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Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis
Positive-real_function
Function with a multiplicative scaling behaviour
In the case of functions of several real variables and real vector spaces, a slightly more general form of homogeneity called positive homogeneity is
Homogeneous_function
and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by
Positive_harmonic_function
Bimodal function
a positive-definite function is, depending on the context, either of two types of function. Let R {\displaystyle \mathbb {R} } be the set of real numbers
Positive-definite_function
Extension of the factorial function
positive integer n {\displaystyle n} . The gamma function can be defined via a convergent improper integral for complex numbers with positive real
Gamma_function
Mathematical function
mathematics, a function of a real variable is a function whose domain is a subset of R {\displaystyle \mathbb {R} } . Many real functions that are often
Function_of_a_real_variable
Real function with secant line between points above the graph itself
mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above or on
Convex_function
Subset of real numbers that are greater than zero
In mathematics, the set of positive real numbers, R > 0 = { x ∈ R ∣ x > 0 } , {\displaystyle \mathbb {R} _{>0}=\left\{x\in \mathbb {R} \mid x>0\right\}
Positive_real_numbers
Mathematical function that outputs real values
mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member
Real-valued_function
Number property of being positive or negative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered
Sign_(mathematics)
Property of a mathematical matrix
{\displaystyle M} with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf {T}}M\mathbf {x} } is positive for every nonzero
Definite_matrix
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Mathematical function with multiple real-number arguments
mathematics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables
Function of several real variables
Function_of_several_real_variables
Polynomial function of degree 3
of the function. The derivative of a cubic function is a quadratic function. A cubic function with real coefficients has either one or three real roots
Cubic_function
Method of mathematical integration
defined on a sub-domain of the real line with respect to the Lebesgue measure. The integral of a positive real function f between boundaries a and b can
Lebesgue_integral
Point where function's value is zero
mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle
Zero_of_a_function
Angle of complex number about real axis
anticlockwise argument with positive sign. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex
Argument_(complex_analysis)
Linear combination of indicator functions of real intervals
mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals
Step_function
Function returning minus 1, zero or plus 1
sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation the sign function is often represented
Sign_function
Topics referred to by the same term
pri, a notation for the scalar projection onto the i-th component Positive-real function, in mathematics Proportional representation, a property of some
PR
Types of special mathematical functions
the upper incomplete gamma function, as defined above for real positive s and x, can be developed into holomorphic functions, with respect both to x and
Incomplete_gamma_function
Uniform restraint of the change in functions
In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle
Uniform_continuity
Association of one output to each input
bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers
Function_(mathematics)
Electric circuit composed of resistors and capacitors
from a given rational function in s. For synthesis to be possible in passive elements, the function must be a positive-real function. To synthesise as an
RC_circuit
Mathematics of real numbers and real functions
Lebesgue integration, and function spaces. Real analysis is also known, especially in older books, as the theory of functions of a real variable, in contrast
Real_analysis
Order-preserving mathematical function
setting of order theory. In calculus, a function f {\displaystyle f} defined on a subset of the real numbers with real values is called monotonic if it is
Monotonic_function
Generalization of a positive-definite matrix
theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first
Positive-definite_kernel
Distance from zero to a number
necessarily positive ( | x | = − x > 0 {\displaystyle |x|=-x>0} ). From an analytic geometry point of view, the absolute value of a real number is that
Absolute_value
Theorem in quantum field theory
quantum field theory, the C-theorem states that there exists a positive real function, C ( g i , μ ) {\displaystyle C(g_{i}^{},\mu )} , depending on the
C-theorem
Mathematical function whose derivative exists
a real or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of
Differentiable_function
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Decomposition of real-valued functions
In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if
Positive_and_negative_parts
Sigmoid shape special function
applications, the function argument is a real number, in which case the function value is also real. In some older texts, the error function is defined without
Error_function
Real numbers with + and - infinity added
behavior is similar to the limit of a function lim x → x 0 f ( x ) {\textstyle \lim _{x\to x_{0}}f(x)} in which the real number x {\displaystyle x} approaches
Extended_real_number_line
Function with a repeating pattern
called a period of the function. If a period P {\displaystyle P} exists, any integer multiple n P {\displaystyle nP} (for a positive integer n {\displaystyle
Periodic_function
Arithmetic operation
except for negative real values of the radicand. This function equals the usual nth root for positive real radicands. For negative real radicands, and odd
