Minimal regular polygons serving as universal covers in R 2

A universal cover is a set K with the property that each set of unit diameter is a subset of a congruent copy of K. It is shown that the smallest regular n-gon, for fixed n 4, which serves as an universal cover in R ² is the smallest regular n-gon covering a Reuleaux triangle of unit width.     

On divergence of series of exponents representing functions regular in convex polygons

We prove that, on a convex polygon, there exist functions from the Smirnov classE whose series of exponents diverge in the metric of the spaceE. Similar facts are established for the convergence almost everywhere on the boundary of a polygon, for the uniform convergence on a closed polygon, and for the pointwise convergence at noncorner points of the boundary.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 443–445, April, 1994.     

Approximation of functions regular in convex polygons by exponential polynomials

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 4, pp. 446–452, July–August, 1988.     

On approximation of functions regular in convex polygons by exponential polynomials of special form

We construct exponential polynomials of special form which sufficiently well approximate functions that are regular in an open convex polygon and continuous on its closure.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 424–426, March, 1992.     

On the modality of convex polygons

Under two definitions of random convex polygons, the expected modality of a random convex polygon grows without bound as the number of vertices grows. This refutes a conjecture of Aggarwal and Melville.     

On induced matchings as star complements in regular graphs

We determine all the finite regular graphs which have an induced matching or a cocktail party graph as a star complement.     

On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

On the rate of convergence of double series of exponents representing regular functions on products of convex polygons

Estimates exact in order are obtained in the uniform and integral metrics for the deviation of partial sums of double series of exponents that represent functions which are regular on products of convex polygons and either continuous on a product of closed polygons or belonging to the Smirnov class.DeceasedTranslated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1271–1274, September, 1994.     

On p-harmonic maps and convex functions

We prove that, in general, given a p-harmonic map F : M → N and a convex function ${H : N \rightarrow \mathbb{R}}$ , the composition ${H\circ F}$ is not p-subharmonic, if p ≠ 2. This answers in the negative an open question arisen from a paper by Lin and Wei. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the p-harmonic map under suitable assumptions on the manifolds.     

On joins of a clique and a co-clique as star complements in regular graphs

Journal of Algebraic Combinatorics - In this paper we consider r-regular graphs G that admit the vertex set partition such that one of the induced subgraphs is the join of an s-vertex clique and a...     

On uniform approximation of abstract functions on convex polygons

Approximation aggregates of summatory type are constructed for functions f: Q → X, where Q is a convex polygon in ℝ² and X is a real Banach space. An assertion providing an accuracy estimate for an uniform approximation in terms of the modulus of continuity of the second order for f is proved. Differential properties of such aggregates are studied. Bibliography: 7 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 59–77     

On single-valuedness of convex set-valued maps

It is proved that ifX andY are linear spaces andF :X p(Y) is a set-valued map with convex graph such thatF(x) Ø for allx X andF(x ₀) is a singleton for somex ₀, thenF is single-valued and affine. Applications to metric projections and to adjoints of set-valued maps are given.Supported by NSF Grant DMS-9100228.The main result of this paper has been obtained while the second author was visiting the Pennsylvania State University in the framework of the exchange agreement between the Romanian Academy and the National Academy of Sciences of the U.S.A.     

The equidistribution of small points for strongly regular pairs of polynomial maps

In this paper, we prove the equidistribution of periodic points of a regular polynomial automorphism $f : \mathbb{A }^n \rightarrow \mathbb{A }^n$ defined over a number field $K$ : let $f$ be a regular polynomial automorphism defined over a number field $K$ and let $v\in M_K$ . Then there exists an $f$ -invariant probability measure $\mu _{f,v}$ on $\mathrm{Berk }\bigl ( \mathbb{P }^n_\mathbb{C _v} \bigr )$ such that the set of periodic points of $f$ is equidistributed with respect to $\mu _{f,v}$ .     

On G-arc-regular dihedrants and regular dihedral maps

A graph Γ is said to be G-arc-regular if a subgroup $G \le\operatorname{\mathsf{Aut}}(\varGamma)$ acts regularly on the arcs of Γ. In this paper connected G-arc-regular graphs are classified in the case when G contains a regular dihedral subgroup D _2n of order 2n whose cyclic subgroup C _n ≤D _2n of index 2 is core-free in G. As an application, all regular Cayley maps over dihedral groups D _2n , n odd, are classified.     

On a problem of Croft on optimally nested regular polygons

We present a solution for the largest regular m-gon contained in a regular n-gon. We find that the answer depends critically on the coprimality of m and n. We show that the optimal polygons are concentric if and only if gcd(m, n) > 1. Our principal result is a complete solution for the case where m and n share a common divisor. For the case of coprime m and n, we present partial results and a conjecture for the general solution. Our findings subsume some special cases which have previously been published on this problem.