Symbolic Matrix Research Articles

A Toeplitz-Like Operator with Rational Matrix Symbol Having Poles on the Unit Circle: Fredholm Properties

In this paper a definition is given for an unbounded Toeplitzlike operator with rational symbol which has poles on the unit circle. It is shown that the operator is Fredholm if and only if the symbol has no zeroes on the unit circle, and a formula for the index is given as well. Finally, a matrix representation of the operator is discussed.

Finite Characterization of the Coarsest Balanced Coloring of a Network

Balanced colorings of networks correspond to flow-invariant synchrony spaces. It is known that the coarsest balanced coloring is equivalent to nodes having isomorphic infinite input trees, but this condition is not algorithmic. We provide an algorithmic characterization: two nodes have the same color for the coarsest balanced coloring if and only if their [Formula: see text]th input trees are isomorphic, where [Formula: see text] is the number of nodes. Here [Formula: see text] is the best possible. The proof is analogous to that of Leighton’s theorem in graph theory, using the universal cover of the network and the notion of a symbolic adjacency matrix to set up a partition refinement algorithm whose output is the coarsest balanced coloring. The running time of the algorithm is cubic in [Formula: see text].

A Novel Hybrid Secure Method Based on DNA Encoding Encryption and Spiral Scrambling in Chaotic OFDM-PON

This paper proposes a novel hybrid secure method based on improved deoxyribonucleic acid (DNA) encoding encryption and spiral scrambling in chaotic OFDM-PON for enhancing the physical-layer security. In the improved DNA encoding encryption, the odd-even cross-bit DNA encoding, base scrambling and base-level substitution are determined by chaotic sequences. For each binary and base, the selected encoding rules and base scrambling methods are dynamically changing, which enhances the robustness against malicious attacks by attackers. In the spiral scrambling process, the QAM symbol matrix is divided into several blocks, and these blocks are scrambled. In the scrambling of the plural matrix, the position where the spiral starts and the orientation and direction of the traversal are also controlled by the chaotic sequences. By employing DNA encoding encryption and spiral scrambling, a key space of $\sim \!{10^{135}}$ can be achieved in the multi-fold encryption of the proposed scheme, which can improve the physical-layer security. The encrypted 16-QAM OFDM data are successfully transmitted over a 60-km SSMF in OFDM-PON. The simulation results demonstrate that it has better BER performance at the BER of ${10^{ - 3}}$ than other schemes. The proposed encryption method can effectively protect data from attacks by eavesdroppers or illegal users.

On the incomplete Srivastava’s triple hypergeometric matrix functions

The paper proposes to introduce incomplete Srivastava’s triple hypergeometric matrix functions through application of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, reduction formula, recursion formula, recurrence relation and differentiation formula of the incomplete Srivastava’s triple hypergeometric matrix functions.

Solving CNLS problems using Levenberg-Marquardt algorithm: A new fitting strategy combining limits and a symbolic Jacobian matrix

The Levenberg-Marquardt algorithm (LMA) is generally used to solve diverse complex nonlinear least square (CNLS) problems and is one of the most used algorithms to extract equivalent electrochemical circuit (EEC) parameters from electrochemical impedance spectroscopy (EIS) data. It is a well-known fact that the convergence properties of the algorithm can be boosted by applying limits on EEC parameter values. However, when EEC parameter values are low (i.e., of the order of magnitude of 10−4 or smaller), the applied limits increase the first derivatives approximation errors which occur when using a numerical Jacobian matrix. In this work, we discuss the importance of the Jacobian matrix in LMA and propose a design of a new EIS fitting engine. The new engine is based on a novel fitting scheme using limits and a symbolic Jacobian matrix instead of the numerical one, i.e. a strategy that has not yet been reported in any EIS study. We show that using a symbolic Jacobian matrix the algorithm convergence is superior to the one with a numerical Jacobian matrix. We also investigate how to improve poor convergence properties when we still have to use a numerical Jacobian matrix when analytic derivatives are not available.

On a weighted linear matroid intersection algorithm by Deg-Det computation

In this paper, we address the weighted linear matroid intersection problem from the computation of the degree of the determinants of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature: Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian eliminations for constructing residual graphs.

A tessitura da pessoa: o sistema de nomeação Wapichana

A maioria das sociedades indígenas das Terras Baixas da América do Sul privilegia uma reflexão sobre a corporalidade não apenas na elaboração de suas cosmologias, mas também quando elaboram um idioma simbólico focal para a pessoa. A proposta deste artigo consiste em problematizar como o sistema de nominação utilizado pelos Wapichana da aldeia Malacacheta, atual estado de Roraima, enfeixa processos de comunicação do corpo com o mundo. Nominar implica fabricar corpos e tal prática é suplementada pela decoração, afirmação e ritualização dos corpos. O fundamento epistemológico dessa produção converge para os postulados de pessoa, corpo e corporalidade. Trata-se de um estudo onomástico, com enfoque qualitativo, cujo objetivo é mostrar como os Wapichana moldam o corpo por meio de palavras, gestos simbólicos, pinturas, fumigações, ornamentos, rituais e visões cosmogônicas, de modo que o corpo pode ser pensado como matriz de símbolos culturais e configura um objeto de significação social.

