Congruence conditions, parcels, and Tutte polynomials of graphs and matroids

Let $G$ be a matrix and $M(G)$ be the matroid defined by linear dependence on the set $E$ of column vectors of $G.$ Roughly speaking, a parcel is a subset of pairs $(f,g)$ of functions defined on $E$ to an Abelian group $A$ satisfying a coboundary condition (that $f-g$ is a flow over $A$ relative to $G$) and a congruence condition (that the size of the supports of $f$ and $g$ satisfy some congruence condition modulo an integer). We prove several theorems of the form: a linear combination of sizes of parcels, with coefficients roots of unity, equals an evaluation of the Tutte polynomial of $M(G)$ at a point $(\lambda-1,x-1)$ on the complex hyperbola $(\lambda - 1)(x-1) = |A|.

A Novel Planar Fractal Antenna with CPW-Feed for Multiband applications

In this paper, a multiband antenna using a novel fractal design is presented. The antenna structure is formed by inscribing a hexagonal slot within a circle. This base structure is then scaled and arranged within the hexagon along its sides without touching the outer structure. The proposed CPW fed, low profile antenna offers good performance in the 1.65-2.59 GHz, 4.16-4.52 GHz and 5.54-6.42 GHz bands and is suitable for GSM 1800/1900, Bluetooth, IMT advanced system and upper WLAN applications. The antenna has been fabricated on a substrate of height 1.6mm and er=4.4 and simulation and experimental results are found to be in good agreement

Exact inflationary solutions in exponential gravity

We consider a modified gravity model of the form $f(R,\phi)=R e^{h(\phi)R}$, where the strong gravity corrections are taken to all orders and $\phi$ is a self-interacting massless scalar field. We show that the conformal transformation of this model to Einstein frame leads to non-canonical kinetic term and negates the advantage of the Einstein frame. We obtain exact solutions for the background in the Jordan frame without performing conformal transformations and show that the model leads to inflation with exit. We obtain scalar and tensor power-spectrum in Jordan frame and show that the model leads to red-tilt. We discuss the implications of the same in the light of cosmological observations.Comment: 17 pages, 4 figures, 2 table

Inflation with f(R,ϕ)f(R,\phi) in Jordan frame

We consider an $f(R)$ action that is non-minimally coupled to a massive scalar field. The model closely resembles scalar-tensor theory and by conformal transformation can be transformed to Einstein frame. To avoid the ambiguity of the frame dependence, we obtain an exact analytical solution in Jordan frame and show that the model leads to a period of accelerated expansion with an exit. Further, we compute the scalar and tensor power spectrum for the model and compare them with observations.Comment: 10 pages, 3 figure