Tables of integrals

Mathematica--A System for Doing Mathematics by Computer.

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

phinoethane) seemed mainly to stabilize intermediates, and often the catalytic reactions were slower when dppe was used instead of the most common monodentate triphenylphosphine. We will briefly review the history of “ligand effects” in catalysis before discussing a range of reactions for which a notable effect has been observed. It has taken quite some time before the positive effect that bidentates can have on selectivities and rates of catalytic reactions was fully recognized. Most of the established examples of bite angle effects involve diphosphine ligands. Therefore, many important catalysts containing a chelate ligand such as bipyridine and diimine will fall outside the scope of this review. The connecting bridge in these bidentates does play a dominant role in the performance of these catalysts, but systematic studies have not been published. The effects of phosphine ligands in catalysis have been known for quite some time. One of the first reports involves the use of triphenylphosphine in the “Reppe” chemistry, the reactions of alkynes, alcohols, and carbon monoxide.1 It was found that formation of acrylic esters was much more efficient using NiBr2(PPh3)2 than NiBr2 without ligand. In the commercial system, though, a phosphine-free catalyst is used. While the reaction was not yet understood mechanistically, the use of phosphines in catalysis attracted the attention of the petrochemical industry worldwide. An early example of a phosphine ligand modified catalytic process is the Shell process for alkene hydroformylation using a cobalt catalyst containing a trialkylphoshine.2 The reaction requires higher temperatures, but it leads to more linear product as compared to the unmodified catalyst. The general mechanism of the hydroformylation reaction has been known for a long time.3 Hydrocyanation as used by Du Pont is another early example of an industrially applied catalytic reaction employing ligands.4 It is a nickel-catalyzed reaction in which aryl phosphite ligands are used for the production of adiponitrile. The development of this process has played a key role in the introduction of the now very common study of “ligand effects” in the field of homogeneous catalysis by organometallic complexes.5 While several industries were working on new homogeneous catalysts, important contributions to the new field were made in academia in the early 1960s with the appearance of the first phosphinemodified hydrogenation catalysts. An early example of a phosphine-free ruthenium catalyst was published by Halpern.6 Triphenylphosphine-modified platinumtin catalysts for the hydrogenation of alkenes were reported by Cramer from Du Pont in 1963.7 In the same year Breslow (Hercules) included a few phosFigure 1. Bite angle: The ligand-metal-ligand angle of bidentate ligands. 2741 Chem. Rev. 2000, 100, 2741−2769

Ligand Bite Angle Effects in Metal-catalyzed C−C Bond Formation

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.

https://iopscience.iop.org/article/10.1088/1751-8113/42/36/365303/pdf

Transitionless quantum driving

For a classical Hamiltonian H=(1/2) p2+V(q,t) with an arbitrary time‐dependent potential V(q,t), exact invariants that can be expressed as series in positive powers of  p, I(q,p,t)=∑∞n=0pnfn(q,t), are examined. The method is based on direct use of the equation dI/dt=∂I/∂t +[I,H] =0. A recursion relation for the coefficients fn(q,t) is obtained. All potentials that admit an invariant quadratic in p are found and, for those potentials, all invariants quadratic in p are determined. The feasibility of extending the analysis to find invariants that are polynomials in p of higher degree than quadratic is discussed. The systems for which invariants quadratic in p have been found are transformed to autonomous systems by a canonical transformation.

A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians

Symmetry Lie algebras of nth order ordinary differential equations

In the cosmological scenario in $f(T)$ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f(T)$-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.

Cosmological Solutions of $f(T)$ Gravity

We present a short history of the Ermakov equation with an emphasis on its discovery by thewest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the east. We present the modern context of the Ermakov equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete math., 2 (2008), 123-145) for an english translation of Ermakov's original paper.

/pdf/the-ermakov-equation-a-commentary-hvld2o9773.pdf

The Ermakov equation: A commentary

We analyze a relativistic model for superdense stars proposed by Tikekar [R. Tikekar, J. Math. Phys. 31, 2454 (1990)]. In this model the hypersurfaces generated by {t=const} have the geometry of the 3‐spheroid which gives the solutions a clear geometrical characterization. The solution of the Einstein field equations is reduced to integrating a second order ordinary differential equation. New classes of solutions are presented by restricting the choice of the spheroidal parameter K so that polynomial solutions are admitted for the first solution and then we find that there exists another solution which is a product of polynomials and algebraic functions. A remarkable feature of the class of solutions generated is that they are expressible completely in terms of polynomials and algebraic functions. Some physical aspects of the solutions are briefly considered. We regain the Tikekar solution as a special case when K=−7. As another example we explicitly present the form of the solution when K=−2.

P. G. L. Leach

Papers

A direct approach to finding exact invariants for one‐dimensional time‐dependent classical Hamiltonians

Symmetry Lie algebras of nth order ordinary differential equations

Cosmological Solutions of $f(T)$ Gravity

The Ermakov equation: A commentary

Exact solutions for the Tikekar superdense star