scholarly journals Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

2018 ◽  
Vol 14 (09) ◽  
pp. 2473-2516 ◽  
Author(s):  
Thomas Garrity ◽  
Peter Mcdonald
Keyword(s):  
Continued Fractions ◽  
Singular Function ◽  
Question Mark ◽  
Basic Properties ◽  
Naturally Occurring ◽  
Almost Everywhere ◽  
Multidimensional Continued Fractions ◽  
One To One ◽  
Almost All ◽  
Minkowski Question Mark Function

The Minkowski question mark function [Formula: see text] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus [Formula: see text] is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

scholarly journals Podophyllotoxin: History, Recent Advances and Future Prospects

Biomolecules ◽  
2021 ◽  
Vol 11 (4) ◽  
pp. 603
Author(s):  
Zinnia Shah ◽  
Umar Farooq Gohar ◽  
Iffat Jamshed ◽  
Aamir Mushtaq ◽  
Hamid Mukhtar ◽  
...  
Keyword(s):  
Structure Optimization ◽  
Cytokine Storm ◽  
Chemotherapeutic Drug ◽  
Future Prospects ◽  
Naturally Occurring ◽  
Pharmacologically Active ◽  
Almost All ◽  
Pharmacologically Active Compounds ◽  
Synthetic Derivatives ◽  
Related Compounds

Podophyllotoxin, along with its various derivatives and congeners are widely recognized as broad-spectrum pharmacologically active compounds. Etoposide, for instance, is the frontline chemotherapeutic drug used against various cancers due to its superior anticancer activity. It has recently been redeveloped for the purpose of treating cytokine storm in COVID-19 patients. Podophyllotoxin and its naturally occurring congeners have low bioavailability and almost all these initially discovered compounds cause systemic toxicity and development of drug resistance. Moreover, the production of synthetic derivatives that could suffice for the clinical limitations of these naturally occurring compounds is not economically feasible. These challenges demanded continuous devotions towards improving the druggability of these drugs and continue to seek structure-optimization strategies. The discovery of renewable sources including microbial origin for podophyllotoxin is another possible approach. This review focuses on the exigency of innovation and research required in the global R&D and pharmaceutical industry for podophyllotoxin and related compounds based on recent scientific findings and market predictions.

Naturally occurring heterocyclic anticancer compounds

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shashi Kiran Misra ◽  
Kamla Pathak
Keyword(s):  
Anticancer Agents ◽  
Topoisomerase I ◽  
Heterocyclic Ring ◽  
Cytotoxic Action ◽  
Extrinsic Pathway ◽  
Anticancer Compounds ◽  
Halichondrin B ◽  
Naturally Occurring ◽  
Microbial Infections ◽  
Almost All

Abstract Naturally occurring heterocyclic scaffolds are key ingredients for the development of various therapeutics employed for biomedical applications. Heterocyclic pharmacophores are widely disseminated and have been befallen in almost all categories of drugs for the alleviation of myriad ailments including diabetes, neurodegenerative, psychiatric, microbial infections, disastrous cancers etc. Countless fused heterocyclic anticancerous templates are reported to display antimetabolite, antioxidant, antiproliferative, cytostatic etc. pharmacological actions via targeting different signaling pathways (cell cycle, PI-3kinase/Akt, p53, caspase extrinsic pathway etc.), overexpressive receptors (EGRF, HER2, EGF, VEGF etc.) and physiological enzymes (topoisomerase I and II, cyclin dependent kinase etc.). A compiled description on various natural sources (plants, microbes, marine) containing anticancer agents comprising heterocyclic ring specified with presence of nitrogen (vincristine, vinblastine, indole-3-carbinol, meridianins, piperine, lamellarins etc.), oxygen (paclitaxel, halichondrin B, quercetin, myricetin, kaempferol etc.) and sulphur atoms (brugine, fucoidan, carrageenan etc.) are displayed here along with their molecular level cytotoxic action and therapeutic applications.

The aqueous chemistry of uranium minerals. Part I. Divalent cation zippeïte

1979 ◽  
Vol 43 (328) ◽  
pp. 539-541 ◽  
Author(s):  
David F. Haacke ◽  
Peter A. Williams
Keyword(s):  
Metal Ion ◽  
Metal Speciation ◽  
Divalent Metal ◽  
Free Energies ◽  
Naturally Occurring ◽  
Solution Studies ◽  
Chemical Studies ◽  
Divalent Metal Ion ◽  
Almost All ◽  
End Members

SynopsisFree energies of formation of divalent metal ion zippeïtes, M2(UO2)6(SO4)3(OH)10 ·nH2O, M = Mg,Co,Ni,Zn have been determined from solution studies and metal speciation calculations, in water. It is found that in the compounds, the number of molecules of water of crystallization is equal to 8. This is at variance with a previous report (Frondel et al., 1976), but it has been found that some at least of the water content of zippeite is either nonessential or very loosely bound in the structure. Based on the octahydrate formulation, values are −3506, −12695, −12683 and −12870±4 kJ mol−1 for the Mg,Co,Ni and Zn end-members, respectively. Almost all of the differences in the values are accounted for by those values for the metal ions alone with the exception of Znzippeïte where a discrepancy of some 22 kJ mol−1 is found. Even this value is small however, and the chemical studies indicate that extensive mutual solid solution between all end members is to be expected. These findings agree perfectly with observations on the composition of naturally occurring zippeite minerals of this group.

