Relevant bibliographies by topics / Algebraic Circuits

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Algebraic Circuits'

Author: Grafiati
Published: 11 January 2025

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Journal articles on the topic "Algebraic Circuits"

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Limaye, Nutan, Srikanth Srinivasan, and Sébastien Tavenas. "Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits." Communications of the ACM 67, no. 2 (January 25, 2024): 101–8. http://dx.doi.org/10.1145/3611094.

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An Algebraic Circuit for a multivariate polynomial P is a computational model for constructing the polynomial P using only additions and multiplications. It is a syntactic model of computation, as opposed to the Boolean Circuit model, and hence lower bounds for this model are widely expected to be easier to prove than lower bounds for Boolean circuits. Despite this, we do not have superpolynomial lower bounds against general algebraic circuits of depth 3 (except over constant-sized finite fields) and depth 4 (over any field other than F 2 ), while constant-depth Boolean circuit lower bounds have been known since the early 1980s. In this paper, we prove the first superpolynomial lower bounds against algebraic circuits of all constant depths over all fields of characteristic 0. We also observe that our super-polynomial lower bound for constant-depth circuits implies the first deterministic sub-exponential time algorithm for solving the Polynomial Identity Testing (PIT) problem for all small-depth circuits using the known connection between algebraic hardness and randomness.
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Sopena, Alejandro, Max Hunter Gordon, Diego García-Martín, Germán Sierra, and Esperanza López. "Algebraic Bethe Circuits." Quantum 6 (September 8, 2022): 796. http://dx.doi.org/10.22331/q-2022-09-08-796.

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The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-12 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.
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Sarangapani, P., T. Thiessen, and W. Mathis. "Differential Algebraic Equations of MOS Circuits and Jump Behavior." Advances in Radio Science 10 (October 2, 2012): 327–32. http://dx.doi.org/10.5194/ars-10-327-2012.

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Abstract. Many nonlinear electronic circuits showing fast switching behavior exhibit jump effects which occurs when the state space of the electronic system contains a fold. This leads to difficulties during the simulation of these systems with standard circuit simulators. A method to overcome these problems is by regularization, where parasitic inductors and capacitors are added at the suitable locations. However, the transient solution will not be reliable if this regularization is not done in accordance with Tikhonov's Theorem. A geometric approach is taken to overcome these problems by explicitly computing the state space and jump points of the circuit. Until now, work has been done in analyzing example circuits exhibiting this behavior for BJT transistors. In this work we apply these methods to MOS circuits (Schmitt trigger, flip flop and multivibrator) and present the numerical results. To analyze the circuits we use the EKV drain current model as equivalent circuit model for the MOS transistors.
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DUMITRIU, LUCIA, and MIHAI IORDACHE. "NUMERICAL STEADY-STATE ANALYSIS OF NONLINEAR ANALOG CIRCUITS DRIVEN BY MULTITONE SIGNALS." International Journal of Bifurcation and Chaos 17, no. 10 (October 2007): 3595–601. http://dx.doi.org/10.1142/s0218127407019421.

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Widely-separated time scales appear in many electronic circuits, making traditional analysis difficult, even impossible, if the circuits are highly nonlinear. This paper presents a new version of the modified nodal method in two time variables for the analysis of the circuit with widely separated time scales. By applying this approach the differential algebraic equations (DAE) describing the nonlinear analog circuits driven by multitone signals are transformed into multitime partial differential equations (MPDEs). In order to solve MPDEs, associated resistive discrete equivalent circuits (companion circuits) for the dynamic circuit elements are used. The boundary conditions are detailed and simulation results are presented.
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Saima Yasin. "Analysis of Reachable and Positive Electrical Circuits Modelled by differential Algebraic System." Mathematical Sciences and Applications 3, no. 1 (June 30, 2024): 1–17. http://dx.doi.org/10.52700/msa.v3i1.23.

