A major area in neuroscience is the study of how the brain processes spatial information. Neurons in the brain represent external stimuli via neural codes. These codes often arise from regions of space called receptive fields: each neuron fires at a high rate precisely when the animal is in the corresponding receptive field. Much research in this area has focused on understanding what features ...

Using the notation of [3], we will consider the power {pλ[X]}λ, Schur {sλ[X]}λ, monomial {mλ[X]}λ, homogeneous {hλ[X]}λ, elementary {eλ[X]}λ and forgotten {fλ[X]}λ bases for the symmetric functions. We will often appeal to [3] for proofs of identities relating these bases. These bases are all indexed by partitions, non-increasing sequences of non-negative integers. The i entry of the partition ...

Abstract Bidiagonal factorizations for the change of basis matrices between monomial and Newton polynomial bases are obtained. The total positivity these is characterized in terms sign nodes bases. It shown that computations to high relative accuracy algebraic problems related can be achieved whenever have same sign. Stirling considered particular cases matrices, then collocation Wronskian Touc...

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems t...

Taylor presented an explicit resolution for arbitrary monomial ideals. Later, Lyubeznik found that already a subcomplex defines a resolution. We show that the Taylor resolution may be obtained by repeated application of the Schreyer Theorem from the theory of Gröbner bases, whereas the Lyubeznik resolution is a consequence of Buchberger’s chain criterion. Finally, we relate Fröberg’s contractin...

Abstract We study symmetries of bases and spanning sets in finite element exterior calculus, using representation theory. want to know which vector-valued spaces have invariant under permutation vertex indices. The permutations indices correspond the symmetry group simplex. That is represented on simplicial by pullback action. determine a natural notion invariance sufficient conditions dimensio...

It is known that differences of symmetric functions corresponding to various bases are nonnegative on the orthant exactly when partitions defining them comparable in dominance order. The only exception case homogeneous where it implies nonnegativity difference functions. was conjectured by Cuttler, Greene, and Skandera 2011 converse also holds, as cases monomial , elementary, power-sum, Schur b...