HK multiplicity, FF-threshold and the Paley-Wiener theorem

For a given algebraically closed field $k$ of characteristic $p>0$ we consider the set ${\mathcal C}_k$, of graded isomorphism classes of {\em standard graded pairs} $(R, I)$, where $R$ is a standard graded ring over the field and $I$ is a graded ideal of finite colength. Here we give a ring homomorphism \Pi:\Z[{\mathcal C}_k] \longrightarrow H(\C)[X], where H(\C) denotes the ring of entire functions. The related entire function and the homomorphism $\Pi$ keep track of the two positive characteristic invariants, $e_{HK}(R, I)$ and $c^I({\bf m})$ of the ring: (1) composing the map $\Pi$ with the evaluation map at $z=0$ gives a ring homomorphism $\Pi_e:\Z[{\mathcal C}_k] \longrightarrow \R[X]$ which sends $$(R,I) \to e_{HK}(R^0, IR^0)+ e_{HK}(R^1, IR^1)X+\cdots + e_{HK}(R^d, IR^d)X^d,$$ where $R^i$ is the union of $i$ dimensional components of $R$ and $e_{HK}(R^i, IR^i)$ is the HK multiplicity of the pair $(R^i, IR^i)$, and in particular the top coefficient is $e_{HK}(R, I)$. (2) If, in addition, $R$ is a two dimensional ring or \mbox {Proj~R} is strongly $F$-regular, then the Fourier transform ${\widehat f}_{R, I}$ belongs to the Paley-Wiener class of the real number, namely the $F$-threshold $c^I_{\bf m}(R)$ of the maximal ideal ${\bf m}$.Comment: 16 page

Website Extractor for Protecting the Content of the site

The previously performed research has focused on quick and effective production of wrapping units, the enlargement of operating components for preservation of wrapping has collected little attentiveness. This is considered to be a vital experimentation issue for this web related information usually varied in number of ways that stopped the specified units particularly from taking out the information in correct manner. So, we are conducting a study on effective technique that brings out information which is in unstructured manner to structural information specifically from web kind of network. The wrapper verification system is going to identify whether it is extracting right information or not because the websites keeps on changing. The specified Verification working frame undergoes automation process going to recover information by making user of Dimensional Reduction methods from varies in the specific Web kind of source part by recognizing information on specified Web related pages of information. Hence after put in wrapped information to one specific Class Classifying unit in corresponding Numerical characteristics for keep away from categorization issue. Therefore finally, the information of outcome put in specific Top-K corresponding query for give best rank related on possibility score values. The specified Wrapping unit validation system depends on one-specific class categorization methods to beat the previously considered weaknesses to recognize the issue by examining both the specific signature and the classifying outcome. If there are sufficiently mislabelled slot units, a method to locate a specific type going to be explored

Recent Approaches of Ocular Disease and Its Herbal Product Treatment: An Updates

One of the most common causes of permanent vision loss is glaucoma. Damage to the optic nerve and retinal ganglion cells, the root cause of visual field impairment, is typically the result of elevated intraocular pressure. Some patients\u27 illness worsens even when treated with eye medications that reduce intraocular pressure. The pathophysiology of glaucoma is not only thought to involve mechanical and vascular dysfunctions of the eye, but also oxidative stress, neuroinflammation, and excitotoxicity. Therefore, natural compounds with antioxidant and anti-inflammatory qualities may offer a different strategy for treating glaucoma. This review summarizes the most up-to-date preclinical and clinical research on natural compounds that have been proven to contain neuroprotective characteristics for retinal ganglion cells and, consequently, may be useful in the treatment of glaucoma. Baicalein, forskolin, marijuana, ginsenoside, resveratrol, and hesperidin are all effective at lowering intraocular pressure. On the other hand, antioxidant, anti-inflammatory, and anti-apoptosis mechanisms are among those through which Ginkgo biloba, More research is needed to determine whether natural items can be used as a safe and effective treatment for glaucoma

Regulating fake news in transnational jurisdiction: a comparative study of Brazil and Argentina

Many oppressive governments have passed cyber laws, cracking down on free speech, and used less traditional tactics to restrict people’s capacity to speak freely and in public. This is particularly true in nations where the government regulates media and information flow due to closed information systems. Laws regulating fake news clash with citizens’ free speech and expression rights. The paper gives an overview of the hurdles in regulating transnational cases of fake news. In cases of cross-border jurisdiction, it becomes vital to examine International legal standards, such as international agreements and international institutions governing fake news. In this paper, we look at two case studies, one from Argentina and one from Brazil, to see how these countries have dealt with the issue of fake news in cases involving transnational jurisdictions. The paper concludes with the observation that various governments employ a variety of approaches and policies in order to combat fake news