Relevant bibliographies by topics / Q-Analogs

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Academic literature on the topic 'Q-Analogs'

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Published: 11 January 2025

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Journal articles on the topic "Q-Analogs"

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Al-Omari, Shrideh, Wael Salameh, and Hamzeh Zureigat. "Convolution Theorem for p,q-Gamma Integral Transforms and Their Application to Some Special Functions." Symmetry 16, no. 7 (July 11, 2024): 882. http://dx.doi.org/10.3390/sym16070882.

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This article introduces (p,q)-analogs of the gamma integral operator and discusses their expansion to power functions, (p,q)-exponential functions, and (p,q)-trigonometric functions. Additionally, it validates other findings concerning (p,q)-analogs of the gamma integrals to unit step functions as well as first- and second-order (p,q)-differential operators. In addition, it presents a pair of (p,q)-convolution products for the specified (p,q)-analogs and establishes two (p,q)-convolution theorems.
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Sagan, Bruce E. "Congruence properties of q-analogs." Advances in Mathematics 95, no. 1 (September 1992): 127–43. http://dx.doi.org/10.1016/0001-8708(92)90046-n.

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Braun, Michael, and Jan Reichelt. "q-Analogs of Packing Designs." Journal of Combinatorial Designs 22, no. 7 (October 15, 2013): 306–21. http://dx.doi.org/10.1002/jcd.21376.

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Al-Omari, Shrideh, and Wael Salameh. "On (p,q)-Analogs of the α-th Fractional Fourier Transform and Some (p,q)-Generalized Spaces." Symmetry 16, no. 10 (October 3, 2024): 1307. http://dx.doi.org/10.3390/sym16101307.

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In this article, the (p,q)-analogs of the α-th fractional Fourier transform are provided, along with a discussion of their characteristics in specific classes of (p,q)-generalized functions. Two spaces of infinitely (p,q)-differentiable functions are defined by introducing two (p,q)-differential symmetric operators. The (p,q)-analogs of the α-th fractional Fourier transform are demonstrated to be continuous and linear between the spaces under discussion. Next, theorems pertaining to specific convolutions are established. This leads to the establishment of multiple symmetric identities, which in turn requires the construction of (p,q)-generalized spaces known as (p,q)-Boehmians. Finally, in addition to deriving the inversion formulas, the generalized (p,q)- analogs of the α-th fractional Fourier transform are introduced, and their general properties are discussed.
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Zabrocki, Michael. "q-Analogs of symmetric function operators." Discrete Mathematics 256, no. 3 (October 2002): 831–53. http://dx.doi.org/10.1016/s0012-365x(02)00350-3.

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Ivanov, V. N. "Interpolation Analogs of Schur Q-Functions." Journal of Mathematical Sciences 131, no. 2 (November 2005): 5495–507. http://dx.doi.org/10.1007/s10958-005-0422-6.

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Bergeron, Nantel, and Michael Zabrocki. "q and q,t-analogs of non-commutative symmetric functions." Discrete Mathematics 298, no. 1-3 (August 2005): 79–103. http://dx.doi.org/10.1016/j.disc.2004.08.044.

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Al-Omari, Shrideh, Wael Salameh, and Sharifah Alhazmi. "Some Estimates for Certain q-analogs of Gamma Integral Transform Operators." Symmetry 16, no. 10 (October 15, 2024): 1368. http://dx.doi.org/10.3390/sym16101368.

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The aim of this work is to examine some q-analogs and differential properties of the gamma integral operator and its convolution products. The q-gamma integral operator is introduced in two versions in order to derive pertinent conclusions regarding the q-exponential functions. Also, new findings on the q-trigonometric, q-sine, and q-cosine functions are extracted. In addition, novel results for first and second-order q-differential operators are established and extended to Heaviside unit step functions. Lastly, three crucial convolution products and extensive convolution theorems for the q-analogs are also provided.
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Alsoboh, Abdullah, Ala Amourah, Maslina Darus, and Carla Amoi Rudder. "Studying the Harmonic Functions Associated with Quantum Calculus." Mathematics 11, no. 10 (May 9, 2023): 2220. http://dx.doi.org/10.3390/math11102220.

