259 results on '"Górska, K."'
Search Results
2. The Havriliak-Negami and Jurlewicz-Weron-Stanislavsky relaxation models revisited: memory functions based study
- Author
-
Górska, K., Horzela, A., and Penson, K. A.
- Subjects
Mathematical Physics ,Condensed Matter - Other Condensed Matter - Abstract
We provide a review of theoretical results concerning the Havriliak-Negami (HN) and the Jurlewicz-Weron-Stanislavsky (JWS) dielectric relaxation models. We derive explicit forms of functions characterizing relaxation phenomena in the time domain - the relaxation, response and probability distribution functions. We also explain how to construct and solve relevant evolution equations within these models. These equations are usually solved by using the Schwinger parametrization and the integral transforms. Instead, in this work we replace it by the powerful Efros theorem. That allows one to relate physically admissible solutions to the memory-dependent evolution equations with phenomenologically known spectral functions and, from the other side, with the subordination mechanism emerging from a stochastic analysis of processes underpinning considered relaxation phenomena. Our approach is based on a systematic analysis of the memory-dependent evolution equations. It exploits methods of integral transforms, operational calculus and special functions theory with the completely monotone and Bernstein functions. Merging analytic and stochastic methods enables us to give a complete classification of the standard functions used to describe the large class of the relaxation phenomena and to explain their properties.
- Published
- 2023
- Full Text
- View/download PDF
3. Volterra-Prabhakar derivative of distributed order and some applications
- Author
-
Górska, K., Pietrzak, T., Sandev, T., and Tomovsky, {Ž}.
- Subjects
Mathematical Physics - Abstract
The paper studies the exact solution of two kinds of generalized Fokker-Planck equations in which the integral kernels are given either by the distributed order function $k_{1}(t) = \int_{0}^{1} t^{-\mu}/\Gamma(1- \mu) d\mu$ or the distributed order Prabhakar function $k_{2}(\alpha, \gamma; \lambda; t) = \int_{0}^{1} e^{-\gamma}_{\alpha, 1 - \mu}(\lambda; t) d\mu$, where the Prabhakar function is denoted as $e^{-\gamma}_{\alpha, 1 - \mu}(\lambda; t)$. Both of these integral kernels can be called the fading memory functions and are the Stieltjes functions. It is also shown that their Stieltjes character is enough to ensure the non-negativity of the mean square values and higher even moments. The odd moments vanish. Thus, the solution of generalized Fokker-Planck equations can be called the probability density functions. We introduce also the Volterra-Prabhakar function and its generalization which are involved in the definition of $k_{2}(\alpha, \gamma; \lambda; t)$ and generated by it the probability density function $p_2(x, t)$.
- Published
- 2022
- Full Text
- View/download PDF
4. Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution
- Author
-
Penson, K. A., Górska, K., Horzela, A., and Duchamp, G. H. E.
- Subjects
Mathematics - Combinatorics ,Mathematical Physics - Abstract
We investigate the combinatorial sequences $A(M, n)$ introduced by W. G. Brown (1964) and W. T. Tutte (1980) appearing in enumeration of convex polyhedra. Their formula is $$A(M, n) = \frac{2 (2M+3)!}{(M+2)! M!}\,\frac{(4n+2M+1)!}{n! (3n + 2M + 3)!} $$ with $n, M =0, 1, 2, \ldots$, and we conceive it as Hausdorff moments, where $M$ is a parameter and $n$ enumerates the moments. We solve exactly the corresponding Hausdorff moment problem: $A(M, n) = \int_{0}^{R} x^{n} W_{M}(x) d x$ on the natural support $(0, R)$, $R = 4^{4}/3^{3}$, using the method of inverse Mellin transform. We provide explicitly the weight functions $W_{M}(x)$ in terms of the Meijer G-functions $G_{4, 4}^{4, 0}$, or equivalently, the generalized hypergeometric functions ${_{3}F_{2}}$ (for $M=0, 1$) and ${_{4}F_{3}}$ (for $M \geq 2$). For $M = 0, 1$, we prove that $W_{M}(x)$ are non-negative and normalizable, thus they are probability distributions. For $M \geq 2$, $W_{M}(x)$ are signed functions vanishing on the extremities of the support. By rephrasing this problem entirely in terms of Meijer G representations we reveal an integral relation which directly furnishes $W_M(x)$ based on ordinary generating function of $A(M, n)$ as an input. All the results are studied analytically as well as graphically.
- Published
- 2022
- Full Text
- View/download PDF
5. Non-Debye relaxations: The ups and downs of the stretched exponential vs Mittag-Leffler's matchings
- Author
-
Górska, K., Horzela, A., and Penson, K. A.
- Subjects
Condensed Matter - Mesoscale and Nanoscale Physics ,Mathematical Physics - Abstract
Experimental data collected to provide us with information on the course of dielectric relaxation phenomena are got according to two distinct schemes: one can measure either the time decay of depolarization current or use methods of the broadband dielectric spectroscopy. Both sets of data are usually fitted by time or frequency dependent elementary functions which in turn may be analytically transformed among themselves using the Laplace transform and compared each other. This leads to the question on comparability of results got using just mentioned experimental procedures. If we would like to do that in the time domain we have to go beyond widely accepted Kohlrausch-Williams-Watts approximation and get acquainted with description using the Mittag-Leffler functions. To convince the reader that the latter is not difficult to understand we propose to look at the problem from the point of view of objects sitting in the heart of stochastic processes approach to relaxation. These are the characteristic exponents which are read out from the standard non-Debye frequency dependent patterns. Characteristic functions appear to be expressed in terms of elementary functions which asymptotic analysis is simple. This opens new possibility to compare behavior of functions used to describe non-Debye relaxations. Results of such done comparison are fully confirmed by calculations which use the powerful apparatus of the Mittag-Leffler functions., Comment: 14 pages, 3 figures
- Published
- 2022
- Full Text
- View/download PDF
6. General approach to stochastic resetting
- Author
-
Singh, R. K., Gorska, K., and Sandev, T.
