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general notion of a monotonic function. The terms nondecreasing and nonincreasing are often used in place of increasing and decreasing in order to avoid...

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept...

double edges or loops. The degree sequence is a list of numbers in nonincreasing order indicating the number of edges incident to each vertex in the...

analysis, it suffices to bound the function of interest with continuous nonincreasing positive definite functions. In other words, when a function belongs...

\infty }a_{n}=\sup _{n}a_{n}} ; if ( a n ) {\displaystyle (a_{n})} is nonincreasing, then lim n → ∞ a n = inf n a n {\displaystyle \lim _{n\to \infty }a_{n}=\inf...

k α k α k ′ {\displaystyle \lambda _{k}\alpha _{k}\alpha _{k}'} is nonincreasing for increasing k {\displaystyle k} , whereas the elements of α k {\displaystyle...

overpartitions of n: an overpartition of n is an ordered sequence of nonincreasing integers that sum to n, where the first occurrence of each integer may...

minimal optimal speeds are nonincreasing in p. If the maximizer is unique, s ∗ ( p ) {\displaystyle s^{\ast }(p)} is nonincreasing in p. Thus, in cases where...

F t } {\displaystyle \left\{Z_{t},{\mathcal {F}}_{t}\right\}} is a nonincreasing predictable sequence (Note that if { X t } {\displaystyle \left\{X_{t}\right\}}...

{\displaystyle \lim _{x\to c}M(x)} both exist as they feature nondecreasing and nonincreasing functions of x {\displaystyle x} , respectively. Consider a sequence...

differentiable on (−1, 0), f ′ {\displaystyle f^{\prime }} is a nondecreasing or nonincreasing function on (−1, 0), and f ′ ( 0 + ) = ( ln ⁡ a ) f ′ ( 0 − )  or  f...

case. Let {a1, a2,...} be a sequence of real numbers that is either nonincreasing or nondecreasing, and let {b1, b2,...} be a sequence of real or complex...

physics. The NAIRU acronym would better be expressed as NIIRU (the nonincreasing inflation rate of unemployment) because it is the unemployment rate...

argument of Fred Galvin shows that the quantity inside this limit is nonincreasing for n ≥ k, and so the limit exists. When β is monotone, its packing...

Note that H δ d ( S ) {\displaystyle H_{\delta }^{d}(S)} is monotone nonincreasing in δ {\displaystyle \delta } since the larger δ {\displaystyle \delta...

above states that a pair of nonnegative integer sequences a and b with nonincreasing a is bigraphic if and only if the conjugate partition b ∗ {\displaystyle...

then the function v p ( R ) / V ( R ) {\displaystyle v_{p}(R)/V(R)} is nonincreasing. This can be generalized to any lower bound on the Ricci curvature (not...

The sequence ( A n ) {\displaystyle \left(A_{n}\right)} is said to be nonincreasing if A n + 1 ⊆ A n {\displaystyle A_{n+1}\subseteq A_{n}} for each n ...

{\displaystyle W(-x)=W(x)} . Monotone: W ( x ) {\displaystyle W(x)} is a nonincreasing function for x ≥ 0 {\displaystyle x\geq 0} . Bounded support: W ( x...

Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every nonincreasing sequence a1, ..., am−1, am = 0, one can find a matrix A such that the...

\rho } , S α ( ρ ) {\displaystyle S_{\alpha }(\rho )} is a continuous, nonincreasing function of the parameter α {\displaystyle \alpha } . A weak version...

)   {\displaystyle (\ -1)^{d-2}\psi ^{-1,(d-2)}(t;\theta )\ } is a nonincreasing and convex function. The following tables highlight the most prominent...

nondecreasing sequence and S n + ( T − T n ) {\displaystyle S_{n}+(T-T_{n})} is nonincreasing. Given m , n > N {\displaystyle m,n>N} then both S n , S m {\displaystyle...

sequence of Lyndon words, in such a way that the words in the sequence are nonincreasing lexicographically. The final Lyndon word in this sequence is the lexicographically...

function π u ( x , Z ) {\displaystyle \pi _{u}(x,Z)} is a monotonically nonincreasing function of x {\displaystyle x} . Here, π u ( x , Z ) {\displaystyle...

Equivalently, u ∗ {\displaystyle u^{*}} is the unique radial and radially nonincreasing function, whose strict sublevel sets are open and have the same measure...

rectangular items. First, the algorithm sorts the items by order of nonincreasing height. Then, starting at position ( 0 , 0 ) {\displaystyle (0,0)} ...

Let γ1, ..., γk+ℓ be the exponents α1, ..., αk, β1, ..., βℓ sorted in nonincreasing order. Then α ⊕ β is defined as α ⊕ β = ωγ1 + ⋯ + ωγk+ℓ. The natural...

and will decrease with the number of active firms in the market, and nonincreasing in the degree of symmetry between them. [clarification needed] This...

( X , B , μ ) {\displaystyle (X,{\mathcal {B}},\mu )} is a monotone nonincreasing function, and it is supported on [ 0 , μ ( X ) ] {\displaystyle [0,\mu...

{\displaystyle Ih\geq 0} . Continuity If h n {\displaystyle h_{n}} is a nonincreasing sequence (i.e. h 1 ≥ ⋯ ≥ h k ≥ ⋯ {\displaystyle h_{1}\geq \cdots \geq...

the contemplated S {\displaystyle S} 's. Clearly, this function is nonincreasing and reaches log ⁡ | { x } | = 0 {\displaystyle \log \left|\{x\}\right|=0}...

>s\}.} We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) of f {\displaystyle f} as the function f ∗ : [ 0 , ∞...

{\displaystyle \pi _{i,j}} (with positive integer indices i and j) that is nonincreasing in both indices. This means that π i , j ≥ π i , j + 1 {\displaystyle...

the Shortest Weighted Processing Time rule of Smith: sequence jobs in nonincreasing order of the priority index w i p i {\displaystyle w_{i}p_{i}} . The...

on classical Lorentz spaces and Hardy's inequality with weights for nonincreasing functions". Transactions of the American Mathematical Society. 320 (2):...

_{0}\left(|z|\right)dz} such that μ 0 ( r ) {\displaystyle \mu _{0}(r)} is nonincreasing on ( 0 , ∞ ) {\displaystyle (0,\infty )} . It is possible that μ ( {...

(a_{n},b_{n}))} be a finite list of nonnegative integers that is in nonincreasing lexicographical order and let ( a i , b i ) {\displaystyle (a_{i},b_{i})}...

"backwards" Euclidean heat kernel over the evolving hypersurface is always nonincreasing.[H90] He later extended his formula to allow for general codimension...

distance between the corresponding solutions to the heat equation is nonincreasing for all time, thus: the set of totally geodesic maps in each homotopy...

nonnegative integer sequences ( a , b ) {\displaystyle (a,b)} with nonincreasing a {\displaystyle a} is digraphic if and only if vector a ∗ {\displaystyle...