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https://en.wikipedia.org/wiki/Symmetric%20difference | In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is .
The symmetric difference of the sets A and B is commonly denoted by (traditionally, ), , or .
It can be viewed as a form of addition modulo 2.
The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse. The power set of any set becomes a Boolean ring, with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
Properties
The symmetric difference is equivalent to the union of both relative complements, that is:
The symmetric difference can also be expressed using the XOR operation ⊕ on the predicates describing the two sets in set-builder notation:
The same fact can be stated as the indicator function (denoted here by ) of the symmetric difference, being the XOR (or addition mod 2) of the indicator functions of its two arguments: or using the Iverson bracket notation .
The symmetric difference can also be expressed as the union of the two sets, minus their intersection:
In particular, ; the equality in this non-strict inclusion occurs if and only if and are disjoint sets. Furthermore, denoting and , then and are always disjoint, so and partition . Consequently, assuming intersection and symmetric difference as primitive operations, the union of two sets can be well defined in terms of symmetric difference by the right-hand side of the equality
.
The symmetric difference is commutative and associative:
The empty set is neutral, and every set is its own inverse:
Thus, the power set of any set X becomes an abelian group under the symmetric difference operation. (More generally, any field of sets forms a group with the symmetric difference as operation.) A group in which every element is its own inverse (or, equivalently, in which every element has order 2) is sometimes called a Boolean group; the symmetric difference provides a prototypical example of such groups. Sometimes the Boolean group is actually defined as the symmetric difference operation on a set. In the case where X has only two elements, the group thus obtained is the Klein four-group.
Equivalently, a Boolean group is an elementary abelian 2-group. Consequently, the group induced by the symmetric difference is in fact a vector space over the field with 2 elements Z2. If X is finite, then the singletons form a basis of this vector space, and its dimension is therefore equal to the number of elements of X. This construction is used in graph theory, to define the cycle space of a graph.
From the property of the inverses in a Boolean group, it follows that the symmetric difference of two repeated symmetric differences is equivalent to |
https://en.wikipedia.org/wiki/Harmonic%20progression | Harmonic progression may refer to:
Chord progression in music
Harmonic progression (mathematics)
Sequence (music) |
https://en.wikipedia.org/wiki/Rayleigh%20distribution | In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom.
The distribution is named after Lord Rayleigh ().
A Rayleigh distribution is often observed when the overall magnitude of a vector in the plane is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed in two dimensions.
Assuming that each component is uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution.
A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically distributed Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.
Definition
The probability density function of the Rayleigh distribution is
where is the scale parameter of the distribution. The cumulative distribution function is
for
Relation to random vector length
Consider the two-dimensional vector which has components that are bivariate normally distributed, centered at zero, and independent. Then and have density functions
Let be the length of . That is, Then has cumulative distribution function
where is the disk
Writing the double integral in polar coordinates, it becomes
Finally, the probability density function for is the derivative of its cumulative distribution function, which by the fundamental theorem of calculus is
which is the Rayleigh distribution. It is straightforward to generalize to vectors of dimension other than 2.
There are also generalizations when the components have unequal variance or correlations (Hoyt distribution), or when the vector Y follows a bivariate Student t-distribution (see also: Hotelling's T-squared distribution).
Suppose is a random vector with components that follows a multivariate t-distribution. If the components both have mean zero, equal variance, and are independent, the bivariate Student's-t distribution takes the form:
Let be the magnitude of . Then the cumulative distribution function (CDF) of the magnitude is:
where is the disk defined by:
Converting to polar coordinates leads to the CDF becoming:
Finally, the probability density function (PDF) of the magnitude may be derived:
In the limit as , the Rayleigh distribution is recovered because:
Properties
The raw moments are given by:
where is the gamma function.
The mean of a Rayleigh random variable is thus :
The standard deviation of a Rayleigh random variable is:
The variance of a Rayleigh random variable is :
The mode is and the maximum pdf is
The skewness is given by:
The excess kurtosis is given by:
The characteristic function is given by:
whe |
https://en.wikipedia.org/wiki/ECM | ECM may refer to:
Economics and commerce
Engineering change management
Equity capital markets
Error correction model, an econometric model
European Common Market
Mathematics
Elliptic curve method
European Congress of Mathematics
Science and medicine
Ectomycorrhiza
Electron cloud model
Engineered Cellular Magmatics
Erythema chronicum migrans
Extracellular matrix
Sport
European Championships Management
Technology
Electrochemical machining
Electronic contract manufacturing
Electronic countermeasure
Electronically commutated motor
Energy conservation measure
Engine control module
Enterprise content management
Error correction mode
Other uses
Editio Critica Maior, a critical edition of the Greek New Testament
ECM Records, a record label
ECM Real Estate Investments, a defunct real estate developer based in Luxembourg
Edinburgh City Mission, a Christian organization in Scotland
Elektrani na Severna Makedonija (), a power company in North Macedonia
Episcopal Conference of Malawi
Every Child Matters, a UK Government initiative for children
Every Child Ministries, a Christian charity for African children
Exceptional case-marking, in linguistics
Energy Corrected Milk, in farming |
https://en.wikipedia.org/wiki/Concave%20function | In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any and in the interval and for any ,
A function is called strictly concave if
for any and .
For a function , this second definition merely states that for every strictly between and , the point on the graph of is above the straight line joining the points and .
A function is quasiconcave if the upper contour sets of the function are convex sets.
Properties
Functions of a single variable
A differentiable function is (strictly) concave on an interval if and only if its derivative function is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.
Points where concavity changes (between concave and convex) are inflection points.
If is twice-differentiable, then is concave if and only if is non-positive (or, informally, if the "acceleration" is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by .
If is concave and differentiable, then it is bounded above by its first-order Taylor approximation:
A Lebesgue measurable function on an interval is concave if and only if it is midpoint concave, that is, for any and in
If a function is concave, and , then is subadditive on . Proof:
Since is concave and , letting we have
For :
Functions of n variables
A function is concave over a convex set if and only if the function is a convex function over the set.
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
Any local maximum of a concave function is also a global maximum. A strictly concave function will have at most one global maximum.
Examples
The functions and are concave on their domains, as their second derivatives and are always negative.
The logarithm function is concave on its domain , as its derivative is a strictly decreasing function.
Any affine function is both concave and convex, but neither strictly-concave nor strictly-convex.
The sine function is concave on the interval .
The function , where is the determinant of a nonnegative-definite matrix B, is concave.
Applications
Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.
In expected utility theory for choice under uncertainty, cardinal utility functions of risk aver |
https://en.wikipedia.org/wiki/Curve%20sketching | In geometry, curve sketching (or curve tracing) are techniques for producing a rough idea of overall shape of a plane curve given its equation, without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features.
Basic techniques
The following are usually easy to carry out and give important clues as to the shape of a curve:
Determine the x and y intercepts of the curve. The x intercepts are found by setting y equal to 0 in the equation of the curve and solving for x. Similarly, the y intercepts are found by setting x equal to 0 in the equation of the curve and solving for y.
Determine the symmetry of the curve. If the exponent of x is always even in the equation of the curve then the y-axis is an axis of symmetry for the curve. Similarly, if the exponent of y is always even in the equation of the curve then the x-axis is an axis of symmetry for the curve. If the sum of the degrees of x and y in each term is always even or always odd, then the curve is symmetric about the origin and the origin is called a center of the curve.
Determine any bounds on the values of x and y.
If the curve passes through the origin then determine the tangent lines there. For algebraic curves, this can be done by removing all but the terms of lowest order from the equation and solving.
Similarly, removing all but the terms of highest order from the equation and solving gives the points where the curve meets the line at infinity.
Determine the asymptotes of the curve. Also determine from which side the curve approaches the asymptotes and where the asymptotes intersect the curve.
Equate first and second derivatives to 0 to find the stationary points and inflection points respectively. If the equation of the curve cannot be solved explicitly for x or y, finding these derivatives requires implicit differentiation.
Newton's diagram
Newton's diagram (also known as Newton's parallelogram, after Isaac Newton) is a technique for determining the shape of an algebraic curve close to and far away from the origin. It consists of plotting (α, β) for each term Axαyβ in the equation of the curve. The resulting diagram is then analyzed to produce information about the curve.
Specifically, draw a diagonal line connecting two points on the diagram so that every other point is either on or to the right and above it. There is at least one such line if the curve passes through the origin. Let the equation of the line be qα+pβ=r. Suppose the curve is approximated by y=Cxp/q near the origin. Then the term Axαyβ is approximately Dxα+βp/q. The exponent is r/q when (α, β) is on the line and higher when it is above and to the right. Therefore, the significant terms near the origin under this assumption are only those lying on the line and the others may be ignored; it produces a simple approximate equation for the curve. There may be several such diagonal lines, each corresponding to one or more branches of the c |
https://en.wikipedia.org/wiki/SciPy | SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing.
SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and engineering.
SciPy is also a family of conferences for users and developers of these tools: SciPy (in the United States), EuroSciPy (in Europe) and SciPy.in (in India). Enthought originated the SciPy conference in the United States and continues to sponsor many of the international conferences as well as host the SciPy website.
The SciPy library is currently distributed under the BSD license, and its development is sponsored and supported by an open community of developers. It is also supported by NumFOCUS, a community foundation for supporting reproducible and accessible science.
Components
The SciPy package is at the core of Python's scientific computing capabilities. Available sub-packages include:
cluster: hierarchical clustering, vector quantization, K-means
constants: physical constants and conversion factors
fft: Discrete Fourier Transform algorithms
fftpack: Legacy interface for Discrete Fourier Transforms
integrate: numerical integration routines
interpolate: interpolation tools
io: data input and output
linalg: linear algebra routines
misc: miscellaneous utilities (e.g. example images)
ndimage: various functions for multi-dimensional image processing
ODR: orthogonal distance regression classes and algorithms
optimize: optimization algorithms including linear programming
signal: signal processing tools
sparse: sparse matrices and related algorithms
spatial: algorithms for spatial structures such as k-d trees, nearest neighbors, Convex hulls, etc.
special: special functions
stats: statistical functions
weave: tool for writing C/C++ code as Python multiline strings (now deprecated in favor of Cython)
Data structures
The basic data structure used by SciPy is a multidimensional array provided by the NumPy module. NumPy provides some functions for linear algebra, Fourier transforms, and random number generation, but not with the generality of the equivalent functions in SciPy. NumPy can also be used as an efficient multidimensional container of data with arbitrary datatypes. This allows NumPy to seamlessly and speedily integrate with a wide variety of databases. Older versions of SciPy used Numeric as an array type, which is now deprecated in favor of the newer NumPy array code.
History
In the 1990s, Python was extended to include an array type for numerical computing called Numeric (This package was eventually replaced by Travis Oliphant who wrote NumPy in 2006 as a blending of Numeric and Numarray which had been started in 2001). As of 2000, there was a growing number of extension modules and increasing interest in creating a complete environment for scientific and technical computing. In 2001, Travis Oliphant |
https://en.wikipedia.org/wiki/Deviation | Deviation may refer to:
Mathematics and engineering
Allowance (engineering), an engineering and machining allowance is a planned deviation between an actual dimension and a nominal or theoretical dimension, or between an intermediate-stage dimension and an intended final dimension.
Deviation (statistics), the difference between the value of an observation and the mean of the population in mathematics and statistics
Standard deviation, which is based on the square of the difference
Absolute deviation, where the absolute value of the difference is used
Relative standard deviation, in probability theory and statistics is the absolute value of the coefficient of variation
Deviation of a local ring in mathematics
Deviation of a poset in mathematics
Frequency deviation, the maximum allowed "distance" in FM radio from the nominal frequency a station broadcasts at
Magnetic deviation, the error induced in compasses by local magnetic fields
Albums
Deviation (Jayne County album), 1995
Deviation (Béla Fleck album), 1984
Deviate (album), a 1998 album by Kill II This
Other uses
Bid‘ah, Islamic term for innovations and deviations acts or groups from orthodox Islamic law (Sharia).
Deviance (sociology), a behavior that is a recognized violation of social norms
Deviation (1971 film), a horror film
Deviation (2006 film), a short film
Deviation, a 2012 British thriller film starring Danny Dyer
Deviation (law) is a departure from a contract or a ship's course, thus breaching the contract
Deviationism, an expressed belief which is not in accordance with official party doctrine
A work of art in the online community DeviantArt
See also
Deviance (disambiguation)
Deviant (disambiguation)
Devious (disambiguation) |
https://en.wikipedia.org/wiki/Differential%20operator | In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).
This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.
Definition
Given a nonnegative integer m, an order- linear differential operator is a map from a function space to another function space that can be written as:
where is a multi-index of non-negative integers, , and for each , is a function on some open domain in n-dimensional space. The operator is interpreted as
Thus for a function :
The notation is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives.
The polynomial p obtained by replacing D by variables in P is called the total symbol of P; i.e., the total symbol of P above is:
where The highest homogeneous component of the symbol, namely,
is called the principal symbol of P. While the total symbol is not intrinsically defined, the principal symbol is intrinsically defined (i.e., it is a function on the cotangent bundle).
More generally, let E and F be vector bundles over a manifold X. Then the linear operator
is a differential operator of order if, in local coordinates on X, we have
where, for each multi-index α, is a bundle map, symmetric on the indices α.
The kth order coefficients of P transform as a symmetric tensor
whose domain is the tensor product of the kth symmetric power of the cotangent bundle of X with E, and whose codomain is F. This symmetric tensor is known as the principal symbol (or just the symbol) of P.
The coordinate system xi permits a local trivialization of the cotangent bundle by the coordinate differentials dxi, which determine fiber coordinates ξi. In terms of a basis of frames eμ, fν of E and F, respectively, the differential operator P decomposes into components
on each section u of E. Here Pνμ is the scalar differential operator defined by
With this trivialization, the principal symbol can now be written
In the cotangent space over a fixed point x of X, the symbol defines a homogeneous polynomial of degree k in with values in .
Fourier interpretation
A differential operator P and its symbol appear naturally in connection with the Fourier transform as follows. Let ƒ be a Schwartz function. Then by the inverse Fourier transform,
This exhibits P as a Fourier multiplier. A more general class of functions p(x,ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the pseudo-differential operators.
Examples
The differential operator is elliptic if its symbol is invertible; that is for each nonzero the bundle map is |
https://en.wikipedia.org/wiki/Edward%20Tufte | Edward Rolf Tufte (; born March 14, 1942), sometimes known as "ET", is an American statistician and professor emeritus of political science, statistics, and computer science at Yale University. He is noted for his writings on information design and as a pioneer in the field of data visualization.
Biography
Edward Rolf Tufte was born in 1942 in Kansas City, Missouri, to Virginia Tufte (1918–2020) and Edward E. Tufte (1912–1999). He grew up in Beverly Hills, California, where his father was a longtime city official, and he graduated from Beverly Hills High School. He received a BS and MS in statistics from Stanford University and a PhD in political science from Yale. His dissertation, completed in 1968, was titled The Civil Rights Movement and Its Opposition. He was then hired by Princeton University's Woodrow Wilson School, where he taught courses in political economy and data analysis while publishing three quantitatively inclined political science books.
In 1975, while at Princeton, Tufte was asked to teach a statistics course to a group of journalists who were visiting the school to study economics. He developed a set of readings and lectures on statistical graphics, which he further developed in joint seminars he taught with renowned statistician John Tukey, a pioneer in the field of information design. These course materials became the foundation for his first book on information design, The Visual Display of Quantitative Information.
After negotiations with major publishers failed, Tufte decided to self-publish Visual Display in 1982, working closely with graphic designer Howard Gralla. He financed the work by taking out a second mortgage on his home. The book quickly became a commercial success and secured his transition from political scientist to information expert.
On March 5, 2010, President Barack Obama appointed Tufte to the American Recovery and Reinvestment Act's Recovery Independent Advisory Panel "to provide transparency in the use of Recovery-related funds".
Work
Tufte is an expert in the presentation of informational graphics such as charts and diagrams, and is a fellow of the American Statistical Association. He has held fellowships from the Guggenheim Foundation and the Center for Advanced Study in the Behavioral Sciences.
Information design
Tufte's writing is important in such fields as information design and visual literacy, which deal with the visual communication of information. He coined the word chartjunk to refer to useless, non-informative, or information-obscuring elements of quantitative information displays. Tufte's other key concepts include what he calls the lie factor, the data-ink ratio, and the data density of a graphic.
He uses the term "data-ink ratio" to argue against using excessive decoration in visual displays of quantitative information. In Visual Display, Tufte explains, "Sometimes decoration can help editorialize about the substance of the graphic. But it is wrong to distort the data measu |
https://en.wikipedia.org/wiki/Archimedean%20property | In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.
The property, typically construed, states that given two positive numbers and , there is an integer such that . It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder.
The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.
An algebraic structure in which any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean.
A structure which has a pair of non-zero elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean.
For example, a linearly ordered group that is Archimedean is an Archimedean group.
This can be made precise in various contexts with slightly different formulations.
For example, in the context of ordered fields, one has the axiom of Archimedes which formulates this property, where the field of real numbers is Archimedean, but that of rational functions in real coefficients is not.
History and origin of the name of the Archimedean property
The concept was named by Otto Stolz (in the 1880s) after the ancient Greek geometer and physicist Archimedes of Syracuse.
The Archimedean property appears in Book V of Euclid's Elements as Definition 4:
Because Archimedes credited it to Eudoxus of Cnidus it is also known as the "Theorem of Eudoxus" or the Eudoxus axiom.
Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs.
Definition for linearly ordered groups
Let and be positive elements of a linearly ordered group G.
Then is infinitesimal with respect to (or equivalently, is infinite with respect to ) if, for any natural number , the multiple is less than , that is, the following inequality holds:
This definition can be extended to the entire group by taking absolute values.
The group is Archimedean if there is no pair such that is infinitesimal with respect to .
Additionally, if is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to .
If is infinitesimal with respect to , then is an infinitesimal element.
Likewise, if is infinite with respect to , then is an infinite element.
The algebraic structure is Archimedean if it has no infinite elements and no infinitesimal elements.
Ordered fields
Ordered fields have some additional properties:
The ra |
https://en.wikipedia.org/wiki/Archimedean%20group | In abstract algebra, a branch of mathematics, an Archimedean group is a linearly ordered group for which the Archimedean property holds: every two positive group elements are bounded by integer multiples of each other. The set R of real numbers together with the operation of addition and the usual ordering relation between pairs of numbers is an Archimedean group. By a result of Otto Hölder, every Archimedean group is isomorphic to a subgroup of this group. The name "Archimedean" comes from Otto Stolz, who named the Archimedean property after its appearance in the works of Archimedes.
Definition
An additive group consists of a set of elements, an associative addition operation that combines pairs of elements and returns a single element,
an identity element (or zero element) whose sum with any other element is the other element, and an additive inverse operation such that the sum of any element and its inverse is zero.
A group is a linearly ordered group when, in addition, its elements can be linearly ordered in a way that is compatible with the group operation: for all elements x, y, and z, if x ≤ y then x + z ≤ y + z and z + x ≤ z + y.
The notation na (where n is a natural number) stands for the group sum of n copies of a.
An Archimedean group (G, +, ≤) is a linearly ordered group subject to the following additional condition, the Archimedean property: For every a and b in G which are greater than 0, it is possible to find a natural number n for which the inequality b ≤ na holds.
An equivalent definition is that an Archimedean group is a linearly ordered group without any bounded cyclic subgroups: there does not exist a cyclic subgroup S and an element x with x greater than all elements in S. It is straightforward to see that this is equivalent to the other definition: the Archimedean property for a pair of elements a and b is just the statement that the cyclic subgroup generated by a is not bounded by b.
Examples of Archimedean groups
The sets of the integers, the rational numbers, and the real numbers, together with the operation of addition and the usual ordering (≤), are Archimedean groups. Every subgroup of an Archimedean group is itself Archimedean, so it follows that every subgroup of these groups, such as the additive group of the even numbers or of the dyadic rationals, also forms an Archimedean group.
Conversely, as Otto Hölder showed, every Archimedean group is isomorphic (as an ordered group) to a subgroup of the real numbers. It follows from this that every Archimedean group is necessarily an abelian group: its addition operation must be commutative.
Examples of non-Archimedean groups
Groups that cannot be linearly ordered, such as the finite groups, are not Archimedean. For another example, see the p-adic numbers, a system of numbers generalizing the rational numbers in a different way to the real numbers.
Non-Archimedean ordered groups also exist; the ordered group (G, +, ≤) defined as follows is not Archimedean. Let the |
https://en.wikipedia.org/wiki/Zero%20of%20a%20function | In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function , is a member of the domain of such that vanishes at ; that is, the function attains the value of 0 at , or equivalently, is the solution to the equation . A "zero" of a function is thus an input value that produces an output of 0.
A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial of degree two, defined by has the two roots (or zeros) that are 2 and 3.
If the function maps real numbers to real numbers, then its zeros are the -coordinates of the points where its graph meets the x-axis. An alternative name for such a point in this context is an -intercept.