- Subjects
Condensed Matter - Statistical Mechanics - Abstract
We address the effect of stochastic resetting on diffusion and subdiffusion process. For diffusion we find that MSD relaxes to a constant only when the distribution of reset times possess finite mean and variance. In this case, the leading order contribution to the PDF of a Gaussian propagator under resetting exhibits a cusp independent of the specific details of the reset time distribution. For subdiffusion we derive the PDF in Laplace space for arbitrary resetting protocol. Resetting at constant rate allows evaluation of the PDF in terms of H-function. We analyze the steady state and derive the rate function governing the relaxation behavior. For a subdiffusive process the steady state could exist even if the distribution of reset times possesses only finite mean.
- Published
- 2022
- Full Text
- View/download PDF
7. Non-Debye relaxations: The characteristic exponent in the excess wings model
- Author
-
Górska, K., Horzela, A., and Pogány, T. K.
- Subjects
Condensed Matter - Statistical Mechanics ,Mathematical Physics - Abstract
The characteristic (Laplace or L\'evy) exponents uniquely characterize infinitely divisible probability distributions. Although of purely mathematical origin they appear to be uniquely associated with the memory functions present in evolution equations which govern the course of such physical phenomena like non-Debye relaxations or anomalous diffusion. Commonly accepted procedure to mimic memory effects is to make basic equations time smeared, i.e., nonlocal in time. This is modeled either through the convolution of memory functions with those describing relaxation/diffusion or, alternatively, through the time smearing of time derivatives. Intuitive expectations say that such introduced time smearings should be physically equivalent. This leads to the conclusion that both kinds of so far introduced memory functions form a "twin" structure familiar to mathematicians for a long time and known as the Sonine pair. As an illustration of the proposed scheme we consider the excess wings model of non-Debye relaxations, determine its evolution equations and discuss properties of the solutions.
- Published
- 2021
- Full Text
- View/download PDF
8. Integral decomposition for the solutions of the generalized Cattaneo equation
- Author
-
Górska, K.
- Subjects
Condensed Matter - Statistical Mechanics ,Mathematical Physics - Abstract
We present the integral decomposition for the fundamental solution of the generalized Cattaneo equation with both time derivatives smeared through convoluting them with some memory kernels. For power-law kernels $t^{-\alpha}$, $\alpha\in(0,1]$ this equation becomes the time fractional one governed by the Caputo derivatives which highest order is 2. To invert the solutions from the Fourier-Laplace domain to the space-time domain we use analytic methods based on the Efross theorem and find out that solutions looked for are represented by integral decompositions which tangle the fundamental solution of the standard Cattaneo equation with non-negative and normalizable functions being uniquely dependent on the memory kernels. Furthermore, the use of methodology arising from the theory of complete Bernstein functions allows us to assign such constructed integral decompositions the interpretation of subordination. This fact is preserved in two limit cases built into the generalized Cattaneo equations, i.e., either the diffusion or the wave equations. We point out that applying the Efross theorem enables us to go beyond the standard approach which usually leads to the integral decompositions involving the Gaussian distribution describing the Brownian motion. Our approach clarifies puzzling situation which takes place for the power-law kernels $t^{-\alpha}$ for which the subordination based on the Brownian motion does not work if $\alpha\in(1/2,1]$.
- Published
- 2021
- Full Text
- View/download PDF
9. The generalized telegraph equation with moving harmonic source: Solvability using the integral decomposition technique and wave aspects
- Author
-
Pietrzak, T., Horzela, A., and Górska, K.
- Published
- 2024
- Full Text
- View/download PDF
10. Non-Debye relaxations: smeared time evolution, memory effects, and the Laplace exponents
- Author
-
Górska, K., Horzela, A., and Pogány, T. K.
- Subjects
Condensed Matter - Statistical Mechanics - Abstract
The non-Debye, \textit{i.e.,} non-exponential, behavior characterizes a large plethora of dielectric relaxation phenomena. Attempts to find their theoretical explanation are dominated either by considerations rooted in the stochastic processes methodology or by the so-called \textsl{fractional dynamics} based on equations involving fractional derivatives which mimic the non-local time evolution and as such may be interpreted as describing memory effects. Using the recent results coming from the stochastic approach we link memory functions with the Laplace (characteristic) exponents of infinitely divisible probability distributions and show how to relate the latter with experimentally measurable spectral functions characterizing relaxation in the frequency domain. This enables us to incorporate phenomenological knowledge into the evolution laws. To illustrate our approach we consider the standard Havriliak-Negami and Jurlewicz-Weron-Stanislavsky models for which we derive well-defined evolution equations. Merging stochastic and fractional dynamics approaches sheds also new light on the analysis of relaxation phenomena which description needs going beyond using the single evolution pattern. We determine sufficient conditions under which such description is consistent with general requirements of our approach.
- Published
- 2021
- Full Text
- View/download PDF
11. The Volterra type equations related to the non-Debye relaxation
- Author
-
Górska, K. and Horzela, A.
