Polynomial Norms

In this paper, we study polynomial norms, i.e. norms that are the $d^{\text{th}}$ root of a degree-$d$ homogeneous polynomial $f$. We first show that a necessary and sufficient condition for $f^{1/d}$ to be a norm is for $f$ to be strictly convex, or equivalently, convex and positive definite. Though not all norms come from $d^{\text{th}}$ roots of polynomials, we prove that any norm can be approximated arbitrarily well by a polynomial norm. We then investigate the computational problem of testing whether a form gives a polynomial norm. We show that this problem is strongly NP-hard already when the degree of the form is 4, but can always be answered by testing feasibility of a semidefinite program (of possibly large size). We further study the problem of optimizing over the set of polynomial norms using semidefinite programming. To do this, we introduce the notion of r-sos-convexity and extend a result of Reznick on sum of squares representation of positive definite forms to positive definite biforms. We conclude with some applications of polynomial norms to statistics and dynamical systems

Implicit norms

Robert Brandom has developed an account of conceptual content as instituted by social practices. Such practices are understood as being implicitly normative. Brandom proposed the idea of implicit norms in order to meet some requirements imposed by Wittgenstein’s remarks on rule-following: escaping the regress of rules on the one hand, and avoiding mere regular behavior on the other. Anandi Hattiangadi has criticized this account as failing to meet such requirements. In what follows, I try to show how the correct understanding of sanctions and the expressivist reading of the issue can meet these challenges

Generalized Induced Norms

Let ||.|| be a norm on the algebra M_n of all n-by-n matrices over the complex field C. An interesting problem in matrix theory is that "are there two norms ||.||_1 and ||.||_2 on C^n such that ||A||=max{||Ax||_2: ||x||_1=1} for all A in M_n. We will investigate this problem and its various aspects and will discuss under which conditions ||.||_1=||.||_2.Comment: 8 page

Operator-Valued Norms

We introduce two kinds of operator-valued norms. One of them is an $L(H)$-valued norm. The other one is an $L(C(K))$-valued norm. We characterize the completeness with respect to a bounded $L(H)$-valued norm. Furthermore, for a given Banach space $\textbf{B}$, we provide an $L(C(K))$-valued norm on $\textbf{B}$. and we introduce an $L(C(K))$-valued norm on a Banach space satisfying special properties.Comment: 8 page

Interpretivism and norms

This article reconsiders the relationship between interpretivism about belief and normative standards. Interpretivists have traditionally taken beliefs to be fixed in relation to norms of interpretation. However, recent work by philosophers and psychologists reveals that human belief attribution practices are governed by a rich diversity of normative standards. Interpretivists thus face a dilemma: either give up on the idea that belief is constitutively normative or countenance a context-sensitive disjunction of norms that constitute belief. Either way, interpretivists should embrace the intersubjective indeterminacy of belief