Symbolic Analyzer for Large Lumped and Distributed Networks

SummaryKnowing the Network Function (NF) pertaining to a certain input–output couple individuated inside a linear electrical network (LEN) is crucial, as it provides a synthetic and effective description of the input–output relationship and allows one to derive several key features of the subject system as well as to devise proper strategies to modify them.From a mathematical point of view, an NF may be expressed as the product of a real number, called the gain constant (GC), by a real rational function of the complex variables, whose numerator and denominator are called the zero polynomial (ZP) and the characteristic polynomial (CP), respectively, and may be obtained by calculating the determinants of proper matrices. However, when a design‐oriented symbolic hand analysis is needed, calculating the said polynomials in this manner may turn out to be tediously cumbersome; and above all, such a route may fail to illuminate the way that their coefficients depend on the circuit elements. The same may be said of numerical computer methods, which are suited to providing a global description of the NF in the frequency domain rather than analytical expressions of its coefficients. In this paper, a method is proposed that permits one to obtain the ZP without calculating any determinants. With respect to other methods that achieve the same goal, the present one does not require prior calculation of the CP or a nullator‐norator pair (NNP) to be compulsorily inserted in the subject circuit. Instead, it is based on applying the Andreani‐Mattisson extension (AME) of the Cochrun–Grabel algorithm (CGA) for the calculation of the CP to a circuit obtained from the assigned one by inserting in the latter a proper dependent source. The nature of such new circuit allows the worker to employ the classical tools of LEN theory so as to make the analysis as simple as possible and often performable by straightforward inspection, without the need for possibly involved equilibrium equations. Moreover, it is shown how the same technique can be applied to a properly modified version of the original network in order to calculate the CP as well. In this case (not compulsorily but if judged convenient for simplifying the calculations), a NNP may be employed in conjunction with the dependent source. The proposed method once again allows one to exploit the classic tools of LEN theory so as to considerably, if not drastically, simplify the analysis associated with the direct application of the AME to the original network. Finally, the calculation of the GC (that completes the knowledge of the whole NF under study) as well as circuit degeneracy are discussed and practical examples of application of the proposed techniques are given. Copyright © 2013 John Wiley & Sons, Ltd.

Graph-based symbolic technique is a viable tool for calculating the behavior or the characterization of an analog circuit. Traditional symbolic analysis tools typically are used to calculate the behavior or the characteristic of a circuit in terms of symbolic parameters (Gielen et al., 1994). The introduction of determinant decision diagrams based symbolic analysis technique allows exact symbolic analysis of much larger analog circuits than any other existing approaches (Shi & Tan, 2000; 2001). Furthermore, with hierarchical symbolic representations (Tan et al., 2005; Tan & Shi, 2000), exact symbolic analysis via DDD graphs essentially allows the analysis of arbitrarily large analog circuits. Some recent advancement in DDD ordering technique and variants of DDD allow even larger analog circuits to be analyzed (Shi, 2010a;b). Once the circuit’s small-signal characteristics are presented by DDDs, the evaluation of DDDs, whose CPU time is proportional to the sizes of DDDs, will give exact numerical values. However, with large networks, the DDD size can be huge and the resulting evaluation can be very time consuming.

“Parallel Training Considered Harmful?”: Comparing series-parallel and parallel feedforward network training

Symbolic analysis of time series is extended to systems with inputs, in order to obtain input/output symbolic models to be used for control policy design. For that, the notion of symbolic word is broadened to possibly include past input values. Then, a model is derived in the form of a controlled Markov chain, i.e., transition probabilities are conditioned on the control value. The quality of alternative models with different word length and alphabet size is assessed by means of an indicator based on Shannon entropy. A control problem is formulated, with the goal of confining the system output in a smaller domain with respect to that of the uncontrolled case. Solving this problem (by means of a suitable numerical method) yields the relevant control policy, as well as an estimate of the probability distribution of the output of the controlled system. Three examples of application (based on the analysis of time series synthetically generated by the logistic map, the Lorenz system, and an epidemiological model) are presented and used to discuss the features and limitations of the method.

The generalization of Bode’s sensitivity analysis technique for all types of the transfer functions and circuit elements is presented in the paper. The proposed formulae for first- and second-order symbolic sensitivity calculation provide the compact size of obtained expression and have the advantages of cancellation-free sum-of-product terms and matrix-free computation. This is achieved by means of the concept of high order summative cofactors and the generalized parameter extraction method. The proposed technique is implemented in symbolic circuit analysis program Cirsym. Illustrative example on symbolic sensitivity circuit analysis and comparison of the presented technique with the transimpedance method and the method based on the modified Coates flow-graph are given.

