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Journal articles on the topic 'Q-space'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 February 2022

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1

Samuel, Christian. "On spaces of operators on $C(Q)$ spaces ($Q$ countable metric space)." Proceedings of the American Mathematical Society 137, no. 03 (September 11, 2008): 965–70. http://dx.doi.org/10.1090/s0002-9939-08-09635-4.

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2

Dipper, Richard, and Gordon James. "q-Tensor Space and q-Weyl Modules." Transactions of the American Mathematical Society 327, no. 1 (September 1991): 251. http://dx.doi.org/10.2307/2001842.

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3

Dipper, Richard, and Gordon James. "$q$-tensor space and $q$-Weyl modules." Transactions of the American Mathematical Society 327, no. 1 (January 1, 1991): 251–82. http://dx.doi.org/10.1090/s0002-9947-1991-1012527-1.

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4

Bakhyt, A., and N. T. Tleukhanova. "The problem of trigonometric Fourier series multipliers of classes in λp,q spaces." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 100, no. 4 (December 30, 2020): 17–25. http://dx.doi.org/10.31489/2020m4/17-25.

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In this article, we consider weighted spaces of numerical sequences λp,q, which are defined as sets of sequences a = {ak}^∞_k=1, for which the norm ||a||λp,q :=\sum^∞_k=1|ak|^q k^(q/p −1)^1/q<∞ is finite. In the case of non-increasing sequences, the norm of the space λp,q coincides with the norm of the classical Lorentz space lp,q. Necessary and sufficient conditions are obtained for embeddings of the space λp,q into the space λp1,q1. The interpolation properties of these spaces with respect to the real interpolation method are studied. It is shown that the scale of spaces λp,q is closed in the relative real interpolation method, as well as in relative to the complex interpolation method. A description of the dual space to the weighted space λp,q is obtained. Specifically, it is shown that the space is reflective, where p', q' are conjugate to the parameters p and q. The paper also studies the properties of the convolution operator in these spaces. The main result of this work is an O’Neil type inequality. The resulting inequality generalizes the classical Young-O’Neil inequality. The research methods are based on the interpolation theorems proved in this paper for the spaces λp,q.

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5

Ferahtia, N., and S. E. Allaoui. "A generalization of a localization property of Besov spaces." Carpathian Mathematical Publications 10, no. 1 (July 3, 2018): 71–78. http://dx.doi.org/10.15330/cmp.10.1.71-78.

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The notion of a localization property of Besov spaces is introduced by G. Bourdaud, where he has provided that the Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$, with $s\in\mathbb{R}$ and $p,q\in[1,+\infty]$ such that $p\neq q$, are not localizable in the $\ell^{p}$ norm. Further, he has provided that the Besov spaces $B^{s}_{p,q}$ are embedded into localized Besov spaces $(B^{s}_{p,q})_{\ell^{p}}$ (i.e., $B^{s}_{p,q}\hookrightarrow(B^{s}_{p,q})_{\ell^{p}},$ for $p\geq q$). Also, he has provided that the localized Besov spaces $(B^{s}_{p,q})_{\ell^{p}}$ are embedded into the Besov spaces $B^{s}_{p,q}$ (i.e., $(B^{s}_{p,q})_{\ell^{p}}\hookrightarrow B^{s}_{p,q},$ for $p\leq q$). In particular, $B_{p,p}^{s}$ is localizable in the $\ell^{p}$ norm, where $\ell^{p}$ is the space of sequences $(a_{k})_{k}$ such that $\|(a_{k})\|_{\ell^{p}}<\infty$. In this paper, we generalize the Bourdaud theorem of a localization property of Besov spaces $B^{s}_{p,q}(\mathbb{R}^{n})$ on the $\ell^{r}$ space, where $r\in[1,+\infty]$. More precisely, we show that any Besov space $B^{s}_{p,q}$ is embedded into the localized Besov space $(B^{s}_{p,q})_{\ell^{r}}$ (i.e., $B^{s}_{p,q}\hookrightarrow(B^{s}_{p,q})_{\ell^{r}},$ for $r\geq\max(p,q)$). Also we show that any localized Besov space $(B^{s}_{p,q})_{\ell^{r}}$ is embedded into the Besov space $B^{s}_{p,q}$ (i.e., $(B^{s}_{p,q})_{\ell^{r}}\hookrightarrow B^{s}_{p,q},$ for $r\leq\min(p,q)$). Finally, we show that the Lizorkin-Triebel spaces $F^{s}_{p,q}(\mathbb{R}^{n})$, where $s\in\mathbb{R}$ and $p\in[1,+\infty)$ and $q\in[1,+\infty]$ are localizable in the $\ell^{p}$ norm (i.e., $F^{s}_{p,q}=(F^{s}_{p,q})_{\ell^{p}}$).