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Using the derivative operators’ q-analogs values, a wide variety of holomorphic function subclasses, q-starlike, and q-convex functions have been researched and examined. With the aid of fundamental ideas from the theory of q-calculus operators, we describe new q-operators of harmonic function Hϱ,χ;qγF(ϖ) in this work. We also define a new harmonic function subclass related to the Janowski and q-analog of Le Roy-type functions Mittag–Leffler functions. Several important properties are assigned to the new class, including necessary and sufficient conditions, the covering Theorem, extreme points, distortion bounds, convolution, and convex combinations. Furthermore, we emphasize several established remarks for confirming our primary findings presented in this study, as well as some applications of this study in the form of specific outcomes and corollaries.
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Zhou, Xia, Tianxin Cai, and David M. Bradley. "Signed $q$-analogs of Tornheim's double series." Proceedings of the American Mathematical Society 136, no. 08 (April 8, 2008): 2689–98. http://dx.doi.org/10.1090/s0002-9939-08-09208-3.

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Dissertations / Theses on the topic "Q-Analogs"

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Leclere, Ludivine. "q-analogues des nombres réels et des matrices unimodulaires : aspects algébriques, combinatoires et analytiques." Electronic Thesis or Diss., Reims, 2024. http://www.theses.fr/2024REIMS019.

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Ce travail est consacré à l'étude des q-analogues de nombre réels. La q-déformation d'un nombre rationnel utilisée est une fraction rationnelle à coefficients entiers qui a été introduite par Sophie Morier-Genoud et Valentin Ovsienko en 2019. Il s'agit dans un premier temps de préciser les propriétés algébriques et de donner des interprétations combinatoires des q-rationnels. On manipule les différentes notions liées aux nombres rationnels: les fractions continues, les matrices de PSL(2,Z), les continuants d'Euler, les triangulations de polygones et le graphe de Farey, et leurs versions q-déformées. Les traces des q-matrices de PSL(2,Z) obtenues sont étudiées et interprétées dans le modèle combinatoire de triangulations d'anneaux. Dans un second temps, nous nous intéressons à la q-déformation des nombres irrationnels et plus particulièrement des irrationnels quadratiques. Nous obtenons une formule explicite permettant d'écrire les q-irrationnels quadratiques. On s'intéresse aux rayons de convergence des séries de Laurent obtenues à partir des q-déformations des nombres réels. Enfin, nous introduisons un second paramètre pour obtenir des (q,t)-déformations de nombres. Ces dernières sont étudiées sous un aspect combinatoire pour affiner les interprétations dans les modèles déjà présentes mais également dans les graphes en serpents
This work is devoted to the study of q-analogs of real numbers. The q-deformation of a rational number that we use is a rational funtion with integer coefficientswhich was introduced by Sophie Morier-Genoud and Valentin Ovsienko in 2019. The first step is to elaborate algebraic properties and to give combinatorial interpretations of the q-rationals. We use different notions linked to rational numbers: continued fractions, PSL(2,Z) matrices, Euler continuants, polygon's triangulations and the Farey graph, and their q-deformed versions.The traces of the q-matrices of PSL(2,Z) that we obtained are studied and interpreted in the combinatorial model of triangulation of annulus. In a second stage, we focus on the q-deformations of irrational real numbers, and more precisely on quadratic irrational real numbers. We obtain an explicit formula to describe q-deformed quadratic irrationals. We give estimate for the radii of convergence of the Laurent series obtained from the q-deformations of real numbers. Finally, we introduce a second parameter to obtain (q, t)-deformations of the rationals. The latter is studied in its combinatorial aspect, in the models already described but also in terms of snake graphs
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Briggs, Karen Sue. "Q-analogues and P, Q-analogues of rook numbers and hit numbers and their extensions /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2003. http://wwwlib.umi.com/cr/ucsd/fullcit?p3091348.

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Ringrose, C. J. "The q-analogue of Van Der Corput's method." Thesis, University of Oxford, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355796.

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ZELIKSON, SHMUEL. "Cristallisation et q-analogues dans la categorie o." Paris 6, 1997. http://www.theses.fr/1997PA066744.