Solution of an equation
Every equation in the unknown may be rewritten as
by regrouping all the terms in the left-hand side. It follows that the solutions of such an equation are exactly the zeros of the function . In other words, a "zero of a function" is precisely a "solution of the equation obtained by equating the function to 0", and the study of zeros of functions is exactly the same as the study of solutions of equations.
Polynomial roots
Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).
Fundamental theorem of algebra
The fundamental theorem of algebra states that every polynomial of degree has complex roots, counted with their multiplicities. The non-real roots of polynomials with real coefficients come in conjugate pairs. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.
Computing roots
Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).
Zero set
In various areas of mathematics, the zero set of a function is the set of all its zeros. More precisely, if is a real-valued function (or, more generally, a function ta |
https://en.wikipedia.org/wiki/Size%20%28disambiguation%29 | Size is the concept of how big or small something is. It may also refer to:
In statistics (hypothesis testing), the size of the test refers to the rate of false positives, denoted by α
File size, in computing
Magnitude (mathematics), magnitude or size of a mathematical object
Magnitude of brightness or intensity of a star or an earthquake as measured on a logarithmic scale
In mathematics there are, in addition to the dimensions mentioned above (equal if there is an isometry), various other concepts of size for sets:
measure (mathematics), a systematic way to assign to each suitable subset a number
cardinality (equal if there is a bijection), of a set is a measure of the "number of elements of the set"
for well-ordered sets: ordinal number (equal if there is an order-isomorphism)
Resizing (fiction), a theme in fiction, in particular in fairy tales, fantasy, and science fiction
Sizing, or size, a filler or glaze
Size (surname)
Clothing size, the label sizes used for garments sold off-the-shelf
Size (band), a Mexican punk rock band
See also |
https://en.wikipedia.org/wiki/Hilbert%20matrix | In linear algebra, a Hilbert matrix, introduced by , is a square matrix with entries being the unit fractions
For example, this is the 5 × 5 Hilbert matrix:
The entries can also be defined by the integral
that is, as a Gramian matrix for powers of x. It arises in the least squares approximation of arbitrary functions by polynomials.
The Hilbert matrices are canonical examples of ill-conditioned matrices, being notoriously difficult to use in numerical computation. For example, the 2-norm condition number of the matrix above is about 4.8.
Historical note
introduced the Hilbert matrix to study the following question in approximation theory: "Assume that , is a real interval. Is it then possible to find a non-zero polynomial P with integer coefficients, such that the integral
is smaller than any given bound ε > 0, taken arbitrarily small?" To answer this question, Hilbert derives an exact formula for the determinant of the Hilbert matrices and investigates their asymptotics. He concludes that the answer to his question is positive if the length of the interval is smaller than 4.
Properties
The Hilbert matrix is symmetric and positive definite. The Hilbert matrix is also totally positive (meaning that the determinant of every submatrix is positive).
The Hilbert matrix is an example of a Hankel matrix. It is also a specific example of a Cauchy matrix.
The determinant can be expressed in closed form, as a special case of the Cauchy determinant. The determinant of the n × n Hilbert matrix is
where
Hilbert already mentioned the curious fact that the determinant of the Hilbert matrix is the reciprocal of an integer (see sequence in the OEIS), which also follows from the identity
Using Stirling's approximation of the factorial, one can establish the following asymptotic result:
where an converges to the constant as , where A is the Glaisher–Kinkelin constant.
The inverse of the Hilbert matrix can be expressed in closed form using binomial coefficients; its entries are
where n is the order of the matrix. It follows that the entries of the inverse matrix are all integers, and that the signs form a checkerboard pattern, being positive on the principal diagonal. For example,
The condition number of the n × n Hilbert matrix grows as .
Applications
The method of moments applied to polynomial distributions results in a Hankel matrix, which in the special case of approximating a probability distribution on the interval [0, 1] results in a Hilbert matrix. This matrix needs to be inverted to obtain the weight parameters of the polynomial distribution approximation.
References
Further reading
. Reprinted in
Numerical linear algebra
Approximation theory
Matrices
Determinants |
https://en.wikipedia.org/wiki/Cosh | Cosh may refer to:
People
Chris Cosh (born 1959), American football coach
John Cosh (1915–2005), British rheumatologist
Science, technology, and mathematics
cosh (mathematical function), hyperbolic cosine, a mathematical function with notation cosh(x)
-COSH, a representation of the thiocarboxylic acid functional group in chemistry
Chlorpromazine, an antipsychotic drug
ChromeOS Shell, an operating system designed by Google
Weaponry
Baton (law enforcement)
Club (weapon)
See also
COSHH (Control of Substances Hazardous to Health Regulations 2002), a set of UK regulations
Kosh (disambiguation)
Chemical cosh, describing a sedative drug
Cosh Boy, a 1953 British film
Harry and Cosh, a British children's television series |
https://en.wikipedia.org/wiki/E%20%28theorem%20prover%29 | E is a high-performance theorem prover for full first-order logic with equality. It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated into other theorem provers and it has been among the best-placed systems in several theorem proving competitions. E is developed by Stephan Schulz, originally in the Automated Reasoning Group at TU Munich, now at Baden-Württemberg Cooperative State University Stuttgart.
System
The system is based on the equational superposition calculus. In contrast to most other current provers, the implementation actually uses a purely equational paradigm, and simulates non-equational inferences via appropriate equality inferences. Significant innovations include shared term rewriting (where many possible equational simplifications are carried out in a single operation), several efficient term indexing data structures for speeding up inferences, advanced inference literal selection strategies, and various uses of machine learning techniques to improve the search behaviour. Since version 2.0, E supports many-sorted logic.
E is implemented in C and portable to most UNIX variants and the Cygwin environment. It is available under the GNU GPL.
Competitions
The prover has consistently performed well in the CADE ATP System Competition, winning the CNF/MIX category in 2000 and finishing among the top systems ever since. In 2008 it came in second place. In 2009 it won second place in the FOF (full first order logic) and UEQ (unit equational logic) categories and third place (after two versions of Vampire) in CNF (clausal logic). It repeated the performance in FOF and CNF in 2010, and won a special award as "overall best" system. In the 2011 CASC-23 E won the CNF division and achieved second places in UEQ and LTB.
Applications
E has been integrated into several other theorem provers. It is, with Vampire, SPASS, CVC4, and Z3, at the core of Isabelle's Sledgehammer strategy. E also is the reasoning engine in SInE and LEO-II and used as the clausification system for iProver.
Applications of E include reasoning on large ontologies, software verification, and software certification.
References
External links
E home page
E's developer
Free software programmed in C
Free theorem provers
Unix programming tools |
https://en.wikipedia.org/wiki/Finite%20geometry | A finite geometry is any geometric system that has only a finite number of points.
The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries.
Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the non-Desarguesian planes. Similar results hold for other kinds of finite geometries.
Finite planes
The following remarks apply only to finite planes.
There are two main kinds of finite plane geometry: affine and projective.
In an affine plane, the normal sense of parallel lines applies.
In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms.
Finite affine planes
An affine plane geometry is a nonempty set X (whose elements are called "points"), along with a nonempty collection L of subsets of X (whose elements are called "lines"), such that:
For every two distinct points, there is exactly one line that contains both points.
Playfair's axiom: Given a line and a point not on , there exists exactly one line containing such that
There exists a set of four points, no three of which belong to the same line.
The last axiom ensures that the geometry is not trivial (either empty or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry.
The simplest affine plane contains only four points; it is called the affine plane of order 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but als |
https://en.wikipedia.org/wiki/Jordan%20curve%20theorem | In topology, the Jordan curve theorem asserts that every Jordan curve (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen. However, this notion has been overturned by Thomas C. Hales and others.
Definitions and the statement of the Jordan theorem
A Jordan curve or a simple closed curve in the plane R2 is the image C of an injective continuous map of a circle into the plane, φ: S1 → R2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval into the plane. It is a plane curve that is not necessarily smooth nor algebraic.
Alternatively, a Jordan curve is the image of a continuous map φ: [0,1] → R2 such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective. The first two conditions say that C is a continuous loop, whereas the last condition stipulates that C has no self-intersection points.
With these definitions, the Jordan curve theorem can be stated as follows:
In contrast, the complement of a Jordan arc in the plane is connected.
Proof and generalizations
The Jordan curve theorem was independently generalized to higher dimensions by H. Lebesgue and L. E. J. Brouwer in 1911, resulting in the Jordan–Brouwer separation theorem.
The proof uses homology theory. It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homology groups of Y = Rn+1 \ X are as follows:
This is proved by induction in k using the Mayer–Vietoris sequence. When n = k, the zeroth reduced homology of Y has rank 1, which means that Y has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is X. A further generalization was found by J. W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of Rn+1 and the reduced cohomology of its complement. If X is an n-dimensional compact connected submanifold of Rn+1 (or Sn+1) without boundary, its complement has 2 connected components.
There is a strengthening of the Jordan curve theorem, called the Jordan |
https://en.wikipedia.org/wiki/Thales%27s%20theorem | In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.
History
Babylonian mathematicians knew this for special cases before Greek mathematicians proved it.
Thales of Miletus (early 6th century BC) is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxographers based on hearsay and speculation. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) documenting Pamphila's (1st century AD) statement that Thales "was the first to inscribe in a circle a right-angle triangle".
Thales was claimed to have traveled to Egypt and Babylonia, where he is supposed to have learned about geometry and astronomy and thence brought their knowledge to the Greeks, along the way inventing the concept of geometric proof and proving various geometric theorems. However, there is no direct evidence for any of these claims, and they were most likely invented speculative rationalizations. Modern scholars believe that Greek deductive geometry as found in Euclid's Elements was not developed until the 4th century BC, and any geometric knowledge Thales may have had would have been observational.
The theorem appears in Book II of Euclid's Elements () as proposition 31: "In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less segment greater than a right angle; further the angle of the greater segment is greater than a right angle, and the angle of the less segment is less than a right angle."
Dante's Paradiso (canto 13, lines 101–102) refers to Thales's theorem in the course of a speech.
Proof
First proof
The following facts are used: the sum of the angles in a triangle is equal to 180° and the base angles of an isosceles triangle are equal.
Since , and are isosceles triangles, and by the equality of the base angles of an isosceles triangle, and .
Let and . The three internal angles of the triangle are , , and . Since the sum of the angles of a triangle is equal to 180°, we have
Q.E.D.
Second proof
The theorem may also be proven using trigonometry: Let , , and . Then is a point on the unit circle . We will show that forms a right angle by proving that and are perpendicular — that is, the product of their slopes is equal to −1. We calculate the slopes for and :
Then we show that their product equals −1:
Note the use of the Pythagorean trigonometric identity
Third proof
Let be a triangle in a circle where is a diameter in |
https://en.wikipedia.org/wiki/Synthetic%20geometry | Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms.
The term "synthetic geometry" was coined only after the 17th century, and the introduction by René Descartes of the coordinate method, which was called analytic geometry. So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones.
According to Felix Klein
Synthetic geometry is that which studies figures as such, without recourse to formulae, whereas analytic geometry consistently makes use of such formulae as can be written down after the adoption of an appropriate system of coordinates.
The first systematic approach for synthetic geometry is Euclid's Elements. However, it appeared at the end of the 19th century that Euclid's postulates were not sufficient for characterizing geometry. The first complete axiom system for geometry was given only at the end of the 19th century by David Hilbert. At the same time, it appeared that both synthetic methods and analytic methods can be used to build geometry. The fact that the two approches are equivalent has been proved by Emil Artin in his book Geometric Algebra.
Because of this equivalence, the distinction between synthetic and analytic geometry is no more in use, except at elementary level, or for geometries that are not related to any sort of numbers, such as some finite geometries and non-Desarguesian geometry.
Logical synthesis
The process of logical synthesis begins with some arbitrary but definite starting point. This starting point is the introduction of primitive notions or primitives and axioms about these primitives:
Primitives are the most basic ideas. Typically they include both objects and relationships. In geometry, the objects are things such as points, lines and planes, while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs, the point being that the primitive terms are just empty placeholders and have no intrinsic properties.
Axioms are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have.
From a given set of axioms, synthesis proceeds as a carefully constructed logical argument. When a significant result is proved rigorously, it becomes a theorem.
Properties of axiom sets
There is no fixed axiom set for geometry, as more than one consistent set ca |
https://en.wikipedia.org/wiki/Nonprobability%20sampling | Sampling is the use of a subset of the population to represent the whole population or to inform about (social) processes that are meaningful beyond the particular cases, individuals or sites studied. Probability sampling, or random sampling, is a sampling technique in which the probability of getting any particular sample may be calculated. In cases where external validity is not of critical importance to the study's goals or purpose, researchers might prefer to use nonprobability sampling. Nonprobability sampling does not meet this criterion. Nonprobability sampling techniques are not intended to be used to infer from the sample to the general population in statistical terms. Instead, for example, grounded theory can be produced through iterative nonprobability sampling until theoretical saturation is reached (Strauss and Corbin, 1990).
Thus, one cannot say the same on the basis of a nonprobability sample than on the basis of a probability sample. The grounds for drawing generalizations (e.g., propose new theory, propose policy) from studies based on nonprobability samples are based on the notion of "theoretical saturation" and "analytical generalization" (Yin, 2014) instead of on statistical generalization.
Researchers working with the notion of purposive sampling assert that while probability methods are suitable for large-scale studies concerned with representativeness, nonprobability approaches are more suitable for in-depth qualitative research in which the focus is often to understand complex social phenomena (e.g., Marshall 1996; Small 2009). One of the advantages of nonprobability sampling is its lower cost compared to probability sampling. Moreover, the in-depth analysis of a small-N purposive sample or a case study enables the "discovery" and identification of patterns and causal mechanisms that do not draw time and context-free assumptions.
Nonprobability sampling is often not appropriate in statistical quantitative research, though, as these assertions raise some questions — how can one understand a complex social phenomenon by drawing only the most convenient expressions of that phenomenon into consideration? What assumption about homogeneity in the world must one make to justify such assertions? Alas, the consideration that research can only be based in statistical inference focuses on the problems of bias linked to nonprobability sampling and acknowledges only one situation in which a nonprobability sample can be appropriate — if one is interested only in the specific cases studied (for example, if one is interested in the Battle of Gettysburg), one does not need to draw a probability sample from similar cases (Lucas 2014a).
Nonprobability sampling is however widely used in qualitative research. Examples of nonprobability sampling include:
Convenience, haphazard or accidental sampling – members of the population are chosen based on their relative ease of access. To sample friends, co-workers, or shoppers at a single mal |
https://en.wikipedia.org/wiki/Systematic%20sampling | In survey methodology, systematic sampling is a statistical method involving the selection of elements from an ordered sampling frame. The most common form of systematic sampling is an equiprobability method.
In this approach, progression through the list is treated circularly, with a return to the top once the list ends. The sampling starts by selecting an element from the list at random and then every kth element in the frame is selected, where k, is the sampling interval (sometimes known as the skip): this is calculated as:
where n is the sample size, and N is the population size.
Using this procedure each element in the population has a known and equal probability of selection (also known as epsem). This makes systematic sampling functionally similar to simple random sampling (SRS). However, it is not the same as SRS because not every possible sample of a certain size has an equal chance of being chosen (e.g. samples with at least two elements adjacent to each other will never be chosen by systematic sampling). It is, however, much more efficient (if the variance within a systematic sample is more than the variance of the population).
Systematic sampling is to be applied only if the given population is logically homogeneous, because systematic sample units are uniformly distributed over the population. The researcher must ensure that the chosen sampling interval does not hide a pattern. Any pattern would threaten randomness.
Example: Suppose a supermarket wants to study buying habits of their customers, then using systematic sampling they can choose every 10th or 15th customer entering the supermarket and conduct the study on this sample.
This is random sampling with a system. From the sampling frame, a starting point is chosen at random, and choices thereafter are at regular intervals. For example, suppose you want to sample 8 houses from a street of 120 houses. 120/8=15, so every 15th house is chosen after a random starting point between 1 and 15. If the random starting point is 11, then the houses selected are 11, 26, 41, 56, 71, 86, 101, and 116. As an aside, if every 15th house was a "corner house" then this corner pattern could destroy the randomness of the sample.
If, more frequently, the population is not evenly divisible (suppose you want to sample 8 houses out of 125, where 125/8=15.625), should you take every 15th house or every 16th house? If you take every 16th house, 8*16=128, there is a risk that the last house chosen does not exist. On the other hand, if you take every 15th house, 8*15=120, so the last five houses will never be selected. The random starting point should instead be selected as a non-integer between 0 and 15.625 (inclusive on one endpoint only) to ensure that every house has an equal chance of being selected; the interval should now be non-integral (15.625); and each non-integer selected should be rounded up to the next integer. If the random starting point is 3.6, then the houses selected are 4, 20, 35 |
https://en.wikipedia.org/wiki/Clifford%20A.%20Pickover | Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research Center in Yorktown, New York, where he was editor-in-chief of the IBM Journal of Research and Development. He has been granted more than 700 U.S. patents, is an elected Fellow for the Committee for Skeptical Inquiry, and is author of more than 50 books, translated into more than a dozen languages.
Life, education and career
He received his PhD in 1982 from Yale University's Department of Molecular Biophysics and Biochemistry, where he conducted research on X-ray scattering and protein structure. Pickover graduated first in his class from Franklin and Marshall College, after completing the four-year undergraduate program in three years.
Pickover was elected as a Fellow for the Committee for Skeptical Inquiry for his "significant contributions to the general public's understanding of science, reason, and critical inquiry through their scholarship, writing, and work in the media." Other Fellows have included Carl Sagan and Isaac Asimov. He has been awarded almost 700 United States patents, and his The Math Book was winner of the 2011 Neumann Prize.
He joined IBM at the Thomas J. Watson Research Center in 1982, as a member of the speech synthesis group and later worked on the design-automation workstations. For much of his career, Pickover has published technical articles in the areas of scientific visualization, computer art, and recreational mathematics.
He is currently an associate editor for the scientific journal Computers and Graphics and is an editorial board member for Odyssey and Leonardo. He is also the Brain-Strain columnist for Odyssey magazine, and, for many years, he was the Brain-Boggler columnist for Discover magazine.
Pickover has received more than 100 IBM invention achievement awards, three research division awards, and four external honor awards.
Work
Pickover's primary interest is in finding new ways to expand creativity by melding art, science, mathematics, and other seemingly disparate areas of human endeavor.
In The Math Book and his companion book The Physics Book, Pickover explains that both mathematics and physics "cultivate a perpetual state of wonder about the limits of thoughts, the workings of the universe, and our place in the vast space-time landscape that we call home."
Pickover is an inventor with over 700 patents, the author of puzzle calendars, and puzzle contributor to magazines geared to children and adults. His Neoreality and Heaven Virus science-fiction series explores the fabric of reality and religion.
Pickover is author of hundreds of technical papers in diverse fields, ranging from the creative visualizations of fossil seashells, genetic sequences, cardiac and speech sounds, and virtual caverns and lava lamps, to fractal and mathematically based studies. |
https://en.wikipedia.org/wiki/Position | Position often refers to:
Position (geometry), the spatial location (rather than orientation) of an entity
Position, a job or occupation
Position may also refer to:
Games and recreation
Position (poker), location relative to the dealer
Position (team sports), a player role within a team
Human body
Human position, the spatial relation of the human body to itself and the environment
Position (obstetrics), the orientation of a baby prior to birth
Positions of the feet in ballet
Position (music), the location of the hand on a musical instrument
Proprioception, the sense of the relative position of neighbouring parts of the body
Asana (yoga), the location and posture of the body while practicing yoga
Sex position, the arrangement of bodies during sexual intercourse
Humanities, law, economics and politics
Philosophical theory, a belief or set of beliefs about questions in philosophy
Position (finance), commitments in a financial marketplace
Social position, the position of an individual in a society and culture
Political position within a political spectrum
Science and mathematics
Position (vector), a mathematical identification of relative location
Position in positional notation of mathematical operations
Other uses
The Position, a novel by Meg Wolitzer
Positions (book), a book by Jacques Derrida
Positions (album), a 2020 album by Ariana Grande
"Positions" (song), the album's title track
The Positions, 2015 album by Gang of Youths
See also
Location (disambiguation)
Positioning (disambiguation) |
https://en.wikipedia.org/wiki/Ineffable%20cardinal | In the mathematics of transfinite numbers, an ineffable cardinal is a certain kind of large cardinal number, introduced by . In the following definitions, will always be a regular uncountable cardinal number.
A cardinal number is called almost ineffable if for every (where is the powerset of ) with the property that is a subset of for all ordinals , there is a subset of having cardinality and homogeneous for , in the sense that for any in , .
A cardinal number is called ineffable if for every binary-valued function , there is a stationary subset of on which is homogeneous: that is, either maps all unordered pairs of elements drawn from that subset to zero, or it maps all such unordered pairs to one. An equivalent formulation is that a cardinal is ineffable if for every sequence such that each ,
there is such that is stationary in .
More generally, is called -ineffable (for a positive integer ) if for every there is a stationary subset of on which is -homogeneous (takes the same value for all unordered -tuples drawn from the subset). Thus, it is ineffable if and only if it is 2-ineffable.
