Real tropicalization and negative faces of the Newton polytope

In this work, we explore the relation between the tropicalization of a real semi-algebraic set $S = \{ f_1 < 0, \dots , f_k < 0\}$ defined in the positive orthant and the combinatorial properties of the defining polynomials $f_1, \dots, f_k$. We describe a cone that depends only on the face structure of the Newton polytopes of $f_1, \dots ,f_k$ and the signs attained by these polynomials. This cone provides an inner approximation of the real tropicalization, and it coincides with the real tropicalization if $S = \{ f < 0\}$ and the polynomial $f$ has generic coefficients. Furthermore, we show that for a maximally sparse polynomial $f$ the real tropicalization of $S = \{ f < 0\}$ is determined by the outer normal cones of the Newton polytope of $f$ and the signs of its coefficients. Our arguments are valid also for signomials, that is, polynomials with real exponents defined in the positive orthant

The logic of where and while in the 13th and 14th centuries.

Medieval analyses of molecular propositions include many non-truthfunctional connectives in addition to the standard modern binary connectives (conjunction, disjunction, and conditional). Two types of non-truthfunctional molecular propositions considered by a number of 13th- and 14th-century authors are temporal and local propositions, which combine atomic propositions with ‘while’ and ‘where’. Despite modern interest in the historical roots of temporal and tense logic, medieval analyses of ‘while’ propositions are rarely discussed in modern literature, and analyses of ‘where’ propositions are almost completely overlooked. In this paper we introduce 13th- and 14th-century views on temporal and local propositions, and connect the medieval theories with modern temporal and spatial counterparts

On the Number of Real Zeros of Random Sparse Polynomial Systems

Consider a random system $\mathfrak{f}_1(x)=0,\ldots,\mathfrak{f}_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $\mathfrak{f}_k$ has a prescribed set of exponent vectors in a set $A_k\subseteq \mathbb{Z}^n$ of size $t_k$. Assuming that the coefficients of the $\mathfrak{f}_k$ are independent Gaussian of any variance, we prove that the expected number of zeros of the random system in the positive orthant is bounded from above by $4^{-n} \prod_{k=1}^n t_k(t_k-1)$. This result is a probabilisitc version of Kushnirenko's conjecture; it provides a bound that only depends on the number of terms and is independent of their degree.Comment: 26 pages. Different original titl

The Logic of Where and While in the 13th and 14th Centuries

Medieval analyses of molecular propositions include many non-truthfunctional connectives in addition to the standard modern binary connectives (conjunction, disjunction, and conditional). Two types of non-truthfunctional molecular propositions considered by a number of 13th- and 14th-century authors are temporal and local propositions, which combine atomic propositions with ‘while’ and ‘where’. Despite modern interest in the historical roots of temporal and tense logic, medieval analyses of ‘while’ propositions are rarely discussed in modern literature, and analyses of ‘where’ propositions are almost completely overlooked. In this paper we introduce 13th- and 14th-century views on temporal and local propositions, and connect the medieval theories with modern temporal and spatial counterparts

Neural mechanisms of predatory aggression in rats-implications for abnormal intraspecific aggression

Our recent studies showed that brain areas that are activated in a model of escalated aggression overlap with those that promote predatory aggression in cats. This finding raised the interesting possibility that the brain mechanisms that control certain types of abnormal aggression include those involved in predation. However, the mechanisms of predatory aggression are poorly known in rats, a species that is in many respects different from cats. To get more insights into such mechanisms, here we studied the brain activation patterns associated with spontaneous muricide in rats. Subjects not exposed to mice, and those which did not show muricide were used as controls. We found that muricide increased the activation of the central and basolateral amygdala, and lateral hypothalamus as compared to both controls; in addition, a ventral shift in periaqueductal gray activation was observed. Interestingly, these are the brain regions from where predatory aggression can be elicited, or enhanced by electrical stimulation in cats. The analysis of more than 10 other brain regions showed that brain areas that inhibited (or were neutral to) cat predatory aggression were not affected by muricide. Brain activation patterns partly overlapped with those seen earlier in the cockroach hunting model of rat predatory aggression, and were highly similar with those observed in the glucocorticoid dysfunction model of escalated aggression. These findings show that the brain mechanisms underlying predation are evolutionarily conservative, and indirectly support our earlier assumption regarding the involvement of predation-related brain mechanisms in certain forms of escalated social aggression in rats

Thomas rotation and Thomas precession

Exact and simple calculation of Thomas rotation and Thomas precessions along a circular world line is presented in an absolute (coordinate-free) formulation of special relativity. Besides the simplicity of calculations the absolute treatment of spacetime allows us to gain a deeper insight into the phenomena of Thomas rotation and Thomas precession.Comment: 20 pages, to appear in Int. J. Theo. Phy