- Subjects
Mathematical Physics - Abstract
We investigate a possibility to describe the non-Debye relaxation processes using the Volterra-type equations with kernels given by the Prabhakar functions with the upper parameter $\nu$ being negative. Proposed integro-differential equations mimic the fading memory effects and are explicitly solved using the umbral calculus and the Laplace transform methods. Both approaches lead to the same results valid for admissible domain of the parameters $\alpha$, $\mu$ and $\nu$ characterizing the Prabhakar function. For the special case $\alpha\in (0,1]$, $\mu=0$ and $\nu=-1$ we recover the Cole-Cole model, in general having a residual polarization. We also show that our scheme gives results equivalent to those obtained using the stochastic approach to relaxation phenomena merged with integral equations involving kernels given by the Prabhakar functions with the positive upper parameter.
- Published
- 2021
- Full Text
- View/download PDF
12. Non-Debye relaxations: two types of memories and their Stieltjes character
- Author
-
Górska, K. and Horzela, A.
- Subjects
Mathematical Physics - Abstract
We show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semiaxis. Using only this property it can be shown that the response and relaxation functions are nonnegative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function $M(t)$ which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes based approach to the relaxation phenomena gives possibility to identify the memory function $M(t)$ with the Laplace (L\'evy) exponent of some infinitely divisible stochastic process and to introduce its partner memory $k(t)$. Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process., Comment: This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory
- Published
- 2021
- Full Text
- View/download PDF
13. The generalized Cattaneo (telegrapher's) equation and corresponding random walks
- Author
-
Górska, K., Horzela, A., Lenzi, E. K., Pagnini, G., and Sandev, T.
- Subjects
Condensed Matter - Statistical Mechanics ,Mathematical Physics - Abstract
The various types of generalized Cattaneo, called also telegrapher's equation, are studied. We find conditions under which solutions of the equations considered so far can be recognized as probability distributions, \textit{i.e.} are normalizable and non-negative on their domains. Analysis of the relevant mean squared displacements enables us to classify diffusion processes described by such obtained solutions and to identify them with either ordinary or anomalous super- or subdiffusion. To complete our study we analyse derivations of just considered examples the generalized Cattaneo equations using the continuous time random walk and the persistent random walk approaches.
- Published
- 2020
- Full Text
- View/download PDF
14. On the Sheffer-type polynomials related to the Mittag-Leffler functions: applications to fractional evolution equations
- Author
-
Górska, K., Horzela, A., Penson, K. A., and Dattoli, G.
- Subjects
Mathematical Physics - Abstract
We present two types of polynomials related to the Mittag-Leffler function namely the fractional Hermite polynomial and the Mittag-Leffler polynomial. The first modifies the Hermite polynomial and the second one is a refashioned Laguerre polynomial. The fractional Hermite and the Mittag-Leffler polynomials are used to solve {the Cauchy problems for} the fractional Fokker-Planck equation where the fractional derivative is taken in the Caputo sense with respect to time and/or space. The generating functions of these two kinds of polynomials are also calculated and they indicate that these polynomials belong to the Sheffer type.
- Published
- 2019
15. Some results on the complete monotonicity of the Mittag-Leffler functions of Le Roy type
- Author
-
Górska, K., Horzela, A., and Garrappa, R.
- Subjects
Mathematics - Classical Analysis and ODEs - Abstract
The paper by R. Garrappa, S. Rogosin, and F. Mainardi, entitled {\em On a generalized three-parameter Wright function of the Le Roy type} and published in [Fract. Calc. Appl. Anal. {\bf 20} (2017) 1196-1215], ends up leaving the open question concerning the range of the parameters $\alpha, \beta$ and $\gamma$ for which Mittag-Leffler functions of Le Roy type $F_{\alpha, \beta}^{(\gamma)}$ are completely monotonic. Inspired by the 1948 seminal H. Pollard's paper which provides the proof of the complete monotonicity of the one parameter Mittag-Leffler function, the Pollard approach is used to find the Laplace transform representation of $F_{\alpha, \beta}^{(\gamma)}$ for integer $\gamma = n$ and rational $0 < \alpha \leq 1/n$. In this way it is possible to show that Mittag-Leffler functions of Le Roy type are completely monotone for $\alpha = 1/n$ and $\beta \geq (n+1)/(2n)$ as well as for rational $0 < \alpha \leq 1/2$, $\beta = 1$ and $n=2$. For further integer values of $n$ the complete monotonicity is tested numerically for rational $0< \alpha < 1/n$ and various choices of $\beta$. The obtained results suggest that for the complete monotonicity the condition $\beta \geq (n+1)/(2n)$ holds for any value of $n$.
- Published
- 2019
- Full Text
- View/download PDF
16. Volterra-Prabhakar function of distributed order and some applications
- Author
-
Górska, K., Pietrzak, T., Sandev, T., and Tomovski, Ž.
- Published
- 2023
- Full Text
- View/download PDF
17. Can Umbral and $q$-calculus be merged?
- Author
-
Dattoli, G., Germano, B., Górska, K., and Martinelli, M. R.
- Subjects
Mathematics - Classical Analysis and ODEs - Abstract
The $q$-calculus is reformulated in terms of the umbral calculus and of the associated operational formalism. We show that new and interesting elements emerge from such a restyling. The proposed technique is applied to a different formulations of $q$ special functions, to the derivation of integrals involving ordinary and $q$-functions and to the study of $q$-special functions and polynomials.
- Published
- 2019
18. A note on paper 'Anomalous relaxation model based on the fractional derivative with a Prabhakarlike kernel' [Z. Angew. Math. Phys. (2019) 70:42]
- Author
-
Górska, K., Horzela, A., and Pogány, T. K.
