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Excited states of radical anions derived from the photoreduction of stable organic molecules are suggested to serve as potent reductants. However, excited states of these species are too short‐lived to allow bimolecular quenching processes. Recently, the singlet excited state of Meisenheimer complexes, which possess a long‐lived excited state, was identified as the competent species for the reduction of challenging organic substrates (−2.63 V vs. SCE, saturated calomel electrode). To produce reasonably stable and simply accessible different Meisenheimer complexes, the addition of nBuLi to readily available aromatic heterocycles was investigated, and the photoreactivity of the generated species was studied. In this paper, we present the straightforward preparation of a family of powerful photoreductants (*Eox<−3 V vs. SCE in their excited states, determined by DFT and time‐dependent TD‐DFT calculations; DFT, density functional theory) that can induce dehalogenation of electron‐rich aryl chlorides and to form C−C bond through radical cyclization. Photophysical analyses and computational studies in combination with experimental mechanistic investigations demonstrate the ability of the adduct to act as a strong electron donor under visible light irradiation. [ABSTRACT FROM AUTHOR]

Excited states of radical anions derived from the photoreduction of stable organic molecules are suggested to serve as potent reductants. However, excited states of these species are too short‐lived to allow bimolecular quenching processes. Recently, the singlet excited state of Meisenheimer complexes, which possess a long‐lived excited state, was identified as the competent species for the reduction of challenging organic substrates (−2.63 V vs. SCE, saturated calomel electrode). To produce reasonably stable and simply accessible different Meisenheimer complexes, the addition of nBuLi to readily available aromatic heterocycles was investigated, and the photoreactivity of the generated species was studied. In this paper, we present the straightforward preparation of a family of powerful photoreductants (*Eox<−3 V vs. SCE in their excited states, determined by DFT and time‐dependent TD‐DFT calculations; DFT, density functional theory) that can induce dehalogenation of electron‐rich aryl chlorides and to form C−C bond through radical cyclization. Photophysical analyses and computational studies in combination with experimental mechanistic investigations demonstrate the ability of the adduct to act as a strong electron donor under visible light irradiation. [ABSTRACT FROM AUTHOR]

In this work, two novel acridone-based photoinitiators were designed and synthesized for the free radical polymerization of acrylates with a light-emitting diode emitting at 405 nm. These acridone derivatives were employed as mono-component Type II photoinitiators and as multicomponent photoinitiating systems in the presence of an iodonium salt or an amine synergist (EDB) in which they achieved excellent polymerization initiating abilities and high final conversions of the acrylate group. Photoinitiation mechanisms through which reactive species are produced were investigated employing different complementary techniques including steady-state photolysis, steady-state fluorescence, cyclic voltammetry, UV–visible absorption spectroscopy, and electron spin resonance spectroscopy. Finally, these molecules were also used in the direct laser writing process for the fabrication of 3D objects. [ABSTRACT FROM AUTHOR]

A new meso‐naphthalenyl BODIPY was designed for the efficient generation of long‐lived triplet excited state under irradiation with green light. The new, heavy atom‐free organic chromophore was employed in a benchmark example of dual palladium and photoredox catalysis, highlighting the applications of BODIPY dyes as photoredox catalysts. [ABSTRACT FROM AUTHOR]

A system of 4 nonlinearly-coupled Ordinary Differential Equations has been recently introduced to investigate the evolution of human respiratory virus epidemics. In this paper we prove that some explicit solutions of that system can be obtained by algebraic operations, provided the parameters of the model satisfy certain constraints. [ABSTRACT FROM AUTHOR]

