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Extending Blaschke and Lebesgue’s classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width *D*. We prove an essentially optimal stability version of this statement in each of the three types of surfaces of constant curvature. In addition, we summarize the fundamental properties of convex bodies of constant width in spaces of constant curvature, and provide a characterization in the hyperbolic case in terms of horospheres.

Let P be a set of n points in general position in the plane. Let *R* be a set of points disjoint from P such that for every *x, y € P* the line through *x* and *y* contains a point in *R*. We show that if *c* in the plane, then *P* has a special property with respect to the natural group structure on *c*. That is, *P* is contained in a coset of a subgroup *H* of c of cardinality at most |*R*|.

We use the same approach to show a similar result in the case where each of *B* and *G* is a set of n points in general position in the plane and every line through a point in *B* and a point in *G* passes through a point in *R*. This provides a partial answer to a problem of Karasev.

The bound

Let *E, G* be Fréchet spaces and *F* be a complete locally convex space. It is observed that the existence of a continuous linear not almost bounded operator *T* on *E* into *F* factoring through *G* causes the existence of a common nuclear Köthe subspace of the triple (*E, G, F*). If, in addition, *F* has the property (*y*), then (*E, G, F*) has a common nuclear Köthe quotient.

In this paper we study the sum *n*, and {*n _{p}
*} is a sequence of integers indexed by primes. Under certain assumptions we show that the aforementioned sum is

In this paper we derive new inequalities involving the generalized Hardy operator. The obtained results generalized known inequalities involving the Hardy operator. We also get new inequalities involving the classical Hardy–Hilbert inequality.

The bipartite domination number of a graph is the minimum size of a dominating set that induces a bipartite subgraph. In this paper we initiate the study of this parameter, especially bounds involving the order, the ordinary domination number, and the chromatic number. For example, we show for an isolate-free graph that the bipartite domination number equals the domination number if the graph has maximum degree at most 3; and is at most half the order if the graph is regular, 4-colorable, or has maximum degree at most 5.

This study proposes a new family of continuous distributions, called the Gudermannian generated family of distributions, based on the Gudermannian function. The statistical properties, including moments, incomplete moments and generating functions, are studied in detail. Simulation studies are performed to discuss and evaluate the maximum likelihood estimations of the model parameters. The regression model of the proposed family considering the heteroscedastic structure of the scale parameter is defined. Three applications on real data sets are implemented to convince the readers in favour of the proposed models.

Let [ · ] be the fioor function. In this paper, we show that when 1 < c < 37/36, then every sufficiently large positive integer *N* can be represented in the form

where p_{1}, p_{2}, p_{3} are primes close to squares.

In this article, we study a family of subgraphs of the Farey graph, denoted as *ℱ _{N}
* for every

*N*∈ ℕ. We show that

*ℱ*is connected if and only if

_{N}*N*is either equal to one or a prime power. We introduce a class of continued fractions referred to as

*ℱ*-continued fractions for each

_{N}*N*> 1. We establish a relation between

*ℱ*-continued fractions and certain paths from infinity in the graph

_{N}*ℱ*. Using this correspondence, we discuss the existence and uniqueness of

_{N}*ℱ*-continued fraction expansions of real numbers.

_{N}Given a finite point set *P* in the plane, a subset S⊆P is called an *island* in *P* if conv(S) ⋂ *P = S*. We say that S ⊂ *P* is a *visible island* if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any *k ≥* 3 and *l* ≥ 4, there is an integer *n = n(k, l*), such that every finite set of at least *n* points in the plane contains *l* collinear points or *k* pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.