Asymptotic analysis

In mathematical analysis, asymptotic analysis known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f as n becomes large. If f = n2 + 3n as n becomes large, the term 3n becomes insignificant compared to n2; the function f is said to be "asymptotically equivalent to n2, as n → ∞". This is written symbolically as f ~ n2, read as "f is asymptotic to n2". An example of an important asymptotic result is the prime number theorem. Let π denote the prime-counting function, i.e. π is the number of prime numbers that are less than or equal to x. The theorem states that π ∼ x ln ⁡ x. Formally, given functions f and g, we define a binary relation f ∼ g if and only if lim x → ∞ f g = 1; the symbol ~ is the tilde. The relation is an equivalence relation on the set of functions of x; the domain of f and g can be any set for which the limit is defined: e.g. real numbers, complex numbers, positive integers. The same notation is used for other ways of passing to a limit: e.g. x → 0, x ↓ 0, |x| → 0.

The way of passing to the limit is not stated explicitly, if it is clear from the context. Although the above definition is common in the literature, it is problematic if g is zero infinitely as x goes to the limiting value. For that reason, some authors use an alternative definition; the alternative definition, in little-o notation, is that f ~ g if and only if f − g = o. This definition is equivalent to the prior definition if g is not zero in some neighbourhood of the limiting value. If f ∼ g and a ∼ b under some mild conditions, the following hold. F r ∼ g r, for every real r log ⁡ ∼ log ⁡ f × a ∼ g × b f / a ∼ g / b Such properties allow asymptotically-equivalent functions to be exchanged in many algebraic expressions. Factorial n! ∼ 2 π n n —this is Stirling's approximationPartition functionFor a positive integer n, the partition function, p, gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered. P ∼ 1 4 n 3 e π 2 n 3 Airy functionThe Airy function, Ai, is a solution of the differential equation y" − xy = 0.

Ai ⁡ ∼ e − 2 3 x 3 2 2 π x 1 / 4 Hankel functions H α ∼ 2 π z e i H α ∼ 2 π z e − i ( z − 2 π α − π 4

Rob, Velike Lašče

Rob is a settlement in the Municipality of Velike Lašče in Slovenia. The area is part of the traditional region of Lower Carniola and is now included in the Central Slovenia Statistical Region. Rob was attested in historical sources as Rab in 1463, 1467, 1484; the name is derived from the common noun rob'edge'. The village is located at the edges of the Rute Plateau and Mačkovec Plateau, where they meet the alluvial valley of the Rašica River and the Mišja Valley; the local parish church, built on a hill north of the village, is dedicated to the Nativity of Mary and belongs to the Roman Catholic Archdiocese of Ljubljana. It was built in 1845 on the site of an earlier church. Rob on Geopedia Media related to Rob, Velike Lašče at Wikimedia Commons

Jahnavi Harrison

Jahnavi Harrison is a British musician known for her kirtan call-and-response singing and her co-founding of Kirtan London, her 2015 album Like a River to the Sea, her appearances as a presenter on BBC Radio 4's Something Understood programme. Jahnavi Harrison grew up in the Hare Krishna community at Bhaktivedanta Manor, where her father is a priest, she trained in Bharatanatyam dance. She gained a bachelor's degree in linguistics and creative writing from Middlesex University in 2009. Harrison has appeared on BBC Radio 2's "Pause for Thought" on the Chris Evans Breakfast Show, she has presented several of BBC Radio 4's Something Understood programmes. She helps to run Kirtan London, has contributed performances to its "Mantra Lounge" at Neal's Yard, London, she teaches Kirtan at Bhaktivedanta College. She works in New London. Harrison's 2015 album Like a River to the Sea, sung in Sanskrit, contains Bhakti yoga songs and mantras in kirtan call-and-response style, set to modern melodies and arrangements.

The instruments used include both the traditional mrdanga harmonium. The title is a reference to a prayer of Queen Kunti, an Indian saint, who asks the god Krishna to draw her thoughts to him, "just as the river forever flows to the sea". McKenna Rowe, reviewing Like a River to the Sea for LA Yoga, wrote that she was "moved and stunned by the beauty of the instruments and arrangements" of the songs, she found the album "a satisfying masterpiece", not only for people who like devotional music. Reviewing the album for Pulse magazine, Sanjeevini Dutta noted that kirtan was "the sound track" to Harrison's childhood, she called it "a first album of astonishing ripeness and sweetness," one that drew the listener "to a profound inner space," yet staying in contact with "life lived full of joys and heartbreak."Amardeep Dhillon, in Songlines magazine, called the music pleasant but unsurprising, the tracks being "soothing and uncluttered, with Harrison's violin weaving in between Celtic and Karnatic strains".

He approved of Harrison's retention of kirtan's communal feeling. He found the title song and the "haunting" Ceili in Braj! Musically the most interesting tracks. In his view the album succeeds through the undoubted "depth of feeling and love that come through". Harrison appeared in "Bhaja Govindam" on Madi Das's 2015 charity album Bhakti Without Borders, nominated for a Grammy award. Harrison has appeared on the Mantra Lounge Volumes from Radha Krishna Records. In 2012 she was presented with a UK Youth Achievement award for her work with sacred music. Website Dandavats review of Like a River to the Sea Mindrolling podcast with Raghu Markus