A totally ineffable cardinal is a cardinal that is -ineffable for every . If is -ineffable, then the set of -ineffable cardinals below is a stationary subset of .
Every -ineffable cardinal is -almost ineffable (with set of -almost ineffable below it stationary), and every -almost ineffable is -subtle (with set of -subtle below it stationary). The least -subtle cardinal is not even weakly compact (and unlike ineffable cardinals, the least -almost ineffable is -describable), but -ineffable cardinals are stationary below every -subtle cardinal.
A cardinal κ is completely ineffable if there is a non-empty such that
- every is stationary
- for every and , there is homogeneous for f with .
Using any finite > 1 in place of 2 would lead to the same definition, so completely ineffable cardinals are totally ineffable (and have greater consistency strength). Completely ineffable cardinals are -indescribable for every n, but the property of being completely ineffable is .
The consistency strength of completely ineffable is below that of 1-iterable cardinals, which in turn is below remarkable cardinals, which in turn is below ω-Erdős cardinals. A list of large cardinal axioms by consistency strength is available in the section below.
See also
List of large cardinal properties
References
.
Large cardinals |
https://en.wikipedia.org/wiki/Gumbel%20distribution | In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory, which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type. This article uses the Gumbel distribution to model the distribution of the maximum value. To model the minimum value, use the negative of the original values.
The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution). It is related to the Gompertz distribution: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.
In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed random variables has a logistic distribution.
The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on his original papers describing the distribution.
Definitions
The cumulative distribution function of the Gumbel distribution is
Standard Gumbel distribution
The standard Gumbel distribution is the case where and with cumulative distribution function
and probability density function
In this case the mode is 0, the median is , the mean is (the Euler–Mascheroni constant), and the standard deviation is
The cumulants, for n > 1, are given by
Properties
The mode is μ, while the median is and the mean is given by
,
where is the Euler–Mascheroni constant.
The standard deviation is hence
At the mode, where , the value of becomes , irrespective of the value of
If are iid Gumbel random variables with parameters then is also a Gumbel random variable with parameters .
If are iid random variables such that has the same distribution as for all natural numbers , then is necessarily Gumbel distributed with scale parameter (actually it suffices to consider just two distinct values of k>1 which are coprime).
Related distributions
If has a Gumbel distribution, then the conditional distribution of Y = −X given that Y is positive, or equivalently given that X is |
https://en.wikipedia.org/wiki/Computational%20neuroscience | Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is a branch of neuroscience which employs mathematics, computer science, theoretical analysis and abstractions of the brain to understand the principles that govern the development, structure, physiology and cognitive abilities of the nervous system.
Computational neuroscience employs computational simulations to validate and solve mathematical models, and so can be seen as a sub-field of theoretical neuroscience; however, the two fields are often synonymous. The term mathematical neuroscience is also used sometimes, to stress the quantitative nature of the field.
Computational neuroscience focuses on the description of biologically plausible neurons (and neural systems) and their physiology and dynamics, and it is therefore not directly concerned with biologically unrealistic models used in connectionism, control theory, cybernetics, quantitative psychology, machine learning, artificial neural networks, artificial intelligence and computational learning theory;
although mutual inspiration exists and sometimes there is no strict limit between fields, with model abstraction in computational neuroscience depending on research scope and the granularity at which biological entities are analyzed.
Models in theoretical neuroscience are aimed at capturing the essential features of the biological system at multiple spatial-temporal scales, from membrane currents, and chemical coupling via network oscillations, columnar and topographic architecture, nuclei, all the way up to psychological faculties like memory, learning and behavior. These computational models frame hypotheses that can be directly tested by biological or psychological experiments.
History
The term 'computational neuroscience' was introduced by Eric L. Schwartz, who organized a conference, held in 1985 in Carmel, California, at the request of the Systems Development Foundation to provide a summary of the current status of a field which until that point was referred to by a variety of names, such as neural modeling, brain theory and neural networks. The proceedings of this definitional meeting were published in 1990 as the book Computational Neuroscience. The first of the annual open international meetings focused on Computational Neuroscience was organized by James M. Bower and John Miller in San Francisco, California in 1989. The first graduate educational program in computational neuroscience was organized as the Computational and Neural Systems Ph.D. program at the California Institute of Technology in 1985.
The early historical roots of the field can be traced to the work of people including Louis Lapicque, Hodgkin & Huxley, Hubel and Wiesel, and David Marr. Lapicque introduced the integrate and fire model of the neuron in a seminal article published in 1907, a model still popular for artificial neural networks studies because of its simplicity (see a recent review).
About 40 years |
https://en.wikipedia.org/wiki/Near%20field | Near field may refer to:
Near-field (mathematics), an algebraic structure
Near-field region, part of an electromagnetic field
Near field (electromagnetism)
Magnetoquasistatic field, the magnetic component of the electromagnetic near field
Near-field communication (NFC) using the magnetic component of the electromagnetic near field (magnetoquasistatic field)
See also
Near-field magnetic induction communication, a technique for deliberately limited-range communication between devices
Near-field communication (NFC), a set of application protocols based on this
Near-field optics
Near-field scanning optical microscope |
https://en.wikipedia.org/wiki/Weakly%20compact%20cardinal | In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.)
Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1.
The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
Equivalent formulations
The following are equivalent for any uncountable cardinal κ:
κ is weakly compact.
for every λ<κ, natural number n ≥ 2, and function f: [κ]n → λ, there is a set of cardinality κ that is homogeneous for f.
κ is inaccessible and has the tree property, that is, every tree of height κ has either a level of size κ or a branch of size κ.
Every linear order of cardinality κ has an ascending or a descending sequence of order type κ.
κ is -indescribable.
κ has the extension property. In other words, for all U ⊂ Vκ there exists a transitive set X with κ ∈ X, and a subset S ⊂ X, such that (Vκ, ∈, U) is an elementary substructure of (X, ∈, S). Here, U and S are regarded as unary predicates.
For every set S of cardinality κ of subsets of κ, there is a non-trivial κ-complete filter that decides S.
κ is κ-unfoldable.
κ is inaccessible and the infinitary language Lκ,κ satisfies the weak compactness theorem.
κ is inaccessible and the infinitary language Lκ,ω satisfies the weak compactness theorem.
κ is inaccessible and for every transitive set of cardinality κ with κ , , and satisfying a sufficiently large fragment of ZFC, there is an elementary embedding from to a transitive set of cardinality κ such that , with critical point κ.
A language Lκ,κ is said to satisfy the weak compactness theorem if whenever Σ is a set of sentences of cardinality at most κ and every subset with less than κ elements has a model, then Σ has a model. Strongly compact cardinals are defined in a similar way without the restriction on the cardinality of the set of sentences.
Properties
Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact cardinals are Mahlo cardinals, and the set of Mahlo cardinals less than a given weakly compact cardinal is stationary.
If is weakly compact, then there are chains of well-founded elementary end-extensions of of arbitrary length .p.6
Weakly compact cardinals remain weakly compact in . Assuming V = L, a cardinal is weakly compact iff it is 2-stationary.
See also
List of large cardinal properties
Refere |
https://en.wikipedia.org/wiki/Survey%20methodology | Survey methodology is "the study of survey methods".
As a field of applied statistics concentrating on human-research surveys, survey methodology studies the sampling of individual units from a population and associated techniques of survey data collection, such as questionnaire construction and methods for improving the number and accuracy of responses to surveys. Survey methodology targets instruments or procedures that ask one or more questions that may or may not be answered.
Researchers carry out statistical surveys with a view towards making statistical inferences about the population being studied; such inferences depend strongly on the survey questions used. Polls about public opinion, public-health surveys, market-research surveys, government surveys and censuses all exemplify quantitative research that uses survey methodology to answer questions about a population. Although censuses do not include a "sample", they do include other aspects of survey methodology, like questionnaires, interviewers, and non-response follow-up techniques. Surveys provide important information for all kinds of public-information and research fields, such as marketing research, psychology, health-care provision and sociology.
Overview
A single survey is made of at least a sample (or full population in the case of a census), a method of data collection (e.g., a questionnaire) and individual questions or items that become data that can be analyzed statistically. A single survey may focus on different types of topics such as preferences (e.g., for a presidential candidate), opinions (e.g., should abortion be legal?), behavior (smoking and alcohol use), or factual information (e.g., income), depending on its purpose. Since survey research is almost always based on a sample of the population, the success of the research is dependent on the representativeness of the sample with respect to a target population of interest to the researcher. That target population can range from the general population of a given country to specific groups of people within that country, to a membership list of a professional organization, or list of students enrolled in a school system (see also sampling (statistics) and survey sampling). The persons replying to a survey are called respondents, and depending on the questions asked their answers may represent themselves as individuals, their households, employers, or other organization they represent.
Survey methodology as a scientific field seeks to identify principles about the sample design, data collection instruments, statistical adjustment of data, and data processing, and final data analysis that can create systematic and random survey errors. Survey errors are sometimes analyzed in connection with survey cost. Cost constraints are sometimes framed as improving quality within cost constraints, or alternatively, reducing costs for a fixed level of quality. Survey methodology is both a scientific field and a profession, meanin |
https://en.wikipedia.org/wiki/Catenoid | In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler.
Soap film attached to twin circular rings will take the shape of a catenoid. Because they are members of the same associate family of surfaces, a catenoid can be bent into a portion of a helicoid, and vice versa.
Geometry
The catenoid was the first non-trivial minimal surface in 3-dimensional Euclidean space to be discovered apart from the plane. The catenoid is obtained by rotating a catenary about its directrix. It was found and proved to be minimal by Leonhard Euler in 1744.
Early work on the subject was published also by Jean Baptiste Meusnier. There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.
The catenoid may be defined by the following parametric equations:
where and and is a non-zero real constant.
In cylindrical coordinates:
where is a real constant.
A physical model of a catenoid can be formed by dipping two circular rings into a soap solution and slowly drawing the circles apart.
The catenoid may be also defined approximately by the stretched grid method as a facet 3D model.
Helicoid transformation
Because they are members of the same associate family of surfaces, one can bend a catenoid into a portion of a helicoid without stretching. In other words, one can make a (mostly) continuous and isometric deformation of a catenoid to a portion of the helicoid such that every member of the deformation family is minimal (having a mean curvature of zero). A parametrization of such a deformation is given by the system
for , with deformation parameter , where:
corresponds to a right-handed helicoid,
corresponds to a catenoid, and
corresponds to a left-handed helicoid.
References
Further reading
External links
Catenoid – WebGL model
Euler's text describing the catenoid at Carnegie Mellon University
Calculating the surface area of a Catenoid
Minimal Surface of Revolution
Geometry
Minimal surfaces
de:Minimalfläche#Das Katenoid |
https://en.wikipedia.org/wiki/Ultraproduct | The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal.
For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this.
Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.
Definition
The general method for getting ultraproducts uses an index set a structure (assumed to be non-empty in this article) for each element (all of the same signature), and an ultrafilter on
For any two elements and of the Cartesian product
declare them to be , written or if and only if the set of indices on which they agree is an element of in symbols,
which compares components only relative to the ultrafilter
This binary relation is an equivalence relation on the Cartesian product
The is the quotient set of with respect to and is therefore sometimes denoted by
or
Explicitly, if the -equivalence class of an element is denoted by
then the ultraproduct is the set of all -equivalence classes
Although was assumed to be an ultrafilter, the construction above can be carried out more generally whenever is merely a filter on in which case the resulting quotient set is called a .
When is a principal ultrafilter (which happens if and only if contains its kernel ) then the ultraproduct is isomorphic to one of the factors.
And so usually, is not a principal ultrafilter, which happens if and only if is free (meaning ), or equivalently, if every cofinite subsets of is an element of
Since every ultrafilter on a finite set is principal, the index set is consequently also usually infinite.
The ultraproduct acts as a filter product space where elements are equal if they are equal only at the filtered components (non-filtered components are ignored under the equivalence).
One may define a finitely additive measure on the index set by saying if and otherwise. Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set. The ultraproduct is the set of equivalence classes thus generated.
Finitary operations on the Cartesian product are defined pointwise (for example, if is a binary function then |
https://en.wikipedia.org/wiki/Survey | Survey may refer to:
Statistics and human research
Statistical survey, a method for collecting quantitative information about items in a population
Survey (human research), including opinion polls
Spatial measurement
Surveying, the technique and science of measuring positions and distances on Earth
Types and methods
Photogrammetry, a method of collecting information using aerial photography and satellite images
Cadastral surveyor, used to document land ownership, by the production of documents, diagrams, plats, and maps
Dominion Land Survey, the method used to divide most of Western Canada into one-square-mile sections for agricultural and other purposes
Public Land Survey System, a method used in the United States to survey and identify land parcels
Survey township, a square unit of land, six miles (~9.7 km) on a side, done by the U.S. Public Land Survey System
Construction surveying, the locating of structures relative to a reference line, used in the construction of buildings, roads, mines, and tunnels
Deviation survey, used in the oil industry to measure a borehole's departure from the vertical
Archaeological field survey, the collection of information by archaeologists prior to excavation
Geospatial survey organizations
Survey of India, India's central agency in charge of mapping and surveying
Ordnance Survey, a national mapping agency for Great Britain
U.S. National Geodetic Survey, performing geographic surveys as part of the U.S. Department of Commerce
United States Coast and Geodetic Survey, a former surveying agency of the United States Government
Geological surveys
Geological survey, an investigation of the subsurface of the ground to create a geological map or model
Types
Cave survey, the three-dimensional mapping of caverns
Geophysical survey, the systematic collection of geophysical data for spatial studies
Hydrographic survey, the gathering of information about navigable waters for the purposes of safe navigation of vessels
Soil survey, the mapping of the properties and varieties of soil in a given area
Geological survey organizations
British Geological Survey, a body which carries out geological surveys and monitors the UK landmass
United States Geological Survey, a government scientific research agency which studies the landscape of the United States
Astronomical surveys
Astronomical survey, imaging or mapping regions of the sky
Durchmusterung, a German word for a systematic survey of objects or data, generally used in astronomy
Redshift survey, an astronomical survey of a section of the sky to calculate the distance of objects from Earth
Other types of survey
Field survey or field research
Site survey, inspection of an area where work is proposed
Vessel safety survey, required for ships
Survey article, a scholarly publication to summarize an area of research
See also
Land survey (disambiguation)
Surveyor (disambiguation)
Survey says (disambiguation) |
https://en.wikipedia.org/wiki/Martingale%20%28probability%20theory%29 | In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all prior values.
History
Originally, martingale referred to a class of betting strategies that was popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins their stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double their bet after every loss so that the first win would recover all previous losses plus win a profit equal to the original stake. As the gambler's wealth and available time jointly approach infinity, their probability of eventually flipping heads approaches 1, which makes the martingale betting strategy seem like a sure thing. However, the exponential growth of the bets eventually bankrupts its users due to finite bankrolls. Stopped Brownian motion, which is a martingale process, can be used to model the trajectory of such games.
The concept of martingale in probability theory was introduced by Paul Lévy in 1934, though he did not name it. The term "martingale" was introduced later by , who also extended the definition to continuous martingales. Much of the original development of the theory was done by Joseph Leo Doob among others. Part of the motivation for that work was to show the impossibility of successful betting strategies in games of chance.
Definitions
A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) X1, X2, X3, ... that satisfies for any time n,
That is, the conditional expected value of the next observation, given all the past observations, is equal to the most recent observation.
Martingale sequences with respect to another sequence
More generally, a sequence Y1, Y2, Y3 ... is said to be a martingale with respect to another sequence X1, X2, X3 ... if for all n
Similarly, a continuous-time martingale with respect to the stochastic process Xt is a stochastic process Yt such that for all t
This expresses the property that the conditional expectation of an observation at time t, given all the observations up to time , is equal to the observation at time s (of course, provided that s ≤ t). The second property implies that is measurable with respect to .
General definition
In full generality, a stochastic process taking values in a Banach space with norm is a martingale with respect to a filtration and probability measure if
Σ∗ is a filtration of the underlying probability space (Ω, Σ, );
Y is adapted to the filtration Σ∗, i.e., for each t in the index set T, the random variable Yt is a Σt-measurable function;
for each t, Yt lies in the Lp space L1(Ω, Σt, ; S), i.e.
for all s and t with s < t and all F ∈ Σs,
where χF denotes the indicator function of the event F. In Grimmett and S |
https://en.wikipedia.org/wiki/Building%20%28mathematics%29 | In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Buildings were initially introduced by Jacques Tits as a means to understand the structure of exceptional groups of Lie type. The more specialized theory of Bruhat–Tits buildings (named also after François Bruhat) plays a role in the study of -adic Lie groups analogous to that of the theory of symmetric spaces in the theory of Lie groups.
Overview
The notion of a building was invented by Jacques Tits as a means of describing simple algebraic groups over an arbitrary field. Tits demonstrated how to every such group one can associate a simplicial complex with an action of , called the spherical building of . The group imposes very strong combinatorial regularity conditions on the complexes that can arise in this fashion. By treating these conditions as axioms for a class of simplicial complexes, Tits arrived at his first definition of a building. A part of the data defining a building is a Coxeter group , which determines a highly symmetrical simplicial complex , called the Coxeter complex. A building is glued together from multiple copies of , called its apartments, in a certain regular fashion. When is a finite Coxeter group, the Coxeter complex is a topological sphere, and the corresponding buildings are said to be of spherical type. When is an affine Weyl group, the Coxeter complex is a subdivision of the affine plane and one speaks of affine, or Euclidean, buildings. An affine building of type is the same as an infinite tree without terminal vertices.
Although the theory of semisimple algebraic groups provided the initial motivation for the notion of a building, not all buildings arise from a group. In particular, projective planes and generalized quadrangles form two classes of graphs studied in incidence geometry which satisfy the axioms of a building, but may not be connected with any group. This phenomenon turns out to be related to the low rank of the corresponding Coxeter system (namely, two). Tits proved a remarkable theorem: all spherical buildings of rank at least three are connected with a group; moreover, if a building of rank at least two is connected with a group then the group is essentially determined by the building.
Iwahori–Matsumoto, Borel–Tits and Bruhat–Tits demonstrated that in analogy with Tits' construction of spherical buildings, affine buildings can also be constructed from certain groups, namely, reductive algebraic groups over a local non-Archimedean field. Furthermore, if the split rank of the group is at least three, it is essentially determined by its building. Tits later reworked the foundational aspects of the theory of buildings using the notion of a chamber system, encoding the building solely in terms of adjacency properties of simplices of maximal dimensi |
https://en.wikipedia.org/wiki/Realm%20%28disambiguation%29 | A realm is the dominion of a king or queen; a kingdom. Realm may also more broadly refer to everything which falls within a certain set of parameters.
Realm may also refer to:
Maths and science
biogeographic realm, the largest scale bio-geographic division of the Earth's surface
A hyperplane in geometry
Domain (biology), the highest taxonomic rank of life, also called realm
Realm (virology), the highest taxonomic rank of viruses
Religion
Realm, an English translation for two terms in Buddhist cosmology:
Trailokya, or three realms
Six realms (gati)
Plane (esotericism)
Information technology
Realm (database), an object database and platform created primarily for mobile devices maintained by Realm Inc.
A URL pattern in OpenID protocol, for which the OpenID authentication is valid
An ID for an instance of a server software- HTTP transmits Realm when answering to a Basic access authentication request to distinguish different areas on server
A scope of operation in networking or in security — as in Active Directory realms
Arts and media
Realms (album), by Cindy Wilson, 2023
Realm (magazine), a British coffee-table magazine
Realm Media, formerly Serial Box, an American audio entertainment company
"Realms", a song by Hawkwind from Space Bandits
The Realm (film), a 2018 Spanish thriller film
Games
Realm (World of Warcraft), a server cluster for playing World of Warcraft
Realm (video game), a 1996 platform shooter
Realms (video game), a 1991 real-time strategy game by Graftgold
The Realm Online, one of the first massively multiplayer online role-playing games
Other uses
Commonwealth realm, one of the states of the Commonwealth of Nations that recognize Queen Elizabeth II as monarch
German Reich, or German Realm, the nation state of the German people
The Realm (company), a surfing products and clothing company
See also
Realmz, a 1994 fantasy adventure
Relm (disambiguation) |
https://en.wikipedia.org/wiki/Ethical%20calculus | An ethical calculus is the application of mathematics to calculate issues in ethics.
Scope
Generally, ethical calculus refers to any method of determining a course of action in a circumstance that is not explicitly evaluated in one's ethical code.
A formal philosophy of ethical calculus is a development in the study of ethics, combining elements of natural selection, self-organizing systems, emergence, and algorithm theory. According to ethical calculus, the most ethical course of action in a situation is an absolute, but rather than being based on a static ethical code, the ethical code itself is a function of circumstances. The optimal ethic is the best possible course of action taken by an individual with the given limitations.
While ethical calculus is, in some ways, similar to moral relativism, the former finds its grounds in the circumstance while the latter depends on social and cultural context for moral judgment. Ethical calculus would most accurately be regarded as a form of dynamic moral absolutism.
