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Very recently, the authors in [5] proposed the exponential-type operator connected with x 4 3 and studied its convergence estimates. In the present research, we extend the study and obtain the general form of its ๐‘-th order moment; ๐‘ โˆˆ โ„• โˆช {0}. Further, we establish the simultaneous approximation for the operator under consideration.

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A bi-cyclic 4-polytope in โ„4 was introduced by Z. Smilansky as the convex hull of evenly spaced points on a generalized trigonometric moment curve in โ„4. We present combinatorial geometric conditions that yield the face lattices of a class of such 4-polytopes.

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A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body ๐พ, the areas of the maximum (resp. minimum) area convex ๐‘›-gons inscribed (resp. circumscribed) in ๐พ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex ๐‘›-gons by disk-๐‘›-gons, obtained as the intersection of ๐‘› closed Euclidean unit disks. It has been proved recently that if ๐ถ is the unit disk of a normed plane, then the same properties hold for the area of ๐ถ-๐‘›-gons circumscribed about a ๐ถ-convex disk ๐พ and for the perimeters of ๐ถ-๐‘›-gons inscribed or circumscribed about a ๐ถ-convex disk ๐พ, but for a typical origin-symmetric convex disk ๐ถ with respect to Hausdorff distance, there is a ๐ถ-convex disk ๐พ such that the sequence of the areas of the maximum area ๐ถ-๐‘›-gons inscribed in ๐พ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of ๐ถ.

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Koppรกny Encz
,
Mรกrton Marits
,
Benedek Vรกli
, and
Mรกtรฉ Weisz

A ๐‘ž-graph with ๐‘’ edges and ๐‘› vertices is defined as an ๐‘’ ร— ๐‘› matrix with entries from {0, โ€ฆ , ๐‘ž}, such that each row of the matrix (called a ๐‘ž-edge) contains exactly two nonzero entries. If ๐ป is a ๐‘ž-graph, then ๐ป is said to contain an ๐‘ -copy of the ordinary graph ๐น, if a set ๐‘† of ๐‘ž-edges can be selected from ๐ป such that their intersection graph is isomorphic to ๐น, and for any vertex ๐‘ฃ of ๐‘† and any two incident edges ๐‘’, ๐‘“ โˆˆ ๐‘† the sum of the entries of ๐‘’ and ๐‘“ is at least ๐‘ . The extremal number ex(๐‘›, ๐น, ๐‘ž, ๐‘ ) is defined as the maximal number of edges in an ๐‘›-vertex ๐‘ž-graph such that it does not contain contain an ๐‘ -copy of the forbidden graph ๐น.

In the present paper, we reduce the problem of finding ex(๐‘›, ๐น, ๐‘ž, ๐‘ž + 1) for even ๐‘ž to the case ๐‘ž = 2, and determine the asymptotics of ex(๐‘›, ๐ถ2๐‘˜+1, ๐‘ž, ๐‘ž + 1).

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Studia Scientiarum Mathematicarum Hungarica
Authors:
David Conlon
,
Jacob Fox
,
Xiaoyu He
,
Dhruv Mubayi
,
Andrew Suk
, and
Jacques Verstraรซte

For positive integers ๐‘›, ๐‘Ÿ, ๐‘  with ๐‘Ÿ > ๐‘ , the set-coloring Ramsey number ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) is the minimum ๐‘ such that if every edge of the complete graph ๐พ๐‘ receives a set of ๐‘  colors from a palette of ๐‘Ÿ colors, then there is guaranteed to be a monochromatic clique on ๐‘› vertices, that is, a subset of ๐‘› vertices where all of the edges between them receive a common color. In particular, the case ๐‘  = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) which imply that ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) = 2ฮ˜(๐‘›๐‘Ÿ) if ๐‘ /๐‘Ÿ is bounded away from 0 and 1. The upper bound extends an old result of Erdล‘s and Szemerรฉdi, who treated the case ๐‘  = ๐‘Ÿ โˆ’ 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

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We consider a function from the Euclidean three space whose zero set is the image of the standard cuspidal edge. The composition of a parametrized singular surface in the three space with this function provides an approximation of the surface by the standard cuspidal edge. Taking a look at singularities of this composition, we study various approximations of singular surfaces like the cross cap, the generalized cuspidal edge and the swallowtail by standard cuspidal edges.

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Based on Peterโ€™s work from 2003, quadrilaterals can be characterized in the following way: โ€œamong all quadrilaterals with given side lengths ๐‘Ž, ๐‘, ๐‘ and ๐‘‘, those of the largest possible area are exactly the cyclic onesโ€. In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties.

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In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight ๐œ” on the unit disc are given, including the growth, the higher order or free derivative descriptions, and integral characterizations of functions in the space.

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We present examples of multiplicative semigroups of positive reals (Beurlingโ€™s generalized integers) with gaps bounded from below.

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In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

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