In this paper, the concepts of $L$-concave structures, concave $L$-interior operators and concave $L$-neighborhood systems are introduced. It is shown that the category of $L$-concave spaces and the category of concave $L$-interior spaces are isomorphic, and they are both isomorphic to the category of concave $L$-neighborhood systems whenever $L$ is a completely distributive lattice. Also, it i...

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert’s method we show that for any integers d and r such that 4 ≤ r ≤ 2d − 2d, there is a non-singular hyperbolic curve of degree 2d in R with exactly r line segments in the boundary of its convex hull. We also give a complete classification of possib...

Let k be a locally compact topological field of positive characteristic. Let L be a cocompact discrete additive subgroup of k. Let U be an open compact additive subgroup of k. Let l, u and a be elements of k, with a nonzero. We study the behavior of the product ∏ 06=x∈(l+L)∩a(u+U) x as a varies, using tools from local class field theory and harmonic analysis. Typically ratios of such products o...

The notion of a k-convex ∆-support function for a toric variety X(∆) is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the ∆-support function ψL. Equivalently L is proved to be k-jet ample if and only if the restriction at each invariant curve has degree at least k.

A Polish group G is called a group of quasi-invariance or a QI-group, if there exist a locally compact group X and a probability measure μ on X such that 1) there exists a continuous monomorphism of G to X, and 2) for each g ∈ X either g ∈ G and the shift μg is equivalent to μ or g 6∈ G and μg is orthogonal to μ. It is proved that G is a σ-compact subset of X. We show that there exists a quotie...

We prove that any standard parabolic subgroup of any Artin group is convex with respect to the standard generating set.

In the frame of fractional calculus, term convexity is primarily utilized to address several challenges in both pure and applied research. The main focus objective this review paper present Hermite–Hadamard (H-H)-type inequalities involving a variety classes convexities pertaining integral operators. Included various are classical convex functions, m-convex r-convex (α,m)-convex (α,m)-geometric...

‎For a homogeneous spaces ‎$‎G/H‎$‎, we show that the convolution on $L^1(G/H)$ is the same as convolution on $L^1(K)$, where $G$ is semidirect product of a closed subgroup $H$ and a normal subgroup $K $ of ‎$‎G‎$‎. ‎Also we prove that there exists a one to one correspondence between nondegenerat $ast$-representations of $L^1(G/H)$ and representations of ...

A Kleinian group Γ is a discrete subgroup of PSL2(C). When non-elementary, such a group possesses a unique non-empty minimal closed invariant subset ΛΓ of the Riemann sphere, called the limit set. A Kleinian group acts properly discontinuously on the complement ∆Γ of ΛΓ and so this set is called the domain of discontinuity. Such a group is said to be convex co-compact if it acts co-compactly on...

A Kleinian group Γ is a discrete subgroup of PSL2(C). When non-elementary, such a group possesses a unique non-empty minimal closed invariant subset ΛΓ of the Riemann sphere, called the limit set. A Kleinian group acts properly discontinuously on the complement ∆Γ of ΛΓ and so this set is called the domain of discontinuity. Such a group is said to be convex co-compact if it acts co-compactly on...