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An addendum to "amplification, arbitrage, and the tensor power trick"

In one of the earliest posts on this blog, I talked about the ability to “arbitrage” a disparity of symmetry in an inequality, and in particular to “amplify” such an inequality into a stronger one. 1,981 more words


On the sign patterns of entrywise positivity preservers in fixed dimension

Apoorva Khare and I have just uploaded to the arXiv our paper “On the sign patterns of entrywise positivity preservers in fixed dimension“. This paper explores the relationship between positive definiteness of Hermitian matrices, and entrywise operations on these matrices. 1,165 more words


Schur convexity and positive definiteness of the even degree complete homogeneous symmetric polynomials

The complete homogeneous symmetric polynomial of variables and degree can be defined as

thus for instance


One can also define all the complete homogeneous symmetric polynomials of variables simultaneously by means of the generating function… 545 more words


Quantitative continuity estimates

Suppose is a continuous (but nonlinear) map from one normed vector space to another . The continuity means, roughly speaking, that if are such that is small, then is also small (though the precise notion of “smallness” may depend on or , particularly if is not known to be… 1,896 more words


Generalisations of the limit functional

Suppose one has a bounded sequence of real numbers. What kinds of limits can one form from this sequence?

Of course, we have the usual notion of limit , which in this post I will refer to as the… 1,421 more words


Notes on nilcharacters and their symbols

A sequence of complex numbers is said to be quasiperiodic if it is of the form

for some real numbers and continuous function . For instance, linear phases such as (where ) are examples of quasiperiodic sequences; the top order coefficient (modulo ) can be viewed as a “frequency” of the integers, and an element of the… 4,730 more words