Industrial analog circuits are usually designed using numerical simulation tools. To obtain a deeper circuit understanding, symbolic analysis techniques can additionally be applied. Approximation methods which reduce the complexity of symbolic expressions are needed in order to handle industrial-sized problems. This paper describes aspects of the field of symbolic analog circuit analysis. Some state-of-the-art simplification algorithms for linear and nonlinear circuits are presented. The basic ideas behind the different techniques are described and two application examples for the linear and nonlinear case will be demonstrated.

Network partition methods for use in symbolic analysis by interpolation

In this paper, we briefly discuss the dynamic and functional approach to computer symbolic tensor analysis. The ccgrg package for Wolfram Language/Mathematica is used to illustrate this approach. Some examples of applications are attached.

This article surveys recent investigations of real sequences (d 1,..., d n ) which arise in the theory of measurement from considerations of uniqueness for numerical representations of qualitative relations on finite sets. The sequences we discuss arise from measurement problems which include measurement of subjective probability, extensive measurement, difference measurement, and additive conjoint measurement. The measurement problems lead to sequences with fascinating combinatorial and number-theoretic properties.The unifying mathematical framework under which we analyze uniqueness of measurement in these diverse areas involves the analysis of the sequences (d 1,...,d n)as the solutions of finite systems of linear equations. Different applications are translated into different restrictions on the types of linear equations that are admissible for each area.Two primary concerns of measurement theory are involved in the work being surveyed: (1) axioms for the qualitative relation that are necessary and sufficient, or at least sufficient, for unique represent ability; (2) the structure of sets of unique solutions. The latter concern leads to combinatorial and number-theoretic problems involving characterizations of unique solutions, counts of numbers of unique solutions, and extreme-value questions. Definitive results and presently open problems are described for the areas covered by the basic theory.KeywordsSubjective ProbabilityIndependent EquationRegular SequenceExtensive MeasurementFibonacci SequenceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A foundational area of mathematics, group theory has broad applications in many scientific domains. An integrated review of its use in cryptography and chemistry is presented in this work. Group theory simplifies structural predictions in chemistry by making it easier to analyze molecular symmetries, molecular orbitals, and spectroscopy. Group theory in cryptography provides the security underpinnings of contemporary encryption techniques, especially through discrete logarithm issues, homomorphisms, and permutation groups. We examine group-theoretical isomorphisms, homomorphism stability, and representations using both recent and classic literature, emphasizing their dual applicability in the information and biological sciences. Opportunities for interdisciplinary advances, such as improved chemical modeling and post-quantum cryptography protocols, are suggested by the synergy between these domains.

One of the persistent challenges in computer vision has been tracking objects under varying lighting conditions. In this paper we present a method for estimation of 3D motion of a rigid object from a monocular video sequence under arbitrary changes in the illumination conditions under which the video was captured. This is achieved by alternately estimating motion and illumination parameters using a generative model for integrating the effects of motion, illumination and structure within a unified mathematical framework. The motion is represented in terms of translation and rotation of the object centroid, and the illumination is represented using a spherical harmonics linear basis. The method does not assume any model for the variation of the illumination conditions - lighting can change slowly or drastically. For the multi-camera tracking scenario, we propose a new photometric constraint that is valid over the overlapping field of view between two cameras. This is similar in nature to the well-known epipolar constraint, except that it relates the photometric parameters, and can provide an additional constraint for illumination invariant multi-camera tracking. We demonstrate the effectiveness of our tracking algorithm on single and multi-camera video sequences under severe changes of lighting conditions.

