On Universal Eigenvalues and Eigenvectors of Hypermatrices

A cubic hypermatrix of order $d$ can be considered as a structure matrix of a tensor with covariant order $r$ and contra-variant order $s=d-r$. Corresponding to this matrix expression of the hypermatrix, an eigenvector $x$ with respect to an eigenvalue $\lambda$ is proposed, called the universal eigenvector and eigenvalue of the hypermatrix. According to the action of tensors, if $x$ is decomposable, it is called a universal hyper-(UH-)eigenvector. Particularly, if all decomposed components are the same, $x$ is called a universal diagonal hyper (UDH-)eigenvector, which covers most of existing definitions of eigenvalue/eigenvector of hypermatrices. Using Semi-tensor product (STP) of matrices, the properties of universal eigenvalues/eigenvectors are investigated. Algorithms are developed to calculate universal eigenvalues/eigenvectors for hypermatrices. Particular efforts have been put on UDH- eigenvalues/eigenvectors, because they cover most of the existing eigenvalues/eigenvectors for hypermatrices. Some numerical examples are presented to illustrate that the proposed technique is universal and efficient

Semi-Tensor Product of Hypermatrices with Application to Compound Hypermatrices

The semi-tensor product (STP) of matrices is extended to the STP of hypermatrices. Some basic properties of the STP of matrices are extended to the STP of hypermatrices. The hyperdeterminant of hypersquares is introduced. Some algebraic and geometric structures of matrices are extended to hypermatrices. Then the compound hypermatrix is proposed. The STP of hypermatrix is used to compound hypermatrix. Basic properties are proved to be available for compound hypermatrix.Comment: 8 page

Global Controllability Criteria and Motion Planning of Regular Affine Systems With Drifts

In this article, we give a condition for the global controllability of affine nonlinear control systems with drifts on Euclidean spaces. Under regularity assumptions, the condition is necessary and sufficient in the codimension-1 and codimension-2 cases, and holds for systems of higher codimensions under mild restrictions. We then investigate motion planning problems for codimension-1 affine systems, and give proof of the global existence of the lift to control curves for certain drifted systems using the homotopy continuation method

Synaptic Neurotransmission Depression in Ventral Tegmental Dopamine Neurons and Cannabinoid-Associated Addictive Learning

Drug addiction is an association of compulsive drug use with long-term associative learning/memory. Multiple forms of learning/memory are primarily subserved by activity- or experience-dependent synaptic long-term potentiation (LTP) and long-term depression (LTD). Recent studies suggest LTP expression in locally activated glutamate synapses onto dopamine neurons (local Glu-DA synapses) of the midbrain ventral tegmental area (VTA) following a single or chronic exposure to many drugs of abuse, whereas a single exposure to cannabinoid did not significantly affect synaptic plasticity at these synapses. It is unknown whether chronic exposure of cannabis (marijuana or cannabinoids), the most commonly used illicit drug worldwide, induce LTP or LTD at these synapses. More importantly, whether such alterations in VTA synaptic plasticity causatively contribute to drug addictive behavior has not previously been addressed. Here we show in rats that chronic cannabinoid exposure activates VTA cannabinoid CB1 receptors to induce transient neurotransmission depression at VTA local Glu-DA synapses through activation of NMDA receptors and subsequent endocytosis of AMPA receptor GluR2 subunits. A GluR2-derived peptide blocks cannabinoid-induced VTA synaptic depression and conditioned place preference, i.e., learning to associate drug exposure with environmental cues. These data not only provide the first evidence, to our knowledge, that NMDA receptor-dependent synaptic depression at VTA dopamine circuitry requires GluR2 endocytosis, but also suggest an essential contribution of such synaptic depression to cannabinoid-associated addictive learning, in addition to pointing to novel pharmacological strategies for the treatment of cannabis addiction