The problem is to estimate the Euclidean norm of the series

| SUM {k from 0 to infty} (I - A)^k A w_k |

where matrix A such that 0 <= A <= I

and vectors w_k such that | w_k | <= 1

]]>Celluloid, the first synthetic plastic material, developed in the 1860s and 1870s from a homogeneous colloidal dispersion of nitrocellulose and camphor. A tough, flexible, and moldable material that is resistant to water, oils, and dilute acids and capable of low-cost production in a variety of colours, celluloid was made into toiletry articles, novelties, photographic film, and many other mass-produced goods. Its popularity began to wane only toward the middle of the 20th century, following the introduction of plastics based on entirely synthetic polymers.

Some historians trace the invention of celluloid to English chemist Alexander Parkes, who in 1856 was granted the first of several patents on a plastic material that he called Parkesine. Parkesine plastics were made by dissolving nitrocellulose (a flammable nitric ester of cotton or wood cellulose) in solvents such as alcohol or wood naphtha and mixing in plasticizers such as vegetable oil or camphor (a waxy substance originally derived from the oils of the Asian camphor tree, Cinnamonum camphora). In 1867 Parkes’s business partner, Daniel Spill, patented Xylonite, a more-stable improvement upon Parkesine. Spill went on to found the Xylonite Company (later the British Xylonite Company Ltd.), which produced molded objects such as chess pieces from his material.

In the United States, meanwhile, inventor and industrialist John Wesley Hyatt produced a plastic that was more commercially successful by mixing solid nitrocellulose, camphor, and alcohol under pressure. The solid solution was kneaded into a doughlike mass to which colouring agents could be added either in the form of dyes for transparent colours or as pigments for opaque colours. The coloured mass was rolled, sheeted, and then pressed into blocks. After seasoning, the blocks were sliced; at this point they could be further fabricated, or the sheeting and pressing process could be repeated for various mottled and variegated effects. The plastic, which softened at the temperature of boiling water, could be heated and then pressed into innumerable shapes, and at room temperature it could be sawed, drilled, turned, planed, buffed, and polished. In 1870 Hyatt and his brother Isaiah acquired the first of many patents on this material, registering it under the trade name Celluloid in 1873. The Hyatts’ Celluloid Manufacturing Company produced celluloid for fabrication into a multitude of products, including combs, brush handles, piano keys, and eyeglass frames. In all these applications celluloid was marketed as an affordable and practical substitute for natural materials such as ivory, tortoiseshell, and horn. Beginning in the 1880s celluloid acquired one of its most prominent uses as a substitute for linen in detachable collars and cuffs for men’s clothing. Over the years a number of competing plastics were introduced under such fanciful names as Coraline, Ivoride, and Pyralin, and celluloid became a generic term.

In 1882 John H. Stevens, a chemist at the Celluloid Manufacturing Company, discovered that amyl acetate was a suitable solvent for diluting celluloid. This allowed the material to be made into a clear, flexible film, which other researchers such as Henry Reichenbach of the Eastman Company (later Eastman Kodak Company) further processed into film for still photography and later for motion pictures. Despite its flammability and tendency to discolour and crack with age, celluloid was virtually unchallenged as the medium for motion pictures until the 1930s, when it began to be replaced by cellulose-acetate safety film.

Other disadvantages of celluloid were its tendency to soften under heat and its unsuitability for new, efficient fabrication processes such as injection molding. In the 1920s and 1930s celluloid began to be replaced in most of its applications by more versatile materials such as cellulose acetate, Bakelite, and the new vinyl polymers. By the end of the 20th century, its only unique application of note was in table-tennis balls. Early celluloid objects have become collector’s items and museum artifacts, valued as specimens of an artificial plastic based on naturally occurring raw materials.

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#7643. What does the term 'Cacophobia' mean?

#7644. What does the term 'Spectrophobia' mean?

]]>#4831. The population of a city is 500,000. The rate of increase is 20% per annum. Find the population at the start of the third year.

]]>#1521. What does the medical term 'Scarification' mean?

]]>Good attempt!

742. The difference between compound interest and simple interest at the rate of 25% per annum is $910. Find the amount of money lent.

]]>It's great to hear that things are going well for you.

I'll illustrate what's happening with an example.

I've chosen the function

When we differentiate we get

and second derivative

Here are the graphs of (black) the function; (red) the first derivative; (green) the second derivative.

I've put the three graphs one after the other, lined up in the x direction but one on top of the other in the y direction. For this reason don't worry about the numbering on the axes; you don't need it. But the changes in gradient for the function are aligned correctly with the first and second derivatives.

The function is a cubic. When you differentiate it you get a quadratic. The quadratic has two zeros and these line up with the turning points on the cubic. When you differentiate again you are finding the gradient function of dy/dx, and you get a linear function.

On the (black) function graph you can see the two turning points; the first a maximum; the second a minimum; but let's say we don't know this yet. Just to the left of a maximum the function has positive gradient; just to the right it is negative. At the turning point the gradient is zero of course.

You can see this on the red graph. Just to the left of the function's turning point the red graph is above the axis so it's values are positive. The red graph crosses its axis at the turning point and after that it has negative values. In a similar way the next turning point (a minimum) has gradient negative to the left of the turning point; zero at the turning point; then positive.

So you can tell if a turning point is a maximum by looking to see if the first derivative goes from positive, through zero, to negative. And a minimum will have first derivative that goes from negative, through zero, to positive.

One way to investigate this is to draw the graphs; another is to calculate gradients just left and right of the turning point. But the second derivative gives a quick way to find out without needing to do any graph drawing. Here's how:

Function has a maximum. Gradient function goes from positive, through zero to negative. So its gradient function must be negative at that point as dy/dx has a reducing gradient.