This paper considers the scenario of the (1+1) evolutionary algorithm (EA) and randomized local search (RLS) with memory. Previously explored solutions are stored in memory until an improvement in fitness is obtained; then the stored information is discarded. This results in two new algorithms: (1+1) EA-m (with a raw list and hash table option) and RLS-m+ (and RLS-m if the function is a priori known to be unimodal). These two algorithms can be regarded as very simple forms of tabu search. Rigorous theoretical analysis of the expected time to find the globally optimal solutions for these algorithms is conducted for both unimodal and multimodal functions. A unified mathematical framework, involving the new concept of spatially invariant neighborhood, is proposed. Under this framework, both (1+1) EA with standard uniform mutation and RLS can be considered as particular instances and in the most general cases, all functions can be considered to be unimodal. Under this framework, it is found that for unimodal functions, the improvement by memory assistance is always positive but at most by one half. For multimodal functions, the improvement is significant; for functions with gaps and another hard function, the order of growth is reduced; for at least one example function, the order can change from exponential to polynomial. Empirical results, with a reasonable fitness evaluation time assumption, verify that (1+1) EA-m and RLS-m+ are superior to their conventional counterparts. Both new algorithms are promising for use in a memetic algorithm. In particular, RLS-m+ makes the previously impractical RLS practical, and surprisingly, does not require any extra memory in actual implementation.

Credit scoring models serve as pivotal instruments for lenders and financial institutions, facilitating the assessment of creditworthiness. Traditional models, while instrumental, grapple with challenges related to efficiency and subjectivity. The advent of machine learning heralds a transformative era, offering data-driven solutions that transcend these limitations. This research delves into a comprehensive analysis of various machine learning algorithms, emphasizing their mathematical underpinnings and their applicability in credit score classification. A comprehensive evaluation is conducted on a range of algorithms, including logistic regression, decision trees, support vector machines, and neural networks, using publicly available credit datasets. Within the research, a unified mathematical framework is introduced, which encompasses preprocessing techniques and critical algorithms such as Particle Swarm Optimization (PSO), the Light Gradient Boosting Model, and Extreme Gradient Boosting (XGB), among others. The focal point of the investigation is the LIME (Local Interpretable Model-agnostic Explanations) explainer. This study offers a comprehensive mathematical model using the LIME explainer, shedding light on its pivotal role in elucidating the intricacies of complex machine learning models. This study’s empirical findings offer compelling evidence of the efficacy of these methodologies in credit scoring, with notable accuracies of 88.84%, 78.30%, and 77.80% for the Australian, German, and South German datasets, respectively. In summation, this research not only amplifies the significance of machine learning in credit scoring but also accentuates the importance of mathematical modeling and the LIME explainer, providing a roadmap for practitioners to navigate the evolving landscape of credit assessment.

Single-cell sequencing technologies that simultaneously generate multimodal cellular profiles present opportunities for improved understanding of cell heterogeneity in tissues. How the multimodal information can be integrated to obtain a common cell type identification, however, poses a computational challenge. Multilayer graphs provide a natural representation of multi-omic single-cell sequencing datasets, and finding cell clusters may be understood as a multilayer graph partition problem. We introduce two spectral algorithms on multilayer graphs, spectral clustering on multilayer graphs and the weighted locally linear (WLL) method, to cluster cells in multi-omic single-cell sequencing datasets. We connect these algorithms through a unifying mathematical framework that represents each layer using a Hamiltonian operator and a mixture of its eigenstates to integrate the multiple graph layers, demonstrating in the process that the WLL method is a rigorous multilayer spectral graph theoretic reformulation of the popular Seurat weighted nearest neighbor (WNN) algorithm. Implementing our algorithms and applying them to a CITE-seq dataset of cord blood mononuclear cells yields results similar to the Seurat WNN analysis. Our work thus extends spectral methods to multimodal single-cell data analysis. The code used in this study can be found at https://github.com/jssong-lab/sc-spectrum. All public data used in the article are accurately cited and described in Materials and Methods and in Supplementary Information. Supplementary data are available at Bioinformatics online.

In this paper, we provide an analysis to the performance of optical time-wavelength code-division multiple-access (OTW-CDMA) network when the system is working above the nominal transmission rate limit imposed by the passive encoding-decoding operation. We address the problem of overlapping in such a system and how it can directly affect the bit error rate (BER). A unified mathematical framework is presented under the assumption of one coincidence sequences with non-repeating wavelengths. A closed form expression of the multiple access interference limited BER is provided as a function of different system parameters. Results show that the performance of OTW-CDMA system may be critically affected when working above the nominal limit; an event that may happen when the network operates at high transmission rate. In addition, the impact of the derived error probability on the performance of two newly proposed MAC protocols, the S-ALOHA and the R3T, is also investigated. It is shown that for low transmission rates, the S-ALOHA is better than the R3T; while the R3T is better at very high transmission rates. However, in general it is postulated that the R3T protocol suffers a higher delay mainly because of the presence of additional modes.