An elementary proof of the Loomis–Whitney theorem

Loomis and Whitney proved an inequality between volume and areas of projections of an open set in n-dimensional space related to the isoperimetric inequality. They reduced the problem to a combinatorial theorem proved by a repeated use of Hölder inequality. In this paper we prove a general inequality between real numbers which easily implies the combinatorial theorem of Loomis and Whitney.

     

An elementary proof of the Adelson—Panjer recursion formula

We present a new proof of the Adelson—Panjer recursion formula for computing discrete Compound Poisson densities. It uses only elementary probability theory and elementary arithmetic.     

An elementary proof of Tverberg’s theorem

A new proof is suggested for Tverberg’s familiar theorem saying that an arbitrary set of q: (d + 1)(p − 1) + 1 points in ℝ ^d can be split into p parts such that their convex hulls have a nonempty intersection. Bibliography: 9 titles.     

An elementary,illustrative proof of the Rado–Horn theorem

The Rado–Horn theorem provides necessary and sufficient conditions for when a family of vectors can be partitioned into a fixed number of linearly independent sets. Such partitions exist if and only if every subfamily of the vectors satisfies the so-called Rado–Horn inequality. In this paper we provide an elementary proof of the Rado–Horn theorem as well as results for the redundant case. Previous proofs give no information about how to actually partition the vectors; we use ideas present in our proof to find subfamilies of vectors which may be used to construct a kind of “optimal” partition.     

An elementary proof of a distribution relation¶on elliptic curves

We give another elementary proof of a certain identity of elliptic functions arising from the K-theory of elliptic curves and Wildeshaus's generalisation of Zagier's conjectures. This proof consists of a calculation with the q-expansions, and is offered in the hope that its more explicit flavour may be generalised to other situations. Received: 7 December 1999 / Revised version: 3 July 2000     

A new proof of Euler's formula for ζ(2n)

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ζ(2n).     

An elementary proof of Ramanujan’s circular summation formula and its generalizations

In this paper, we give a completely elementary proof of Ramanujan’s circular summation formula of theta functions and its generalizations given by S.H. Chan and Z.-G. Liu, who used the theory of elliptic functions. In contrast to all other proofs, our proofs are elementary. An application of this summation formula is given.     

Ghirlanda–Guerra identities and ultrametricity: An elementary proof in the discrete case

In this Note we give another proof of the fact that a random overlap array, which satisfies the Ghirlanda–Guerra identities and whose elements take values in a finite set, is ultrametric with probability one. The new proof bypasses random change of density invariance principles for directing measures of such arrays and, in addition to the Dovbysh–Sudakov representation, is based only on elementary algebraic consequences of the Ghirlanda–Guerra identities.     

An alternative proof of the Aubin–Lions lemma

Using a generalized version of the Weyl–Riesz criterion for compactness of subsets of Lebesgue–Bochner spaces, we present in this short note an alternative proof of a result by J. Simon [4] that extends the classical result by J.P. Aubin and J.L. Lions on compact embeddings in Lebesgue–Bochner spaces to the non-reflexive Banach space case.     

On the measure of irrationality of the number π

A new estimate of the measure of irrationality of the number π is obtained. The previous result (M. Hata, 1993) is improved by means of another integral construction.     

An elementary approach to the polynomial τ-functions of the KP Hierarchy

We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 23–36, January, 1999.     

Fast Computation of ζ(3) and of Some Special Integrals Using the Ramanujan Formula and Polylogarithms

The application of the Fast E-function Evaluation (FEE) method to fast calculation of the value of (3) and of some special integrals on the basis of the Ramanujan formula and its generalization is considered.This revised version was published online in October 2005 with corrections to the Cover Date.     

An elementary proof of the Karush–Kuhn–Tucker theorem in normed linear spaces for problems with a finite number of inequality constraints

We present an elementary proof of the Karush–Kuhn–Tucker theorem for the problem with a finite number of nonlinear inequality constraints in normed linear spaces under the linear independence constraint qualification. Most proofs in the literature rely on advanced concepts and results such as the convex separation theorem and Farkas, lemma. By contrast, the proofs given in this article, including a proof of the lemma, employ only basic results from linear algebra. The lemma derived in this article represents an independent theoretical result.     

Characteristic Foliations of Spheres Embedded in the Standard Overtwisted Structure (R3, ζ1)

The characteristic foliation of a sphere embedded in the standard tight contact structure (R³, ₀) is unique up to isotopy. We show that any Morse-Smale foliation on the sphere with null Euler class, is, up to isotopy, the characteristic foliation of a sphere embedded in the standard overtwisted contact structure (R³, ₁). We thus have a new way of looking at the two standard structures as opposites in the world of contact structures.     

Sums of the error term function in the mean square for ζ(s)

Sums of the form are investigated, where is the error term in the mean square formula for . The emphasis is on the case k = 1, which is more difficult than the corresponding sum for the divisor problem. The analysis requires bounds for the irrationality measure of e^2πm and for the partial quotients in its continued fraction expansion. Authors’ addresses: Y. Bugeaud, Université Louis Pasteur, Mathématiques, 7 rue René Descartes, F-67084 Strasbourg cedex, France; A. Ivić, Katedra Matematike RGF-a, Universitet u Beogradu, Đušina 7, 11000 Beograd, Serbia     

Another elementary proof that p(11n+6)≡0 (mod 11)

In this paper we give an elementary proof of the partition congruence $p(11n+6)\equiv 0\ (\operatorname{mod}\ 11)$ , using only Euler’s Pentagonal Number Theorem and Jacobi’s Identity for $(q;q)^{3}_{\infty}$ .     

A simple and elementary proof of the Karush–Kuhn–Tucker theorem for inequality-constrained optimization

We present an elementary proof of the Karush–Kuhn–Tucker Theorem for the problem with nonlinear inequality constraints and linear equality constraints. Most proofs in the literature rely on advanced optimization concepts such as linear programming duality, the convex separation theorem, or a theorem of the alternative for systems of linear inequalities. By contrast, the proof given here uses only basic facts from linear algebra and the definition of differentiability.