Exponentiation
Smooth approximation of one-hot arg max
The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution
Softmax_function
Constructing a strictly convex compact surface with specified Gaussian curvature
specified. More precisely, the input to the problem is a strictly positive real function ƒ defined on a sphere, and the surface that is to be constructed
Minkowski_problem
Form of continuity for functions
In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion
Absolute_continuity
Mathematical function with convex lower level sets
mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the set of
Quasiconvex_function
Mathematical functions
that has a positive real part. This defines a single valued analytic function, which is defined everywhere, except for non-positive real values of the
Inverse_hyperbolic_functions
Mathematical function such that every output has at least one input
real domain X such that x2 = y. The natural logarithm function ln : (0, +∞) → R is a surjective and even bijective (mapping from the set of positive real
Surjective_function
Fundamental trigonometric functions
allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic
Sine_and_cosine
Distance from a point to the boundary of a set
The function has positive values at points x inside Ω, it decreases in value as x approaches the boundary of Ω where the signed distance function is zero
Signed_distance_function
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Product of a number by itself
a real function called the square function or the squaring function. Its domain is the whole real line, and its image is the set of nonnegative real numbers
Square_(algebra)
Piecewise function that clamps its input to be non-negative
The ramp function is a unary real function, whose graph is shaped like a ramp. It can be expressed by numerous definitions, for example "0 for negative
Ramp_function
Mathematical function
{2}{2k-1}}=-\gamma -2\ln 2+2H_{2n}-H_{n}.} If the real part of z is positive then the digamma function has the following integral representation due to
Digamma_function
Second-order partial differential equation
a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side
Laplace's_equation
Functions such that f(–x) equals f(x) or –f(x)
In mathematics, an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain
Even_and_odd_functions
Indicator function of rational numbers
}(x+T)=\mathbf {1} _{\mathbb {Q} }(x)} . The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods
Dirichlet_function
Special mathematical functions defined on the surface of a sphere
properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit
Spherical_harmonics
Smooth and compactly supported function
function. Start with any smooth function c : R → R {\displaystyle c:\mathbb {R} \to \mathbb {R} } that vanishes on the negative reals and is positive
Bump_function
Describes approximate behavior of a function
presentation of many analytic inequalities. For functions defined on positive real numbers or positive integers, a more restrictive and somewhat conflicting
Big_O_notation
Differentiable function whose derivative is not Riemann integrable
In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function V defined on the real line R with the following curious combination
Volterra's_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most
Sigmoid_function
Specific values of a multivalued function
there are two such values, the one with positive real part. A simple example is given by the square root function: every nonzero complex number has two
Principal_value
Mathematical theorem
In real analysis, a branch of mathematics, Bernstein's theorem, named after Sergei Bernstein, states that every real-valued function on the half-line
Bernstein's theorem on monotone functions
Bernstein's_theorem_on_monotone_functions
Linear map or polynomial function of degree one
When the function is of only one variable, it is of the form f ( x ) = a x + b , {\displaystyle f(x)=ax+b,} where a and b are constants, often real numbers
Linear_function
Polynomial function of degree two
quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic. The graph of a real single-variable quadratic function is
Quadratic_function
Logarithm to the base of the mathematical constant e
multi-valued function: see complex logarithm for more. The natural logarithm function, if considered as a real-valued function of a positive real variable
Natural_logarithm
Function that is continuous everywhere but differentiable nowhere
mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere
Weierstrass_function
Function that is holomorphic on the whole complex plane
an entire function. If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments
Entire_function
Complex-differentiable (mathematical) function
analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex
Holomorphic_function
Mathematical function, inverse of an exponential function
a function from the reals to the positive reals. Let b be a positive real number not equal to 1 and let f(x) = b x. It is a standard result in real analysis
Logarithm
Indicator function of positive numbers
one for positive arguments. Different conventions concerning the value H(0) are in use. It is an example of the general class of step functions, all of
Heaviside_step_function
Design technique for linear electrical circuits
extracted from the function leaving a remainder of another PRF called a minimum positive-real function, or just minimum function. For example, the minimum
Network_synthesis
Conjecture on zeros of the zeta function
problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics In
Riemann_hypothesis
Largest and smallest value taken by a function at a given point
set of real numbers, have no minimum or maximum. In statistics, the corresponding concept is the sample maximum and minimum. A real-valued function f defined
Maximum_and_minimum
