Pointwise monotonicity of heat kernels

In this paper we present an elementary proof of a pointwise radial monotonicity property of heat kernels that is shared by the Euclidean spaces, spheres and hyperbolic spaces. The main result was discovered by Cheeger and Yau in 1981 and rediscovered in special cases during the last few years. It deals with the monotonicity of the heat kernel from special points on revolution hypersurfaces. Our proof hinges on a non straightforward but elementary application of the parabolic maximum principle. As a consequence of the monotonicity property, we derive new inequalities involving classical special functions.

An Elementary Proof to eγ log log n > Ψ(n)/n for any n > 30

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

On two relations characterizing the golden ratio

V.G. Zhuravlev found two relations associated with the golden ratio: $\tau=\frac{1+\sqrt{5}}{2}$: $[([i\tau]+1)\tau]=[i\tau^2]+1$ and $[[i\tau]\tau]+1=[i\tau^2]$. We give a new elementary proof of these relations and show that they give a characterization of the golden ratio. Further we consider satisfability of our relations for finite sets of $i$-s and establish some forcing property for this situation.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

An Elementary Proof to the Riemann Hypothesis

Let Ƥ be the set of all primes, Ψ/(n) = nIIn∈Ƥ,p|n (1 + 1/Ƥ) be the Dedekind psi function, we unconditionally show that eγ log log n > Ψ(n)/n for any n > 30, where γ if Euler's constant.

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

Bipolar behavior of submodular, law-invariant capacities

In the case of a submodular, law-invariant capacity, we provide an entirely elementary proof of a result of Marinacci [M. Marinacci, Upper probabilities and additivity, Sankhyā Ser. A 61 1999, no. 3, 358–361]. As a corollary, we also show that the anticore of a continuous submodular, law-invariant nonatomic capacity has a dichotomous nature: either it is one-dimensional or it is infinite-dimensional. The results have implications for the use of such capacities in financial and economic applications.