Hadamard Multiplication Research Articles

On the Hadamard multiplication theorem in ℂ n

The Hadamard product of two holomorphic functions given by Taylor series ∑ α a α z α and ∑ α b α z α , defined on neighborhoods Ω 1 and Ω 2 of the origin, is the function whose Taylor series is ∑ α a α b α z α . For n = 1 , it is known that the Hadamard product extends to a holomorphic function on the so-called *-product of Ω 1 and Ω 2 . In this paper, we generalize this result to the case of several complex variables.

On a Banach algebra of entire functions with a weighted Hadamard multiplication

New algebraic-analytic properties of a previously studied Banach algebra A(p) of entire functions are established. For a given fixed sequence (p(n))n≥0 of positive real numbers, such that limn→∞⁡p(n)1n=∞, the Banach algebra A(p) is the set of all entire functions f such that f(z)=∑n=0∞fˆ(n)zn (z∈C), where the sequence (fˆ(n))n≥0 of Taylor coefficients of f satisfies fˆ(n)=O(p(n)−1) for n→∞, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication ⁎ with weight given by p (i.e., (f⁎g)(z)=∑n=0∞p(n)fˆ(n)gˆ(n)zn for all z∈C), and the norm ‖f‖=supn≥0⁡p(n)|fˆ(n)|. The following results are shown:•The Topological stable rank of A(p) is 1.•The Bass stable rank of A(p) is 1.•A(p) is a Hermite ring.•A(p) is not a projective-free ring.•Idempotents in A(p) are described.•Exponentials in A(p) are described, and it is shown that every invertible element of A(p) has a logarithm, so that the first Čech cohomology group H1(M(A(p)),Z) with integer coefficients of the maximal ideal space M(A(p)) is trivial.•A generalised necessary and sufficient ‘corona-type condition’ on the matricial data (A,b) with entries from A(p) is given for the solvability of Ax=b with x also having entries from A(p).•The Krull dimension of A(p) is infinite.•A(p) is neither Artinian nor Noetherian.•A(p) is coherent.•The special linear group over A(p) is generated by elementary matrices.

Belief reliability index determination method based on group decision‐making

AbstractFor newly developing systems, the determination of the reliability index is an important task in the system demonstration process that determines both the system design goal and the verification requirement. However, due to the insufficient data at the beginning of the system life cycle, the problem of epistemic uncertainty cannot be ignored. Researchers have proposed several belief reliability metrics and analysis methods to quantify the epistemic uncertainty of a system, but to the best knowledge of the authors, no relevant studies have been performed on system reliability index determination. In this paper, we propose a belief reliability index determination method based on group decision‐making. Using a similar system‐based reliability index determination method, this method comprehensively considers the demands of users and the capabilities of contractors. In addition, the Delphi method and Hadamard multiplication convex combination are combined to effectively solve the epistemic uncertainty quantification problem. The Delphi method is used to find the conflicts caused by multi‐expert scoring, and the Hadamard multiplication convex combination can integrate the scoring matrix of all experts, which makes the results more reasonable. Finally, a Low‐Earth‐orbit (LEO) satellite communication system is used to verify the effectiveness of our belief reliability index determination method.

Multipliers on Spaces of Holomorphic Functions

We consider multipliers on the space of holomorphic functions of one variable H(Omega ),,Omega subset mathbb {C} open, that is, linear continuous operators for which all monomials are eigenvectors. If zero belongs to Omega and Omega is a domain these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Hadamard multiplication operators are multipliers. In the case of Runge open sets we represent all multipliers via a kind of multiplicative convolutions with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. We also identify which topology should be put on the subspace of analytic functionals in order for that isomorphism to become a topological isomorphism, when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on H(Omega ). We provide one more representation of multipliers via suitable germs of holomorphic functions with Laurent or Taylor coefficients equal to the eigenvalues of the operator. We also discuss a special case, namely, when Omega is convex.

Effect of Hadamard multiplication on bloom filter and double bloom filter transformations

AbstractIris biometric is the most common and widely accepted biometric authentication method due to its high accuracy. The iris biometric template should be protected to overcome security and privacy attacks. The two standard iris biometric template protection methods are bloom filter and double bloom filter‐based feature transformations. In this article, we first permuted and then applied the p‐order Hadamard product on the iris template before the bloom filter and double bloom filter‐based transformation to enhance the security. The performance and security are measured using the CASIAv1 iris benchmark database to study the effect of Hadamard multiplication on bloom and double bloom filter‐based transformation. The Hadamard multiplication on the iris template increases the security of the transformed iris template and slightly degrades the matching performance.

