To date, the theoretical development of q-calculus has rested on a non-uniform basis. Generally, the bulky Gasper-Rahman notation was used, but the published works on q-calculus looked different ...depending on where and by whom they were written. This confusion of tongues not only complicated the theoretical development but also contributed to q-calculus remaining a neglected mathematical field. This book overcomes these problems by introducing a new and interesting notation for q-calculus based on logarithms.For instance, q-hypergeometric functions are now visually clear and easy to trace back to their hypergeometric parents. With this new notation it is also easy to see the connection between q-hypergeometric functions and the q-gamma function, something that until now has been overlooked.The book covers many topics on q-calculus, including special functions, combinatorics, and q-difference equations. Apart from a thorough review of the historical development of q-calculus, this book also presents the domains of modern physics for which q-calculus is applicable, such as particle physics and supersymmetry, to name just a few.?
We first introduce a notation for multiple (
≥ 3)
-hypergeometric functions, where negative values of summation indices are allowed. Then we extend the notation for
-Horn functions to include tilde ...values corresponding to powers of 2. Karlssons reduction formulas are correspondingly
-deformed by using these notations. A formula for sums of inverse
-shifted factorials is used to find further formulas. The second part of the paper is devoted to convergence aspects for
-Horn functions and ’abnormal’
-Horn functions. It turns out that some simple estimates for convergence can be made in the
-case, these are then supplemented with tables of numerical values. It is shown that the convergence regions are significantly increased in the
-case, and we compare with convergence regions in the ordinary case.
The purpose of this article is to study how q-real numbers can be used for computations of convergence regions, q-integral representations of certain multiple triple q-Lauricella functions. The ...corresponding q-difference equations are also given without proof. In the process, we slightly improve Exton’s original formulas. We also survey the current attempts to generalize the above functions to triple and quadruple hypergeometric functions. Finally, we compute some q-analogues of transformation formulas for Horn functions.
Treatment-free remission (TFR) is a new and significant goal of chronic myeloid leukemia management. TFR should be considered for patients in stable deep molecular response (DMR) after careful ...discussion in the shared decision-making process. Second-generation tyrosine kinase inhibitors (TKIs) improve the speed of response and the incidence of DMR. Treatment may be changed to a more active TKI to improve the depth of response in selected patients who have not reached DMR. Stem cell persistence is associated with active immune surveillance and activation of BCR-ABL1-independent pathways, eg, STAT3, JAK1/2, and BCL2. Ongoing studies aim to prove the efficacy of maintenance therapies targeting these pathways after TKI discontinuation.
The purpose of this article is to introduce a new complete multiple q-hypergeometric symbolic calculus, which leads to q-Euler integrals and a very similar canonical system of q-difference equations ...for multiple q-hypergeometric functions. q-analogues of recurrence formulas in Horns paper and Borngässers thesis lead to a more exact way to find these Frobenius solutions. To find the right formulas, the parameters in q-shifted factorials can be changed to negative integers, which give no extra q-factors. In proving these q-formulas, in the limit q→1 we obtain versions of the paper by Debiard and Gaveau for the solution of differential or q-difference equations. The paper is also a correction of some of the statements in the paper by Debiard and Gaveau, e.g., the Euler integrals and other solutions to differential equations for Appell functions, also without references to page numbers in the standard work of Appell and Kampé de Fériet. Sometimes the q-binomial theorem is used to simplify q-integral formulas. By the Horn method, we find another solution to the Appell Φ1 function partial differential equation, which was not mentioned in the thesis by Le Vavasseur 1893.
The Horn–Karlsson approach to find convergence regions is applied to find convergence regions for triple q-hypergeometric functions. It turns out that the convergence regions are significantly ...increased in the q-case; just as for q-Appell and q-Lauricella functions, additions are replaced by Ward q-additions. Mostly referring to Krishna Srivastava 1956, we give q-integral representations for these functions.