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This research paper undertakes an exploration of novel concepts within the domain of electrical circuits modeled by matrices. The study is centered around circuits encompassing elements such as resistances, coils, and voltage sources, with a specific focus on elucidating aspects of positivity and reachability. A distinctive approach is employed through the introduction of additional inductors, aimed at enhancing the circuit’s functionality. Employing mathematical analysis, this research endeavor advances an extensive comprehension of these circuits, shedding light on their behavior even when subject to specific constraints. As a consequence, this inquiry contributes to the revelation of noteworthy insights about the operational dynamics of circuits.
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Agrawal, Manindra, Sumanta Ghosh, and Nitin Saxena. "Bootstrapping variables in algebraic circuits." Proceedings of the National Academy of Sciences 116, no. 17 (April 11, 2019): 8107–18. http://dx.doi.org/10.1073/pnas.1901272116.

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We show that for the blackbox polynomial identity testing (PIT) problem it suffices to study circuits that depend only on the first extremely few variables. One needs only to consider size-s degree-s circuits that depend on the firstlog○c svariables (where c is a constant and composes a logarithm with itself c times). Thus, the hitting-set generator (hsg) manifests a bootstrapping behavior—a partial hsg against very few variables can be efficiently grown to a complete hsg. A Boolean analog, or a pseudorandom generator property of this type, is unheard of. Our idea is to use the partial hsg and its annihilator polynomial to efficiently bootstrap the hsg exponentially w.r.t. variables. This is repeated c times in an efficient way. Pushing the envelope further we show that (i) a quadratic-time blackbox PIT for 6,913-variate degree-s size-s polynomials will lead to a “near”-complete derandomization of PIT and (ii) a blackbox PIT for n-variate degree-s size-s circuits insnδtime, forδ<1/2, will lead to a near-complete derandomization of PIT (in contrast,sntime is trivial). Our second idea is to study depth-4 circuits that depend on constantly many variables. We show that a polynomial-time computable,O(s1.49)-degree hsg for trivariate depth-4 circuits bootstraps to a quasipolynomial time hsg for general polydegree circuits and implies a lower bound that is a bit stronger than that of Kabanets and Impagliazzo [Kabanets V, Impagliazzo R (2003)Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing STOC ’03].
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Ozawa, Kazuhiro, Kaoru Hirota, LászlóT Kóczy, and Ken Ōmori. "Algebraic fuzzy flip-flop circuits." Fuzzy Sets and Systems 39, no. 2 (January 1991): 215–26. http://dx.doi.org/10.1016/0165-0114(91)90214-b.

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Tykhovod, Sergii, and Ihor Orlovskyi. "Development and Research of Method in the Calculation of Transients in Electrical Circuits Based on Polynomials." Energies 15, no. 22 (November 15, 2022): 8550. http://dx.doi.org/10.3390/en15228550.

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Long electromagnetic transients occur in electrical systems because of switching and impulse actions As a result, the simulation time of such processes can be long, which is undesirable. Simulation time is significantly increased if the circuit in the study is complex, and also if this circuit is described by a rigid system of state equations. Modern requests of design engineers require an increase in the speed of calculations for realizing a real-time simulation. This work is devoted to the development of a unified spectral method for calculating electromagnetic transients in electrical circuits based on the representation of solution functions by series in algebraic and orthogonal polynomials. The purpose of the work is to offer electrical engineers a method that can significantly reduce the time for modeling transients in electrical circuits. Research methods. Approximation of functions by orthogonal polynomials, numerical methods for integrating differential equations, matrix methods, programming and theory of electrical circuits. Obtained results. Methods for calculating transients in electrical circuits based on the approximation of solution functions by series in algebraic polynomials as well as in the Chebyshev, Hermite and Legendre polynomials, have been developed and investigated. The proposed method made it possible to convert integro-differential equations of state into linear algebraic equations for images of time-dependent functions. The developed circuit model simplifies the calculation method. The images of true current functions are interpreted as direct currents in the proposed equivalent circuit. A computer program for simulating the transient process in an electrical circuit was developed on the basis of the described methods. The performed comparison of methods made it possible to choose the best method and a way to use it. The advantages of the presented method over other known methods are to reduce the simulation time of electromagnetic transients (for the considered examples by more than 6 times) while ensuring the required accuracy. The calculation of the process in the circuit over a long time interval showed a decrease and stabilization of errors, which indicates the prospects for using research methods for calculating complex electrical circuits.
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Siegal-Gaskins, Dan, Elisa Franco, Tiffany Zhou, and Richard M. Murray. "An analytical approach to bistable biological circuit discrimination using real algebraic geometry." Journal of The Royal Society Interface 12, no. 108 (July 2015): 20150288. http://dx.doi.org/10.1098/rsif.2015.0288.

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Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem—a tool from nineteenth-century real algebraic geometry—to comparing ‘functionally equivalent’ bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh–Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis.
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Erata, Ferhat, Chuanqi Xu, Ruzica Piskac, and Jakub Szefer. "Quantum Circuit Reconstruction from Power Side-Channel Attacks on Quantum Computer Controllers." IACR Transactions on Cryptographic Hardware and Embedded Systems 2024, no. 2 (March 12, 2024): 735–68. http://dx.doi.org/10.46586/tches.v2024.i2.735-768.

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The interest in quantum computing has grown rapidly in recent years, and with it grows the importance of securing quantum circuits. A novel type of threat to quantum circuits that dedicated attackers could launch are power trace attacks. To address this threat, this paper presents first formalization and demonstration of using power traces to unlock and steal quantum circuit secrets. With access to power traces, attackers can recover information about the control pulses sent to quantum computers. From the control pulses, the gate level description of the circuits, and eventually the secret algorithms can be reverse engineered. This work demonstrates how and what information could be recovered. This work uses algebraic reconstruction from power traces to realize two new types of single trace attacks: per-channel and total power attacks. The former attack relies on per-channel measurements to perform a brute-force attack to reconstruct the quantum circuits. The latter attack performs a single-trace attack using Mixed-Integer Linear Programming optimization. Through the use of algebraic reconstruction, this work demonstrates that quantum circuit secrets can be stolen with high accuracy. Evaluation on 32 real benchmark quantum circuits shows that our technique is highly effective at reconstructing quantum circuits. The findings not only show the veracity of the potential attacks, but also the need to develop new means to protect quantum circuits from power trace attacks. Throughout this work real control pulse information from real quantum computers is used to demonstrate potential attacks based on simulation of collection of power traces.
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Dissertations / Theses on the topic "Algebraic Circuits"

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König, Daniel [Verfasser]. "Parallel evaluation of algebraic circuits / Daniel König." Siegen : Universitätsbibliothek der Universität Siegen, 2017. http://d-nb.info/1138836931/34.

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Pade, Jonas. "Analysis and waveform relaxation for a differential-algebraic electrical circuit model." Doctoral thesis, Humboldt-Universität zu Berlin, 2021. http://dx.doi.org/10.18452/23044.