- Subjects
Mathematical Physics - Abstract
Inspired by the article "Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel" (Z. Angew. Math. Phys. (2019) 70:42) which authors D. Zhao and HG. Sun studied the integro-differential equation with the kernel given by the Prabhakar function $e^{-\gamma}_{\alpha, \beta}(t, \lambda)$ we provide the solution to this equation which is complementary to that obtained up to now. Our solution is valid for effective relaxation times which admissible range extends the limits given in \cite[Theorem 3.1]{DZhao2019} to all positive values. For special choices of parameters entering the equation itself and/or characterizing the kernel the solution comprises to known phenomenological relaxation patterns, e.g. to the Cole-Cole model (if $\gamma = 1, \beta=1-\alpha$) or to the standard Debye relaxation.
- Published
- 2019
- Full Text
- View/download PDF
19. On the complete monotonicity of the three parameter generalized Mittag-Leffler function $E_{\alpha, \beta}^{\gamma}(-x)$
- Author
-
Górska, K., Horzela, A., Lattanzi, A., and Pogány, T. K.
- Subjects
Mathematical Physics - Abstract
Using the Bernstein theorem we give a simple proof of the complete monotonicity of the three parameter generalized Mittag-Leffler function $E_{\alpha, \beta}^{\gamma}(-x)$ for $x \geq 0$ and suitably adjusted parameters $\alpha$, $\beta$ and $\gamma$
- Published
- 2018
- Full Text
- View/download PDF
20. Pharmacological Inhibition of Chitotriosidase (CHIT1) as a Novel Therapeutic Approach for Sarcoidosis
- Author
-
Dymek B, Sklepkiewicz P, Mlacki M, Güner NC, Nejman-Gryz P, Drzewicka K, Przysucha N, Rymaszewska A, Paplinska-Goryca M, Zagozdzon A, Proboszcz M, Krzemiński Ł, von der Thüsen JH, Górska K, Dzwonek K, Zasłona Z, Dobrzanski P, and Krenke R
- Subjects
chitinase ,oatd-01 ,granuloma ,macrophages ,interstitial lung disease ,Pathology ,RB1-214 ,Therapeutics. Pharmacology ,RM1-950 - Abstract
Barbara Dymek,1,2 Piotr Sklepkiewicz,1 Michal Mlacki,1 Nazan Cemre Güner,1 Patrycja Nejman-Gryz,3 Katarzyna Drzewicka,1 Natalia Przysucha,3 Aleksandra Rymaszewska,1 Magdalena Paplinska-Goryca,3 Agnieszka Zagozdzon,1 Małgorzata Proboszcz,3 Łukasz Krzemiński,1 Jan H von der Thüsen,4 Katarzyna Górska,3 Karolina Dzwonek,1 Zbigniew Zasłona,1 Pawel Dobrzanski,1 Rafał Krenke3 1Molecure SA, Warsaw, 02-089, Poland; 2Postgraduate School of Molecular Medicine, Medical University of Warsaw, Warsaw, 02-097, Poland; 3Department of Internal Medicine, Pulmonary Diseases and Allergy, Medical University of Warsaw, Warsaw, 02-097, Poland; 4Department of Pathology, Erasmus Medical Center, Rotterdam, 3015 GD, the NetherlandsCorrespondence: Barbara Dymek, Żwirki i Wigury 101, Warsaw, 02-089, Poland, Tel +48 22 552 67 24, Email b.dymek@molecure.comIntroduction: Sarcoidosis is a systemic disease of unknown etiology characterized by granuloma formation in the affected tissues. The pathologically activated macrophages are causatively implicated in disease pathogenesis and play important role in granuloma formation. Chitotriosidase (CHIT1), macrophage-derived protein, is upregulated in sarcoidosis and its levels correlate with disease severity implicating CHIT1 in pathology.Methods: CHIT1 was evaluated in serum and bronchial mucosa and mediastinal lymph nodes specimens from sarcoidosis patients. The therapeutic efficacy of OATD-01 was assessed ex vivo on human bronchoalveolar lavage fluid (BALF) macrophages and in vivo in the murine models of granulomatous inflammation.Results: CHIT1 activity was significantly upregulated in serum from sarcoidosis patients. CHIT1 expression was restricted to granulomas and localized in macrophages. Ex vivo OATD-01 inhibited pro-inflammatory mediators’ production (CCL4, IL-15) by lung macrophages. In the acute model of granulomatous inflammation in mice, OATD-01 showed anti-inflammatory effects reducing the percentage of neutrophils and CCL4 concentration in BALF. In the chronic model, inhibition of CHIT1 led to a decrease in the number of organized lung granulomas and the expression of sarcoidosis-associated genes.Conclusion: In summary, CHIT1 activity was increased in sarcoidosis patients and OATD-01, a first-in-class CHIT1 inhibitor, demonstrated efficacy in murine models of granulomatous inflammation providing a proof-of-concept for its clinical evaluation in sarcoidosis.Keywords: chitinase, OATD-01, granuloma, macrophages, interstitial lung disease
- Published
- 2022
21. Composition law for the Cole-Cole relaxation and ensuing evolution equations
- Author
-
Górska, K., Horzela, A., and Lattanzi, A.
- Subjects
Condensed Matter - Mesoscale and Nanoscale Physics ,Mathematical Physics - Abstract
Physically natural assumption says that the any relaxation process taking place in the time interval $[t_{0},t_{2}]$, $t_{2} > t_{0}\ge 0$ may be represented as a composition of processes taking place during time intervals $[t_{0}, t_{1}]$ and $[t_{1},t_{2}]$ where $t_{1}$ is an arbitrary instant of time such that $t_{0} \leq t_{1} \leq t_{2}$. For the Debye relaxation such a composition is realized by usual multiplication which claim is not valid any longer for more advanced models of relaxation processes. We investigate the composition law required to be satisfied by the Cole-Cole relaxation and find its explicit form given by an integro-differential relation playing the role of the time evolution equation. The latter leads to differential equations involving fractional derivatives, either of the Caputo or the Riemann-Liouville senses, which are equivalent to the special case of the fractional Fokker-Planck equation satisfied by the Mittag-Leffler function known to describe the Cole-Cole relaxation in the time domain.
