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alternate case: positive real numbers

Lehmer mean (837 words) [view diff] exact match in snippet view article find links to article

mathematics, the Lehmer mean of a tuple x {\displaystyle x} of positive real numbers, named after Derrick Henry Lehmer, is defined as: L p ( x ) = ∑

{\displaystyle x_{i}} and coefficients c k {\displaystyle c_{k}} are positive real numbers, and the exponents a i k {\displaystyle a_{ik}} are real numbers

1 , u 2 , … , u n {\displaystyle u_{1},u_{2},\dots ,u_{n}} and positive real numbers v 1 , v 2 , … , v n {\displaystyle v_{1},v_{2},\dots ,v_{n}} : (

example of a topology on the set S := R+ \ Z+, i.e. the set of all positive real numbers that are not positive whole numbers. The open sets in this topology

one can take exponents from any strictly increasing sequence of positive real numbers, and the same result holds. Szász showed that for complex number

David Silverman. It is played by two players on a given set S of positive real numbers. Before play starts, a threshold T and penalty ν are chosen with

The identric mean of two positive real numbers x, y is defined as: I ( x , y ) = 1 e ⋅ lim ( ξ , η ) → ( x , y ) ξ ξ η η ξ − η = lim ( ξ , η ) → ( x

numbers, the real cube root is not the principal cube root. For positive real numbers, the principal cube root is the real cube root. If y is any cube

or negative. Such a polynomial must have coefficients that are positive real numbers. The term is sometimes restricted to polynomials whose roots have

In mathematics, a sequence of positive real numbers ( s 1 , s 2 , . . . ) {\displaystyle (s_{1},s_{2},...)} is called superincreasing if every element

the number of digits right of the decimal point. Truncation of positive real numbers can be done using the floor function. Given a number x ∈ R + {\displaystyle

In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g0 = x and

x_{1},\dots ,x_{n}} are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is M p ( x 1 , … , x

write down arbitrary (not necessarily in X {\displaystyle X} ) positive real numbers x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } such

homomorphism from the set of invertible operators to the set of positive real numbers. The Fuglede−Kadison determinant of an operator A {\displaystyle

states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic

in honour of Franz Mertens. This definition can be extended to positive real numbers as follows: M ( x ) = M ( ⌊ x ⌋ ) . {\displaystyle M(x)=M(\lfloor

\lambda =1/\theta } ⁠ In each of these forms, both parameters are positive real numbers. The distribution has important applications in various fields,

[X]^{2}/\operatorname {E} [Y].} Example 1. Nesbitt's inequality. For positive real numbers a , b , c : {\displaystyle a,b,c:} a b + c + b a + c + c a + b ≥

→ R is a surjective and even bijective (mapping from the set of positive real numbers to the set of all real numbers). Its inverse, the exponential function

1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} are positive real numbers. Then 0 < n 1 x 1 + 1 x 2 + ⋯ + 1 x n ≤ x 1 x 2 ⋯ x n n ≤ x 1 +

can be generalized to positive real numbers (not necessarily irrational) and negative integers as well: if positive real numbers r {\displaystyle r} and

real exponential function and hence satisfies eln x = x for all positive real numbers x. Complex logarithm functions can be constructed by explicit formulas

function of a metric space, but instead of taking values in the positive real numbers, it takes values in a group of reflections, called the Weyl group

^{2}}{2}}+2\zeta (3)\right)z^{2}+O(z^{3}).} When restricted to the positive real numbers, the gamma function is a strictly logarithmically convex function

real values (like the determinant or matrix trace). Working with positive real-numbers brings several advantages: If the estimator of a single parameter

{ωn} clearly tends toward zero, by well-known properties of the positive real numbers. This fact can be used to prove, rigorously, that the convergents

example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential

{\displaystyle f} is said to be continuous if there is a function of all positive real numbers ε {\displaystyle \varepsilon } and x ∈ X {\displaystyle x\in X}

original on 2019-08-10. Retrieved 2019-08-09. The parallel sum of two positive real numbers x:y = [(1/x) + (1/y)]−1 arises in electrical circuit theory as the

{\displaystyle \lambda =(\lambda _{1},\dots ,\lambda _{n})} of positive real numbers (the widths of the subintervals), satisfying ∑ i = 1 n λ i = 1.