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Die Hauptthemen dieser Arbeit sind einerseits eine tiefgehende Analyse von nichtlinearen differential-algebraischen Gleichungen (DAEs) vom Index 2, die aus der modifizierten Knotenanalyse (MNA) von elektrischen Schaltkreisen hervorgehen, und andererseits die Entwicklung von Konvergenzkriterien für Waveform Relaxationsmethoden zum Lösen gekoppelter Probleme. Ein Schwerpunkt in beiden genannten Themen ist die Beziehung zwischen der Topologie eines Schaltkreises und mathematischen Eigenschaften der zugehörigen DAE. Der Analyse-Teil umfasst eine detaillierte Beschreibung einer Normalform für Schaltkreis DAEs vom Index 2 und Abschätzungen, die für die Sensitivität des Schaltkreises bezüglich seiner Input-Quellen folgen. Es wird gezeigt, wie diese Abschätzungen wesentlich von der topologischen Position der Input-Quellen im Schaltkreis abhängen. Die zunehmend komplexen Schaltkreise in technologischen Geräten erfordern oftmals eine Modellierung als gekoppeltes System. Waveform relaxation (WR) empfiehlt sich zur Lösung solch gekoppelter Probleme, da sie auf die Subprobleme angepasste Lösungsmethoden und Schrittweiten ermöglicht. Es ist bekannt, dass WR zwar bei Anwendung auf gewöhnliche Differentialgleichungen konvergiert, falls diese eine Lipschitz-Bedingung erfüllen, selbiges jedoch bei DAEs nicht ohne Hinzunahme eines Kontraktivitätskriteriums sichergestellt werden kann. Wir beschreiben allgemeine Konvergenzkriterien für WR auf DAEs vom Index 2. Für den Fall von Schaltkreisen, die entweder mit anderen Schaltkreisen oder mit elektromagnetischen Feldern verkoppelt sind, leiten wir außerdem hinreichende topologische Konvergenzkriterien her, die anhand von Beispielen veranschaulicht werden. Weiterhin werden die Konvergenzraten des Jacobi WR Verfahrens und des Gauss-Seidel WR Verfahrens verglichen. Simulationen von einfachen Beispielsystemen zeigen drastische Unterschiede des WR-Konvergenzverhaltens, abhängig davon, ob die Konvergenzbedingungen erfüllt sind oder nicht.
The main topics of this thesis are firstly a thorough analysis of nonlinear differential-algebraic equations (DAEs) of index 2 which arise from the modified nodal analysis (MNA) for electrical circuits and secondly the derivation of convergence criteria for waveform relaxation (WR) methods on coupled problems. In both topics, a particular focus is put on the relations between a circuit's topology and the mathematical properties of the corresponding DAE. The analysis encompasses a detailed description of a normal form for circuit DAEs of index 2 and consequences for the sensitivity of the circuit with respect to its input source terms. More precisely, we provide bounds which describe how strongly changes in the input sources of the circuit affect its behaviour. Crucial constants in these bounds are determined in terms of the topological position of the input sources in the circuit. The increasingly complex electrical circuits in technological devices often call for coupled systems modelling. Allowing for each subsystem to be solved by dedicated numerical solvers and time scales, WR is an adequate method in this setting. It is well-known that while WR converges on ordinary differential equations if a Lipschitz condition is satisfied, an additional convergence criterion is required to guarantee convergence on DAEs. We present general convergence criteria for WR on higher index DAEs. Furthermore, based on our results of the analysis part, we derive topological convergence criteria for coupled circuit/circuit problems and field/circuit problems. Examples illustrate how to practically check if the criteria are satisfied. If a sufficient convergence criterion holds, we specify at which rate of convergence the Jacobi and Gauss-Seidel WR methods converge. Simulations of simple benchmark systems illustrate the drastically different convergence behaviour of WR depending on whether or not the circuit topological convergence conditions are satisfied.
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Tavenas, Sébastien. "Bornes inférieures et supérieures dans les circuits arithmétiques." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2014. http://tel.archives-ouvertes.fr/tel-01066752.

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La complexité arithmétique est l'étude des ressources nécessaires pour calcu- ler des polynômes en n'utilisant que des opérations arithmétiques. À la fin des années 70, Valiant a défini (de manière semblable à la complexité booléenne) des classes de polynômes. Les polynômes, ayant des circuits de taille polyno- miale, considérés faciles forment la classe VP. Les sommes exponentielles de ces derniers correpondent alors à la classe VNP. L'hypothèse de Valiant est la conjecture que VP ̸= VNP.Bien que cette conjecture soit encore grandement ouverture, cette dernière semble toutefois plus accessible que son homologue booléen. La structure algé- brique sous-jacente limite les possibilités de calculs. En particulier, un résultat important du domaine assure que les polynômes faciles peuvent aussi être cal- culés efficacement en paralèlle. De plus, quitte à autoriser une augmentation raisonnable de la taille, il est possible de les calculer avec une profondeur de calcul bornée par une constante. Comme ce dernier modèle est très restreint, de nombreuses bornes inférieures sont connues. Nous nous intéresserons en premier temps à ces résultats sur les circuits de profondeur constante.Bürgisser a montré qu'une conjecture (la τ-conjecture) qui borne supérieu- rement le nombre de racines de certains polynômes univariés, impliquait des bornes inférieures en complexité arithmétique. Mais, que se passe-t-il alors, si on essaye de réduire, comme précédemment, la profondeur du polynôme consi- déré? Borner le nombre de racines réelles de certaines familles de polynômes permetterait de séparer VP et VNP. Nous étudierons finalement ces bornes su- périeures sur le nombre de racines réelles.
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Jain, Divyanshu. "MVHAM an extension of the homotopy analysis method for improving convergence of the multivariate solution of nonlinear algebraic equations as typically encountered in analog circuits /." Cincinnati, Ohio : University of Cincinnati, 2007. http://www.ohiolink.edu/etd/view.cgi?acc%5Fnum=ucin1194974755.

