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See searches and references containing FUNCTIONAL SQUARE-ROOT!FUNCTIONAL SQUARE-ROOT
Function that, applied twice, gives another function
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition
Functional_square_root
Number whose square is a given number
In mathematics, a square root of a number x is a number y such that y2=x; in other words, a number y whose square (the result of multiplying the number
Square_root
Functional square root of an exponential
In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f such that f composed with itself
Half-exponential_function
Operation on mathematical functions
distribution of a function of a random variable Functional decomposition Functional square root Functional equation Higher-order function Infinite compositions
Function_composition
Algorithms for calculating square roots
Square root algorithms compute the non-negative square root S of a positive real number S. Since all square roots of natural numbers, other than of perfect
Square_root_algorithms
Topics referred to by the same term
group active from 2003 to 2009 rin(), one representation of the functional square root of sin() Rin (album) (凛, "Dignified") 2017 Rin, an English language
Rin
Matrix B such that B² equals a given matrix A
mathematics, the square root of a matrix extends the notion of square root from numbers to matrices. A matrix B is said to be a square root of A if the matrix
Square_root_of_a_matrix
Branch of mathematical analysis
{\displaystyle {\sqrt {D}}=D^{\frac {1}{2}}} as an analogue of the functional square root for the differentiation operator, that is, an expression for some
Fractional_calculus
Result of repeatedly applying a mathematical function
equation Functional square root Abel equation Böttcher's equation Infinite compositions of analytic functions Flow (mathematics) Tetration Functional equation
Iterated_function
Branch of functional analysis
instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2. Using the functional calculus for larger classes of functions
Borel_functional_calculus
German mathematician
exponential; on the base of this Abel function, he constructed the functional square root of the exponential function as a half-iteration of the exponential
Hellmuth_Kneser
Arithmetic operation, inverse of nth power
number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree
Nth_root
Polygonal curve made from right triangles
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles
Spiral_of_Theodorus
Mathematical operation with only one operand
exist as well, for example, in the case of the square root, a horizontal bar extending the square root sign over the argument can indicate the extent
Unary_operation
Equation for fixed point of functional composition
been constructed; in effect, the entire orbit. For instance, the functional square root is h1/2(x) = Ψ−1(s1/2 Ψ(x)), so that h1/2(h1/2(x)) = h(x), and so
Schröder's_equation
Theory allowing one to apply mathematical functions to mathematical operators
a commutative algebra. For example, the Borel functional calculus can rigorously define the "square root" of the (negative) Laplacian operator − Δ {\displaystyle
Functional_calculus
Length of a line segment
The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the
Euclidean_distance
the problem in 1953. Kato asked whether the square roots of certain elliptic operators, defined via functional calculus, are analytic. The full statement
Kato's_conjecture
Algorithm for finding zeros of functions
Aitken's delta-squared process Bisection method Euler method Fast inverse square root Fisher scoring Gradient descent Integer square root Kantorovich theorem
Newton's_method
Figurate number
all other strategies". By analogy with the square root of x, one can define the (positive) triangular root of x as the number n such that Tn = x: n =
Triangular_number
Completing the square Abel–Ruffini theorem Bring radical Binomial theorem Blossom (functional) Root of a function nth root (radical) Surd Square root Methods
List_of_polynomial_topics
Arithmetic operation
have the form ωk=e2kπi/n with k coprime with n. The unique primitive square root of unity is −1; the primitive fourth roots of unity are i and −i. The
Exponentiation
Human tooth
mesial, while the occlusal table remains squared. Viewed from the buccal the buccal cusp is centered over the root as in the three cusp variety. Viewed from
Mandibular_second_premolar
Length in a vector space
the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the
Norm_(mathematics)
Statistical property
statistic is the standard deviation of its sampling distribution. It is the square root of the variance of an estimator of a parameter, as in the standard error
Standard_error
Association of one output to each input
the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of −1; while, when
Function_(mathematics)
Indicator for how well data points fit a line or curve
calculated as the square-root of the sum of squares of residuals (SSR): norm of residuals=SSres=‖e‖. Similarly, the reduced chi-square is calculated as
Coefficient_of_determination
Dimensionality reduction technique
operator R=ΣXX−1/2ΣXYΣYY−1/2. Since ΣXX and ΣYY are compact operators, the square root of the auto-covariance operator of ℒ2 processes may not be invertible
Functional_correlation
Mathematical inequality relating inner products and norms
