Orbits in a stochastic Goodwin–Lotka–Volterra model

This paper examines the cycling behavior of a deterministic and a stochastic version of the economic interpretation of the Lotka–Volterra model, the Goodwin model. We provide a characterization of orbits in the deterministic highly non-linear model. We then study a stochastic version, with Brownian noise introduced via a heterogeneous productivity factor. Existence conditions for a solution to the system are provided. We prove that the system produces cycles around a unique equilibrium point in finite time for general volatility levels, using stochastic Lyapunov techniques for recurrent domains. Numerical insights are provided.     

Linear codes associated to skew-symmetric determinantal varieties

In this article we consider linear codes coming from skew-symmetric determinantal varieties, which are defined by the vanishing of minors of a certain fixed size in the space of skew-symmetric matrices. In odd characteristic, the minimum distances of these codes are determined and a recursive formula for the weight of a general codeword in these codes is given.     

Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

Let k be an algebraically closed field and A the polynomial algebra in r variables with coefficients in k. In case the characteristic of k is 2, Carlsson [9] conjectured that for any DG-A-module M of dimension N as a free A-module, if the homology of M is nontrivial and finite dimensional as a k-vector space, then $2^r\le N$. Here we state a stronger conjecture about varieties of square-zero upper triangular $N\times N$ matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N<8$ or $r<3$ without any restriction on the characteristic of k. As a consequence, we obtain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N>4$ and $r=2$.     

Norms and lower bounds of some matrix operators on Fibonacci weighted difference sequence space

Norm of an operator T:X→Y is the best possible value of U satisfying the inequality and lower bound for T is the value of L satisfying the inequality where ‖.‖_X and ‖.‖_Y are the norms on the spaces X and Y, respectively. The main goal of this paper is to compute norms and lower bounds for some matrix operators from the weighted sequence space ?_p(w) into a new space called as Fibonacci weighted difference sequence space. For this purpose, we firstly introduce the Fibonacci difference matrix and the space consisting of sequences whose ‐transforms are in .     

An optimal perturbation bound

Cai and Zhang establish separate perturbation bounds for distances with spectral and Frobenius norms (Cai T, Zhang A. Rate‐optimal perturbation bounds for singular subspaces with applications to high‐dimensional statistics. The Annals of Statistics. 2018; Vol. 46, No. 1: 60?89). We extend their theorem to each unitarily invariant norm. It turns out that our estimation is optimal as well.