On the Expansion of Group-Based Lifts

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\cdots{},v_k$ and each edge $(u,v)$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been studied as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are: (1) There is a constant $c_1$ such that for every $k\geq 2^{c_1nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph with $H$ being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This can be viewed as an analogue of the well-known no-expansion result for abelian Cayley graphs. (2) A uniform random lift in a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues also bounded by $\lambda+O(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. In particular, there is a constant $c_2$ such that for every $k\leq 2^{c_2n/d^2}$, there exists a lift $H$ of every Ramanujan graph in a cyclic group of order $k$ with $H$ being almost Ramanujan. We use this to design a quasi-polynomial time algorithm to construct almost Ramanujan expanders deterministically. The existence of expanding lifts in cyclic groups of order $k=2^{O(n/d^2)}$ can be viewed as a lower bound on the order $k_0$ of the largest abelian group that produces expanding lifts. Our results show that the lower bound matches the upper bound for $k_0$ (upto $d^3$ in the exponent)

W+W−W^+W^- production in Large extra dimension model at next-to-leading order in QCD at the LHC

We present next-to-leading order QCD corrections to production of two $W$ bosons in hadronic collisions in the extra dimension ADD model. Various kinematical distributions are obtained to order $\alpha_s$ in QCD by taking into account all the parton level subprocesses. We estimate the impact of the QCD corrections on various observables and find that they are significant. We also show the reduction in factorization scale uncertainty when ${\cal O}(\alpha_s)$ effects are included.Comment: Journal versio

Biodiversity Conservation in India: A Review

India is one of the 34 Mega biodiversity hotspots of the world. It is home for threatened and endemic species that have immense ecological and commercial value. Due to increased human population and overexploitation of natural resources biodiversity is under threat worldwide. Threats to species are principally due to decline and fragmentation of their habitat. Biodiversity, as measured by the number of plant and vertebrate species, is greatest in the Western Ghats and North East in India.Biodiversity has several values such as economical, ecological, ethical, medicinal, aesthetical, social and many more. The present need of the hour is the sustainable use of biodiversity. Inventory only will identify the key issues of management of biodiversity which include a continuing process of searching and re-examining the early findings. Conservation of biodiversity is being done in the form of various legislations, the establishment of the protected area, Zoos and botanical gardens, gene Bank, seed bank etc. In this paper, the overview of Biodiversity and Its types, values, the status of biodiversity in India, causes of threats and various steps to be taken for the conservation of biodiversity have been discussed