Ethical calculus is not to be confused with ethics in mathematics or ethics of quantification which study the moral questions coming from mathematical practice and quantification in society.
Examples
Francis Hutcheson devoted a section of his 1725 work Inquiry into the Original of our ideas and Beauty and Virtue to "an attempt to introduce a Mathematical Calculation in subjects of Morality". Formulas included:
M = B × A
where,
M is the moral importance of any agent
B is the benevolence of the agent
A is the ability of the agent
Another example is the felicific calculus formulated by utilitarian philosopher Jeremy Bentham for calculating the degree or amount of pleasure that a specific action is likely to cause. Bentham, an ethical hedonist, believed the moral rightness or wrongness of an action to be a function of the amount of pleasure or pain that it produced. The felicific calculus could, in principle at least, determine the moral status of any considered act.
See also
Ethics
Felicific calculus
Formal ethics
Moral absolutism
Morality
Science of morality
References
Ethical principles
Utilitarianism |
https://en.wikipedia.org/wiki/Precedence | Precedence may refer to:
Message precedence of military communications traffic
Order of precedence, the ceremonial hierarchy within a nation or state
Precedence (mathematics) for defining the order of operations in a computation
Precedence Entertainment, a defunct American game publisher
Precedence (solitaire), a solitaire card game which uses two decks of playing cards
Precedence, a brand of SPECT/CT scanner manufactured by Philips
See also
Precedent, a legal case establishing a principle to be adhered to in subsequent rulings |
https://en.wikipedia.org/wiki/Deconvolution | In mathematics, deconvolution is the operation inverse to convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. Due to the measurement error of the recorded signal or image, it can be demonstrated that the worse the signal-to-noise ratio (SNR), the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.
The foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949). The book was based on work Wiener had done during World War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of weather forecasting and economics.
Description
In general, the objective of deconvolution is to find the solution f of a convolution equation of the form:
Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with a filter or distortion function g, before we recorded it. Usually, h is a distorted version of f and the shape of f can't be easily recognized by the eye or simpler time-domain operations. The function g represents the impulse response of an instrument or a driving force that was applied to a physical system. If we know g, or at least know the form of g, then we can perform deterministic deconvolution. However, if we do not know g in advance, then we need to estimate it. This can be done using methods of statistical estimation or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations.
There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters:
Raw deconvolution
When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing the Fourier transform of the recorded signal h and the system response function g, you get H and G, with G as the transfer function. Using to the Convolution theorem,
where F is the estimated Fourier transform of f. Finally, the inverse Fourier transform of the function F is taken to find the estimated deconvolved signal f. Note that G is at the denominator and could amplify elements of the error model if present.
Deconvolution with noise
In physical measurements, the situation is usually closer to
In this case ε is noise that has entered our recorded signal. If a noisy signal or image is assumed to be noiseless, the statistical estimate of g will be incorrect. In turn, the estimate of ƒ will also be inc |
https://en.wikipedia.org/wiki/Range%20of%20a%20function | In mathematics, the range of a function may refer to either of two closely related concepts:
The codomain of the function
The image of the function
Given two sets and , a binary relation between and is a (total) function (from to ) if for every in there is exactly one in such that relates to . The sets and are called domain and codomain of , respectively. The image of is then the subset of consisting of only those elements of such that there is at least one in with .
Terminology
As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the codomain. More modern books, if they use the word "range" at all, generally use it to mean what is now called the image. To avoid any confusion, a number of modern books don't use the word "range" at all.
Elaboration and example
Given a function
with domain , the range of , sometimes denoted or , may refer to the codomain or target set (i.e., the set into which all of the output of is constrained to fall), or to , the image of the domain of under (i.e., the subset of consisting of all actual outputs of ). The image of a function is always a subset of the codomain of the function.
As an example of the two different usages, consider the function as it is used in real analysis (that is, as a function that inputs a real number and outputs its square). In this case, its codomain is the set of real numbers , but its image is the set of non-negative real numbers , since is never negative if is real. For this function, if we use "range" to mean codomain, it refers to ; if we use "range" to mean image, it refers to .
In many cases, the image and the codomain can coincide. For example, consider the function , which inputs a real number and outputs its double. For this function, the codomain and the image are the same (both being the set of real numbers), so the word range is unambiguous.
See also
Bijection, injection and surjection
Essential range
Notes and references
Bibliography
Functions and mappings
Basic concepts in set theory |
https://en.wikipedia.org/wiki/Time-scale%20calculus | In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems. It has applications in any field that requires simultaneous modelling of discrete and continuous data. It gives a new definition of a derivative such that if one differentiates a function defined on the real numbers then the definition is equivalent to standard differentiation, but if one uses a function defined on the integers then it is equivalent to the forward difference operator.
History
Time-scale calculus was introduced in 1988 by the German mathematician Stefan Hilger. However, similar ideas have been used before and go back at least to the introduction of the Riemann–Stieltjes integral, which unifies sums and integrals.
Dynamic equations
Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations. The general idea is to prove a result for a dynamic equation where the domain of the unknown function is a so-called time scale (also known as a time-set), which may be an arbitrary closed subset of the reals. In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.
The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus. Dynamic equations on a time scale have a potential for applications such as in population dynamics. For example, they can model insect populations that evolve continuously while in season, die out in winter while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a non-overlapping population.
Formal definitions
A time scale (or measure chain) is a closed subset of the real line . The common notation for a general time scale is .
The two most commonly encountered examples of time scales are the real numbers and the discrete time scale .
A single point in a time scale is defined as:
Operations on time scales
The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point , respectively. Formally:
(forward shift/jump operator)
(backward shift/jump operator)
The graininess is the distance from a point to the closest point on the right and is given by:
For a right-dense , and .
For a left-dense ,
Classification of points
For any , is:
left dense if
right dense if
left scattered if
right scattered if
dense if both left dense and right dense
isolated if both left scatte |
https://en.wikipedia.org/wiki/Module%20%28mathematics%29 | In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.
Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication.
Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
Introduction and definition
Motivation
In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization. In commutative algebra, both ideals and quotient rings are modules, so that many arguments about ideals or quotient rings can be combined into a single argument about modules. In non-commutative algebra, the distinction between left ideals, ideals, and modules becomes more pronounced, though some ring-theoretic conditions can be expressed either about left ideals or left modules.
Much of the theory of modules consists of extending as many of the desirable properties of vector spaces as possible to the realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces; for instance, not all modules have a basis, and even those that do, free modules, need not have a unique rank if the underlying ring does not satisfy the invariant basis number condition, unlike vector spaces, which always have a (possibly infinite) basis whose cardinality is then unique. (These last two assertions require the axiom of choice in general, but not in the case of finite-dimensional spaces, or certain well-behaved infinite-dimensional spaces such as Lp spaces.)
Formal definition
Suppose that R is a ring, and 1 is its multiplicative identity.
A left R-module M consists of an abelian group and an operation such that for all r, s in R and x, y in M, we have
The operation · is called scalar multiplication. Often the symbol · is omitted, but in this article we use it and reserve juxtaposition for multiplication in R. One may write RM to emphasize that M is a left R-module. A right R-module MR is defined similarly in terms of an operation .
Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be unital.
An (R,S)-bimodule is an abelian group together with both a left scalar |
https://en.wikipedia.org/wiki/Vector%20bundle | In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space in such a way that these vector spaces fit together to form another space of the same kind as (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over .
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space such that for all in : in this case there is a copy of for each in and these copies fit together to form the vector bundle over . Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.
Definition and first consequences
A real vector bundle consists of:
topological spaces (base space) and (total space)
a continuous surjection (bundle projection)
for every in , the structure of a finite-dimensional real vector space on the fiber
where the following compatibility condition is satisfied: for every point in , there is an open neighborhood of , a natural number , and a homeomorphism
such that for all in ,
for all vectors in , and
the map is a linear isomorphism between the vector spaces and .
The open neighborhood together with the homeomorphism is called a local trivialization of the vector bundle. The local trivialization shows that locally the map "looks like" the projection of on .
Every fiber is a finite-dimensional real vector space and hence has a dimension . The local trivializations show that the function is locally constant, and is therefore constant on each connected component of . If is equal to a constant on all of , then is called the rank of the vector bundle, and is said to be a vector bundle of rank . Often the definition of a vector bundle includes that the rank is well defined, so that is constant. Vector bundles of rank 1 are called line bundles, while those of rank 2 are less common |
https://en.wikipedia.org/wiki/Fiber%20bundle | In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber.
In the trivial case, is just and the map is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.
Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping as projection) to is called a section of Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber .
History
In topology, the terms fiber (German: Faser) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1933, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space was given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.
The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau, Whitney, Norman Steenrod, Charles Ehresmann, Jean-Pierre Serre, and others.
Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.
Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, that is a fiber bundle whose fiber is a sphere of arbitrary dimension.
Formal definition
A fiber bundle is a structure where and are topological spaces and is a continuous surjection satisfying a local triviality condition |
https://en.wikipedia.org/wiki/Exponential%20map%20%28Riemannian%20geometry%29 | In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself. The (pseudo) Riemannian metric determines a canonical affine connection, and the exponential map of the (pseudo) Riemannian manifold is given by the exponential map of this connection.
Definition
Let be a differentiable manifold and a point of . An affine connection on allows one to define the notion of a straight line through the point .
Let be a tangent vector to the manifold at . Then there is a unique geodesic :[0,1] → satisfying with initial tangent vector . The corresponding exponential map is defined by . In general, the exponential map is only locally defined, that is, it only takes a small neighborhood of the origin at , to a neighborhood of in the manifold. This is because it relies on the theorem of existence and uniqueness for ordinary differential equations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
Properties
Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and goes in that direction, for a unit time. Since v corresponds to the velocity vector of the geodesic, the actual (Riemannian) distance traveled will be dependent on that. We can also reparametrize geodesics to be unit speed, so equivalently we can define expp(v) = β(|v|) where β is the unit-speed geodesic (geodesic parameterized by arc length) going in the direction of v. As we vary the tangent vector v we will get, when applying expp, different points on M which are within some distance from the base point p—this is perhaps one of the most concrete ways of demonstrating that the tangent space to a manifold is a kind of "linearization" of the manifold.
The Hopf–Rinow theorem asserts that it is possible to define the exponential map on the whole tangent space if and only if the manifold is complete as a metric space (which justifies the usual term geodesically complete for a manifold having an exponential map with this property). In particular, compact manifolds are geodesically complete. However even if expp is defined on the whole tangent space, it will in general not be a global diffeomorphism. However, its differential at the origin of the tangent space is the identity map and so, by the inverse function theorem we can find a neighborhood of the origin of TpM on which the exponential map is an embedding (i.e., the exponential map is a local diffeomorphism). The radius of the largest ball about the origin in TpM that can be mapped diffeomorphically via expp is called the injectivity radius of M at p. The cut locus of the exponential map is, roughly speaking, the set of all points where the exponential map fails to have a unique minimum.
An important property of the exponential map is the following lemma of Gauss ( |
https://en.wikipedia.org/wiki/Ricci%20curvature | In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.
The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe.
Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bilinear form . Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of the Laplacian in the analysis of functions; in this analogy, the Riemann curvature tensor, of which the Ricci curvature is a natural by-product, would correspond to the full matrix of second derivatives of a function. However, there are other ways to draw the same analogy.
In three-dimensional topology, the Ricci tensor contains all of the information which in higher dimensions is encoded by the more complicated Riemann curvature tensor. In part, this simplicity allows for the application of many geometric and analytic tools, which led to the solution of the Poincaré conjecture through the work of Richard S. Hamilton and Grigory Perelman.
In differential geometry, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. This is since lower bounds on the Ricci tensor can be successfully used in studying the length functional in Riemannian geometry, as first shown in 1941 via Myers's theorem.
One common source of the Ricci tensor is that it arises whenever one commutes the covariant derivative with the tensor Laplacian. This, for instance, explains its presence in the Bochner formula, which is used ubiquitously in Riemannian geometry. For example, this formula explains why the gradient estimates due to Shing-Tung Yau (and their developments such as the Cheng-Yau and Li-Yau inequalities) nearly always depend on a lower bound for the Ricci curvature.
In 2007, John Lott, Karl-Theodor Sturm, and Cedric Villani demonstrated decisively that lower bounds on Ricci curvature can be understood entirely in terms of the metric space structure of a Riemannian manifold, together with its volume form. This established a deep link between Ricci curvature and Wasserstein geometry and optimal transport, which is presently |
https://en.wikipedia.org/wiki/Minimal%20surface | In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum.
Definitions
Minimal surfaces can be defined in several equivalent ways in . The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the crossroads of several mathematical disciplines, especially differential geometry, calculus of variations, potential theory, complex analysis and mathematical physics.
Local least area definition: A surface is minimal if and only if every point p ∈ M has a neighbourhood, bounded by a simple closed curve, which has the least area among all surfaces having the same boundary.
This property is local: there might exist regions in a minimal surface, together with other surfaces of smaller area which have the same boundary. This property establishes a connection with soap films; a soap film deformed to have a wire frame as boundary will minimize area.
Variational definition: A surface is minimal if and only if it is a critical point of the area functional for all compactly supported variations.
This definition makes minimal surfaces a 2-dimensional analogue to geodesics, which are analogously defined as critical points of the length functional.
Mean curvature definition: A surface is minimal if and only if its mean curvature is equal to zero at all points.
A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures. Additionally, this makes minimal surfaces into the static solutions of mean curvature flow. By the Young–Laplace equation, the mean curvature of a soap film is proportional to the difference in pressure between the sides. If the soap film does not enclose a region, then this will make its mean curvature zero. By contrast, a spherical soap bubble encloses a region which has a different pressure from the exterior region, and as such does not have zero mean curvature.
Differential equation definition: A surface is minimal if and only if it can be locally expressed as the graph of a solution of
The partial differential equation in this definition was originally found in 1762 by Lagrange, and Jean Baptiste Meusnier discovered in 1776 that it implied a vanishing mean curvature.
Energy d |
https://en.wikipedia.org/wiki/Extreme%20value%20theorem | In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed interval , then must attain a maximum and a minimum, each at least once. That is, there exist numbers and in such that:
The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function on the closed interval is bounded on that interval; that is, there exist real numbers and such that:
This does not say that and are necessarily the maximum and minimum values of on the interval which is what the extreme value theorem stipulates must also be the case.
The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
History
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. Both proofs involved what is known today as the Bolzano–Weierstrass theorem.
Functions to which the theorem does not apply
The following examples show why the function domain must be closed and bounded in order for the theorem to apply. Each fails to attain a maximum on the given interval.
defined over is not bounded from above.
defined over is bounded but does not attain its least upper bound .
defined over is not bounded from above.
defined over is bounded but never attains its least upper bound .
Defining in the last two examples shows that both theorems require continuity on .
Generalization to metric and topological spaces
When moving from the real line to metric spaces and general topological spaces, the appropriate generalization of a closed bounded interval is a compact set. A set is said to be compact if it has the following property: from every collection of open sets such that , a finite subcollection can be chosen such that . This is usually stated in short as "every open cover of has a finite subcover". The Heine–Borel theorem asserts that a subset of the real line is compact if and only if it is both closed and bounded. Correspondingly, a metric space has the Heine–Borel property if every closed and bounded set is also compact.
The concept of a continuous function can likewise be generalized. Given topological spaces , a function is said to be continuous if for every open set , is also open. Given these definitions, continuous functions can be shown to preserve compactness:
Theorem. If are topological spaces, is a continuous function, and is compact, then is also compact.
In particular, if , then this theorem implies that is closed and bounded for any compact set , which in turn implies that at |
https://en.wikipedia.org/wiki/Uniqueness%20quantification | In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1". For example, the formal statement
may be read as "there is exactly one natural number such that ".
Proving uniqueness
The most common technique to prove the unique existence of a certain object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, and ) must be equal to each other (i.e. ).
For example, to show that the equation has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds:
To establish the uniqueness of the solution, one would then proceed by assuming that there are two solutions, namely and , satisfying . That is,
Then since equality is a transitive relation,
Subtracting 2 from both sides then yields
which completes the proof that 3 is the unique solution of .
In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.
An alternative way to prove uniqueness is to prove that there exists an object satisfying the condition, and then to prove that every object satisfying the condition must be equal to .
Reduction to ordinary existential and universal quantification
Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula to mean
which is logically equivalent to
An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is
Another equivalent definition, which has the advantage of brevity, is
Generalizations
The uniqueness quantification can be generalized into counting quantification (or numerical quantification). This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.
Uniqueness depends on a notion of equality. Loosening this to some coarser equivalence relation yields quantification of uniqueness up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in category theory are defined to be unique up to isomorphism.
The exclamation mark can be also used as a separate quantification symbol, so , where . E.g. it can be safely used in the replacement axiom, instead of .
See also
Essentiall |
https://en.wikipedia.org/wiki/Root%20system | In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory (such as singularity theory). Finally, root systems are important for their own sake, as in spectral graph theory.
Definitions and examples
As a first example, consider the six vectors in 2-dimensional Euclidean space, R2, as shown in the image at the right; call them roots. These vectors span the whole space. If you consider the line perpendicular to any root, say β, then the reflection of R2 in that line sends any other root, say α, to another root. Moreover, the root to which it is sent equals α + nβ, where n is an integer (in this case, n equals 1). These six vectors satisfy the following definition, and therefore they form a root system; this one is known as A2.
Definition
Let E be a finite-dimensional Euclidean vector space, with the standard Euclidean inner product denoted by . A root system in E is a finite set of non-zero vectors (called roots) that satisfy the following conditions:
The roots span E.
The only scalar multiples of a root that belong to are itself and .
For every root , the set is closed under reflection through the hyperplane perpendicular to .
(Integrality) If and are roots in , then the projection of onto the line through is an integer or half-integer multiple of .
An equivalent way of writing conditions 3 and 4 is as follows:
For any two roots , the set contains the element
For any two roots , the number is an integer.
Some authors only include conditions 1–3 in the definition of a root system. In this context, a root system that also satisfies the integrality condition is known as a crystallographic root system. Other authors omit condition 2; then they call root systems satisfying condition 2 reduced. In this article, all root systems are assumed to be reduced and crystallographic.
In view of property 3, the integrality condition is equivalent to stating that β and its reflection σα(β) differ by an integer multiple of α. Note that the operator
defined by property 4 is not an inner product. It is not necessarily symmetric and is linear only in the first argument.
The rank of a root system Φ is the dimension of E.
Two root systems may be combined by regarding the Euclidean spaces they span as mutually orthogonal subspaces of a common Euclidean space. A root system which does not arise from such a combination, su |
https://en.wikipedia.org/wiki/Weight%20%28representation%20theory%29 | In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space.
Motivation and general concept
Given a set S of matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists a basis of V consisting of simultaneous eigenvectors of all elements of S. Each of these common eigenvectors v ∈ V defines a linear functional on the subalgebra U of End(V) generated by the set of endomorphisms S; this functional is defined as the map which associates to each element of U its eigenvalue on the eigenvector v. This map is also multiplicative, and sends the identity to 1; thus it is an algebra homomorphism from U to the base field. This "generalized eigenvalue" is a prototype for the notion of a weight.
The notion is closely related to the idea of a multiplicative character in group theory, which is a homomorphism χ from a group G to the multiplicative group of a field F. Thus χ: G → F× satisfies χ(e) = 1 (where e is the identity element of G) and
for all g, h in G.
Indeed, if G acts on a vector space V over F, each simultaneous eigenspace for every element of G, if such exists, determines a multiplicative character on G: the eigenvalue on this common eigenspace of each element of the group.
The notion of multiplicative character can be extended to any algebra A over F, by replacing χ: G → F× by a linear map χ: A → F with:
for all a, b in A. If an algebra A acts on a vector space V over F to any simultaneous eigenspace, this corresponds an algebra homomorphism from A to F assigning to each element of A its eigenvalue.
If A is a Lie algebra (which is generally not an associative algebra), then instead of requiring multiplicativity of a character, one requires that it maps any Lie bracket to the corresponding commutator; but since F is commutative this simply means that this map must vanish on Lie brackets: χ([a,b])=0. A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ([x, y])=0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra [g,g] and hence descends to a weight on the abelian Lie algebra g/[g,g]. Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuti |
https://en.wikipedia.org/wiki/Computer%20mathematics | Computer mathematics may refer to:
Automated theorem proving, the proving of mathematical theorems by a computer program
Symbolic computation, the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects
Computational science, constructing numerical solutions and using computers to analyze and solve scientific and engineering problems
Theoretical computer science, collection of topics of computer science and mathematics that focuses on the more abstract and mathematical aspects of computing |
https://en.wikipedia.org/wiki/Mathematics%20and%20architecture | Mathematics and architecture are related, since, as with other arts, architects use mathematics for several reasons. Apart from the mathematics needed when engineering buildings, architects use geometry: to define the spatial form of a building; from the Pythagoreans of the sixth century BC onwards, to create forms considered harmonious, and thus to lay out buildings and their surroundings according to mathematical, aesthetic and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to minimise wind speeds around the bases of tall buildings.
