In this paper, the singular-value decomposition theory of complex matrices is explored to study constantly curved 2-spheres minimal in both CPn and hyperquadric CPn. The moduli space all those noncongruent ones introduced, which can be described by certain symmetric modulo an appropriate group action. Using description, many examples, such as holomorphic higher degree, nonhomogenous constant cu...

We prove that if M is an o-minimal structure whose underlying order is dense then Th(M) does not interpret the theory of an infinite discretely ordered structure. We also make a conjecture concerning the class of the theory of an infinite discretely ordered o-minimal structure. Introduction. In [9], Świerczkowski proves that Th(〈ω,<〉) is not interpretable (with parameters) in RCF (the theory of...

Let β > 1 be an algebraic number. The relations between the coefficient vector of its minimal polynomial and the digits of the Rényi β-expansion of unity are investigated in terms of the germ of curve associated with β, which is constructed from the Salem parametrization, and the Parry Upper function fβ(z). If β is a Parry number, the Parry Upper function fβ(z) is simply related to the dynamica...

Certificates to a linear algebra computation are additional data structures for each output, which can be used by a—possibly randomized— verification algorithm that proves the correctness of each output. Wiedemann’s algorithm projects the Krylov sequence obtained by repeatedly multiplying a vector by a matrix to obtain a linearly recurrent sequence. The minimal polynomial of this sequence divid...

several tables have been given due to a − minimal sets. our main aim in this paper is tocomplete these tables by employing several examples.

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbe...

Neurons perform computations, and convey the results of those computations through the statistical structure of their output spike trains. Here we present a practical method, grounded in the information-theoretic analysis of prediction, for inferring a minimal representation of that structure and for characterizing its complexity. Starting from spike trains, our approach finds their causal stat...

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is