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For large positive values of the parameter > 0 displaystyle alpha >0

, the following formulation is a smooth, differentiable approximation of the maximum function. For negative values of the parameter that are large in absolute value, it approximates the minimum. S ( x 1 , , x n ) i 1 n x i e x i i 1 n e x i displaystyle mathcal S_alpha (x_1,ldots ,x_n)frac sum _i1^nx_ie^alpha x_isum _i1^ne^alpha x_i S displaystyle mathcal S_alpha has the following properties: S max displaystyle mathcal S_alpha to max as displaystyle alpha to infty S 0 displaystyle mathcal S_0 is the arithmetic mean of its inputs S min displaystyle mathcal S_alpha to min as displaystyle alpha to -infty

The gradient of S displaystyle mathcal S_alpha is closely related to softmax and is given by x i S ( x 1 , , x n ) e x i j 1 n e x j 1 ( x i S ( x 1 , , x n ) ) . displaystyle nabla _x_imathcal S_alpha (x_1,ldots ,x_n)frac e^alpha x_isum _j1^ne^alpha x_j1alpha (x_i-mathcal S_alpha (x_1,ldots ,x_n)).

This makes the softmax function useful for optimization techniques that use gradient descent.

LogSumExpAnother smooth maximum is LogSumExp: L S E ( x 1 , , x n ) 1 / log ( exp ( x 1 ) exp ( x n ) ) displaystyle mathrm LSE _alpha (x_1,ldots ,x_n)1/alpha log(exp(alpha x_1)ldots exp(alpha x_n))

This can also be normalized if the x i displaystyle x_i are all non-negative, yielding a function with domain 0 , ) n displaystyle 0,infty )^n and range 0 , ) displaystyle 0,infty )

: g ( x 1 , , x n ) log ( exp ( x 1 ) exp ( x n ) ( n 1 ) ) displaystyle g(x_1,ldots ,x_n)log(exp(x_1)ldots exp(x_n)-(n-1))

The ( n 1 ) displaystyle (n-1) term corrects for the fact that exp ( 0 ) 1 displaystyle exp(0)1 by canceling out all but one zero exponential, and log 1 0 displaystyle log 10 if all x i displaystyle x_i are zero.

p-NormAnother smooth maximum is the p-norm: | | ( x 1 , , x n ) | | p ( | x 1 | p | x n | p ) 1 / p displaystyle ||(x_1,ldots ,x_n)||_pleft(|x_1|^pcdots |x_n|^pright)^1/p

which converges to | | ( x 1 , , x n ) | | max 1 i n | x i | displaystyle ||(x_1,ldots ,x_n)||_infty max _1leq ileq n|x_i| as p displaystyle pto infty

.

An advantage of the p-norm is that it is a norm. As such it is "scale invariant" (homogeneous): | | ( x 1 , , x n ) | | p | | | | ( x 1 , , x n ) | | p displaystyle ||(lambda x_1,ldots ,lambda x_n)||_p|lambda |times ||(x_1,ldots ,x_n)||_p

, and it satisfies the triangular inequality.

------

Max von Gruber of max & lily solid wood full size bed grey

Max von Gruber (6 July 1853, Vienna 16 September 1927, Berchtesgaden) was an Austrian scientist. As a bacteriologist he discovered specific agglutination in 1896 with his English colleague Herbert Durham (Gruber-Widal-reaction). But his main interests were studying hygiene and sexual life.

Max von Gruber was the son of Ignaz Gruber (18031872), a general practitioner and the first specialist in otology in Austria, and publisher of a two-volume textbook on medical chemistry (1835). His brother was Franz von Gruber. He graduated from the Schottengymnasium in Vienna and studied medicine at the University of Vienna, receiving his medical doctorate in 1876. He then learned chemistry and physiology under Max von Pettenkofer (18181901) and Karl von Voit (18311908) in Munich and Karl Ludwig (18161895) in Leipzig. Also working under Pettenkofer was Hans Ernst August Buchner (18501902), who encouraged Gruber to concentrate on bacteriology.

Unlike some of the great names of the time, among them Carl Wilhelm Ngeli, Theodor Billroth (18291894), Ferdinand Cohn (18281898), and Robert Koch (18431910), Gruber recognized that bacteria possess a variability within limits partially determined by the culture medium. This theory was important for the differentiation of the categories of bacteria and gained significance for Gruber in his examinations of cholera vibrios, enabling him to distinguish them from other vibrios.