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Nous presentons un resume rapide de la theorie de cristallisation pour les modules de dimension finie de l'algebre enveloppante quantifiee u q(g). Nous construisons d'operateurs cristallins sur les modules de verma de u q(g), ayant des proprietes de compatibilite avec la forme de shapovalov et avec la tensorisation a gauche par des modules de dimension finie, generalisant celles des operateurs cristallins sur les modules de dimension finie. En particulier, nous introduisons la classe des semi-cristaux, incluant celle des cristaux, et admettant un produit tensoriel. Les restrictions sur cette construction impliquent qu'elle est probablement unique. Nous demontrons que la generalisation directe aux modules de verma de la construction des bases cristallines de kashiwara est impossible. Il est possible toutefois, de definir des fonctions i, fonctions caracterisant un semi-cristal en tant qu'ensemble, pour ces modules, de maniere compatible avec la tensorisation a gauche par des modules de dimension finie, et les inclusions entre modules de verma correspondant a l'ordre de bruhat faible. Ces fonctions permettent l'introduction de nouveaux q-analogues multiples de la multiplicite, les caracteres cristallins. Les resultats pointent vers l'existence d'une formule de weyl pour ces caracteres. Nous considerons ensuite la construction algebrique par r. K. Brylinsky des q-analogues de lusztig. Ces analogues ne sont connus qu'essentiellement pour les poids dominants d'un module simple de dimension finie. Nous presentons une methode de calcul de ces analogues pour n'importe quel poids se basant sur la l'existence d'une base p. B. W. Special du o-dual d'un module de verma, ainsi que sur la combinatoire du modele de chemins de littelmann. Nous l'appliquons pour calculer les polynomes de brylinsky pour g sl 3. Nous conjecturons que les coefficients de ces polynomes sont invariants sous l'action du groupe de weyl. Nous calculons leur terme principal lorsque g est simplement lacee, ce qui implique dans ce cas la validite de la conjecture pour les u (g) modules simples de dimension finie dont les espaces poids sont de dimension 1.
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Yin, Jingbin. "A q-analogue of spanning trees : nilpotent transformations over finite fields." Thesis, Massachusetts Institute of Technology, 2009. http://hdl.handle.net/1721.1/50270.

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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2009.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 67).
The main result of this work is a q-analogue relationship between nilpotent transformations and spanning trees. For example, nilpotent endomorphisms on an n-dimensional vector space over Fq is a q-analogue of rooted spanning trees of the complete graph Kn. This relationship is based on two similar bijective proofs to calculate the number of spanning trees and nilpotent transformations, respectively. We also discuss more details about this bijection in the cases of complete graphs, complete bipartite graphs, and cycles. It gives some refinements of the q-analogue relationship. As a corollary, we find the total number of nilpotent transformations with some restrictions on Jordan block sizes.
by Jingbin Yin.
Ph.D.
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Gregory, James Philip. "A q-discrete Analogue of the Third Painlevé Equation and its Linear Problem." Thesis, The University of Sydney, 2016. http://hdl.handle.net/2123/15470.

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In this thesis we investigate the rational and Riccati type special solutions for particular parameter values of a q-discrete analogue of the third Painlevé equation, with rational surface A(1) 5 and affine Weyl group (A2 + A1)(1). The general solutions of this equation are highly transcendental in nature. We work closely with an associated system of discrete linear equations, which we refer to as ‘the linear problem.’We demonstrate that the linear problem can be solved both in terms of q-Gamma functions and series expansions for different parameter values of our discrete Painlevé equation. By developing a Schlesinger transformation, which transforms a series expansion of the linear problem for one parameter value to another series expansion of the linear problem for another parameter value, we are able to develop determinantal representations of the rational and Riccati type special solutions. These determinantal forms appear different to those discovered previously by Kajiwara. This technique has only been used to develop the determinantal forms of two other continuous and discrete Painlevé equations and hence the results presented here further indicate its potential.
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Derycke, Henri. "Combinatoire dans des stabilisations du modèle du tas de sable sur la grille Z²." Thesis, Bordeaux, 2018. http://www.theses.fr/2018BORD0327/document.

