Stable subnorms revisited

Let A be a finite-dimensional, power-associative algebra over a field F, either R or C, and let S, a subset of A, be closed under scalar multiplication. A real-valued function f defined on S, shall be called a subnorm if f(a) > 0 for all 0 not equal a is an element of S, and f(alpha a) = |alpha| f(a) for all a is an element of S and alpha is an element of F. If in addition, S is closed under raising to powers, then a subnorm f shall be called stable if there exists a constant sigma > 0 so that f(a(m)) less than or equal to sigma f(a)(m) for all a is an element of S and m = 1, 2, 3.... The purpose of this paper is to provide an updated account of our study of stable subnorms on subsets of finite-dimensional, power-associative algebras over F. Our goal is to review and extend several of our results in two previous papers, dealing mostly with continuous subnorms on closed sets

Multiplicativity factors for seminorms

AbstractLet A be an algebra and let S be a seminorm on A. In this paper we study multiplicativity factors for S, i.e., constants μ > 0 for which S(xy) ⩽ μS(x) S(y) for all x, y∈A. We begin by investigating these factors in terms of the kernel of S. We then specialize our study to function algebras and to seminorms generated by the sup norm, where we provide a convenient characterization of multiplicativity factors

Homotonic Algebras

An algebra $\mathcal{A}$ of real or complex valued functions defined on a set $\mathbf{T}$ shall be called \textit{homotonic} if $\mathcal{A}$ is closed under forming of absolute values, and for all $f$ and $g$ in $\mathcal{A}$, the product $f\times g$ satisfies $|f\times g|\le|f|\times|g|$. Our main purpose in this paper is two-fold: To show that the above definition is equivalent to an earlier definition of homotonicity, and to provide a simple inequality which characterizes sub-multiplicativity and strong stability for weighted sup norms on homotonic algebras.Comment: 8 page

Wear characteristics of ultra-hard cutting tools when machining austempered ductile iron

Nodularised Ductile Cast Iron, when subjected to heat treatment processes - austenitising and austempering produces Austempered Ductile Iron (ADI). The microstructure of ADI also known as &quot;ausferrite&quot; consists of ferrite, austenite and graphite nodules. Machining ADI using conventional techniques is often a problematic issue due to the microstructural phase transformation from austenite to martensite during machining. This paper evaluates the wear characteristics of ultra hard cutting tools when machining ADI and its effect on machinability. Machining trials consist of turning ADI (ASTMGrade3) using two sets of PCBN tools with 90% and 50% CBN content and two sets of ceramics tools; Aluminium Oxide Titanium Carbide and Silicon Carbide - whisker reinforced Ceramic. The cutting parameters chosen are categorized as roughing and finishing conditions; the roughing condition comprises of constant cutting speed (425 m/min) and depth of cut (2mm) combined with variable feed rates of 0.1, 0.2, 0.3 and 0.4mm/rev. The finishing condition comprises of constant cutting speed (700 m/min) and depth of cut (0.5mm) combined with variable feed rates of 0.1, 0.2, 0.3 and 0.4mm/rev. The benchmark condition to evaluate the performance of the cutting tools was tool wear evaluation, surface texture analysis and cutting force analysis. The paper analyses thermal softening of the workpiece by the tool and its effect on the shearing mechanism under rough and finish machining conditions in term of lower cutting forces and enhanced surface texture of the machined part.<br /

Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number

In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201

Stable seminorms revisited

A seminorm S on an algebra A is called stable if for some constant σ > 0 , S(x^k) ≤ σS(x)^k for all x ∈ A and all k = 1, 2, 3,.... We call S strongly stable if the above holds with σ = 1 . In this note we use several known and new results to shed light on the concepts of stability. In particular, the interrelation between stability and similar ideas is discussed