromorphic functions in terms of a linear complex operator and a special class of Hurwitz-Lerch-Zeta functions. Hence, several of its geometric attributes are deduced. Furthermore, the paper highlights the different fascinating advantages and applications of various new geometric subclasses in relation to the subordination and inclusion theorems.

Using the operator $\frak{D}_{q,\varrho}^m(\lambda,l)$, we introduce the subclasses $\frak{Y}^{*m}_{q,\varrho}(l,\lambda,\gamma)$ and $\frak{K}^{*m}_{q,\varrho}(l,\lambda,\gamma)$ of normalized analytic functions. Among the results investigated for each of these function classes, we derive some subordination results involving the Hadamard product of the associated functions. The interesting consequences of some of these subordination results are also discussed. Also, we derive integral means results for these classes.

In the present paper, we determine necessary and sufficient conditions for the subclass T(α, b) of analytic functions associated with Pascal distribution. Further, we consider properties of a special function related to Pascal distribution series. Several corollaries and consequences of the main results are also considered.

Let H be a Hilbert space. In this paper we show among others that, if f, g are continuous on the interval I with 0

Using the Salagean q-differential operator, we investigate a novel subclass of analytic functions in the open unit disc, and we use the Hadamard product to provide some inclusion relations. Furthermore, the coefficient conditions, convolution properties, and applications of the q-fractional calculus operators are investigated for this class of functions. In addition, we extend the Miller and Mocanu inequality to the q-theory of analytic functions. [ABSTRACT FROM AUTHOR]

The selection of a demand forecasting method is critical for companies aiming to avoid manufacturing overproduction or shortages in pursuit of sustainable development. Various qualitative and quantitative criteria with different weights must be considered during the evaluation of a forecasting method. The qualitative criteria and criteria weights are usually assessed in linguistic terms. Aggregating these various criteria and linguistic weights for evaluating and selecting demand forecasting methods in sustainable manufacturing is a major challenge. This paper proposes an extension of fuzzy elimination and choice translating reality (ELECTRE) I to resolve this problem. In the proposed method, fuzzy weighted ratings are defuzzified with the signed distance to develop a crisp ELECTRE I model. Moreover, an extension to ELECTRE I is developed by suggesting an extended modified discordance matrix and a closeness coefficient for ranking alternatives. The proposed extension can overcome the problem of information loss, which can lead to incorrect ranking results when using the Hadamard product to combine concordance and modified discordance matrices. A comparison is conducted to show the advantage of the proposed extension. Finally, a numerical example is used to demonstrate the feasibility of the proposed method. Furthermore, a numerical comparison is made to display the advantage of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, the present study employs the utilization of the concepts pertaining to (x , y) -symmetrical functions, Janowski type functions, and q-calculus in order to establish a novel subclass within the open unit disk. Specifically, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore specific characteristics of the class S ˜ q x , y (α , β , λ) , thereby further scrutinizing the convolution properties of these newly defined classes. [ABSTRACT FROM AUTHOR]

In this paper, we establish some theorems in order to evaluate certain single and double integral formulas associated with the generalized M-series. The results are generalizations and unification of formulas for a fairly general class of special functions with a focus on the Mittag-Leffler function and hypergeometric function. By specializing the parameters, some new or known integrals are also investigated as corollaries in terms of generalized hypergeometric function. Further, applying the Hadamard product of two analytic functions, we have represented all our main findings in the Hadamard product of two known functions. [ABSTRACT FROM AUTHOR]

The objective of the current paper is to find the necessary and sufficient conditions for Miller–Ross-type Poisson distribution series to be in the classes S T * (γ , β) and K T (γ , β) of analytic functions with negative coefficients. Furthermore, we investigate several inclusion properties of the class Y σ (V , W) associated of the operator I α , c ε defined by this distribution. We also take into consideration an integral operator connected to series of Miller–Ross-type Poisson distributions. Special cases of the main results are also considered. [ABSTRACT FROM AUTHOR]

The standard simplex in |$\mathbb{R}^{n}$|⁠ , also known as the probability simplex, is the set of nonnegative vectors whose entries sum up to 1. It frequently appears as a constraint in optimization problems that arise in machine learning, statistics, data science, operations research and beyond. We convert the standard simplex to the unit sphere and thus transform the corresponding constrained optimization problem into an optimization problem on a simple, smooth manifold. We show that Karush-Kuhn-Tucker points and strict-saddle points of the minimization problem on the standard simplex all correspond to those of the transformed problem, and vice versa. So, solving one problem is equivalent to solving the other problem. Then, we propose several simple, efficient and projection-free algorithms using the manifold structure. The equivalence and the proposed algorithm can be extended to optimization problems with unit simplex, weighted probability simplex or |$\ell _{1}$| -norm sphere constraints. Numerical experiments between the new algorithms and existing ones show the advantages of the new approach. Open source code is available at https://github.com/DanielMckenzie/HadRGD. [ABSTRACT FROM AUTHOR]

Using a large number of drones poses the challenge of correct, transparent, and high-performance modeling of their co-operation as a swarm. The authors develop and present an example of transitioning from a description of the control system of a single drone to the behavior of a swarm of identical drones based on the Hadamard algebra (product). It is shown that this approach increases model performance tenfold and allows the modeling of swarm systems with a very large number of units. [ABSTRACT FROM AUTHOR]

Based on the binary product described by any Latin square, the Hadamard quasigroup product is introduced in this paper as a natural generalization of the classical Hadamard product of matrices. The successive iteration of this new product is endowed with a cyclic behaviour that enables one to define a pair of new isomorphism invariants of Latin squares. Of particular interest is the set of Latin squares for which this iteration preserves the Latin square property, which requires the existence of successive localized Latin transversals within the Latin square under consideration. In order to enumerate and classify, up to isomorphism, these Latin squares, we propose a computational algebraic geometry approach based on the computation of reduced Gröbner bases. To illustrate this point, we obtain the classification of the sought Latin squares, for order up to six, by using the open computer algebra system for polynomial computations Singular.

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions ΤJ(αi). Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f∈ΤJ(αi). In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass.