Relevant bibliographies by topics / Polynomial product

Academic literature on the topic 'Polynomial product'

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Published: 9 March 2023
Last updated: 11 March 2023

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Journal articles on the topic "Polynomial product"

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Szilágyi, Zsolt. "On Chern classes of the tensor product of vector bundles." Acta Universitatis Sapientiae, Mathematica 14, no. 2 (December 1, 2022): 330–40. http://dx.doi.org/10.2478/ausm-2022-0022.

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Abstract We present two formulas for Chern classes (polynomial) of the tensor product of two vector bundles. In the first formula the Chern polynomial of the product is expressed as determinant of a polynomial in a matrix variable involving the Chern classes of the first bundle with Chern classes of the second bundle as coefficients. In the second formula the total Chern class of the tensor product is expressed as resultant of two explicit polynomials. Finally, formulas for the total Chern class of the second symmetric and the second alternating products are deduced.
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Sastre, Jorge, and Javier Ibáñez. "Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method." Mathematics 9, no. 14 (July 7, 2021): 1600. http://dx.doi.org/10.3390/math9141600.

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Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.
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GAN, C. S. "The complete product of annihilatingly unique digraphs." International Journal of Mathematics and Mathematical Sciences 2005, no. 9 (2005): 1327–31. http://dx.doi.org/10.1155/ijmms.2005.1327.

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LetGbe a digraph withnvertices and letA(G)be its adjacency matrix. A monic polynomialf(x)of degree at mostnis called an annihilating polynomial ofGiff(A(G))=0.Gis said to be annihilatingly unique if it possesses a unique annihilating polynomial. Difans and diwheels are two classes of annihilatingly unique digraphs. In this paper, it is shown that the complete product of difan and diwheel is annihilatingly unique.
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Koelink, H. T. "Addition Formula For Big q-Legendre Polynomials From The Quantum Su(2) Group." Canadian Journal of Mathematics 47, no. 2 (April 1, 1995): 436–48. http://dx.doi.org/10.4153/cjm-1995-024-8.

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AbstractFrom Koornwinder's interpretation of big q-Legendre polynomials as spherical elements on the quantum SU(2) group an addition formula is derived for the big g-Legendre polynomial. The formula involves Al-Salam-Carlitz polynomials, little q-Jacobi polynomials and dual q-Krawtchouk polynomials. For the little q-ultraspherical polynomials a product formula in terms of a big q-Legendre polynomial follows by q-integration. The addition and product formula for the Legendre polynomials are obtained when q tends to 1.
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Knor, Martin, and Niko Tratnik. "A New Alternative to Szeged, Mostar, and PI Polynomials—The SMP Polynomials." Mathematics 11, no. 4 (February 13, 2023): 956. http://dx.doi.org/10.3390/math11040956.

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Szeged-like topological indices are well-studied distance-based molecular descriptors, which include, for example, the (edge-)Szeged index, the (edge-)Mostar index, and the (vertex-)PI index. For these indices, the corresponding polynomials were also defined, i.e., the (edge-)Szeged polynomial, the Mostar polynomial, the PI polynomial, etc. It is well known that, by evaluating the first derivative of such a polynomial at x=1, we obtain the related topological index. The aim of this paper is to introduce and investigate a new graph polynomial of two variables, which is called the SMP polynomial, such that all three vertex versions of the above-mentioned indices can be easily calculated using this polynomial. Moreover, we also define the edge-SMP polynomial, which is the edge version of the SMP polynomial. Various properties of the new polynomials are studied on some basic families of graphs, extremal problems are considered, and several open problems are stated. Then, we focus on the Cartesian product, and we show how the (edge-)SMP polynomial of the Cartesian product of n graphs can be calculated using the (weighted) SMP polynomials of its factors.
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DIAO, Y., G. HETYEI, and K. HINSON. "TUTTE POLYNOMIALS OF TENSOR PRODUCTS OF SIGNED GRAPHS AND THEIR APPLICATIONS IN KNOT THEORY." Journal of Knot Theory and Its Ramifications 18, no. 05 (May 2009): 561–89. http://dx.doi.org/10.1142/s0218216509007075.

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It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman's Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simple. We show that this Tutte polynomial of a signed tensor product of signed graphs may be expressed in terms of the Tutte polynomials of the original signed graphs by using a simple substitution rule. Our result enables us to compute the Jones polynomials of some large non-alternating knots. The combinatorics used to prove our main result is similar to Tutte's original way of counting "activities" and specializes to a new, perhaps simpler proof of the known formulas for the ordinary Tutte polynomial of the tensor product of unsigned graphs or matroids.
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González, Manuel, and Joaquí M. Gutiérrez. "Polynomial Grothendieck properties." Glasgow Mathematical Journal 37, no. 2 (May 1995): 211–19. http://dx.doi.org/10.1017/s0017089500031116.

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AbstractA Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.
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Jiang, Xue, and Kai Cui. "The Representation of D-Invariant Polynomial Subspaces Based on Symmetric Cartesian Tensors." Axioms 10, no. 3 (August 19, 2021): 193. http://dx.doi.org/10.3390/axioms10030193.

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Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).
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Chen, Lin-An, Tzong Shi Lee, and Wenyaw Chan. "Tensor product polynomial splines." Communications in Statistics - Theory and Methods 26, no. 9 (January 1997): 2093–111. http://dx.doi.org/10.1080/03610929708832036.

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Hammerlindl, Andy. "Polynomial global product structure." Proceedings of the American Mathematical Society 142, no. 12 (August 15, 2014): 4297–303. http://dx.doi.org/10.1090/s0002-9939-2014-12255-6.

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Dissertations / Theses on the topic "Polynomial product"

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Wise, Steven M. "POLSYS_PLP: A Partitioned Linear Product Homotopy Code for Solving Polynomial Systems of Equations." Thesis, Virginia Tech, 1998. http://hdl.handle.net/10919/36933.

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Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. This thesis describes the theory behind and performance of the new code POLSYS_PLP, which consists of Fortran 90 modules for finding all isolated solutions of a complex coefficient polynomial system of equations by a probability-one homotopy method. The package is intended to be used in conjunction with HOMPACK90, and makes extensive use of Fortran 90 derived data types to support a partitioned linear product (PLP) polynomial system structure. PLP structure is a generalization of m-homogeneous structure, whereby each component of the system can have a different m-homogeneous structure. POLSYS_PLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different PLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.
Master of Science
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MATSUMOTO, KOHJI. "ON THE MEAN SQUARE OF THE PRODUCT OF ζ(s) AND A DIRICHLET POLYNOMIAL." Rikkyo Daigaku, 2004. http://hdl.handle.net/2237/20071.

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Wang, Ting. "Algorithms for parallel and sequential matrix-chain product problem." Ohio : Ohio University, 1997. http://www.ohiolink.edu/etd/view.cgi?ohiou1184355429.

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Araaya, Tsehaye. "The Symmetric Meixner-Pollaczek polynomials." Doctoral thesis, Uppsala University, Department of Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3501.

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The Symmetric Meixner-Pollaczek polynomials are considered. We denote these polynomials in this thesis by pn(λ)(x) instead of the standard notation pn(λ) (x/2, π/2), where λ > 0. The limiting case of these sequences of polynomials pn(0) (x) =limλ→0 pn(λ)(x), is obtained, and is shown to be an orthogonal sequence in the strip, S = {z ∈ ℂ : −1≤ℭ (z)≤1}.

From the point of view of Umbral Calculus, this sequence has a special property that makes it unique in the Symmetric Meixner-Pollaczek class of polynomials: it is of convolution type. A convolution type sequence of polynomials has a unique associated operator called a delta operator. Such an operator is found for pn(0) (x), and its integral representation is developed. A convolution type sequence of polynomials may have associated Sheffer sequences of polynomials. The set of associated Sheffer sequences of the sequence pn(0)(x) is obtained, and is found

to be ℙ = {{pn(λ) (x)} =0 : λ ∈ R}. The major properties of these sequences of polynomials are studied.

The polynomials {pn(λ) (x)}∞n=0, λ < 0, are not orthogonal polynomials on the real line with respect to any positive real measure for failing to satisfy Favard’s three term recurrence relation condition. For every λ ≤ 0, an associated nonstandard inner product is defined with respect to which pn(λ)(x) is orthogonal.

Finally, the connection and linearization problems for the Symmetric Meixner-Pollaczek polynomials are solved. In solving the connection problem the convolution property of the polynomials is exploited, which in turn helps to solve the general linearization problem.

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DE, PICCOLI ALESSANDRO. "OPTIMIZED REPRESENTATIONS IN CRYPTOGRAPHIC PRIMITIVES." Doctoral thesis, Università degli Studi di Milano, 2022. http://hdl.handle.net/2434/932549.

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Il lavoro di tesi si focalizza sull'ottimizzazione di primitive crittografiche sia dal punto di vista teorico che da quello pratico. Riguardo il punto di vista teorico sarà analizzato il problema dell'accelerazione degli algoritmi di moltiplicazione polinomiale, ampiamente impiegati in Crittografia Post-Quantum, come, ad esempio, NTRU e McEliece. Quest'ultimo, in particolare, utilizza campi di Galois e loro estensioni, i cui elementi possono essere rappresentati in forma polinomiale. Saranno dunque esposte nuove tecniche che permettono una riduzione del numero di porte logiche e verranno presentati i risultati sperimentali della loro applicazione all'implementazione del cifrario McEliece attualmente candidato come nuovo standard Post-Quantum al NIST. Dal punto di vista pratico, questo lavoro di tesi, si focalizza sull’ottimizzazione di attacchi alla prima pre-immagine dell'algoritmo di hash SHA-1 basati su SAT solvers. Nessuna delle rappresentazioni testate ha mostrato una particolare efficienza in termini di velocità di risoluzione. Al contrario, un'accurata scelta di valori ha permesso di raggiungere un nuovo stato dell'arte, rivelando al contempo la debolezza di alcune pre-immagini.
This work focuses on optimization of cryptographic primitives both in theory and in applications. From a theoretical point of view, it addresses the problem of speeding up the polynomial multiplication used in Post-Quantum cryptosystems such as NTRU and McEliece. In particular, the latter extensively uses Galois fields whose elements can be represented in polynomial form. After presenting the reduction of the number of gates for polynomial multiplication through new techniques, in this work experimental results of such techniques applied to the current implementation of McEliece will be presented. From a practical point of view, this work focuses on the optimization of a SAT solver-based preimage attack against SHA-1 and on its strength. None of the tested representations of SHA-1 seems to be competitive in terms of resolution. On the contrary, an accurate choice of some pre-image bits allows one to reach a better state of art, revealing meanwhile the weakness of some pre-images.
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Tsang, Chiu-yin, and 曾超賢. "Finite Blaschke products versus polynomials." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2012. http://hub.hku.hk/bib/B4784971X.

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The objective of the thesis is to compare polynomials and finite Blaschke products, and demonstrate that they share many similar properties and hence we can establish a dictionary between these two kinds of finite maps for the first time. The results for polynomials were reviewed first. In particular, a special kind of polynomials was discussed, namely, Chebyshev polynomials, which can be defined by the trigonometric cosine function cos ?. Also, a complete classification for two polynomials sharing a set was given. In this thesis, some analogous results for finite Blaschke products were proved. Firstly, Chebyshev-Blaschke products were introduced. They can be defined by re- placing the trigonometric cosine function cos z by the Jacobi cosine function cd(u; ? ). They were shown to have several similar properties of Chebyshev polynomials, for example, both of them share the same monodromy, satisfy some differential equations and solve some minimization problems. In addition, some analogous results about two finite Blaschke products sharing a set were proved, based on Dinh's and Pakovich's ideas. Moreover, the density of prime polynomials was investigated in two different ways: (i) expressing the polynomials of degree n in terms of the zeros and the leading coefficient; (ii) expressing the polynomials of degree n in terms of the coefficients. Also, the quantitative version of the density of composite polynomials was developed and a density estimate on the set of composite polynomials was given. Furthermore, some analogous results on the the density of prime Blaschke products were proved.
published_or_final_version
Mathematics
Doctoral
Doctor of Philosophy
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CAMPOS, Suene Ferreira. "Teorema sobre o produto tensorial em característica positiva." Universidade Federal de Campina Grande, 2008. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1207.

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Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-22T13:41:27Z No. of bitstreams: 1 SUENE FERREIRA CAMPOS - DISSERTAÇÃO PPGMAT 2008..pdf: 741113 bytes, checksum: 7fc13ffd22412553f540977137401f24 (MD5)
Made available in DSpace on 2018-07-22T13:41:27Z (GMT). No. of bitstreams: 1 SUENE FERREIRA CAMPOS - DISSERTAÇÃO PPGMAT 2008..pdf: 741113 bytes, checksum: 7fc13ffd22412553f540977137401f24 (MD5) Previous issue date: 2008-12
Neste trabalho apresentamos um estudo sobre o comportamento das identidades polinomiais dos produtos tensoriais de álgebras T-primas sobre corpos infinitos com diferentes características. Mais precisamente, apresentamos o Teorema sobre Produto Tensorial (TPT), descrito por Kemer para corpos de característica zero, e verificamos a sua validade sobre corpos infinitos com característica positiva. Incialmente, a partir de resultados apresentados por Azevedo e Koshlukov, estudamos os T-ideais das álgebras M1,1(G) eG⊗G, para corpos infinitos com característica zero e característicap > 2. Aqui, G = G0⊕G1 é a álgebra de Grassmann de dimensão infinita eM1,1(G) é a subálgebra de M2(G) que consiste das matrizes de ordem 2 que têm na diagonal principal entradas emG0 e na diagonal secundária entradas emG1. Em seguida, utilizando métodos introduzidos por Regev e desenvolvidos por Azevedo, Fidélis e Koshlukov, verificamos a validade do TPT para corpos de característica positiva, quando o mesmo é restrito a polinômios multilineares. Finalmente, apresentamos alguns resultados obtidos por Alves, Azevedo, Fidélis e Koshlukov, que comprovam que o TPT é falso quando o corpo base é infinito e tem característicap>2.
In this work we present a study about the behavior of polynomial identities of tensor products of T-prime T-ideals over infinite fields of different characteristics. More precisely, we present the Tensor Product Theorem (TPT), described by Kemer for fields of characteristic zero, and verify its validity over infinite fields with positive characteristic. First, based on results of Azevedo and Koshlukov, we study the Tideals of the algebrasM1,1(G) eG⊗G, for infinite fields of characteristic zero and characteristicp>2. Here,G=G0 ⊕G1 is the Grassmann algebra of infinite dimension andM1,1(G) is the subalgebras ofM2(G) consisting of matrices of order2 which main diagonal entries are inG0 and the secondary diagonal entries are inG1. Second, using methods introduced by Regev and developed by Azevedo, Fidélis and Koshlukov, we verify the validity of the TPT for fields of positive characteristic, when it is restricted to multilinear polynomials. Finally, we present some results of Alves, Azevedo, Fidelis and Koshlukov, which show that the TPT is false when the basis field is infinite and has characteristicp>2.
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Masetti, Masha. "Product Clustering e Machine Learning per il miglioramento dell'accuratezza della previsione della domanda: il caso Comer Industries S.p.A." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2021.

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I lunghi lead time della catena di fornitura cinese dell’azienda Comer Industries S.p.A la obbligano ad ordinare i materiali con sei mesi di anticipo, data in cui spesso i clienti non sono consapevoli dei quantitativi di materiale che necessiteranno. Al fine di rispondere ai clienti mantenendo l’alto livello di servizio garantito storicamente da Comer Industries, risulta essenziale ordinare il materiale basandosi sulle previsioni della domanda. Tuttavia, attualmente le previsioni non sono sufficientemente accurate. L’obiettivo di questa ricerca è individuare un possibile metodo per incrementare l’accuratezza delle previsioni della domanda. Potrebbe, al fine del miglioramento della forecast accuracy, incidere positivamente l’utilizzo dell’Intelligenza Artificiale? Per rispondere alla domanda di ricerca, si sono implementati l’algoritmo K-Means e l’algoritmo Gerarchico in Visual Basic Application al fine di dividere i prodotti in cluster sulla base dei componenti comuni. Successivamente, si sono analizzati gli andamenti della domanda. Implementando differenti algoritmi di Machine Learning su Google Colaboratory, si sono paragonate le accuratezze ottenute e si è individuato un algoritmo di previsione ottimale per ciascun profilo di domanda. Infine, con le previsioni effettuate, si è potuto identificare con il K-means un miglioramento dell’accuracy di circa il 54,62% rispetto all’accuratezza iniziale ed un risparmio del 47% dei costi per il mantenimento del safety stock, mentre con il Clustering Gerarchico si è rilevato un miglioramento dell’accuracy del 11,15% ed un risparmio del 45% dei costi attuali. Si è, pertanto, concluso che la clusterizzazione dei prodotti potrebbe apportare un contributo positivo all’accuratezza delle previsioni. Inoltre, si è osservato come il Machine Learning potrebbe costituire lo strumento ideale per individuare le soluzioni ottimali sia all’interno degli algoritmi di Clustering sia all’interno dei metodi previsionali.
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Magnin, Loïck. "Two-player interaction in quantum computing : cryptographic primitives & query complexity." Thesis, Paris 11, 2011. http://www.theses.fr/2011PA112275/document.

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Cette thèse étudie deux aspects d'interaction entre deux joueurs dans le modèle du calcul et de la communication quantique.Premièrement, elle étudie deux primitives cryptographiques quantiques, des briques de base pour construire des protocoles cryptographiques complexes entre deux joueurs, comme par exemple un protocole d'identification. La première primitive est la ``mise en gage quantique". Cette primitive ne peut pas être réalisée de manière inconditionnellement sûre, mais il possible d'avoir une sécurité lorsque les deux parties sont soumis à certaines contraintes additionnelles. Nous étudions cette primitive dans le cas où les deux joueurs sont limités à l'utilisation d'états et d'opération gaussiennes, un sous-ensemble de la physique quantique central en optique, donc parfaitement adapté pour la communication via fibres optiques. Nous montrons que cette restriction ne permet malheureusement pas la réalisation de la mise en gage sûre. Pour parvenir à ce résultat, nous introduisons la notion de purification intrinsèque, qui permet de contourner l'utilisation du théorème de Uhlman, en particulier dans le cas gaussien. Nous examinons ensuite une primitive cryptographique plus faible, le ``tirage faible à pile ou face'', dans le modèle standard du calcul quantique. Carlos Mochon a donné une preuve d'existence d'un tel protocole avec un biais arbitrairement petit. Nous donnons une interprétation claire de sa preuve, ce qui nous permet de la simplifier et de la raccourcir grandement.La seconde partie de cette thèse concerne l'étude de méthodes pour prouver des bornes inférieures dans le modèle de la complexité en requête. Il s'agit d'un modèle de complexité central en calcul quantique dans lequel de nombreux résultats majeurs ont été obtenus. Dans ce modèle, un algorithme ne peut accéder à l'entrée uniquement en effectuant des requêtes sur chacun des bits de l'entrée. Nous considérons une extension de ce modèle dans lequel un algorithme ne calcule pas une fonction, mais doit générer un état quantique. Cette généralisation nous permet de comparer les différentes méthodes pour prouver des bornes inférieures dans ce modèle. Nous montrons d'abord que la méthode par adversaire ``multiplicative" est plus forte que la méthode ``additive". Nous montrons ensuite une réduction de la méthode polynomiale à la méthode multiplicative, ce qui permet de conclure à la supériorité de la méthode par adversaire multiplicative sur toutes les autres méthodes. Les méthodes par adversaires sont en revanche souvent difficiles à utiliser car elles nécessite le calcul de normes de matrices de très grandes tailles. Nous montrons comment l'étude des symétries d'un problème simplifie grandement ces calculs. Enfin, nous appliquons ces formules pour prouver la borne inférieure optimale du problème INDEX-ERASURE un problème de génération d'état quantique lié au célèbre problème GRAPH-ISOMORPHISM
This dissertation studies two different aspects of two-player interaction in the model of quantum communication and quantum computation.First, we study two cryptographic primitives, that are used as basic blocks to construct sophisticated cryptographic protocols between two players, e.g. identification protocols. The first primitive is ``quantum bit commitment''. This primitive cannot be done in an unconditionally secure way. However, security can be obtained by restraining the power of the two players. We study this primitive when the two players can only create quantum Gaussian states and perform Gaussian operations. These operations are a subset of what is allowed by quantum physics, and plays a central role in quantum optics. Hence, it is an accurate model of communication through optical fibers. We show that unfortunately this restriction does not allow secure bit commitment. The proof of this result is based on the notion of ``intrinsic purification'' that we introduce to circumvent the use of Uhlman's theorem when the quantum states are Gaussian. We then examine a weaker primitive, ``quantum weak coin flipping'', in the standard model of quantum computation. Mochon has showed that there exists such a protocol with arbitrarily small bias. We give a clear and meaningful interpretation of his proof. That allows us to present a drastically shorter and simplified proof.The second part of the dissertation deals with different methods of proving lower bounds on the quantum query complexity. This is a very important model in quantum complexity in which numerous results have been proved. In this model, an algorithm has restricted access to the input: it can only query individual bits. We consider a generalization of the standard model, where an algorithm does not compute a classical function, but generates a quantum state. This generalization allows us to compare the strength of the different methods used to prove lower bounds in this model. We first prove that the ``multiplicative adversary method'' is stronger than the ``additive adversary method''. We then show a reduction from the ``polynomial method'' to the multiplicative adversary method. Hence, we prove that the multiplicative adversary method is the strongest one. Adversary methods are usually difficult to use since they involve the computation of norms of matrices with very large size. We show how studying the symmetries of a problem can largely simplify these computations. Last, using these principles we prove the tight lower bound of the INDEX-ERASURE problem. This a quantum state generation problem that has links with the famous GRAPH-ISOMORPHISM problem
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Piah, Abd Rahni bin Mt. "Construction of smooth closed surfaces by piecewise tensor product polynomials." Thesis, University of Dundee, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.295312.

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Books on the topic "Polynomial product"

1

Positive polynomials and product type actions of compact groups. Providence, R.I., USA: American Mathematical Society, 1985.

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Kalnins, E. G. Tensor products of special unitary and oscillator algebras. Hamilton, N.Z: University of Waikato, 1992.

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D, Alpay, Fuhrmann Paul Abraham, Arazy J, Frazho Arthur E. 1950-, Olshevsky Vadim 1961-, Clancey Kevin 1944-, Davidson Kenneth R, et al., eds. Spectral Theory in Inner Product Spaces and Applications: 6th Workshop on Operator Theory in Krein Spaces and Operator Polynomials, Berlin, December 2006. Basel: Birkhäuser Basel, 2009.

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International Conference on p-Adic Functional Analysis (11th 2010 Université Blaise Pascal). Advances in non-Archimedean analysis: Eleventh International Conference on p-Adic Functional Analysis, July 5-9 2010, Université Blaise Pascal, Clermont-Ferrand, France. Edited by Araujo-Gomez Jesus 1965-, Diarra B. (Bertin) 1944-, and Escassut Alain. Providence, R.I: American Mathematical Society, 2011.

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Succi, Sauro. Lattice Boltzmann for reactive flows. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780199592357.003.0026.

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The dynamics of reactive flows lies at the heart of several important applications, such as combustion, heterogeneous catalysis, pollutant conversion, pattern formation in biology and many others. In general, LB is well suited to describe reaction-diffusion applications with flowing species. This chapter provides the basic guidelines to include reactive phenomena within the LBE formalism. Reactive flows obey the usual fluid equations, augmented with a reactive source term, accounting for species transformations due to chemical reactions. Such term comes typically in the form of a polynomial product of the mass densities of the reacting species.
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Khoruzhenko, Boris, and Hans-Jurgen Sommers. Characteristic polynomials. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.19.

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This article considers characteristic polynomials and reviews a few useful results obtained in simple Gaussian models of random Hermitian matrices in the presence of an external matrix source. It first considers the products and ratio of characteristic polynomials before discussing the duality theorems for two different characteristic polynomials of Gaussian weights with external sources. It then describes the m-point correlation functions of the eigenvalues in the Gaussian unitary ensemble and how they are deduced from their Fourier transforms U(s1, … , sm). It also analyses the relation of the correlation function of the characteristic polynomials to the standard n-point correlation function using the replica and supersymmetric methods. Finally, it shows how the topological invariants of Riemann surfaces, such as the intersection numbers of the moduli space of curves, may be derived from averaged characteristic polynomials.
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van Moerbeke, Pierre. Determinantal point processes. Edited by Gernot Akemann, Jinho Baik, and Philippe Di Francesco. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198744191.013.11.

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This article presents a list of algebraic, combinatorial, and analytic mechanisms that give rise to determinantal point processes. Determinantal point processes have been used in random matrix theory (RMT) since the early 1960s. As a separate class, determinantal processes were first used to model fermions in thermal equilibrium and the term ‘fermion’ point processes were adopted. The article first provides an overview of the generalities associated with determinantal point processes before discussing loop-free Markov chains, that is, the trajectories of the Markov chain do not pass through the same point twice almost surely. It then considers the measures given by products of determinants, namely, biorthogonal ensembles. An especially important subclass of biorthogonal ensembles consists of orthogonal polynomial ensembles. The article also describes L-ensembles, a general construction of determinantal point processes via the Fock space formalism, dimer models, uniform spanning trees, Hermitian correlation kernels, and Pfaffian point processes.
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Invitation to Nonlinear Algebra. American Mathematical Society, 2021.

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Book chapters on the topic "Polynomial product"

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Pin, Jean-Eric, and Pascal Weil. "Polynomial closure and unambiguous product." In Automata, Languages and Programming, 348–59. Berlin, Heidelberg: Springer Berlin Heidelberg, 1995. http://dx.doi.org/10.1007/3-540-60084-1_87.

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Cancellieri, Giovanni. "Binomial Product Generator LDPC Block Codes." In Polynomial Theory of Error Correcting Codes, 545–80. Cham: Springer International Publishing, 2014. http://dx.doi.org/10.1007/978-3-319-01727-3_11.

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de Groote, Philippe. "The Non-associative Lambek Calculus with Product in Polynomial Time." In Lecture Notes in Computer Science, 128–39. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48754-9_14.

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Yamaguchi, Yasushi. "Detection and Computation of Degenerate Normal Vectors on Tensor Product Polynomial Surfaces." In Geometric Modeling: Theory and Practice, 102–13. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/978-3-642-60607-6_8.

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Çakçak, Emrah. "A Note on the Minimal Polynomial of the Product of Linear Recurring Sequences." In Finite Fields and Applications, 57–69. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-56755-1_6.

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Averbuch, Amir, Shmuel Winograd, and Zvi Galil. "Classification of all the minimal bilinear algorithms for computing the coefficients of the product of two polynomials modulo a polynomial." In Automata, Languages and Programming, 31–39. Berlin, Heidelberg: Springer Berlin Heidelberg, 1986. http://dx.doi.org/10.1007/3-540-16761-7_52.

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Feng, Weiming, Heng Guo, Mark Jerrum, and Jiaheng Wang. "A simple polynomial-time approximation algorithm for the total variation distance between two product distributions." In Symposium on Simplicity in Algorithms (SOSA), 343–47. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2023. http://dx.doi.org/10.1137/1.9781611977585.ch30.

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Denker, Manfred, and Stefan-M. Heinemann. "Polynomial Skew Products." In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, 175–89. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-56589-2_8.

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Titi, Jihad, and Jürgen Garloff. "Symbolic-Numeric Computation of the Bernstein Coefficients of a Polynomial from Those of One of Its Partial Derivatives and of the Product of Two Polynomials." In Computer Algebra in Scientific Computing, 583–99. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-60026-6_34.

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Borwein, Peter. "Products of Cyclotomic Polynomials." In Computational Excursions in Analysis and Number Theory, 43–52. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-0-387-21652-2_6.

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Conference papers on the topic "Polynomial product"

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Anshelevich, Michael. "Product-type non-commutative polynomial states." In Noncommutative Harmonic Analysis with Applications to Probability II. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2010. http://dx.doi.org/10.4064/bc89-0-2.

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Condo, Carlo, Francois Leduc-Primeau, Gabi Sarkis, Pascal Giard, and Warren J. Gross. "Stall pattern avoidance in polynomial product codes." In 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP). IEEE, 2016. http://dx.doi.org/10.1109/globalsip.2016.7905932.

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Makous, Walter, David R. Williams, and Donald I. A. MacLeod. "Nonlinear transformation in human vision." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1985. http://dx.doi.org/10.1364/oam.1985.thy3.

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We have measured the form of a nonlinear transformation of retinal illuminance in the human visual system by measuring the detectability of the distortions it produces and approximating the transformation with orthogonal polynomials. Superimposing two interference fringes, of frequencies f1 and f2, produces a neural distortion product that can be more easily detected than the fringes themselves. The amplitudes of difference frequencies produced at (2f2 − f1) and (f 2 − f1) were measured separately by finding the contrast of superimposed fringes necessary to produce a threshold grating of 10 cpd. These thresholds, along with the threshold for a 10 cpd fringe, determine a set of simultaneous equations that yield the coefficients of the polynomials. As no distortion products of spatial frequency (mf2 – f1) could be detected when m > 2, we approximated the transformation with a third-order polynomial. The function that results depends somewhat on mean retinal illuminance, but it has the shape of a simple, saturating function that is more linear, and saturates more abruptly, than the function V = l/(l + s). A tenfold change of mean illumination changes the sale of intensity by slightly less than tenfold.
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Hajer, B., and B. B. Naceur. "Homogeneous Lyapunov functions for polynomial systems: a Tensor product approach." In 2007 IEEE International Conference on Control and Automation. IEEE, 2007. http://dx.doi.org/10.1109/icca.2007.4376694.

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Xiangjiu Che, Shiying Zong, Na Che, and Zhanheng Gao. "The product calculation of linear polynomial and B-spline curve." In 2009 IEEE 10th International Conference on Computer-Aided Industrial Design & Conceptual Design. IEEE, 2009. http://dx.doi.org/10.1109/caidcd.2009.5375452.

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Du, Chengyuan, and Lixin Gao. "Stability Analysis of Polynomial Nonlinear Systems via Semi-tensor Product Method." In 2020 7th International Conference on Information Science and Control Engineering (ICISCE). IEEE, 2020. http://dx.doi.org/10.1109/icisce50968.2020.00266.

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Lin, Shing-Hong, Thomas F. Krile, and John F. Walkup. "Optical polynomial processing based on the bilinear transform." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1986. http://dx.doi.org/10.1364/oam.1986.thl1.

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With the motivation to extend linear/bilinear operators to a more general class of nonlinear operators via a Volterra series (polynomial) approximation, we look into optical polynomial implementations using a factored representation so that presently known bilinear techniques can be employed. Since there are two inputs and one kernel in the generalized bilinear transform, the two inputs act as polynomial input variables and the elements in the kernel represent coefficients of the quadratic polynomial. Thus higher polynomial processing can be realized by iterating the bilinear transform. A dual-LCLV system is proposed to form such a quadratic polynomial. With electronic or optical feedback, a general optical polynomial processor is achievable. The work has been extended to perform bipolar complex analog and binary digital polynomial operations. The analog operations are performed by using separate parallel channels for real/imaginary and positive/negative numbers and making use of triple matrix-matrix product processing. Systolic and wavefront processors and a triple product processor are used to implement binary digital polynomial processing.
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Kang, Chang Woo, Soroush Abbaspour, and Massoud Pedram. "Buffer sizing for minimum energy-delay product by using an approximating polynomial." In the 13th ACM Great Lakes Symposium. New York, New York, USA: ACM Press, 2003. http://dx.doi.org/10.1145/764808.764838.

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Balakrishnan, U. "Product of polynomial values at integral points and some of its applications." In INTERNATIONAL CONFERENCE ON PHOTONICS, METAMATERIALS & PLASMONICS: PMP-2019. AIP Publishing, 2019. http://dx.doi.org/10.1063/1.5120908.

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Shieh, Meng-Dar, and Hsin-En Fang. "Using Support Vector Regression in the Study of Product Form Images." In ASME 2008 International Mechanical Engineering Congress and Exposition. ASMEDC, 2008. http://dx.doi.org/10.1115/imece2008-69150.

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In this paper, Support Vector Regression (SVR) training models using three different kernels: polynomial, Radial Basis Function (RBF), and mixed kernels, are constructed to demonstrate the training performance of unarranged data obtained from 32 virtual 3-D computer models. The 32 samples used as input data for training the three SVR models are represented by the coordination value sets of points extracted from 3-D models built by the 3-D software according to the shapes of 32 actual hairdryer products. To train the SVR model, an adjective (streamline) is used to evaluate all the 32 samples by 37 subjects. Then the scores of all the subjects are averaged to be the target values of the training models. In addition, a technique called k-fold cross-validation (C-V) is used to find the optimal parameter combination for optimizing the SVR models. The performance of the SVR using these three kernels to estimate the product image values is determined by the values of the Root Mean Square Error (RMSE). The results show that the optimal SVR model using the polynomial kernel performed better than the one using the RBF kernel. However, it is important to note that the mixed kernel had the best performance of the three. It is also shown in this study that the single RBF has a local characteristic and cannot process the broadly distributed data well. It can, however, be used to improve the power of the SVR by combining with the polynomial kernel.
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