An extremal theorem of Riemannian geometry

An exact upper estimate for the volume of a tubular neighborhood of a smooth submanifold N of a complete Riemann space M depending upon the volume of N and lower bound for the sectional curvatures of M is given. If N is a closed geodesic, then the equality is attained in the estimate if and only if M is a generalized lens space.Translated from Matematicheskie Zametki, Vol. 19, No. 5, pp. 795–804, May, 1976.The author thanks Yu. D. Burago for help and guidance.     

Four-dimensional superconformal index reloaded

We consider the four-dimensional N≥1 superconformal index and its generalization to the lens space. We discuss reductions of the latter to the three-dimensional N≥2 sphere partition function, the threedimensional N≥2 superconformal index, and the two-dimensional N≥(2, 2) sphere partition function. We apply these reductions to a class of four-dimensional N=1 superconformal field theories dual to toric Calabi-Yau manifolds, and we find surprising connections with integrable spin chains and hyperbolic geometry. We comment on the problem of classifying infrared fixed points of four-dimensional and threedimensional supersymmetric gauge theories.     

Robots,CAM and O.R.

Microelectronics have greatly influenced the nature of manufacturing technology and systems. New factories make growing use of microcomputers and microprocessors in robots, C.N.C. machine tools and flexible manufacturing systems. The result is an increasing trend towards fully integrated computer-based manufacturing systems. As a consequence, O.R. software has a newer, more direct role in the modern factory than ever before. Fully integrated CAM systems are implemented by multi-disciplinary teams of engineers and management scientists. The O.R. practitioner can make a major contribution, for example in layout optimisation and plant simulations.     

Existence, Uniqueness, and Construction of the Solution of a System of Ordinary Functional Differential Equations, with Application to the Design of Perfectly Focusing Symmetric Lenses

In the design of perfectly focusing symmetric lenses, one isled, in a natural way, to a set offunctional differential equations;that is, differential equations involving composites of unknownfunctions, with initial conditions prescribed on the lens axis.This paper concentrateson those features of the equations whichmake them uniquely solvable. They are: (i) a contractivenessproperty of the equations near the axis; (ii) a uniform retardationin the arguments of thecomposite functions away from the axis.The second and third sections of this paper generalize and formalizethese properties and provide proofs of existence, uniqueness,and continuous dependence on the data for solutions of suchgeneralized systems of functional differential equations. Becauseof the lens context which motivates our study, the problem inwhich the contractiveness property (i) above holds is calledthe ‘local’ problem, and the problem in which thearguments of composite functions are uniformly retarded is calledthe ‘global’ problem. In the final section of thepaper we apply the general results of the preceding sectionsto prove existence and uniqueness of perfectly focusing symmetriclenses up to distances from the lens axis at which various typesof breakdown, discussed in the text, may occur.     

Description of light focusing by a spherical lens using diffraction integral method

We describe here the propagation of light through a plano-convex spherical lens using the Stratton-Chu vector diffraction integrals. The values of the electric and magnetic fields on the two lens surfaces and at a point outside the lens are calculated by repeated use of the diffraction integrals and the intensity distribution in the focal region of the lens is determined. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

本刊英文版Vol.35(2019),No.7论文摘要（英文）

正Rearrangements in Carnot Groups Juan J.MANFREDI Virginia N.VERA DE SERIO Abstract In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups.We define rearrangement with respect to a given family of     

N.M.R. and I.R. studies of 1, 1, 1 trifluoroacetone in proton donor solvents

Interactions of 1, 1, 1 trifluoroacetone with proton donor solvents of the type RH [where R=HO, CH₃S, CH₃CH₂CH₂CH₂S, CH₃O, (CH₃)₂N, (C₂H₅)₂N] have been investigated using NMR and infra-red techniques. Evidence of the formation of addition products of the type CF₃.C(OH)R.CH₃ where OH group is hydrogen-bonded with the fluorine of the CF₃ group has been obtained.     

Using Multibody Systems for the Investigation of Dynamic Aberrations in High Precision Optics

The coupling of multibody system dynamics and optical simulations by using ray tracing through moving lens systems will be summarized for both rigid and flexible lens systems. Furthermore, a method will be introduced for efficient simulations in the time domain. This method provides a direct integration of the optical simulation into the equations of motion of a multibody system. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO($n$) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.     

“Grandi” lenti in un reticolo di rette

A buffon problem is studied for a “thin lens” in the plane, a convex test body made by the union of two circular segments both less than or equal to a semicircle. The case in which the lens is “small” compared with the distance between the parallel lines of the Buffon lattice has been treated in [4]. The instance investigated in this paper is the one of a “large” lens, i.e. of a lens which can have multiple intersections with the Buffon lattice.