We introduce most of the concepts for
-Lie algebras in a way independent of the base field
. Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar ...definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple
-Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case
= 1. The computations of sample
-roots for certain well-known
-Lie groups contain an extra
-addition, and consequently, for most of the quantities which are
-deformed, we add a prefix
in the respective name. Important examples are the
-Cartan subalgebra and the
-Cartan Killing form. We introduce the concept
homogeneous spaces in a formal way exemplified by the examples
and
with corresponding
Lie groups and
-geodesics. By introducing a
-deformed semidirect product, we can define exact sequences of
-Lie groups and some other interesting
-homogeneous spaces. We give an example of the corresponding
-Iwasawa decomposition for SL
(2).
Socioeconomic disparities are associated with differences in cognitive development. The extent to which this translates to disparities in brain structure is unclear. We investigated relationships ...between socioeconomic factors and brain morphometry, independently of genetic ancestry, among a cohort of 1,099 typically developing individuals between 3 and 20 years of age. Income was logarithmically associated with brain surface area. Among children from lower income families, small differences in income were associated with relatively large differences in surface area, whereas, among children from higher income families, similar income increments were associated with smaller differences in surface area. These relationships were most prominent in regions supporting language, reading, executive functions and spatial skills; surface area mediated socioeconomic differences in certain neurocognitive abilities. These data imply that income relates most strongly to brain structure among the most disadvantaged children.
B-lymphoid transcription factors, such as PAX5 and IKZF1, are critical for early B-cell development, yet lesions of the genes encoding these transcription factors occur in over 80% of cases of ...pre-B-cell acute lymphoblastic leukaemia (ALL). The importance of these lesions in ALL has, until now, remained unclear. Here, by combining studies using chromatin immunoprecipitation with sequencing and RNA sequencing, we identify a novel B-lymphoid program for transcriptional repression of glucose and energy supply. Our metabolic analyses revealed that PAX5 and IKZF1 enforce a state of chronic energy deprivation, resulting in constitutive activation of the energy-stress sensor AMPK. Dominant-negative mutants of PAX5 and IKZF1, however, relieved this glucose and energy restriction. In a transgenic pre-B ALL mouse model, the heterozygous deletion of Pax5 increased glucose uptake and ATP levels by more than 25-fold. Reconstitution of PAX5 and IKZF1 in samples from patients with pre-B ALL restored a non-permissive state and induced energy crisis and cell death. A CRISPR/Cas9-based screen of PAX5 and IKZF1 transcriptional targets identified the products of NR3C1 (encoding the glucocorticoid receptor), TXNIP (encoding a glucose-feedback sensor) and CNR2 (encoding a cannabinoid receptor) as central effectors of B-lymphoid restriction of glucose and energy supply. Notably, transport-independent lipophilic methyl-conjugates of pyruvate and tricarboxylic acid cycle metabolites bypassed the gatekeeper function of PAX5 and IKZF1 and readily enabled leukaemic transformation. Conversely, pharmacological TXNIP and CNR2 agonists and a small-molecule AMPK inhibitor strongly synergized with glucocorticoids, identifying TXNIP, CNR2 and AMPK as potential therapeutic targets. Furthermore, our results provide a mechanistic explanation for the empirical finding that glucocorticoids are effective in the treatment of B-lymphoid but not myeloid malignancies. Thus, B-lymphoid transcription factors function as metabolic gatekeepers by limiting the amount of cellular ATP to levels that are insufficient for malignant transformation.
The purpose of this article is to continue the study of q, omega-special functions in the spirit of Wolfgang Hahn from the previous papers by Annaby et al. and Varma et al., with emphasis on q, ...omega-Apostol Bernoulli and Euler polynomials, Ward-omega numbers and multiple q, omega-power sums. Like before, the q, omega-module for the alphabet of q, omega-real numbers plays a crucial role, as well as the q, omega-rational numbers and the Ward-omega numbers. There are many more formulas of this type, and the deep symmetric structure of these formulas is described in detail.
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