- Published
- 2018
- Full Text
- View/download PDF
22. Coherence, squeezing and entanglement -- an example of peaceful coexistence
- Author
-
Górska, K., Horzela, A., and Szafraniec, F. H.
- Subjects
Mathematical Physics ,Mathematics - Complex Variables - Abstract
After exhaustive inspection of bosonic coherent states appearing in physical literature two of us, Horzela and Szafraniec, came in 2012 to the reasonably general definition which relies exclusively on reproducing kernels. The basic feature of coherent states, which is the resolution of the identity, is preserved though it now achieves advantageous form of the Segal-Bargmann transform. It turns out that the aforesaid definition is not only extremely economical but also puts under a common umbrella typical coherent states as well as those which are squeezed and entangled. We examine the case here on the groundwork of holomorphic Hermite polynomials in one and two variables. An interesting side of this story is how some limit procedure allows disentangling., Comment: Contribution to the Proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama", November 14-18th, 2016, Centre International de Rencontres Scientifiques, Luminy, Marseille, France
- Published
- 2017
- Full Text
- View/download PDF
23. Mittag-Leffler function and fractional differential equations
- Author
-
Górska, K., Lattanzi, A., and Dattoli, G.
- Subjects
Mathematical Physics ,Condensed Matter - Statistical Mechanics - Abstract
We adopt a procedure of operational-umbral type to solve the $(1+1)$-dimensional fractional Fokker-Planck equation in which time fractional derivative of order $\alpha$ ($0 < \alpha < 1$) is in the Riemann-Liouville sense. The technique we propose merges well documented operational methods to solve ordinary FP equation and a redefinition of the time by means of an umbral operator. We show that the proposed method allows significant progress including the handling of operator ordering.
- Published
- 2017
24. Holomorphic Hermite polynomials in two variables
- Author
-
Górska, K., Horzela, A., and Szafraniec, F. H.
- Subjects
Mathematics - Classical Analysis and ODEs - Abstract
Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$.
- Published
- 2017
25. The Havriliak-Negami relaxation and its relatives: the response, relaxation and probability density functions
- Author
-
Górska, K., Horzela, A., Bratek, Ł., Penson, K. A., and Dattoli, G.
- Subjects
Condensed Matter - Statistical Mechanics ,Condensed Matter - Materials Science - Abstract
We study functions related to the experimentally observed Havriliak-Negami dielectric relaxation pattern in the frequency domain $\sim[1+(i\omega\tau_{0})^{\alpha}]^{-\beta}$ with $\tau_{0}$ being some characteristic time. For $\alpha = l/k< 1$ ($l$ and $k$ positive integers) and $\beta > 0$ we furnish exact and explicit expressions for response and relaxation functions in the time domain and suitable probability densities in their "dual" domain. All these functions are expressed as finite sums of generalized hypergeometric functions, convenient to handle analytically and numerically. Introducing a reparameterization $\beta = (2-q)/(q-1)$ and $\tau_{0} = (q-1)^{1/\alpha}$ $(1 < q < 2)$ we show that for $0 < \alpha < 1$ the response functions $f_{\alpha, \beta}(t/\tau_{0})$ go to the one-sided L\'{e}vy stable distributions when $q$ tends to one. Moreover, applying the self-similarity property of the probability densities $g_{\alpha, \beta}(u)$, we introduce two-variable densities and show that they satisfy the integral form of the evolution equation.
- Published
- 2016
26. Relativistic Heat Equation via L\'{e}vy stable distributions: Exact Solutions
- Author
-
Penson, K. A., Górska, K., Horzela, A., and Dattoli, G.
- Subjects
Mathematical Physics - Abstract
We introduce and study an extension of the heat equation relevant to relativistic energy formula involving square root of differential operators. We furnish exact solutions of corresponding Cauchy (initial) problem using the operator formalism invoking one-sided L\'{e}vy stable distributions. We note a natural appearance of Bessel polynomials which allow one the obtention of closed form solutions for a number of initial conditions. The resulting relativistic diffusion is slower than the non-relativistic one, although it still can be termed a normal one. Its detailed statistical characterization is presented in terms of exact evaluation of arbitrary moments and is compared with the non-relativistic case.
- Published
- 2016
- Full Text
- View/download PDF
27. Theory of relativistic heat polynomials and one-sided L\'evy distributions
- Author
-
Dattoli, G., Górska, K., Horzela, A., Penson, K. A., and Sabia, E.
- Subjects
Mathematical Physics - Abstract
The theory of pseudo-differential operators is a powerful tool to deal with differential equations involving differential operators under the square root sign. These type of equations are pivotal elements to treat problems in anomalous diffusion and in relativistic quantum mechanics. In this paper we report on new and unsuspected links between fractional diffusion, quantum relativistic equations and particular families of polynomials, linked to the Carlitz family, and playing the role of relativistic heat polynomials. We introduce generalizations of these polynomial families and point out their specific use for the solutions of problems of practical importance., Comment: Typos corrected, references added
- Published
- 2016
- Full Text
- View/download PDF
28. The stretched exponential behavior and its underlying dynamics. The phenomenological approach
- Author
-
Górska, K., Horzela, A., Penson, K. A., Dattoli, G., and Duchamp, G. H. E.
