Given an integer *k* ≧ 2 and a real number γ ∈ [0; 1], which graphs of edge density γ contain the largest number of *k*-edge stars? For *k* = 2 Ahlswede and Katona proved that asymptotically there cannot be more such stars than in a clique or in the complement of a clique (depending on the value of γ). Here we extend their result to all integers *k* ≧ 2.

It is proved that the set of all idempotent operations defined on a given set forms a Menger algebra which can be characterized by its densely embedded *v*-ideal. We also describe automorphisms of this algebra.

A ring *R* has the (*A*)-property (resp., strong (*A*)-property) if every finitely generated ideal of *R* consisting entirely of zero divisors (resp., every finitely generated ideal of *R* generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (*A*) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring *R*[*x*] and rings whose total ring of quotients are von Neumann regular. Let *f* : *A* → *B* be a ring homomorphism and *J* be an ideal of *B*. In this paper, we investigate when the (*A*)-property and strong (*A*)-property are satisfied by the amalgamation of rings denoted by *A* ⋈^{f}
*J*, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (*A*)-rings that are not strong (*A*)-rings, (*A*)-rings that are not Noetherian and (*A*)-rings whose total ring of quotients are not Von Neumann regular rings.

In this paper, a class *S*
_{s}(*q*) of close-to-convex functions is considered. Among the results studied for this class are its various characteristic properties such as the radius of convexity, certain bounds and coeffcient estimates. A suffcient condition for a function f to be in the class *S*
_{s}(*q*), is also obtained.

Multiplicative inverse transversals of regular semigroups were introduced by Blyth and McFadden in 1982. Since then, regular semigroups with an inverse transversal and their generalizations, such as regular semigroups with an orthodox transversal and abundant semigroups with an ample transversal, are investigated extensively in literature. On the other hand, restriction semigroups are generalizations of inverse semigroups in the class of non-regular semigroups. In this paper we initiate the investigations of E-semiabundant semigroups by using the ideal of "transversals". More precisely, we first introduce multiplicative restriction transversals for E-semiabundant semigroups and obtain some basic properties of E-semiabundant semigroups containing a multiplicative restriction transver- sal. Then we provide a construction method for E-semiabundant semigroups containing a multiplicative restriction transversal by using the Munn semigroup of an admissible quadruple and a restriction semigroup under some natural conditions. Our construction is similar to Hall's spined product construction of an orthodox semigroup. As a corollary, we obtain a new construction of a regular semigroup with a multiplicative inverse transversal and an abundant semigroup having a multiplicative ample transversal, which enriches the corresponding results obtained by Blyth-McFadden and El-Qallali, respectively.

In the Euclidean plane, the Erdős-Mordell inequality indicates that the sum of distances of an interior point of a triangle *T* to its vertices is larger than or equal to twice the sum of distances to the sides of *T*. We extend this theorem to arbitrary (normed or) Minkowski planes, and we generalize in an analogous way some other related inequalities, e.g. referring to polygons. We also derive Minkowskian analogues of Erdős-Mordell inequalities for tetrahedra and *n*-dimensional simplices. Finally, some related inequalities are obtained which additionally involve total edge-lengths of simplices.

We analyze the strong polarized partition relation with respect to several cardinal characteristics and forcing notions of the reals. We prove that random reals (as well as the existence of real-valued measurable cardinals) yield downward negative polarized relations.

Clean integral 2 × 2 matrices are characterized. Up to similarity the strongly clean matrices are completely determined and large classes of uniquely clean matrices are found. In particular, classes of uniquely clean matrices which are not strongly clean are found.

This paper is devoted to study the following Schrödinger-Poisson system
$$\{\begin{array}{l}-\Delta u+\left(\lambda a\left(x\right)+b\left(x\right)\right)u+K\left(x\right)\varphi u=f\left(u\right),x\in {\mathbb{R}}^{3},\\ -\Delta \varphi =4\pi K\left(x\right){u}^{2},x\in {\mathbb{R}}^{3},\end{array}$$
where λ is a positive parameter, *a* ∈ *C*(R^{3},R^{+}) has a bounded potential well Ω = *a*
^{−1}(0), *b* ∈ *C*(R^{3}, R) is allowed to be sign-changing, *K* ∈ *C*(R^{3}, R^{+}) and *f* ∈ *C*(R, R). Without the monotonicity of *f*(*t*)=/|*t*|^{3} and the Ambrosetti-Rabinowitz type condition, we establish the existence and exponential decay of positive multi-bump solutions of the above system for $$\lambda \ge \overline{\Lambda}$$, and obtain the concentration of a family of solutions as λ →+∞, where $$\overline{\Lambda}>0$$ is determined by terms of *a*, *b*, *K* and *f*. Our results improve and generalize the ones obtained by C. O. Alves, M. B. Yang [3] and X. Zhang, S. W. Ma [38].

Let *R* be a ring with an endomorphism *σ* and *F* ∪ {0} the free monoid generated by *U* = {*u*
_{1}, ..., *u _{t}*} with 0 added, and

*M*=

*F*∪ {0}/(

*I*) where

*I*is the set of certain monomial in

*U*such that

*M*

^{n}= 0, for some

*n*. Then we can form the non-semiprime skew monoid ring

*R*[

*M*; σ]. An element

*a*∈

*R*is uniquely strongly clean if

*a*has a unique expression as

*a*=

*e*+

*u*, where

*e*is an idempotent and

*u*is a unit with

*ea*=

*ae*. We show that a σ-compatible ring

*R*is uniquely clean if and only if

*R*[

*M*; σ] is a uniquely clean ring. If

*R*is strongly π-regular and uniquely strongly clean, then

*R*[

*M*; σ] is uniquely strongly clean. It is also shown that idempotents of

*R*[

*M*; σ] (and hence the ring

*R*[

*x*; σ]=(

*x*

^{n})) are conjugate to idempotents of

*R*and we apply this to show that

*R*[

*M*; σ] over a projective-free ring

*R*is projective-free. It is also proved that if

*R*is semi-abelian and σ(

*e*) =

*e*for each idempotent

*e*∈

*R*, then

*R*[

*M*; σ] is a semi-abelian ring.