A regioselective vinylation of aromatic and aliphatic aldehydes promoted by the merging of photoredox and nickel catalysis is here reported. A comprehensive investigation on the reaction conditions allowed the disclosure of a valid and reproducible protocol based on a nickel‐mediated reductive coupling approach under visible light irradiation. The employment of 3CzClIPN (2,4,6‐tris(carbazol‐9‐yl)‐5‐chloro‐isophthalonitrile) as the photocatalyst and Hantzsch's ester as the sacrificial organic reductant replace the use of boron‐, silicon‐ or zinc‐based reducing agents, making this method a worthy alternative to the already known protocols. The developed mild reaction conditions allow the access to a wide range of substituents decorating both the aldehyde and the alkyne. Moreover, careful photophysical investigations shed light on the mechanism of the reaction. [ABSTRACT FROM AUTHOR]

Organochromium(III) species are multipurpose nucleophiles used in the synthesis of complex organic molecules due to their high functional group tolerance and extraordinary chemoselectivity for aldehydes. The preparation of organochromium(III) species starting from organic halides requires the use of a stoichiometric amount of chromium(II) salts or a catalytic amount of chromium(III) salts in the presence of stoichiometric reductants (such as Mn(0)). Recently, radicals formation from readily available alkenes, followed by their trapping with stoichiometric or catalytic amount of chromium(II) salts were reported in photoredox conditions for the generation of Cr(III) organometallic species. In this paper we disclose a real Nozaki‐Hiyama reaction (NH) in which photoredox conditions enable the preparation of Cr(III) allyl reagents starting from available allyl halides and stable Cr(III) salts. [ABSTRACT FROM AUTHOR]

Here we report a practical, highly enantioselective photoredox allylation of aldehydes mediated by chiral nickel complexes with commercially available allyl acetate as the allylating agent. The methodology allows the clean stereoselective allylation of aldehydes in good to excellent yields and up to 93 % e.e. using a catalytic amount of NiCl2(glyme) in the presence of the chiral aminoindanol‐derived bis(oxazoline) as the chiral ligand. The photoredox system is constituted by the organic dye 3DPAFIPN and a Hantzsch's ester as the sacrificial reductant. The reaction proceeds under visible‐light irradiation (blue LEDs, 456 nm) at 8–12 °C. Compared to other published procedures, no metal reductants (such as Zn or Mn), additives (e.g. CuI) or air‐sensitive Ni(COD)2 are necessary for this reaction. Accurate DFT calculations and photophysical experiments have clarified the mechanistic picture of this stereoselective allylation reaction. [ABSTRACT FROM AUTHOR]

Here we report a practical, highly enantioselective photoredox allylation of aldehydes mediated by chiral nickel complexes with commercially available allyl acetate as the allylating agent. The methodology allows the clean stereoselective allylation of aldehydes in good to excellent yields and up to 93 % e.e. using a catalytic amount of NiCl2(glyme) in the presence of the chiral aminoindanol‐derived bis(oxazoline) as the chiral ligand. The photoredox system is constituted by the organic dye 3DPAFIPN and a Hantzsch's ester as the sacrificial reductant. The reaction proceeds under visible‐light irradiation (blue LEDs, 456 nm) at 8–12 °C. Compared to other published procedures, no metal reductants (such as Zn or Mn), additives (e.g. CuI) or air‐sensitive Ni(COD)2 are necessary for this reaction. Accurate DFT calculations and photophysical experiments have clarified the mechanistic picture of this stereoselective allylation reaction. [ABSTRACT FROM AUTHOR]

The construction of a hybrid metal-organo-photoredox catalyst based on the conjugation of an imidazolidinone organocatalyst and Ir(ppy)2(bipy) (ppy = 2-phenylpyridine, bipy = bipyridine) is described. The introduction of the desired organocatalyst into the bipyridine moiety is quite modular, allowing the preparation of different hybrid photocatalysts, and is realized though a simple click reaction. The hybrid photocatalysts obtained were employed in the benchmark photoredox alkylation of aldehydes. Remarkably, the conjugation of a first-generation MacMillan catalyst produces an active and stereoselective hybrid photoredox catalyst. [ABSTRACT FROM AUTHOR]

