Related papers: The Schur Algorithm in Terms of System Realization
This paper introduces the application of the asynchronous iterations theory within the framework of the primal Schur domain decomposition method. A suitable relaxation scheme is designed, which asynchronous convergence is established under…
Schur-Weyl duality is a powerful tool in representation theory which has many applications to quantum information theory. We provide a generalization of this duality and demonstrate some of its applications. In particular, we use it to…
Two possible realizations of the formal neuron are considered as quantum system. The first type complies with classical system. The second type vastly increases the possible problems.
We develop the theory of minimal realizations and factorizations of rational functions where the coefficient space is a ring of the type introduced in our previous work, the scaled quaternions, which includes as special cases the…
Let $E$ be a $W^{\ast}$-correspondence over a von Neumann algebra $M$ and let $H^{\infty}(E)$ be the associated Hardy algebra. If $\sigma$ is a faithful normal representation of $M$ on a Hilbert space $H$, then one may form the dual…
In this paper, we prove the following. First, every square matrix whose entries are multivariable rational functions over a field $\mathbb{F}$ has a Bessmertny\u{i} realization, i.e., is the Schur complement of an affine linear square…
We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one…
Mutual coherence is a measure of similarity between two opinions. Although the notion comes from philosophy, it is essential for a wide range of technologies, e.g., the Wahl-O-Mat system. In Germany, this system helps voters to find…
The main purpose of this article is to describe potential benefits and applications of the SP theory, a unique attempt to simplify and integrate ideas across artificial intelligence, mainstream computing and human cognition, with…
This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the…
An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of…
The Wigner function of quantum systems is an effective instrument to construct the approximate classical description of the systems for which the classical approximation is possible. During the last time, the Wigner function formalism is…
Solving symbolic reasoning problems that require compositionality and systematicity is considered one of the key ingredients of human intelligence. However, symbolic reasoning is still a great challenge for deep learning models, which often…
This is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. Functional analysis takes us beyond finite matrices, allowing us to work with infinite sets of…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
This paper presents the main features of a system that aims to transform regular expressions into shorter equivalent expressions. The system is also capable of computing other operations useful for simplification, such as checking the…
In this vision paper, we explore the challenges and opportunities of a form of computation that employs an empirical (rather than a formal) approach, where the solution of a computational problem is returned as empirically most likely…
Higher-level cognition depends on the ability to learn models of the world. We can characterize this at the computational level as a structure-learning problem with the goal of best identifying the prevailing causal relationships among a…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
Coherent state theory is shown to reproduce three categories of representations of the spectrum generating algebra for an algebraic model: (i) classical realizations which are the starting point for geometric quantization; (ii) induced…