The minimal slope conjecture, which was proposed by K. S. Kedlaya, asserts that two irreducible overconvergent F-isocrystals on a smooth variety are isomorphic to each other if both constitutions of filtrations other. We affirmatively solve the conjecture for curves and $$\overline{{\mathbb {Q}}}_p$$ -F-isocrystals varieties over finite fields.

This paper is a reflection on the impacts of Coronavirus pandemic social life and draws Nietzsche’s views pessimism, will to power affirmation. The question that lies at its centre what it means experience with an affirmative or life-negating attitude. It aims open up space for discussion how actually can possibly be experienced affirmatively. In order do so, first all provides outline analysis...

In the monograph [2], the authors define the operator spectrum σ op (A) of a band-dominated operator A (these terms are defined below) and prove that A is Fredholm if and only if all of the operators in σ op (A) are invertible with uniformly bounded inverses. They also ask whether the uniform boundedness condition can in fact be dispensed with. In this note we answer this question affirmatively.

We prove that the homotopy class of a Morin mapping f : P p → Q with p − q odd contains a cusp mapping. This affirmatively solves a strengthened version of the Chess conjecture [5],[3]. Also, in view of the Saeki-Sakuma theorem [10] on the Hopf invariant one problem and Morin mappings, this implies that a manifold P p with odd Euler characteristic does not admit Morin mappings into R for p ≥ 2k...

In 1985 Fischer, Lynch, and Paterson (FLP) proved that consensus is not attainable in asynchronous message passing systems with failures. In the following years, Loui and Abu-Amara, and Herlihy proved that solving consensus in an asynchronous shared memory system is harder than implementing a read/write register. However, it was not known whether read/ write registers are implementable in async...

The inverse degree r(G) of a finite graph G = (V,E) is defined as r(G) = ∑ v∈V 1 d(v) , where d(v) is the degree of vertex v. In Discrete Math. 310 (2010), 940-946, Mukwembi posed the following conjecture: Let G be a connected chemical graph with diameter diam(G) and inverse degree r(G). Then diam(G) ≤ 12 5 r(G) +O(1). In this paper, we settle the conjecture affirmatively.