How to succeed at Witten diagram recursions without really trying

Witten diagrams are basic objects for studying dynamics in AdS space, and also play key roles in the analytic functional bootstrap. However, these diagrams are notoriously hard to evaluate, making it extremely difficult to search for recursion relations among them. In this note, we present simple methods to obtain recursion relations for exchange Witten diagrams from conformal block recursion relations. We discover a variety of new relations, including the dimensional reduction formulae for exchange Witten diagrams. In particular, we find a five-term recursion relation relating exchange Witten diagrams in d and d − 2 dimensions. This gives the holographic analogue of a similar formula for conformal blocks due to Parisi-Sourlas supersymmetry. We also extend the analysis to two-point functions in CFTs with conformal boundaries, and obtain similar results.

q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding [Formula: see text]-supercongruence. Similar [Formula: see text]-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the [Formula: see text]-Lucas numbers.

THE BLUE ALIEN IN KOI MIL GAYA FILM: POPULAR LITERATURE

<p>The purpose of this research is to<strong> </strong>describe<strong> </strong>the blue alien as the phenomenon Alien’s representation in science fiction of Bollywood and Bollywood’s action in bringing outer space alien to Indian culture that is represented in<em> Koi Mil Gaya </em>film. This research uses qualitative research. The researcher needs popular literature by Ida Rochani Adi to get what the author is willing to share her readers. It is also a way to the researcher to investigate why the author choose alien as the new character, and is there popular culture inside the character evidences the effects and goals of the author in creating a story. The analysis reveals that the alien’s representation of Bollywood’s science fiction, and Bollywood’s action in bringing outer space alien to India culture. The conclusion shows there are similar formula in each Bollywood science fiction in alien’s representation and Bollywood action in bringing outer space alien to India culture is influenced by 3 factors, there are: Hollywood influence, Ancient India influence, and popular news in India. The researcher uses the symbol to analyze the blue alien as the representation of Lord Krishna. It is Hindu mythology. Hindu mythology is popular culture in India belief. It is appropriate with the researcher’s assumption that the blue alien has correlation with India culture. In conclusion, the alien which has blue skin is the appearance of Lord Krishna.</p>

Formulating the Building Blocks for National Cyberpower

Cyber threats pose a growing risk to national security for all nations; cyberpower is consequently becoming an increasingly prominent driver in the attainment of national security for any state. This paper investigates the national cyberpower environment by analysing the elements of cyberspace as part of national security. David Jablonsky (1997) distinguishes between natural and social determinants of power in his discussion of national power. Also, Jablonsky refers to Ray Cline's formula (Cline, 1993) to determine a rough estimate of “perceived” national power by focusing primarily on a state's capacity to wage war. In this paper, the formula for Perceived Power (PP) will be adapted for use in cyberspace to create a similar formula for Perceived Cyberpower (PCP) that focuses primarily on a state's capacity for cyberwarfare. Military cyberpower is one of the critical elements of cyberpower. The paper also discusses how to operationalise military cyberpower.

Building Blocks and Measurement of National Cyberpower

Cyberspace and cyber threats are increasingly recognized to pose a significant risk to a state's security. Cyberpower is central to national power and thus a driver towards the attainment of national security. The authors decompose national cyberpower by analyzing the elements of cyberspace as part of national security. David Jablonsky distinguishes between natural and social determinants of power in his discussion of national power and refers to Ray Cline's formula to determine a rough estimate of “perceived” national power by focusing primarily on a state's capacity to wage war. The authors present an adaptation of the formula for perceived power for use in cyberspace to create a similar formula for perceived cyberpower that focuses primarily on a state's capacity for cyberwarfare. Military cyberpower is one of the critical elements of cyberpower but little attention has been paid to this concept in the literature. In this chapter, concepts such as cyber effectiveness and the operationalization of military cyberpower are also addressed.