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6

Yaying, Taja, Bipan Hazarika, and Mohammad Mursaleen. "On Generalized p , q -Euler Matrix and Associated Sequence Spaces." Journal of Function Spaces 2021 (June 19, 2021): 1–14. http://dx.doi.org/10.1155/2021/8899960.

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In this study, we introduce new BK -spaces b s r , t p , q and b ∞ r , t p , q derived by the domain of p , q -analogue B r , t p , q of the binomial matrix in the spaces ℓ s and ℓ ∞ , respectively. We study certain topological properties and inclusion relations of these spaces. We obtain a basis for the space b s r , t p , q and obtain Köthe-Toeplitz duals of the spaces b s r , t p , q and b ∞ r , t p , q . We characterize certain classes of matrix mappings from the spaces b s r , t p , q and b ∞ r , t p , q to space μ ∈ ℓ ∞ , c , c 0 , ℓ 1 , b s , c s , c s 0 . Finally, we investigate certain geometric properties of the space b s r , t p , q .

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7

Othman, Hakeem A. "q-SPACE (q-TIME)-DEFORMATION OF THE CONTINUITY EQUATION." International Journal of Research -GRANTHAALAYAH 9, no. 8 (August 31, 2021): 185–92. http://dx.doi.org/10.29121/granthaalayah.v9.i8.2021.4177.

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A q-space (q-time)-deformation of the continuity equations are introduced using the q-derivative (or Jackson derivative). By quantum calculus, we solve such equations. The free cases are discussed separately.

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8

Feng, Xiaogao, Shengjin Huo, and Shuan Tang. "Universal Teichmüller spaces and F(p,q,s) space." Annales Academiae Scientiarum Fennicae Mathematica 42 (February 2017): 105–18. http://dx.doi.org/10.5186/aasfm.2017.4208.

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9

Furqan, Salman, Hüseyin Işık, and Naeem Saleem. "Fuzzy Triple Controlled Metric Spaces and Related Fixed Point Results." Journal of Function Spaces 2021 (May 20, 2021): 1–8. http://dx.doi.org/10.1155/2021/9936992.

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In this study, we introduce fuzzy triple controlled metric space that generalizes certain fuzzy metric spaces, like fuzzy rectangular metric space, fuzzy rectangular b -metric space, fuzzy b -metric space, and extended fuzzy b -metric space. We use f , g , h , three noncomparable functions as follows: m q μ , η , t + s + w ≥ m q μ , ν , t / f μ , ν ∗ m q ν , ξ , s / g ν , ξ ∗ m q ξ , η , w / h ξ , η . We prove Banach fixed point theorem in the settings of fuzzy triple controlled metric space that generalizes Banach fixed point theorem for aforementioned spaces. An example is presented to support our main results. We also apply our technique to the uniqueness for the solution of an integral equation.

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10

Markoski, Gjorgji, and Abdulla Buklla. "On product of spaces of quasicomponents." Filomat 31, no. 20 (2017): 6307–11. http://dx.doi.org/10.2298/fil1720307m.

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We use a characterization of quasicomponents by continuous functions to obtain the well known theorem which states that product of quasicomponents Qx,Qy of topological spaces X,Y, respectively, gives quasicomponent in the product space X x Y. If spaces X,Y are locally-compact, paracompact and Haussdorf, then we prove that the space of quasicomponents of the product Q(XxY) is homeomorphic with the product space Q(X) x Q(Y), so these two spaces have the same topological properties.

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11

Zhang, Yujing, and Kaiyun Wang. "Bounded sobriety and k-bounded sobriety of Q-cotopological spaces." Filomat 33, no. 7 (2019): 2095–106. http://dx.doi.org/10.2298/fil1907095z.