The interplay among the time-evolution of the coefficients y m t and the zeros x n t of a generic time-dependent (monic) polynomial provides a convenient tool to identify certain classes of solvable dynamical systems. Recently, this tool has been extended to the case of nongeneric polynomials characterized by the presence, for all time, of a single double zero; subsequently, significant progress has been made to extend this finding to the case of polynomials featuring a single zero of arbitrary multiplicity. In this paper, we introduce an approach suitable to deal with the most general case, i.e., that of a nongeneric time-dependent polynomial with an arbitrary number of zeros each of which features, for all time, an arbitrary (time-independent) multiplicity. We then focus on the special case of a polynomial of degree 4 featuring only 2 different zeros, and by using a recently introduced additional twist of this approach, we thereby identify many new classes of solvable dynamical systems of the following type: x ̇ n = P n x 1 , x 2 , n = 1,2 , with P n x 1 , x 2 being two polynomials in the two variables x 1 t and x 2 t . [ABSTRACT FROM AUTHOR]

The synthesis, optical resolution through derivatization, characterization, and utilization of a new helical bis(phosphinite gold) complex derived from HELIXOL are described. By using an efficient cobalt catalyst, neither irradiation nor high catalytic loading was required to access the helicene of interest. The latter was phosphorylated to afford a new chiral ligand that was studied. Especially, treatment with AuCl(SMe2) gives a dinuclear gold complex that was used in cycloisomerization reactions. The synthesis, optical resolution through derivatization, characterization and utilization of a new helical bisphosphinite–gold complex derived from HELIXOL are described. [ABSTRACT FROM AUTHOR]

Recently a convenient technique to relate the time evolution of the N zeros of a time-dependent polynomial p N z ; t of degree N in the (complex) variable z to the time evolution of its N coefficients has been exploited to identify large classes of dynamical systems solvable by algebraic operations. These models also include N-body problems that evolve in the complex plane (or, equivalently, in the real Cartesian plane) according to systems of nonlinearly-coupled equations of motion of Newtonian type (“accelerations equal forces”). Many of these models feature remarkable properties: for instance, they are Hamiltonian and integrable and/or multiply periodic or even isochronous (featuring completely periodic solutions with a fixed period largely independent of the initial data), or asymptotically isochronous (featuring isochrony only up to corrections vanishing in the remote future). In this paper, an analogous technique is introduced that focuses instead on the time evolution of the N zeros of an appropriate class of time-dependent rational functions R N z ; t , thereby opening large vistas of new dynamical systems solvable by algebraic operations and featuring remarkable properties. A few examples are reported. [ABSTRACT FROM AUTHOR]

The article focuses on the simple extensions of the Classical Lotka-Volterra Prey–Predator Model. Topics include the system of two evolution ordinary differential equations (ODEs) stated to be characterizing this model, and the Lotka-Volterra Prey–Predator Model stated to be describing the time evolution of two interacting populations via two simple, nonlinearly coupled ODEs.

It has been recently demonstrated that it is possible to construct isochronous cosmologies, extending to general relativity a result valid for non-relativistic Hamiltonian systems. In this paper, we review these findings and we discuss the Newtonian limit of these isochronous spacetimes, showing that it reproduces the analogous findings in the context of non-relativistic dynamics. [ABSTRACT FROM AUTHOR]

Dual photoredox catalysis is modifying the approach to sustainable metal catalysis based on metals in low oxidation state, as the use of stoichiometric metal as reductants are avoided. In this study, we showcase the potential use of vanadium in a dual photoredox process, allowing us to effectively control the formation of V(II) and its corresponding chemical reactions. A diastereoselective pinacol coupling with aromatic aldehydes is described. The process shows broad scope, and interestingly, unprotected salicylic aldehydes are compatible with the reaction protocol. Additionally, among all the ligands investigated we found that inexpensive bipyridines control the diastereoselection of the reaction favoring the d,l chiral compound (dr>20:1 for most of the aldehydes investigated). To promote the reaction and avoid an unproductive and not stereoselective radical coupling of the generated ketyl radical, the use of [DMQA]+, a stable organic dye, was found crucial. Furthermore, this dye possesses the remarkable capability of efficiently capturing light in the red segment of the visible spectrum, effectively raising efficiency of the protocol. The utilization of red light carries notable advantages in terms of penetration and safety, making it particularly significant. Moreover, careful photophysical investigations were able to give indications about the mechanism of the reaction. [ABSTRACT FROM AUTHOR]

