Color blindness (color vision deficiency) is the decreased ability to see color or differences in color. It can impair tasks such as selecting ripe fruit, choosing clothing, and reading traffic lights. Color blindness may make some academic activities more difficult. However, issues are generally minor, and the colorblind automatically develop adaptations and coping mechanisms. People with total color blindness (achromatopsia) may also be uncomfortable in bright environments and have decreased visual acuity.

The most common cause of color blindness is an inherited problem or variation in the functionality of one or more of the three classes of cone cells in the retina, which mediate color vision. Males are more likely to be color blind than females, because the genes responsible for the most common forms of color blindness are on the X chromosome. Non-color-blind females can carry genes for color blindness and pass them on to their children. Color blindness can also result from physical or chemical damage to the eye, the optic nerve, or parts of the brain. Screening for color blindness is typically done with the Ishihara color test.

There is no cure for color blindness. Diagnosis may allow an individual, or their parents/teachers to actively accommodate the condition. Special lenses such as EnChroma glasses or X-chrom contact lenses may help people with red–green color blindness at some color tasks, but they do not grant the wearer "normal color vision". Mobile apps can help people identify colors.

Red–green color blindness is the most common form, followed by blue–yellow color blindness and total color blindness. Red–green color blindness affects up to 1 in 12 males (8%) and 1 in 200 females (0.5%). The ability to see color also decreases in old age. In certain countries, color blindness may make people ineligible for certain jobs, such as those of aircraft pilots, train drivers, crane operators, and people in the armed forces. The effect of color blindness on artistic ability is controversial, but a number of famous artists are believed to have been color blind.

**Details**

Colour blindness is the inability to distinguish one or more of the three colours red, green, and blue. Most people with colour vision problems have a weak colour-sensing system rather than a frank loss of colour sensation. In the retina (the light-sensitive layer of tissue that lines the back and sides of the eyeball), humans have three types of cones (the visual cells that function in the perception of colour). One type absorbs light best in wavelengths of blue-violet and another in the wavelengths of green. The third type is most sensitive to longer wavelengths—more sensitive to red. Normal colour vision, when all three cone types are functioning correctly, is known as trichromacy (or trichromatism).

**Types of colour blindness**

There are several different types of colour blindness, which may be subdivided generally into dichromacy (dichromatism), when only two cone types are functional, and monochromacy (monochromatism), when none or only one type of cone receptor is functional. Dichromatic individuals are ordinarily unable to distinguish between red and green. Blindness to red is known as protanopia, a state in which the red cones are absent, leaving only the cones that absorb blue and green light. Blindness to green is known as deuteranopia, wherein green cones are lacking and blue and red cones are functional. Some persons experience anomalous dichromatic conditions, which involve only minor reductions or weaknesses in colour sensitivity. In protanomaly, for example, sensitivity to red is reduced as a result of abnormalities in the red cone photopigment. In deuteranomaly, in which sensitivity to green is reduced, the green cones are functionally limited. Two forms of blue-yellow colour blindness are known: tritanopia (blindness to blue, usually with the inability to distinguish between blue and yellow), which occurs when blue cones are absent; and tritanomaly (reduced sensitivity to blue), which arises from the abnormal function of blue cones.

Monochromacy, or complete colour blindness, is sometimes accompanied by deficiencies in visual acuity. Such conditions are rare and include achromatopsia (or rod monochromacy; the complete absence of functional cone photopigments) and cone monochromacy (when two of the three cone types are nonfunctional).

**Inherited and acquired colour blindness**

Hereditary red-green colour blindness occurs mainly in males and Caucasian persons, with about 8 percent of men and 0.5 percent of women of European ancestry inheriting the conditions. Its predominance in males is due to the fact that red-green colour blindness is a gender-linked recessive characteristic, carried on the X chromosome. Hence, the trait for red-green colour blindness is passed from mother to son, from mother to daughter, or from mother and father to daughter. A son who inherits the trait from a carrier mother will be red-green colour blind (males inherit only one X chromosome, directly from the mother). A daughter who inherits the trait from a carrier mother (with a normal father) will have normal colour vision but be a carrier of the trait. A daughter who inherits the trait from both her mother and her father will be red-green colour blind.

