We prove Trudinger-type inequalities for variable Riesz potentials I alpha(& centerdot;),tau f of functions in Musielak-Orlicz-Morrey spaces L Phi,kappa,theta(X) over bounded non-doubling metric measure spaces X, under conditions on 'I which are weaker than those considered in the previous paper by Hurri-Syrja & uml;nen and the authors in 2023. We also discuss the case when 'I is the double-phase functional with variable exponents.
Motivated by the well-known graphical method, we give a geometric characterization of optimal linear programs. Our condition and approach rely on the tools of convex analysis. Among the applications, we present the strong duality theorem and revisit Farkas' lemma, as well. Although the auxiliary tools are well-known or follow from highly nontrivial results, we present their (independent) proof in order to keep the paper self-contained.
In this paper, we investigate a generalized Randers metric with a special it-form. Precisely, following the pullback approach to global Finsler geometry, we start with a Finsler metric (M, L) that admits a concurrent it-vector field, then we consider the Randers change Le = L + B, where B is the associated 1-form. We study some of the geometric objects and properties attached to (M, Le). We prove that the corresponding metric tensor of Le is nondegenerate without any conditions on the 1-form B. By calculating the relation between the attached geodesic sprays of L and Le, we establish that the change Le = L + B preserves the geodesics of (M, L), that is, the change is projective. Moreover, under a certain condition, this change preserves the deviation tensor, the (v)h-torsion tensor, the (h)h-curvature tensor and the curvature tensor of the Barthel connection. We provide an example of a Finsler metric with some details that admits a concurrent vector field together with the associated it-form.
Let 0 =/ D subset of R-n be a convex set. We say that the function f : D -> R is weakly k-affine if, for any affinely independent system of k+1 points x(0), x(1), ... , x(k) is an element of D, there exist lambda(0), lambda(1), ... , lambda(k) = 1 and (lambda(0)x(0 )+ lambda(1)x(1)+& centerdot;& centerdot;& centerdot;+lambda(k)x(k)) = lambda(0)f(x(0)) +lambda(1)f(x(1)) +& centerdot;& centerdot;& centerdot;+lambda(k)f(x(k)). Our main result is that any continuous, weakly 2-affine function is necessarily affine. It is a well-known result that a continuous, weakly 1-affine function is affine. However, in the paper, we will show that if continuity is replaced by several weaker regularity conditions, then this implication fails to hold. We also introduce a new concept of generalized convexity, namely the class of weakly k-convex sets, which turns out to be naturally related to weakly k-affine functions. We present results concerning a subclass of continuous, weakly k-affine functions for k >= 3, as well.
We answer two alienation questions involving the cosine and sine addition formulas on a semigroup.~The findings in both cases reveal a large degree of dependence between these two functional equations.