Function has a minimum. Gradient function goes from negative, through zero, to positive. So its gradient function must be positive at that point as dy/dx has an increasing gradient.

In my example

so the turning points are at x = - √ (1/3) and + √ (1/3) At x = - √ (1/3) the second derivative is negative => this turning point is a minimum.At x = + √ (1/3) the second derivative is postive => this turning point is a maximum.

Note: I don't even have to do the full calculation here; I just need to know if the second derivative is positive or negative at each turning point; so it's a quick way to tell.

Bob

]]>#7817. A man bought a television set for $800 and spent $400 for it. Find his gain percent.

]]>#3677. What does the noun *ethnocentrism* mean?

#3678. What does the noun *ethnology* mean?

Prince Louis-Victor de Broglie (Louis Victor Pierre Raymond de Broglie, 7th duc de Broglie) of the French Academy, Permanent Secretary of the Academy of Sciences, and Professor at the Faculty of Sciences at Paris University, was born at Dieppe (Seine Inférieure) on 15th August, 1892, the son of Victor, Duc de Broglie and Pauline d’Armaillé. After studying at the Lycée Janson of Sailly, he passed his school-leaving certificate in 1909. He applied himself first to literary studies and took his degree in history in 1910. Then, as his liking for science prevailed, he studied for a science degree, which he gained in 1913. He was then conscripted for military service and posted to the wireless section of the army, where he remained for the whole of the war of 1914-1918. During this period he was stationed at the Eiffel Tower, where he devoted his spare time to the study of technical problems. At the end of the war Louis de Broglie resumed his studies of general physics. While taking an interest in the experimental work carried out by his elder brother, Maurice, and co-workers, he specialized in theoretical physics and, in particular, in the study of problems involving quanta. In 1924 at the Faculty of Sciences at Paris University he delivered a thesis Recherches sur la Théorie des Quanta (Researches on the quantum theory), which gained him his doctor’s degree. This thesis contained a series of important findings which he had obtained in the course of about two years. The ideas set out in that work, which first gave rise to astonishment owing to their novelty, were subsequently fully confirmed by the discovery of electron diffraction by crystals in 1927 by Davisson and Germer; they served as the basis for developing the general theory nowadays known by the name of wave mechanics, a theory which has utterly transformed our knowledge of physical phenomena on the atomic scale.

After the maintaining of his thesis and while continuing to publish original work on the new mechanics, Louis de Broglie took up teaching duties. On completion of two year’s free lectures at the Sorbonne he was appointed to teach theoretical physics at the Institut Henri Poincaré which had just been built in Paris. The purpose of that Institute is to teach and develop mathematical and theoretical physics. The incumbent of the chair of theoretical physics at the Faculty of Sciences at the University of Paris since 1932, Louis de Broglie runs a course on a different subject each year at the Institut Henri Poincaré, and several of these courses have been published. Many French and foreign students have come to work with him and a great deal of doctorate theses have been prepared under his guidance.

Between 1930 and 1950, Louis de Broglie’s work has been chiefly devoted to the study of the various extensions of wave mechanics: Dirac’s electron theory, the new theory of light, the general theory of spin particles, applications of wave mechanics to nuclear physics, etc. He has published numerous notes and several papers on this subject, and is the author of more than twenty-five books on the fields of his particular interests.

Since 1951, together with young colleagues, Louis de Broglie has resumed the study of an attempt which he made in 1927 under the name of the theory of the double solution to give a causal interpretation to wave mechanics in the classical terms of space and time, an attempt which he had then abandoned in the face of the almost universal adherence of physicists to the purely probabilistic interpretation of Born, Bohr, and Heisenberg. Back again in this his former field of research, he has obtained a certain number of new and encouraging results which he has published in notes to Comptes Rendus de l’Académie des Sciences and in various expositions.

After crowning Louis de Broglie’s work on two occasions, the Academie des Sciences awarded him in 1929 the Henri Poincaré medal (awarded for the first time), then in 1932, the Albert I of Monaco prize. In 1929 the Swedish Academy of Sciences conferred on him the Nobel Prize for Physics “for his discovery of the wave nature of electrons”. In 1952 the first Kalinga Prize was awarded to him by UNESCO for his efforts to explain aspects of modern physics to the layman. In 1956 he received the gold medal of the French National Scientific Research Centre. He has made major contributions to the fostering of international scientific co-operation.

Elected a member of the Academy of Sciences of the French Institute in 1933, Louis de Broglie has been its Permanent Secretary for the mathematical sciences since 1942. He has been a member of the Bureau des Longitudes since 1944. He holds the Grand Cross of the Légion d’Honneur and is an Officer of the Order of Leopold of Belgium. He is an honorary doctor of the Universities of Warsaw, Bucharest, Athens, Lausanne, Quebec, and Brussels, and a member of eighteen foreign academies in Europe, India, and the U.S.A.

Louis de Broglie died on March 19, 1987.

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I am a big fan of functional programming myself.

Have you taken a look at Haskell? I suspect that you might find this language way more interesting and elegant!

]]>Well done!

CG#123. ABCD is a rectangle formed by the points A (-1,-1), B (-1,4), C (5,4), and D (5,-1). P, Q, R, and S are the mid-points of AB, BC, CD, and DA respectively. Is the quadrilateral PQRS a square, a rectangle, or a rhombus? Justify your answer.

]]>This is an algorithm that test if one number is prime and if the number it´s not prime returns the biggest factor of that number.

Are you going to keep it a secret and profit off of it or are you going to enlighten our poor souls as to what it is?

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