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Thesis (M.S.)--University of Cincinnati, 2007.
Advisor: Harold W. Carter. Title from electronic thesis title page (viewed Feb. 18, 2008). Includes abstract. Keywords: Homotopy Analysis Method; Solution of Nonlinear Algebraic Equations; Convergence. Includes bibliographical references.
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Lagarde, Guillaume. "Contributions to arithmetic complexity and compression." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC192/document.

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Cette thèse explore deux territoires distincts de l’informatique fondamentale : la complexité et la compression. Plus précisément, dans une première partie, nous étudions la puissance des circuits arithmétiques non commutatifs, qui calculent des polynômes non commutatifs en plusieurs indéterminées. Pour cela, nous introduisons plusieurs modèles de calcul, restreints dans leur manière de calculer les monômes. Ces modèles en généralisent d’autres, plus anciens et largement étudiés, comme les programmes à branchements. Les résultats sont de trois sortes. Premièrement, nous donnons des bornes inférieures sur le nombre d’opérations arithmétiques nécessaires au calcul de certains polynômes tels que le déterminant ou encore le permanent. Deuxièmement, nous concevons des algorithmes déterministes fonctionnant en temps polynomial pour résoudre le problème du test d’identité polynomiale. Enfin, nous construisons un pont entre la théorie des automates et les circuits arithmétiques non commutatifs, ce qui nous permet de dériver de nouvelles bornes inférieures en utilisant une mesure reposant sur le rang de la matrice dite de Hankel, provenant de la théorie des automates. Une deuxième partie concerne l’analyse de l’algorithme de compression sans perte Lempel-Ziv. Pourtant très utilisé, sa stabilité est encore mal établie. Vers la fin des années 90s, Jack Lutz popularise la question suivante, connue sous le nom de « one-bit catastrophe » : « étant donné un mot compressible, est-il possible de le rendre incompressible en ne changeant qu’un seul bit ? ». Nous montrons qu’une telle catastrophe est en effet possible. Plus précisément, en donnant des bornes optimales sur la variation de la taille de la compression, nous montrons qu’un mot « très compressible » restera toujours compressible après modification d’un bit, mais que certains mots « peu compressibles » deviennent en effet incompressibles
This thesis explores two territories of computer science: complexity and compression. More precisely, in a first part, we investigate the power of non-commutative arithmetic circuits, which compute multivariate non-commutative polynomials. For that, we introduce various models of computation that are restricted in the way they are allowed to compute monomials. These models generalize previous ones that have been widely studied, such as algebraic branching programs. The results are of three different types. First, we give strong lower bounds on the number of arithmetic operations needed to compute some polynomials such as the determinant or the permanent. Second, we design some deterministic polynomial-time algorithm to solve the white-box polynomial identity problem. Third, we exhibit a link between automata theory and non-commutative arithmetic circuits that allows us to derive some old and new tight lower bounds for some classes of non-commutative circuits, using a measure based on the rank of a so-called Hankel matrix. A second part is concerned with the analysis of the data compression algorithm called Lempel-Ziv. Although this algorithm is widely used in practice, we know little about its stability. Our main result is to show that an infinite word compressible by LZ’78 can become incompressible by adding a single bit in front of it, thus closing a question proposed by Jack Lutz in the late 90s under the name “one-bit catastrophe”. We also give tight bounds on the maximal possible variation between the compression ratio of a finite word and its perturbation—when one bit is added in front of it
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Remscrim, Zachary (Zachary N. ). "Algebraic methods in pseudorandomness and circuit complexity." Thesis, Massachusetts Institute of Technology, 2016. http://hdl.handle.net/1721.1/106089.

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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2016.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 93-96).
In this thesis, we apply tools from algebra and algebraic geometry to prove new results concerning extractors for algebraic sets, AC⁰-pseudorandomness, the recursive Fourier sampling problem, and VC dimension. We present a new construction of an extractor which works for algebraic sets defined by polynomials over F₂ of substantially higher degree than the previous state-of-the-art construction. We exhibit a collection of natural functions that behave pseudorandomly with regards to AC⁰ tests. We also exactly determine the F₂-polynomial degree of the recursive Fourier sampling problem and use this to provide new partial results towards a circuit lower bound for this problem. Finally, we answer a question posed in [MR15] concerning VC dimension, interpolation degree and the Hilbert function.
by Zachary Remscrim.
Ph. D.
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Ramponi, Marco. "Clifford index and gonality of curves on special K3 surfaces." Thesis, Poitiers, 2017. http://www.theses.fr/2017POIT2317/document.

