**Summary**

A shipyard (also called a dockyard or boatyard) is a place where ships are built and repaired. These can be yachts, military vessels, cruise liners or other cargo or passenger ships. Dockyards are sometimes more associated with maintenance and basing activities than shipyards, which are sometimes associated more with initial construction. The terms are routinely used interchangeably, in part because the evolution of dockyards and shipyards has often caused them to change or merge roles.

Countries with large shipbuilding industries include Australia, Brazil, China, Croatia, Denmark, Finland, France, Germany, India, Ireland, Italy, Japan, the Netherlands, Norway, the Philippines, Poland, Romania, Russia, Singapore, South Korea, Sweden, Taiwan, Turkey, the United Arab Emirates, Ukraine, the United Kingdom, the United States and Vietnam. The shipbuilding industry is more fragmented in Europe than in Asia where countries tend to have fewer, larger companies. Many naval vessels are built or maintained in shipyards owned or operated by the national government or navy.

Shipyards are constructed near the sea or tidal rivers to allow easy access for their ships. The United Kingdom, for example, has shipyards on many of its rivers.

The site of a large shipyard will contain many specialised cranes, dry docks, slipways, dust-free warehouses, painting facilities and extremely large areas for fabrication of the ships. After a ship's useful life is over, it makes its final voyage to a shipbreaking yard, often on a beach in South Asia. Historically shipbreaking was carried on in drydock in developed countries, but high wages and environmental regulations have resulted in movement of the industry to developing regions.

**Details**

A shipyard is a shore establishment for building and repairing ships. The shipbuilding facilities of the ancient and medieval worlds reached a culmination in the math of Venice, a shipyard in which a high degree of organization produced an assembly-line technique, with a ship’s fittings added to the completed hull as it was floated past successive docks. In 18th-century British shipyards, the hull was towed to a floating stage called a sheer hulk, where it received its masts and rigging. Modern ships also are launched incomplete.

Typically, a shipyard has a limited number of building berths, sloping down toward the waterway, with large adjacent working areas. Plates and sections are delivered to a point distant from the berth and converge toward the berth as they are assembled into components and subassemblies, which are ultimately welded together. Very large ships are often built in deep drydocks because of the greater convenience in lowering large components. When the hull is complete, water is admitted and the ship floated to the fitting-out basin.

**History**

The world's earliest known dockyards were built in the Harappan port city of Lothal circa 2400 BC in Gujarat, India. Lothal's dockyards connected to an ancient course of the Sabarmati river on the trade route between Harappan cities in Sindh and the peninsula of Saurashtra when the present-day surrounding Kutch desert formed a part of the Arabian Sea.

Lothal engineers accorded high priority to the creation of a dockyard and a warehouse to serve the purposes of naval trade. The dock was built on the eastern flank of the town, and is regarded by archaeologists as an engineering feat of the highest order. It was located away from the main current of the river to avoid silting, but provided access to ships at high tide as well.

The name of the ancient Greek city on the Gulf of Corinth, Naupactus, means "shipyard" (combination of the Greek words "ship, boat"; and "builder, fixer"). Naupactus' reputation in this field extends to the time of legend, in which it is depicted as the place where the Heraclidae built a fleet to invade the Peloponnesus.

In the Spanish city of Barcelona, the Drassanes shipyards were active from at least the mid-13th century until the 18th century, although at times they served as a barracks for troops as well as an math. During their time of operation the Drassanes were continuously changed, rebuilt and modified, but two original towers and part of the original eight construction-naves remain today. The site is currently a maritime museum.

From the 14th century, several hundred years before the Industrial Revolution, ships were the first items to be manufactured in a factory - in the Venice math of the Venetian Republic in present-day Italy. The math apparently mass-produced nearly one ship every day using pre-manufactured parts and assembly lines. At its height in the 16th century the enterprise employed 16,000 people.

Spain built component ships of the Great Armada of 1588 at ports such as Algeciras or Málaga.

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I came across a link here.

In the relevant topic.

Interesting!

]]>In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.

The law was first formulated by Joseph-Louis Lagrange in 1773, followed by Carl Friedrich Gauss in 1835, both in the context of the attraction of ellipsoids. It is one of Maxwell's four equations, which forms the basis of classical electrodynamics. Gauss's law can be used to derive Coulomb's law, and vice versa.

**Qualitative description**

In words, Gauss's law states that

*The net electric flux through any hypothetical closed surface is equal to*

Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss's law for magnetism and Gauss's law for gravity. In fact, any inverse-square law can be formulated in a way similar to Gauss's law: for example, Gauss's law itself is essentially equivalent to the inverse-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity.

The law can be expressed mathematically using vector calculus in integral form and differential form; both are equivalent since they are related by the divergence theorem, also called Gauss's theorem. Each of these forms in turn can also be expressed two ways: In terms of a relation between the electric field E and the total electric charge, or in terms of the electric displacement field D and the free electric charge.

]]>Решений можно записать два. Одно когда взаимно простые....

Сумма их имеет вид.

И когда нет...

Ну и дальше всё совсем просто... взаимно простые когда скобки некоторые равны 1... ну или -1

]]>imcute,

1. Use appropriate case when you answer.

2. Avoid abbreviations like btw, ty etc.

Oh ok.