Then, given c {\displaystyle c} and α {\displaystyle \alpha } two positive real numbers, if f ∼ G G ( a , d , p ) {\displaystyle f\sim GG(a,d,p)} , then

mean. It was introduced by Kenneth B. Stolarsky in 1975. For two positive real numbers x, y the Stolarsky Mean is defined as: S p ( x , y ) = lim ( ξ

square roots of negative numbers. For example, if x and y are both positive real numbers, the following chain of equalities appears reasonable at first glance:

inequality to a power n > 0 (equiv., −n < 0), when a and b are positive real numbers: 0 ≤ a ≤ b ⇔ 0 ≤ an ≤ bn. 0 ≤ a ≤ b ⇔ a−n ≥ b−n ≥ 0. Taking the

have value ⁠π/2⁠ whenever the numbers 3, 5, 7… are replaced by positive real numbers such that the sum of their reciprocals is less than 1. In the example

measured in grams is a function from this collection of weight to positive real numbers. The term "weight function", an allusion to this example, is used

real values (like the determinant or matrix trace). Working with positive real numbers brings several advantages: If the estimator of a single parameter

function is strictly logarithmically convex when restricted to the positive real numbers. In fact, by the Bohr–Mollerup theorem, this property can be used

sometimes a branch cut is used; in this case removing the non-positive real numbers from the domain of the function and eliminating π {\displaystyle

n ) n ∈ N {\displaystyle (\lambda _{n})_{n\in \mathbb {N} }} of positive real numbers λ n > 0 {\displaystyle \lambda _{n}>0} such that x ¯ = lim n → ∞

problems with the extension of these definitions to bases that are not positive real numbers. For example, a negative real number has a real nth root, which

}}({\vec {p}},{\vec {x}},l)} , where the v l {\displaystyle v_{l}} are positive real numbers chosen as hyperparameters. The style loss is based on the Gram matrices

convergence. Specifically, Abel's test states that if a sequence of positive real numbers ( a n ) {\displaystyle (a_{n})} is decreasing monotonically (or

particular, with multiplication. This means that, if a and b are positive real numbers such that a = sup x ∈ A x {\displaystyle a=\sup _{x\in A}x} and

neural ODEs, however, layers form a continuous family indexed by positive real numbers. Specifically, the function h : R ≥ 0 → R {\displaystyle h:\mathbb

} When a {\displaystyle a} and r {\displaystyle r} are positive real numbers, this is equivalent to taking the geometric mean of the partial

{\displaystyle X} and r > 0 {\displaystyle r>0} ranges over the positive real numbers, forms a basis for a topology on X {\displaystyle X} that is called

coordinates for a point of the Teichmüller space of S consist of 3g − 3 positive real numbers called the lengths and 3g − 3 real numbers called the twists. A

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 . {\displaystyle

{x}{y}}+{\sqrt {\frac {y}{z}}}+{\sqrt[{3}]{\frac {z}{x}}}} for all positive real numbers x, y and z. Suppose we wish to find the minimal value of this function

{\displaystyle f(z)={\sqrt {z}}\,} be the usual square root function on positive real numbers. One may extend its domain to a neighbourhood of z = 1 {\displaystyle

\{M_{i}\}} in H converges to M in H if there are a sequence of positive real numbers ϵ i {\displaystyle \epsilon _{i}} converging to 0, and a sequence

norm topology of continuous functions on a closed interval in the positive real numbers. This is a generalization of the Stone–Weierstrass theorem. Paul

(PM)^{2}}={(MY)^{2}\cdot (PM)^{2}},} hence MX = MY, since MX, MY, and PM are all positive, real numbers. Thus, M is the midpoint of XY. Other proofs exist, including one

c=f_{n}(c^{n})=f_{n}(b^{n})=b} holds, as the root function on the positive real numbers is an inverse function to the function z ↦ z n {\displaystyle z\mapsto