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Nous allons étudier les propriétés des courbes algébriques sur des surfaces K3 spéciales, du point de vue de la théorie de Brill-Noether.La démonstration de Lazarsfeld du théorème de Gieseker-Petri a mis en lumière l'importance de la théorie de Brill-Noether des courbes admettant un plongement dans une surface K3. Nous allons donner une démonstration détaillée de ce résultat classique, inspirée par les idées de Pareschi. En suite, nous allons décrire le théorème de Green et Lazarsfeld, fondamental pour tout notre travail, qui établit le comportement de l'indice de Clifford des courbes sur les surfaces K3.Watanabe a montré que l'indice de Clifford de courbes sur certaines surfaces K3, admettant un recouvrement double des surfaces de del Pezzo, est calculé en utilisant les involutions non-symplectiques. Nous étudions une situation similaire pour des surfaces K3 avec un réseau de Picard isomorphe à U(m), avec m>0 un entier quelconque. Nous montrons que la gonalité et l'indice de Clifford de toute courbe lisse sur ces surfaces, avec une seule exception déterminée explicitement, sont obtenus par restriction des fibrations elliptiques de la surface. Ce travail est basé sur l'article suivant :M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355–362, 2016.Knutsen et Lopez ont étudié en détail la théorie de Brill-Noether des courbes sur les surfaces d'Enriques. En appliquant leurs résultats, nous allons pouvoir calculer la gonalité et l'indice de Clifford de toute courbe lisse sur les surfaces K3 qui sont des recouvrements universels d'une surface d'Enriques. Ce travail est basé sur l'article suivant :M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315–322, 2017
We study the properties of algebraic curves lying on special K3 surfaces, from the viewpoint of Brill-Noether theory.Lazarsfeld's proof of the Gieseker-Petri theorem has revealed the importance of the Brill-Noether theory of curves which admit an embedding in a K3 surface. We give a proof of this classical result, inspired by the ideas of Pareschi. We then describe the theorem of Green and Lazarsfeld, a key result for our work, which establishes the behaviour of the Clifford index of curves on K3 surfaces.Watanabe showed that the Clifford index of curves lying on certain special K3 surfaces, realizable as a double covering of a smooth del Pezzo surface, can be determined by a direct use of the non-simplectic involution carried by these surfaces. We study a similar situation for some K3 surfaces having a Picard lattice isomorphic to U(m), with m>0 any integer. We show that the gonality and the Clifford index of all smooth curves on these surfaces, with a single, explicitly determined exception, are obtained by restriction of the elliptic fibrations of the surface. This work is based on the following article:M. Ramponi, Gonality and Clifford index of curves on elliptic K3 surfaces with Picard number two, Archiv der Mathematik, 106(4), p. 355-362, 2016.Knutsen and Lopez have studied in detail the Brill-Noether theory of curves lying on Enriques surfaces. Applying their results, we are able to determine and compute the gonality and Clifford index of any smooth curve lying on the general K3 surface which is the universal covering of an Enriques surface. This work is based on the following article:M. Ramponi, Special divisors on curves on K3 surfaces carrying an Enriques involution, Manuscripta Mathematica, 153(1), p. 315-322, 2017
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Reich, Sebastian. "Differential-algebraic equations and applications in circuit theory." Universität Potsdam, 1992. http://opus.kobv.de/ubp/volltexte/2010/4664/.