A technique is introduced which allows to generate--starting from any solvable discrete-time dynamical system involving N time-dependent variables--new, generally nonlinear, generations of discrete-time dynamical systems, also involving N time-dependent variables and being as well solvable by algebraic operations (essentially by finding the N zeros of explicitly known polynomials of degree N). The dynamical systems constructed using this technique may also feature large numbers of arbitrary constants, and they need not be autonomous. The solvable character of these models allows to identify special cases with remarkable time evolutions: for instance, isochronous or asymptotically isochronous discrete-time dynamical systems. The technique is illustrated by a few examples. [ABSTRACT FROM AUTHOR]

Recently a simple differential algorithm to compute all the zeros of a generic polynomial was introduced. In this paper an analogous, but finite-difference, algorithm is introduced and discussed. At the end of the paper a minor generalization of the differential algorithm is also mentioned. [ABSTRACT FROM AUTHOR]

A technique to manufacture solvable variants of the 'goldfish' many-body problem is introduced, and several many-body problems yielded by it are identified and discussed, including cases featuring multiperiodic or isochronous dynamics. [ABSTRACT FROM AUTHOR]

The notion of generations of monic polynomials such that the coefficients of each polynomial of the next generation coincide with the zeros of a polynomial of the current generation is introduced, and its relevance to the identification of endless sequences of new solvable many-body problems 'of goldfish type' is demonstrated. [ABSTRACT FROM AUTHOR]

Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer (N ≥ 2). These models are characterized by nonlinear Newtonian ("accelerations equal forces") equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods T/p with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future. [ABSTRACT FROM AUTHOR]

Various functional equations satisfied by one or two ( N × N)-matrices $${\mathbf{F}(z) }$$ and $${\mathbf{G}(z) }$$ depending on the scalar variable z are investigated, with N an arbitrary positive integer. Some of these functional equations are generalizations to the matrix case ( N > 1) of well-known functional equations valid in the scalar ( N = 1) case, such as $${\mathbf{F}(x) \, \mathbf{F}(y) = \, \mathbf{F}(x y) \, \rm and \, \mathbf{G}({\it x}) \, \mathbf{G}({\it y}) = \mathbf{G}({\it x+y}) }$$ ; others-such as $${\mathbf{G}(y) \, \mathbf{F}(x) = \mathbf{F}(x) \, \mathbf{G}(xy) }$$ -possess nontrivial solutions only in the matrix case ( N > 1), namely their scalar ( N = 1) counterparts only feature quite trivial solutions. It is also pointed out that if two ( N × N)-matrices $${\mathbf{F}(x) \, \rm and \, \mathbf{G}({\it y})}$$ satisfy the triplet of functional equations written above-and nontrivial examples of such matrices are exhibited- then they also satisfy an endless hierarchy of matrix functional relations involving an increasing number of scalar independent variables, the first items of which read $${\mathbf{F}(x_{1}) \, \mathbf{G}(y_{1}) \, \mathbf{F} (x_{2}) = \mathbf{F}(x_{1} x_{2}) \, \mathbf{G } (x_{2} y_{1}) \, \rm and \, \mathbf{G}({\it y}_{1}) \, \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it y}_{2}) = \mathbf{F} ({\it x}_{1}) \, \mathbf{G} ({\it x}_{1} {\it y}_{1}+{\it y}_{2}) }$$ . [ABSTRACT FROM AUTHOR]

In this paper we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the q -Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted. [ABSTRACT FROM AUTHOR]