1 , r 2 , r 3 , … {\displaystyle r_{1},r_{2},r_{3},\ldots } of positive real numbers such that B i ⊆ r i B {\displaystyle B_{i}\subseteq r_{i}B} for

multiplicative group action of the scaling transformations on the positive real numbers. Another class of simple examples comes from statistical estimation

{\displaystyle {\sqrt {a}}{\sqrt {b}}={\sqrt {ab}},} which is valid for positive real numbers a and b, and was also used in complex number calculations with one

abundant numbers is given by the following monotonic mapping from the positive real numbers. Let e p ( ε ) = ⌊ ln ⁡ ( 1 + p − 1 p 1 + ε − p ) ln ⁡ p ⌋ {\displaystyle

Gallagher. Let x {\displaystyle x} and Q {\displaystyle Q} be any two positive real numbers with x 1 / 2 log − A ⁡ x ≤ Q ≤ x 1 / 2 . {\displaystyle x^{1/2}\log

used of these uses the gamma function, which can be defined for positive real numbers as the integral Γ ( z ) = ∫ 0 ∞ x z − 1 e − x d x . {\displaystyle

x = e. (See figure at right) x−x Unique global maximum over the positive real numbers at x = 1/e. x3/3 − x First derivative x2 − 1 and second derivative

multiplicative characters. Consider the multiplicative group of positive real numbers (R+,·). Then functions fu : (R+,·) → C such that fu(a) = au, where

{\displaystyle {\mathcal {P}}} and r {\displaystyle r} ranges over the positive real numbers is a subbasis at the origin for the topology induced by P {\displaystyle

composite numbers is given by the following monotonic mapping from the positive real numbers. Let e p ( x ) = ⌊ 1 p x − 1 ⌋ {\displaystyle e_{p}(x)=\left\lfloor

\infty }x_{n}.} Let ( x n ) {\displaystyle (x_{n})} be a sequence of positive real numbers converging to l {\displaystyle l} and define a n := log ⁡ ( x 1

{\displaystyle \{x\in \mathbb {R} \mid x>0\}} is the set of all strictly positive real numbers, which can be written in interval notation as ( 0 , ∞ ) {\displaystyle

poisson-gamma or Tweedie distribution is continuous over the strictly positive real numbers, with a mass at zero. For any set of independent random variables

n = [ a , n ] {\displaystyle I_{n}=[a,n]} a bounded interval of positive real numbers. For x, y belonging to I, define x ⋆ n y {\displaystyle x\star _{n}y}

numbers and r 1 , … , r n {\displaystyle r_{1},\dots ,r_{n}} are positive real numbers. Seminormed spaces and topological groups In a seminormed space

x\in X.} Suppose a {\displaystyle a} and b {\displaystyle b} are positive real numbers and q , p 1 , … , p n {\displaystyle q,p_{1},\ldots ,p_{n}} are

points of branches of the curve as the parameter φ ranges over the positive real numbers. For any (x, y) generated in this way, dividing x by y cancels the

distribution resembles a normal distribution. More specifically, for large positive real numbers κ {\displaystyle \kappa } , f ( x ∣ μ , κ ) ≈ 1 σ 2 π exp ⁡ [ −

\left(y_{i}\right)_{i=1}^{\infty }} in H {\displaystyle H} and positive real numbers ( λ i ) i = 1 ∞ {\displaystyle \left(\lambda _{i}\right)_{i=1}^{\infty

given by: The nonzero complex numbers for a complex place p The positive real numbers for a real place p in S, and all nonzero real numbers for p not

{\displaystyle f} is R + {\displaystyle \mathbb {R} ^{+}} (the set of all positive real numbers and zero). The preimage of { 4 , 9 } {\displaystyle \{4,9\}} under

{X}}} , let { a N } {\displaystyle \{a_{N}\}} be a sequence of positive real numbers such that lim N a N = ∞ {\displaystyle \lim _{N}a_{N}=\infty }

