Narrow Results

The intersection of geometric methods and signal processing has led to significant advancements in various fields, including physics, engineering, and mathematics. In recent years, the biquaternion domain, a generalization of complex numbers with a geometric interpretation, has emerged as a promising tool for signal analysis. This paper introduces the Biquaternion Linear Canonical Stockwell Transform (BiQLCST), a novel fusion of advanced mathematical frameworks that offers enhanced signal analysis capabilities and unveils new uncertainty principles, bridging the gap between complex transformations and uncertainty analysis in high-dimensional spaces. First, we established the various fundamental properties, including linearity, shift, modulation, parity, orthogonality relation, reconstruction formula and Plancherel’s theorem. Heisenberg uncertainty principle associated with Biquaternion Linear Canonical Stockwell Transform (BiQLCST) is also established. Toward the end, some potential applications of the BiQLCST are presented.

In order to give numerical characterizations of the notion of "mutual orthogonality", we introduce two kinds of family of positive definite matrices for a unitary u in a finite von Neumann algebra M. They are arising from u naturally depending on the decompositions of M. One corresponds to the tensor product decomposition and the other does to the crossed product decomposition. By using the von Neumann entropy for these positive definite matrices, we characterize the notion of mutual orthogonality between subalgebras.

We comment on some apparently weak points in the novel strategies recently developed by various authors aiming at a proof of the Riemann hypothesis. After noting the existence of relevant previous papers where similar tools have been used, we refine some of these strategies. It is not clear at the moment if the problems we point out here can be resolved rigorously, and thus a proof of the RH be obtained, along the lines proposed. However, a specific suggestion of a procedure to overcome the encountered difficulties is made, what constitutes a step towards this goal.

In switching algebra, there exist standard forms of Boolean functions such as the disjunctive or conjunctive form. This paper discusses the theory of the extended standard forms of Boolean functions. In addition to the four existing standard forms, two further forms are introduced and thus expanded to six basic forms. On the one hand, the existence of the extended forms is presented and on the other hand new formulas and equations are illustrated. Equations relating to the resolution and/or solution of conjunction/disjunctions are detailed and proven to be valid. In addition, equations for conversion between forms are exemplified. Finally, certain formal relations between the basic forms that are valid under certain conditions are featured.

Well-designed software systems, with providers only modules, have been rigorously obtained by algebraic procedures from the software Laplacian Matrices or their respective Modularity Matrices. However, a complete view of the whole software system should display, besides provider relationships, also consumer relationships. Consumers may have two different roles in a system: either internal or external to modules. Composite modules, including both providers and internal consumers, are obtained from the joint providers and consumers Laplacian matrix, by the same spectral method which obtained providers only modules. The composite modules are integrated into a whole Software System by algebraic connectors. These algebraic connectors are a minimal Occam’s razor set of consumers external to composite modules, revealed through iterative splitting of the Laplacian matrix by Fiedler eigenvectors. The composite modules, of the respective standard Modularity Matrix for the whole software system, also obey linear independence of their constituent vectors, and display block-diagonality. The spectral method leading to composite modules and their algebraic connectors is illustrated by case studies. The essential novelty of this work resides in the minimal Occam’s razor set of algebraic connectors — another facet of Brooks’ Propriety principle leading to Conceptual Integrity of the whole Software System — within Linear Software Models, the unified algebraic theory of software modularity.

Although noise-based logic shows potential advantages of reduced power dissipation and the ability of large parallel operations with low hardware and time complexity the question still persist: Is randomness really needed out of orthogonality? In this Letter, after some general thermodynamical considerations, we show relevant examples where we compare the computational complexity of logic systems based on orthogonal noise and sinusoidal signals, respectively. The conclusion is that in certain special-purpose applications noise-based logic is exponentially better than its sinusoidal version: Its computational complexity can be exponentially smaller to perform the same task.

It is well known that the lattice of subspaces of a vector space over a field is modular. We investigate under which conditions this lattice is orthocomplemented with respect to the orthogonality operation. Using this operation, we define closed subspaces of a vector space and study the lattice of these subspaces. In particular, we investigate when this lattice is modular or orthocomplemented. Finally, we introduce splitting subspaces as special closed subspaces and we prove that the poset of splitting subspaces and the poset of projections are isomorphic orthomodular posets. The vector spaces under consideration are of arbitrary dimension and over arbitrary fields.