Quadratic approximation in ℚp

2014 ◽  
Vol 11 (01) ◽  
pp. 193-209 ◽  
Author(s):  
Yann Bugeaud ◽  
Tomislav Pejković
Keyword(s):  
Prime Number ◽  
Continued Fractions ◽  
Quadratic Approximation ◽  
Almost All

Let p be a prime number. Let w2 and [Formula: see text] denote the exponents of approximation defined by Mahler and Koksma, respectively, in their classifications of p-adic numbers. It is well-known that every p-adic number ξ satisfies [Formula: see text], with [Formula: see text] for almost all ξ. By means of Schneider's continued fractions, we give explicit examples of p-adic numbers ξ for which the function [Formula: see text] takes any prescribed value in the interval (0, 1].

scholarly journals On the endomorphism near-ring of a free group

1980 ◽  
Vol 23 (1) ◽  
pp. 103-121 ◽  
Author(s):  
R. Warwick Zeamer
Keyword(s):  
Wreath Product ◽  
Free Group ◽  
The Other ◽  
Finite Support ◽  
Finite Product ◽  
Prime Factorization ◽  
Infinite Rank ◽  
One To One ◽  
Almost All ◽  
Countably Infinite

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

scholarly journals On the affirmative solution to Salem’s problem

Analysis ◽  
2019 ◽  
Vol 39 (4) ◽  
pp. 135-149
Author(s):  
Semyon Yakubovich
Keyword(s):  
Linear Combination ◽  
Special Functions ◽  
Type Problem ◽  
Classical Analysis ◽  
Question Mark ◽  
Affirmative Solution ◽  
Minkowski Question Mark Function ◽  
Stieltjes Transforms

Abstract The Salem problem to verify whether Fourier–Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier–Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier–Stieltjes coefficients of a power {m\in\mathbb{N}} of the Minkowski question mark function are derived.

scholarly journals Reduced hæmatin and hæmochromogen

1929 ◽  
Vol 105 (735) ◽  
pp. 112-130 ◽  
Keyword(s):  
Carbon Monoxide ◽  
Helix Pomatia ◽  
Nitrogen Compound ◽  
Strong Solutions ◽  
Ion Concentration ◽  
Free Oxygen ◽  
Hydrogen Ion Concentration ◽  
Naturally Occurring ◽  
Chemical Combination ◽  
Almost All

Hæmochromogen, which originally was known only as an artificial degradation product of hæmoglobin, has since been found to occur in almost all organisms. Helicorubin, a hæmochromogen occurring in the intestinal fluid of snails (such as Helix pomatia ) was described by Krukenberg (1884), and its properties examined by Dhéré (1917). Cytochrome has been shown by Keilin (1925) to be a mixture of at least two, possibly three, hæmochromogens. Just as hæmoglobin differs in its behaviour towards oxygen from any of the other compounds of hæmatin at present known, so most of the naturally occurring hæmochromogens differ in properties from any of the artificial hæmochromogens. Ordinary hæmochromogen, obtained directly from hæmoglobin, combines with carbon monoxide to give CO-hæmochromogen, and is rapidly oxidised under all conditions by free oxygen to hæmatin. The hæmochromogens of cytochrome do not combine with carbon monoxide, and with the exception of component b (Keilin, 1929) are not rapidly oxidised by free oxygen over definite ranges of hydrogen-ion concentration. Originally hæmochromogen was only a name for the spectrum of reduced hæmatin under special conditions. Zeynek (1920) had shown by an analysis of solid pyridine hæmochromogen that it contains 2·2 molecules of pyridine. Anson and Mirsky (1925) were the first to show that this particular type of hæmatin spectrum is due to chemical combination of reduced hæmatin with substances containing nitrogen, and that each nitrogen compound gave its characteristic hæmochromogen. They showed that these compounds were dissociable in solution, and also that the various nitrogen compounds were in equilibrium with the hæmochromogen according to their relative affinities. Late I (1926) measured the minimum quantity of pyridine required to change completely the spectrum of reduced hæmatin into hæmochromogen, using strong solutions of hæmatin, and found that 2 molecules per molecule of reduced hæmatin were required, which agreed with Zeynek's analysis.

Sign in / Sign up

Export Citation Format

Share Document