Mathematics
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Functions f ( x ) = a x are shown for several values of a . e is the unique value of a , such that the derivative of f ( x ) = a x at the point x = 0 is equal to 1. The blue curve illustrates this case, e x . For comparison, functions 2 x (dotted curve) and 4 x (dashed curve) are shown; they are not tangent to the line of slope 1 (red).
e (mathematical constant)
Tetration - Wikipedia, the free encyclopedia
, for n = 1, 2, 3 ..., showing convergence to the infinite power tower between the two dots. Infinite power tower for bases In mathematics , tetration (also known as hyper-4 ) is an iterated exponential and is the next hyper operator after exponentiation . The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- (four) and iteration . Tetration is used for the notation of very large numbers .
Derivative - Wikipedia, the free encyclopedia
The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. In calculus , a branch of mathematics , the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity . The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. For a real-valued function of a single real variable, the derivative at a point equals the slope of the tangent line to the graph of the function at that point.
A parabola obtained as the intersection of a cone with a plane parallel to a straight line on its surface. In mathematics , a parabola ( / p ə ˈ r æ b ə l ə / ; plural parabolae or parabolas , from the Greek παραβολή ) is a conic section , created from the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. Another way to generate a parabola is to examine a point (the focus ) and a line (the directrix ) on a plane.
Parabola - Wikipedia, the free encyclopedia
Cardioid - Wikipedia, the free encyclopedia
A cardioid given as the envelope of circles whose centers lie on a given circle and which pass through a fixed point on the given circle. A cardioid (from the Greek καρδία "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp . It is also a type of sinusoidal spiral , and an inverse curve of the parabola with the focus as the center of inversion. [ 1 ] The name was coined by de Castillon in 1741 [ 2 ] but had been the subject of study decades beforehand. [ 3 ] Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk.
Stanford Lectures
The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant , cotangent , secant , sine , and tangent ). Let be an angle measured counterclockwise from the x -axis along the arc of the unit circle . Then is the horizontal coordinate of the arc endpoint. The common schoolbook definition of the cosine of an angle
Cosine -- from Wolfram MathWorld
The sine and cosine functions are related in multiple ways. The derivative of is . Also they are out of phase by 90°: = .
Sine - Wikipedia, the free encyclopedia
Tangent - Wikipedia, the free encyclopedia
Tangent plane to a sphere In geometry , the tangent line (or simply the tangent ) to a plane curve at a given point is the straight line that "just touches" the curve at that point—that is, coincides with the curve at that point without crossing to the other side of the curve. More precisely, a straight line is said to be a tangent of a curve y = f ( x ) at a point x = c on the curve if the line passes through the point ( c , f ( c )) on the curve and has slope f ' ( c ) where f ' is the derivative of f . A similar definition applies to space curves and curves in n -dimensional Euclidean space . As it passes through the point where the tangent line and the curve meet, or the point of tangency, the tangent line is "going in the same direction" as the curve, and in this sense it is the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.For compact 2-dimensional surfaces without boundary , if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere (usually just called a sphere). The Poincaré conjecture asserts that the same is true for 3-dimensional spaces. By contrast, neither of the two colored loops on this torus can be continuously tightened to a point. A torus is not homeomorphic to a sphere. In mathematics , the Poincaré conjecture (
Solution of the Poincaré conjecture - Wikipedia, the free encyclopedia
The Kepler conjecture , named after the 17th-century German astronomer Johannes Kepler , is a mathematical conjecture about sphere packing in three-dimensional Euclidean space . It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing ( face-centered cubic ) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%. In 1998 Thomas Hales , following an approach suggested by Fejes Tóth (1953) , announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations.
Kepler conjecture - Wikipedia, the free encyclopedia
Fractional quantum Hall effect - Wikipedia, the free encyclopedia
The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2D electrons shows precisely quantised plateaus at fractional values of . It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. Its discovery and explanation were recognized by the 1998 Nobel Prize in Physics .Quantum nonlocality - Wikipedia, the free encyclopedia
Quantum nonlocality is the phenomenon by which measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics . Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory . Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem .The holographic principle is a property of quantum gravity and string theories which states that the description of a volume of space can be thought of as encoded on a boundary to the region—preferably a light-like boundary like a gravitational horizon . First proposed by Gerard 't Hooft , it was given a precise string-theory interpretation by Leonard Susskind [ 1 ] who combined his ideas with previous ones of 't Hooft and Charles Thorn . [ 1 ] [ 2 ] As pointed out by Raphael Bousso , [ 3 ] Thorn observed in 1978 that string theory admits a lower dimensional description in which gravity emerges from it in what would now be called a holographic way. In a larger and more speculative sense, the theory suggests that the entire universe can be seen as a two-dimensional information structure "painted" on the cosmological horizon , such that the three dimensions we observe are only an effective description at macroscopic scales and at low energies .