Related papers: The $abc$ Conjecture implies the Weak Diversity Co…

Brou\'{e}'s Abelian Defect Conjecture predicts interesting derived equivalences between derived categories of modular representations of finite groups. We investigate a generalization of Brou\'{e}'s Conjecture to ring spectrum coefficients…

In this article we introduce the notion of weak identities in a group and study their properties. We show that weak identities have some similar properties to ordinary ones. We use this notion to prove that any finitely generated solvable…

An algebra with identities $[a,b]c=2a(bc)-2b(ac), a[b,c]=2(ab)c-2(ac)b$ is called weak Leibniz. We show that weak Leibniz operad is self-dual and is not Koszul. We establish that polarization of any weak Leibniz algebra is transposed…

We present the universal theory of weak crossed biproducts, and we prove that every weak projection of weak bialgebras induces an example of this crossed structure. As an example, we give the construction of a weak projection of a weak…

We continue our investigation on pcf with weak form of choice. Characteristically we assume DC + P(Y) when looking and prod_{s in Y} delta_s. We get more parallel of theorems on pcf.

Fundamental discrepancy between first order logic and statistical inference (global versus local properties of universe) is shown to be the obstacle for integration of logic and probability in L.p. logic of Bacchus. To overcome the…

We prove that number fields with arbitrary degree but weak ramification satisfy the Leopoldt conjecture on the l-adic rank of the group of units

We show that the existence of a weakly compact cardinal over the Zermelo-Fraenkel's set theory is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued…

The celebrated union-closed conjecture is concerned with the cardinalities of various subsets of the Boolean $d$-cube. The cardinality of such a set is equivalent, up to a constant, to its measure under the uniform distribution, so we can…

The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than $2^{\alpeh_0}$, so under the Continuum Hypothesis such a family is necessarily countable. However, it is…

We investigate a combinatorial game on $\omega_1$ and show that mild large cardinal assumptions imply that every normal ideal on $\omega_1$ satisfies a weak version of precipitousness. As an application, we show that that the…

We prove three main conjectures of Berkovich and Uncu (Ann. Comb. 23 (2019) 263--284) on the inequalities between the numbers of partitions of $n$ with bounded gap between largest and smallest parts for sufficiently large $n$. Actually our…

In this paper, we refute some conjectures of Marques and Lengyel concerning the $p$-adic valuations of members of the Tribonacci sequence.

We show that the Volume Conjecture for polyhedra implies a weak version of the Stoker Conjecture; in turn we prove that this weak version of the Stoker conjecture implies the Stoker conjecture. The main tool used is an extension of a result…

We prove the Fargues-Rapoport conjecture for p-adic period domains in the non-basic case with minuscule cocharacter. More precisely, we give a group theoretical criterion for the cases when the admissible locus and weakly admissible locus…

We consider a generalisation of the Basilica group to all odd primes: the $p$-Basilica groups acting on the $p$-adic tree. We show that the $p$-Basilica groups have the $p$-congruence subgroup property but not the congruence subgroup…

The Axiom of Dependent Choice $\mathsf{DC}$ and the Axiom of Countable Choice $\mathsf{AC}_\omega$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\mathsf{DC}(X)$ asserts that any total binary relation on…

The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c)…

We discuss the ACC conjecture and the LSC conjecture for minimal log discrepancies of generalized pairs. We prove that some known results on these two conjectures for usual pairs are still valid for generalized pairs. We also discuss the…