(0.1) F∞(u, U) := ‖|Du|‖L∞(U) among all such functions. Here |Du| is the Euclidean length of the gradient Du of u. We will also be interested in the “Lipschitz constant” functional as well. If K is any subset of IR and u : K → IR, its least Lipschitz constant is denoted by (0.2) Lip(u,K) := inf {L ∈ IR : |u(x)− u(y)| ≤ L|x− y| ∀ x, y ∈ K} . Of course, inf ∅ = +∞. Likewise, if any definition suc...

We prove that a preference relation which is continuous on every straight line has a utility representation if its domain is a convex subset of a finite dimensional vector space. Our condition on the domain of a preference relation is stronger than Eilenberg (1941) and Debreu (1959, 1964), but our condition on the continuity of a preference relation is strictly weaker than theirs. JEL classific...

A subset X in the k-dimensional Euclidean space R that contains n points (elements) is called an n-point isosceles set if every triplet of points selected from them forms an isosceles triangle. In this paper, we show that there exist exactly two 11-point isosceles sets up to isomorphism and that the maximum cardinality of isosceles sets in R is 11 .

In 1981 Walter Rudin exhibited a totally real embedding of the Klein bottle into C2. We show that the polynomially convex hull of Rudin’s Klein bottle contains an open subset of C2. We also describe another totally real Klein bottle in C2 whose hull has topological dimension equal to three.

a subgroup $h$ is said to be $nc$-supplemented in a group $g$ if there exists a subgroup $kleq g$ such that $hklhd g$ and $hcap k$ is contained in $h_{g}$, the core of $h$ in $g$. we characterize the supersolubility of finite groups $g$ with that every maximal subgroup of the sylow subgroups is $nc$-supplemented in $g$.

suppose that $h$ is a subgroup of $g$‎, ‎then $h$ is said to be‎ ‎$s$-permutable in $g$‎, ‎if $h$ permutes with every sylow subgroup of‎ ‎$g$‎. ‎if $hp=ph$ hold for every sylow subgroup $p$ of $g$ with $(|p|‎, ‎|h|)=1$)‎, ‎then $h$ is called an $s$-semipermutable subgroup of $g$‎. ‎in this paper‎, ‎we say that $h$ is partially $s$-embedded in $g$ if‎ ‎$g$ has a normal subgroup $t$ such that $ht...

This paper investigates the complexity of steepest descent algorithms for two classes of discrete convex functions, M-convex functions and L-convex functions. Simple tie-breaking rules yield complexity bounds that are polynomials in the dimension of the variables and the size of the effective domain. Combination of the present results with a standard scaling approach leads to an efficient algor...

Abstract In this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ Lp -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML"><m...

Let L be a commutative Moufang loop (CML) with multiplication group M, and let F(L), F(M) be the Frattini subgroup and Frattini subgroup of L and M respectively. It is proved that F(L) = L if and only if F(M) = M and is described the structure of this CLM. Constructively it is defined the notion of normalizer for subloops in CML. Using this it is proved that if F(L) 6= L then L satisfies the no...