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In this paper, we extend bounded sobriety and k-bounded sobriety to the setting of Q-cotopological spaces, whereQis a commutative and integral quantale. The main results are: (1) The category BSobQ-CTop of all bounded sober Q-cotopological spaces is a full reflective subcategory of the category SQ-CTop of all stratified Q-cotopological spaces; (2) We present the relationships among Hausdorff, T1, sobriety, bounded sobriety and k-bounded sobriety in the setting ofQ-cotopological spaces; (3) For a linearly ordered quantale Q, a topological space X is bounded (resp., k-bounded) sober if and only if the corresponding Q-cotopological space ?Q(X) is bounded (resp., k-bounded) sober, where ?Q : Top ? SQ-CTop is the well-known Lowen functor in fuzzy topology.

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12

Castillo, Jesús M. F., Manuel Gonzáles, Anatolij M. Plichko, and David Yost. "Twisted properties of banach spaces." MATHEMATICA SCANDINAVICA 89, no. 2 (December 1, 2001): 217. http://dx.doi.org/10.7146/math.scand.a-14339.

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If $\mathcal P$, $\mathcal Q$ are two linear topological properties, say that a Banach space $X$ has the property $\mathcal P$-by-$\mathcal Q$ (or is a $\mathcal P$-by-$\mathcal Q$ space) if $X$ has a subspace $Y$ with property $\mathcal P$ such that the corresponding quotient $X/Y$ has property $\mathcal Q$. The choices $\mathcal P,\mathcal Q \in\{\hbox{separable, reflexive}\}$ lead naturally to some new results and new proofs of old results concerning weakly compactly generated Banach spaces. For example, every extension of a subspace of $L_1(0,1)$ by a WCG space is WCG. They also give a simple new example of a Banach space property which is not a 3-space property but whose dual is a 3-space property.

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13

Sharma, Sanjay, Drema Lhamu, and Sunil Kumar Singh. "Characterization of weighted function spaces in terms of wavelet transforms." Boletim da Sociedade Paranaense de Matemática 37, no. 4 (January 9, 2018): 69–82. http://dx.doi.org/10.5269/bspm.v37i4.36226.

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In this paper, we have characterized a weighted function space $ B_{\omega,\psi}^{p,q}, ~ 1\leq p,q<\infty$ in terms of wavelet transform and shown that the norms on the spaces $B_{\omega,\psi}^{p,q}$ and $\bigwedge_\omega^{p,q}$ (the space defined in terms of differences $\triangle_x$) are equivalent.

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14

Cerchiai, B. L., and J. Wess. "$q$ -deformed Minkowski space based on a $q$ -Lorentz algebra." European Physical Journal C 5, no. 3 (September 1998): 553–66. http://dx.doi.org/10.1007/s100529800868.

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15

Öztop, S. "Multipliers on some weightedLp-spaces." International Journal of Mathematics and Mathematical Sciences 23, no. 9 (2000): 651–56. http://dx.doi.org/10.1155/s016117120000096x.

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LetGbe a locally compact abelian group with Haar measuredx, and letωbe a symmetric Beurling weight function onG(Reiter, 1968). In this paper, using the relations betweenpiandqi, where1<pi, qi<∞,pi≠qi(i=1,2), we show that the space of multipliers fromLωp(G)to the spaceS(q′1,q′2,ω−1), the space of multipliers fromLωp1(G)∩Lωp2(G)toLωq(G)and the space of multipliersLωp1(G)∩Lωp2(G)toS(q′1,q′2,ω−1).

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16

Poria, Anirudha, and Jitendriya Swain. "Hilbert space valued Gabor frames in weighted amalgam spaces." Advances in Pure and Applied Mathematics 10, no. 4 (October 1, 2019): 377–94. http://dx.doi.org/10.1515/apam-2018-0067.

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AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

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17

WACHTER, HARTMUT. "ANALYSIS ON q-DEFORMED QUANTUM SPACES." International Journal of Modern Physics A 22, no. 01 (January 10, 2007): 95–164. http://dx.doi.org/10.1142/s0217751x07034155.

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A q-deformed version of classical analysis is given to quantum spaces of physical importance, i.e. Manin plane, q-deformed Euclidean space in three or four dimensions, and q-deformed Minkowski space. The subject is presented in a rather complete and self-contained way. All relevant notions are introduced and explained in detail. The different possibilities to realize the objects of q-deformed analysis are discussed and their elementary properties are studied. In this manner attention is focused on star products, q-deformed tensor products, q-deformed translations, q-deformed partial derivatives, dual pairings, q-deformed exponentials, and q-deformed integration. The main concern of this work is to show that these objects fit together in a consistent framework, which is suitable to formulate physical theories on quantum spaces.

