Subalgebras of Matrix Algebras Generated by Companion Matrices

Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index in terms of the resultant of $f$ and $g$. If $R$ is a commutative ring with identity we determine when $R=M_n(R)$, in which case a presentation for $M_n(R)$ in terms of $C$ and $D$ is given

A Closed Formula for the Product in Simple Integral Extensions

Let $\xi$ be an algebraic number and let $\alpha,\beta\in \mathbb Q[\xi]$. An explicit formula for the coordinates of the product $\alpha\beta$ is given in terms of the coordinates of $\alpha$ and $\beta$ and the companion matrix of the minimal polynomial of $\xi$. The formula as well as its proof extend to fairly general simple integral extensions

Mouse Cognition-Related Behavior in the Open-Field: Emergence of Places of Attraction

Spatial memory is often studied in the Morris Water Maze, where the animal's spatial orientation has been shown to be mainly shaped by distal visual cues. Cognition-related behavior has also been described along “well-trodden paths”—spatial habits established by animals in the wild and in captivity reflecting a form of spatial memory. In the present study we combine the study of Open Field behavior with the study of behavior on well-trodden paths, revealing a form of locational memory that appears to correlate with spatial memory. The tracked path of the mouse is used to examine the dynamics of visiting behavior to locations. A visit is defined as either progressing through a location or stopping there, where progressing and stopping are computationally defined. We then estimate the probability of stopping at a location as a function of the number of previous visits to that location, i.e., we measure the effect of visiting history to a location on stopping in it. This can be regarded as an estimate of the familiarity of the mouse with locations. The recently wild-derived inbred strain CZECHII shows the highest effect of visiting history on stopping, C57 inbred mice show a lower effect, and DBA mice show no effect. We employ a rarely used, bottom-to-top computational approach, starting from simple kinematics of movement and gradually building our way up until we end with (emergent) locational memory. The effect of visiting history to a location on stopping in it can be regarded as an estimate of the familiarity of the mouse with locations, implying memory of these locations. We show that the magnitude of this estimate is strain-specific, implying a genetic influence. The dynamics of this process reveal that locations along the mouse's trodden path gradually become places of attraction, where the mouse stops habitually

Knots: Attractive Places with High Path Tortuosity in Mouse Open Field Exploration

When introduced into a novel environment, mammals establish in it a preferred place marked by the highest number of visits and highest cumulative time spent in it. Examination of exploratory behavior in reference to this “home base” highlights important features of its organization. It might therefore be fruitful to search for other types of marked places in mouse exploratory behavior and examine their influence on overall behavior

When Too Much Is Not Enough: Obsessive-Compulsive Disorder as a Pathology of Stopping, Rather than Starting

Background: In obsessive-compulsive disorder (OCD), individuals feel compelled to repeatedly perform security-related behaviors, even though these behaviours seem excessive and unwarranted to them. The present research investigated two alternative ways of explaining such behavior: (1) a dysfunction of activation—a starting problem—in which the level of excitation in response to stimuli suggesting potential danger is abnormally strong; versus (2) a dysfunction of termination— a stopping problem—in which the satiety-like process for shutting down security-related thoughts and actions is abnormally weak. Method: In two experiments, 70 patients with OCD (57 with washing compulsions, 13 with checking compulsions) and 72 controls were exposed to contamination cues—immersing a hand in wet diapers —and later allowed to wash their hands, first limited to 30 s and then for as long as desired. The intensity of activation of security motivation was measured objectively by change in respiratory sinus arrythmia. Subjective ratings (e.g., contamination) and behavioral measures (e.g., duration of hand washing) were also collected. Results: Compared to controls, OCD patients with washing compulsions did not differ significantly in their levels of initial activation to the threat of contamination; however, they were significantly less able to reduce this activation by engaging in the corrective behavior of hand-washing. Further, the deactivating effect of hand-washing in OCD patients with checkin