Let both functions be defined on some unbounded subset of the positive real numbers, and g ( x ) {\displaystyle g(x)} be non-zero (often, but not necessarily

onto (surjective). However, if the codomain is restricted to the positive real numbers R + ≡ ( 0 , ∞ ) {\displaystyle \mathbb {R} ^{+}\equiv \left(0,\infty

number fields. Main source: An infinite set of trivial solutions in positive real numbers is given by x = y . {\displaystyle x=y.} Nontrivial solutions can

{\displaystyle a=(a_{1},\dots ,a_{n})} define the "a-mean" [a] of positive real numbers x1, ..., xn by [ a ] = 1 n ! ∑ σ x σ 1 a 1 ⋯ x σ n a n , {\displaystyle

and b 1 , b 2 ⋯ b n {\displaystyle b_{1},b_{2}\cdots b_{n}} are positive real numbers with a 1 + a 2 ⋯ + a n = a {\displaystyle a_{1}+a_{2}\cdots +a_{n}=a}

{ a j } j ∈ N {\displaystyle \{a_{j}\}_{j\in \mathbb {N} }} be positive real numbers with ∑ j a j < ∞ {\displaystyle \sum _{j}a_{j}<\infty } . Define

spaces, though they lack some of the properties that logarithms of positive real numbers possess. The continuum hypothesis (CH) states that there are no

product is at least K {\displaystyle K} . When defined over the positive real numbers, S {\displaystyle S} has infinitely many minimal elements of the

generalization of Schur's inequality is the following: Suppose a,b,c are positive real numbers. If the triples (a,b,c) and (x,y,z) are similarly sorted, then the

_{n=1}^{\infty }\left({\frac {4n^{2}}{4n^{2}-1}}\right).} The product of positive real numbers ∏ n = 1 ∞ a n {\displaystyle \prod _{n=1}^{\infty }a_{n}} converges

{\displaystyle x\in X} is a vector and a , c > 0 {\displaystyle a,c>0} are positive real numbers such that p ( x ) + a c   <   inf k ∈ K p ( x + a k ) {\displaystyle

∞ {\displaystyle M=\{M_{k}\}_{k=0}^{\infty }} be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions CM([a,b]) is defined

two points are sometimes called the foci. Let d1, d2 be non-equal positive real numbers. Let C be the internal division point of AB in the ratio d1 : d2

graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property: Twice any label is the sum of the

is strictly larger as the Legendre transform is only defined for positive real numbers. See this article for example. Let F denote a cumulative distribution

induce the same topology, which happens if and only if there exist positive real numbers C {\displaystyle C} and D {\displaystyle D} such that for all x

{\displaystyle pX+qY\leq m} and graphically by a budget line in the positive real numbers. Assuming strongly monotonic preferences, only bundles that are

properties. In fact, since R is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real

\left|x\right|^{r}+\left|y\right|^{s}+\left|z\right|^{t}=1} where r, s, and t are positive real numbers that determine the main features of the superquadric. Namely: less

numbers count the (0,1) matrices for which all eigenvalues are positive real numbers. The proof is bijective: a matrix A is an adjacency matrix of a

elements of an infinite continued fraction consist entirely of positive real numbers, the determinant formula can easily be applied to demonstrate when

root function is holomorphic everywhere except on the set of non-positive real numbers (on strictly negative reals it is not even continuous). The above

For example, the real numbers form a group for addition, and the positive real numbers form a group for multiplication. The exponential function x ↦ e

in the n-body problem of celestial mechanics, for any choice of positive real numbers m1, ..., mn as the masses? Partially resolved. Proved for almost

{\displaystyle H_{\rm {v}}^{px}=1/H_{\rm {s}}^{xp}} and that, both being positive real numbers, ln ⁡ H s x p = − ln ⁡ ( 1 / H s x p ) = − ln ⁡ H v p x {\displaystyle

set = R + n   . {\displaystyle \mathbb {R} _{+}^{n}\ .} (a set of positive real numbers, the consumer cannot preference negative amount of commodities)

|v_{k}\rangle ),} where p k > 0 {\displaystyle {\sqrt {p_{k}}}>0} are positive real numbers, r ψ {\displaystyle r_{\psi }} is the Schmidt rank of | ψ ⟩ {\displaystyle