Blue-yellow colour blindness, by contrast, is an autosomal dominant disorder and therefore is not gender-linked and requires only one copy of the defective gene from either parent to be expressed. Achromatopsia is an autosomal recessive disorder, occurring only when two copies of the defective gene (one from each parent) have been inherited. Persons who inherit colour blindness may show symptoms at birth (congenital colour blindness), or they may become symptomatic later, in childhood or adulthood.

Acquired colour blindness is usually of the blue-yellow type and ranges from mild to severe. Often it is associated with chronic disease, such as macular degeneration, glaucoma, diabetes mellitus, retinitis pigmentosa, or Alzheimer disease. Certain drugs and chemicals can also cause acquired colour blindness.

]]>

In 1851, George Gabriel Stokes derived an expression, now known as Stokes law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

**Statement of the law**

The force of viscosity on a small sphere moving through a viscous fluid is given by:

where:

is the frictional force – known as Stokes' drag – acting on the interface between the fluid and the particle

is the dynamic viscosity (some authors use the symbol )

R is the radius of the spherical object

v is the flow velocity relative to the object.

In SI units,

is given in newtons in Pa·s , R in meters, and v in m/s.Stokes' law makes the following assumptions for the behavior of a particle in a fluid:

* Laminar flow

* Spherical particles

* Homogeneous (uniform in composition) material

* Smooth surfaces

* Particles do not interfere with each other.

Particles do not interfere with each other.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

**Applications**

Stokes' law is the basis of the falling-sphere viscometer, in which the fluid is stationary in a vertical glass tube. A sphere of known size and density is allowed to descend through the liquid. If correctly selected, it reaches terminal velocity, which can be measured by the time it takes to pass two marks on the tube. Electronic sensing can be used for opaque fluids. Knowing the terminal velocity, the size and density of the sphere, and the density of the liquid, Stokes' law can be used to calculate the viscosity of the fluid. A series of steel ball bearings of different diameters are normally used in the classic experiment to improve the accuracy of the calculation. The school experiment uses glycerine or golden syrup as the fluid, and the technique is used industrially to check the viscosity of fluids used in processes. Several school experiments often involve varying the temperature and/or concentration of the substances used in order to demonstrate the effects this has on the viscosity. Industrial methods include many different oils, and polymer liquids such as solutions.

The importance of Stokes' law is illustrated by the fact that it played a critical role in the research leading to at least three Nobel Prizes.

Stokes' law is important for understanding the swimming of microorganisms and sperm; also, the sedimentation of small particles and organisms in water, under the force of gravity.

In air, the same theory can be used to explain why small water droplets (or ice crystals) can remain suspended in air (as clouds) until they grow to a critical size and start falling as rain (or snow and hail). Similar use of the equation can be made in the settling of fine particles in water or other fluids.

For molecules Stokes' law is used to define their Stokes radius and diameter.

The CGS unit of kinematic viscosity was named "stokes" after his work.

]]>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.

]]>

This site is "Maths Is Fun". So the field is wide open.

Bob

]]>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 number

(spelled out as "pi") is a mathematical constant that is the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. The number π appears in many formulas across mathematics and physics. It is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22/7 are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an equation involving only sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found.For thousands of years, mathematicians have attempted to extend their understanding of

, sometimes by computing its value to a high degree of accuracy. Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematicians approximated π to seven digits, while Indian mathematicians made a five-digit approximation, both using geometrical techniques. The first computational formula forπ, based on infinite series, was discovered a millennium later. The earliest known use of the Greek letter to represent the ratio of a circle's circumference to its diameter was by the Welsh mathematician William Jones in 1706.The invention of calculus soon led to the calculation of hundreds of digits of

, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of to many trillions of digits. These computations are motivated by the development of efficient algorithms to calculate numeric series, as well as the human quest to break records. The extensive computations involved have also been used to test supercomputers.Because its definition relates to the circle,

is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses and spheres. It is also found in formulae from other topics in science, such as cosmology, fractals, thermodynamics, mechanics, and electromagnetism. In modern mathematical analysis, it is often instead defined without any reference to geometry; therefore, it also appears in areas having little to do with geometry, such as number theory and statistics. The ubiquity of π makes it one of the most widely known mathematical constants inside and outside of science. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.]]>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.

]]>Tungsten, or wolfram, is a chemical element with the symbol W and atomic number 74. Tungsten is a rare metal found naturally on Earth almost exclusively as compounds with other elements. It was identified as a new element in 1781 and first isolated as a metal in 1783. Its important ores include scheelite and wolframite, the latter lending the element its alternate name.

The free element is remarkable for its robustness, especially the fact that it has the highest melting point of all known elements barring carbon (which sublimes at normal pressure), melting at 3,422 °C (6,192 °F; 3,695 K). It also has the highest boiling point, at 5,930 °C (10,710 °F; 6,200 K). Its density is 19.25 grams per cubic centimetre, comparable with that of uranium and gold, and much higher (about 1.7 times) than that of lead. Polycrystalline tungsten is an intrinsically brittle and hard material (under standard conditions, when uncombined), making it difficult to work. However, pure single-crystalline tungsten is more ductile and can be cut with a hard-steel hacksaw.

Tungsten occurs in many alloys, which have numerous applications, including incandescent light bulb filaments, X-ray tubes, electrodes in gas tungsten arc welding, superalloys, and radiation shielding. Tungsten's hardness and high density make it suitable for military applications in penetrating projectiles. Tungsten compounds are often used as industrial catalysts.

Tungsten is the only metal in the third transition series that is known to occur in biomolecules, being found in a few species of bacteria and archaea. However, tungsten interferes with molybdenum and copper metabolism and is somewhat toxic to most forms of animal life.

]]>Why can't we take the second derivative of it ?

My solution would probably be to rotate it 45° counter-clockwise,

take the second derivative

find the relation it has with the unaltered equation

and transform it back.

Is there something out there, which is more handy ?

]]>The discussion on this point is accepted with open arms in the community; the community answers the question with proof of the existence of certain other numbers on a variety of numerics.

a = 7, b = 5 , n = 5 175 = 1 x 7 x 5 x 5

a = 2, b = 8, n = 8 128 = 1 x 2 x 8 x 8

a = 3, b = 5, n = 9 135 = 1 x 3 x 5 x 9

a = 4, b = 4, n = 9 144 = 1 x 4 x 4 x 9

Starting with the acceptance that several digits can exist in the pattern discussed in the problem proves an abundant variable available that you can find. Though there was some difficulty in understanding the equation, the community did find the solution keeping the context that the number can be in any range and giving the result lies the numbers in the statement.]]>

Glad to be of help

Bob

]]>The discussion on the forum starts with the comments on the statement of his occasional mathematician doing it for fun. The argument proceeds with the complexity of the question and the terms related to the octal and tetra vertices. Further, it was pointed out that this solid doesn't represent any regular solid and does not obey Euler's relation. To understand all this and your calculation, I had to draw 2D images, called Schlegel diagrams, where I made the drawing and placed vertices with nine edges, then I ended up with the F = 6, which Euler's relation would predict.]]>

Roraborealis wrote:I saw taht on a seicnce pgmramroe on the tivoeielsn! Vrey odd idened.

While we're on about science and letters, count the number of F's in this paragraph:

One funny Friday, fifteen people flew to Fellgate zoo. Frankly, a few of the people found out that five of the animals were feeling ill. They thought it was cruel and demanded their full money back.

There are 14.

Correct! 3 Uppercase and 11 lower case Fs.

]]>