On an assumption of geometric foundation of numbers

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying.     

Circle geometry in affine Cayley-Klein planes

Some theorems from inversive and Euclidean circle geometry are extended to all affine Cayley-Klein planes. In particular, we obtain an analogue to the first step of Clifford’s chain of theorems, a statement related to Napoleon’s theorem, extensions of Wood’s theorem on similar-perspective triangles and of the known fact that the three radical axes of three given circles are parallel or have a point in common. For proving these statements, we use generalized complex numbers. Supported by a grant D01-761/24.10.06 from the Ministry of Education and Sciences, and by a grant 108/2007 from Sofia University.     

On the constitutive relations for isotropic and transversely isotropic materials

In this work the strain and stress spaces constitutive relations for isotropic and transversely isotropic softening materials are developed. The loading surface is considered in the strain space and the normality rule; the stress relaxation is proportional to the gradient of the loading surface, is adopted. It is found that the strain space plasticity theory allows us to describe the hardening, perfectly plastic and softening materials more accurately. The validity of the strain space constitutive relation for transversely isotropic materials are confirmed by comparing with the experimental data for fiber reinforced composite materials. Some numerical examples in two and three dimensional elasto-plastic problems for various loading–unloading conditions are presented, and give a very good agreement with the existing results.     

On the quantitative steinitz theorem in the plane

We prove that for anyk ≥ 4, any setX of points in the plane, and any pointP ε interior conv(X), there is a subsetY (X of at mostk points such that if conv(X) contains a disk with radiusr aroundP, then conv(Y) contains a disk with radius [cos(2/(k + 1))π]/[cos(1/(k + 1))π]r aroundP. This generalizes the quantitative Steinitz theorem in the plane and proves a conjecture of Bárány and Heppes. Dedicated to Professor Dr. Jürgen Flachsmeyer on the occasion of his sixtieth birthday     

On the Alexandroff decomposition theorem

We prove an Alexandroff decomposition type theorem, which extends a decomposition theorem proved in [de LUCIA, P.—MORALES, P.: Decomposition of group-valued measures in orthoalgebras, Fund. Math. 158 (1998), 109–124].      

On the generalizations of Denjoy-Wolff theorem

We consider generalizations of Denjoy-Wolff theorem on strongly pseudo-convex domains. Beardon [3] gave a general Denjoy-Wolff theorem in hyperboli domain in the complex plane. Our main results are generalizations of Beardon's result in a higher dimensional setting.     

On the Trotter-Kato theorem in a variable space

In this paper, the proof of a Trotter-Kato type theorem in a variable Banach space is given and some special cases and examples are considered.     

ON THE CENTRAL LIMIT THEOREM IN PRODUCT SPACES

ＯＮＴＨＥＣＥＮＴＲＡＬＬＩＭＩＴＴＨＥＯＲＥＭＩＮＰＲＯＤＵＣＴＳＰＡＣＥＳＳＵＺＨＯＮＧＧＥＮＡｂｓｔｒａｃｔ：ＳｕｐｐｏｓｅｔｈａｔＥａｎｄＦａｒｅｓｅｐａｒａｂｌｅＢａｎａｃｈｓｐａｃｅｓ，ＸａｎｄＹａｒｅｉｎｄｅｐｅｎｄｅｎｔｓｙｍｍｅｔｒ...     

An analog of the Baum-Katz theorem for weakly dependent random variables

In this paper we study the limiting behavior of sums of dependent random variables under a strong mixing condition. We obtain conditions for which an analog of the Baum-Katz theorem holds and cite an example showing their optimality. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 360–368, March, 2000.     

On the exact constant in the quantitative Steinitz theorem in the plane

We determine the maximal value ofr with the following property. If the convex hull of a set inR ² contains a unit circleB, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radiusr concentric withB. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed aroundB. Imre Bárány was partially supported by Hungarian National Science Foundation Grant Nos. 1907 and 1909. Aladár Heppes was partially supported by Hungarian National Science Foundation Grant No. 2583.     

On extensions of the domain invariance theorem

If f maps continuously a compact subset X of Rⁿ into Rⁿ and x is a point whose distance from the boundary ∂X is greater than double diameter of the fibres of the points in f(∂X) then f(x) is in the interior of f(X). This theorem extends some results due to Borsuk and Sitnikov.     

On the Mazur-Ulam theorem in non-Archimedean fuzzy normed spaces

We prove that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space.     

On the Zudilin-Rivoal theorem

We propose a new method for proving the Zudilin-Rivoal theorem stating, in particular, that the sequence of values of the Dirichlet beta function at even natural points contains infinitely many irrational values. For polylogarithms, we use Hermite—Padé approximations of the first type, invariant with respect to the Klein group. Quantitative additions to this theorem are obtained.     

Some subsets of points in the plane associated to truncated Reed-Muller codes with good parameters

We gives some examples of subsets of points in the projective plane associated to truncated generalized projective Reed-Muller codes with good parameters, of dimensions 6 and 10 over GF(7), GF(8) and GF(9).     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.