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Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.
Die mathematische Modellierung technisch physikalischer Systeme wie elektrische Netzwerke, führt häufig auf ein System von Differentialgleichungen und nichtlinearen impliziten Gleichungen sogenannten Algebrodifferentialgleichungen (ADGL). Es zeigt sich, daß die numerischen und analytischen Eigenschaften von ADGL durch den Index des Problems charakterisiert werden können. Insbesondere können die bekannten Integrationsformeln von Gear im allgemeinen nur auf ADGL mit dem Index eins angewendet werden. In diesem Beitrag wird eine geometrische Interpretation von ADGL mit einem höheren Index gegeben sowie auf Probleme im Zusammenhang mit derartigen ADGL an Hand verschiedener Beispiele hingewiesen.
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Bächle, Simone. "Numerical solution of differential-algebraic systems arising in circuit simulation." [S.l.] : [s.n.], 2007. http://opus.kobv.de/tuberlin/volltexte/2007/1524.

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Grenet, Bruno. "Représentations des polynômes, algorithmes et bornes inférieures." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2012. http://tel.archives-ouvertes.fr/tel-00770148.

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Abstract:
La complexité algorithmique est l'étude des ressources nécessaires -- le temps, la mémoire, ... -- pour résoudre un problème de manière algorithmique. Dans ce cadre, la théorie de la complexité algébrique est l'étude de la complexité algorithmique de problèmes de nature algébrique, concernant des polynômes.Dans cette thèse, nous étudions différents aspects de la complexité algébrique. D'une part, nous nous intéressons à l'expressivité des déterminants de matrices comme représentations des polynômes dans le modèle de complexité de Valiant. Nous montrons que les matrices symétriques ont la même expressivité que les matrices quelconques dès que la caractéristique du corps est différente de deux, mais que ce n'est plus le cas en caractéristique deux. Nous construisons également la représentation la plus compacte connue du permanent par un déterminant. D'autre part, nous étudions la complexité algorithmique de problèmes algébriques. Nous montrons que la détection de racines dans un système de n polynômes homogènes à n variables est NP-difficile. En lien avec la question " VP = VNP ? ", version algébrique de " P = NP ? ", nous obtenons une borne inférieure pour le calcul du permanent d'une matrice par un circuit arithmétique, et nous exhibons des liens unissant ce problème et celui du test d'identité polynomiale. Enfin nous fournissons des algorithmes efficaces pour la factorisation des polynômes lacunaires à deux variables.
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Books on the topic "Algebraic Circuits"

1

Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, and Antonio García Ríos. Algebraic Circuits. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-54649-5.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. Arithmetic and Algebraic Circuits. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9.

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Krishnaswamy, Smita. Design, Analysis and Test of Logic Circuits Under Uncertainty. Dordrecht: Springer Netherlands, 2013.

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Barg, Alexander, and O. R. Musin. Discrete geometry and algebraic combinatorics. Providence, Rhode Island: American Mathematical Society, 2014.

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Tsfasman, M. A. Algebraic geometry codes: Basic notions. Providence, R.I: American Mathematical Society, 2007.

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International Conference Arithmetic, Geometry, Cryptography and Coding Theory (14th 2013 Marseille, France). Algorithmic arithmetic, geometry, and coding theory: 14th International Conference, Arithmetic, Geometry, Cryptography, and Coding Theory, June 3-7 2013, CIRM, Marseille, France. Edited by Ballet Stéphane 1971 editor, Perret, M. (Marc), 1963- editor, and Zaytsev, Alexey (Alexey I.), 1976- editor. Providence, Rhode Island: American Mathematical Society, 2015.

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Thornton, Mitchell Aaron. Spectral techniques in VLSI CAD. Boston: Kluwer Academic Publishers, 2001.

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Alta.) WIN (Conference) (2nd 2011 Banff. Women in Numbers 2: Research directions in number theory : BIRS Workshop, WIN2 - Women in Numbers 2, November 6-11, 2011, Banff International Research Station, Banff, Alberta, Canada. Edited by David Chantal 1964-, Lalín Matilde 1977-, and Manes Michelle 1970-. Providence, Rhode Island: American Mathematical Society, 2013.

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Viamontes, George F. Quantum Circuit Simulation. Dordrecht: Springer Science+Business Media B.V., 2009.

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Stanković, Radomir S. From Boolean logic to switching circuits and automata: Towards modern information technology. Berlin: Springer Verlag, 2011.