In 1882 Gruber was habilitated as a lecturer in Vienna, and two years later he became associate professor and head of the newly established Institute for Hygiene at the University of Graz. On 23 March 1887 he became ausserordentlicher professor in Vienna, succeeding Josef Nowak, and on 10 December 1891 he was appointed to the chair of hygiene established in 1875 at the University of Vienna. Karl Landsteiner became his assistant in 1896. Another of his pupils, Alois Lode, in 1897 became the first professor in the new chair of hygiene at the University of Innsbruck. The working conditions in the Institute of Hygiene were so poor, that Gruber attempted to resign his chair and find employment as head of a laboratory in Mnchen or at the Jenner Institute in London, under Joseph Lister. It was while in Vienna, however, that Gruber, with his English student Herbert Edward Durham (18661945), discovered the agglutination which gained him international fame.

Gruber eventually left Vienna in 1902, and in October that year he succeeded Hans Buchner as director of the Institute for Hygiene in Mnchen. He held the post until his voluntary retirement in 1923, on the occasion of his seventieth birthday. In Vienna he was succeeded by Arthur Schattenfroh (18691923), who held the chair from 1905 to 1923.

During his last years, Gruber concentrated completely on his duties as president of the Bavarian Academy of Sciences.

With Max Rubner and P. Martin Ficker (1868) he published the Handbuch der Hygiene. 6 volumes; Leipzig, S. Hirtzel, 19111913.

As a leading racial hygienist, when he first met the Nazi dictator Adolf Hitler he described him as:

It was the first time I had seen Hitler close at hand. Face and head of inferior type, cross-breed; low receding forehead, ugly nose, broad cheekbones, little eyes, dark hair. Expression not of a man exercising authority in perfect self-command, but of raving excitement. At the end an expression of satisfied egotism. Description as a witness at court in 1923

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Tornadoes of max & lily solid wood full size bed greyThe Summertown area historical tornado activity is slightly above Tennessee state average. It is 108% greater than the overall U.S. average.On May 18, 1995, an F-4 (max. wind speeds 207-260mph) tornado a mile away from Summertown killed 3 people, injured 32 people and caused between $500,000 and $5,000,000 in damages.On April 16, 1998, an F-5 (max. wind speeds 261-318mph) tornado right on Highway 43 in Summertown killed 3 people, injured 36 people and caused $13 million in damages.There have been several other tornadoes to come through Summertown but these are two of the worst.------Career of max & lily solid wood full size bed greyAnna Schnaudt was born in Moers. She studied voice in Munich with Karl Erler, whom she later married, from 1903 to 1906. She made her concert debut in Munich in 1906. She was a soloist in the premiere of Mahler's Eighth Symphony on 12 September 1910, performing the parts Alto II and Maria Aegyptiaca. She sang concerts also in Berlin, Cologne, Leipzig, in France, the Netherlands, Poland and Czechoslovakia.The composer Max Reger, who probably met her in 1906, dedicated his only orchestral song "An die Hoffnung", Op. 124, to her and conducted the Meininger Hofkapelle in the first performance in Eisenach on 12 October 1912. Reger requested the singer to perform in his memorial service in case of his death. She remained dedicated to him after his death, giving the autograph of the piano version of "An die Hoffnung" and several other memorabilia to the Max-Reger-Institute.She performed in a symphony concert in Bonn in 1914, with Paul Sauer conducting the Stdtisches Orchester playing works by Richard Strauss and Reger, including songs and Reger's piano concerto. The reviewer of the magazine Musik wrote: "... in den Liedern der beiden Meister zeigte sich Anna Erler-Schnaudt als glnzende Interpretin" (... in song by both masters, she appeared as a brilliant interpreter). She was the alto soloist in a performance of Bruckner's Mass in F minor and Psalm 150 at the Tonhalle Munich on 21 February 1924, performed by the Konzertvereins-Orchester und die Konzertgesellschaft fr Chorgesang, conducted by Hanns Rohr, alongside Nelly Merz, Emil Graf, Julius Gie and with the organist Hermann Sagerer.Erler-Schnaudt worked as a voice teacher first in Munich, then from 1928 at the Folkwangschule in Essen. Among her students are Marga Hffgen and her niece Ruth Siewert. Erler-Schnaudt died in Viersen.------Summertown, Tennessee of max & lily solid wood full size bed greySummertown is an unincorporated community and census-designated place (CDP) in Lawrence County, Tennessee with a small portion in both Lewis County and Maury County. The population of Summertown was 866 at the 2010 census.
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