In ancient Egypt, ancient Greece, India, and the Islamic world, buildings including pyramids, temples, mosques, palaces and mausoleums were laid out with specific proportions for religious reasons. In Islamic architecture, geometric shapes and geometric tiling patterns are used to decorate buildings, both inside and outside. Some Hindu temples have a fractal-like structure where parts resemble the whole, conveying a message about the infinite in Hindu cosmology. In Chinese architecture, the tulou of Fujian province are circular, communal defensive structures. In the twenty-first century, mathematical ornamentation is again being used to cover public buildings.
In Renaissance architecture, symmetry and proportion were deliberately emphasized by architects such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio, influenced by Vitruvius's De architectura from ancient Rome and the arithmetic of the Pythagoreans from ancient Greece.
At the end of the nineteenth century, Vladimir Shukhov in Russia and Antoni Gaudí in Barcelona pioneered the use of hyperboloid structures; in the Sagrada Família, Gaudí also incorporated hyperbolic paraboloids, tessellations, catenary arches, catenoids, helicoids, and ruled surfaces. In the twentieth century, styles such as modern architecture and Deconstructivism explored different geometries to achieve desired effects. Minimal surfaces have been exploited in tent-like roof coverings as at Denver International Airport, while Richard Buckminster Fuller pioneered the use of the strong thin-shell structures known as geodesic domes.
Connected fields
The architects Michael Ostwald and Kim Williams, considering the relationships between architecture and mathematics, note that the fields as commonly understood might seem to be only weakly connected, since architecture is a profession concerned with the practical matter of making buildings, while mathematics is the pure study of number and other abstract objects. But, they argue, the two are strongly connected, and have been since antiquity. In ancient Rome, Vitruvius described an architect as a man who knew enough of a range of other disciplines, primarily geometry, to enable him to oversee skilled artisans in all the other necessary areas, such as masons and carpenters. The same applied in the Middle Ages, where graduates learnt arith |
https://en.wikipedia.org/wiki/TZero | TZero may refer to:
AC Propulsion tzero, automobile
T Zero, a collection of stories by Italo Calvino
t0, a symbol used in mathematics referring to the starting point or the beginning of time within a system |
https://en.wikipedia.org/wiki/Truncated%20tetrahedron | In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron at one third of the original edge length.
A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called rectification. The rectification of a tetrahedron produces an octahedron.
A truncated tetrahedron is the Goldberg polyhedron containing triangular and hexagonal faces.
A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.
Area and volume
The area A and the volume V of a truncated tetrahedron of edge length a are:
Densest packing
The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = , as reported by two independent groups using Monte Carlo methods. Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.
Cartesian coordinates
Cartesian coordinates for the 12 vertices of a truncated tetrahedron centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
(+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
(−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
(−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
(+3,−1,−1), (+1,−3,−1), (+1,−1,−3)
Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of:
(0,0,1,2)
Orthogonal projection
Spherical tiling
The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Friauf polyhedron
A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d symmetry) is called a Friauf polyhedron in crystals such as complex metallic alloys. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree dihedral angle on a subset of 6-6 edges. It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2".
Uses
Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with |
https://en.wikipedia.org/wiki/Truncated%20octahedron | In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagons and 6 squares), 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a 6-zonohedron. It is also the Goldberg polyhedron GIV(1,1), containing square and hexagonal faces. Like the cube, it can tessellate (or "pack") 3-dimensional space, as a permutohedron.
The truncated octahedron was called the "mecon" by Buckminster Fuller.
Its dual polyhedron is the tetrakis hexahedron. If the original truncated octahedron has unit edge length, its dual tetrakis hexahedron has edge lengths and .
Construction
A truncated octahedron is constructed from a regular octahedron with side length 3a by the removal of six right square pyramids, one from each point. These pyramids have both base side length (a) and lateral side length (e) of a, to form equilateral triangles. The base area is then a2. Note that this shape is exactly similar to half an octahedron or Johnson solid J1.
From the properties of square pyramids, we can now find the slant height, s, and the height, h, of the pyramid:
The volume, V, of the pyramid is given by:
Because six pyramids are removed by truncation, there is a total lost volume of a3.
Orthogonal projections
The truncated octahedron has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B2 and A2 Coxeter planes.
Spherical tiling
The truncated octahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Coordinates
All permutations of (0, ±1, ±2) are Cartesian coordinates of the vertices of a truncated octahedron of edge length a = √2 centered at the origin. The vertices are thus also the corners of 12 rectangles whose long edges are parallel to the coordinate axes.
The edge vectors have Cartesian coordinates and permutations of these. The face normals (normalized cross products of edges that share a common vertex) of the 6 square faces are , and . The face normals of the 8 hexagonal faces are . The dot product between pairs of two face normals is the cosine of the dihedral angle between adjacent faces, either − or −. The dihedral angle is approximately 1.910633 radians (109.471° ) at edges shared by two hexagons or 2.186276 radians (125.263° ) at edges shared by a hexagon and a square.
Dissection
The truncated octahedron can be dissected into a central octahedron, surrounded by 8 triangular cupolae on each face, and 6 square pyramids above the vertices.
Removing the central octahedron and 2 or 4 triangular cupolae creates two Stewart toroids, with d |
https://en.wikipedia.org/wiki/Fredholm%20operator | In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator T : X → Y between two Banach spaces with finite-dimensional kernel and finite-dimensional (algebraic) cokernel , and with closed range . The last condition is actually redundant.
The index of a Fredholm operator is the integer
or in other words,
Properties
Intuitively, Fredholm operators are those operators that are invertible "if finite-dimensional effects are ignored." The formally correct statement follows. A bounded operator T : X → Y between Banach spaces X and Y is Fredholm if and only if it is invertible modulo compact operators, i.e., if there exists a bounded linear operator
such that
are compact operators on X and Y respectively.
If a Fredholm operator is modified slightly, it stays Fredholm and its index remains the same. Formally: The set of Fredholm operators from X to Y is open in the Banach space L(X, Y) of bounded linear operators, equipped with the operator norm, and the index is locally constant. More precisely, if T0 is Fredholm from X to Y, there exists ε > 0 such that every T in L(X, Y) with ||T − T0|| < ε is Fredholm, with the same index as that of T0.
When T is Fredholm from X to Y and U Fredholm from Y to Z, then the composition is Fredholm from X to Z and
When T is Fredholm, the transpose (or adjoint) operator is Fredholm from to , and . When X and Y are Hilbert spaces, the same conclusion holds for the Hermitian adjoint T∗.
When T is Fredholm and K a compact operator, then T + K is Fredholm. The index of T remains unchanged under such a compact perturbations of T. This follows from the fact that the index i(s) of is an integer defined for every s in [0, 1], and i(s) is locally constant, hence i(1) = i(0).
Invariance by perturbation is true for larger classes than the class of compact operators. For example, when U is Fredholm and T a strictly singular operator, then T + U is Fredholm with the same index. The class of inessential operators, which properly contains the class of strictly singular operators, is the "perturbation class" for Fredholm operators. This means an operator is inessential if and only if T+U is Fredholm for every Fredholm operator .
Examples
Let be a Hilbert space with an orthonormal basis indexed by the non negative integers. The (right) shift operator S on H is defined by
This operator S is injective (actually, isometric) and has a closed range of codimension 1, hence S is Fredholm with . The powers , , are Fredholm with index . The adjoint S* is the left shift,
The left shift S* is Fredholm with index 1.
If H is the classical Hardy space on the unit circle T in the complex plane, then the shift operator with respect to the orthonormal basis of complex exponentials
is the multiplication operator Mφ with the function . More generally, let φ be |
https://en.wikipedia.org/wiki/Truncated%20cube | In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.
If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and 2 + .
Area and volume
The area A and the volume V of a truncated cube of edge length a are:
Orthogonal projections
The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B2 and A2 Coxeter planes.
Spherical tiling
The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated hexahedron centered at the origin with edge length 2ξ are all the permutations of
(±ξ, ±1, ±1),
where ξ = − 1.
The parameter ξ can be varied between ±1. A value of 1 produces a cube, 0 produces a cuboctahedron, and negative values produces self-intersecting octagrammic faces.
If the self-intersected portions of the octagrams are removed, leaving squares, and truncating the triangles into hexagons, truncated octahedra are produced, and the sequence ends with the central squares being reduced to a point, and creating an octahedron.
Dissection
The truncated cube can be dissected into a central cube, with six square cupolae around each of the cube's faces, and 8 regular tetrahedra in the corners. This dissection can also be seen within the runcic cubic honeycomb, with cube, tetrahedron, and rhombicuboctahedron cells.
This dissection can be used to create a Stewart toroid with all regular faces by removing two square cupolae and the central cube. This excavated cube has 16 triangles, 12 squares, and 4 octagons.
Vertex arrangement
It shares the vertex arrangement with three nonconvex uniform polyhedra:
Related polyhedra
The truncated cube is related to other polyhedra and tilings in symmetry.
The truncated cube is one of a family of uniform polyhedra related to the cube and regular octahedron.
Symmetry mutations
This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry, and a series of polyhedra and tilings n.8.8.
Alternated truncation
Truncating alternating vertices of the cube gives the chamfered tetrahedron, i.e. the edge truncation of the tetrahedron.
The truncated triangular trapezohedron is another polyhedron which can be formed from cube edge truncation.
Related polytopes
The truncated cube, is second in a sequence of truncated hypercubes:
Truncated cubical graph
In the mathematical field of graph theory, a truncated cubical graph is the graph of vertices and edges of the |
https://en.wikipedia.org/wiki/Confidence%20interval | In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated confidence level; the 95% confidence level is most common, but other levels, such as 90% or 99%, are sometimes used. The confidence level, degree of confidence or confidence coefficient represents the long-run proportion of CIs (at the given confidence level) that theoretically contain the true value of the parameter; this is tantamount to the nominal coverage probability. For example, out of all intervals computed at the 95% level, 95% of them should contain the parameter's true value.
Factors affecting the width of the CI include the sample size, the variability in the sample, and the confidence level. All else being the same, a larger sample produces a narrower confidence interval, greater variability in the sample produces a wider confidence interval, and a higher confidence level produces a wider confidence interval.
Definition
Let be a random sample from a probability distribution with statistical parameter , which is a quantity to be estimated, and , representing quantities that are not of immediate interest. A confidence interval for the parameter , with confidence level or coefficient , is an interval determined by random variables and with the property:
The number , whose typical value is close to but not greater than 1, is sometimes given in the form (or as a percentage ), where is a small positive number, often 0.05.
It is important for the bounds and to be specified in such a way that as long as is collected randomly, every time we compute a confidence interval, there is probability that it would contain , the true value of the parameter being estimated. This should hold true for any actual and .
Approximate confidence intervals
In many applications, confidence intervals that have exactly the required confidence level are hard to construct, but approximate intervals can be computed. The rule for constructing the interval may be accepted as providing a confidence interval at level if
to an acceptable level of approximation. Alternatively, some authors simply require that
which is useful if the probabilities are only partially identified or imprecise, and also when dealing with discrete distributions. Confidence limits of the form
and
are called conservative; accordingly, one speaks of conservative confidence intervals and, in general, regions.
Desired properties
When applying standard statistical procedures, there will often be standard ways of constructing confidence intervals. These will have been devised so as to meet certain desirable properties, which will hold given that the assumptions on which the procedure relies are true. These desirable properties may be described as: validity, optimality, and invariance.
Of the three, "validity" is most important, followed closely by "optimality". "Invariance" may be considered as a property of the |
https://en.wikipedia.org/wiki/Texas%20Academy%20of%20Mathematics%20and%20Science | The Texas Academy of Mathematics and Science (TAMS) is a two-year residential early entrance college program serving approximately 375 high school juniors and seniors at the University of North Texas. Students are admitted from every region of the state through a selective admissions process. TAMS is a member of the National Consortium for Specialized Secondary Schools of Mathematics, Science and Technology.
History
TAMS was established on June 23, 1987 by the 70th Texas Legislature, in order to provide high school students an opportunity to take advanced coursework in math, science, and engineering. It was designed as a residential program at the University of North Texas for high school-aged students gifted in mathematics and science.
The establishment of this innovative program is from national concern among educators about anticipated shortages of students who would be sufficiently well prepared in mathematical and scientific problem solving. Recognizing that American youth would need to compete in an increasingly technological society, several states including Texas opted to create alternative educational programs that would attract students to the fields of mathematics and science as well as offer bright, motivated young people an accelerated education in these areas of study. TAMS differs from other state-supported residential math and science schools in that the academy offers students the opportunity to complete two years of college concurrently with the last two years of high school.
The first TAMS class arrived on August 22, 1988. This graduating Class of 1990 included 65 students, colloquially known as "TAMSters." The academy has since grown and, in recent years, the graduating classes have been as large as 185 students.
Admissions
TAMS is required by "to identify exceptionally gifted and intelligent high school students at the junior and senior levels and offer them a challenging education to maximize their development".
Applicants are limited to Texas residents in high school. The admissions process for TAMS is holistic and modeled on those of most colleges. Applications are typically accepted between July and mid-May of a student's sophomore year of high school, though some freshmen apply. Multiple criteria are assessed, including middle and high school grades, the rigor of classes taken at school (particularly for mathematics classes), letters of recommendation from teachers and an academic counselor or principal, SAT scores from no later than the January of the applicant's sophomore year, and an essay. Selected applicants are then invited to interview at one of TAMS's "Interview Days." During Interview Day, applicants take algebra diagnostic tests, tour the campus, and attend group interviews. Out of more than 500 students who apply each year, only around 200 are ultimately accepted into the program.
Academics
Due to the selectivity of its admissions process, TAMS has traditionally had a strong academic culture among it |
https://en.wikipedia.org/wiki/Prism%20%28geometry%29 | In geometry, a prism is a polyhedron comprising an polygon base, a second base which is a translated copy (rigidly moved without rotation) of the first, and other faces, necessarily all parallelograms, joining corresponding sides of the two bases. All cross-sections parallel to the bases are translations of the bases. Prisms are named after their bases, e.g. a prism with a pentagonal base is called a pentagonal prism. Prisms are a subclass of prismatoids.
Like many basic geometric terms, the word prism () was first used in Euclid's Elements. Euclid defined the term in Book XI as “a solid figure contained by two opposite, equal and parallel planes, while the rest are parallelograms”. However, this definition has been criticized for not being specific enough in relation to the nature of the bases, which caused confusion among later geometry writers.
Oblique vs right
An oblique prism is a prism in which the joining edges and faces are not perpendicular to the base faces.
Example: a parallelepiped is an oblique prism whose base is a parallelogram, or equivalently a polyhedron with six parallelogram faces.
A right prism is a prism in which the joining edges and faces are perpendicular to the base faces. This applies if and only if all the joining faces are rectangular.
The dual of a right n-prism is a right n-bipyramid.
A right prism (with rectangular sides) with regular n-gon bases has Schläfli symbol { }×{n}. It approaches a cylinder as n approaches infinity.
Special cases
A right rectangular prism (with a rectangular base) is also called a cuboid, or informally a rectangular box. A right rectangular prism has Schläfli symbol { }×{ }×{ }.
A right square prism (with a square base) is also called a square cuboid, or informally a square box.
Note: some texts may apply the term rectangular prism or square prism to both a right rectangular-based prism and a right square-based prism.
Regular prism
A regular prism is a prism with regular bases.
Uniform prism
A uniform prism or semiregular prism is a right prism with regular bases and all edges of the same length.
Thus all the side faces of a uniform prism are squares.
Thus all the faces of a uniform prism are regular polygons. Also, such prisms are isogonal; thus they are uniform polyhedra. They form one of the two infinite series of semiregular polyhedra, the other series being formed by the antiprisms.
A uniform n-gonal prism has Schläfli symbol t{2,n}.
Volume
The volume of a prism is the product of the area of the base by the height, i.e. the distance between the two base faces (in the case of a non-right prism, note that this means the perpendicular distance).
The volume is therefore:
where B is the base area and h is the height.
The volume of a prism whose base is an n-sided regular polygon with side length s is therefore:
Surface area
The surface area of a right prism is:
where B is the area of the base, h the height, and P the base perimeter.
The surface area of a right p |
https://en.wikipedia.org/wiki/Algebraic%20data%20type | In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite type, i.e., a type formed by combining other types.
Two common classes of algebraic types are product types (i.e., tuples and records) and sum types (i.e., tagged or disjoint unions, coproduct types or variant types).
The values of a product type typically contain several values, called fields. All values of that type have the same combination of field types. The set of all possible values of a product type is the set-theoretic product, i.e., the Cartesian product, of the sets of all possible values of its field types.
The values of a sum type are typically grouped into several classes, called variants. A value of a variant type is usually created with a quasi-functional entity called a constructor. Each variant has its own constructor, which takes a specified number of arguments with specified types. The set of all possible values of a sum type is the set-theoretic sum, i.e., the disjoint union, of the sets of all possible values of its variants. Enumerated types are a special case of sum types in which the constructors take no arguments, as exactly one value is defined for each constructor.
Values of algebraic types are analyzed with pattern matching, which identifies a value by its constructor or field names and extracts the data it contains.
Algebraic data types were introduced in Hope, a small functional programming language developed in the 1970s at the University of Edinburgh.
Examples
One of the most common examples of an algebraic data type is the singly linked list. A list type is a sum type with two variants, Nil for an empty list and Cons x xs for the combination of a new element x with a list xs to create a new list. Here is an example of how a singly linked list would be declared in Haskell:
data List a = Nil | Cons a (List a)
or
data [] a = [] | a : [a]
Cons is an abbreviation of construct. Many languages have special syntax for lists defined in this way. For example, Haskell and ML use [] for Nil, : or :: for Cons, respectively, and square brackets for entire lists. So Cons 1 (Cons 2 (Cons 3 Nil)) would normally be written as 1:2:3:[] or [1,2,3] in Haskell, or as 1::2::3::[] or [1,2,3] in ML.
For a slightly more complex example, binary trees may be implemented in Haskell as follows:
data Tree = Empty
| Leaf Int
| Node Int Tree Tree
or
data BinaryTree a = BTNil
| BTNode a (BinaryTree a) (BinaryTree a)
Here, Empty represents an empty tree, Leaf represents a leaf node, and Node organizes the data into branches.
In most languages that support algebraic data types, it is possible to define parametric types. Examples are given later in this article.
Somewhat similar to a function, a data constructor is applied to arguments of an appropriate type, yielding an instance of the data type to which the type constructor belongs. For example, the data constructor Le |
https://en.wikipedia.org/wiki/Dummy%20variable%20%28statistics%29 | In regression analysis, a dummy variable (also known as indicator variable or just dummy) is one that takes a binary value (0 or 1) to indicate the absence or presence of some categorical effect that may be expected to shift the outcome. For example, if we were studying the relationship between biological sex and income, we could use a dummy variable to represent the sex of each individual in the study. The variable could take on a value of 1 for males and 0 for females (or vice versa). In machine learning this is known as one-hot encoding.
Dummy variables are commonly used in regression analysis to represent categorical variables that have more than two levels, such as education level or occupation. In this case, multiple dummy variables would be created to represent each level of the variable, and only one dummy variable would take on a value of 1 for each observation. Dummy variables are useful because they allow us to include categorical variables in our analysis, which would otherwise be difficult to include due to their non-numeric nature. They can also help us to control for confounding factors and improve the validity of our results.
As with any addition of variables to a model, the addition of dummy variables will increases the within-sample model fit (coefficient of determination), but at a cost of fewer degrees of freedom and loss of generality of the model (out of sample model fit). Too many dummy variables result in a model that does not provide any general conclusions.
Dummy variables are useful in various cases. For example, in econometric time series analysis, dummy variables may be used to indicate the occurrence of wars, or major strikes. It could thus be thought of as a Boolean, i.e., a truth value represented as the numerical value 0 or 1 (as is sometimes done in computer programming).
Dummy variables may be extended to more complex cases. For example, seasonal effects may be captured by creating dummy variables for each of the seasons: D1=1 if the observation is for summer, and equals zero otherwise; D2=1 if and only if autumn, otherwise equals zero; D3=1 if and only if winter, otherwise equals zero; and D4=1 if and only if spring, otherwise equals zero. In the panel data fixed effects estimator dummies are created for each of the units in cross-sectional data (e.g. firms or countries) or periods in a pooled time-series. However in such regressions either the constant term has to be removed, or one of the dummies removed making this the base category against which the others are assessed, for the following reason:
If dummy variables for all categories were included, their sum would equal 1 for all observations, which is identical to and hence perfectly correlated with the vector-of-ones variable whose coefficient is the constant term; if the vector-of-ones variable were also present, this would result in perfect multicollinearity, so that the matrix inversion in the estimation algorithm would be impossible. This is refer |
https://en.wikipedia.org/wiki/Unit%20square | In mathematics, a unit square is a square whose sides have length . Often, the unit square refers specifically to the square in the Cartesian plane with corners at the four points ), , , and .