- Subjects
Condensed Matter - Statistical Mechanics ,Mathematical Physics - Abstract
We show that the anomalous diffusion equations with a fractional derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the L\'evy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution operator is presented as integral transforms whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and different initial conditions., Comment: figures and references added
- Published
- 2016
29. Operational versus umbral methods and the Borel transform
- Author
-
Dattoli, G., Di Palma, E., Sabia, E., Górska, K., Horzela, A., and Penson, K. A.
- Subjects
Mathematics - Classical Analysis and ODEs - Abstract
Integro-differential methods, currently exploited in calculus, provide an inexhaustible source of tools to be applied to a wide class of problems, involving the theory of special functions and other subjects. The use of integral transforms of the Borel type and the associated formalism is shown to be a very effective mean, constituting a solid bridge between umbral and operational methods. We merge these different points of view to obtain new and efficient analytical techniques for the derivation of integrals of special functions and the summation of associated generating functions as well., Comment: Corrected version
- Published
- 2015
- Full Text
- View/download PDF
30. Relativistic Wave Equations: An Operational Approach
- Author
-
Dattoli, G., Sabia, E., Górska, K., Horzela, A., and Penson, K. A.
- Subjects
Mathematical Physics ,Quantum Physics - Abstract
The use of operator methods of algebraic nature is shown to be a very powerful tool to deal with different forms of relativistic wave equations. The methods provide either exact or approximate solutions for various forms of differential equations, such as relativistic Schr\"odinger, Klein-Gordon and Dirac. We discuss the free particle hypotheses and those relevant to particles subject to non-trivial potentials. In the latter case we will show how the proposed method leads to easily implementable numerical algorithms.
- Published
- 2015
- Full Text
- View/download PDF
31. Explicit representations for multiscale L\'evy processes, and asymptotics of multifractal conservation laws
- Author
-
Górska, K. and Woyczynski, W. A.
- Subjects
Mathematical Physics ,Mathematics - Probability - Abstract
Nonlinear conservation laws driven by L\'evy processes have solutions which, in the case of supercritical nonlinearities, have an asymptotic behavior dictated by the solutions of the linearized equations. Thus the explicit representation of the latter is of interest in the nonlinear theory. In this paper we concentrate on the case where the driving L\'evy process is a multiscale stable (anomalous) diffusion, which corresponds to the case of multifractal conservation laws considered in [1-4]. The explicit representations, building on the previous work on single-scale problems (see, e.g.,[5]), are developed in terms of the special functions (such as Meijer G functions), and are amenable to direct numerical evaluations of relevant probabilities.
- Published
- 2015
- Full Text
- View/download PDF
32. Effects of Osteopathic Manual Therapy on Hyperinflation in Patients with Chronic Obstructive Pulmonary Disease: A Randomized Cross-Over Study
- Author
-
Maskey-Warzechowska, M., Mierzejewski, M., Gorska, K., Golowicz, R., Jesien, L., Krenke, R., Crusio, Wim E., Series Editor, Lambris, John D., Series Editor, Rezaei, Nima, Series Editor, and Pokorski, Mieczyslaw, editor
- Published
- 2019
- Full Text
- View/download PDF
33. The Volterra type equations related to the non-Debye relaxation
- Author
-
Górska, K. and Horzela, A.
- Published
- 2020
- Full Text
- View/download PDF
34. On the properties of Laplace transform originating from one-sided L\'evy stable laws
- Author
-
Penson, K. A. and Górska, K.
- Subjects
Mathematical Physics ,Condensed Matter - Statistical Mechanics - Abstract
We consider the conventional Laplace transform of $f(x)$, denoted by $\mathcal{L}[f(x); p]~\equiv~F(p)=\int_{0}^{\infty} e^{-p x} f(x) dx$ with ${\rm \mathfrak{Re}}(p) > 0$. For $0 < \alpha < 1$ we furnish the closed form expressions for the inverse Laplace transforms $\mathcal{L}^{-1}[F(p^{\alpha}); x]$ and $\mathcal{L}^{-1}[p^{\alpha-1}F(p^{\alpha}); x]$. In both cases they involve definite integration with kernels which are appropriately rescaled one-sided L\'{e}vy stable probability distribution functions $g_{\alpha}(x)$, $0 < \alpha < 1$, $x > 0$. Since $g_{\alpha}(x)$ are exactly and explicitly known for rational $\alpha$, \textit{i.e.} for $\alpha = l/k$ with $l, k=1, 2, \ldots$, $l < k$, our results extend the known and tabulated case of $\alpha = 1/2$ to any rational $0 < \alpha < 1$. We examine the integral kernels of this procedure as well as the resulting two kinds of L\'{e}vy integral transformations.
- Published
- 2014
- Full Text
- View/download PDF
35. On the Laplace transform of the Fr\'{e}chet distribution
- Author
-
Penson, K. A. and Górska, K.
- Subjects
Mathematics - Probability ,Mathematics - Classical Analysis and ODEs - Abstract
We calculate exactly the Laplace transform of the Fr\'{e}chet distribution in the form $\gamma x^{-(1+\gamma)} \exp(-x^{-\gamma})$, $\gamma > 0$, $0 \leq x < \infty$, for arbitrary rational values of the shape parameter $\gamma$, i.e. for $\gamma = l/k$ with $l, k = 1,2, \ldots$. The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fr\'{e}chet distribution serves as a kernel of Fr\'{e}chet integral transform. It turns out that the Fr\'{e}chet transform of one-sided L\'{e}vy law reproduces the Fr\'{e}chet distribution., Comment: 10 pages, 4 figures; one reference added
- Published
- 2014
- Full Text
- View/download PDF
36. Squeezing: the ups and downs
- Author
-
Gorska, K., Horzela, A., and Szafraniec, F. H.