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Le modèle du tas de sable est un modèle de diffusion discret et isotrope introduit par les physiciens Bak, Tang et Wiesenfeld comme illustration de la criticalité auto-organisée. Pour tout graphe, souvent supposé fini, Dhar a formalisé de nombreuses propriétés simplifiant son analyse. Cette thèse propose des études de ce modèle sur la grille bidimensionnelle usuelle et certains de ses sous-graphes également infinis que sont les bandes bi-infinies de hauteur finie. Des approximations du comportement de la pile de sable peuvent se rapprocher de certains modèles de bootstrap percolation avec un support de stabilisation rectangulaire. Les lois sur son demi-périmètre peuvent se décrire à l’aide de statistiques sur les permutations. Un sous-produit de ce travail fait apparaître une différence de deux séries génératrices comptant des permutations selon deux statistiques mahoniennes classiques dont est extrait un polynôme à coefficients entiers et surtout positifs. La suite de cette thèse revisite dans le cadre de ces graphes infinis, des structures jusque-là bien définies uniquement dans le cas des graphes finis, notamment la récurrence. Dans le modèle sur une bande de hauteur finie H, l’existence donnée par Járai et Lyons d’automates finis reconnaissant les configurations récurrentes lues colonne par colonne est étendu par une construction explicite d’automates avec un nombre moindre d’états, se rapprochant de la conjecture de Gamlin. Dans une seconde approche, l’étude se concentre sur les configurations sur la grille entière qui sont périodiques dans les deux directions. Le puits, un sommet du graphe garantissant la terminaison de la stabilisation, est placé à l’infini dans une direction de pente rationnelle. Ceci permet à la fois de préserver la bipériodicité et de proposer une forme affaiblie du critère de Dhar caractérisant ainsi par un algorithme effectif les configurations récurrentes. Ces configurations récurrentes bipériodiques sont des candidates naturelles pour être les éléments de sous-groupes finis de l’éventuel groupe du tas de sable sur la grille. Des éléments de construction de cette loi de groupe donnent expérimentalement quelques sous-groupes finis
The sandpile model is a discrete model for diffusion of grains on a graph introduced by physicists Bak, Tang and Wiesenfeld as an illustration for self-organised criticality. For any finite graph, Dhar identified many of its numerous structures which simplify its analysis. This thesis focus on the usual square lattice and its subgraphs which are strips of height H, both notions of infinite graphs. Approximations on the behaviour of the stabilisation of a large stack of grains at the origin of the square lattice lead to some random distribution of grains, which stabilisation is connected to some models of bootstrap percolation where modified vertices by this stabilisation forms a rectangle. The laws of the half-perimeter of this rectangle are described by statistics on permutations. As a byproduct, the difference between the generating functions over some permutations of two classical mahonian statistics on permutations appears to mainly be a polynomial with coefficients which are integers and especially positive. Then, this thesis visits in the case of the studied infinite graphs some well-defined structures on finite graphs, in particular the recurrence. In the model on an horizontal strip of height H, we extend the existence of finite automata recognizing recurrent configurations read column by column presented by Járai and Lyons to new automata with significantly less states and these numbers are closer to a conjecture due to Gamlin. An implementation leads to explicit automata for heights 3 and 4 while up to now only the case 2 was obtained by hand. In a second approach, we consider the configurations on the twodimensional square lattice which are periodic in two directions. We suggest to place the sink ensuring that the stabilisation ends at infinity in a direction of rational slope which allows to preserve biperiodicity and a weaker form of Dhar criterion for recurrent configurations. Hence we obtain an effective algorithm defining recurrent configurations among the biperiodic and stable configurations. These biperiodic and recurrent configurations are natural candidates for being the elements of finite subgroups of the hypothetical group on configurations of the sandpile model on the square lattice. We discuss some notions allowing the definition of the law of such a group and experimentally provide some finite subgroups
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Gillespie, Jason Michael. "A Combinatorial Proof of the Positivity of the Lusztig q-Analogue of Weight Multiplicity for Rank 2 Lie Algebras." Diss., Virginia Tech, 2003. http://hdl.handle.net/10919/11071.

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We prove the positivity of Lusztig's q-analogue of weight multiplicity in a purely combinatorial way for rank 2 Lie algebras. Each summand in the polynomial can be interpreted as a linear combination of positive roots. We prove that all negative coefficients are cancelled in the polynomial. Further, the analysis of the root systems allows us to state formulae for every coefficient in Lusztig's q-analogue for rank 2 Lie algebras.
Ph. D.
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Costal, Flavia Soares Louro. "Benfotiamina e Mito Q protegem ilhotas pancreáticas de rato em cultura dos efeitos pró-apoptóticos dos produtos finais de glicação avançada (AGEs)." Universidade de São Paulo, 2012. http://www.teses.usp.br/teses/disponiveis/5/5135/tde-24052012-161456/.

