• Title/Summary/Keyword: Quantum calculus

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NEW QUANTUM VARIANTS OF SIMPSON-NEWTON TYPE INEQUALITIES VIA (α, m)-CONVEXITY

  • Saad Ihsan Butt;Qurat Ul Ain;Huseyin Budak
    • Korean Journal of Mathematics
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    • v.31 no.2
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    • pp.161-180
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    • 2023
  • In this article, we will utilize (α, m)-convexity to create a new form of Simpson-Newton inequalities in quantum calculus by using q𝝔1-integral and q𝝔1-derivative. Newly discovered inequalities can be transformed into quantum Newton and quantum Simpson for generalized convexity. Additionally, this article demonstrates how some recently created inequalities are simply the extensions of some previously existing inequalities. The main findings are generalizations of numerous results that already exist in the literature, and some fundamental inequalities, such as Hölder's and Power mean, have been used to acquire new bounds.

Multinomial Probability Distribution and Quantum Deformed Algebras

  • Fridolin Melong
    • Kyungpook Mathematical Journal
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    • v.63 no.3
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    • pp.463-484
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    • 2023
  • An examination is conducted on the multinomial coefficients derived from generalized quantum deformed algebras, and on their recurrence relations. The 𝓡(p, q)-deformed multinomial probability distribution and the negative 𝓡(p, q)-deformed multinomial probability distribution are constructed, and the recurrence relations are determined. From our general result, we deduce particular cases that correspond to quantum algebras considered in the literature.

EXTRACTING LINEAR FACTORS IN FEYNMAN'S OPERATIONAL CALCULI : THE CASE OF TIME DEPENDENT NONCOMMUTING OPERATORS

  • Ahn, Byung-Moo
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.3
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    • pp.573-587
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    • 2004
  • Disentangling is the essential operation of Feynman's operational calculus for noncommuting operators. Thus formulas which simplify this operation are central to the subject. In a recent paper the procedure for 'extracting a linear factor' has been established in the setting of Feynman's operational calculus for time independent operators $A_1, ... , A_n$ and associated probability measures ${\mu}_1,..., {\mu}_n$. While the setting just described is natural in many circumstances, it is not natural for evolution problems. There the measures should not be restricted to probability measures and it is worthwhile to allow the operators to depend on time. The main purpose for this paper is to extend the procedure for extracting a linear factor to this latter setting. We should mention that Feynman's primary motivation for developing an operational calculus for noncommuting operators came from a desire to describe the evolution of certain quantum systems.m systems.

FRACTIONAL EULER'S INTEGRAL OF FIRST AND SECOND KINDS. APPLICATION TO FRACTIONAL HERMITE'S POLYNOMIALS AND TO PROBABILITY DENSITY OF FRACTIONAL ORDER

  • Jumarie, Guy
    • Journal of applied mathematics & informatics
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    • v.28 no.1_2
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    • pp.257-273
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    • 2010
  • One can construct a theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function. In this framework, it seems of interest to generalize some useful classical mathematical tools, so that they are more suitable in fractional calculus. After a short background on fractional calculus based on modified Riemann Liouville derivative, one summarizes some definitions on probability density of fractional order (for the motive), and then one introduces successively fractional Euler's integrals (first and second kind) and fractional Hermite polynomials. Some properties of the Gaussian density of fractional order are exhibited. The fractional probability so introduced exhibits some relations with quantum probability.

A CHARACTERIZATION OF GIBBS MEASURES ON /$R \times W_{0,0})^{Z^{\nu}}$ VIA STOCHASTIC CALCULUS

  • Lim, Hye-Young;Park, Yong-Moon;Yoo, Hyun-Jae
    • Communications of the Korean Mathematical Society
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    • v.9 no.3
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    • pp.711-730
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    • 1994
  • We consider Gibbs measures on $(R \times W_{0,0})^{Z^\nu}, W_{0,0} = {\omega \in C[0,1] : \omega(0) = \omega(1)}$, which are associated to an interaction between particles in lattice boson systems (quantum unbounded spin systems). In [4], the Gibbs measures were introduced in the study of equilibrium states of interacting lattice boson systems and were characterized by means of the equilibrium conditions. In this paper we utilize the techniques of the stochastic calculus of variations and the infinite dimensional Ito integral to derive stochastic equations which we call the equilibrium equations. We show that under appropriate conditions the equilibrium conditions and the equilibrium equations are equivalent. The lattice boson systems with superstable and regular interactions, which we studied in [4], are typical examples.

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COHERENT SATE REPRESENTATION AND UNITARITY CONDITION IN WHITE NOISE CALCULUS

  • Obata, Nobuaki
    • Journal of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.297-309
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    • 2001
  • White noise distribution theory over the complex Gaussian space is established on the basis of the recently developed white noise operator theory. Unitarity condition for a white noise operator is discussed by means of the operator symbol and complex Gaussian integration. Concerning the overcompleteness of the exponential vectors, a coherent sate representation of a white noise function is uniquely specified from the diagonal coherent state representation of the associated multiplication operator.

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BESSEL-WRIGHT TRANSFORM IN THE SETTING OF QUANTUM CALCULUS

  • Karoui, Ilyes;Dhaouadi, Lazhar;Binous, Wafa;Haddad, Meniar
    • Korean Journal of Mathematics
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    • v.29 no.2
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    • pp.253-266
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    • 2021
  • This work is devoted to the study of a q-harmonic analysis related to the q-analog of the Bessel-Wright integral transform [6]. We establish some important properties of this transform and we focalise our attention in studying the associated transmutation operator.

Univalent Functions Associated with the Symmetric Points and Cardioid-shaped Domain Involving (p,q)-calculus

  • Ahuja, Om;Bohra, Nisha;Cetinkaya, Asena;Kumar, Sushil
    • Kyungpook Mathematical Journal
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    • v.61 no.1
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    • pp.75-98
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    • 2021
  • In this paper, we introduce new classes of post-quantum or (p, q)-starlike and convex functions with respect to symmetric points associated with a cardiod-shaped domain. We obtain (p, q)-Fekete-Szegö inequalities for functions in these classes. We also obtain estimates of initial (p, q)-logarithmic coefficients. In addition, we get q-Bieberbachde-Branges type inequalities for the special case of our classes when p = 1. Moreover, we also discuss some special cases of the obtained results.