Related papers: The Fundamental Theorem of Algebra: From The Four …
Elementary proofs of unique factorization in rings of arithmetic functions using a simple variant of Euclid's proof for the fundamental theorem of arithmetic.
We prove sharp Landau- and Schottky-type theorems for analytic functions which omit the $n$-th roots of unity. The proofs are based on a sharp lower bound for the Poincar\'e metric of the complex plane punctured at the roots of unity.
We give an explicit upper bound for the algebraic degree and an explicit lower bound for the absolute value of the minimum of a polynomial function on a compact connected component of a basic closed semialgebraic set when this minimum is…
Derivations are linear operators which satisfy the Leibniz rule, while integrations are linear operators which satisfy the Rota-Baxter rule. In this paper, we introduce the notion of an FTC-pair, which consists of an algebra and module with…
We introduce a framework for proving statements about linear operators by verification of ideal membership in a free algebra. More specifically, arbitrary first-order statements about identities of morphisms in preadditive semicategories…
We apply to operator algebra theory a monotone selection principle which apparently escaped attention (of operator algebra theorists) so far. This principle relates to the basic order theoretic characterisation of von Neumann algebras given…
Recently, Todorov and Wilf independently realized that de Branges' original proof of the Bieberbach and Milin conjectures and the proof that was later given by Weinstein deal with the same special function system that de Branges had…
An elementary introduction to the principles of algebraic surgery.
Based on Bergman's Lemma on centralizers, we obtain a sharp lower degree bound for nonconstant elements in a subalgebra generated by two elements of a free associative algebra over an arbitrary field.
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
In this paper, we prove that the algebra of an \'etale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes…
We prove that there exists essentially one {\it minimal} differential algebra of distributions $\A$, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l'impossibilit\'e de la multiplication des…
In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of…
This is an introduction to the algebras $A\subset B(H)$ that the linear operators $T:H\to H$ can form, once a complex Hilbert space $H$ is given. Motivated by quantum mechanics, we are mainly interested in the von Neumann algebras, which…
It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories…
We generalize several recognizability theorems for free single-sorted algebras to the field of many-sorted algebras and provide, in a uniform way and without using neither regular tree grammars nor tree automata, purely algebraic proofs of…
It is shown that the *-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that…
Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem,…
L. Weinstein's brilliant short proof of de Branges's Theorem is made even shorter by using computer algebra.
It is shown that with appropriate boundary conditions, a real function satisfying the differential equation $f'(x) = f(x+a)$ has all known properties of the sine function. A number of elementary derivations are presented including proofs…