We define the generalized basic hypergeometric polynomial of degree N in terms of the generalized basic hypergeometric function, by choosing one of its parameters to allow the termination of the series after a finite number of summands. In this paper, we obtain a set of nonlinear algebraic equations satisfied by the N zeros of the polynomial. Moreover, we obtain an N × N matrix M defined in terms of the zeros of the polynomial, which, in turn, depend on the parameters of the polynomial. The eigenvalues of this remarkable matrix M are given by neat expressions that depend only on some of the parameters of the polynomial; that is, the matrix M is isospectral. Moreover, in case the parameters that appear in the expressions for the eigenvalues of M are rational, the matrix M has rational eigenvalues, a Diophantine property. [ABSTRACT FROM AUTHOR]

The synthesis of silodosin, an antidysuria drug, has been accomplished starting from commercially available indoline. The synthetic strategy is based on CuI-catalysed C-C arylation, regioselective cyanation, and diastereoselective reductive amination. The enantiopure compound was obtained by selective crystallisation of a diasteroisomeric mixture. [ABSTRACT FROM AUTHOR]

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period T yet mimic with good approximation, or even exactly, any given many-body problem (within a quite large class, encompassing most of nonrelativistic physics) over times $\widetilde{T}$ which may also be arbitrarily large (but of course such that $\widetilde{T}< T$). In this paper we review and further explore the possibility to extend this finding to a general relativity context, so that it becomes relevant for cosmology. [ABSTRACT FROM AUTHOR]

Two synthetic approaches to boehmeriasin A are described. A gram scale racemic preparation is accompanied by an efficient preparation of both the pure enantiomers using the conformationally stable 2-piperidin-2-yl acetaldehyde as starting material. The anti-proliferative activity in three cancer cell lines (CEM, HeLa and L1210) and two endothelial cell lines (HMEC-1, BAEC) indicates promising activity at the nanomolar range. Topoisomerases and SIRT2 are identified as biological targets and the experimental data has been supported by docking studies. [ABSTRACT FROM AUTHOR]

Two square matrices of (arbitrary) order N are introduced. They are defined in terms of N arbitrary numbers zn, and of an arbitrary additional parameter (a respectively q), and provide finite-dimensional representations of the two operators acting on a function f (z) as follows: [f (z + a) - f (z)]/a respectively [ f (q z) - f (z)]/[(q - 1)z]. These representations are exact--in a sense explained in the paper--when the function f (z) is a polynomial in z of degree less than N. This formalism allows to transform difference equations valid in the space of polynomials of degree less than N into corresponding matrix-vector equations. As an application of this technique, several remarkable square matrices of order N are identified, which feature explicitly N arbitrary numbers zn, or the N zeros of polynomials belonging to the Askey and q-Askey schemes. Several of these findings have a Diophantine character. [ABSTRACT FROM AUTHOR]

In this paper, we provide properties-which are, to the best of our knowledge, new-of the zeros of the polynomials belonging to the Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted. [ABSTRACT FROM AUTHOR]

Cover Feature: A Photoredox Nozaki-Hiyama Reaction Catalytic in Chromium (Eur. J. Org. Keywords: Aldehydes; Allylbromide; Blue LED; Nozaki-Hiyama Reaction Photoredox catalysis EN Aldehydes Allylbromide Blue LED Nozaki-Hiyama Reaction Photoredox catalysis 1 1 1 08/09/22 20220805 NES 220805 B The Cover Feature b celebrates Japanese chemists Prof. Hitosi Nozaki and Prof. Tamejiro Hiyama, who discovered the Cr-mediated allyl addition to aldehydes. This work reports a catalytic photoredox version of the Nozaki-Hiyama reaction, without any problem related to the use of stoichiometric Mn or Zn as reducing agents....Mondai arimasen!. [Extracted from the article]