Ricci curvature for parametric statistics via optimal transport

We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, which are based on Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

Global hypoellipticity for strongly invariant operators

In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator P with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that P is globally hypoelliptic. As an application, we obtain the characterization of global hypoellipticity on compact Lie groups and examples on the sphere and the torus. We also investigate relations between the global hypoellipticity of P and global subelliptic estimates.

Subshifts, λ-graph bisystems and C⁎-algebras

We introduce a notion of λ-graph bisystem that consists of a pair (L−,L+) of two labeled Bratteli diagrams L−,L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale groupoids constructed from (L−,L+). We study structure of the C⁎-algebras, and show that they are universal unital unique C⁎-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.

Wavelet Channel Coding: An Algebraic Approach

In this paper, the wavelet channel coding (WCC) is revisited in a more general analysis embracing flat real wavelet matrices derived from a Haar matrix and complex input symbols. WCC encoding and decoding are algebraically described and a probability distribution of wavelet symbols is formulated. Signal constellations for transmission of wavelet symbols are proposed and the constellation average energy is deduced from probability generating functions of the wavelet symbols. System performance over a flat Rayleigh channel is analyzed and compared with symbol-by-symbol detecting systems and diversity two space-time block coding (STBC) systems. Simulation results show that WCC presents better performance than ordinary symbol-by-symbol detecting systems, particularly at higher signal-to-noise ratios for higher spectral efficiencies, and STBC systems for spectral efficiency of 1 bit/s/Hz.

Incremental backstepping robust fault-tolerant control with improved IHSTD for RLVs

Aiming at unknown disturbances/uncertainties, partial effectiveness loss fault (PELF) and stuck failure (SF) of the actuator, a composite robust fault-tolerant control strategy based on incremental backstepping (IBS) is proposed for a reusable launch vehicle (RLV) during re-entry. By converting PELF to disturbances/uncertainties, this paper presents an incremental form of disturbance observer based on an improved inverse hyperbolic sine tracking differentiator (IHSTD) to compensate these interference terms originally ignored in the IBS design process. Furthermore, a failure symbol matrix is set to control the on-off states of the reaction control system of the RLV to make up for the missing torque of the actuator SF, which can strengthen the fault-tolerance capability of the control system. The simulation results show that the tracking effect of the proposed method on the attitude-angle commands is better than traditional backstepping with disturbance observer, and the presented control allocation strategy is capable of timely resolving the actuator SF problem to ensure stability of flight.

Социальная память и травматический образ прошлого: социологический дискурс

The paper analyzes the possibilities of the methodology of sociological sciences, allowing to study social memory based on trauma. The empirical basis of the study was the materials of in-depth interviews that the author conducted in 2018 among relatives of missing during the armed phase of the Ossetian-Ingush conflict of 1992. During the interviewing victims, the study of individual traumas of relatives of missing have been built within the following logical chain: trauma as a loss – trauma as a symbolic matrix – trauma as a consolidating event. Considerable attention is paid to the characterization of approaches to the analysis of social memory, methodological aspects of its construction by people who have experienced psychological trauma. The key concepts of social memory are presented both at the individual and collective levels of victims.

Channel Magnitude-Based MIMO With Energy Detection for Internet of Things Applications

In this article, three low complexity multiple-input multiple-output (MIMO) receivers using energy detection and multilevel amplitude-shift keying (ASK) for Internet of Things (IoT) applications are described. To circumvent the inherent nonlinearity of the noncoherent MIMO system, an equivalent linear system model for the channel is introduced. This allows the proposed receivers to be configured analogously to coherent-based approaches. The tradeoff is that increased numbers of receive antennas need to be used but the overall complexity is decreased compared to existing noncoherent approaches. The receivers can also easily be integrated with energy harvesting systems. The three receiver structures considered are zero forcing (ZF), ZF with ordered successive interference cancellation (ZF-OSIC), and weighted envelope detection (WENVD), and their performance in flat Rayleigh fading channels with multilevel ASK is investigated. For the WENVD receiver, the error performance under the condition of high signal-to-noise ratio is also analyzed, and closed form approximations of the conditional (conditioned on the channel matrix) symbol vector error probability (SEP) and the average of the conditional SEP are derived. It is found that the ZF receiver has the lowest complexity and the WENVD receiver the highest; however, error performance comparison through numerical results reveals that the WENVD receiver has the best performance and the ZF-OSIC receiver has the next best. When combined with multiport energy harvesting technology the receivers have significant potential for use in IoT.

Proving the world more imaginary?