)/{\sqrt {2}}} . Where α , β {\displaystyle \alpha ,\beta } are positive real numbers to be chosen later with α 2 + β 2 = 1 {\displaystyle \alpha ^{2}+\beta

These different conventions differ by elementary terms that are positive real numbers. Kramer, K.; Tunnell, J. (1982). "Elliptic curves and local ϵ-factors"

asymptotically positive. (the domain of logarithms is the set of all positive real numbers.) Although L'Hôpital's rule applies to both 0 / 0 {\displaystyle

numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry). In algebraic geometry, the dimension of an

(n+1)}{\varphi (n)}},\;\;n=1,2,\ldots \right\}} is dense in the positive real numbers. They also proved that the set { φ ( n ) n , n = 1 , 2 , … } {\displaystyle

{\displaystyle f:S\to \mathbb {R} } be any map. If there exist positive real numbers a {\displaystyle a} and b {\displaystyle b} such that 0 ≥ inf s

_{i}(x_{i}^{2}+y_{i}^{2}),} where α , β {\displaystyle \alpha ,\beta } are positive real numbers which are independent of which atom we are looking at but depend

non-decreasing function δ ( ε ) {\displaystyle \delta (\varepsilon )} of positive real numbers ε {\displaystyle \varepsilon } such that Δ t < δ ( ε ) {\displaystyle

{1}{4}}+{\frac {1}{4}}}}={\frac {3}{1.5}}=2\,.} The harmonic mean H of the positive real numbers x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} is

{\displaystyle \lambda _{i}} , representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the

(1/2+it)|^{2k}\,dt=O(T^{\varepsilon })} for all positive integers k and all positive real numbers ε. This has been proved for k = 1 or 2, but the case k = 3 seems

\sigma _{Y}(n-1)^{1/2}.} It has been shown that for a sample {yi} of positive real numbers, σ y 2 ≤ 2 y max ( A − H ) , {\displaystyle \sigma _{y}^{2}\leq

generalized from the domain N {\displaystyle \mathbb {N} } uniquely to positive real numbers only due to their recurrence relation and one given function-value

) ≥ f ( a ) + f ( b ) {\displaystyle f(a+b)\geq f(a)+f(b)} for positive real numbers a {\displaystyle a} and b {\displaystyle b} . Proof Since f {\displaystyle

be arbitrarily slow in the following sense: for any sequence of positive real numbers ( a n ) n ∈ N {\displaystyle (a_{n})_{n\in \mathbb {N} }} decreasing

convergent to 0 {\displaystyle 0} if there exists a sequence of positive real numbers r ∙ = ( r i ) i = 1 ∞ {\displaystyle r_{\bullet }=(r_{i})_{i=1}^{\infty

of radius r {\displaystyle r} if and only if there exist three positive real numbers a {\displaystyle a} , b {\displaystyle b} , and c = b a {\displaystyle

center)? Answer this question for all positive integers n and all positive real numbers r. (Certain special cases can be easy to compute.) Andrews, D. and

\delta (t)={\frac {1}{a}}\ \delta \!\left({\frac {t}{a}}\right)} for positive real numbers a {\displaystyle a} , it follows that: Ш   T ⁡ ( t ) = 1 T Ш ( t

w=(w_{n})_{n=1}^{\infty }\in c_{0}\setminus \ell _{1}} be a sequence of positive real numbers satisfying 1 = w 1 ≥ w 2 ≥ w 3 ≥ ⋯ {\displaystyle 1=w_{1}\geq w_{2}\geq

with the usual definition of the arithmetic–geometric mean for positive real numbers. See The Arithmetic-Geometric Mean of Gauss by David A. Cox. Milne

y. If a {\displaystyle a} and b {\displaystyle b} are distinct positive real numbers then 1 2 min ( a , b ) < | a ∥ b | < min ( a , b ) . {\displaystyle

foam faces (and hence the spin network edges) were labelled by positive real numbers as opposed to the half-integer labels of SU(2) spin networks. These

the angle between sides a and b is a right angle. For any three positive real numbers a, b, and c such that a2 + b2 = c2, there exists a triangle with

