Related papers: A proof of Smale's mean value conjecture
The notion of Ces\`aro stable function is generalized by introducing Ces\`aro mean of type $(b-1;c)$ which give rise to a new concept of generalized Ces\`aro stable function. As an application of generalized Ces\`aro stable functions we…
In this article we study the arithmetic mean value $\Sigma(n)$ of the square roots of the first $n$ integers. For this quantity, we develop an asymptotic expression, and derive a formula for its integer part which has been conjectured…
In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is…
This is a survey on Sarnak's Conjecture
The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…
In this paper, we give a proof of the DDVV conjecture which is a pointwise inequality involving the scalar curvature, the normal scalar curvature and the mean curvature on a submanifold of a real space form. Furthermore we solved the…
A simple proof of Atanassov's Conjecture is presented. Atanassov's Conjecture is a generalization of Sperner's Lemma, a lemma which has been used to prove Brouwer's Fixed Point Theorem, among other fixed point theorems. The proof of…
A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] +…
In this paper we establish an exponential covering theorem implying a conjecture formulated by A. Zygmund circa 1935 whose three-dimensional case was obtained by the first named author in 1978.
We will give a simple proof of the ambiguous class number formula.
We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's…
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
In this note, while giving an overview of the state of art of the well known Hadamard conjecture, which is more than a century old and now it has been established by using the methods given in the two papers by Mohan et al [6,7].
The estimation of a probability p from repeated Bernoulli trials is considered in this paper. A sequential approach is followed, using a simple stopping rule. A closed-form expression and an upper bound are obtained for the mean absolute…
Mathematical proofs should be paired with formal proofs, whenever feasible.
In this paper we prove Shapiro's 1958 Conjecture on exponential polynomials, assuming Schanuel's Conjecture.
We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's…
We give new proofs on Arnold Chord Conjecture and Weinstein Conjecture in M\times C which generalizes the previous works.
This paper presents a simple proof of Dekel (1986)'s representation theorem for betweenness preferences. The proof is based on the separation theorem.