Type of function in linear algebra
sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm, on a vector space is a real-valued function with
Sublinear_function
for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable t (often
List_of_Laplace_transforms
Geometric representation of the complex numbers
real axis, from −1 to 1, and obtain a sheet on which g(z) is a single-valued function. Alternatively, the cut can run from z = 1 along the positive real
Complex_plane
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and
Thomae's_function
Monochrome light beam whose amplitude envelope is a Gaussian function
{\mathsf {p}}\geq -|m|} is real-valued, Γ(x) is the gamma function and 1F1(a, b; x) is a confluent hypergeometric function. Some subfamilies of hypergeometric-Gaussian
Gaussian_beam
Mathematical function
rectangle, and the Jacobi elliptic functions will all be real valued on the real line. Since the Jacobi elliptic functions are doubly periodic in u {\displaystyle
Jacobi_elliptic_functions
Special function defined by an integral
logarithmic integral has an integral representation defined for all positive real numbers x ≠ 1 by the definite integral li ( x ) = ∫ 0 x d t ln t
Logarithmic_integral_function
Number whose square is a given number
principal square root function is thus defined using the non-positive real axis as a branch cut. If z {\displaystyle z} is a non-negative real number (which happens
Square_root
All numbers between two given numbers
by a positive or negative infinity symbol. The set of all positive real numbers is an interval in this sense, denoted (0, ∞); the set of all real numbers
Interval_(mathematics)
Theorem in mathematics
) {\displaystyle Z(s)} is a positive-real function (PRF) then R ( s ) {\displaystyle R(s)} is a PRF for all real, positive values of k {\displaystyle k}
Richards'_theorem
Analytic function in mathematics
large positive real numbers. In the following, N(T) is the total number of real zeros and N0(T) the total number of zeros of odd order of the function ζ(1/2
Riemann_zeta_function
Strong form of uniform continuity
the function is called a contraction. In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant
Lipschitz_continuity
Special function defined by an integral
for small values. For positive real values of the argument, E 1 {\displaystyle E_{1}} can be bracketed by elementary functions as follows: 1 2 e − x ln
Exponential_integral
Number with a real and an imaginary part
the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers
Complex_number
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Solution of a confluent hypergeometric equation
du.} thus M(a, a+b, it) is the characteristic function of the beta distribution. For a with positive real part U can be obtained by the Laplace integral
Confluent hypergeometric function
Confluent_hypergeometric_function
Real-valued mathematical function
In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change;
Rvachev_function
Concept in real analysis
discontinuous). Given a subset S ⊆ R {\displaystyle S\subseteq \mathbb {R} } , a real function f : S → R {\displaystyle f:S\to \mathbb {R} } is said to be continuously
Continuously differentiable function of a single real variable
Continuously_differentiable_function_of_a_single_real_variable
Class of mathematical expression
Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose
Division_by_zero
Mathematical functions which are smooth but not analytic
In real analysis, a smooth function is infinitely differentiable at each point in its domain, while a real analytic function is, at each point in its
Non-analytic_smooth_function
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Theorem in mathematics
complex-valued functions of a complex variable. It generalizes to functions from n-tuples (of real or complex numbers) to n-tuples, and to functions between
Inverse_function_theorem
Mathematical concept
an example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input
Inverse_function
Mathematical function
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form f ( x ) = exp ( − x 2 ) {\displaystyle f(x)=\exp(-x^{2})}
Gaussian_function
Mathematical function with no sudden changes
formed by all positive real numbers { x ∣ x > 0 } {\displaystyle \{x\mid x>0\}} . These rules imply that every polynomial function is continuous everywhere
Continuous_function
Electrical network theorem
rational one-port network from its polynomial function, a condition now known to be a positive-real function, and the reverse problem of which networks were
Foster's_reactance_theorem
S-shaped curve
x_{0}} is the x {\displaystyle x} value of the function's midpoint. The logistic function has domain the real numbers, the limit as x → − ∞ {\displaystyle
Logistic_function
Theorem
positive). One of the most important theorems of complex analysis is that holomorphic functions are analytic and vice versa. (A holomorphic function at
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Special mathematical function
polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent
Polylogarithm
Type of random mathematical object
{R} ^{d}} , this is achieved by introducing a locally integrable positive function λ : R d → [ 0 , ∞ ) {\displaystyle \lambda \colon \mathbb {R} ^{d}\to
Poisson_point_process
Hungarian-American mathematician (1923-2005)
studied existence of electronic filters corresponding to given positive-real functions. In 1949 they proved a fundamental theorem of filter synthesis
Raoul_Bott
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
Surname or Lastname
English
English : variant of Dale (from the Old Kentish form del) or a habitational name from Deal in Kent, named with this word.Americanized spelling of German Diel or Diehl.Dutch (de Ruyter) : variant spelling (17th century) of De Ruiter
Boy/Male
Hindu, Indian
Positive
Female
English
English name derived from the vocabulary word, TEAL means "blue-green" or "teal duck."
