In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.

In the study of certain prime Noetherian rings it is natural to consider the set C of elements which are regular modulo all height-1 prime ideals of R. For R commutative, this set C is simply the set of units. In general this is not the case, though with certain additional conditions we can state non-commutative versions of the Principal Ideal Theorem.

The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus 4 with unique trigonal structure.

If K is a field with a non-trivial non-Archimedean absolute value (multiplicative norm) | |, we describe all non-Archimedean K-algebra norms on the polynomial algebra K[X1, . . . , Xr] which extend | |.

We answer a question of Skalski and Sołan (2016) about inner faithfulness of the Curran’s map of extending a quantum increasing sequence to a quantum permutation. Roughly speaking, we find a inductive setting in which the inner faithfulness of Curran’s map can be boiled down to inner faithfulness of similar map for smaller algebras and then rely on inductive generation result for quantum permutation groups of Brannan, Chirvasitu and Freslon (2018).

In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

We prove that a semigroup generated by finitely many truncated convolution operators on Lp[0, 1] with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n=3 and n=4, we provide an explicit definition of this cup product and prove its basic properties.

We classify all non-affine Hopf algebras H over an algebraically closed field k of characteristic zero that are integral domains of Gelfand–Kirillov dimension two and satisfy the condition Ext1H(k, k) ≠ 0. The affine ones were classified by the authors in 2010 (Goodearl and Zhang, J. Algebra324 (2010), 3131–3168).

In [12], Loy and Miller proved that a locally compact, eudoxian TR group is algebraically and order-theoretically (and hence, topologically) isomorphic to a finite product of copies of the real numbers. In [18], Wirth used their result to describe the subgroup of a locally compact TR group generated by the compact neighbourhoods of zero. The proof of Loy and Miller relied heavily on a result of Mackey (cf. [10], p. 390) and either the finite-dimensional case of the Choquet-Kendall Theorem (cf. [15], pp. 9–10) or the representation theory of Kakutani (cf. [11], Appendix). Below we use only elementary topological results and order-theoretic arguments and a theorem of Conrad [4] to characterize all non-secular, locally compact TRL groups (Theorem 3). Our proof of Theorem 3 allows us to deduce algebraically the theorems both of Loy and Miller and of Wirth, in both cases without appealing to the theorem of Conrad.

Let $G$ be a finite $p$-group, where $p$ is an odd prime number, $H$ a subgroup of $G$ and s$\theta\in \hbox{\rm Irr}(H)$ an irreducible character of $H$. Assume also that $|G:H|=p^2$. Then the character $\theta^G$ of $G$ induced by $\theta$ is either a multiple of an irreducible character of $G$, or has at least $\frac{p\,{+}\,1}{2}$ distinct irreducible constituents.

Recently, P. Kaplan and K. S. Williams [10] considered (as an example) the representation of primes by binary quadratic forms of discriminant –768. These forms fall into 4 genera, each consisting of two classes. In particular, they considered the forms

F=3X2+642 and G = 12X2+12XY+19Y2.

It follows from genus theory (as explained in [10]) that every prime p ≡ 19 mod 24 is represented by exactly one of the forms F and G. Based on numerical data, they conjectured that a prime p ≡ 19 mod 24 is represented by

where

Vo = 2, V1 = -4, Vn+2=-4Vn+1 -Vn (n∨0).

The lower radical of a module type. For a ring R with unit, the module type t(R) was defined in [6] as follows: t(0) = 0; t(R) = d if every free R-module has invariant rank; t(R) = (c, k) for integers c, k ≧ 1 if every free R-module of rank < c has invariant rank, while a free module of rank h ≧ c has rank h + nk for any integer n ≧ 0. The module types form a lattice under the ordering 0 < (c, k) < d and (c', k') ≦ (c, k) if and only if c ' ≦ c and k' Ⅰ k. Two of the basic theorems on types are:

A. [6; Theorem 2, p. 115] If R → R' is a unit-preserving homomorphism, then t(R')≦ t(R).

In [3], two open problems were whether either of the diophantine equations

with n ∈ Z and f a prime number, is solvable if ω > 3 and 3 √ ω, but in this paper we allow f to be any (rational) integer and also 3 | ω. Equations of this form and more general ones can effectively be solved [5] with an advanced method based on analytical results, but the search limits are usually of enormous size. Here both equations (1) are norm equations in K (√–3): N(a + bp) = fw with p = (√–1 + –3)/2 which makes it possible to treat them arithmetically.

We classify all possible JSJ decompositions of doubles of free groups of rank two, and we also compute the Makanin–Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup separable.

In [10], C. Sundberg uses a clever argument involving an idea of Davie and Jewell [13] to prove an isomorphism theorem for a very general class of operators. A related spectral inclusion theorem is an immediate consequence of the proof of this result, as Sundberg points out. He goes on to list several well known examples that are applications of his main result and remarks that the proof of the McDonald–Sundberg theorem (c.f. [9]) can now be considerably simplified. The purpose of this note is to give further evidence of the utility of the criterion established in [10]. Here and throughout X denotes a compact Hausdorff space and A is a function algebra on X. The Shilovboundary of A is the minimal closed subset ∂(A) of X with the property that

.

Linear spaces on which both an order and a topology are defined and related in various ways have been studied for some time now. Given an order on a linear space it is sometimes possible to define a useful topology using the order and linear structure. In this note we focus on a special type of space called a linear lattice and determine those lattice properties which are both necessary and sufficient for the existence of a classical norm, called an M-norm, for the lattice. This result is a small step in a program to determine which intrinsic order properties of an ordered linear space are necessary and sufficient for the existence of various given types of topologies for the space. This study parallels, in a certain sense, the study of purely topological spaces to determine intrinsic properties of a topology which make it metrizable and the study of the relation between order and topology on spaces which have no algebraic structure, or. algebraic structures other than a linear one.

We give a short and constructive proof of the general (multi-dimensional) Implicit Function Theorem (IFT), using infinitesimal (i.e. nonstandard) methods to implement our basic intuition about the result. Here is the statement of the IFT, quoted from [4];

Theorem. Let A ⊂ ℝn × ℝmbe an open set and let F:A → ℝ be a function of class Cp (p≥1). Suppose that (xO, yO) ε A with F(xO, yO) = 0 (xO ε ℝn, yO ε ℝm) and that the Jacobian determinantis not zero at (xO, yO). Then there is an open neighbourhood U of xO and a unique function f:U→ ℝmwith

F(x, f(x)) = 0

for all x ε U. Moreover, f is of class Cp.

A closer look at Laguerre and Meixner polynomials shows that they interact, with a more active role played by the latter. We intend here to expound on this development of the story up to some level of abstraction.

We study the spectral decomposition with respect to the Jacobi operator, J, of spherical immersions and characterize those with a simple decomposition in terms of the Finite Chen-type submanifolds. As a consequence, we give an application to the inverse problem for J.