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18

Riahi, Anis, Amine Ettaieb, Wathek Chammam, and Ziyad Ali Alhussain. "An Analytic Characterization of p , q -White Noise Functionals." Journal of Mathematics 2020 (December 8, 2020): 1–8. http://dx.doi.org/10.1155/2020/6319138.

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In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

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19

Dobrev, V. K. "New q-minkowski space-time and q-Maxwell equations hierarchy from q-conformal invariance." Physics Letters B 341, no. 2 (December 1994): 133–38. http://dx.doi.org/10.1016/0370-2693(94)90301-8.

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20

Merali, Zeeya. "Q&A: Space-time visionary." Nature 515, no. 7526 (November 2014): 196–97. http://dx.doi.org/10.1038/515196a.

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21

Hoffman, Jascha. "Q&A: The space poet." Nature 479, no. 7374 (November 2011): 477. http://dx.doi.org/10.1038/479477a.

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22

King, Martin D., John Houseman, Simon A. Roussel, Nicholas Van Bruggen, Stephen R. Williams, and David G. Gadian. "q-Space imaging of the brain." Magnetic Resonance in Medicine 32, no. 6 (December 1994): 707–13. http://dx.doi.org/10.1002/mrm.1910320605.

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23

Baete, Steven H., Stephen Yutzy, and Fernando E. Boada. "Radial q-space sampling for DSI." Magnetic Resonance in Medicine 76, no. 3 (September 12, 2015): 769–80. http://dx.doi.org/10.1002/mrm.25917.

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24

Cowen, Ron. "Q&A: The space crusader." Nature 509, no. 7502 (May 2014): 562. http://dx.doi.org/10.1038/509562a.

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25

Shamsi, M., M. Ardakani, and I. F. Blake. "Mixed-Q linear space-time codes." IEEE Transactions on Communications 54, no. 5 (May 2006): 849–57. http://dx.doi.org/10.1109/tcomm.2006.873990.

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26

Alavi, S. A. "One-Dimensional Potentials in q Space." Chinese Physics Letters 20, no. 1 (December 9, 2002): 8–11. http://dx.doi.org/10.1088/0256-307x/20/1/303.

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27

Hoffman, Jascha. "Q&A: The space entrepreneur." Nature 461, no. 7266 (October 2009): 885. http://dx.doi.org/10.1038/461885a.

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28

Wu, Yan, and Yi Qi. "Some notes on Q,0 space." Journal of Mathematical Analysis and Applications 506, no. 1 (February 2022): 125498. http://dx.doi.org/10.1016/j.jmaa.2021.125498.

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29

Wang, Heping, and Xuebo Zhai. "Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 05 (September 2014): 1461012. http://dx.doi.org/10.1142/s0219691314610128.

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In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the Lq space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the Lq space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the Lq space for 1 ≤ q < ∞. Also, in the Lq space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

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30

MA, Congbian, and Guoxi Zhao. "An Interpolation Theorem for Quasimartingales in Noncommutative Symmetric Spaces." Advances in Mathematical Physics 2021 (January 30, 2021): 1–5. http://dx.doi.org/10.1155/2021/6678150.

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Let E be a separable symmetric space on 0 , ∞ and E M the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E M and obtain the following interpolation result: let E ^ M be the space of all bounded E M -quasimartingales and 1 < p < p E < q E < q < ∞ . Then, there exists a symmetric space F on 0 , ∞ with nontrivial Boyd indices such that E ^ M = L ^ p M , L ^ q M F , K .

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31

Dutriaux, Antoine, and Dimitri Gurevich. "The Maxwell operator on q-Minkowski space and q-hyperboloid algebras." Journal of Physics A: Mathematical and Theoretical 41, no. 31 (June 30, 2008): 315203. http://dx.doi.org/10.1088/1751-8113/41/31/315203.

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32

Li, Songxiao. "On an integral-type operator from the Bloch space into the QK(p,q) space." Filomat 26, no. 2 (2012): 331–39. http://dx.doi.org/10.2298/fil1202331l.

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Let n be a positive integer, 1 ? H(D) and ? be an analytic self-map of D. The boundedness and compactness of the integral operator (Cn ?,1 f )(z) = ?z 0 f (n)(?(?))1(?)d? from the Bloch and little Bloch space into the spaces QK(p, q) and QK,0(p, q) are characterized.