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Book chapters on the topic "Algebraic Circuits"

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, and Antonio García Ríos. "Basic Algebraic Circuits." In Intelligent Systems Reference Library, 159–215. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-54649-5_4.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Basic Algebraic Circuits." In Arithmetic and Algebraic Circuits, 379–457. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_9.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Basic Arithmetic Circuits." In Arithmetic and Algebraic Circuits, 77–131. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_2.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Galois Fields GF(pn)." In Arithmetic and Algebraic Circuits, 515–50. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_11.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Multiplication." In Arithmetic and Algebraic Circuits, 257–85. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_6.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Special Functions." In Arithmetic and Algebraic Circuits, 317–77. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_8.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Number Systems." In Arithmetic and Algebraic Circuits, 1–75. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_1.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Residue Number Systems." In Arithmetic and Algebraic Circuits, 133–72. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_3.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Addition and Subtraction." In Arithmetic and Algebraic Circuits, 221–56. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_5.

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Lloris Ruiz, Antonio, Encarnación Castillo Morales, Luis Parrilla Roure, Antonio García Ríos, and María José Lloris Meseguer. "Two Galois Fields Cryptographic Applications." In Arithmetic and Algebraic Circuits, 551–65. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-67266-9_12.

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Conference papers on the topic "Algebraic Circuits"

1

Andrews, Robert, and Avi Wigderson. "Constant-Depth Arithmetic Circuits for Linear Algebra Problems." In 2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS), 2367–86. IEEE, 2024. http://dx.doi.org/10.1109/focs61266.2024.00138.

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Agrawal, Manindra, Sumanta Ghosh, and Nitin Saxena. "Bootstrapping variables in algebraic circuits." In STOC '18: Symposium on Theory of Computing. New York, NY, USA: ACM, 2018. http://dx.doi.org/10.1145/3188745.3188762.

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Man, Ka Lok, Abhinav Asthana, K. Kapoor Hemangee, Tomas Krilavicius, and Jian Chang. "Process algebraic specification of DI circuits." In 2010 International SoC Design Conference (ISOCC 2010). IEEE, 2010. http://dx.doi.org/10.1109/socdc.2010.5682885.

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Ben-Asher, Yosi, and Gady Haber. "Efficient parallel solutions of linear algebraic circuits." In the eleventh annual ACM symposium. New York, New York, USA: ACM Press, 1999. http://dx.doi.org/10.1145/305619.305644.

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Rao, Vikas, Haden Ondricek, Priyank Kalla, and Florian Enescu. "Algebraic Techniques for Rectification of Finite Field Circuits." In 2021 IFIP/IEEE 29th International Conference on Very Large Scale Integration (VLSI-SoC). IEEE, 2021. http://dx.doi.org/10.1109/vlsi-soc53125.2021.9606976.

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Limaye, Nutan, Srikanth Srinivasan, and Sebastien Tavenas. "Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits." In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2022. http://dx.doi.org/10.1109/focs52979.2021.00083.

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Dutta, Pranjal, Prateek Dwivedi, and Nitin Saxena. "Demystifying the border of depth-3 algebraic circuits." In 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2022. http://dx.doi.org/10.1109/focs52979.2021.00018.

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Calvino, Alessandro Tempia, and Giovanni De Micheli. "Algebraic and Boolean Methods for SFQ Superconducting Circuits." In 2024 29th Asia and South Pacific Design Automation Conference (ASP-DAC). IEEE, 2024. http://dx.doi.org/10.1109/asp-dac58780.2024.10473899.

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Grigoraş, Victor, and Carmen Grigoraş. "Nonlinear System with Adaptive Algebraic Function." In 2023 International Symposium on Signals, Circuits and Systems (ISSCS). IEEE, 2023. http://dx.doi.org/10.1109/isscs58449.2023.10190933.

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Hiroaki Yoshida, Makoto Ikeda, and Kunihiro Asada. "An algebraic approach for transistor circuit synthesis." In 2005 12th IEEE International Conference on Electronics, Circuits and Systems - (ICECS 2005). IEEE, 2005. http://dx.doi.org/10.1109/icecs.2005.4633590.

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Reports on the topic "Algebraic Circuits"

1

Barnett, Janet Heine. Applications of Boolean Algebra: Claude Shannon and Circuit Design. Washington, DC: The MAA Mathematical Sciences Digital Library, July 2013. http://dx.doi.org/10.4169/loci004000.

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