Lavoro eseguito col contributo parziale del M.U.R.S.T.     

Averaging in parabolic systems,subjected to weakly dependent random perturbations. The L1-approach

One considers an averaging method in equations of parabolic type, situated under the action of centered, weakly dependent random perturbations so that their integrals, normalized in an appropriate manner, satisfy S. N. Bernshtein's exponential estimate. For normalized fluctuations of the solution of the initial equation relative to the solution of the averaged equation, which turns out to be deterministic, one has established S. N. Bernshtein's exponential estimates. On the basis of the obtained inequalities, for an arbitrary prescribed confidence level, one can indicate a confidence band, whose bounds are determined by the solving of the averaged equation, which contains the solution of the initial problem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 167–172, February, 1991.     

On the knot complement problem for non-hyperbolic knots

This paper explicitly provides two exhaustive and infinite families of pairs (M,k), where M is a lens space and k is a non-hyperbolic knot in M, which produces a manifold homeomorphic to M, by a non-trivial Dehn surgery. Then, we observe the uniqueness of such knot in such lens space, the uniqueness of the slope, and that there is no preserving homeomorphism between the initial and the final M's. We obtain further that Seifert fibered knots, except for the axes, and satellite knots are determined by their complements in lens spaces. An easy application shows that non-hyperbolic knots are determined by their complement in atoroidal and irreducible Seifert fibered 3-manifolds.     

On L.N.Shevrin‘s Problem

In this short paper, our purpose is to give an affirmative answer to a problem raised by L. N. Shevrin in [1] for some special cases.     

Travel Time Tomography

We survey some results on travel time tomography. The question is whether we can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as geometry problems, the boundary rigidity problem and the lens rigidity problem. The boundary rigidity problem is whether we can determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points. The lens rigidity problem problem is to determine a Riemannian metric of a Riemannian manifold with boundary by measuring for every point and direction of entrance of a geodesic the point of exit and direction of exit and its length. The linearization of these two problems is tensor tomography. The question is whether one can determine a symmetric two-tensor from its integrals along geodesics. We emphasize recent results on boundary and lens rigidity and in tensor tomography in the partial data case, with further applications.     

高维非均匀核N.N.估计的强一致收敛速度

本文讨论了高维空间中非均匀核Ｎ．Ｎ．Ｗ估计的一致强收敛强度。在密度数的条件与［１，２］相同时，得到了比［１，２］更好的收敛速度，就其收敛的主要部分而言已无可改进。由于均匀核Ｎ．Ｎ．估计是非均匀核Ｎ．Ｎ．估计的特例，从而大大拓广了Ｎ．Ｎ．估计的理论价值和应用范围。     

Sagitta,Lenses, and Maximal Volume

We give a characterization of critical points that allows us to define a metric invariant on all Riemannian manifolds M with a lower sectional curvature bound and an upper radius bound. We show there is a uniform upper volume bound for all such manifolds with an upper bound on this invariant. We generalize results by Grove and Petersen by showing any such M that has volume sufficiently close to this upper bound is homeomorphic to the standard sphere \(S^{n}\) or a standard lens space \(S^n/{\mathbb {Z}}_m\) where \(m\in \{2,3,\ldots \}\) is no larger than an a priori constant.     

On Countable-Dimensional Spaces with the Menger Property, Rational Dimension and a Question of S. D. Iliadis

 This paper deals with the class of spaces which are countable unions of zero-dimensional sets and with the larger class of Haver’s C-spaces. All spaces are assumed to be separable and metrizable. We are concerned with various aspects of universality of these classes when they are combined with the covering analogue for σ-compactness defined by Menger and the rational dimension introduced by Menger and N?beling. A solution of a problem of S. D. Iliadis [16] concerning universal spaces for rational dimension will result. Received 11 September 1998; in revised form 12 January 1999     

Minimal triangulations for an infinite family of lens spaces

The notion of a layered triangulation of a lens space was definedby Jaco and Rubinstein, and unless the lens space is L(3,1),a layered triangulation with the minimal number of tetrahedrawas shown to be unique and termed its minimal layered triangulation.This paper proves that for each n 2, the minimal layered triangulationof the lens space L(2n, 1) is its unique minimal triangulation.More generally, the minimal triangulations (and hence the complexity)are determined for an infinite family of lens spaces containingthe lens space of the form L(2n, 1). Received May 16, 2008.