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33

Martínez, Ángel D., and Daniel Spector. "An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces." Advances in Nonlinear Analysis 10, no. 1 (December 20, 2020): 877–94. http://dx.doi.org/10.1515/anona-2020-0157.

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Abstract It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality $$\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}$$ for all $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any $\beta \in (0,N], \; {\text{where}} \; \Omega \subset \mathbb{R}^N, \mathcal{H}^{\beta}_{\infty}$ is the Hausdorff content, LN/α,q(Ω) is a Lorentz space with q ∈ (1,∞], q' = q/(q − 1) is the Hölder conjugate to q, and Iαf denotes the Riesz potential of f of order α ∈ (0, N).

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34

Jevtic, Miroljub, and Miroslav Pavlovic. "On the solid hull of the Hardy-Lorentz space." Publications de l'Institut Math?matique (Belgrade) 85, no. 99 (2009): 55–61. http://dx.doi.org/10.2298/pim0999055j.

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35

Saleem, Naeem, Hüseyin Işık, Salman Furqan, and Choonkil Park. "Fuzzy double controlled metric spaces and related results." Journal of Intelligent & Fuzzy Systems 40, no. 5 (April 22, 2021): 9977–85. http://dx.doi.org/10.3233/jifs-202594.

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In this paper, we introduce the concept of fuzzy double controlled metric space that can be regarded as the generalization of fuzzy b-metric space, extended fuzzy b-metric space and controlled fuzzy metric space. We use two non-comparable functions α and β in the triangular inequality as: M q ( x , z , t α ( x , y ) + s β ( y , z ) ) ≥ M q ( x , y , t ) ∗ M q ( y , z , s ) . We prove Banach contraction principle in fuzzy double controlled metric space and generalize the Banach contraction principle in aforementioned spaces. We give some examples to support our main results. An application to existence and uniqueness of solution for an integral equation is also presented in this work.

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36

El-Sayed Ahmed, A., and M. A. Bakhit. "Composition operators on some holomorphic Banach function spaces." MATHEMATICA SCANDINAVICA 104, no. 2 (June 1, 2009): 275. http://dx.doi.org/10.7146/math.scand.a-15098.

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In this paper, we study composition operators on some Möbius invariant Banach function spaces like Bloch and $F(p,q,s)$ spaces. We give a Carleson measure characterization on $F(p,q,s)$ spaces, then we use this Carleson measure characterization of the compact compositions on $F(p,q,s)$ spaces to show that every compact composition operator on $F(p,q,s)$ spaces is compact on a Bloch space. Also, we give conditions to clarify when the converse holds.

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37

Wulan, Hasi, and Jizhen Zhou. "QKtype spaces of analytic functions." Journal of Function Spaces and Applications 4, no. 1 (2006): 73–84. http://dx.doi.org/10.1155/2006/910813.

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For a nondecreasing functionK:[0,8)?[0,8)and0<p<8,-2<q<8, we introduceQK(p,q), aQKtype space of functions analytic in the unit disk and study the characterizations ofQK(p,q). Necessary and sufficient conditions onKsuch thatQK(p,q)become some known spaces are given.

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38

Jevtic, Miroljub. "Multipliers." Filomat 27, no. 7 (2013): 1277–83. http://dx.doi.org/10.2298/fil1307277j.

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We describe the multiplier spaces (Hp,q,?,H?), and (Hp,q,?,H?,v,?), where Hp,q,? are mixed norm spaces of analytic functions in the unit disk D and H? is the space of bounded analytic functions in D. We extend some results from [7] and [3], particularly Theorem 4.3 in [3].

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39

Wess, Julius. "q-Deformed Phase Space and its Lattice Structure." International Journal of Modern Physics A 12, no. 28 (November 10, 1997): 4997–5005. http://dx.doi.org/10.1142/s0217751x97002656.

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Quantum groups lead to an algebraic structure that can be realized on quantum spaces. These are noncommutative spaces that inherit a well-defined mathematical structure from the quantum group symmetry. In turn such quantum spaces can be interpreted as noncommutative configuration spaces for physical systems. We study the noncommutative Euclidean space that is based on the quantum group SO q(3).

