On the topologies induced by a cone

Let $A$ be a commutative and unital $\mathbb{R}$-algebra, and $M$ be an Archimedean quadratic module of $A$. We define a submultiplicative seminorm $\|\cdot\|_M$ on $A$, associated with $M$. We show that the closure of $M$ with respect to $\|\cdot\|_M$-topology is equal to the closure of $M$ with respect to the finest locally convex topology on $A$. We also compute the closure of any cone in $\|\cdot\|_M$-topology. Then we omit the Archimedean condition and show that there still exists a lmc topology associated to $M$, pursuing the same properties

Lower Bounds for a Polynomial on a basic closed semialgebraic set using geometric programming

$f,g_1,...,g_m$ be elements of the polynomial ring $\mathbb{R}[x_1,...,x_n]$. The paper deals with the general problem of computing a lower bound for $f$ on the subset of $\mathbb{R}^n$ defined by the inequalities $g_i\ge 0$, $i=1,...,m$. The paper shows that there is an algorithm for computing such a lower bound, based on geometric programming, which applies in a large number of cases. The algorithm extends and generalizes earlier algorithms of Ghasemi and Marshall, dealing with the case $m=0$, and of Ghasemi, Lasserre and Marshall, dealing with the case $m=1$ and $g_1= M-(x_1^d+\cdots+x_n^d)$. Here, $d$ is required to be an even integer $d \ge \max\{2,\deg(f)\}$. The algorithm is implemented in a SAGE program developed by the first author. The bound obtained is typically not as good as the bound obtained using semidefinite programming, but it has the advantage that it is computable rapidly, even in cases where the bound obtained by semidefinite programming is not computable

Curved Corner Contribution to the Entanglement Entropy in an Anisotropic Spacetime

We study the holographic entanglement entropy of anisotropic and nonconformal theories that are holographically dual to geometries with hyperscaling violation, parameterized by two parameters $z$ and $\theta$. In the vacuum state of a conformal field theory, it is known that the entanglement entropy of a kink region contains a logarithmic universal term which is only due to the singularity of the entangling surface. But, we show that the effects of the singularity as well as anisotropy of spacetime on the entanglement entropy exhibit themselves in various forms depending on $z$ and $\theta$ ranges. We identify the structure of various divergences that may be appear in the entanglement entropy, specially those which give rise to a universal contribution in the form of the logarithmic or double logarithmic terms. In the range $z>1$, for values $z=2k/(2k-1)$ with some integer $k$ and $\theta=0$, Lifshitz geometry, we find a double logarithmic term. In the range $0<z$, for values $\theta=1-2n|z-1|$ with some integer $n$ we find a logarithmic term.Comment: 19 pages, 2 figs; v2: introduction and conclusion expanded, refs adde

Entanglement entropy of singular surfaces under relevant deformations in holography

In the vacuum state of a CFT, the entanglement entropy of singular surfaces contains a logarithmic universal term which is only due to the singularity of the entangling surface. We consider the relevant perturbation of a three dimensional CFT for singular entangling surface. We observe that in addition to the universal term due to the entangling surface, there is a new logarithmic term which corresponds to a relevant perturbation of the conformal field theory with a coefficient depending on the scaling dimension of the relevant operator. We also find a new power law divergence in the holographic entanglement entropy. In addition, we study the effect of a relevant perturbation in the Gauss-Bonnet gravity for a singular entangling surface. Again a logarithmic term shows up. This new term is proportional to both the dimension of the relevant operator and the Gauss-Bonnet coupling. We also introduce the renormalized entanglement entropy for a kink region which in the UV limit reduces to a universal positive finite term.Comment: 21 pages. v2: 30 pages, title changed, one section regarding the renormalization added, minor corrections in text and equation