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A perda da função das células beta acelera a deterioração do controle metabólico em pessoas com diabetes tipo 2. Além da lipo- e da glicotoxicidade, os AGEs parecem contribuir para esse processo, promovendo a apoptose das ilhotas pancreáticas. Em outros tecidos, os AGEs interagem com seu receptor específico (RAGE), produzindo espécies reativas de oxigênio (ROS) e ativando o NF-kB. Para investigar o efeito temporal dos AGEs sobre a apoptose de ilhotas, bem como o potencial de compostos antioxidantes para diminuir danos causados pelos AGEs, ilhotas pancreáticas de ratos foram tratadas durante 24, 48, 72, 96 e 120 h com AGEs gerados a partir de co-incubação de albumina de soro bovino (BSA) com Dgliceraldeído (GAD, 5 mg/mL) ou tampão fostato (controle). A apoptose foi avaliada pela quantificação do DNA fragmentado (ELISA), atividade de caspase 3 e detecção da permeabilidade da membrana mitocondrial (MitoProbe JC-1). O estresse oxidativo foi avaliado pela detecção de espécies de oxigênio (Image-iT LIVE Green) e a atividade da NADPH oxidase foi mensurada pelo método de quimioluminescência da lucigenina. A expressão dos genes Bax, Bcl2 e Nfkb1 foi avaliada por reação em cadeia da polimerase quantitativa após transcrição reversa (RT-qPCR). Em um dos tempos em que foi detectado o aumento da apoptose, o efeito de dois compostos antioxidantes foi avaliado: benfotiamina (350 M), uma vitamina B1 lipossolúvel, e Mito Q (1 M), um derivado da ubiquinona com alvo seletivo para a mitocôndria. Em 24 e 48 h, os AGES promoveram um aumento do índice de apoptose em relação ao controle, concomitantemente com o aumento na expresssão do gene Bcl2 (gene anti-apoptótico) e uma redução na expressão do gene Nfkb1. Em contraste, após 72, 96 h e 120 h, os AGEs promoveram um aumento do índice de apoptose em comparação com a condição de controle, concomitantemente com uma diminuição na expressão do gene Bcl2 e um aumento na expressão do gene Nfkb1. Em 24 h, os AGEs promoveram uma diminuição do conteúdo de ROS nas ilhotas, enquanto que nos tempos de 48 e 72 h, os AGEs promoveram um efeito oposto. A benfotiamina e o Mito Q foram capazes de diminuir o índice de apoptose e o estresse oxidativo de ilhotas expostas aos AGEs por 72 h. Em conclusão, os AGEs exerceram um duplo efeito em cultura de ilhotas pancreáticas, sendo de proteção contra a apoptose após exposição curta, mas pró-apoptótica após exposição prolongada. O Mito Q e e a benfotiamina merecem ser adicionalmente estudados como drogas com o potencial de oferecer proteção às ilhotas pancreáticas em condições de hiperglicemia crônica
Loss of beta cell function hastens the deterioration of metabolic control in people with type 2 diabetes. Besides lipo- and glucotoxicity, AGEs seem to contribute to this process by promoting islet apoptosis. In other tissues, AGEs interact with their specific receptors (RAGE) and elicit reactive oxygen species (ROS) generation and NF-kB activation. In order to investigate the temporal effect of AGEs on islet apoptosis as well as the potential of antioxidant compounds to decrease islet damage caused by AGEs, rat pancreatic islets were treated for 24, 48, 72, 96 and 120 h with either AGEs generated from co-incubation of bovine serum albumin (BSA) with D-glyceraldehyde (GAD, 5 mg/mL) or phosphate-buffered saline (PBS, control). Apoptosis was evaluated by quantification of DNA fragmentation (ELISA), caspase-3 enzyme activity and detection of mitochondrial permeability transition (MitoProbe JC-1). Oxidative stress was evaluated by oxygen species detection (Image-iT LIVE Green) and the activity of NADPH oxidase was measured by the lucigenin-enhanced chemiluminescence method. The expression of the genes Bax, Bcl2 and Nfkb1 was evaluated by reverse transcription real-time quantitative polymerase chain reaction (RT-qPCR). In one of the time points at which increased apoptosis was detected, the effect of two antioxidant compounds was evaluated: benfotiamine (350 M), a liposoluble vitamin B1, and Mito Q (1 M), a derivative of ubiquinone targeted to mitochondria. In 24 and 48 h, AGEs elicited a significant decrease in the apoptosis rate in comparison to the control condition concomitantly with a significant increase in the RNA expression of the antiapoptotic gene Bcl2 and a significant decrease in the Nfkb1 RNA expression. In contrast, after 72 and 96 h, AGEs promoted a significant increase in the apoptosis rate in comparison to the control condition concomitantly with a significant decrease in Bcl2 RNA expression and a significant increase in Nfkb1 RNA expression. In 24 h, AGEs elicited a significant decrease in the islet content of ROS while after 48 and 72 h, AGEs promoted an opposite effect. Benfotiamine and Mito Q were able to decrease the apoptosis rate and the ROS content in islets exposed to AGEs for 72 h. In conclusion, AGEs exerted a dual effect in cultured pancreatic islets, being protective against apoptosis after short exposition but proapoptotic after prolonged exposition. Mito Q and benfotiamine deserve further evaluation as drugs that could offer islet protection in conditions of chronic hyperglycemia
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Smith, Craig. "Quantum groups at q=0, a Tannakian reconstruction theorem for IndBanach spaces, and analytic analogues of quantum groups." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:e28b7312-3a47-473c-ac76-7d69db175309.

