An equilateral four point property which characterizes generalized euclidean spaces

Various characterizations of generalized euclidean spaces among complete metric spaces which contain a metric line joining each two points make use of euclidean four point properties, which require that every quadruple from a suitably chosen class of quadruples of points of the metric space be isometric with a quadruple of euclidean points. The present paper shows if every quadruple which contains an equilateral triple and a linear triple is embeddable, the space is generalized euclidean.     

On combinatorial properties of spheres in euclidean spaces

For λ>√2 there exists a triangle-free graphG such that for nod canG be imbedded into thed-dimensional unit sphere with two points joined if and only if their distance is >λ.     

An alternating extragradient method with non euclidean projections for saddle point problems

In this work we analyze a first order method especially tailored for smooth saddle point problems, based on an alternating extragradient scheme. The proposed method is based on three successive projection steps, which can be computed also with respect to non Euclidean metrics. The stepsize parameter can be adaptively computed, so that the method can be considered as a black-box algorithm for general smooth saddle point problems. We develop the global convergence analysis in the framework of non Euclidean proximal distance functions, under mild local Lipschitz conditions, proving also the \(\mathcal {O}(\frac{1}{k})\) rate of convergence on the primal–dual gap. Finally, we analyze the practical behavior of the method and its effectiveness on some applications arising from different fields.     

Continuity properties of decomposable probability measures on euclidean spaces

It is shown that every full e^A decomposable probability measure on R^k, where A is a linear operator all of whose eigenvalues have negative real part, is either absolutely continuous with respect to Lebesgue measure or continuous singular with respect to Lebesgue measure. This result is used to characterize the continuity properties of random variables that are limits of solutions of certain stochastic difference equations.     

A new class of four-point properties which characterize euclidean spaces

Characterizations of generalized euclidean spaces by means of euclidean four-point properties.state that every metric space which is complete, and which contains a metric line joining each two of its points is a generalized euclidean space if and only if each quadruple from a certain class of quadruples of the space is congruent with a quadruple of points in a euclidean space. It is known that it suffices to consider only quadruples containing a linear triple, or quadruples in which one of the linear points is a metric midpoint of the other two. Another class of four-point properties involves quadruples which contain a linear triple and a point equidistant from two of the linear points. The present paper presents three characterizations of euclidean spaces based on four-point properties in which the embedded quadruples contain a linear triple and some three of the distances determined by the four points are equal.     

On the mean square of the remainder for the euclidean lattice point counting problem

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal Γ-system has zero naive entropy in both senses, if Γ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.     

On euclidean domains

We consider Euclidean domains and their groups of units. Let K(a,b) be the set of remainders in the division of a by b. If Card K(a,b) = 1 for any a and b from a Euclidean domain R, then R is known to be isomorphic to the ring of polynomials over some field, see [4], [5]. On the other hand, the condition Card K(a,b) = 2 for any a and b implies that R is isomorphic to the ring Z of integers, see [2]. We give characterization of Euclidean domains and their groups of units under some other conditions on K(a,b).     

Extremal properties of nondifferentiable convex functions on euclidean sets of combinations with repetitions

A general approach is suggested for studying extremal properties of nondifferentiable convex functions on Euclidean combinatorial sets. On the basis of this approach, by solving the linear optimization problem on a set of combinations with repetitions, we obtain estimates of minimum values of convex and strongly convex objective functions in optimization problems on sets of combinations with repetitions and establish sufficient conditions for the existence of the corresponding minima. Kharkov Institute of Radioelectronics, Kharkov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 680–691, June, 1994.     

The euclidean bianchi group

The Bianchi Groups Γ_d are PSL₂(0_d) where 0_d is the ring of integers in the quadratic imaginary number field Q√-d with d a positive square-free rational integer. If d=1,2,3,7,11 0_d has a Euclidean algorithm and the corresponding groups are called the Euclidean Bianchi Groups. The group Γ₁ is the Picard Group has been studied independently. Here the subgroup structure of the remaining Euclidean Bianchi Groups is investigated. We show that they have a unique normal subgroup of index in if(n,6)=1 among other results. We also classify the abelian and nilpotent subgroups and discuss the structure of both congruence and non-congruence subgroups.     

Separating subadditive euclidean functionals

The classical Beardwood‐Halton‐Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form for the minimum length of a Traveling Salesperson Tour throuh n random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such Euclidean functionals on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through n random points in was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2‐factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation. We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch‐and‐bound algorithms based on these natural lower bounds must take nearly exponential () time to solve the TSP to optimality, even in average case. This is the first average‐case superpolynomial lower bound for these branch‐and‐bound algorithms (a lower bound as strong as was not even been known in worst‐case analysis). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 375–403, 2017