Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.Comment: Restructured the presentation in order to be the extended version of a paper that will shortly appear in SIAM Journal on Computin

The effects of enhanced Z penguins on lepton polarizations in B→Xsℓ+ℓ−B \to X_s \ell^+ \ell^-

The sensitivity of the $B \to \pi K$ mode to electroweak penguins and the recent experimental data for the $B \to \pi \pi$, $\pi K$ modes has given rise to what is known as the ``$B \to \pi K$ puzzle''. Recently it has been observed that this {\sl puzzle} can be resolved by considering the new physics which can enter via $Z^0$ penguins. In this note we analyze the effect of this enhanced $Z^0$ penguins on the lepton polarization asymmetries of $b \to s \ell^+ \ell^-$.Comment: Published versio

The Hard X-ray emission of the blazar PKS 2155--304

The synchrotron peak of the X-ray bright High Energy Peaked Blazar (HBL) PKS 2155$-$304 occurs in the UV-EUV region and hence its X-ray emission (0.6--10 keV) lies mostly in the falling part of the synchrotron hump. We aim to study the X-ray emission of PKS 2155$-$304 during different intensity states in 2009$-$2014 using XMM$-$Newton satellite. We studied the spectral curvature of all of the observations to provide crucial information on the energy distribution of the non-thermal particles. Most of the observations show curvature or deviation from a single power-law and can be well modeled by a log parabola model. In some of the observations, we find spectral flattening after 6 keV. In order to find the possible origin of the X-ray excess, we built the Multi-band Spectral Energy distribution (SED). We find that the X-ray excess in PKS 2155--304 is difficult to fit in the one zone model but, could be easily reconciled in the spine/layer jet structure. The hard X-ray excess can be explained by the inverse Comptonization of the synchrotron photons (from the layer) by the spine electrons.Comment: 14 pages, 7 Figures, Accepted for publication in Ap

Intermediation and Value Creation in an Incomplete Market: Implications for Securitization

This paper studies the impact of financial innovations on real investment decisions. We model an incomplete market economy comprised of firms, investors and an intermediary. The firms face unique investment opportunities that are not spanned by the traded securities in the financial market, and thus, cannot be priced uniquely using the no-arbitrage principle. The specific innovation we consider is securitization; the intermediary buys claims from the firms that are fully backed by cash flows from the new projects, pools these claims together, and then issues tranches of secondary securities to the investors. We first derive necessary and sufficient conditions under which pooling provides value enhancement and the prices paid to the firms are acceptable to them compared to the no-investment option or the option of forming alternative pools. We find that there is a unique pool that is sustainable, and may or may not consist of all projects in the intermediary’s consideration set. We then determine the optimal design of tranches, fully backed by the asset pool, to be sold to different investor classes. We determine the general structure of the tranches. The new securities created by the intermediary could have up to three components, one that is a marketable claim, one that represents the arbitrage opportunities available in the market due to special ability to design and sell securities to a subset of investors, and a third component that is the rest of the asset pool which is sold at a price which does not exceed arbitrage based bounds to investors. The presence of these three components in the tranching solution has direct bearing upon the size of the asset pool, and therefore value creation due to financing additional projects

Intermediation and Value Creation in an Incomplete Market: Implications for Securitization

This paper studies the impact of financial innovations on real investment decisions. We model an incomplete market economy comprised of firms, investors and an intermediary. The firms face unique investment opportunities that are not spanned by the securities traded in the financial market, and thus, cannot be priced uniquely using the no-arbitrage principle. The specific innovation we consider is securitization: the intermediary buys claims from the firms that are fully backed by cash flows from the new projects, pools these claims together, and then issues tranches of secondary securities to the investors. We first derive necessary and sufficient conditions under which pooling provides value enhancement from the new projects that are undertaken, and the prices paid to the firms are acceptable to them compared to the no-investment option or the option of forming alternative pools. We find that there is a unique pool that is sustainable, and which may or may not consist of all projects in the intermediary’s consideration set

Spiders (Araneae) from Agricultural fields near foothill of Satpura Mountain ranges of Amravati District, Maharashtra, India.

This paper deals with the study of spider distribution in agricultural fields adjoining the Satpura Mountain Ranges of the Amravati district. The total collection of spiders comprises 12 families, 37 genera and 76 species. The Araneidae and Thomisidae families were dominant, followed by Saltisidae from the selected agricultural area. Lycosidae and Tetragnathidae were also found in a decreasing number of species due to their specific habitation. The analysis of guild structure revealed six feeding guilds. Orb web weavers and Ambushers constituted the dominant feeding guild representing 33% and 22% of the total collection respectively. Stalkers and ground runners represent 20% and 17% and foliage hunters and sheet web weavers represent 5% and 3% respectively. A new species of genus Sassacus (Beetle like spider), family Saltisidae was found for the first time in the Vidarbha region. The richness of spider diversity is a sign of the overall biodiversity which is a useful indicator of the species richness of the agro-ecosystem.
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