Elementary proofs of recent congruences for overpartitions wherein non-overlined parts are not divisible by 6

This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps \(F(x,y)\) defined on a finite-dimensional Euclidean space. There are no hypotheses on the continuity of the partial derivatives of \(F\). The proof employs the mean-value theorem, the intermediate-value theorem, Darboux’s property (the intermediate-value property for derivatives), and determinants theory. The proof avoids compactness arguments, fixed-point theorems, and Lebesgue’s measure. A stronger than the classical version of the Inverse Function Theorem is also shown. Two illustrative examples are given.

We give an elementary direct proof of the following property: if for a discrete group G G some l p ( G ) {l_p}(G) -space ( 1 > p > ∞ ) (1 > p > \infty ) is an algebra, then all elements of G G have uniformly bounded order.

A sphere that is tangent to all four face-planes (i.e., the affine hulls of the faces) of a tetrahedron is called a tangent sphere of the tetrahedron. Two tangent spheres are called neighboring if exactly one face-plane separates them. Grace’s theorem states that for a pair of neighboring tangent spheres S and T of a tetrahedron there is a unique sphere such that (1) passes through the three vertices of the tetrahedron lying on the face-plane that separates S and T, and (2) is either externally tangent to both S, T or internally tangent to both S, T. It seems that this theorem is not widely known, and that no elementary proof has been given. The purpose of this article is to present an elementary and direct proof of this theorem in the case of a trirectangular tetrahedron, and to obtain several further results in this direction. Among them is also the confirmation of a slightly generalized form of Grace’s theorem.

An elementary proof of double Greibach normal form

It is pointed out that in many one-sided testing situations for a real-valued parameter θ, the monotonicity of the power function hinges on the stochastic order of the underlying family of distributions [Fθ] rather than on the stronger property of monotone likelihood ratio of the family. An elementary proof, accessible to students of introductory probability and statistics, is presented.

We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpiński carpet is greater than 2, no proof of which in this generality had previously been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uniform distribution) in this case, proved first by M. Hino in [Probab. Theory Related Fields 132 (2005), no. 2, 265–290].

We present a direct and elementary proof that all the solutions of the Painleve Equations I, II and IV are meromorphic functions on the whole complex plane. The proof uses some ideas from the existing proofs but applies the ideas in a different setting.

Some Rapidly Growing Functions ,

We give an elementary proof that, in its domain of definition, the time-p map of a scalar, autonomous holomorphic, complex differential equation, is itself holomorphic. This result is used by Sverdlove [1] when considering limit cycles in complex holomorphic differential equations. However no proof or reference for the result is given in [1]. Although this result must be well established, a proof does not appear to be readily accessible in the reference literature.

"An Elementary Proof That Primes are Scarce." The American Mathematical Monthly, 77(4), pp. 396–397

In this paper we will give a short and elementary proof that critical relations unfold transversally in the space of rational maps.

Let M be an Abelian W*-algebra of operators on a Hilbert space H. Let M0 be the set of all linear, closed, densely defined transformations in H which commute with every unitary operator in the commutant M’ of M. A well known result of R. Pallu de Barriere states that if ɸ is a normal positive linear functional on M, then ɸ is of the form T → (Tx, x) for some x in H, where T is in M. An elementary proof of this result is given, using only those properties which are consequences of the fact that ReM is a Dedekind complete Riesz space with plenty of normal integrals. The techniques used lead to a natural construction of the class M0, and an elementary proof is given of the fact that a positive self-adjoint transformation in M0 has a unique positive square root in M0. It is then shown that when the algebraic operations are suitably defined, then M0 becomes a commutative algebra. If ReM0 denotes the set of all self-adjoint elements of M0, then it is proved that ReM0 is Dedekind complete, universally complete Riesz spaces which contains ReM as an order dense ideal. A generalization of the result of R. Pallu de la Barriere is obtained for the Riesz space ReM0 which characterizes the normal integrals on the order dense ideals of ReM0. It is then shown that ReM0 may be identified with the extended order dual of ReM, and that ReM0 is perfect in the extended sense. Some secondary questions related to the Riesz space ReM are also studied. In particular it is shown that ReM is a perfect Riesz space, and that every integral is normal under the assumption that every decomposition of the identity operator has non-measurable cardinal. The presence of atoms in ReM is examined briefly, and it is shown that ReM is finite dimensional if and only if every order bounded linear functional on ReM is a normal integral.

(1976). An Elementary Proof that the Unit Disc is a Swiss Cheese. The American Mathematical Monthly: Vol. 83, No. 7, pp. 552-554.

(1984). An Elementary Proof That the Hilbert Cube is Compact. The American Mathematical Monthly: Vol. 91, No. 9, pp. 563-564.

Publisher Summary The purpose of this chapter is pure iconoclasm, the focus being on some rapidly growing functions. When the mathematician says “large,” the logician is sure to think “small.” The first cliche usually resorted to in discussions of largeness is the Skewes number. The Skewes number has toppled from its position of supremacy. In 1955, Skewes showed how to lower the bound if one still assumed the Riemann Hypothesis, but he saved his reputation by obtaining the even larger upper bound. The Ketonen–Solovay elementary proof, like elementary proofs of theorems of analytic number theory, is somewhat longer than the Paris-Harrington proof. However, a bit of the flavor of their proof by showing that H(x + 1, x, x) eventually majorizes all functions F n for finite n. The Ketonen–Solovay elementary proof, like elementary proofs of theorems of analytic number theory, is somewhat longer than the Paris–Harrington proof. The proponents of rapid growth do not stop here, but they seek ever more rapidly growing functions and ever more powerful principles to produce such functions.