number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality
Cauchy–Schwarz_inequality
Davenport–Schmidt theorem Irrational number Square root of two Quadratic irrational Integer square root Algebraic number Pisot–Vijayaraghavan number
List_of_number_theory_topics
Measure of variation in statistics
set or probability distribution is the square root of its variance (the variance being the average of the squared deviations from the mean). A useful property
Standard_deviation
Ancient algorithm for generating prime numbers
optimal trial division algorithm uses all prime numbers not exceeding its square root, whereas the sieve of Eratosthenes produces each composite from its prime
Sieve_of_Eratosthenes
Scientific theory
Properties of the Pauli potential for the square root of the electron density and the kinetic energy density functional". Physical Review A. 38: 625. doi:10
Orbital-free density functional theory
Orbital-free_density_functional_theory
Species of rodent
population can reach densities of up to 2,600 individuals per square kilometre. The giant root-rat has thick fur and large teeth, and creates large networks
Giant_root-rat
Approximation method in statistics
Least-squares spectral analysis Measurement uncertainty Orthogonal projection Proximal gradient methods for learning Quadratic loss function Root mean
Least_squares
Arithmetical calculations using only the human brain
specific types of mental calculation, such as addition, multiplication, square root or calendar reckoning. The first Mental Calculation World Cup took place
Mental_calculation
Number with a real and an imaginary part
nonzero real number, whether positive or negative, squares to a positive number, taking the square root of a negative number was historically considered
Complex_number
Study of the origin and evolution of words
consonants of the root word. For example: unhappy, happily, and unhappily are all derivatives of the root word happy. The terms root and derivative are
Etymology
Design pattern in functional programming to build generic types
In functional programming, monads are a way to structure computations as a sequence of steps, where each step produces a value, plus some extra information
Monad (functional programming)
Monad_(functional_programming)
partition Irrational number irrationality of log23 irrationality of the square root of 2 Mathematical induction sum identity Power rule differential of xn
List_of_mathematical_proofs
Physical law
distance by a factor of 1.4 (the square root of 2), and to double illumination, reduce the distance to 0.7 (square root of 1/2). When the illuminant is
Inverse-square_law
Function of a polynomial's coefficients that gives information about its roots
distinct non-real complex conjugates. The discriminant appears under the square root in the quadratic formula. More generally, the discriminant is a polynomial
Discriminant
Number of arguments required by a function
floor, ceiling, fractional part, sign, absolute value, square root (the principal square root), complex conjugate (unary of "one" complex number, that
Arity
MRI procedure that measures brain activity by detecting associated changes in blood flow
Functional magnetic resonance imaging or functional MRI (fMRI) measures brain activity by detecting changes associated with blood flow. This technique
Functional magnetic resonance imaging
Functional_magnetic_resonance_imaging
Akra–Bazzi method (asymptotic analysis) Bisection method (root finding) Brent's method (root finding) Condorcet method (voting systems) Coombs' method
List of mathematics-based methods
List_of_mathematics-based_methods
Topological group with compact topology
root system A n − 1 {\displaystyle A_{n-1}} The odd spin groups Spin ( 2 n + 1 ) {\displaystyle \operatorname {Spin} (2n+1)} correspond to the root
Compact_group
Repetition of a process
an example of an iterative method. Manual calculation of a number's square root is a common use and a well-known example. In computing, iteration is
Iteration
Office building in Chicago, Illinois
000 square feet (93,000 m2) of new office space and manage nearly one fifth of the office space in the Loop. Daniel Burnham and John Wellborn Root met
Monadnock_Building
Sum of elements on the main diagonal
multiple; in other words: If f {\displaystyle f} is a linear functional on the space of square matrices that satisfies f ( x y ) = f ( y x ) , {\displaystyle
Trace_(linear_algebra)
Type of mathematical expression
or "name". It was derived from the term binomial by replacing the Latin root bi- with the Greek poly-. That is, it means a sum of many terms (many monomials)
Polynomial
When a Gauss sum is the square root of a prime number, multiplied by a root of unity
is a root of unity, then the character must be quadratic. It is a result in number theory determining cases where a Gauss sum is the square root of a
Chowla–Mordell_theorem
Genus of flowering plants
perennial plant with a taproot, so the greens can be repeatedly harvested if the root remains in the ground. Raw dandelion greens contain high amounts of vitamins
Taraxacum
Metrics in magnetic resonance imaging
BOLD signal within a voxel or region of interest. ALFF measures the square root of the power spectrum within a predefined low-frequency band (commonly
Amplitude of low frequency fluctuations
Amplitude_of_low_frequency_fluctuations