We define the generalized hypergeometric polynomial of degree N as follows: P N ( α 1 , . . . , α p ; β 1 , . . . , β q ; z ) = ∑ m = 0 N [ ( − N ) m ( α 1 ) m ⋅ ⋅ ⋅ ( α p ) m z N − m m ! ( β 1 ) m ⋅ ⋅ ⋅ ( β q ) m ] = z N F q p + 1 ( − N , α 1 , . . . , α p ; β 1 , . . . , β q ; 1 / z ) . Here N is an arbitrary positive integer, p and q are arbitrary nonnegative integers, the p + q parameters α j and β k are arbitrary (“generic”, possibly complex) numbers, ( α ) m is the Pochhammer symbol and F q p + 1 ( α 0 , α 1 , . . . , α p ; β 1 , . . . , β q ; z ) is the generalized hypergeometric function. In this paper we obtain a set of N nonlinear algebraic equations satisfied by the N zeros ζ n of this polynomial. We moreover manufacture an N × N matrix L ̲ in terms of the 1 + p + q parameters N , α j , β k characterizing this polynomial, and of its N zeros ζ n , and we show that it features the N eigenvalues λ m = m ∏ k = 1 q ( − β k + 1 − m ) , m = 1 , . . . , N . These N eigenvalues depend only on the q parameters β k , implying that the N × N matrix L ̲ is isospectral for variations of the p parameters α j ; and they clearly are integer (or rational ) numbers if the q parameters β k are themselves integer (or rational ) numbers: a nontrivial Diophantine property. [ABSTRACT FROM AUTHOR]

A class of N-body problems is identified, characterized by second-order discrete-time evolution equations determining the motion in the complex z-plane of an arbitrary number N of points zn ≡ zn (l), where l = 0,±1,±2,... is the discrete-time independent variable. Both these equations of motion, and the solution of their initialvalue problem, only involve algebraic operations: finding the zeros of explicitly known polynomials of degree N in z, finding the eigenvectors and eigenvalues of explicitly known N × N matrices. These models feature an arbitrarily large number of arbitrary parameters ("coupling constants"). [ABSTRACT FROM AUTHOR]

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period T yet mimic with good approximation, or even exactly, any given many-body problem (within a large, physically relevant, class) over times which may also be arbitrarily large (but of course such that ). Purpose and scope of this paper is to explore the possibility to extend this finding to a general relativity context. For simplicity we restrict our consideration to the case of homogeneous and isotropic metrics and show that, via an approach analogous to that used for the nonrelativistic many-body problem, a class of homogeneous and isotropic cyclic solutions of Einstein's equations may be obtained. For these solutions the duration of the cycles does not depend on the initial conditions, so we call these models isochronous cosmologies. We give a physical interpretation of such metrics and in particular we show that they may behave arbitrarily closely, or even identically, to the Friedman-Robertson-Walker solutions of Einstein's equations for an arbitrarily long time (of course shorter than their period, which can also be assigned arbitrarily), so that they may reproduce all the satisfactory phenomenological features of the standard cosmological Λ-CDM model in a portion of their cycle; while these isochronous cosmologies may be geodesically complete and therefore singularity-free. [ABSTRACT FROM AUTHOR]

Abstract: A linear second-order ODE featuring an arbitrary number of free parameters is identified, all solutions of which are polynomials. [Copyright &y& Elsevier]

The introduction of a hydrophobic group at position 7 of 9-fluorenone-2-carboxylic acid generates new tubulin binders, the design of which is suggested by modeling studies. The synthesis is based on the use of 2,7-dibromo-fluorenone as starting material. The antiproliferative activity on two different cell lines, fluorescent microscopy, flow cytometry, and sedimentation assay tests confirmed the supposed mechanism. [ABSTRACT FROM AUTHOR]

