SUMMARY / RELATED TOPICS

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function f on a set E if, given any arbitrarily small positive number ϵ, a number N can be found such that each of the functions f N, f N + 1, f N + 2, … differ from f by no more than ϵ at every point x in E. Described in an informal way, if f n converges to f uniformly the rate at which f n approaches f is "uniform" throughout its domain in the following sense: in order to determine how large n needs to be to guarantee that f n falls within a certain distance ϵ of f, we do not need to know the value of x ∈ E in question — there is a single value of N = N independent of x, such that choosing n to be larger than N will suffice; the difference between uniform convergence and pointwise convergence was not appreciated early in the history of calculus, leading to instances of faulty reasoning. The concept, first formalized by Karl Weierstrass, is important because several properties of the functions f n, such as continuity, Riemann integrability, with additional hypotheses, differentiability, are transferred to the limit f if the convergence is uniform, but not if the convergence is not uniform.

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Standard notions of convergence did not exist at the time, Cauchy handled convergence using infinitesimal methods; when put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions; the term uniform convergence was first used by Christoph Gudermann, in an 1838 paper on elliptic functions, where he employed the phrase "convergence in a uniform way" when the "mode of convergence" of a series ∑ n = 1 ∞ f n is independent of the variables ϕ and ψ.

While he thought it a "remarkable fact" when a series converged in this way, he did not give a formal definition, nor use the property in any of his proofs. Gudermann's pupil Karl Weierstrass, who attended his course on elliptic functions in 1839–1840, coined the term gleichmäßig konvergent which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently, similar concepts were articulated by Philipp Ludwig von Seidel and George Gabriel Stokes. G. H. Hardy compares the three definitions in his paper "Sir George Stokes and the concept of uniform convergence" and remarks: "Weierstrass's discovery was the earliest, he alone realized its far-reaching importance as one of the fundamental ideas of analysis." Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others. We first define uniform convergence for real-valued functions, although the concept is generalized to functions mapping to metric spaces and, more uniform spaces.

Suppose E is a n ∈ N is a sequence of real-valued functions on it. We say the sequence n ∈ N is uniformly convergent on E with limit f: E → R if for every ϵ > 0, {\displaystyle \epsi

Gilad Hochman

Gilad Hochman is an Israeli classical music composer. He was born to an Odessa native father and a Paris native mother and resides in Berlin, Germany. Hochman began his musical life at the age of 6, he started composing at the age of 9 and at 18 graduated from the Herzeli'ya Music Conservatory, where he studied under composer Ilya Heifets and pianist Mark Shaviner. In 2007 Hochman graduated with honors from the Buchman-Mehta School of Music at Tel Aviv University, studying composition under Gil Shohat and theoretical subjects under some of Israel's leading senior musicians. Hochman's music was described as “written with a true artist’s hand”, “the highpoint of an exciting performance”, “sheds new light on customary expressions in music” and as “original, fascinating and colorful” by the Israeli Prime Minister Award committee in 2007, he was defined as “one of Israel’s most important composers” in an article concerning “composer Gilad Hochman’s meteoric career", as "a rising star in the classical music world" and as an "already well known classical composer" by the BBC.

The New York Times commented: "perhaps most impressive was Mr. Hochman’s gift for assembling musical gestures that come across as psychologically revealing, whether it’s players’ finishing each other's sentences, the neurotic instability suggested by a microtonal wobble on a held note or the freezing self-doubt of a painfully quiet passage."At the age of 24 Hochman became the youngest composer to be awarded the prestigious Israeli Prime Minister Award for his artistic work. At age 22 he was the youngest to be appointed composer in residence by one of Israel’s most known orchestras, Ra’anana Symphonette, he founded and artistically directed the Arco String Ensemble and the New Sounds concert series of the Israel Composers’ League and in 2007 won a merit certificate from the city of Ra'anana. In 2017 he was awarded the S&R Washington Award for his musical work. Hochman's oeuvre includes a range of compositions for solo instruments, chamber music and orchestras which reflects a variety of aesthetic approaches.

His music is, on the one hand, a continuation of classical music's development, yet on the other hand he puts a great emphasis on themes relating to the Jewish tradition and his Israeli origin. His diverse body of works includes Whom My Soul Loveth for mixed choir; the piece explores feeling of human yearning and is based upon a single verse from the biblical Solomon's Song of Songs. In this work Hochman defines, in an original fashion, the place of the individual cello vis-à-vis the choir surrounding it, creating a fascinating dialogue that shifts between a whispered prayer and dramatic, powerfully expressive climaxes; the uniqueness of this work does not derive from the unconventional ensemble, but more from Hochman's sensitive treatment of the melodic phrases, the gradual construction of the dramatic progression and the way in which the distinctive artistic goal is reached in the end. This composition was commissioned by the Chorbiennale of Aachen, Germany in year 2009. Hochman's music is performed by leading musicians and music institutes worldwide.

It was played, among others, by Oriol Ensemble, the Deutsches Kammerorchester Berlin, Synergy Ensemble, Musica Nova Ensemble, XelmYa Ensemble – Berlin, Ensemble Meitar, Ensemble Sirenot, the Aachener Kammerchor, Studium Chorale, Tel Aviv Soloists Ensemble and the Israel Chamber Orchestra. It was performed at the Heidelberg Biennale for New Music, Aachen International Chorbiennale, Israel Music Fest, the Israeli Schubertiade, Musica Sacra Festival - Maastricht and in dozens of different concert series and in music academies like Sibelius Academy, UdK, Guildhall School of Music and Rubin. Notable performances of Hochman's compositions include two retrospective concerts in Germany, at the Hochschule für Musik Mainz in November 2009 and at Bochum Art Museum in April 2011, an orchestral debut concert at the Berlin Philharmonie, a debut concert at Rachmaninov Hall of the Moscow Tchaikovsky Conservatory in February 2013 and an orchestral debut at London's St James's Church, Piccadilly of his Fantasia Concertante for mandolin and string orchestra.

In October 2015 Hochman's orchestral piece “Suspended Reality” was played in a concert tour by the Israel Sinfonietta Beer Sheva and the Philharmonie der Nationen under conductor Justus Frantz, starting with its premiere at the German Parliament and continuing with concerts at the Alte Oper Frankfurt, Wuppertal's Stadthalle and Tel Aviv's Opera House, among others. Music of Israel Official Website

Jeffrey Ettinger

Jeffrey Martin Ettinger is an American corporate executive and former CEO of Hormel. He earned his Bachelor of Juris Doctor from the University of California, Los Angeles, he served as a law clerk to judge Arthur Alarcon of the United States Court of Appeals for the Ninth Circuit. He has been with Hormel since 1989, fulfilling roles such as senior corporate attorney and president of Jennie-O, he served as General Counsel of Comar Marketing. He served on the American Meat Institute Board of Directors, Grocery Manufacturers Association Board of Directors, the Minnesota Business Partnership Board of Directors, he assumed the role of President, CEO, Chairman of the Board of Directors for Hormel in 2005 and retired as CEO effective October 30, 2016. He continued to serve as the company's Chairman of the Board until he retired on November 20, 2017