Mathematical function, denoted exp(x) or e^x
operation Gaussian function Half-exponential function, a compositional square root of an exponential function Lambert W function#Solving equations – Multivalued
Exponential_function
Field of anatomy dedicated to human teeth
cells into the deeper tissues forms Hertwig's Epithelial Root Sheath, which determines the root shape of the tooth. The dental papilla contains cells that
Dental_anatomy
Number that can be used to count certain kinds of binary trees
OEIS), and where the constant given by the part of the expression in the square root is approximately 0.3188 (sequence A245651 in the OEIS). Young & Yung
Wedderburn–Etherington_number
Relation between sides of a right triangle
by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive
Pythagorean_theorem
L-functions satisfy a number of analytic properties, including an important functional equation. The setting is in the generality of a connected quasi-split
Langlands–Shahidi_method
Four-dimensional number system
The norm of a quaternion (the square root of the product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding
Quaternion
Type of vector space in math
fruitful era for functional analysis. Apart from the classical Euclidean vector spaces, examples of Hilbert spaces include spaces of square-integrable functions
Hilbert_space
Matrix of second derivatives
mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function,
Hessian_matrix
Statistical technique
Frobenius norm, the square root of the sum of the squares of all entries in a matrix and so equivalently the square root of the sum of squares of the lengths
Total_least_squares
Video-sharing platform
a new multiplayer feature for that service, supporting multiplayer functionality across desktop and mobile devices. As of December 2024[update] the Playables
YouTube
Purely functional data structure
science, a finger tree is a purely functional data structure that can be used to efficiently implement other functional data structures. A finger tree gives
Finger_tree
Every polynomial has a real or complex root
rely on −1 having a square root. If we take a field where −1 has no square root, and every polynomial of degree n ∈ I has a root, where I is any fixed
Fundamental theorem of algebra
Fundamental_theorem_of_algebra
Algorithm for analyzing noisy data streams
likely signal. In cases that are most computationally straightforward, root mean square deviation can be used as the decision criterion for the lowest error
Maximum likelihood sequence estimation
Maximum_likelihood_sequence_estimation
Extension to the x86 instruction set by AMD
square root approximation PFRCPIT1 – Packed floating-point reciprocal, first iteration step PFRSQIT1 – Packed floating-point reciprocal square root,
3DNow!
Electric current that periodically reverses direction
DC component) is assumed. The RMS voltage is the square root of the mean over one cycle of the square of the instantaneous voltage. For an arbitrary periodic
Alternating_current
Measure of the size of the ring of integers
field, and occurs in several important analytic formulas such as the functional equation of the Dedekind zeta function of K {\displaystyle K} , and the
Discriminant of an algebraic number field
Discriminant_of_an_algebraic_number_field
Arithmetic mean is greater than or equal to geometric mean
e. x = y. The AM–GM inequality then follows from taking the positive square root of both sides and then dividing both sides by 2. For a geometrical interpretation
AM–GM_inequality
Branch of mathematical analysis
hypercomplex numbers determines the function theory. Functions such as square root of a matrix, matrix exponential, and logarithm of a matrix are basic
Hypercomplex_analysis
Group of symmetries of an n-dimensional hypercube
four are reflections. The group operation is functional composition; for example, first reflecting the square across the horizontal axis, then reflecting
Hyperoctahedral_group
Infinite integer series where the next number is the sum of the two preceding it
{x+2x^{2}}{1-x-x^{2}}}} Φ ( x ) {\displaystyle \Phi (x)} satisfies the functional equation Φ ( x ) − Φ ( − 1 x ) = 2 {\displaystyle \Phi (x)-\Phi \!\left(-{\frac
Lucas_number
Mathematical concept
and so, invertible. The inverse function here is called the (positive) square root function and is denoted by x↦x. The following table shows several standard
Inverse_function
Distance from zero to a number
difference. See § Distance below. Since the square root symbol represents the unique positive square root, when applied to a positive number, it follows
Absolute_value
Quotient of leaf area to leaf dry mass
SI units of square centimeters per kilogram (cm²/kg). The inverse of SLA is leaf mass per area (LMA), with SI units of kilogram per square centimeters
Specific_leaf_area
Transformation in quantum mechanics
uses two. The advantage of this representation is that the complicated square root terms of the Holstein–Primakoff transformation are avoided. The Schwinger
Schwinger boson representation
Schwinger_boson_representation
MIPS microprocessor
four functional units, an adder, a multiplier, divide unit and square root unit. The adder and multiplier are pipelined, but the divide and square root units
R10000
Removal of limitations from iOS devices
Sealed System Volume (SSV) to grant root access (UID 0) and sandbox escapes without modifying the read-only root filesystem. This allows for the installation
