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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. 1,167 more words

Math.CA

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

and

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

Expository

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

Expository

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

Expository

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

Expository

Yves Meyer wins the 2017 Abel Prize

Just a short post to note that Norwegian Academy of Science and Letters has just announced that the 2017 Abel prize has been awarded to Yves Meyer… 352 more words

Math.CA

Math 246A, Notes 4: singularities of holomorphic functions

In the previous set of notes we saw that functions that were holomorphic on an open set enjoyed a large number of useful properties, particularly if the domain was simply connected. 11,838 more words

Math.CA