In this paper, we investigate the 2-adic valuation of the Stirling numbers S(n, k) of the second kind. We show that v2(S(2n + 1, k + 1)) = s2(n) − 1 for any positive integer n, where s2(n) is the sum of binary digits of n. This confirms a conjecture of Amdeberhan, Manna and Moll. We show also that v2(S(4i, 5)) = v2(S(4i + 3, 5)) if and only if i 6≡ 7 (mod 32). This proves another conjecture of ...

In this paper, we study the conjecture of Kühnel and Lutz, who state that a combinatorial triangulation of the product of two spheres S×S with j ≥ i is tight if and only if it has exactly i+2j+4 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when ...

The sensitivity of a Boolean function f is the maximum, over all inputs x, of the number of sensitive coordinates of x (namely the number of Hamming neighbors of x with different f -value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree poly(s). The best known u...

Let S(n) be the set of all polynomials of degree n with all roots in the unit disk, and define d(P ) to be the maximum of the distances from each of the roots of a polynomial P to that root’s nearest critical point. In this notation, Sendov’s conjecture asserts that d(P ) ≤ 1 for every P ∈ S(n). Define P ∈ S(n) to be locally extremal if d(P ) ≥ d(Q) for all nearby Q ∈ S(n), and note that identi...

It is well known that if D is a bounded derivation on a Banach algebra A and if s is an element of A satisfying [s; Ds] = 0 then Ds must be quasinilpotent. The unbounded Kleinecke-Shirokov conjecture states that the same result holds even if D is unbounded. As yet, the conjecture has neither been proved nor has a case been found where it fails. If there is a counterexample we obtain a further r...

One denotes by S(d) the collection of all torsion primes of degree d. The well known strong uniform boundedness conjecture states that for every d, the set S(d) is nite. The main results of [KaMa-92] summarize as follows: for all d, S(d) is of density zero, and for d 8, S(d) is nite. It should be mentioned that the original conjecture is not restricted to prime levels, but in [KaMa-92] it is sh...

We prove the 0-(co)homology part of the conjecture on the cup-products on tangent cohomology in the Tsygan formality [Sh2]. We discuss its applications to the Duflo formula. A short introduction The Tsygan formality conjecture for chains [Ts] was proven in the author’s work [Sh2] by an explicit construction of suitable Kontsevich-type integrals. This paper is a further development of ideas of [...

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorie...

Ž . For a ring R with identity, define Unip R to be the group of upper-triangular n matrices over R all of whose diagonal entries are 1. For i 1, 2, . . . , n 1, let Si denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and Ž . Ž . i 1 st column. Then for any integer m including m 0 , it is easy to see that Ž . the S generate Unip Z mZ . Reiner gave relations among th...

Bazzoni’s Conjecture states that the Prüfer domain R has finite character if and only if R has the property that an ideal of R is finitely generated if and only if it is locally principal. In [4] the authors use the language and results from the theory of lattice-ordered groups to show that the conjecture is true. In this article we supply a purely ring theoretic proof. 1. Bazzoni’s Conjecture ...