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This paper contains some general existence theorems for critical points of a continuously differentiable functional I on a real Banach space. The strongest results are for the case in which I is even. Applications are given to partial differential and integral equations.
We prove some matrix monotonicity and matrix convexity properties for functions derived from operator connections. As a consequence, we prove that a certain product of matrix monotone functions is a matrix monotone function. Similar results are proved for matrix convex functions. These results include some known results for matrix monotone and matrix convex functions.
We investigate the logic L(αα) which allows the second-order quantifier “ααs’ meaning “for almost all countable sets s.” We prove Replica Ghd Wholesale Completeness, Compactness, and Omitting Types Theorems and develop a Gentzen-style proof theory for this logic, as well as for the infinitary version LA(αα). Relations with various sublogics like L(Q) are discussed.
A proper circular-arc graph is a graph that has an intersection model formed by a family of overlapping arcs on some circle in which no arc contains another. A unit circular-arc graph is *Ghd Iv Straighteners Uk* a graph that has an intersection model formed by a family of unit-length arcs on some circle. This paper gives structure theorems for proper circular-arc graphs and for unit circular-arc graphs.