{\displaystyle \mathbb {C} } contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:

\mathbb {C} } and let M n {\displaystyle M_{n}} be a sequence of positive real numbers such that | f n ( x ) | ≤ M n {\displaystyle |f_{n}(x)|\leq M_{n}}

impedance is a quantity measured in ohms which represents resistance as positive real numbers in the usual manner, and also represents reactance as positive and

Marc Chamberland. He showed that the conjecture does not hold for positive real numbers since there are infinitely many fixed points, as well as orbits

this arg {\displaystyle \arg } is 0 (i.e. all three products are positive real numbers), and by which Ptolemy's theorem A B ¯ ⋅ C D ¯ + A D ¯ ⋅ B C ¯ =

X} is the set of possible outcomes. E.g., it may be the set of positive real numbers, representing the possible annual gross domestic product. It is

and ‖ ⋅ ‖ ′ {\displaystyle \|\cdot \|'} on Rn you can always find positive real numbers α , β > 0 {\displaystyle \alpha ,\beta >0} , such that α ⋅ ‖ x ‖

{\displaystyle {\widehat {p\,}}^{k}} not exceeding x, x runs over the positive real numbers, and Λ ( n ) {\displaystyle \Lambda (n)} is the von Mangoldt function

two function, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse. Alternatively, it may be defined

a surreal number that is bigger than zero, but smaller than all positive real numbers. set membership symbol ∈ is based on ε Ϝ is sometimes used to represent

be shown to be invalid by using a specific value of the set of positive real numbers for A: Value [ A ∨ ¬ A ] = Value [ A ] ∪ Value [ ¬ A ] = Value [

distance between the zeros of ζ (⁠1/2⁠ + it) on intervals of large positive real numbers. In the following, N(T) is the total number of real zeros and N0(T)

discoveries was the notion of the arithmetic-geometric mean (AGM) of two positive real numbers. He discovered its relation to elliptic integrals in the years 1798–1799

{\displaystyle L_{p}} , where p = 2. The contraharmonic mean of a set of positive real numbers is defined as the arithmetic mean of the squares of the numbers

positive number as its base. The same is true for the logarithm of positive real numbers as long as the logarithm base is positive and not 1. Irrational

{\displaystyle \sigma (T_{0})} is a discrete, countable subset of positive real numbers, and all the eigenspaces are finite-dimensional. Since F {\displaystyle

{\displaystyle rU,} where r > 0 {\displaystyle r>0} ranges over the positive real numbers, induces a topological vector space topology on X . {\displaystyle

characterized by a fixed rate of occurrence Log-normal distribution, for positive real numbers that are assumed to grow exponentially, such as incomes or prices

)-regularity]. Suppose δ j , d j {\displaystyle \delta _{j},d_{j}} are positive real numbers and r ≥ 1 {\displaystyle r\geq 1} is an integer. G ( j ) {\displaystyle

when the arguments of the logarithm and the square root are non-positive real numbers. The principal value of the square root is thus defined outside

is a formal product of a fractional ideal of K and a vector of positive real numbers with components indexed by the infinite places of K. A replete divisor

cumulative distribution function of a probability measure on the positive real numbers. The distribution is therefore defined, and any of its multiples

({\boldsymbol {wB}})} . Suppose that the Borel transform converges for all positive real numbers to a function growing sufficiently slowly that the following integral

, … {\displaystyle r_{1},r_{2},\ldots } can be any sequence of positive real numbers that converges to 0 {\displaystyle 0} in R {\displaystyle \mathbb

{\displaystyle \epsilon _{1}} and ϵ 2 {\displaystyle \epsilon _{2}} are positive real numbers that control the squareness of the shape. The surface of the superellipsoid