Male
English
English surname transferred to forename use, derived from an Old English byname, Red, READ means "red-headed or ruddy-complexioned."Â
Boy/Male
Tamil
Positive, Suitable
Male
English
Variant spelling of English Neil, NEAL means "champion."
Boy/Male
Hindu
Positive, Suitable
Boy/Male
Tamil
Real
Boy/Male
Hindu, Indian
Positive Thinking
Girl/Female
Tamil
Real
Surname or Lastname
English
English : nickname for a person with red hair or a ruddy complexion, from Middle English re(a)d ‘red’.English : topographic name for someone who lived in a clearing, from an unattested Old English rīed, r̄d ‘woodland clearing’.English : Read in Lancashire, the name of which is a contracted form of Old English rǣghēafod, from rǣge ‘female roe deer’, ‘she-goat’ + hēafod ‘head(land)’; Rede in Suffolk, so called from Old English hrēod ‘reeds’; or Reed in Hertfordshire, so called from an Old English ryhð ‘brushwood’.English : A family called Read were established in America in the early 18th century by John Read, who was born in Dublin, sixth in descent from Sir Thomas Read of Berkshire, England. His son, George Read (1733–98), was one of the signers of the Declaration of Independence, and as a lawyer helped frame the Constitution.
Boy/Male
Tamil
Real
Surname or Lastname
English, Spanish, and Portuguese
English, Spanish, and Portuguese : nickname for a loyal or trustworthy person, from Old French leial, Spanish and Portuguese leal ‘loyal’, ‘faithful (to obligations)’, Latin legalis, from lex, ‘law’, ‘obligation’ (genitive legis).
Boy/Male
Indian
Positive Power
Boy/Male
Hindu, Indian, Tamil
Positive Energy
Boy/Male
Hindu
Positive, Suitable
Boy/Male
Tamil
Positive, Suitable
Girl/Female
English
The bird teal; also the blue-green color.
Female
Greek
Variant spelling of Greek Rhea, REAH means "ease, flow."
Girl/Female
Indian
Real
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
Boy/Male
Gujarati, Hindu, Indian, Japanese, Kannada, Malayalam, Marathi, Sanskrit, Telugu
Issue; Name of the Great Marathi Worrier
Boy/Male
French, Hindu, Indian, Japanese
Gets What He Wants; God will Nourish
Girl/Female
Arabic
Life
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Marathi
Wish; Desired
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi
Camphor
Boy/Male
Muslim/Islamic
Oath right hand, right wing (of the army)
Girl/Female
Australian, Finnish, German
Beautiful Victory
Girl/Female
Arabic, Swahili
Woman; Life
Boy/Male
Australian, British, English
The West Meadow
Boy/Male
Hindu, Indian, Tamil
Strongest Man
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
POSITIVE REAL-FUNCTION
a.
Definitely laid down; explicitly stated; clearly expressed; -- opposed to implied; as, a positive declaration or promise.
a.
Electro-positive.
v. i.
To affix one's seal, or a seal.
imp. & p. p.
of Read
n.
See Rial, an old English coin.
v. t.
To sprinkle with, or as with, meal.
v. t.
To place in the rear; to secure the rear of.
a.
Hence: Positive; metallic; basic; -- distinguished from negative, nonmetallic, or acid.
a.
Hence: Not admitting of any doubt, condition, qualification, or discretion; not dependent on circumstances or probabilities; not speculative; compelling assent or obedience; peremptory; indisputable; decisive; as, positive instructions; positive truth; positive proof.
a.
Having a real position, existence, or energy; existing in fact; real; actual; -- opposed to negative.
a.
True; genuine; not artificial, counterfeit, or factitious; often opposed to ostensible; as, the real reason; real Madeira wine; real ginger.
a.
Royal; regal; kingly.
a.
Actually being or existing; not fictitious or imaginary; as, a description of real life.
n.
A Spanish coin. See Real.
v. t.
To close by means of a seal; as, to seal a drainpipe with water. See 2d Seal, 5.
a.
Having the power of direct action or influence; as, a positive voice in legislation.
n.
The positive degree or form.
a.
Corresponding with the original in respect to the position of lights and shades, instead of having the lights and shades reversed; as, a positive picture.
a.
Pertaining to things fixed, permanent, or immovable, as to lands and tenements; as, real property, in distinction from personal or movable property.
n.
The positive plate of a voltaic or electrolytic cell.