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This thesis is divided into the following three parts. A categorical reconstruction of crystals and quantum groups at q = 0. The quantum co-ordinate algebra Aq(𝔤) associated to a KacMoody Lie algebra g forms a Hopf algebra whose comodules are direct sums of finite dimensional irreducible Uq(𝔤) modules. In Part I we investigate whether an analogous result is true when q = 0. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over ℤ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig's quantum group at v = ∞. A Tannakian Reconstruction Theorem for IndBanach Spaces. Classically, Tannaka-Krein duality allows us to reconstruct a (co)algebra from its category of representation. In Part II we present an approach that allows us to generalise this theory to the setting of Banach spaces. This leads to several interesting applications in the directions of analytic quantum groups, bounded cohomology and Galois descent. A large portion of Part II is dedicated to such examples. On analytic analogues of quantum groups. In Part III we present a new construction of analytic analogues of quantum groups over non-Archimedean fields and construct braided monoidal categories of their representations. We do this by constructing analytic Nichols algebras and use Majid's double-bosonisation construction to glue them together. We then go on to study the rigidity of these analytic quantum groups as algebra deformations of completed enveloping algebras through bounded cohomology. This provides the first steps towards a p-adic Drinfel'd-Kohno Theorem, which should relate this work to Furusho's p-adic Drinfel'd associators. Finally, we adapt these constructions to working over Archimedean fields.
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Books on the topic "Q-Analogs"

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Koekoek, Roelof, Peter A. Lesky, and René F. Swarttouw. Hypergeometric Orthogonal Polynomials and Their q-Analogues. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-05014-5.

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Peter, Lesky, and Swarttouw, René F. (René Franc̦ois), 1964-, eds. Hypergeometric orthogonal polynomials and their q-analogues. Heidelberg: Springer, 2010.

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Derivation of q-analogs for the radial Schrödinger equation in N space-dimensions. Commack, NY: Nova Science Publishers, 1997.

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Koekoek, Roelof, Peter A. Lesky, René F. Swarttouw, and Tom H. Koornwinder. Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, 2012.

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From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory. University of California Press, 2023.

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Darrigol, Olivier. From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory. University of California Press, 2021.

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Darrigol, Olivier. From C-Numbers to Q-Numbers: The Classical Analogy in the History of Quantum Theory. University of California Press, 2021.

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From c-numbers to q-numbers: The classical analogy in the history of quantum theory. Berkeley: University of California Press, 1992.

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Coopersmith, Jennifer. Hamiltonian Mechanics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198743040.003.0007.

