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## Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) *modular* on a set *X* is a function *w*:(0,*∞*)×*X*×*X*→[0,*∞*] satisfying, for all *x*,*y*,*z*∈*X*, the following three properties: *x*=*y* if and only if *w*(*λ*,*x*,*y*)=0 for all *λ*>0; *w*(*λ*,*x*,*y*)=*w*(*λ*,*y*,*x*) for all *λ*>0; *w*(*λ*+*μ*,*x*,*y*)≤*w*(*λ*,*x*,*z*)+*w*(*μ*,*y*,*z*) for all *λ*,*μ*>0. We show that, given *x*0∈*X*, the set *X**w*={*x*∈*X*:lim*λ*→*∞**w*(*λ*,*x*,*x*0)=0} is a metric space with metric

The notion of a metric modular on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces and Orlicz spaces, were recently introduced and studied by the author [Chistyakov: Dokl. Math. 73(1):32–35, 2006 and Nonlinear Anal. 72(1):1–30, 2010]. In this chapter we present yet one more application of the metric modulars theory to the existence of fixed points of modular contractive maps in modular metric spaces. These are related to contracting generalized average velocities rather than metric distances, and the successive approximations of fixed points converge to the fixed points in the modular sense, which is weaker than the metric convergence. We prove the existence of solutions to a Carathéodory-type differential equation with the right-hand side from the Orlicz space. Metric modular, Modular convergence, Modular contraction, Fixed point, Mapping of finite f-variation, Carathéodory-type differential equation

Given two points *a*=(*a*_{1},…,*a**n*) and *b*=(*b*_{1},…,*b**n*) from *R**n* with *a*<*b* componentwise and a map *f* from the rectangle into a metric semigroup *M*=(*M*,*d*,+), we study properties of the *total variation* of *f* on introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of bounded variation of several variables, in: Real Analysis Exchange 27th Summer Symposium, Opava, Czech Republic, 2003, pp. 217–222] such as the additivity, generalized triangle inequality and sequential lower semicontinuity. This extends the classical properties of C. Jordan's total variation (*n*=1) and the corresponding properties of the total variation in the sense of Hildebrandt [T.H. Hildebrandt, Introduction to the Theory of Integration, Academic Press, 1963] (*n*=2) and Leonov [A.S. Leonov, On the total variation for functions of several variables and a multidimensional analog of Helly's selection principle, Math. Notes 63 (1998) 61–71] (*n*∈*N*) for real-valued functions of *n* variables.

Similarity searching has a vast range of applications in various fields of computer science. Many methods have been proposed for exact search, but they all suffer from the curse of dimensionality and are, thus, not applicable to high dimensional spaces. Approximate search methods are considerably more efficient in high dimensional spaces. Unfortunately, there are few theoretical results regarding the complexity of these methods and there are no comprehensive empirical evaluations, especially for non-metric spaces. To fill this gap, we present an empirical analysis of data structures for approximate nearest neighbor search in high dimensional spaces. We provide a comparison with recently published algorithms on several data sets. Our results show that small world approaches provide some of the best tradeoffs between efficiency and effectiveness in both metric and non-metric spaces.

Given a=(a1,…,an), b=(b1,…,bn)∈Rn with ab componentwise and a map f from the rectangle Iab=[a1,b1]×⋯×[an,bn] into a metric semigroup M=(M,d,+), denote by TV(f,Iab) the Hildebrandt–Leonov total variation of f on Iab, which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}j∈Nof maps fromIabinto M is such that the closure in M of the set{fj(x)}j∈Nis compact for eachx∈IabandC≡supj∈NTV(fj,Iab)is finite, then there exists a subsequence of{fj}j∈N, which converges pointwise onIabto a map f such thatTV(f,Iab)⩽C. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M∗.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.