Abstract Proving statements about linear operators expressed in terms of identities often leads to finding elements certain form noncommutative polynomial ideals. We illustrate this by examples coming from actual operator and discuss relevant algorithmic methods for such polynomials based on Gröbner bases. In particular, we present algorithms computing the intersection a two-sided ideal with on...

Multi-class classification algorithms are very widely used, but we argue that they are not always ideal from a theoretical perspective, because they assume all classes are characterised by the data, whereas in many applications, training data for some classes may be entirely absent, rare, or statistically unrepresentative. We evaluate onesided classifiers as an alternative, since they assume th...

We study the state complexity of regular operations in the class of ideal languages. A language L ⊆ Σ∗ is a right (left) ideal if it satisfies L = LΣ∗ (L = Σ∗L). It is a two-sided ideal if L = Σ∗LΣ∗, and an all-sided ideal if L = Σ∗ L, the shuffle of Σ∗ with L. We prefer the term “quotient complexity” instead of “state complexity”, and we use derivatives to calculate upper bounds on quotient co...

A right ideal (left ideal, two-sided ideal) is a non-empty language L over an alphabet Σ such that L = LΣ∗ (L = Σ∗L, L = Σ∗LΣ∗). Let k = 3 for right ideals, 4 for left ideals and 5 for two-sided ideals. We show that there exist sequences (Ln | n > k) of right, left, and two-sided regular ideals, where Ln has quotient complexity (state complexity) n, such that Ln is most complex in its class und...

It has been established by Inoue that a complex locally C*-algebra with a dense ideal posesses a bounded approximate identity which belonges to that ideal. It has been shown by Fritzsche that if a unital complex locally C*-algebra has an unbounded element then it also has a dense one-sided ideal. In the present paper we obtain analogues of the aforementioned results of Inoue and Fritzsche for r...

We use the venerable “fooling set” method to prove new lower bounds on the quantum communication complexity of various functions. Let f : X × Y → {0, 1} be a Boolean function, fool1(f) its maximal fooling set size among 1-inputs, Q1(f) its one-sided-error quantum communication complexity with prior entanglement, and NQ(f) its nondeterministic quantum communication complexity (without prior enta...

A string is Abelian square-free if it contains no Abelian squares; that is, adjacent substrings which are permutations of each other. An Abelian square-free string is maximal if it cannot be extended to the left or right by concatenating alphabet symbols without introducing an Abelian square. We construct Abelian square-free finite strings which are maximal by modifying a construction of Zimin....

We introduce the notion of n -homoderivation on a ring id="M3"> ℜ and show that semiprime id="M4"> must have nontrivial central ideal if it admits an appropriate id="M5"> which is centralizing some one-sided ideal. Under similar hypotheses, we prove commutativity in prime rings.

Abstract When reporting tests of significance, researchers might claim to have conducted a two-sided test when in fact they two one-sided tests. Mark Rubin explains the confusion and how avoid it.

The term ‘possible’, in Aristotle’s view, is ambiguous. It has two senses known as one-sided possibility and two-sided possibility (or contingency). Being two-sided possible means being neither impossible nor necessary, and being one-sided possible simply means being not impossible. Aristotle defines the two senses in Prior Analytics 1.13 (32a18–21). This definition is followed by a controversi...