Sustainability research has set itself the double-challenge of uncovering the complexity of a globally, locally and historically unsustainable development path, and of contributing to a search process for more sustainable development paths for humanity. A small number of researchers involved in this area have suggested “that maybe the challenge of sustainability isn’t to prove the world more real […] but to prove the world more imaginary” (Robinson as quoted in Taylor 2012, n. p.). Taking up this invocation of the imaginary, the article investigates some imaginaries and imagination of sustainability at play in sustainability research. Four relatively distinct approaches to sustainability research are identified, characterized and differentiated: “triple bottom-line”, “sustainability transformation”, “holistic healing/biophilia”, and “culture of qualitative complexity”. They each develop a specific focus, are nourished by partly different imaginaries and develop their imaginations in distinct directions. In this article, imagination is understood as an individual and social, perceptive and creative process by which we shape realities in our encounters with the world; whereas the imaginary is understood as a deep symbolic matrix that enables our access to the world. Imaginaries are not just made up and imposed on the world by the humans, but the result of an imaginative encounter with the human and other-than-human world. Focused attention on imagination and imaginaries not only allows to observe the area of sustainability research through a differentiating perspective that helps understand certain contrasting and/or shared features across different approaches to sustainability research. This focused attention also bears a potentially instrumental value for inter- and transdisciplinary sustainability research itself, because it encourages sustainability researchers to further reflect on the importance, modalities and different framings of creative and reflective approaches to futures-oriented research agendas. The creative exercise of the imagination is not only at the core of “anticipatory competences” (Wiek et al. 2011, p. 7) for sustainability, but also at the core of percipience to nature-culture’s dynamic complexity. In this respect, sustainability research needs to develop its self-reflexivity beyond discourse-rational approaches to narratives, with a deeper understanding of both embodied cognition and of culture. Reflection on, and radically imaginative work with both dominant and alternative imaginaries that sustainability researchers operate from, such as the four imaginaries discussed in this article, are a precondition to any movement beyond institutional path-dependency to a globally unsustainable development.

A method of inserting symbol matrix on the workpiece sketch and assembling a device

In the process of designing the technological devices for orientation and fastening for semifinished parts, there is a close connection between the technological device designer and the process technologist, the one representing on the operating plans, for each operation, the orientation and fixing of the piece in the device by means of the symbols information. The work comes in support of the process technologist because all this activity becomes fully automated. Each symbol in the information symbol matrix (SI), specific for the design of technological devices, also contains information (name, number of degrees of freedom taken over, type of surface, etc.), which can be extracted by the Excel-AutoCAD link, for processing them. This paper presents concretely, inserting symbols on the surfaces of the loose piece as an example and how to form a control device using Tool Palettes.

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in L 2 ⁢ ( ℝ n ) {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space L 2 ⁢ ( ℝ n ; ℂ 2 n ) {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})} . Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

PAPR and security in OFDM-PON via optimum block dividing with dynamic key and 2D-LASM.

An approach for peak-to-average-power ratio (PAPR) reduction and security improvement in an orthogonal frequency division multiplexing passive optical network (OFDM-PON) is proposed by using optimum block dividing with 2-dimensional logistic adjusted sine map (2D-LASM) and dynamic key assignment technique. One frame of OFDM signal can be regarded as a symbol matrix. All the divisors of length and width of matrix can be calculated out, and each divisor of length and width can be arbitrarily combined to divide the matrix into blocks. A 4D hyperchaotic system is applied to generate a cipher book for 2D-LASM. And we assign different dynamic key groups from the cipher book for 2D-LASM to encrypt different block dividing situations. Different encrypted divisor blocks can obtain different values of PAPR. The optimum dividing situation is obtained by calculating out the minimum value of PAPR (VPAPR). The values of optimum encryption signal (VPAPR-op), the original signal (VPAPR-or) and the optimization ratio (η) are gradually equal to 146, 269 and 1.82, respectively, with the number of quadrature amplitude modulation (QAM) symbols in each subcarrier increasing. Simulating 1000 sets of 120×200 QAM symbols, the distribution of η approximately meets the Rayleigh distribution and its central distribution is around 1.7. The processing time decreases with the number of QAM symbols increasing in each block. The performance of PAPR reduction is more than 3 dB between the secure optimum signal and the original signal. In addition, the hyper-threading technique with two algorithms is applied to improve the performance of the encryption method. They promote the processing time by 38.6% and 50%, respectively. Finally, a 22.06 Gb/s optimum encryption OFDM signal transmits through a back-to-back (BTB) system and a 25-km standard single mode fiber (SSMF). These experimental results verify that the proposed approach is a promising candidate for solving both of PAPR reduction and security improvement in access network systems.

On the incomplete second Appell hypergeometric matrix functions

ABSTRACT In this paper, we introduce the incomplete second Appell hypergeometric matrix functions in terms of the incomplete Pochhammer matrix symbols. We also derive certain properties such as matrix differential equation, integral formula, recurrence relation, recursion formula and differentiation formula of the incomplete second Appell hypergeometric matrix functions. This enriches the theory of special matrix functions. The results listed here are believed to be new.

Operator Scaling: Theory and Applications

In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over $${\mathbb {Q}}$$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (Ivanyos et al. in Comput Complex 26(3):717–763, 2017) (whether or not randomization is allowed). The algorithm efficiently solves the “word problem” for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004), who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits 2004). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp–Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular, we explain how some of these sources played an important role in the development of Gurvits’ algorithm and in our analysis of it here.