Super-Resolution TOA and AOA Estimation for OFDM Radar Systems Based on Compressed Sensing

This article presents a compressed sensing-based time of arrival (TOA) and angle-of-arrival (AOA) estimation algorithm for orthogonal frequency division multiplexing (OFDM) radar systems. The algorithm is designed for noncooperative targets based on a uniform linear array using a cyclic prefix (CP) added OFDM signal. The algorithm makes three key technical contributions. First, the algorithm adopts the CP-based OFDM signal for the radar TOA/AOA estimation to suppress the multitarget interference and the impact of time delay on the subcarrier orthogonality. Second, this article exploits the structure of the CP-OFDM radar signal model to construct the optimization problem of the joint TOA/AOA recovery. The super-resolution AOA estimation is obtained by using a redundant dictionary containing much more basis than the number of antennas. Third, the algorithm proposes an efficient way to solve the optimization by utilizing the properties of the circulant matrix, fast Fourier transform, and Hadamard multiplication. The simulation results indicate the effectiveness of the proposed algorithm.

On the theory of interpolation of functions on sets of matrices with the Hadamard multiplication

This article is devoted to the problem of interpolation of functions defined on sets of matrices with multiplication in the sense of Hadamard and is mainly an overview. It contains some known information about the Hadamard matrix multiplication and its properties. For functions defined on sets of square and rectangular matrices, various interpolation polynomials of the Lagrange type, containing both the operation of matrix multiplication in the Hadamard sense and the usual matrix product, are given. In the case of analytic functions defined on sets of square matrices with the Hadamard multiplication, some analogues of the Lagrange type trigonometric interpolation formulas are considered. Matrix analogues of splines and the Cauchy integral are given on sets of matrices with the Hadamard multiplication. Some of its applications in the theory of interpolation are considered. Theorems on the convergence of some Lagrange interpolation processes for analytic functions defined on a set of matrices with multiplication in the Hadamard sense are proved. The results obtained are based on the application of some well-known provisions of the theory of interpolation of scalar functions. Data presentation is illustrated by a number of examples.

Optimum Extrapolation Techniques for Two-Dimensional Antenna Array Tapered Beamforming

Optimizing antenna arrays is essential for achieving efficient beamforming with very low sidelobe level (SLL) where adopting tapered window functions is one of the straightforward efficient techniques for achieving this goal. Recently, two-dimensional (2D) beamforming has been extensively required for many applications; therefore, this paper proposes two extrapolation techniques applied to one-dimensional (1D) tapered functions to efficiently feed 2D antenna arrays using cross-linear and adaptive radial tapering techniques. The first proposed 2D cross-linear tapering technique determines the 2D tapering coefficients by Hadamard multiplication of two right-angled grids of repeated 1D functions, while the second proposed adaptive radial tapering technique locates the antenna element in the 2D array in terms of its radial distance with respect to the array center, then converts this distance to an element index in a virtual 1D tapering window to determine the element weighting value. The adaptive radial tapering technique is optimized for achieving the minimum SLLs. The two proposed techniques are analyzed and discussed, where it is found that the adaptive radial tapering provides deeper SLLs compared to the cross-linear tapering technique. The two extrapolation techniques are examined for four window functions including triangular (Bartlett), Hamming, cosine-square, and Blackman windows, and the simulation results show that for extrapolating the Blackman window using adaptive radial tapering, a –50 dB SLL can be achieved which is independent on the array size, while cross-linear tapering provides –35 dB and –41 dB SLLs for and antenna arrays, respectively.

Contribution to the Hadamard multiplication theorem

In this article we define a binary linear operator T for holomorphic functions in given open sets \(A\) and \(B\) in the complex plane under certain additional assumptions. It coincides with the classical Hadamard product of holomorphic functions in the case where \(A\) and \(B\) are the unit disk. We show that the operator T exists provided \(A\) and \(B\) are simply connected domains containing the origin. Moreover, T is determined explicitly by means of an integral form. To this aim we prove an alternative representation of the star product \(A*B\) of any sets \(A,B\subset\mathbb{C}\) containing the origin. We also touch the problem of holomorphic extensibility of Hadamard product.

On symmetric association schemes and associated quotient-polynomial graphs

Let $\Gamma$ denote an undirected, connected, regular graph with vertex set $X$, adjacency matrix $A$, and ${d+1}$ distinct eigenvalues. Let ${\mathcal A}={\mathcal A}(\Gamma)$ denote the subalgebra of Mat$_X({\mathbb C})$ generated by $A$. We refer to ${\mathcal A}$ as the {\it adjacency algebra} of $\Gamma$. In this paper we investigate algebraic and combinatorial structure of $\Gamma$ for which the adjacency algebra ${\mathcal A}$ is closed under Hadamard multiplication. In particular, under this simple assumption, we show the following: (i) ${\mathcal A}$ has a standard basis $\{I,F_1,\ldots,F_d\}$; (ii) for every vertex there exists identical distance-faithful intersection diagram of $\Gamma$ with $d+1$ cells; (iii) the graph $\Gamma$ is quotient-polynomial; and (iv) if we pick $F\in \{I,F_1,\ldots,F_d\}$ then $F$ has $d+1$ distinct eigenvalues if and only if span$\{I,F_1,\ldots,F_d\}=$span$\{I,F,\ldots,F^d\}$. We describe the combinatorial structure of quotient-polynomial graphs with diameter $2$ and $4$ distinct eigenvalues. As a consequence of the technique from the paper we give an algorithm which computes the number of distinct eigenvalues of any Hermitian matrix using only elementary operations. When such a matrix is the adjacency matrix of a graph $\Gamma$, a simple variation of the algorithm allow us to decide wheter $\Gamma$ is distance-regular or not. In this context, we also propose an algorithm to find which distance-$i$ matrices are polynomial in $A$, giving also these polynomials.