]]>y=mx+c

y-c=mx+c-c

y-c=mx

(y-c)/m=(mx)/m

(y-c)/m=x

PROOF

Solve;

1) 15=2x+3

x=(y-c)/m

x=(15-3)/2

x=12/2

x=6 CORRECT

**20th and 21st centuries**

In 1910, the Indian mathematician Srinivasa Ramanujan found several rapidly converging infinite series of

, includingwhich computes a further eight decimal places of

with each term in the series. His series are now the basis for the fastest algorithms currently used to calculate . Even using just the first term givesFrom the mid-20th century onwards, all calculations of

have been done with the help of calculators or computers.In 1944, D. F. Ferguson, with the aid of a mechanical desk calculator, found that William Shanks had made a mistake in the 528th decimal place, and that all succeeding digits were incorrect.

In the early years of the computer, an expansion of π to 100000 decimal places was computed by Maryland mathematician Daniel Shanks (no relation to the aforementioned William Shanks) and his team at the United States Naval Research Laboratory in Washington, D.C. In 1961, Shanks and his team used two different power series for calculating the digits of

. For one, it was known that any error would produce a value slightly high, and for the other, it was known that any error would produce a value slightly low. And hence, as long as the two series produced the same digits, there was a very high confidence that they were correct. The first 100,265 digits of π were published in 1962. The authors outlined what would be needed to calculate to 1 million decimal places and concluded that the task was beyond that day's technology, but would be possible in five to seven years.In 1989, the Chudnovsky brothers computed π to over 1 billion decimal places on the supercomputer IBM 3090 using the following variation of Ramanujan's infinite series of

:Records since then have all been accomplished using the Chudnovsky algorithm. In 1999, Yasumasa Kanada and his team at the University of Tokyo computed

to over 200 billion decimal places on the supercomputer HITACHI SR8000/MPP (128 nodes) using another variation of Ramanujan's infinite series of . In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000, a 64-node supercomputer with 1 terabyte of main memory, to calculate to roughly 1.24 trillion digits in around 600 hours (25 days).]]>I resonate with what you wrote. There is a coherence and elegance in math that points to something spiritual behind it and beyond it, something relevant to the foundation and structure of life, the universe, and everything. Mathematical patterns and equations are not inventions of men’s minds but discoveries of reality. We need a heart as well as a brain to fully appreciate math.

]]>I'm not sure my brother would be happy to be described as lazy

Mycroft did his analysis sitting in an armchair. Physically not much but mentally a tour de force.

Bob

Oopsies! I didn't think of it that way.

Then your brother, who's in the University of Edinburgh, is working on a mathematical fact checker?

That sounds really awesome.

]]>In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. If 0 is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid 1700s.

The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

**Definition**

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series

where

denotes the factorial of n. In the more compact sigma notation, this can be written aswhere

denotes the nth derivative of f evaluated at the point a. (The derivative of order zero of f is defined to be f itself and and 0! are both defined to be 1.)When a = 0, the series is also called a Maclaurin series.

]]>In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle.

**Standard version of the theorem**

If a real-valued function f is continuous on a proper closed interval [a, b], differentiable on the open interval (a, b), and f (a) = f (b), then there exists at least one c in the open interval (a, b) such that

This version of Rolle's theorem is used to prove the mean value theorem, of which Rolle's theorem is indeed a special case. It is also the basis for the proof of Taylor's theorem.

**History**

Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. The name "Rolle's theorem" was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846.

The theorem, that derivatives are zero at a maximum, was stated by Bhaskara II in his Siddhanta Shiromani, completed in 1150. Bhaskara II was part of the Kerala school of mathematics, and also theorized the derivatives of trigonometric functions.

]]>When p=7,

No solution up to n=10,000

]]>**Summary**

Möbius strip is a a one-sided surface that can be constructed by affixing the ends of a rectangular strip after first having given one of the ends a one-half twist. This space exhibits interesting properties, such as having only one side and remaining in one piece when split down the middle. The properties of the strip were discovered independently and almost simultaneously by two German mathematicians, August Ferdinand Möbius and Johann Benedict Listing, in 1858.

**Details**

In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip.

As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides. It has only a single boundary curve.

Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a ruled surface by a line segment rotating in a rotating plane, with or without self-crossings. A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a developable surface or be folded flat; the flattened Möbius strips include the trihexaflexagon. The Sudanese Möbius strip is a minimal surface in a hypersphere, and the Meeks Möbius strip is a self-intersecting minimal surface in ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Mobius strip, the cross-cap, have a circular boundary. A Möbius strip without its boundary, called an open Möbius strip, can form surfaces of constant curvature. Certain highly-symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip.

The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes appear opposite each other. Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in social choice theory. In popular culture, Möbius strips appear in artworks by M. C. Escher, Max Bill, and others, and in the design of the recycling symbol. Many architectural concepts have been inspired by the Möbius strip, including the building design for the NASCAR Hall of Fame. Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. The canons of J. S. Bach have been analyzed using Möbius strips. Many works of speculative fiction feature Möbius strips; more generally, a plot structure based on the Möbius strip, of events that repeat with a twist, is common in fiction.

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For even n, the double factorial is

and for odd n it is

For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.

The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as

1, 2, 8, 48, 384, 3840, 46080, 645120,...

The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as

1, 3, 15, 105, 945, 10395, 135135,...

The term odd factorial is sometimes used for the double factorial of an odd number.

]]>The logarithm of 2 in other bases is obtained with the formula

The common logarithm in particular is

The inverse of this number is the binary logarithm of 10:

.By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.

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