By investigating the behavior of two solvable isochronous N-body problems in the immediate vicinity of their equilibria, functional equations satisfied by the para-Jacobi polynomial $${p_{N} \left(0, 1; \gamma; x \right)}$$ and by the Jacobi polynomial $${P_{N}^{\left(-N-1,-N-1 \right)} \left(x \right)}$$ (or, equivalently, by the Gegenbauer polynomial $${C_{N}^{-N-1/2}\left( x \right) }$$) are identified, as well as Diophantine properties of the zeros and coefficients of these polynomials. [ABSTRACT FROM AUTHOR]

Matrices ... and ... of arbitrary rank N, given by simple expressions in terms of the N zeros of certain Laguerre or para-Jacobi polynomials of degree N, feature a Diophantine property. In the Laguerre case, this property states that the 2N zeros of the polynomial p2N (λ) = det [λ2+ λ... +...]are all integers; indeed we conjecture that det [λ2+ λ... + ... ] = ΠNk=1 (λ2 - k2). The results in the para-Jacobi case are somewhat analogous; they refer to the zeros of the general solution of the ODE generally characterizing Jacobi polynomials, in the special case in which this solution is a para-Jacobi polynomial featuring an additional, arbitrary parameter. [ABSTRACT FROM AUTHOR]

The answer to the question posed in the title is affirmative. Indeed, while the second-orderODE (1-x2 )p"+[&bgr;-&agr;-(&agr;+&bgr;+2)x]p'+N(N+&agr;+&bgr;+1)p = 0 for generic values of the two parameters&agr; and &bgr;(and N a non-negative integer) identifies uniquely (up to an overall multiplicative constant characterizing its normalization) the Jacobi polynomial P(&agr;,&bgr;) N (x) as its polynomial solution--if instead&agr;= -N + m and &bgr;= -N + n, with m and n being two non-negative integers satisfying the restriction m + n < N, then the general solution of this ODE is a polynomial of degree N, depending--in addition of course to the three parameters N, m and n--on two arbitrary parameters (i.e. on an additional one besides the overall multiplicative parameter characterizing its normalization [ABSTRACT FROM AUTHOR]

It is shown how, given an arbitrary dynamical system, other systems can be manufactured which are isochronous (periodic in all their degrees of freedom with an arbitrarily assigned fixed period T) and whose dynamics coincides exactly with that of the original system over a fraction 1 − σ of that period where σ is a number that can be arbitrarily assigned in the range 0 < σ < 1. The treatment is in the context of autonomous dynamical systems whose equations of motion feature infinitely differentiable functions of the dependent variables. It is argued that this possibility suggests that it would be useful to devise measures of the degree of complexity of a dynamical system which are associated with finite portions of its time evolution-hence going beyond the standard characterizations of the chaotic behavior of a dynamical system (via its Lyapunov coefficients or its ergodic properties), requiring its observation over infinite time. [ABSTRACT FROM AUTHOR]

A simple algorithm to compute all the zeros of a generic polynomial is proposed. [ABSTRACT FROM AUTHOR]

A new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an isochronous behavior in the remote future. [ABSTRACT FROM AUTHOR]

We take advantage of the simple approach, recently discussed, which associates to (solvable) matrix equations (solvable) dynamical systems interpretable as (interesting) many-body problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. Starting from a solvable matrix evolution equation, we obtain the corresponding many-body model and note that in one case the auxiliary variables can be altogether eliminated, obtaining thereby an (also Hamiltonian) extension of the “goldfish” model. The solvability of this novel model, and of its isochronous variant, is exhibited. A related, as well solvable, model, is also introduced, as well as its isochronous variant. Finally, the small oscillations of the isochronous models around their equilibrium configurations are investigated, and from their isochronicity certain diophantine relations are evinced. [ABSTRACT FROM AUTHOR]

A simple approach is discussed which associates to (solvable) matrix equations (solvable) dynamical systems, generally interpretable as (interesting) many-body problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. We then focus on cases in which the auxiliary variables can be altogether eliminated, reobtaining thereby (via this unified approach) well-known solvable many-body problems, and moreover a (solvable) extension of the “goldfish” model.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