Cartesian coordinates
In a Cartesian coordinate system with coordinates , a unit square is defined as a square consisting of the points where both and lie in a closed unit interval from to .
That is, a unit square is the Cartesian product , where denotes the closed unit interval.
Complex coordinates
The unit square can also be thought of as a subset of the complex plane, the topological space formed by the complex numbers.
In this view, the four corners of the unit square are at the four complex numbers , , , and .
Rational distance problem
It is not known whether any point in the plane is a rational distance from all four vertices of the unit square.
See also
Unit circle
Unit cube
Unit sphere
References
External links
1 (number)
Types of quadrilaterals
Squares in number theory |
https://en.wikipedia.org/wiki/Baker%E2%80%93Campbell%E2%80%93Hausdorff%20formula | In mathematics, the Baker–Campbell–Hausdorff formula is the solution for to the equation
for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in and and iterated commutators thereof. The first few terms of this series are:
where "" indicates terms involving higher commutators of and . If and are sufficiently small elements of the Lie algebra of a Lie group , the series is convergent. Meanwhile, every element sufficiently close to the identity in can be expressed as for a small in . Thus, we can say that near the identity the group multiplication in —written as —can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.
If and are sufficiently small matrices, then can be computed as the logarithm of , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that can be expressed as a series in repeated commutators of and .
Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.
History
The formula is named after Henry Frederick Baker, John Edward Campbell, and Felix Hausdorff who stated its qualitative form, i.e. that only commutators and commutators of commutators, ad infinitum, are needed to express the solution. An earlier statement of the form was adumbrated by Friedrich Schur in 1890 where a convergent power series is given, with terms recursively defined. This qualitative form is what is used in the most important applications, such as the relatively accessible proofs of the Lie correspondence and in quantum field theory. Following Schur, it was noted in print by Campbell (1897); elaborated by Henri Poincaré (1899) and Baker (1902); and systematized geometrically, and linked to the Jacobi identity by Hausdorff (1906). The first actual explicit formula, with all numerical coefficients, is due to Eugene Dynkin (1947). The history of the formula is described in detail in the article of Achilles and Bonfiglioli and in the book of Bonfiglioli and Fulci.
Explicit forms
For many purposes, it is only necessary to know that an expansion for in terms of iterated commutators of and exists; the exact coefficients are often irrelevant. (See, for example, the discussion of the relationship between Lie group and Lie algebra homomorphisms in Section 5.2 of Hall's book, where the precise coefficients play no role in the argument.) A remarkably direct existence proof was given by Martin Eichler, see also the "Existence results" section below.
In other cases, one may need detailed information about and it is therefore desirable to compute as explicitly as possible. |
https://en.wikipedia.org/wiki/Rhombicosidodecahedron | In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
It has 20 regular triangular faces, 30 square faces, 12 regular pentagonal faces, 60 vertices, and 120 edges.
Names
Johannes Kepler in Harmonices Mundi (1618) named this polyhedron a rhombicosidodecahedron, being short for truncated icosidodecahedral rhombus, with icosidodecahedral rhombus being his name for a rhombic triacontahedron.<ref>Harmonies Of The World by Johannes Kepler, Translated into English with an introduction and notes by E. J. Aiton, A. M. Duncan, "J. V. Field, 1997, (page 123)</ref> There are different truncations of a rhombic triacontahedron into a topological rhombicosidodecahedron: Prominently its rectification (left), the one that creates the uniform solid (center), and the rectification of the dual icosidodecahedron (right), which is the core of the dual compound.
Dimensions
For a rhombicosidodecahedron with edge length a, its surface area and volume are:
Geometric relations
If you expand an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, or do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron, with a square for each edge of either.
Alternatively, if you expand each of five cubes by moving the faces away from the origin the right amount and rotating each of the five 72° around so they are equidistant from each other, without changing the orientation or size of the faces, and patch the pentagonal and triangular holes in the result, you get a rhombicosidodecahedron. Therefore, it has the same number of squares as five cubes.
Two clusters of faces of the bilunabirotunda, the lunes (each lune featuring two triangles adjacent to opposite sides of one square), can be aligned with a congruent patch of faces on the rhombicosidodecahedron. If two bilunabirotundae are aligned this way on opposite sides of the rhombicosidodecahedron, then a cube can be put between the bilunabirotundae at the very center of the rhombicosidodecahedron.
The rhombicosidodecahedron shares the vertex arrangement with the small stellated truncated dodecahedron, and with the uniform compounds of six or twelve pentagrammic prisms.
The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.
Twelve of the 92 Johnson solids are derived from the rhombicosidodecahedron, four of them by rotation of one or more pentagonal cupolae: the gyrate, parabigyrate, metabigyrate, and trigyrate rhombicosidodecahedron. Eight more can be constr |
https://en.wikipedia.org/wiki/Parallel%20transport | In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.
The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, vice versa, parallel transport is the local realization of a connection.
As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose–Singer theorem makes explicit this relationship between the curvature and holonomy.
Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.
Parallel transport on a vector bundle
Let M be a smooth manifold. Let E→M be a vector bundle with covariant derivative ∇ and γ: I→M a smooth curve parameterized by an open interval I. A section of along γ is called parallel if
By example, if is a tangent space in a tangent bundle of a manifold, this expression means that, for every in the interval, tangent vectors in are "constant" (the derivative vanishes) when an infinitesimal displacement from in the direction of the tangent vector is done.
Suppose we are given an element e0 ∈ EP at P = γ(0) ∈ M, rather than a section. The parallel transport of e0 along γ is the extension of e0 to a parallel section X on γ.
More precisely, X is the unique part of E along γ such that
Note that in any given coordinate patch, (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–Lindelöf theorem guarantees the existence and uniqueness of the solution.
Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve:
from the vector space lying over γ(s) to that over γ(t). This isomorphism is known as the parallel transport map associated to the curve. The isomorphisms between fibers obtained in this way will, in gen |
https://en.wikipedia.org/wiki/Sectional%20curvature | In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.
The sectional curvature determines the curvature tensor completely.
Definition
Given a Riemannian manifold and two linearly independent tangent vectors at the same point, u and v, we can define
Here R is the Riemann curvature tensor, defined here by the convention Some sources use the opposite convention in which case K(u,v) must be defined with in the numerator instead of
Note that the linear independence of u and v forces the denominator in the above expression to be nonzero, so that K(u,v) is well-defined. In particular, if u and v are orthonormal, then the definition takes on the simple form
It is straightforward to check that if are linearly independent and span the same two-dimensional linear subspace of the tangent space as , then So one may consider the sectional curvature as a real-valued function whose input is a two-dimensional linear subspace of a tangent space.
Alternative definitions
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let be a two-dimensional plane in . Let for sufficiently small denote the image under the exponential map at of the unit circle in , and let denote the length of . Then it can be proven that
as , for some number . This number at is the sectional curvature of at .
Manifolds with constant sectional curvature
One says that a Riemannian manifold has "constant curvature " if for all two-dimensional linear subspaces and for all
The Schur lemma states that if (M,g) is a connected Riemannian manifold with dimension at least three, and if there is a function such that for all two-dimensional linear subspaces and for all then f must be constant and hence (M,g) has constant curvature.
A Riemannian manifold with constant sectional curvature is called a space form. If denotes the constant value of the sectional curvature, then the curvature tensor can be written as
for any
Since any Riemannian metric is parallel with respect to its Levi-Civita connection, this shows that the Riemann tensor of any constant-curvature space is also parallel. The Ricci tensor is then given by and the scalar curvature is In particular, any constant-curvature space is Einstein and has constant scalar curvature.
The model examples
Given a positive number define
to be the standard Riemannian structure
to be the sphere with given by the pullback of the stan |
https://en.wikipedia.org/wiki/Scalar%20curvature | In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor.
The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations. Furthermore, this scalar curvature is the Lagrangian density for the Einstein–Hilbert action, the Euler–Lagrange equations of which are the Einstein field equations in vacuum.
The geometry of Riemannian metrics with positive scalar curvature has been widely studied. On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques. Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature. In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.
Definition
Given a Riemannian metric , the scalar curvature S (commonly also R, or Sc) is defined as the trace of the Ricci curvature tensor with respect to the metric:
The scalar curvature cannot be computed directly from the Ricci curvature since the latter is a (0,2)-tensor field; the metric must be used to raise an index to obtain a (1,1)-tensor field in order to take the trace. In terms of local coordinates one can write, using the Einstein notation convention, that:
where are the components of the Ricci tensor in the coordinate basis, and where are the inverse metric components, i.e. the components of the inverse of the matrix of metric components . Based upon the Ricci curvature being a sum of sectional curvatures, it is possible to also express the scalar curvature as
where denotes the sectional curvature and is any orthonormal frame at . By similar reasoning, the scalar curvature is twice the trace of the curvature operator. Alternatively, given the coordinate-based definition of Ricci curvature in terms of the Christoffel symbols, it is possible to express scalar curvat |
https://en.wikipedia.org/wiki/Gaussian%20curvature | In differential geometry, the Gaussian curvature or Gauss curvature of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point:
The Gaussian radius of curvature is the reciprocal of .
For example, a sphere of radius has Gaussian curvature everywhere, and a flat plane and a cylinder have Gaussian curvature zero everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.
Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured “within” or along the surface, not on the way it is isometrically embedded in Euclidean space. This is the content of the Theorema egregium.
Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.
Informal definition
At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these , . The Gaussian curvature is the product of the two principal curvatures .
The sign of the Gaussian curvature can be used to characterise the surface.
If both principal curvatures are of the same sign: , then the Gaussian curvature is positive and the surface is said to have an elliptic point. At such points, the surface will be dome like, locally lying on one side of its tangent plane. All sectional curvatures will have the same sign.
If the principal curvatures have different signs: , then the Gaussian curvature is negative and the surface is said to have a hyperbolic or saddle point. At such points, the surface will be saddle shaped. Because one principal curvature is negative, one is positive, and the normal curvature varies continuously if you rotate a plane orthogonal to the surface around the normal to the surface in two directions, the normal curvatures will be zero giving the asymptotic curves for that point.
If one of the principal curvatures is zero: , the Gaussian curvature is zero and the surface is said to have a parabolic point.
Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.
Relation to geometries
When a surface has a constant zero Gaussian curvature, then it is a developable surface and the geometry of the surface is Euclidean geometry.
When a surface has a constant positive Gaussian curvature, then the geometry of the surface is spherical geometry. Spheres and patches of spheres have this geometry, but there exist other |
https://en.wikipedia.org/wiki/Spoke%E2%80%93hub%20distribution%20paradigm | The spoke–hub distribution paradigm is a form of transport topology optimization in which traffic planners organize routes as a series of "spokes" that connect outlying points to a central "hub". Simple forms of this distribution/connection model contrast with point-to-point transit systems, in which each point has a direct route to every other point, and which modeled the principal method of transporting passengers and freight until the 1970s. Delta Air Lines pioneered the spoke–hub distribution model in 1955, and the concept revolutionized the transportation logistics industry after Federal Express demonstrated its value in the early 1970s. In the late 1970s the telecommunications and information technology sector subsequently adopted this distribution topology, dubbing it the star network network topology.
"Hubbing" involves "the arrangement of a transportation network as a hub-and-spoke model".
Benefits
The hub-and-spoke model, as compared to the point-to-point model, requires fewer routes. For a network of n nodes, only routes are necessary to connect all nodes so the upper bound is , and the complexity is O(n). That compares favourably to the routes, or O(n2), which would be required to connect each node to every other node in a point-to-point network. For example, in a system with 10 destinations, the spoke–hub system requires only 9 routes to connect all destinations, and a true point-to-point system would require 45 routes. However distance traveled per route will necessarily be more than with a point-to-point system (except where the route happens to have no interchange). Therefore, efficiency may be reduced. Conversely, for a same number of aircraft, having fewer routes to fly means each route can be flown more frequently and with higher capacity because the demand for passengers can be resourced from more than just one city (assuming the passengers are willing to change, which will of itself incur its own costs).
Complicated operations, such as package sorting and accounting, can be carried out at the hub rather than at every node, and this leads to economies of scale. As a result of this, spokes are simpler to operate, and so new routes can easily be created.
Drawbacks
Because the model is centralised, day-to-day operations may be relatively inflexible, and changes at the hub, even in a single route, may have unexpected consequences throughout the network. It may be difficult or even impossible to handle occasional periods of high demand between two spokes. As a result of this, route scheduling is complicated for the network operator, since scarce resources must be used carefully to avoid starving the hub and careful traffic analysis and precise timing are required to keep the hub operating efficiently.
In addition, the hub constitutes a bottleneck or single point of failure in the network. The total cargo capacity of the network is limited by the hub's capacity. Delays at the hub (such as from bad weather conditions) can |
https://en.wikipedia.org/wiki/Limit%20of%20a%20function | Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1.
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input.
Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, when is applied to any input sufficiently close to , the output value is forced arbitrarily close to . On the other hand, if some inputs very close to are taken to outputs that stay a fixed distance apart, then we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the concept of limit: roughly, a function is continuous if all of its limits agree with the values of the function. The concept of limit also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
History
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see #(ε, δ)-definition of limit below) to define continuous functions. However, his work was not known during his lifetime.
In his 1821 book , Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of by saying that an infinitesimal change in necessarily produces an infinitesimal change in , while claims that he used a rigorous epsilon-delta definition in proofs. In 1861, Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today. He also introduced the notations and
The modern notation of placing the arrow below the limit symbol is due to Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.
Motivation
Imagine a person walking on a landscape represented by the graph . Their horizontal position is given by , much like the position given by a map of the land or by a global positioning system. Their altitude is given by the coordinate . Suppose they walk towards a position , as they get closer and closer to this point, they will notice that their altitude approaches a specific value . If asked about the altitude corresponding to , they would reply by saying .
What, then, does it mean to say, their altitude is approaching ? It means that their altitude gets nearer and nearer to —except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: they must get within ten meters of . They report b |
https://en.wikipedia.org/wiki/Limit%20of%20a%20sequence | As the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ."
In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers.
History
The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes.
Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of approximations to determine an area or a volume. Archimedes succeeded in summing what is now called a geometric series.
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment."
Pietro Mengoli anticipated the modern idea of limit of a sequence with his study of quasi-proportions in Geometriae speciosae elementa (1659). He used the term quasi-infinite for unbounded and quasi-null for vanishing.
Newton dealt with series in his works on Analysis with infinite series (written in 1669, circulated in manuscript, published in 1711), Method of fluxions and infinite series (written in 1671, published in English translation in 1736, Latin original published much later) and Tractatus de Quadratura Curvarum (written in 1693, published in 1704 as an Appendix to his Optiks). In the latter work, Newton considers the binomial expansion of , which he then linearizes by taking the limit as tends to .
In the 18th century, mathematicians such as Euler succeeded in summing some divergent series by stopping at the right moment; they did not much care whether a limit existed, as long as it could be calculated. At the end of the century, Lagrange in his Théorie des fonctions analytiques (1797) opined that the lack of rigour precluded further development in calculus. Gauss in his etude of hypergeometric series (1813) for the first time rigorously investigated the conditions under which a series converged to a limit.
The modern definition of a limit (for any there exists an index so that ...) was given by Bernard Bolzano (Der binomische Lehrsatz, Prague 1816, which was little noticed at the time), and by Karl Weierstrass in the 1870s.
Real numbers
In the real numbers, a number is the limit of the sequence , if the numbers in the sequence become closer and closer to , and not to any other number.
Examples
If for constant , then .
If |
https://en.wikipedia.org/wiki/Truncated%20cuboctahedron | In geometry, the truncated cuboctahedron or great rhombicuboctahedron is an Archimedean solid, named by Kepler as a truncation of a cuboctahedron. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices, and 72 edges. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated cuboctahedron is a 9-zonohedron. The truncated cuboctahedron can tessellate with the octagonal prism.
Names
There is a nonconvex uniform polyhedron with a similar name: the nonconvex great rhombicuboctahedron.
Cartesian coordinates
The Cartesian coordinates for the vertices of a truncated cuboctahedron having edge length 2 and centered at the origin are all the permutations of:
(±1, ±(1 + ), ±(1 + 2)).
Area and volume
The area A and the volume V of the truncated cuboctahedron of edge length a are:
Dissection
The truncated cuboctahedron is the convex hull of a rhombicuboctahedron with cubes above its 12 squares on 2-fold symmetry axes. The rest of its space can be dissected into 6 square cupolas below the octagons, and 8 triangular cupolas below the hexagons.
A dissected truncated cuboctahedron can create a genus 5, 7, or 11 Stewart toroid by removing the central rhombicuboctahedron, and either the 6 square cupolas, the 8 triangular cupolas, or the 12 cubes respectively. Many other lower symmetry toroids can also be constructed by removing the central rhombicuboctahedron, and a subset of the other dissection components. For example, removing 4 of the triangular cupolas creates a genus 3 toroid; if these cupolas are appropriately chosen, then this toroid has tetrahedral symmetry.
Uniform colorings
There is only one uniform coloring of the faces of this polyhedron, one color for each face type.
A 2-uniform coloring, with tetrahedral symmetry, exists with alternately colored hexagons.
Orthogonal projections
The truncated cuboctahedron has two special orthogonal projections in the A2 and B2 Coxeter planes with [6] and [8] projective symmetry, and numerous [2] symmetries can be constructed from various projected planes relative to the polyhedron elements.
Spherical tiling
The truncated cuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Full octahedral group
Like many other solids the truncated octahedron has full octahedral symmetry - but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of its dual is a fundamental domain of the group.
The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.
The edges of the solid correspond to the 9 |
https://en.wikipedia.org/wiki/Fractal%20dimension | In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured.
It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.
The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1). In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section Examples.
Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created. After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants."
One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake is infinite. No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively by thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is approximately 1.2619.
Introduction
A fractal dimension is an index for characterizing fractal patterns or sets by quantifying their complexity as a ratio of the change in detail to the change in scale. Several types of fractal dimension can be measured theoretically and empirically (see Fig. 2). Fractal dimensions are used to characterize a broad spectrum of objects ranging from the abstract to practical phenomena, including turbulence, river networks, urban growth, human physiology, medicine, and market trends. The essential idea of fractional or fractal dimensions has a long history in mathematics that can be traced back to the 1600s, but the terms fractal and fract |
https://en.wikipedia.org/wiki/Social%20statistics | Social statistics is the use of statistical measurement systems to study human behavior in a social environment. This can be accomplished through polling a group of people, evaluating a subset of data obtained about a group of people, or by observation and statistical analysis of a set of data that relates to people and their behaviors.
Statistics in the social sciences
History
Adolph Quetelet was a proponent of social physics. In his book Physique sociale he presents distributions of human heights, age of marriage, time of birth and death, time series of human marriages, births and deaths, a survival density for humans and curve describing fecundity as a function of age. He also developed the Quetelet Index.
Francis Ysidro Edgeworth published "On Methods of Ascertaining Variations in the Rate of Births, Deaths, and Marriages" in 1885 which uses squares of differences for studying fluctuations and George Udny Yule published "On the Correlation of total Pauperism with Proportion of Out-Relief" in 1895.
A numerical calibration for the fertility curve was given by Karl Pearson in 1897 in his "The Chances of Death, and Other Studies in Evolution" In this book Pearson also uses standard deviation, correlation and skewness for studying humans.
Vilfredo Pareto published his analysis of the distribution of income in Great Britain and Ireland in 1897, this is now known as the Pareto principle.
Louis Guttman proposed that the values of ordinal variables can be represented by a Guttman scale, which is useful if the number of variables is large and allows the use of techniques such as ordinary least squares.
Macroeconomic statistical research has provided stylized facts, which include:
Bowley's law (1937) regarding the proportion between wages and national output
The Phillips curve (1958) regarding the relation between wages and unemployment
Statistics and statistical analyses have become a key feature of social science: statistics is employed in economics, psychology, political science, sociology and anthropology.
Statistical methods in social sciences
Methods and concepts used in quantitative social sciences include:
Research design, survey methodology and survey sampling
Delphi method
Statistical techniques include:
Covariance based methods
Regression analysis
Canonical correlation
Causal analysis
Multilevel models
Factor analysis
Linear discriminant analysis
Path analysis
Structural Equation Modeling
Probability based methods
Probit and logit
Item response theory
Bayesian statistics
Stochastic process
Latent class model
Distance based methods
Cluster analysis
Multidimensional scaling
Methods for categorical data
Classification analysis
Cohort analysis
Usage and applications
Social scientists use social statistics for many purposes, including:
the evaluation of the quality of services available to a group or organization,
analyzing behaviors of groups of people in their environment and special situations,
d |
https://en.wikipedia.org/wiki/Modular%20form | In mathematics, a modular form is a (complex) analytic function on the upper half-plane, , that satisfies:
a kind of functional equation with respect to the group action of the modular group,
and a growth condition.
The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group .
The term "modular form", as a systematic description, is usually attributed to Hecke.
Each modular form is attached to a Galois representation.
