Related papers: Some New Positive Observations

In this note we prove a more general (and topological) version of Gr\"unbaum's conjecture about affine invariant points. As an application of our result we show that, if we consider the action of the group of similarities, Gr\"unbaum's…

Let $a,b$ and $n$ be positive integers with $a>b$. In this note, we prove that $$(2bn+1)(2bn+3){2bn \choose bn}\bigg|3(a-b)(3a-b){2an \choose an}{an\choose bn}.$$ This confirms a recent conjecture of Amdeberhan and Moll.

We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing…

We give an alternative proof of a (former) conjecture of Bj\"orner stating that the matrix expressing face numbers in terms of g numbers is totally non-negative. We briefly discuss the case of simple flag polytopes.

Generalizing a conjecture by De Loera et al., we conjecture that integral generalized permutohedra all have positive Ehrhart coefficients. Berline and Vergne construct a valuation that assigns values to faces of polytopes, which provides a…

We prove certain identities involving Euler and Bernoulli polynomials that can be treated as recurrences. We use these and also other known identities to indicate connection of Euler and Bernoulli numbers with entries of inverses of certain…

In this article we have studied bicomplex valued measurable functions on an arbitrary measurable space. We have established the bicomplex version of Lebesgue's dominated convergence theorem and some other results related to this theorem.…

We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…

The density operator of the arbitrary physical system must be positive definite. Employing the general master equation technique which preserves this property we derive equations of motion for the density operator of an active atom which…

The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation in a family of canonically po-larised…

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$…

This article extends to the vector setting the results of our previous work Kruger et al. (2015) which refined and slightly strengthened the metric space version of the Borwein-Preiss variational principle due to Li and Shi, J. Math. Anal.…

Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…

Interpretations for the q-binomial coefficient evaluated at -q are discussed. A (q,t)-version is established, including an instance of a cyclic sieving phenomenon involving unitary spaces.

The Shub-Smale Tau Conjecture is a hitherto unproven statement (on integer roots of polynomials) whose truth implies both a variant of $P\neq NP$ (for the BSS model over C) and the hardness of the permanent. We give alternative conjectures,…

In this paper, we prove the existence portion of the Bertram-Feinberg-Mukai Conjecture for an infinite family of new cases using degeneration technique. This not only leads to a substantial improvement of known results but also develops…

We define some generalizations of the classical descent and inversion statistics on signed permutations that arise from the work of Sack and Ulfarsson [20] and called after width-k descents and width-k inversionsof type A in Davis's work…

In this paper, we first establish two new Bailey pairs via finding two generalizations of Euler's pentagonal number theorem. Next, we specificize the Bailey lemmas with these two Bailey pairs. As applications, we finally establish some…

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for…

Based on computeralgebra experiments we formulate a refined version of Green's conjecture and a conjecture of Schicho-Schreyer-Weimann which conjecturally also holds in positive characteristic. The experiments are done by using our…