The concept of the spectral rank of an element in a ring is defined, and it is shown to be a genuine generalization of the same concept first studied in the setting of a Banach algebra. Furthermore, we prove that many of the desirable properties of this rank are still valid in the more abstract setting and give several examples to support and motivate the given definition. In particular, we are able to show that a nonzero idempotent of a semiprimitive and additively torsion-free ring is minimal if and only if it has a spectral rank of one. We also discover a precise connection between the spectral rank of an element in a ring and a purely algebraic definition of rank considered only recently by N. Stopar in [Rank of elements of general rings in connection with unit-regularity, J. Pure Appl. Algebra 224 (2020) 106211]. Specifically, we are able to show that if an element $a$ has a finite algebraic rank in a semiprimitive and additively torsion-free ring, then $a$ has the exact same spectral rank. An extra condition under which the converse holds true is also provided, and connections to the socle are identified. Finally, for both of these extended notions of rank considered in the setting of a ring, we prove a generalized Frobenius Inequality.

A simple method of construction of a pair of orthogonal wavelet frames in L₂(ℝ^d) is presented. This is a generalization of one-dimensional case to higher dimension. The construction is based on the well-known Unitary Extension Principle (UEP). The presented method produces the polyphase components of the filters of the wavelet functions, and hence the filters. A pair of orthogonal wavelet frames can be constructed with an extra condition. In the construction, the polyphase matrix is used as opposed to the modulation matrix. This is less restrictive and yields a fewer wavelet functions in the system than in the previously known constructions.

In this paper, we propose a novel integral transform coined as quaternion quadratic-phase wavelet transform (QQPWLT) by invoking the elegant convolution structure associated with the quaternion quadratic-phase Fourier transform. First, we explore some mathematical properties of the QQPWLT, including the orthogonality relation, inversion formula, reproducing kernel and some notable inequalities. Second, we study Heisenberg’s uncertainty principles and the logarithmic uncertainty principle associated with the quadratic-phase wavelet transform in quaternion domain. We culminate our investigation by presenting some illustrative examples.

Supersaturated designs are useful in screening experiments. This paper discusses the topic or multi-level supersaturated design. Two quantities, E(d²) and D_f, are proposed to evaluate the optimality of supersaturated designs. A lower bound of E(d²) is obtained with a necessary condition for achieving it. Some E(d²)-optimal supersaturated designs of 3, 4, and 5 levels are given.

Recall that a Dirichlet character is called imprimitive if it is induced from a character of smaller level, and otherwise it is called primitive. In this paper, we introduce a modification of "inducing to higher level" which causes imprimitive characters to behave primitively, in the sense that the properties of the associated Gauss sum and the functional equation of the attached L-function take on a form usually associated to a primitive character.

In this paper, we first examine the properties of the matrices of the generalized $m$-step Jacobsthal sequence, and we develop a public key cryptosystem using key exchange based on the relationship between inner product and orthogonality. Then we create encryption and decryption algorithms using this key exchange, self-adjoint operator, and generalized $m$-step Jacobsthal sequence, and we provide a detailed example of how encryption and decryption algorithms work. Finally, a security strength and complexity analysis of these algorithms will be made.

Although noise-based logic shows potential advantages of reduced power dissipation and the ability of large parallel operations with low hardware and time complexity the question still persist: Is randomness really needed out of orthogonality? In this Letter, after some general thermodynamical considerations, we show relevant examples where we compare the computational complexity of logic systems based on orthogonal noise and sinusoidal signals, respectively. The conclusion is that in certain special-purpose applications noise-based logic is exponentially better than its sinusoidal version: Its computational complexity can be exponentially smaller to perform the same task.

Discrepancy is a kind of important measure used in experimental design. Recently, a so-called discrete discrepancy has been applied to evaluate the uniformity of factorial designs. In this paper, we review some recent advances on application of the discrete discrepancy to several common experimental designs and summarize some important results.