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Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.
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Darrigol, Olivier. From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory (California Studies in the History of Science). University of California Press, 1993.

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Book chapters on the topic "Q-Analogs"

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Braun, Michael, Michael Kiermaier, and Alfred Wassermann. "q-Analogs of Designs: Subspace Designs." In Network Coding and Subspace Designs, 171–211. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-70293-3_8.

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Buratti, Marco, Michael Kiermaier, Sascha Kurz, Anamari Nakić, and Alfred Wassermann. "2. q-analogs of group divisible designs." In Combinatorics and Finite Fields, edited by Kai-Uwe Schmidt and Arne Winterhof, 21–38. Berlin, Boston: De Gruyter, 2019. http://dx.doi.org/10.1515/9783110642094-002.

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Vyas, Yashoverdhan. "Multidimensional Extensions of Basic (or q-)Analogs of Certain Erdélyi Type Integrals." In Springer Proceedings in Physics, 693–701. Cham: Springer Nature Switzerland, 2024. https://doi.org/10.1007/978-3-031-69146-1_55.

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Nakajima, Hiraku. "t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8." In Representation Theory of Algebraic Groups and Quantum Groups, 257–72. Boston: Birkhäuser Boston, 2010. http://dx.doi.org/10.1007/978-0-8176-4697-4_10.

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Richards, Donald St P. "Analogs and Extensions of Selberg’s Integral." In q-Series and Partitions, 109–37. New York, NY: Springer US, 1989. http://dx.doi.org/10.1007/978-1-4684-0637-5_10.

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Zhao, Jianqiang. "Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota–Baxter Algebras." In Springer Proceedings in Mathematics & Statistics, 259–92. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37031-2_10.

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Saxena, Vinod Prakash, Praveen Agarwal, and Altaf Ahmad Bhat. "Associated Generating Function q and (p,q)-Analogue of I-Function." In I-Function and Its Applications, 232–55. Boca Raton: Chapman and Hall/CRC, 2024. http://dx.doi.org/10.1201/9781003517757-7.

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Han, G. N., A. Randrianarivony, and J. Zeng. "Un autre q-analogue des nombres d’Euler." In The Andrews Festschrift, 139–58. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-56513-7_5.

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Ariki, Susumi. "q-Analogue of a Twisted Group Ring." In Finite Reductive Groups: Related Structures and Representations, 1–13. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-1-4612-4124-9_1.

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Aomoto, Kazuhiko, and Yoshifumi Kato. "A q-analogue of de Rham cohomology." In ICM-90 Satellite Conference Proceedings, 30–62. Tokyo: Springer Japan, 1991. http://dx.doi.org/10.1007/978-4-431-68170-0_2.

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Conference papers on the topic "Q-Analogs"

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Voronov, Th Th, Piotr Kielanowski, Victor Buchstaber, Anatol Odzijewicz, Martin Schlichenmaier, and Theodore Voronov. "Q-manifolds and Higher Analogs of Lie Algebroids." In XXIX WORKSHOP ON GEOMETRIC METHODS IN PHYSICS. AIP, 2010. http://dx.doi.org/10.1063/1.3527417.

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Dennis, Michael L., and Thomas R. Clark. "Optimally biased coherent I/Q analog photonic link." In 2008 Conference on Lasers and Electro-Optics (CLEO). IEEE, 2008. http://dx.doi.org/10.1109/cleo.2008.4551270.

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Solaiman, Maryam, Theodore Mui, Qi Wang, and Phil Mui. "Modeling Unlearning and Relearning with Multi-agent Q-Learning Systems." In 10th International Conference on Computer Science, Engineering and Information Technology, 135–43. Academy & Industry Research Collaboration Center, 2024. http://dx.doi.org/10.5121/csit.2024.141912.

