arXiv subjects

Mehdi Ghasemi

Publications and source records attributed to Mehdi Ghasemi.

23 records · Page 2Linked to original sources

arXiv2012

Closure of the cone of sums of 2d-powers in real topological algebras

Let $R$ be a unitary commutative real algebra and $K\subseteq Hom(R,\mathbb{R})$, closed with respect to the product topology. We consider $R$ endowed with the topology $\mathcal{T}_K$, induced by the family of seminorms $ρ_α(a):=|α(a)|$, for $α\in K$ and $a\in R$. In case $K$ is compact, we also consider the topology induced by $\|a\|_K:=\sup_{α\in K}|α(a)|$ for $a\in R$. If $K$ is Zariski dense, then those topologies are Hausdorff. In this paper we prove that the closure of the cone of sums of 2d-powers, $\sum R^{2d}$, with respect to those two topologies is equal to $Psd(K):=\{a\in R:α(a)\geq 0,\textrm{for all}α\in K\}$. In particular, any continuous linear functional $L$ on the polynomial ring $R=\mathbb{R}[X_1,...,X_n]$ with $L(h^{2d})\ge0$ for each $h\in R$ is integration with respect to a positive Borel measure supported on $K$. Finally we give necessary and sufficient conditions to ensure the continuity of a linear functional with respect to those two topologies.

math.RA↗

arXiv2012

The Moment Problem for Continuous Positive Semidefinite Linear functionals

Let $τ$ be a locally convex topology on the countable dimensional polynomial $\reals$-algebra $\rx:=\reals[X_1,...,X_n]$. Let $K$ be a closed subset of $\reals^n$, and let $M:=M_{\{g_1, ... g_s\}}$ be a finitely generated quadratic module in $\rx$. We investigate the following question: When is the cone $\Pos(K)$ (of polynomials nonnegative on $K$) included in the closure of $M$? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of $M=\sos$ with respect to weighted norm-$p$ topologies. We show that this closure coincides with the cone $\Pos(K)$ where $K$ is a certain convex compact polyhedron.

math.AG↗

arXiv2012

A New Full Adder Cell for Molecular Electronics

Due to high power consumption and difficulties with minimizing the CMOS transistor size, molecular electronics has been introduced as an emerging technology. Further, there have been noticeable advances in fabrication of molecular wires and switches and also molecular diodes can be used for designing different logic circuits. Considering this novel technology, we use molecules as the active components of the circuit, for transporting electric charge. In this paper, a full adder cell based on molecular electronics is presented. This full adder is consisted of resonant tunneling diodes and transistors which are implemented via molecular electronics. The area occupied by this kind of full adder would be much times smaller than the conventional designs and it can be used as the building block of more complex molecular arithmetic circuits.

cs.ET↗

arXiv2011

Closure of the cone of sums of 2d-powers in certain weighted $\ell_1$-seminorm topologies

Berg, Christensen and Ressel prove that the closure of the cone of sums of squares in the ring of real polynomials in the topology induced by the $\ell_1$-norm is equal to the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$. The result is deduced as a corollary of a general result which is valid for any commutative semigroup. In later work Berg and Maserick and also Berg, Christensen and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $\ell_1$-seminorm topology associated to an absolute value. In the present paper we give a new proof of these results which is based on Jacobi's representation theorem. At the same time, we use Jacobi's representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer $d>0$, the closure of the cone of sums of 2d-powers in the ring of real polynomials in the topology induced by the $\ell_1$-norm is equal the cone consisting of all polynomials which are non-negative on the hypercube $[-1,1]^n$.

math.AC↗

arXiv2011

Lower bounds for polynomials using geometric programming

We make use of a result of Hurwitz and Reznick, and a consequence of this result due to Fidalgo and Kovacec, to determine a new sufficient condition for a polynomial $f\in\mathbb{R}[X_1,...,X_n]$ of even degree to be a sum of squares. This result generalizes a result of Lasserre and a result of Fidalgo and Kovacec, and it also generalizes the improvements of these results given in [6]. We apply this result to obtain a new lower bound $f_{gp}$ for $f$, and we explain how $f_{gp}$ can be computed using geometric programming. The lower bound $f_{gp}$ is generally not as good as the lower bound $f_{sos}$ introduced by Lasserre and Parrilo and Sturmfels, which is computed using semidefinite programming, but a run time comparison shows that, in practice, the computation of $f_{gp}$ is much faster. The computation is simplest when the highest degree term of $f$ has the form $\sum_{i=1}^n a_iX_i^{2d}$, $a_i>0$, $i=1,...,n$. The lower bounds for $f$ established in [6] are obtained by evaluating the objective function of the geometric program at the appropriate feasible points.

math.OC↗