)=(AB,\beta )} This right group is the direct product of a group (positive real numbers under multiplication) and a right zero semigroup induced by the

number ωx = { 0, r ωxL | s ωxR }, where r and s range over the positive real numbers. If x < y then ωy is "infinitely greater" than ωx, in that it is

the convolution algebra of the locally compact abelian group of positive real numbers with multiplication. The Mellin transform of the function f ( x

(pitch) of their fundamental, and thus are parametrized by the positive real numbers, R+. Musical tones that differ by an octave (a doubling of frequency)

{\displaystyle f} be a function of compact support that maps into the positive real numbers f : Ω → R 0 + . {\displaystyle f:\Omega \rightarrow \mathbb {R}

and non-negative respectively describe matrices with exclusively positive real numbers as elements and matrices with exclusively non-negative real numbers

{\varepsilon } =(\varepsilon _{0},\varepsilon _{1},\dots )} of positive real numbers, there is a positive integer S {\displaystyle S} such that for every

(a_{n})} with a n → a {\displaystyle a_{n}\to a} and a sequence of positive real numbers ( p n ) {\displaystyle (p_{n})} with 1 p 1 + ⋯ + p n → 0 {\displaystyle

r_{\bullet }=\left(r_{i}\right)_{i=1}^{\infty }\to \infty } of positive real numbers such that the sequence ( r i f i ) i = 1 ∞ {\displaystyle

{\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} are positive real numbers (see List of trigonometric identities). A special case is π = arctan

p>1} and let b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} be positive real numbers. Set S k = ∑ i = 1 k b i {\displaystyle S_{k}=\sum _{i=1}^{k}b_{i}}

{\displaystyle b} , x {\displaystyle x} , and y {\displaystyle y} are positive real numbers and b ≠ 1 {\displaystyle b\neq 1} , and c {\displaystyle c} and

example, a square root calculating function that works only on positive real numbers might be specified as follows: SQRTP(x:real)r:real pre x >=0 post

integral. The volume formula can be generalized even further. For positive real numbers p1, …, pn, define the (p1, …, pn) ball with limit L ≥ 0 to be B

\partial {z_{n}}^{k_{n}}}}} It is possible to define a combination of positive real numbers { r ν   ( ν = 1 , … , n ) } {\displaystyle \{r_{\nu }\ (\nu =1,\dots

)x\subseteq (0,1)K} if and only if there exists a divergent sequence of positive real numbers t 1 , t 2 , t 3 , ⋯ → ∞ {\textstyle t_{1},t_{2},t_{3},\cdots \to

{\displaystyle m_{\bullet }=\left(m_{i}\right)_{i=1}^{\infty }} of positive real numbers such that | t i | ≤ m i {\displaystyle \left|t_{i}\right|\leq m_{i}}

whether the collection of positive integers and the collection of positive real numbers "can be corresponded so that each individual of one collection corresponds

arguments ( R + {\displaystyle \mathbb {R} ^{+}} denotes the set of positive real numbers). For instance, the trading function of the constant product market

a 2 , … , {\displaystyle a_{1},a_{2},\ldots ,} is a sequence of positive real numbers, let c k = a 1 + 1 a 2 + 1 a 3 + 1 ⋱ + 1 a k {\displaystyle c_{k}=a_{1}+{\cfrac

stability. Given a time-series x ( t ) {\displaystyle x(t)} , for any positive real numbers T , τ {\displaystyle T,\tau } , define the real number sequence

power, always producing a positive result, so logb(x) for any two positive real numbers b and x, where b is not equal to 1, is always a unique real number y

following property, known as the Riesz property: There exists two positive real numbers A {\displaystyle A} and B {\displaystyle B} such that for any square

possible to determine a positive quantity δ so that for every pair of positive real numbers a, b with | b − a | < δ {\displaystyle |b-a|<\delta } , every transformed

_{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}x-p_{n}|}=-\tau } for some positive real numbers σ , τ {\displaystyle \sigma ,\tau } , then we can establish an upper

extension to the set of non-zero real numbers by taking limits (for positive real numbers one can use the same rules as above to compute it). This extension