Formulating the Building Blocks for National Cyberpower

Cyber threats pose a growing risk to national security for all nations; cyberpower is consequently becoming an increasingly prominent driver in the attainment of national security for any state. This paper investigates the national cyberpower environment by analysing the elements of cyberspace as part of national security. David Jablonsky (1997) distinguishes between natural and social determinants of power in his discussion of national power. Also, Jablonsky refers to Ray Cline's formula (Cline, 1993) to determine a rough estimate of “perceived” national power by focusing primarily on a state's capacity to wage war. In this paper, the formula for Perceived Power (PP) will be adapted for use in cyberspace to create a similar formula for Perceived Cyberpower (PCP) that focuses primarily on a state's capacity for cyberwarfare. Military cyberpower is one of the critical elements of cyberpower. The paper also discusses how to operationalise military cyberpower.

Taking Community Seriously: Lessons from the Israeli Disengagement Plan

Eminent domain, or the expropriation of private property, is among the most controversial of legal arrangements. The challenges and threats that it poses to private property make it the subject of debate and dispute. Surprisingly, however, most Western jurisdictions embrace a similar formula to address expropriation, both in terms of the purposes that justify such action and the compensation that should be awarded to property owners.This article challenges the prevailing eminent domain formula, according to which, regardless of the circumstances of the expropriation, compensation to the property owner is determined by reference to the market value of the property. By exploring the case of Israel's 2005 disengagement plan, as a result of which 21 residential communities were uprooted by expropriation, this article argues that loss of communality should be taken into account in expropriations that uproot entire communities. However, in order for the legal arrangement to be efficient, fair and, of no less importance, to reflect the values embodied in the right to property, it should be constituted within a normative infrastructure that takes into account the values that the society wishes to endorse, and the inner meaning of these values.

K1 of products of Drinfeld modular curves and special values of L-functions

Beilinson [Higher regulators and values of L-functions, Itogi Nauki i Tekhniki Seriya Sovremennye Problemy Matematiki Noveishie Dostizheniya (Current problems in mathematics), vol. 24 (Vserossiisky Institut Nauchnoi i Tekhnicheskoi Informatsii, Moscow, 1984), 181–238] obtained a formula relating the special value of the L-function of H2 of a product of modular curves to the regulator of an element of a motivic cohomology group, thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves, providing evidence for an analogous conjecture in the case of function fields.

Asymptotics for the Distributions of Subtableaux in Young and Up-Down Tableaux

Let $\mu$ be a partition of $k$, and $T$ a standard Young tableau of shape $\mu$. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of $N$ cells contains $T$ as a subtableau is asymptotic to $f^\mu/k!$ as $N$ goes to infinity, where $f^\mu$ is the number of all tableaux of shape $\mu$. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young tableaux with at most $n$ rows is proportional to $\prod_{1\le i < j\le n}\bigl((\mu_i-i)-(\mu_j-j)\bigr)$; as $n$ goes to infinity, the probabilities approach $f^\mu/k!$ as expected. We have a similar formula for up-down tableaux; the probability approaches $f^\mu/k!$ if $\mu$ has $k$ cells and thus the up-down tableau is actually a standard tableau, and approaches 0 if $\mu$ has fewer than $k$ cells.

A $q$-Analogue of Faulhaber's Formula for Sums of Powers

Let $$ S_{m,n}(q):=\sum_{k=1}^{n}\frac{1-q^{2k}}{1-q^2} \left(\frac{1-q^k}{1-q}\right)^{m-1}q^{\frac{m+1}{2}(n-k)}. $$ Generalizing the formulas of Warnaar and Schlosser, we prove that there exist polynomials $P_{m,k}(q)\in{\Bbb Z}[q]$ such that $$ S_{2m+1,n}(q) =\sum_{k=0}^{m}(-1)^kP_{m,k}(q) \frac{(1-q^n)^{m+1-k}(1-q^{n+1})^{m+1-k}q^{kn}} {(1-q^2)(1-q)^{2m-3k}\prod_{i=0}^{k}(1-q^{m+1-i})}, $$ and solve a problem raised by Schlosser. We also show that there is a similar formula for the following $q$-analogue of alternating sums of powers: $$ T_{m,n}(q):=\sum_{k=1}^{n}(-1)^{n-k} \left(\frac{1-q^k}{1-q}\right)^{m}q^{\frac{m}{2}(n-k)}. $$