Related papers: On a problem by Arens, Goldberg, and Luxemburg

Let $k$ be an algebraically closed field of characteristic $0$ and $A$ a graded $k$-algebra finitely generated in degree $1$. In this paper, for $3$-dimensional quadratic AS-regular algebras except for Type EC, we give a complete list of…

We consider a nonlinear representation of a Lie algebra which is regular on an abelian ideal, we define a normal form which generalizes that defined in [D. Arnal, M. Ben Ammar, M. Selmi, {\rm Normalisation d'une repr\'esentation non…

By the result of Artin--Tate--Van den Bergh, every $3$-dimensional cubic AS-regular algebra A can be expressed as a geometric algebra $A=\mathcal{A}(E,\sigma)$, where $E$ is either $\mathbb{P}^{1}\times \mathbb{P}^{1}$ or a curve of…

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language.…

Let $k$ be a field of characteristic $0$. Using the method of idealization, we show that there is a non-Koszul, quadratic, Artinian, Gorenstein, standard graded $k$-algebra of regularity $3$ and codimension $8$, answering a question of…

The Hahn-Banach theorem states that onto each line in every normed space, there is a unitary projection, and Kadec and Snobar proved (using John's ellipsoid) that onto each $n$-dimensional subspace of any real normed space, there is a…

It is shown that if $\gamma: [a,b] \to S^2$ is $C^3$ with $\det(\gamma, \gamma', \gamma'') \neq 0$, and if $A \subseteq \mathbb{R}^3$ is a Borel set, then $\dim \pi_{\theta} (A) \geq \min\left\{ 2,\dim A, \frac{ \dim A}{2} + \frac{3}{4}…

In this note one tries to venture into a study of some notions, in the context of a (unital) normed algebra, in particular the algebra of operators on a Hilbert space. Namely, one considers ``moving norms'', i.e.\ norming an element minus a…

Classification of AS-regular algebras is one of the main interests in noncommutative algebraic geometry. We say that a $3$-dimensional quadratic AS-regular algebra is of Type EC if its point scheme is an elliptic curve in $\mathbb{P}^{2}$.…

In this paper, we give the explicit structure of $ \otimes^{3} H $ and $ \wedge^{3} H $ where $ H $ is a generalized Heisenberg Lie algebra of rank at most $ 2. $ Moreover, for a non-abelian nilpotent Lie algebra $ L, $ we obtain an upper…

The concept of "table algebra" was introduced by Z Arad anf H. Blau in order to study in a uniform way properties of products of conjugacy classes and of irreducible characters of a finite group, Except for certain cases which remain open,…

We study some formality criteria for differential graded algebras over differential graded operads. This unifies and generalizes other known approaches like the ones by Manetti and Kaledin. In particular, we construct general operadic…

We introduce two-sorted theories in the style of [CN10] for the complexity classes \oplusL and DET, whose complete problems include determinants over Z2 and Z, respectively. We then describe interpretations of Soltys' linear algebra theory…

In this note, we show that for each minimal norm $N(\cdot)$ on the algebra $M_n$ of all $n \times n$ complex matrices, there exist norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on ${\mathbb C}^n$ such that $$N(A)=\max\{\|Ax\|_2: \|x\|_1=1, x\in…

Suppose that G is a finite group and A is a subset of G such that 1_A has algebra norm at most M. Then 1_A is a plus/minus sum of at most L cosets of subgroups of G, and L can be taken to be triply tower in O(M). This is a quantitative…

A finite-dimensional Lie algebra $L$ over a field $F$ is called an $A$-algebra if all of its nilpotent subalgebras are abelian. This is analogous to the concept of an $A$-group: a finite group with the property that all of its Sylow…

The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum…

This paper's central theme is to prove the existence of an n-algebra whose multiplication cannot be expressed employing any binary operation. Furthermore, to prove if two algebras are not isomorphic, this property does not hold for…

This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras…

In this article, we study bounded-below locally finite $\mathbb{Z}$-graded algebras, which are referred to as commonly graded algebras in literature. Commonly graded algebras have almost similar theory as that of connected graded algebras,…