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In this paper, we present a new approach to finding the minimal representation of dc-functions in the sense of Zalgaller. We introduce the notion of maximal pair of convex sets and show that support functions of this pair are the minimal representation of a given dc-function. For this purpose, we study the properties of Minkowski addition and subtraction in the family of closed convex sets sharing their recession cone. We show how to find maximal pair equivalent to a given one. [ABSTRACT FROM AUTHOR]

Having a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions. [ABSTRACT FROM AUTHOR]

A growing crystal can be understood as a series of polyhedra with faces parallel to planes from a fixed finite set. Such series of polyhedra forms a curve, often broken line, in a vector space of virtual polyhedra. In this paper we apply a modified Minkowski addition to study geometry of a cone of all polyhedra with faces parallel the faces of regular octahedron. [ABSTRACT FROM AUTHOR]

The paper examines different kinds of p-convexity of a function g which are sufficient for the existence of a linear functional such that in Theorem 1.13 of Simons in his monograph ‘From Hahn-Banach to monotonicity’, published in Springer lecture notes (2008). We replace sublinearity of p with convexity, the field with Dedekind vector lattice and present -convexity which is also necessary. In Theorem 4.7 we also generalize a result of MM. Neumann from 1991 published in Czech. Mathem. Journal Vol 41 on the Mazur-Orlicz theorem. [ABSTRACT FROM AUTHOR]

We prove that in finite dimensional spaces every ordered median function is the Minkowski dual of a reduced pair of polytopes. This implies a very general theorem on the representation of an ordered median function as a uniquely determined difference of two sublinear functions up to adding and subtracting one and the same arbitrary sublinear function. [ABSTRACT FROM AUTHOR]

In this paper we present growing crystal as a point in finite dimentional space moving inside a cone SG ⊂ ℝm corresponding to the family BG of polyhedra with faces parallel to planes from a fixed finite set G. For this purpose we modify Minkowski addition of convex sets and explain in depth a representation of a growth of certain crystal of NaCl. We also provide a number of illustrations of a respective cone S'G of symmetric polyhedra and the trajectory of our growing crystal which is a broken line. [ABSTRACT FROM AUTHOR]

After a number of events, during World War II in Belarus collaboration of the local people with the German occupation authorities took place. One of the most obvious episodes of collaboration was the establishment and the activity of the Belarusian National Defence. The armed unit was formed at the end of spring and in the beginning of summer 1944 through mobilization and was intended for armed combat with Soviet partisans and the Red Army. The article analyzes the process of establishing the unit, its numbers, the degree of combat readiness and participation in warfare. In summer 1944, after withdrawal of German forces from Belarus some of the soldiers of the Belarusian National Defence turned up at the Western front. [ABSTRACT FROM AUTHOR]

In this article, we study symmetry properties of ordered median functions (S. Nickel and J. Puerto, Location Theory – A Unified Approach, Springer-Verlag, Berlin, Heidelberg, 2005). In particular, we prove that every ordered median function is a DCH-function (V.F. Demyanov and A.M. Rubinov, Quasidifferential Calculus, Optimization Software Inc., Publications Division, New York, 1986; D. Pallaschke and R. Urbański, Pairs of Compact Convex Sets – Fractional Arithmetic with Convex Sets, Mathematics and its Applications, Vol. 548, Kluwer Academic Publishers, Dordrecht, 2002), i.e. can be represented as a difference of two sublinear functions. Moreover, we give a necessary and sufficient condition for an ordered median function to be convex. [ABSTRACT FROM AUTHOR]

In this article we give, for an arbitrary continuous function which is linear on the octants of [image omitted] an explicit representation in terms of a max-min expression. Moreover, we also consider its representation as the difference of two sublinear functions and determine minimal representations in the sense of Pallaschke and Urbanski [D. Pallaschke and R. Urbanski, Pairs of Compact Convex Sets - Fractional Arithmetic with Convex Sets, Vol. 548, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 2002]. [ABSTRACT FROM AUTHOR]

During World War II, many residents of Belarus occupied by Nazi Germany being driven by the great variety of motives were drafted to the local auxiliary police (Schutzmannschaften). The total number of which in General District of Belarus accounted for some 30 thousand people in the years 1941-1944. In 1943 the 13th battalion of 1,000 strong auxiliary police SD (Schutzmannschafts Bataillon der SD 13) was formed, a subordinate formation of the Security Service (SD) SS. In years 1943-1944, this formation participated in numerous anti-partisan campaigns and pacification. After the withdrawal from Belarus in the summer of 1944 it took part in the suppression of Warsaw's Uprising and fought together with the Red Army in Poland and the resistance movements in Poland, Czechoslovakia, and Italy. [ABSTRACT FROM AUTHOR]

In this paper, we determine the cardinality of various subsets of the family of all minimal pairs of compact convex subsets of n (Theorems 2.1 and 2.8). Furthermore, we express minimality properties of a pair of closed bounded convex sets in set-theoretical terms (Theorem 3.3). [ABSTRACT FROM AUTHOR]

In this study, some relationships between quasidifferentiability and weakly subdifferentiability of a function are investigated, and some results on calculating the quasidifferential in terms of weak exhausters are presented. Moreover, under some assumptions weak subdifferentiability of a quasidifferentiable function is proved and the weak subdifferential is obtained by using the quasidifferential. Under some assumptions, it is proved that a positively homogeneous and lower semicontinuous function is quasidifferentiable. It is also shown that directional differentiability and weak subdifferentiability imply quasidifferentiability. Furthermore, a method to evaluate the quasidifferential is given by using reduced weak lower exhausters. [ABSTRACT FROM AUTHOR]

In this work, it is proved that weak exhausters of a positively homogeneous function can be reduced if weak subdifferential/superdifferential can be represented as the sum of a subset of it and weak subdifferential of zero function. It is also shown that weak exhausters can be reduced if this subset is the set of minimal elements of weak subdifferential/superdifferential with respect to weak subdifferential of zero function. At the end, some optimality conditions are given using reduced weak exhausters. [ABSTRACT FROM AUTHOR]