Variations on a theme of rationality of cycles

We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2 ^r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.     

Variations on a theme of Boole and Stein-Weiss

We give an alternative proof of a theorem of Stein and Weiss: The distribution function of the Hilbert transform of a characteristic function of a set E only depends on the Lebesgue measure |E| of such a set. We exploit a rational change of variable of the type used by George Boole in his paper “On the comparison of transcendents, with certain applications to the theory of definite integrals” together with the observation that if two functions f and g have the same Lp norm in a range of exponents p₁<p<p₂ then their distribution functions coincide.     

Variations on a theme of Jost and Pais

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set , , n2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in L²(Ω;dⁿx), , to modified Fredholm determinants associated with operators in L²(∂Ω;dⁿ⁻¹σ), n2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.     

Degree-bounded coloring of graphs: Variations on a theme by brooks

A graph G is degree-bounded-colorable (briefly, db-colorable) if it can be properly vertex-colored with colors 1,2, …, k ≤ Δ(G) such that each vertex v is assigned a color c(v) ≤ v. We first prove that if a connected graph G has a block which is neither a complete graph nor an odd cycle, then G is db-colorable. One may think of this as an improvement of Brooks' theorem in which the global bound Δ(G) on the number of colors is replaced by the local bound deg v on the color at vertex v. Extending the above result, we provide an algorithmic characterization of db-colorable graphs, as well as a nonalgorithmic characterization of db-colorable trees. We briefly examine the problem of determining the smallest integer k such that G is db-colorable with colors 1, 2,…, k. Finally, we extend these results to set coloring. © 1995, John Wiley & Sons, Inc.     

Variations on a theme of Debs and Saint Raymond

A famous theorem of Debs and Saint Raymond states that the complementof a set of first category is of strong multiplicity. We provea theorem which combines this with a result of Rudin which statesthat independent closed sets of strong multiplicity exist. Wealso prove a theorem which combines the theorem of Debs andSaint Raymond with a theorem of Wiener and Wintner which statesthat there exists a measure with singular support whose convolutionsquare is absolutely continuous.     

Variations on the theme of FC-groups

The following is clearly equivalent to the usual definition of FC-group. A group is an FC-group, if each of its cyclic subgroups has only finitely many conjugates. We consider several weaker conditions on the conjugates of cyclic subgroups, the strongest of which we show is equivalent to the FC-condition for many classes of groups.     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

Variations on a theme: introducing new representations of fraction into two classrooms

Abstract

The interplay between generalisations and particular instances—examples—is an essential feature of mathematics teaching and learning. In this paper, we bring together our experiences of personal and classroom mathematics activity, and demonstrate that examples do not always fulfil their intended purpose (to point to generalisations). A distinction is drawn between ‘empirical’ and ‘structural’ generalisation, and the role of generic examples is discussed as a means of supporting the second of these qualities of generalisation.     

Variations on the theme of solvability by radicals

We discuss the problem of representability and nonrepresentability of algebraic functions by radicals. We show that the Riemann surfaces of functions that are the inverses of Chebyshev polynomials are determined by their local behavior near branch points. We find lower bounds on the degrees of equations to which sufficiently general algebraic functions can be reduced by radicals. We also begin to classify rational functions of prime degree whose inverses are representable by radicals. Original Russian Text ? A.G. Khovanskii, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 86–105. To Vladmir Igorevich Arnold, mathematical idol of my generation     

Variations on the theme of the Carlson theorem

Translated from Matematicheskie Zametki, Vol. 52, No. 5, pp. 49–55, November, 1992.     

Variations on the theme of solvability by radicals

We discuss the problem of representability and nonrepresentability of algebraic functions by radicals. We show that the Riemann surfaces of functions that are the inverses of Chebyshev polynomials are determined by their local behavior near branch points. We find lower bounds on the degrees of equations to which sufficiently general algebraic functions can be reduced by radicals. We also begin to classify rational functions of prime degree whose inverses are representable by radicals.     

The Sommerfeld half-plane problem revisited,IV: Variations on a theme of Carlson and Heins

A plane wave is incident upon an infinite set of equally spaced, semi-infinite parallel and staggered plates. The boundary conditions on the plates alternate between the Dirichlet and Neumann ones. This problem is formulated as a pair of coupled Wiener-Hopf integral equations and solved by a method proposed by A. E. Heins in 1950. For the case of specular reflection, that is, a single reflected plane wave, the magnitudes of the reflection coefficient and the transmission coefficients are determined.     

Variations on the theme of slacks-based measure of efficiency in DEA

In DEA, there are typically two schemes for measuring efficiency of DMUs; radial and non-radial. Radial models assume proportional change of inputs/outputs and usually remaining slacks are not directly accounted for inefficiency. On the other hand, non-radial models deal with slacks of each input/output individually and independently, and integrate them into an efficiency measure, called slacks-based measure (SBM). In this paper, we point out shortcomings of the SBM and propose four variants of the SBM model. The original SBM model evaluates efficiency of DMUs referring to the furthest frontier point within a range. This results in the hardest score for the objective DMU and the projection may go to a remote point on the efficient frontier which may be inappropriate as the reference. In an effort to overcome this shortcoming, we first investigate frontier (facet) structure of the production possibility set. Then we propose Variation I that evaluates each DMU by the nearest point on the same frontier as the SBM found. However, there exist other potential facets for evaluating DMUs. Therefore we propose Variation II that evaluates each DMU from all facets. We then employ clustering methods to classify DMUs into several groups, and apply Variation II within each cluster. This Variation III gives more reasonable efficiency scores with less effort. Lastly we propose a random search method (Variation IV) for reducing the burden of enumeration of facets. The results are approximate but practical in usage.     

The Monge problem in $ {\mathbb{R}^d} $: Variations on a theme

In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in \mathbbR^d {\mathbb{R}^d} for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.     

Variations at infinity in contact form geometry

Let v be a nonsingular Morse–Smale vector field in the kernel of a contact form α, with Reeb vector field , defined on M³. We establish that the associated variational problem at infinity defined by the action functional on the stratified space of curves made of -pieces of orbits alternating with -pieces of orbits satisfies the Palais–Smale condition. This result takes a more special form for the standard contact structure of S³. Dedicated to Felix Browder on his eightieth birthday     

Track geometry defect rectification based on track deterioration modelling and derailment risk assessment

Analysing track geometry defects is critical for safe and effective railway transportation. Rectifying the appropriate number, types and combinations of geo-defects can effectively reduce the probability of derailments. In this paper, we propose an analytical framework to assist geo-defect rectification decision making. Our major contributions lie in formulating and integrating the following three data-driven models: (1) A track deterioration model to capture the degradation process of different types of geo-defects; (2) A survival model to assess the dynamic derailment risk as a function of track defect and traffic conditions; (3) An optimization model to plan track rectification activities with two different objectives: a cost-based formulation (CF) and a risk-based formulation (RF). We apply these approaches to solve the optimal rectification planning problem for a real-world railway application. We show that the proposed formulations are efficient as well as effective, as compared with existing strategies currently in practice.