Onto extensions of free groups

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.

Short Character Sums and the Pólya–Vinogradov Inequality

We show in a quantitative way that any odd primitive character χ modulo q of fixed order g ≥ 2 satisfies the property that if the Pólya–Vinogradov inequality for χ can be improved to $$\begin{equation*} \max_{1 \leq t \leq q} \left|\sum_{n \leq t} \chi(n)\right| = o_{q \rightarrow \infty}(\sqrt{q}\log q) \end{equation*}$$ then for any ɛ > 0 one may exhibit cancellation in partial sums of χ on the interval [1, t] whenever $t \gt q^{\varepsilon}$, i.e., $$\begin{equation*} \sum_{n \leq t} \chi(n) = o_{q \rightarrow \infty}(t)\ \text{for all } t \gt q^{\varepsilon}. \end{equation*}$$ We also prove a converse implication, to the effect that if all odd primitive characters of fixed order dividing g exhibit cancellation in short sums then the Pólya–Vinogradov inequality can be improved for all odd primitive characters of order g. Some applications are also discussed.

General Tauberian theorems for the Cesàro integrability of functions

For a locally integrable function f on {[0,\infty)}, we defineF(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!dufor {t>0}. The improper integral {\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be {(C,\alpha)} integrable to L for some {\alpha>-1} if the limit {\lim_{x\to\infty}\sigma_{\alpha}(t)=L} exists. It is known that {\lim_{t\to\infty}F(t)=\ell} implies {\lim_{t\to\infty}\sigma_{\alpha}(t)=\ell} for {\alpha>-1}, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the {(C,\alpha)} integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,\alpha) integrability of functions, Positivity 21 2017, 1, 73–83].

Bell-shaped dose response for a system with no IFFLs

It is well known that the presence of an incoherent feedforward loop (IFFL) in a network may give rise to a steady state non-monotonic dose response. This note shows that the converse implication does not hold. It gives an example of a three-dimensional system that has no IFFLs, yet its dose response is bell-shaped. It also studies under what conditions the result is true for two-dimensional systems, in the process recovering, in far more generality, a result given in the T-cell activation literature.

TAUBERIAN CONDITIONS FOR $q$-CES\`{A}RO INTEGRABILITY

Given a $q$-integrable function $f$ on $[0, \infty)$, we define $s(x)=\int_{0}^{x}f(t)d_qt$ and $\sigma(s(x))=\frac{1}{x}\int _{0}^{x} s(t)d_{q}t$ for $x>0$. It is known that if $\lim _{x \to \infty}s(x)$ exists andis equal to $A$, then $\lim _{x \to \infty}\sigma(s(x))=A$. But the converse of this implication is not true in general. Our goal is to obtain Tauberian conditions imposed on the general control modulo of $s(x)$ under which the converse implication holds. These conditions generalize some previously obtained Tauberian conditions.

Solving an Open Problem About the G-Drazin Partial Order

G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: if A is below B under the G-Drazin partial order then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, we solve completely this problem. We show that the converse, in general, is false and we provide a form to construct counterexamples. We also prove that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.

Ball characterizations in spaces of constant curvature

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

Universality and Almost Decidability

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result—namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable—solves an open problem in [9]. We conclude with some open problems.

The phiS polar decomposition when the cosquare of S is nonderogatory

For $S \in GL_n$, define $\phi_S: M_n \rightarrow M_n$ by $\phi_S(A) = S^{-1}A^TS$. A matrix $A \in M_n$ is $\phi_S$ \textit{orthogonal} if $\phi_S(A) = A^{-1}$; $A$ is $\phi_S$ \textit{symmetric} if $\phi_S(A) = A$; $A$ has a $\phi_S$ \textit{polar decomposition} if $A = ZY$ for some $\phi_S$ orthogonal $Z$ and $\phi_S$ symmetric $Y$. If $A$ has a $\phi_S$ polar decomposition, then $A$ commutes with the cosquare $S^{-T}S$. Conditions under which the converse implication holds for the case where $S^{-T}S$ is nonderogatory, are obtained.

On Fuzzy Ideals ofBL-Algebras

In this paper we investigate further properties of fuzzy ideals of aBL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of aBL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime idealωis a fuzzy irreducible ideal if and only ifω0=1and|Im⁡(ω)|=2. We give the Krull-Stone representation theorem of fuzzy ideals inBL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of aBL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.