On complex and real identifiability of tensors

We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.Comment: To appear on Rivista di Matematica dell'Universit\`a di Parma, Volume 8, Number 2, 2017, pages 367-37

Higher secants of spinor varieties

Let $S_h$ be the even pure spinors variety of a complex vector space $V$ of even dimension $2h$ endowed with a non degenerate quadratic form $Q$ and let $\sigma_k(S_h)$ be the $k$-secant variety of $S_h$. We decribe a probabilistic algorithm which computes the complex dimension of $\sigma_k(S_h)$. Then, by using an inductive argument, we get our main result: $\sigma_3(S_h)$ has the expected dimension except when $h\in \{7,8\}$. Also we provide theoretical arguments which prove that $S_7$ has a defective 3-secant variety and $S_8$ has defective 3-secant and 4-secant varieties.Comment: 23 page

An original and additional mathematical model characterizing a Bayesian approach to decision theory

We propose an original mathematical model according to a Bayesian approach explaining uncertainty from a point of view connected with vector spaces. A parameter space can be represented by means of random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. We observe that metric properties of the notion of -product mathematically fulfill the ones of a coherent prevision of a bivariate random quantity. We introduce fundamental metric expressions connected with transformed random quantities representing changes of origin. We obtain a posterior probability law by applying the Bayes’ theorem into a geometric context connected with a two-dimensional parameter space

Detecting the β\beta-family in iterated algebraic K-theory of finite fields

The Lichtenbaum-Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni-Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and prove a highly nontrivial case this conjecture. In particular, if the $n$-th Greek letter family is detected by a commutative ring spectrum $R$, then we conjecture that the $n+1$-st Greek letter family will be detected by the algebraic K-theory of $R$. We prove this in the case $n=1$ for $R=K(\mathbb{F}_q)_p$ where $p\ge 5$ and $q$ is prime power generator of the units in $\mathbb{Z}/p^2\mathbb{Z}$. In particular, we prove that the commutative ring spectrum $K(K(\mathbb{F}_q)_p)$ detects the $\beta$-family. The method of proof also implies that the $\beta$-family is detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to modular forms satisfying certain congruences.Comment: 28 pages, Comments welcom

Parallel Tempering for the planted clique problem

The theoretical information threshold for the planted clique problem is $2\log_2(N)$, however no polynomial algorithm is known to recover a planted clique of size $O(N^{1/2-\epsilon})$, $\epsilon>0$. In this paper we will apply a standard method for the analysis of disordered models, the Parallel-Tempering (PT) algorithm, to the clique problem, showing numerically that its time-scaling in the hard region is indeed polynomial for the analyzed sizes. We also apply PT to a different but connected model, the Sparse Planted Independent Set problem. In this situation thresholds should be sharper and finite size corrections should be less important. Also in this case PT shows a polynomial scaling in the hard region for the recovery.Comment: 12 pages, 5 figure

Consumption and Habit Formation when Time Horizon is Finite

This paper provides a closed-form solution under labour uncertainty for optimal consumption and the value function in a finite horizon life-cycle model with habit persistence.habit formation, life-cycle consumption, precautionary saving

Parameter estimates for fishes of the upper Paraná river floodplain and Itaipu reservoir (Brazil)

Estimates of the growth (K), natural mortality (M), consumption/biomass (Q/B) rate and trophic level (TL) for 35 species in the upper Paraná river floodplain and the Itaipu reservoir (interconnected ecosystems) are presented. A compilation of these biological statistics is made for comparison purposes and some general trends are briefly discussed