An elementary proof of the Lagrange multiplier theorem in normed linear spaces

We present an elementary proof of the Lagrange multiplier theorem for optimization problems with equality constraints in normed linear spaces. Most proofs in the literature rely on advanced concepts and results, such as the implicit function theorem and the Lyusternik theorem. By contrast, the proof given in this article employs only basic results from linear algebra, the critical-point condition for unconstrained minima and the fact that a continuous function attains its minimum over a closed ball in the finite-dimensional space.     

A penalty function proof of a Lagrange multiplier theorem with application to linear delay systems

Using a penalty function method, a Lagrange multiplier theorem in dual Banach spaces is proved. This theorem is applied to the optimal control of linear, autonomous time-delay systems with function space equality end condition and pointwise control restrictions. Under an additional regularity condition, the resulting Lagrange multiplier can be identified with an element ofL .     

A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by m (respectively, m*) the number of lattice points in the boundary of M (respectively, in the boundary of the dual polygon). Then m + m* = 12.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 117–120.Original Russian Text Copyright © 2005 by D. Repov, M. Skopenkov, M. Cencelj.This revised version was published online in April 2005 with a corrected issue number.     

A short proof of the twelve-point theorem

We present a short elementary proof of the following twelve-point theorem. Let M be a convex polygon with vertices at lattice points, containing a single lattice point in its interior. Denote by m (respectively, m*) the number of lattice points in the boundary of M (respectively, in the boundary of the dual polygon). Then m + m* = 12.     

A short proof of the Wedderburn-Artin theorem

We give a short proof of the Wedderburn–Artin theorem from the viewpoint of completely reducible modules.     

A short proof of Fan's theorem

In this note, we give a new short proof of the following theorem: Let G be a 2-connected graph of order n. If for any two vertices u and v with d(u,v)=2,max{d(u),d(v)}?c/2, then the circumference of G is at least c, where 3?c?n and d(u,v) is the distance between u and v in G.     

A short proof of the Dennis-Schnabel theorem

It is shown on the basis of simple orthogonality relations that all least change secant updates under additional linear constraints can be represented as projected rank one or rank two corrections.     

A short proof of the Arendt-Chernoff-Kato theorem

We give a short new proof of the Arendt-Chernoff-Kato theorem, which characterizes generators of positive C ₀ semigroups in order unit spaces. The proof avoids half-norms and subdifferentials and is based on a sufficient condition for an operator to have positive inverse, which is new even for matrices.     

A short proof of a general multiplier rule

A general multiplier rule obtained by Hestenes is shown to follow directly from the Lagrange multiplier rule by a simple compactness argument. A similar simplification is effected in the proof of Gittleman's extension of Hestenes' rule.     

A short proof of the generalized Bonnet theorem

In three‐dimensional Euclidean space E³, the Bonnet theorem says that a curve on a ruled surface in three‐dimensional Euclidean space, having two of the following properties, has also a third one, namely, it can be a geodesic, that it can be the striction line, and that it cuts the generators under constant angle. In this work, in n dimensional Euclidean space Eⁿ, a short proof of the theorem generalized for (k + 1) dimensional ruled surfaces by Hagen in 4 is given. Copyright © 2013 John Wiley & Sons, Ltd.     

A short proof of the jordan decomposition theorem

We present a short proof of the Jordan Decomposition Theorem