- Subjects
Mathematical Physics ,Mathematics - Functional Analysis ,Mathematics - Operator Algebras ,Mathematics - Spectral Theory - Abstract
We present an operator theoretic side of the story of squeezed states regardless the order of squeezing. For low order, that is for displacement (order 1) and squeeze (order 2) operators, we bring back to consciousness what is know or rather what has to be known by making the exposition as exhaustive as possible. For the order 2 (squeeze) we propose an interesting model of the Segal-Bargmann type. For higher order the impossibility of squeezing in the traditional sense is proved rigorously. Nevertheless what we offer is the state-of-the-art concerning the topic., Comment: 21 pages; improved presentation; it has been published by Proceedings of the Royal Society A
- Published
- 2014
- Full Text
- View/download PDF
37. Photoluminescence decay of silicon nanocrystals and L\'{e}vy stable distributions
- Author
-
Dattoli, G., Gorska, K., Horzela, A., and Penson, K. A.
- Subjects
Condensed Matter - Mesoscale and Nanoscale Physics - Abstract
Recent experiments have shown that photoluminescence decay of silicon nanocrystals can be described by the stretched exponential function. We show here that the associated decay probability rate is the one-sided Levy stable distribution which describes well the experimental data. The relevance of these conclusions to the underlying stochastic processes is discussed in terms of Levy processes., Comment: Improved presentation, references added
- Published
- 2013
- Full Text
- View/download PDF
38. Chitinases and Chitinase-Like Proteins in Obstructive Lung Diseases – Current Concepts and Potential Applications
- Author
-
Przysucha N, Górska K, and Krenke R
- Subjects
asthma ,copd ,ykl-40 ,chit1 ,chitotriosidase ,amcase ,Diseases of the respiratory system ,RC705-779 - Abstract
Natalia Przysucha, Katarzyna Górska, Rafal Krenke Department of Internal Medicine, Pulmonary Diseases and Allergy, Medical University of Warsaw, Warsaw, PolandCorrespondence: Katarzyna GórskaDepartment of Internal Medicine, Pulmonary Diseases and Allergy, Medical University of Warsaw, Banacha 1A, Warsaw 02-097 Email drkpgorska@gmail.comAbstract: Chitinases, enzymes that cleave chitin’s chain to low molecular weight chitooligomers, are widely distributed in nature. Mammalian chitinases belong to the 18-glycosyl-hydrolase family and can be divided into two groups: true chitinases with enzymatic activity (AMCase and chitotriosidase) and chitinase-like proteins (CLPs) molecules which can bind to chitin or chitooligosaccharides but lack enzymatic activity (eg, YKL-40). Chitinases are thought to be part of an innate immunity against chitin-containing parasites and fungal infections. Both groups of these hydrolases are lately evaluated also as chemical mediators or biomarkers involved in airway inflammation and fibrosis. The aim of this article is to present the current knowledge on the potential role of human chitinases and CLPs in the pathogenesis, diagnosis, and course of obstructive lung diseases. We also assessed the potential role of chitinase and CLPs inhibitors as therapeutic targets in chronic obstructive pulmonary disease and asthma.Keywords: asthma, COPD, YKL-40, CHIT1, chitotriosidase, AMCase
- Published
- 2020
39. Squeezed States and Hermite polynomials in a Complex Variable
- Author
-
Ali, S. T., Gorska, K., Horzela, A., and Szafraniec, F. H.
- Subjects
Quantum Physics ,Mathematical Physics - Abstract
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pur. Appl. Math. 14, 187 (1961)]., Comment: 15 pages
- Published
- 2013
- Full Text
- View/download PDF
40. The Higher-Order Heat-Type Equations via signed L\'{e}vy stable and generalized Airy functions
- Author
-
Gorska, K., Horzela, A., Penson, K. A., and Dattoli, G.
- Subjects
Condensed Matter - Statistical Mechanics ,Mathematics - Analysis of PDEs - Abstract
We study the higher-order heat-type equation with first time and M-th spatial partial derivatives, M = 2, 3, ... . We demonstrate that its exact solutions for M even can be constructed with the help of signed Levy stable functions. For M odd the same role is played by a special generalization of Airy Ai function that we introduce and study. This permits one to generate the exact and explicit heat kernels pertaining to these equations. We examine analytically and graphically the spacial and temporary evolution of particular solutions for simple initial conditions., Comment: 11 pages, 6 figures; several typos corrected
- Published
- 2013
- Full Text
- View/download PDF
41. Multidimensional Catalan and related numbers as Hausdorff moments
- Author
-
Gorska, K. and Penson, K. A.
- Subjects
Mathematics - Combinatorics - Abstract
We study integral representation of so-called $d$-dimensional Catalan numbers $C_{d}(n)$, defined by $[\prod_{p=0}^{d-1} \frac{p!}{(n+p)!}] (d n)!$, $d = 2, 3, ...$, $n=0, 1, ...$. We prove that the $C_{d}(n)$'s are the $n$th Hausdorff power moments of positive functions $W_{d}(x)$ defined on $x\in[0, d^d]$. We construct exact and explicit forms of $W_{d}(x)$ and demonstrate that they can be expressed as combinations of $d-1$ hypergeometric functions of type $_{d-1}F_{d-2}$ of argument $x/d^d$. These solutions are unique. We analyse them analytically and graphically. A combinatorially relevant, specific extension of $C_{d}(n)$ for $d$ even in the form $D_{d}(n)=[\prod_{p = 0}^{d-1} \frac{p!}{(n+p)!}] [\prod_{q = 0}^{d/2 - 1} \frac{(2 n + 2 q)!}{(2 q)!}]$ is analyzed along the same lines., Comment: comments added, two new references added
- Published
- 2013
42. Lacunary Generating Functions for the Laguerre Polynomials
- Author
-
Babusci, D., Dattoli, G., Gorska, K., and Penson, K. A.