An estimate is presented of the angular momentum associated with the stochastic cosmic tremor, which has been hypothesized to be caused by universal gravitation and by the granularity of matter, and to be itself the cause of quantization (‘cosmic origin of quantizakion’). If that universal tremor has the spatial coherence which is instrumental in order that the estimated action associated with it have the order of magnitude of Planck's constant h, then the estimated order of magnitude of the angular momentum associated with it also has the same value. We moreover indicate how these findings (originally based on a simplified model of the Universe, as being made up only of particles having the nucleon mass) are affected (in fact, essentially unaffected) by the possible presence in the mass of the Universe of a large component made up of particles much lighter than nucleons (‘dark’, or ‘missing’, mass). [ABSTRACT FROM AUTHOR]

We identify a class of potentials for which the semiclassical estimate N(semi)=(1/π)∫0∞dr √ of −V(r)θ[−V(r)] of the number N of (S-wave) bound states provides a (rigorous) lower limit: N&ges;{{N(semi)}}, where the double braces denote the integer part. Higher partial waves can be included via the standard replacement of the potential V(r) with the effective ℓ-wave potential Vℓ(eff)(r)=V(r)+ℓ(ℓ+1)/r2. An analogous upper limit is also provided for a different class of potentials, which is however quite severely restricted. [Copyright &y& Elsevier]

Focuses on the rational expression of irrational numbers. Mathematical formulae which can be used to convert real numbers to rational numbers and vice versa; Definition of the Champernowne constant; Representation of the Champernowne constant as an infinite series of irrational numbers; Mathematical proof of the representation.

We introduce and investigate the class of central potentials VCIC(g2,μ2,ℓ,R;r)=−g2/R2r/R4ℓ1+1/2ℓ+1r/R2ℓ+12−1+μ2−2, which possess, in the context of nonrelativistic quantum mechanics, a number of ℓ-wave bound states given by the (ℓ-independent!) formula Nℓ(CIC)(g2,μ2)=1/πr/R4ℓ1+1/2ℓ+1r/R2ℓ+12−1+μ2−2, which possess, in the context of nonrelativistic quantum mechanics, a number of ℓ-wave bound states given by the (ℓ-independent!) formula Nℓ(CIC)(g2,μ2)=1/π4ℓ1+1/2ℓ+1r/R2ℓ+12−1+μ2−2, which possess, in the context of nonrelativistic quantum mechanics, a number of ℓ-wave bound states given by the (ℓ-independent!) formula Nℓ(CIC)(g2,μ2)=1/πr/R2ℓ+12−1+μ2−2, which possess, in the context of nonrelativistic quantum mechanics, a number of ℓ-wave bound states given by the (ℓ-independent!) formula Nℓ(CIC)(g2,μ2)=1/π2ℓ+12−1+μ2−2, which possess, in the context of nonrelativistic quantum mechanics, a number of ℓ-wave bound states given by the (ℓ-independent!) formula Nℓ(CIC)(g2,μ2)=1/π√ of g2+μ2−1μ2−1−1arctanμ2−1 √ of μ2−1−1arctanμ2−1 −1arctan√ of μ2−1 . Here g and μ are two arbitrary real parameters, ℓ is the angular momentum quantum number, and the double braces denote of course the integer part. An extension of this class features potentials that possess the same number of ℓ-wave bound states and behave as a/r2 both at the origin (r→0+) and at infinity (r→∞), where a is an additional free parameter. [Copyright &y& Elsevier]

New upper and lower limits are given for the number of S-wave bound states yielded by an attractive (monotonic) potential in the context of the Schrödinger or Klein-Gordon equation. [ABSTRACT FROM AUTHOR]

Reports on the slow paced bomb-grade uranium transaction deal between the United States and Russia Federation government suggested by Thomas Neff of Massachusetts Institute of Technology. Conditions under the deal; Execution of the deal; Influence of Russian ratification on the deal's speed; Reason why there is a need to fast-track the deal.