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We model unlearning by simulating a Q-agent (using the reinforcement learning Qlearning algorithm), representing a real-world learner, playing the game of Nim against different adversarial agents to learn the optimal Nim strategy. When the Q-agent plays against sub-optimal agents, its percentage of optimal moves is decreased, analogous to a person forgetting (“unlearning”) what they have learned previously. To mitigate the effect of this “unlearning”, we experimented with modulating the Q-learning so that minimal learning occurs with untrusted opponents. This trust-based modulation is modeled by observing opponent moves that are different from those that a Q-agent has learned. This model parallels human trust which tends to increase with those whom one agrees with. With this modulated learning, we observe that a Q-agent with a baseline optimal strategy is able to robustly retain previously learned strategy, in some cases achieving a 0.3 difference in accuracy from the unlearning model. We then ran a three-phase simulation where the Qagent played against optimal agents in the first phase, sub-optimal agents in the second “unlearning” phase, and optimal or random agents in the third phase. We found that even after unlearning, the Q-agent was quickly able to relearn most of its knowledge about the optimal strategy for Nim.
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Spreafico, Elen Viviani Pereira, Kênia Cristina Pereira Silva, and Cecı́lia Pereira Andrade. "q-Analogues of Jacobsthal Identities Via Weighted Tilings." In CNMAC 2016 - XXXVI Congresso Nacional de Matemática Aplicada e Computacional. SBMAC, 2017. http://dx.doi.org/10.5540/03.2017.005.01.0225.

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KANEKO, JYOICHI. "FORRESTER'S CONSTANT TERM CONJECTURE AND ITS q–ANALOGUE." In Proceedings of the Nagoya 2000 International Workshop. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810007_0003.

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McFeetors, G., and M. Okoniewski. "Custom Fabricated High-Q Analog Dual-Gap RF MEMS Varactors." In 2006 International Conference on Microwaves, Radar & Wireless Communications. IEEE, 2006. http://dx.doi.org/10.1109/mikon.2006.4345127.

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Grichener, Alex, Balaji Lakshminarayanan, and Gabriel M. Rebeiz. "High-Q RF MEMS capacitor with digital/analog tuning capabilities." In 2008 IEEE MTT-S International Microwave Symposium Digest - MTT 2008. IEEE, 2008. http://dx.doi.org/10.1109/mwsym.2008.4633294.

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Kramer, M. "Intracavity Sub-G Hz-Mod illation of Nanosecond Laser Pulses via Resonator enhanced Four Wave Mixing." In The European Conference on Lasers and Electro-Optics. Washington, D.C.: Optica Publishing Group, 1996. http://dx.doi.org/10.1364/cleo_europe.1996.cmg4.

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The temporal shape of laser pulses emitted from an unstable laser resonator was investigated [1, 2]. The resonator was Q-switched by means of stimulated Brillouin scattering (SBS) in a cell filled with acetone. Besides a change in pulse length from 40 ns to 2-3 ns by a variation of the resonator configuration, sub-GHz-modulations of the 40 ns-pulses were observed (fig. 1a)-c)). In a set of 150 laser shots 28 were strongly modulated, which were taken for analysis in fig.2. Only modulation frequencies which are harmonic multiples of the resonator roundtrip frequency (100 MHz) appeared whereas frequencies of 400 MHz and above 600 MHz were missing. It is remarkable, that the measured frequencies belong to a subharmonic of the Brillouin shift (in acetone 3000 MHz at 1064 nm), whereas the missing frequencies are no subhamtonics of the Brillouin shift. It turns out, that the modulations are caused by a not too low Q-value before the Q-switching of the laser, which enables prelasing. Based on long prelasing pulses a spontaneaous mode locking takes place, which is responsible for the statistical behavior of the modulation frequencies. If the modulation frequency of this mode locking is matched to the Brillouin shift, a four wave mixing process is initiated in the SBS-cell causing a Q-switch of the resonator. In analogy to Brillouin enhanced four wave mixing (BEFWM) this process may be called resonator enhanced four wave mixing (REFWM).
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Lai, Hengwei, Yixiong Li, and Hao Wu. "Analog and Q-ary pulse-position modulation in digital communication system." In International Conference on Intelligent Systems, Communications, and Computer Networks (ISCCN 2022), edited by Tok Wang Ling. SPIE, 2022. http://dx.doi.org/10.1117/12.2652556.

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Zhang, Ming, and Nicolas Llaser. "Design of analog and mixed circuits for resonator's Q-factor measurement." In 2020 27th IEEE International Conference on Electronics, Circuits and Systems (ICECS). IEEE, 2020. http://dx.doi.org/10.1109/icecs49266.2020.9294784.

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