ISAR Imaging Motion Compensation in Low SNR Environments Using Phase Gradient and Filtering Techniques

In a low signal-to-noise ratio (SNR) environment, conventional range profiles-based motion compensation (MoCo) methods become inapplicable for inverse synthetic aperture radar (ISAR) imaging. In this article, a novel method for range profile alignment along with phase adjustment is proposed. The method is applied in the uncompressed phase history domain without the need for a prominent point. Motion errors are estimated from a phase map using only fast Fourier transform (FFT) and Hadamard multiplication operations, making the method suitable for real-time applications. The proposed method strives to aggregate the energy of all scatterers into a single peak by 2-D coherent accumulation incorporating the product of the radar pulse backscattering and its subsequent pulse, hence the reference to phase gradient processing (PGP). This process strengthens SNR and overcomes the problems associated with low signal power, even when the range profiles are submerged in noise. Experimental results are provided to demonstrate the performance of the proposed method compared with the state-of-the-art algorithms.

Fractal Generalized Pascal Matrices

The set of generalized Pascal matrices whose entries are generalized binomial coefficients is regarded as a group with respect to Hadamard multiplication. A special system of matrices is introduced and is used to construct fractal generalized Pascal matrices. The Pascal matrix (triangle) is expanded in theHadamard product of fractal generalized Pascal matrices whose nonzero entries are pk, where p is a fixed prime and k = 0, 1, 2,.... The introduced system of matrices suggests the idea of “zero” generalized Pascal matrices, each of which is the limit case of a certain set of generalized Pascal matrices. “Zero” fractal generalized Pascal matrices, which are exemplified by the Pascal triangle modulo 2, are considered.

A Fast Logic Mapping Algorithm for Multiple-Type-Defect Tolerance in Reconfigurable Nano-Crossbar Arrays

Unlike conventional CMOS circuits, nano-crossbar arrays have considerably high defect rates. Multiple-type defects randomly occur both on crosspoint switches and wires that substantially complicates the design phase of the circuits with an elimination of systematic design choices. In order to overcome this problem, a logic mapping methodology is presented in this paper. A fast heuristic algorithm using pre-mapping logic morphing, defect oriented adaptive sorting, matching with Hadamard multiplication, and backtracking is introduced. The proposed algorithm covers both crosspoint defects including stuck-open and stuck-closed types and wire defects including bridging and broken types. Effects of stuck-closed defects, mostly disregarded in the literature, are studied in depth. In simulations, an industrial benchmark suit is used for obtaining runtime and success rate values of the proposed algorithm in comparison with those of the existing algorithms in the literature. A relative accuracy evaluation is also given in comparison with exact mapping techniques. Finally, the steps of the algorithm that are based on pre-mapping and heuristic matching techniques, are separately justified with experimental results.

On the formation of convex and starlike functions with the form of Hadamard multiplication redefine

On the formation of convex and starlike functions with the form of Hadamard multiplication redefine

$\mathcal{E}'$ as an algebra by multiplicative convolution

We study the algebra $\mathscr{E}'(\mathbb{R}^d)$ equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1y_1,\dots,x_dy_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(\Omega)$, $\Omega$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators on open subsets of $\mathbb{R}_+^d$.

The Hadamard multiplication theorem in several complex variables

The aim of this paper is to generalize the Hadamard multiplication theorem and the notion of the -product of domains to the case of several complex variables. We obtain a result of this kind in a certain special class of domains.

Convolution operators on spaces of holomorphic functions

A class of convolution operators on spaces of holomorphic functions related to the Hadamard multiplication theorem for power series and generalizing infinite order Euler differential operators is introduced and investigated. Emphasis is placed on question

Series transformation formulas of Euler type, Hadamard product of series, and harmonic number identities

The Hadamard multiplication theorem for series is used to establish several Euler-type series transformation formulas. As applications we obtain a number of binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers.

Growth of entire functions of exponential type defined by convolution

Based on the Borel transformation and the Hadamard multiplication theorem on singularities on the convolution of holomorphic functions, results on the growth of entire functions defined by convolution of an entire function of exponential type with a function holomorphic at the origin are obtained.

The α-scalar diagonal stability of block matrices

We generalize two results: Kraaijevanger’s 1991 characterization of diagonal stability via Hadamard products and the block matrix version of the closure of the positive definite matrices under Hadamard multiplication. We restate our generalizations in terms of P α -matrices and α-scalar diagonally stable matrices.