IOS_jailbreaking
Polynomial whose nonzero terms all have the same degree
degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics
Homogeneous_polynomial
Integer side lengths of a right triangle
{\displaystyle (1,1,{\sqrt {2}})} is not a Pythagorean triple because the square root of 2 is not an integer. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle
Pythagorean_triple
Logic-based spreadsheet
formulas involving inequalities and non-invertible functions (such as square root). More generally, we could build spreadsheets with symbolic, rather than
Logical_spreadsheet
Measure of the "size" of linear operators
} then the matrix norm given to a matrix A {\displaystyle A} is the square root of the largest eigenvalue of the matrix A ∗ A {\displaystyle A^{*}A}
Operator_norm
Numbers that evenly divide powers of 60
Robson discuss the calculation of square roots, such as how the Babylonians found an approximation to the square root of 2, perhaps using regular number
Regular_number
{\displaystyle P(A\mid B)} : see "|". √ (square-root symbol) Denotes square root and is read as the square root of. Rarely used in modern mathematics without
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
1999 video game
need an original copy of the game to play it as intended. Fast inverse square root, sometimes referred to as Fast InvSqrt() or by the hexadecimal constant
Quake_III_Arena
Area of mathematics
YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem
Computational_mathematics
N-th root of the product of n numbers
defined as a1a2⋯antn. Hence, the geometric mean of two numbers is the square root of their product, for example with numbers 2 and 8 the geometric
Geometric_mean
Concepts from linear algebra
. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor λ {\displaystyle \lambda } (possibly a negative
Eigenvalues_and_eigenvectors
Variety and variability of life forms
genetic diversity and molecular diversity) functional diversity (which is a measure of the number of functionally disparate species within a population (e
Biodiversity
Dilation theorem
1} ), its defect operator DT is defined to be the (unique) positive square root DT = (I - T*T)½. In the special case that S is an isometry, DS* is a
Sz.-Nagy's_dilation_theorem
In mathematics, invariant of square matrices
scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square matrices and linear transformations
Determinant
Statistical measure of how far values spread from their average
the expected value of the squared deviation from the mean of a random variable. The standard deviation is the square root of the variance. Technically
Variance
Mathematical form
math.illinoisstate.edu. Retrieved 2020-08-16. Clarke, Francis (2013). Functional analysis, calculus of variations and optimal control. Dordrecht: Springer
Product_(mathematics)
City in West Yorkshire, England
Brittonic *Lātēnses (via Late Brittonic Lādēses), composed of the Celtic root *lāt- "violent, boiling" and the borrowed Latin plural derivational suffix
Leeds
Property of a mathematical matrix
{1}{2}},} hence it is also called the positive square root of M . {\displaystyle M.} The non-negative square root should not be confused with other decompositions
Definite_matrix
Integral of the Gaussian function, equal to sqrt(π)
{\displaystyle 2{\sqrt {n}}(\pi /2)(2n-3)!!/(2n-2)!!} [ambiguous] By taking the square root of the Wallis formula, π 2 = ∏ n = 1 ( 2 n ) 2 ( 2 n − 1 ) ( 2 n + 1
Gaussian_integral
2025 Android mobile operating system
expanded or collapsed. Additionally, the picker now includes search functionality, enabling users to search for specific photos and videos within their
Android_16
Mathematical functions of split-complex numbers
> |y| } = U1, is the range of the squaring function and the exponential. Thus it is the domain of the square root and logarithm functions. The other
Motor_variable
Performing order of mathematical operations
The radical symbol √ {\displaystyle \surd } , which signifies a square root, is traditionally extended by a bar (the vinculum) over the radicand;
Order_of_operations
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FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
Boy/Male
British, English
Spear-man
Boy/Male
American, Australian, British, English
Shield Bearer; Knight's Companion
Boy/Male
Anglo Saxon American English Scottish
Steward.
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
Surname or Lastname
English
English : variant of Spear.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English
English : patronymic from Squire.
Boy/Male
English American
Shieldbearer.
Male
Egyptian
, a great functionary.
Boy/Male
Italian
Squire.
Boy/Male
French Latin
A squire.
Boy/Male
English
Shieldbearer.
Male
Swedish
Swedish name derived from Old Norse stúra, STURE means "obstinate."
Boy/Male
American, British, English
Shield Bearer
Girl/Female
British, English
Bless
Male
Egyptian
, an Egyptian functionary.
Male
English
French form of English Stewart, STUART means "house guard; steward." In use by the English and Scottish.
Surname or Lastname
English
English : variant of Squire.
Surname or Lastname
English
English : nickname for a frugal person, from Middle English spare ‘sparing’, ‘frugal’.
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
FUNCTIONAL SQUARE-ROOT
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