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40

Blasco, Oscar. "Multipliers on Spaces of Analytic Functions." Canadian Journal of Mathematics 47, no. 1 (February 1, 1995): 44–64. http://dx.doi.org/10.4153/cjm-1995-003-5.

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AbstractIn the paper we find, for certain values of the parameters, the spaces of multipliers (H(p, q, α), H(s, t, β) and (H(p, q, α), ls), where H(p, q, α) denotes the space of analytic functions on the unit disc such that . As corollaries we recover some new results about multipliers on Bergman spaces and Hardy spaces.

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41

Zhu, Xiangling. "Composition operators and closures of $\mathcal{Q}_K(p,q)$-type spaces in the Logarithmic Bloch space." Bulletin of the Belgian Mathematical Society - Simon Stevin 27, no. 1 (May 2020): 49–60. http://dx.doi.org/10.36045/bbms/1590199303.

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42

Canales-Rodríguez, Erick Jorge, Lester Melie-García, and Yasser Iturria-Medina. "Mathematical description of q-space in spherical coordinates: Exact q-ball imaging." Magnetic Resonance in Medicine 61, no. 6 (June 2009): 1350–67. http://dx.doi.org/10.1002/mrm.21917.

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43

Guliyev, Vagif S., and Yagub Y. Mammadov. "Riesz potential on the Heisenberg group and modified Morrey spaces." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 1 (May 1, 2012): 189–212. http://dx.doi.org/10.2478/v10309-012-0013-8.

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Abstract In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).

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Murugesan, Chinnswamy, and Nagarajan Subramanian. "Properties of $\Gamma^{2}$ defined by a modulus function." Boletim da Sociedade Paranaense de Matemática 31, no. 1 (January 25, 2012): 193. http://dx.doi.org/10.5269/bspm.v31i1.15264.

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In this article, we introduces the generalized difference paranormed double sequence spaces $\Gamma^{2}\left(\Delta^{m}_{\gamma},f,p,q,s\right)$ and $\Lambda^{2} \left(\Delta^{m}_{\gamma},f,p,q,s\right)$ defined over a seminormed sequence space $\left(X,q\right)$

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ATAKISHIYEV, N. M., and A. U. KLIMYK. "DISCRETE COORDINATE REALIZATIONS OF THE q-OSCILLATOR WHEN q>1." Modern Physics Letters A 21, no. 29 (September 21, 2006): 2205–16. http://dx.doi.org/10.1142/s0217732306021578.

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We elaborate on the Macfarlane–Biedenharn q-oscillator when q>1. In this case the position operator Q = a†+a and the momentum operator P = i(a†-a) are symmetric, but not self-adjoint. For this reason, one cannot specify spectra of Q and P. Since these operators have one-parameter families of self-adjoint extensions with different spectra, the common definition of such q-oscillator is not complete. We derive an action of Q and P (as well as of the related Hamiltonian) upon functions given on the corresponding coordinate spaces, on which Q and P are self-adjoint operators. To each self-adjoint extension of Q there corresponds an appropriate coordinate space (a spectrum of this self-adjoint extension). Thus, for every fixed q>1 one obtains a one-parameter family of non-equivalent q-oscillators in their coordinate spaces.

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Pasha Hosseinbor, A., Moo K. Chung, Yu-Chien Wu, Barbara B. Bendlin, and Andrew L. Alexander. "A 4D hyperspherical interpretation of q-space." Medical Image Analysis 21, no. 1 (April 2015): 15–28. http://dx.doi.org/10.1016/j.media.2014.11.013.

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Sorensen, C. M. "Q-space analysis of scattering by dusts." Journal of Quantitative Spectroscopy and Radiative Transfer 115 (January 2013): 93–95. http://dx.doi.org/10.1016/j.jqsrt.2012.09.001.

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Cadavid, A. C., and R. J. Finkelstein. "The q‐Coulomb problem in configuration space." Journal of Mathematical Physics 37, no. 8 (August 1996): 3675–83. http://dx.doi.org/10.1063/1.531594.

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Pennini, F., G. L. Ferri, and A. Plastino. "q-generalization of quantum phase-space representations." Physica A: Statistical Mechanics and its Applications 423 (April 2015): 97–107. http://dx.doi.org/10.1016/j.physa.2014.12.033.

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Majid, S. "Fermionic q-Fock space and braided geometry." Journal of Mathematical Physics 38, no. 9 (September 1997): 4845–53. http://dx.doi.org/10.1063/1.532128.

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