In this paper, Auslander-Reiten triangles are introduced into algebraic topology, and it is proved that their existence characterizes Poincaré duality spaces. Invariants in the form of quivers are also introduced, and Auslander-Reiten triangles and quivers over spheres are computed. The quiver over the d-dimensional sphere turns out to consist of d − 1 components, each isomorphic to ZA∞. So qui...

For self-adjoint operators A,B, a bounded operator J , and a function f : R → C, we obtain bounds in quasi-normed ideals of compact operators for the difference f(A)J − Jf(B) in terms of the operator AJ − JB. The focus is on functions f that are smooth everywhere except for finitely many points. A typical example is the function f(t) = |t|γ with γ ∈ (0, 1). The obtained results are applied to d...

There are only very few natural ways in which arbitrary functions can be combined. One composition operator is override: for arbitrary functions f and g, f > g is the function with domain dom( f) U dom(g) that behaves like f on dom( f) and like g on dom(g) \ dom( f ) . Another operator is update: f[g] has the same domain as f, behaves like f on dom( f) \ dom(g), and like g on dom( f) n dom(g). ...

Considering the family F of contour curves F = {h(x, y) = k) of an (idempotent) aggregation operator h in two variables as a oneparametric family of curves, the differential equation y′ = f(x, y) having F as general solution is associated to h. Properties of h have then be translated to properties of its differential equation. Reciprocally, for a differential equation fulfilling some easy prope...

A specific algebraic realization of the Ginsparg-Wilson relation in the form γ5(γ5D) + (γ5D)γ5 = 2a (γ5D) 2k+2 is discussed, where k stands for a nonnegative integer and k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation. From a view point of algebra, a characteristic property of our proposal is that we have a closed algebraic relation for one unknown operator D, although this...

This de nition clearly coincides with the usual (algebraic) notion of derivation when [A; +, ·] is a ring. However, it can be formally stated for every algebraic structure endowed with two binary operations. In this paper, we will consider the special case in which [A; +, ·] is a lattice, so that + and · are, respectively, the join and the meet operations. These ideas have been introduced and d...

Let k be an arbitrary field, let G be a (smooth) linear algebraic group over k, and let U be a vector group over k on which G acts by automorphisms of algebraic groups. The action of G on U is said to be linear if there is a G-equivariant isomorphism of algebraic groups U ' Lie(U). Suppose that G is connected and that the unipotent radical of G is defined over k. If the G-module Lie(U) is simpl...

Abstract. We consider functions f(A,B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B ∞,1 (R), then we have the following Lipschitz type estimate in the trace norm: ‖f(A1, B1)− f(A2, B2)‖S1 ≤ const(‖A1 −A2‖S1 + ‖B1 −B2‖S1). However, the condition f ∈ B ∞,1 (R) does not imply the Lipschitz ...

and Applied Analysis 3 The role of “compact operators” is replaced by that of “minimal one-sided ideals”. The proof of our results relies on the quasispatiality of the derivation and Banach algebra techniques. This paper is a continuation of 5 . Some definitions and notations can be found in 5 . 2. Preliminaries Throughout this paper, X is a complex Banach space, and X∗ is the topological dual ...