Definition
In general, given a subgroup of finite index, called an arithmetic group, a modular form of level and weight is a holomorphic function from the upper half-plane such that two conditions are satisfied:
Automorphy condition: For any there is the equality
Growth condition: For any the function is bounded for
where and the function is identified with the matrix The identification of such functions with such matrices causes composition of such functions to correspond to matrix multiplication. In addition, it is called a cusp form if it satisfies the following growth condition:
Cuspidal condition: For any the function as
As sections of a line bundle
Modular forms can also be interpreted as sections of a specific line bundle on modular varieties. For a modular form of level and weight can be defined as an element ofwhere is a canonical line bundle on the modular curveThe dimensions of these spaces of modular forms can be computed using the Riemann–Roch theorem. The classical modular forms for are sections of a line bundle on the moduli stack of elliptic curves.
Modular function
A modular function is a function that is invariant with respect to the modular group, but without the condition that be holomorphic in the upper half-plane (among other requirements). Instead, modular functions are meromorphic: they are holomorphic on the complement of a set of isolated points, which are poles of the function.
Modular forms for SL(2, Z)
Standard definition
A modular form of weight for the modular group
is a complex-valued function on the upper half-plane satisfying the following three conditions:
is a holomorphic function on .
For any and any matrix in as above, we have:
is required to be bounded as .
Remarks:
The weight is typically a positive integer.
For odd , only the zero function can satisfy the second condition.
The third condition is also phrased by saying that is "holomorphic at the cusp", a terminology that is explained below. Explicitly, the condition means that there exist some such that , meaning is bounded above some horizontal lin |
https://en.wikipedia.org/wiki/Absorption%20law | In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.
Two binary operations, ¤ and ⁂, are said to be connected by the absorption law if:
a ¤ (a ⁂ b) = a ⁂ (a ¤ b) = a.
A set equipped with two commutative and associative binary operations ("join") and ("meet") that are connected by the absorption law is called a lattice; in this case, both operations are necessarily idempotent (i.e. a a = a and a a = a).
Examples of lattices include Heyting algebras and Boolean algebras, in particular sets of sets with union (∪) and intersection (∩) operators, and ordered sets with min and max operations.
In classical logic, and in particular Boolean algebra, the operations OR and AND, which are also denoted by and , satisfy the lattice axioms, including the absorption law. The same is true for intuitionistic logic.
The absorption law does not hold in many other algebraic structures, such as commutative rings, e.g. the field of real numbers, relevance logics, linear logics, and substructural logics. In the last case, there is no one-to-one correspondence between the free variables of the defining pair of identities.
See also
Absorption (logic)
References
Abstract algebra
Boolean algebra
Theorems in propositional logic
Lattice theory |
https://en.wikipedia.org/wiki/National%20Institute%20of%20Justice | The National Institute of Justice (NIJ) is the research, development and evaluation agency of the United States Department of Justice.
NIJ, along with the Bureau of Justice Statistics (BJS), Bureau of Justice Assistance (BJA), Office of Juvenile Justice and Delinquency Prevention (OJJDP), Office for Victims of Crime (OVC), and other program offices, comprise the Office of Justice Programs (OJP) branch of the Department of Justice.
History
The National Institute of Law Enforcement and Criminal Justice was established on October 21, 1968, under the Omnibus Crime Control and Safe Streets Act of 1968, as a component of the Law Enforcement Assistance Administration (LEAA). In 1978, it was renamed as the National Institute of Justice. Some functions of the LEAA were absorbed by NIJ on December 27, 1979, with passage of the Justice System Improvement Act of 1979. The act, which amended the Omnibus Crime Control and Safe Streets Act of 1968, also led to creation of the Bureau of Justice Statistics. In 1982, the LEAA was succeeded by the Office of Justice Assistance, Research, and Statistics (1982–1984) and then the Office of Justice Programs in 1984.
NIJ was notable among U.S. governmental research organizations because it is headed by a political appointee of the president rather than by a scientist or a member of the civil service. The Presidential Appointment Efficiency and Streamlining Act of 2011 removed the need for Senate confirmation of the NIJ director.
In 2010, the United States National Research Council released a report on reforming the NIJ, and identified issues with its independence, budget, and scientific mission. While it considered making the NIJ separate from its current department, Office of Justice Programs, it recommended retaining the NIJ within the OJP but giving it increased independence and authority through clear qualifications for its director, control over its budget, and a statutory advisory board. It also recommended that the NIJ: (1) a focus on research rather than forensic capacity building activities,(2) increase funding for programs for graduate researchers, (3) increase transparency, and (4) do periodic self-assessments.
Research areas
NIJ is focused on advancing technology for criminal justice application including law enforcement and corrections, forensics, and judicial processes, as well as criminology, criminal justice, and related social science research. Much of this research is facilitated by providing grants to academic institutions, non-profit research organizations, and other entities, as well as collaborating with state and local governments. Areas of social science research include violence against women, corrections, and crime prevention, as well as program evaluation.
Grants for technology development help facilitate research and development of technology and tools for criminal justice application, which is a need that the private sector is otherwise reluctant to meet. NIJ also supports developm |
https://en.wikipedia.org/wiki/255%20%28number%29 | 255 (two hundred [and] fifty-five) is the natural number following 254 and preceding 256.
In mathematics
Its factorization makes it a sphenic number. Since 255 = 28 – 1, it is a Mersenne number (though not a pernicious one), and the fourth such number not to be a prime number. It is a perfect totient number, the smallest such number to be neither a power of three nor thrice a prime.
Since 255 is the product of the first three Fermat primes, the regular 255-gon is constructible.
In base 10, it is a self number.
255 is a repdigit in base 2 (11111111), in base 4 (3333), and in base 16 (FF).
In computing
255 is a special number in some tasks having to do with computing. This is the maximum value representable by an eight-digit binary number, and therefore the maximum representable by an unsigned 8-bit byte (the most common size of byte, also called an octet), the smallest common variable size used in high level programming languages (bit being smaller, but rarely used for value storage). The range is 0 to 255, which is 256 total values.
For example, 255 is the maximum value of
components in the 24-bit RGB color model, since each color channel is allotted eight bits;
any dotted quad in an IPv4 address; and
the alpha blending scale in Delphi (255 being 100% visible and 0 being fully transparent).
The use of eight bits for storage in older video games has had the consequence of it appearing as a hard limit in many video games. For example, in the original The Legend of Zelda game, Link can carry a maximum of 255 rupees. It was often used for numbers where casual gameplay would not cause anyone to exceed the number. However, in most situations it is reachable given enough time. This can cause many other peculiarities to appear when the number wraps back to 0, such as the infamous "kill screen" seen after clearing level 255 of Pac-Man.
This number could be interpreted by a computer as −1 if a programmer is not careful about which 8-bit values are signed and unsigned, and the two's complement representation of −1 in a signed byte is equal to that of 255 in an unsigned byte.
References
Integers |
https://en.wikipedia.org/wiki/Arne%20Beurling | Arne Carl-August Beurling (3 February 1905 – 20 November 1986) was a Swedish mathematician and professor of mathematics at Uppsala University (1937–1954) and later at the Institute for Advanced Study in Princeton, New Jersey. Beurling worked extensively in harmonic analysis, complex analysis and potential theory. The "Beurling factorization" helped mathematical scientists to understand the Wold decomposition, and inspired further work on the invariant subspaces of linear operators and operator algebras, e.g. Håkan Hedenmalm's factorization theorem for Bergman spaces.
He is perhaps most famous for single-handedly decrypting an early version of the German cipher machine Siemens and Halske T52 in a matter of two weeks during 1940, using only pen and paper. This machine's cipher is generally considered to be more complicated than that of the more famous Enigma machine.
Early life
Beurling was born on 3 February 1905 in Gothenburg, Sweden and was the son of the landowner Konrad Beurling and baroness Elsa Raab. After graduating in 1924, he was enrolled at the Uppsala University where he received a Bachelor of Arts degree in 1926 and two years later a Licentiate of Philosophy degree.
Career
Early career
Beurling was assistant teacher at Uppsala University from 1931 to 1933. He received his doctorate in mathematics in 1933 for his dissertation Études sur un problème de majoration. Beurling was a docent of mathematics at Uppsala University from 1933 and then professor of mathematics from 1937 to 1954.
World War II
In the summer of 1940 he single-handedly deciphered and reverse-engineered an early version of the Siemens and Halske T52 also known as the Geheimfernschreiber ("secret teletypewriter") used by Nazi Germany in World War II for sending ciphered messages. The T52 was one of the so-called "Fish cyphers", that, using transposition, created nearly one quintillion (893,622,318,929,520,960) different variations. It took Beurling two weeks to solve the problem using pen and paper. Using Beurling's work, a device was created that enabled Sweden to decipher German teleprinter traffic passing through Sweden from Norway on a cable. In this way, Swedish authorities knew about Operation Barbarossa before it occurred. Since the Swedes would not reveal how this knowledge was attained, the Swedish warning was not treated as credible by Soviets.
This became the foundation for the Swedish National Defence Radio Establishment (FRA). The cypher in the Geheimfernschreiber is generally considered to be more complex than the cypher used in the Enigma machines.
Later life
He was visiting professor at Harvard University from 1948 to 1949. From 1954 he was professor at the Institute for Advanced Study in Princeton, New Jersey, United States, where he took over Albert Einstein's office.
He was the doctoral advisor of Lennart Carleson and Carl-Gustav Esseen.
Personal life
Arne Beurling was first married (1936–40) to Britta Östberg (born 1907), daughter of Henrik |
https://en.wikipedia.org/wiki/Truncated%20icosidodecahedron | In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
It has 62 faces: 30 squares, 20 regular hexagons, and 12 regular decagons. It has the most edges and vertices of all Platonic and Archimedean solids, though the snub dodecahedron has more faces. Of all vertex-transitive polyhedra, it occupies the largest percentage (89.80%) of the volume of a sphere in which it is inscribed, very narrowly beating the snub dodecahedron (89.63%) and small rhombicosidodecahedron (89.23%), and less narrowly beating the truncated icosahedron (86.74%); it also has by far the greatest volume (206.8 cubic units) when its edge length equals 1. Of all vertex-transitive polyhedra that are not prisms or antiprisms, it has the largest sum of angles (90 + 120 + 144 = 354 degrees) at each vertex; only a prism or antiprism with more than 60 sides would have a larger sum. Since each of its faces has point symmetry (equivalently, 180° rotational symmetry), the truncated icosidodecahedron is a 15-zonohedron.
Names
The name great rhombicosidodecahedron refers to the relationship with the (small) rhombicosidodecahedron (compare section Dissection).
There is a nonconvex uniform polyhedron with a similar name, the nonconvex great rhombicosidodecahedron.
Area and volume
The surface area A and the volume V of the truncated icosidodecahedron of edge length a are:
If a set of all 13 Archimedean solids were constructed with all edge lengths equal, the truncated icosidodecahedron would be the largest.
Cartesian coordinates
Cartesian coordinates for the vertices of a truncated icosidodecahedron with edge length 2φ − 2, centered at the origin, are all the even permutations of:
(±, ±, ±(3 + φ)),
(±, ±φ, ±(1 + 2φ)),
(±, ±φ2, ±(−1 + 3φ)),
(±(2φ − 1), ±2, ±(2 + φ)) and
(±φ, ±3, ±2φ),
where φ = is the golden ratio.
Dissection
The truncated icosidodecahedron is the convex hull of a rhombicosidodecahedron with cuboids above its 30 squares, whose height to base ratio is . The rest of its space can be dissected into nonuniform cupolas, namely 12 between inner pentagons and outer decagons and 20 between inner triangles and outer hexagons.
An alternative dissection also has a rhombicosidodecahedral core. It has 12 pentagonal rotundae between inner pentagons and outer decagons. The remaining part is a toroidal polyhedron.
Orthogonal projections
The truncated icosidodecahedron has seven special orthogonal projections, centered on a vertex, on three types of edges, and three types of faces: square, hexagonal and decagonal. The last two correspond to the A2 and H2 Coxeter planes.
Spherical tilings and Schlegel diagrams
The truncated icosidodecahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographi |
https://en.wikipedia.org/wiki/Snub%20dodecahedron | In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most of the 13 Archimedean solids): 12 are pentagons and the other 80 are equilateral triangles. It also has 150 edges, and 60 vertices.
It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. The union of both forms is a compound of two snub dodecahedra, and the convex hull of both forms is a truncated icosidodecahedron.
Kepler first named it in Latin as dodecahedron simum in 1619 in his Harmonices Mundi. H. S. M. Coxeter, noting it could be derived equally from either the dodecahedron or the icosahedron, called it snub icosidodecahedron, with a vertical extended Schläfli symbol and flat Schläfli symbol sr{5,3}.
Cartesian coordinates
Let ξ ≈ be the real zero of the cubic polynomial , where φ is the golden ratio. Let the point p be given by
.
Let the rotation matrices M1 and M2 be given by
and
M1 represents the rotation around the axis (0,1,φ) through an angle of counterclockwise, while M2 being a cyclic shift of (x,y,z) represents the rotation around the axis (1,1,1) through an angle of . Then the 60 vertices of the snub dodecahedron are the 60 images of point p under repeated multiplication by M1 and/or M2, iterated to convergence. (The matrices M1 and M2 generate the 60 rotation matrices corresponding to the 60 rotational symmetries of a regular icosahedron.) The coordinates of the vertices are integral linear combinations of 1, φ, ξ, φξ, ξ2 and φξ2. The edge length equals
Negating all coordinates gives the mirror image of this snub dodecahedron.
As a volume, the snub dodecahedron consists of 80 triangular and 12 pentagonal pyramids.
The volume V3 of one triangular pyramid is given by:
and the volume V5 of one pentagonal pyramid by:
The total volume is
The circumradius equals
The midradius equals ξ. This gives an interesting geometrical interpretation of the number ξ. The 20 "icosahedral" triangles of the snub dodecahedron described above are coplanar with the faces of a regular icosahedron. The midradius of this "circumscribed" icosahedron equals 1. This means that ξ is the ratio between the midradii of a snub dodecahedron and the icosahedron in which it is inscribed.
The triangle–triangle dihedral angle is given by
The triangle–pentagon dihedral angle is given by
Metric properties
For a snub dodecahedron whose edge length is 1, the surface area is
Its volume is
Alternatively, this volume may be written as
where,
Its circumradius is
Its midradius is
There are two inscribed spheres, one touching the triangular faces, and one, slightly smaller, touching the pentagonal faces. Their radii are, respectively:
and
The four positive real roots of the sextic equation in R2
are the circumradii of the snub dodecahedron (U29), great snub icosidodecahedron (U57) |
https://en.wikipedia.org/wiki/Number%20line | In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a point.
The integers are often shown as specially-marked points evenly spaced on the line. Although the image only shows the integers from –3 to 3, the line includes all real numbers, continuing forever in each direction, and also numbers that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.
In advanced mathematics, the number line can be called the real line or real number line, formally defined as the set of all real numbers. It is viewed as a geometric space, namely the real coordinate space of dimension one, or the Euclidean space of dimension one – the Euclidean line. It can also be thought of as a vector space (or affine space), a metric space, a topological space, a measure space, or a linear continuum.
Just like the set of real numbers, the real line is usually denoted by the symbol (or alternatively, , the letter "R" in blackboard bold). However, it is sometimes denoted or in order to emphasize its role as the first real space or first Euclidean space.
History
The first mention of the number line used for operation purposes is found in John Wallis's Treatise of algebra. In his treatise, Wallis describes addition and subtraction on a number line in terms of moving forward and backward, under the metaphor of a person walking.
An earlier depiction without mention to operations, though, is found in John Napier's A description of the admirable table of logarithmes, which shows values 1 through 12 lined up from left to right.
Contrary to popular belief, Rene Descartes's original La Géométrie does not feature a number line, defined as we use it today, though it does use a coordinate system. In particular, Descartes's work does not contain specific numbers mapped onto lines, only abstract quantities.
Drawing the number line
A number line is usually represented as being horizontal, but in a Cartesian coordinate plane the vertical axis (y-axis) is also a number line. According to one convention, positive numbers always lie on the right side of zero, negative numbers always lie on the left side of zero, and arrowheads on both ends of the line are meant to suggest that the line continues indefinitely in the positive and negative directions. Another convention uses only one arrowhead which indicates the direction in which numbers grow. The line continues indefinitely in the positive and negative directions according to the rules of geometry which define a line without endpoints as an infinite line, a line with one endpoint as a ray, and a line with two endpoints as a line segment.
Comparing numbers
If a particular number is farther to the right on the number line than is another number, then the first number is grea |
https://en.wikipedia.org/wiki/Inverse%20function%20theorem | In mathematics, specifically differential calculus, the inverse function theorem gives a sufficient condition for a function to be invertible in a neighborhood of a point in its domain: namely, that its derivative is continuous and non-zero at the point. The theorem also gives a formula for the derivative of the inverse function.
In multivariable calculus, this theorem can be generalized to any continuously differentiable, vector-valued function whose Jacobian determinant is nonzero at a point in its domain, giving a formula for the Jacobian matrix of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds, for differentiable functions between Banach spaces, and so forth.
The theorem was first established by Picard and Goursat using an iterative scheme: the basic idea is to prove a fixed point theorem using the contraction mapping theorem.
Statements
For functions of a single variable, the theorem states that if is a continuously differentiable function with nonzero derivative at the point ; then is injective (or bijective onto the image) in a neighborhood of , the inverse is continuously differentiable near , and the derivative of the inverse function at is the reciprocal of the derivative of at :
It can happen that a function may be injective near a point while . An example is . In fact, for such a function, the inverse cannot be differentiable at , since if were differentiable at , then, by the chain rule, , which implies . (The situation is different for holomorphic functions; see #Holomorphic inverse function theorem below.)
For functions of more than one variable, the theorem states that if is a continuously differentiable function from an open subset of into , and the derivative is invertible at a point (that is, the determinant of the Jacobian matrix of at is non-zero), then there exist neighborhoods of in and of such that and is bijective. Writing , this means that the system of equations has a unique solution for in terms of when . Note that the theorem does not say is bijective onto the image where is invertible but that it is locally bijective where is invertible.
Moreover, the theorem says that the inverse function is continuously differentiable, and its derivative at is the inverse map of ; i.e.,
In other words, if are the Jacobian matrices representing , this means:
The hard part of the theorem is the existence and differentiability of . Assuming this, the inverse derivative formula follows from the chain rule applied to . (Indeed, ) Since taking the inverse is infinitely differentiable, the formula for the derivative of the inverse shows that if is continuously times differentiable, with invertible derivative at the point , then the inverse is also continuously times differentiable. Here is a positive integer or .
There are two variants of the inverse function theorem. Given a continuously differenti |
https://en.wikipedia.org/wiki/Transcendental%20extension | In mathematics, a transcendental extension is a field extension such that there exists an element in the field that is transcendental over the field ; that is, an element that is not a root of any univariate polynomial with coefficients in . In other words, a transcendental extension is a field extension that is not algebraic. For example, are both transcendental extensions of
A transcendence basis of a field extension (or a transcendence basis of over ) is a maximal algebraically independent subset of over Transcendence bases share many properties with bases of vector spaces. In particular, all transcendence bases of a field extension have the same cardinality, called the transcendence degree of the extension. Thus, a field extension is a transcendental extension if and only if its transcendence degree is positive.
Transcendental extensions are widely used in algebraic geometry. For example, the dimension of an algebraic variety is the transcendence degree of its function field. Also, global function fields are transcendental extensions of degree one of a finite field, and play in number theory in positive characteristic a role that is very similar to the role of algebraic number fields in characteristic zero.
Transcendence basis
Zorn's lemma shows there exists a maximal linearly independent subset of a vector space (i.e., a basis). A similar argument with Zorn's lemma shows that, given a field extension L / K, there exists a maximal algebraically independent subset of L over K. It is then called a transcendence basis. By maximality, an algebraically independent subset S of L over K is a transcendence basis if and only if L is an algebraic extension of K(S), the field obtained by adjoining the elements of S to K.
The exchange lemma (a version for algebraically independent sets) implies that if S, S' are transcendence bases, then S and S' have the same cardinality. Then the common cardinality of transcendence bases is called the transcendence degree of L over K and is denoted as or . There is thus an analogy: a transcendence basis and transcendence degree, on the one hand, and a basis and dimension on the other hand. This analogy can be made more formal, by observing that linear independence in vector spaces and algebraic independence in field extensions both form examples of finitary matroids (pregeometries). Any finitary matroid has a basis, and all bases have the same cardinality.
If G is a generating set of L (i.e., L = K(G)), then a transcendence basis for L can be taken as a subset of G. In particular, the minimum cardinality of generating sets of L over K. Also, a finitely generated field extension admits a finite transcendence basis.
If no field K is specified, the transcendence degree of a field L is its degree relative to some fixed base field; for example, the prime field of the same characteristic, or K, if L is an algebraic function field over K.
The field extension L / K is purely transcendental if there is a subse |
https://en.wikipedia.org/wiki/Pseudo-Riemannian%20manifold | In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed.
Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space.
A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.
Introduction
Manifolds
In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an n-dimensional Euclidean space any point can be specified by n real numbers. These are called the coordinates of the point.