- Subjects
Mathematical Physics - Abstract
Symbolic methods of umbral nature play an important and increasing role in the theory of special functions and in related fields like combinatorics. We discuss an application of these methods to the theory of lacunary generating functions for the Laguerre polynomials for which we give a number of new closed form expressions. We present furthermore the different possibilities offered by the method we have developed, with particular emphasis on their link to a new family of special functions and with previous formulations, associated with the theory of quasi monomials.
- Published
- 2013
43. The spherical Bessel and Struve functions and operational methods
- Author
-
Babusci, D., Dattoli, G., Gorska, K., and Penson, K. A.
- Subjects
Mathematics - Classical Analysis and ODEs - Abstract
We review some aspects of the theory of spherical Bessel functions and Struve functions by means of an operational procedure essentially of umbral nature, capable of providing the straightforward evaluation of their definite integrals and of successive derivatives. The method we propose allows indeed the formal reduction of these family of functions to elementary ones of Gaussian type. We study the problem in general terms and present a formalism capable of providing a unifying point of view including Anger and Weber functions too. The link to the multi-index Bessel functions is also briefly discussed.
- Published
- 2012
44. Exact and explicit evaluation of Brezin-Hikami kernels
- Author
-
Gorska, K. and Penson, K. A.
- Subjects
Mathematical Physics ,Condensed Matter - Statistical Mechanics - Abstract
We present exact and explicit form of the kernels $hat{K}(x, y)$ appearing in the theory of energy correlations in the ensembles of Hermitian random matrices with Gaussian probability distribution, see E. Brezin and S. Hikami, Phys. Rev. E 57, 4140 and E 58, 7176 (1998). In obtaining this result we have exploited the analogy with the method of producing exact forms of two-sided, symmetric Levy stable laws, presented by us recently. This result is valid for arbitrary values of parameters in question. We furnish analytical and graphical representations of physical quantities calculated from $hat{K}(x, y)$'s., Comment: Improved presentation: two new figures added, two new appendices added
- Published
- 2012
- Full Text
- View/download PDF
45. Generating Functions for Laguerre Polynomials: New Identities for Lacunary Series
- Author
-
Babusci, D., Dattoli, G., Gorska, K., and Penson, K. A.
- Subjects
Mathematical Physics - Abstract
We present a number of identities involving standard and associated Laguerre polynomials. They include double-, and triple-lacunary, ordinary and exponential generating functions of certain classes of Laguerre polynomials., Comment: The list a number of identities satisfied by standard Laguerre polynomials (3 pages)
- Published
- 2012
46. Holomorphic Hermite polynomials in two variables
- Author
-
Górska, K., Horzela, A., and Szafraniec, F.H.
- Published
- 2019
- Full Text
- View/download PDF
47. Symbolic methods for the evaluation of sum rules of Bessel functions
- Author
-
Babusci, D., Dattoli, G., Gorska, K., and Penson, K. A.
- Subjects
Mathematical Physics ,Mathematics - Classical Analysis and ODEs - Abstract
The use of the umbral formalism allows a significant simplification of the derivation of sum rules involving products of special functions and polynomials. We rederive in this way known sum rules and addition theorems for Bessel functions. Furthermore, we obtain a set of new closed form sum rules involving various special polynomials and Bessel functions. The examples we consider are relevant for applications ranging from plasma physics to quantum optics., Comment: J. Math. Phys. vol. 54, 073501 (2013), 6 pages
- Published
- 2012
- Full Text
- View/download PDF
48. On Mittag-Leffler function and associated polynomials
- Author
-
Babusci, D., Dattoli, G., and Górska, K.
- Subjects
Mathematical Physics - Abstract
The Mittag-Leffler function plays a role of central importance in the theory of fractional derivatives. In this brief note we discuss the properties of this function and its connection with the Wright-Bessel functions and with a new family of associated heat polynomials., Comment: 6 pages, no figures
- Published
- 2012
49. Symbolic calculus and integrals of Laguerre polynomials
- Author
-
Babusci, D., Dattoli, G., and Górska, K.
- Subjects
Mathematics - Classical Analysis and ODEs ,Mathematical Physics ,33C45 - Abstract
An umbral type formalism is used to derive integrals involving products of Laguerre polynomials and other special functions., Comment: 6 pages
- Published
- 2012
50. Levy stable distributions via associated integral transform
- Author
-
Gorska, K. and Penson, K. A.
- Subjects
Mathematical Physics ,Condensed Matter - Statistical Mechanics - Abstract
We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Levy stable probability distributions g_{\alpha}(x), 0 \leq x < \infty, 0 < \alpha < 1. We demonstrate that the knowledge of one such a distribution g_{\alpha}(x) suffices to obtain exactly g_{\alpha^{p}}(x), p=2, 3,... Similarly, from known g_{\alpha}(x) and g_{\beta}(x), 0 < \alpha, \beta < 1, we obtain g_{\alpha \beta}(x). The method is based on the construction of the integral operator, called Levy transform, which implements the above operations. For \alpha rational, \alpha = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for g_{l/k}(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration., Comment: 12 pages, typos removed, references added
- Published
- 2012
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.