An n-dimensional differentiable manifold is a generalisation of n-dimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into n-dimensional Euclidean space.
See Manifold, Differentiable manifold, Coordinate patch for more details.
Tangent spaces and metric tensors
Associated with each point in an -dimensional differentiable manifold is a tangent space (denoted ). This is an -dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point .
A metric tensor is a non-degenerate, smooth, symmetric, bilinear map that assigns a real number to pairs of tangent vectors at each tangent space of the manifold. Denoting the metric tensor by we can express this as
The map is symmetric and bilinear so if are tangent vectors at a point to the manifold then we have
for any real number .
That is non-degenerate means there is no non-zero such that for all .
Metric signatures
Given a metric tensor g on an n-dimensional real manifold, the quadratic form associated with the metric tensor applied to each vector of any orthogonal basis produces n real values. By Sylvester's law of inertia, the number of each positive, negative and zero values produced in this manner are invariants of the metric tensor, independent of the choice of orthogonal basis. The signature of the metric tensor gives these numbers, shown in the same order. A non-degenerate metric tensor has and the signature may be denoted (p, q), where .
Definition
A pseudo-Riemannian manifold is a differentiable manifold equipped with an everywhere non-degenerate, smooth, symmetric metric tensor .
Such a metric is called a pseudo-Riemannian metric. Applied to a vector field, the resulting scalar field value at any point of the manifold can be positive, negative or zero.
The signature of a pseudo-Riemannian metric is , where both p and q are non-negative. The non-degeneracy condition together with continuity implies that p and q remain unchanged throughout the manifold (assu |
https://en.wikipedia.org/wiki/GPR | GPR may refer to:
Science and technology
Gaussian process regression, an interpolation method in statistics
General-purpose register of a microprocessor
G-protein coupled receptor
Ground-penetrating radar
Ground potential rise, in electrical engineering
Other
General practice residency, in dentistry in the United States
Georgia Public Radio, in Georgia, United States
Glider Pilot Regiment of the British Army
GPR index, a stock index of property companies
Grupa na rzecz Partii Robotniczej, the Polish section of the Committee for a Workers' International |
https://en.wikipedia.org/wiki/Algebraic%20independence | In abstract algebra, a subset of a field is algebraically independent over a subfield if the elements of do not satisfy any non-trivial polynomial equation with coefficients in .
In particular, a one element set is algebraically independent over if and only if is transcendental over . In general, all the elements of an algebraically independent set over are by necessity transcendental over , and over all of the field extensions over generated by the remaining elements of .
Example
The two real numbers and are each transcendental numbers: they are not the roots of any nontrivial polynomial whose coefficients are rational numbers. Thus, each of the two singleton sets and is algebraically independent over the field of rational numbers.
However, the set is not algebraically independent over the rational numbers, because the nontrivial polynomial
is zero when and .
Algebraic independence of known constants
Although both and e are known to be transcendental,
it is not known whether the set of both of them is algebraically independent over . In fact, it is not even known if is irrational.
Nesterenko proved in 1996 that:
the numbers , , and , where is the gamma function, are algebraically independent over .
the numbers and are algebraically independent over .
for all positive integers , the number is algebraically independent over .
Lindemann–Weierstrass theorem
The Lindemann–Weierstrass theorem can often be used to prove that some sets are algebraically independent over . It states that whenever are algebraic numbers that are linearly independent over , then are also algebraically independent over .
Algebraic matroids
Given a field extension which is not algebraic, Zorn's lemma can be used to show that there always exists a maximal algebraically independent subset of over . Further, all the maximal algebraically independent subsets have the same cardinality, known as the transcendence degree of the extension.
For every set of elements of , the algebraically independent subsets of satisfy the axioms that define the independent sets of a matroid. In this matroid, the rank of a set of elements is its transcendence degree, and the flat generated by a set of elements is the intersection of with the field . A matroid that can be generated in this way is called an algebraic matroid. No good characterization of algebraic matroids is known, but certain matroids are known to be non-algebraic; the smallest is the Vámos matroid.
Many finite matroids may be represented by a matrix over a field , in which the matroid elements correspond to matrix columns, and a set of elements is independent if the corresponding set of columns is linearly independent. Every matroid with a linear representation of this type may also be represented as an algebraic matroid, by choosing an indeterminate for each row of the matrix, and by using the matrix coefficients within each column to assign each matroid element a linear combination of thes |
https://en.wikipedia.org/wiki/Upper%20half-plane | In mathematics, the upper half-plane, , is the set of points in the Cartesian plane with
. The lower half-plane is defined similarly, by requiring that be negative instead. Each is an example of two-dimensional half-space.
Affine geometry
The affine transformations of the upper half-plane include
shifts , , and
dilations , .
Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes
to .
Proof: First shift the center of to . Then take
and dilate. Then shift the center of .
Definition: .
can be recognized as the circle of radius centered at , and as the polar plot of
.
Proposition: , , and are collinear points.
In fact, is the reflection of the line in the unit circle. Indeed, the diagonal from
to has squared length , so that
is the reciprocal of that length.
Metric geometry
The distance between any two points and in the upper half-plane can be consistently defined as follows: The perpendicular bisector of the segment from to either intersects the boundary or is parallel to it. In the latter case and lie on a ray perpendicular to the boundary and logarithmic measure can be used to define a distance that is invariant under dilation. In the former case and lie on a circle centered at the intersection of their perpendicular bisector and the boundary. By the above proposition this circle can be moved by affine motion to
. Distances on can be defined using the correspondence with points on and logarithmic measure on this ray. In consequence, the upper half-plane becomes a metric space. The generic name of this metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to the set of complex numbers with positive imaginary part:
The term arises from a common visualization of the complex number as the point in the plane endowed with Cartesian coordinates. When the axis is oriented vertically, the "upper half-plane" corresponds to the region above the axis and thus complex numbers for which .
It is the domain of many functions of interest in complex analysis, especially modular forms. The lower half-plane, defined by
is equally good, but less used by convention. The open unit disk (the set of all complex numbers of absolute value less than one) is equivalent by a conformal mapping to (see "Poincaré metric"), meaning that it is usually possible to pass between
and .
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.
The uniformization theorem for surfaces states that the upper half-plane is the universal covering space of surfaces with constant negative Ga |
https://en.wikipedia.org/wiki/Uniformization%20theorem | In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the plane to arbitrary simply connected Riemann surfaces.
Since every Riemann surface has a universal cover which is a simply connected Riemann surface, the uniformization theorem leads to a classification of Riemann surfaces into three types: those that have the Riemann sphere as universal cover ("elliptic"), those with the plane as universal cover ("parabolic") and those with the unit disk as universal cover ("hyperbolic"). It further follows that every Riemann surface admits a Riemannian metric of constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.
The uniformization theorem also yields a similar classification of closed orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant curvature, where the curvature can be taken to be 1 in the elliptic, 0 in the parabolic and -1 in the hyperbolic case.
History
Felix and Henri conjectured the uniformization theorem for (the Riemann surfaces of) algebraic curves. extended this to arbitrary multivalued analytic functions and gave informal arguments in its favor. The first rigorous proofs of the general uniformization theorem were given by and . Paul Koebe later gave several more proofs and generalizations. The history is described in ; a complete account of uniformization up to the 1907 papers of Koebe and Poincaré is given with detailed proofs in (the Bourbaki-type pseudonym of the group of fifteen mathematicians who jointly produced this publication).
Classification of connected Riemann surfaces
Every Riemann surface is the quotient of free, proper and holomorphic action of a discrete group on its universal covering and this universal covering, being a simply connected Riemann surface, is holomorphically isomorphic (one also says: "conformally equivalent" or "biholomorphic") to one of the following:
the Riemann sphere
the complex plane
the unit disk in the complex plane.
For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group ; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.
Classification of closed oriented Riemannian 2-manifolds
On an oriented 2-manifold, a Riemannian metric induces a complex structure using the passage to isothermal coordinates. If the |
https://en.wikipedia.org/wiki/Ricci%20flow | In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be analogous to the diffusion of heat and the heat equation, due to formal similarities in the mathematical structure of the equation. However, it is nonlinear and exhibits many phenomena not present in the study of the heat equation.
The Ricci flow, so named for the presence of the Ricci tensor in its definition, was introduced by Richard Hamilton, who used it through the 1980s to prove striking new results in Riemannian geometry. Later extensions of Hamilton's methods by various authors resulted in new applications to geometry, including the resolution of the differentiable sphere conjecture by Simon Brendle and Richard Schoen.
Following Shing-Tung Yau's suggestion that the singularities of solutions of the Ricci flow could identify the topological data predicted by William Thurston's geometrization conjecture, Hamilton produced a number of results in the 1990s which were directed towards the conjecture's resolution. In 2002 and 2003, Grigori Perelman presented a number of fundamental new results about the Ricci flow, including a novel variant of some technical aspects of Hamilton's program. Perelman's work is now widely regarded as forming the proof of the Thurston conjecture and the Poincaré conjecture, regarded as a special case of the former. It should be emphasized that the Poincare conjecture has been a well-known open problem in the field of geometric topology since 1904. These results by Hamilton and Perelman are considered as a milestone in the fields of geometry and topology.
Mathematical definition
On a smooth manifold , a smooth Riemannian metric automatically determines the Ricci tensor . For each element of , by definition is a positive-definite inner product on the tangent space at . If given a one-parameter family of Riemannian metrics , one may then consider the derivative , which then assigns to each particular value of and a symmetric bilinear form on . Since the Ricci tensor of a Riemannian metric also assigns to each a symmetric bilinear form on , the following definition is meaningful.
Given a smooth manifold and an open real interval , a Ricci flow assigns, to each in the interval , a Riemannian metric on such that .
The Ricci tensor is often thought of as an average value of the sectional curvatures, or as an algebraic trace of the Riemann curvature tensor. However, for the analysis of existence and uniqueness of Ricci flows, it is extremely significant that the Ricci tensor can be defined, in local coordinates, by a formula involving the first and second derivatives of the metric tensor. This makes the Ricci flow into a geometrically-defined partial differential equation. The analysis of the ellipticity of the local coordinate formula provides the foundation for the exi |
https://en.wikipedia.org/wiki/Global%20field | In mathematics, a global field is one of two types of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields:
Algebraic number field: A finite extension of
Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of , the field of rational functions in one variable over the finite field with elements.
An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.
Formal definitions
A global field is one of the following:
An algebraic number field
An algebraic number field F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q.
The function field of an algebraic curve over a finite field
A function field of a variety is the set of all rational functions on that variety. On an algebraic curve (i.e. a one-dimensional variety V) over a finite field, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.
Analogies between the two classes of fields
There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x:
The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for curves over finite fields settled by André Weil in 1940. The terminology may be due to Weil, who wrote his Basic Number Theory (1967) in part to work out the parallelism.
It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example. The analogy was also influential in the development of Iwasawa theory and the Main Conjecture. The proof of the fu |
https://en.wikipedia.org/wiki/Risch%20algorithm | In symbolic computation, the Risch algorithm is a method of indefinite integration used in some computer algebra systems to find antiderivatives. It is named after the American mathematician Robert Henry Risch, a specialist in computer algebra who developed it in 1968.
The algorithm transforms the problem of integration into a problem in algebra. It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral. However, the algorithm does not always succeed in identifying whether or not the antiderivative of a given function in fact can be expressed in terms of elementary functions.
The complete description of the Risch algorithm takes over 100 pages. The Risch–Norman algorithm is a simpler, faster, but less powerful variant that was developed in 1976 by Arthur Norman.
Some significant progress has been made in computing the logarithmic part of a mixed transcendental-algebraic integral by Brian L. Miller.
Description
The Risch algorithm is used to integrate elementary functions. These are functions obtained by composing exponentials, logarithms, radicals, trigonometric functions, and the four arithmetic operations (). Laplace solved this problem for the case of rational functions, as he showed that the indefinite integral of a rational function is a rational function and a finite number of constant multiples of logarithms of rational functions . The algorithm suggested by Laplace is usually described in calculus textbooks; as a computer program, it was finally implemented in the 1960s.
Liouville formulated the problem that is solved by the Risch algorithm. Liouville proved by analytical means that if there is an elementary solution to the equation then there exist constants and functions and in the field generated by such that the solution is of the form
Risch developed a method that allows one to consider only a finite set of functions of Liouville's form.
The intuition for the Risch algorithm comes from the behavior of the exponential and logarithm functions under differentiation. For the function , where and are differentiable functions, we have
so if were in the result of an indefinite integration, it should be expected to be inside the integral. Also, as
then if were in the result of an integration, then only a few powers of the logarithm should be expected.
Problem examples
Finding an elementary antiderivative is very sensitive to details. For instance, the following algebraic function (posted to sci.math.symbolic by Henri Cohen in 1993) has an elementary antiderivative, as Wolfram Mathematica since version 13 shows (however, Mathematica does not use the Risch algorithm to compute this integral):
namely:
But if the c |
https://en.wikipedia.org/wiki/Nilradical%20of%20a%20ring | In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements:
It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals.
In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this.
The nilradical of a Lie algebra is similarly defined for Lie algebras.
Commutative rings
The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element is nilpotent (by commutativity). It can also be characterized as the intersection of all the prime ideals of the ring (in fact, it is the intersection of all minimal prime ideals).
A ring is called reduced if it has no nonzero nilpotent. Thus, a ring is reduced if and only if its nilradical is zero. If R is an arbitrary commutative ring, then the quotient of it by the nilradical is a reduced ring and is denoted by .
Since every maximal ideal is a prime ideal, the Jacobson radical — which is the intersection of maximal ideals — must contain the nilradical. A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
Noncommutative rings
For noncommutative rings, there are several analogues of the nilradical. The lower nilradical (or Baer–McCoy radical, or prime radical) is the analogue of the radical of the zero ideal and is defined as the intersection of the prime ideals of the ring. The analogue of the set of all nilpotent elements is the upper nilradical and is defined as the ideal generated by all nil ideals of the ring, which is itself a nil ideal. The set of all nilpotent elements itself need not be an ideal (or even a subgroup), so the upper nilradical can be much smaller than this set. The Levitzki radical is in between and is defined as the largest locally nilpotent ideal. As in the commutative case, when the ring is Artinian, the Levitzki radical is nilpotent and so is the unique largest nilpotent ideal. Indeed, if the ring is merely Noetherian, then the lower, upper, and Levitzki radical are nilpotent and coincide, allowing the nilradical of any Noetherian ring to be defined as the unique largest (left, right, or two-sided) nilpotent ideal of the ring.
References
Eisenbud, David, "Commutative Algebra with a View Toward Algebraic Geometry", Graduate Texts in Mathematics, 150, Springer-Verlag |
https://en.wikipedia.org/wiki/Unit%20%28ring%20theory%29 | In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that
where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ).
Less commonly, the term unit is sometimes used to refer to the element of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.
Examples
The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit: if , then is a multiplicative inverse of .
In a nonzero ring, the element 0 is not a unit, so is not closed under addition.
A nonzero ring in which every nonzero element is a unit (that is, ) is called a division ring (or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of real numbers is .
Integer ring
In the ring of integers , the only units are and .
In the ring of integers modulo , the units are the congruence classes represented by integers coprime to . They constitute the multiplicative group of integers modulo .
Ring of integers of a number field
In the ring obtained by adjoining the quadratic integer to , one has , so is a unit, and so are its powers, so has infinitely many units.
More generally, for the ring of integers in a number field , Dirichlet's unit theorem states that is isomorphic to the group
where is the (finite, cyclic) group of roots of unity in and , the rank of the unit group, is
where are the number of real embeddings and the number of pairs of complex embeddings of , respectively.
This recovers the example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since .
Polynomials and power series
For a commutative ring , the units of the polynomial ring are the polynomials
such that is a unit in and the remaining coefficients are nilpotent, i.e., satisfy for some N.
In particular, if is a domain (or more generally reduced), then the units of are the units of .
The units of the power series ring are the power series
such that is a unit in .
Matrix rings
The unit group of the ring of matrices over a ring is the group of invertible matrices. For a commutative ring , an element of is invertible if and only if the determinant of is invertible in . In that case, can be given explicitly in terms of the adjugate matrix.
In general
For elements and in a ring , if is invertible, then is invert |
https://en.wikipedia.org/wiki/Cram%C3%A9r%27s%20conjecture | In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his heuristic actually supports the stronger statement
and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven.
Conditional proven results on prime gaps
Cramér gave a conditional proof of the much weaker statement that
on the assumption of the Riemann hypothesis. The best known unconditional bound is
due to Baker, Harman, and Pintz.
In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,
His result was improved by R. A. Rankin, who proved that
Paul Erdős conjectured that the left-hand side of the above formula is infinite, and this was proven in 2014 by Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao, and independently by James Maynard. The two sets of authors improved the result by a factor later that year.
Heuristic justification
Cramér's conjecture is based on a probabilistic model—essentially a heuristic—in which the probability that a number of size x is prime is 1/log x. This is known as the Cramér random model or Cramér model of the primes.
In the Cramér random model,
with probability one. However, as pointed out by Andrew Granville, Maier's theorem shows that the Cramér random model does not adequately describe the distribution of primes on short intervals, and a refinement of Cramér's model taking into account divisibility by small primes suggests that (), where is the Euler–Mascheroni constant. János Pintz has suggested that the limit sup may be infinite, and similarly Leonard Adleman and Kevin McCurley write
As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cramér's conjecture has been called into question [...] It is still probably true that for every constant , there is a constant such that there is a prime between and .
Similarly, Robin Visser writes
In fact, due to the work done by Granville, it is now widely believed that Cramér's conjecture is false. Indeed, there some theorems concerning short intervals between primes, such as Maier's theorem, which contradict Cramér's model.
(internal references removed).
Related conjectures and heuristics
Daniel Shanks conjectured the following asymptotic equality, stronger than Cramér's conjecture, for record gaps:
J.H. Cadwell has proposed the formula for the maximal gaps:
which is |
https://en.wikipedia.org/wiki/Ren%C3%A9%20Maurice%20Fr%C3%A9chet | René Maurice Fréchet (; 2 September 1878 – 4 June 1973) was a French mathematician. He made major contributions to general topology and was the first to define metric spaces. He also made several important contributions to the field of statistics and probability, as well as calculus. His dissertation opened the entire field of functionals on metric spaces and introduced the notion of compactness. Independently of Riesz, he discovered the representation theorem in the space of Lebesgue square integrable functions. He is often referred to as the founder of the theory of abstract spaces.
Biography
Early life
He was born to a Protestant family in Maligny to Jacques and Zoé Fréchet. At the time of his birth, his father was a director of a Protestant orphanage in Maligny and was later in his youth appointed a head of a Protestant school. However, the newly established Third Republic was not sympathetic to religious education and so laws were enacted requiring all education to be secular. As a result, his father lost his job. To generate some income his mother set up a boarding house for foreigners in Paris. His father was able later to obtain another teaching position within the secular system – it was not a job of a headship, however, and the family could not expect as high standards as they might have otherwise.
Maurice attended the secondary school Lycée Buffon in Paris where he was taught mathematics by Jacques Hadamard. Hadamard recognised the potential of young Maurice and decided to tutor him on an individual basis. After Hadamard moved to the University of Bordeaux in 1894, Hadamard continuously wrote to Fréchet, setting him mathematical problems and harshly criticising his errors. Much later Fréchet admitted that the problems caused him to live in a continual fear of not being able to solve some of them, even though he was very grateful for the special relationship with Hadamard he was privileged to enjoy.
After completing high-school Fréchet was required to enroll in military service. This is the time when he was deciding whether to study mathematics or physics – he chose mathematics out of dislike of the chemistry classes he would have had to take otherwise. Thus in 1900 he enrolled to École Normale Supérieure to study mathematics.
He started publishing quite early, having published four papers in 1903. He also published some of his early papers with the American Mathematical Society due to his contact with American mathematicians in Paris—particularly Edwin Wilson.
Middle life
Fréchet served at many different institutions during his academic career. From 1907 to 1908 he served as a professor of mathematics at the Lycée in Besançon, then moved in 1908 to the Lycée in Nantes to stay there for a year. After that he served at the University of Poitiers between 1910 and 1919.
He married in 1908 to Suzanne Carrive (1881–1945) and had four children: Hélène, Henri, Denise and Alain.
First World War
Fréchet was planning to spend a year in th |
https://en.wikipedia.org/wiki/EOF | EOF or Eof may refer to:
Science and technology
Electro-osmotic flow, the motion of liquid induced by an applied potential
Empirical orthogonal functions, in statistics and signal processing
Ethyl orthoformate, an organic compound
Computing
End-of-file, a condition where no more data can be read from a data source
Enterprise Objects Framework, a NeXT object-relational mapping product
EoF, a song editing program for the free Guitar Hero clone Frets on Fire
Ethereum Object Format, an object container standard specifying header, code, data, types
Other uses
Eof (about 701), a swineherd who claimed to have seen a vision of the Virgin Mary in England
End of Fashion, an Australian band
(EOF), an audiovisual work eligible for broadcasting and investment quotas in France |
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