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alternate case: k-1

Standard deviation (8,182 words) [view diff] exact match in snippet view article find links to article

k − 1 + k − 1 k ( x k − A k − 1 ) 2 = Q k − 1 + ( x k − A k − 1 ) ( x k − A k ) {\displaystyle {\begin{aligned}Q_{0}&=0\\Q_{k}&=Q_{k-1}+{\frac {k-1

k = 1 ∞ q k 1 − q k , {\displaystyle \textstyle L(q):=\sum _{k=1}^{\infty }{\frac {q^{k}}{1-q^{k}}},} since 1 F 2 k = 5 ( ψ 2 k 1 − ψ 2 k − ψ 4 k 1 −

distribution is Pr ( X 1 = k 1 , X 2 = k 2 ) = exp ⁡ ( − λ 1 − λ 2 − λ 3 ) λ 1 k 1 k 1 ! λ 2 k 2 k 2 ! ∑ k = 0 min ( k 1 , k 2 ) ( k 1 k ) ( k 2 k ) k ! ( λ

calculating f ( k + 1 , n , p ) f ( k , n , p ) = ( n − k ) p ( k + 1 ) ( 1 − p ) {\displaystyle {\frac {f(k+1,n,p)}{f(k,n,p)}}={\frac {(n-k)p}{(k+1)(1-p)}}}

i p i = a k p k + a k + 1 p k + 1 + a k + 2 p k + 2 + ⋯ {\displaystyle s=\sum _{i=k}^{\infty }a_{i}p^{i}=a_{k}p^{k}+a_{k+1}p^{k+1}+a_{k+2}p^{k+2}+\cdots

k − 1 ] T Y k − 1 ∣ k − 1 F k − 1 C k = M k [ M k + Q k − 1 ] − 1 L k = I − C k Y k ∣ k − 1 = L k M k + C k Q k − 1 C k T y ^ k ∣ k − 1 = L k [ F k −

λ k − 1 ( p k − 1 k ) λ k − 1 ( p k − 2 k ) λ k − 1 ⋯ λ k − 1 ( p 1 k ) λ k − 1 {\displaystyle \lambda _{k}=\lambda _{k-1}(p_{k-1}k)\lambda _{k-1}(p_{k-2}k)\lambda

× ⋯ × ( n − k + 1 ) k × ( k − 1 ) × ⋯ × 1 , {\displaystyle {\binom {n}{k}}={\frac {n\times (n-1)\times \cdots \times (n-k+1)}{k\times (k-1)\times \cdots

K_{\mathrm {m} }} is equal to k + 2 / k + 1 {\displaystyle k_{+2}/k_{+1}} or to k − 1 / k + 1 {\displaystyle k_{-1}/k_{+1}} or to something else. G. E. Briggs

identity can be proved: ∏ k = 1 n sin ⁡ ( 2 k − 1 ) π 4 n = ∏ k = 1 n cos ⁡ ( 2 k − 1 ) π 4 n = 2 2 n {\displaystyle \prod _{k=1}^{n}\sin {\frac {\left(2k-1\right)\pi

case of a perfect prediction (i.e., when p k = 1 {\displaystyle p_{k}=1} and y k = 1 {\displaystyle y_{k}=1} , or p k = 0 {\displaystyle p_{k}=0} and y

( n − k + 1 ) k ( k − 1 ) ( k − 2 ) ⋯ 2 ⋅ 1 . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!\,(n-k)!}}={\frac {n(n-1)(n-2)\cdots (n-k+1)}{k(k-1)(k-2)\cdots

constants defining the International System of Units, the SI, with k = 1.380649×10−23 J K−1. The Boltzmann constant is a proportionality constant between the

among common substances, about 4184 J⋅kg−1⋅K−1 at 20 °C; but that of ice, just below 0 °C, is only 2093 J⋅kg−1⋅K−1. The specific heat capacities of iron,

{\displaystyle R=N_{\rm {A}}k} = 6.02214076×1023 mol−1 × 1.380649×10−23 J⋅K−1 = 8.31446261815324 J⋅K−1⋅mol−1 Since the 2019 revision of the SI, both NA and k are defined

⁡ ( 1 + x ) = ∑ k = 1 ∞ ( − 1 ) k − 1 k x k = x − x 2 2 + x 3 3 − ⋯ , {\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac

0.3 - the value of r {\displaystyle r} , k 1 {\displaystyle k_{1}} and k 2 / k 1 {\displaystyle k_{2}/k_{1}} must be linearly scaled for a different

∑ k = 1 N m k ( y k 2 + z k 2 ) − ∑ k = 1 N m k x k y k − ∑ k = 1 N m k x k z k − ∑ k = 1 N m k x k y k ∑ k = 1 N m k ( x k 2 + z k 2 ) − ∑ k = 1 N m

+k)+(k+1)={\frac {k(k+1)}{2}}+(k+1).} Algebraically, the right hand side simplifies as: k ( k + 1 ) 2 + ( k + 1 ) = k ( k + 1 ) + 2 ( k + 1 ) 2 = ( k + 1 )

e., v k ( x ) = { x − x k − 1 x k − x k − 1  if  x ∈ [ x k − 1 , x k ] , x k + 1 − x x k + 1 − x k  if  x ∈ [ x k , x k + 1 ] , 0  otherwise , {\displaystyle

| x − a | k + 1 ( k + 1 ) ! ≤ M r k + 1 ( k + 1 ) ! {\displaystyle |R_{k}(x)|\leq M{\frac {|x-a|^{k+1}}{(k+1)!}}\leq M{\frac {r^{k+1}}{(k+1)!}}} for all

α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯ ( α − n + 1 ) n ! . {\displaystyle {\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac

x 2 ) x − 1 ∏ k = 1 n k + 1 k + x {\displaystyle x!=\lim _{n\to \infty }\left(n+1+{\frac {x}{2}}\right)^{x-1}\prod _{k=1}^{n}{\frac {k+1}{k+x}}} which

M S = ( k − 1 + k 2 ) / k 1 {\displaystyle K_{M}^{S}=(k_{-1}+k_{2})/k_{1}} and K M P = ( k − 1 + k 2 ) / k − 2 {\displaystyle K_{M}^{P}=(k_{-1}+k_{2})/k_{-2}}

and k ≥ 1, then φ ( p k ) = p k − p k − 1 = p k − 1 ( p − 1 ) = p k ( 1 − 1 p ) . {\displaystyle \varphi \left(p^{k}\right)=p^{k}-p^{k-1}=p^{k-1

k − 1 e − ( x / λ ) k , x ≥ 0 , 0 , x < 0 , {\displaystyle f(x;\lambda ,k)={\begin{cases}{\frac {k}{\lambda }}\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda

k = 1 n k = 1 + 2 + ⋯ + n = n 2 + n ( n + 1 ) 2 = n ( n + 1 ) 2 = ( n + 1 2 ) {\displaystyle \displaystyle {\begin{aligned}T_{n}&=\sum _{k=1}^{n}k=1+2+\dotsb

d t k + a k − 1 d k − 1 x d t k − 1 + ⋯ + a 1 d x d t + a 0 x = 0. {\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots

∑ k = 1 n + 1 a k − 1 x k + ∑ k = 0 n a k x k = ∑ k = 1 n a k − 1 x k + a n x n + 1 + a 0 x 0 + ∑ k = 1 n a k x k = a 0 x 0 + ∑ k = 1 n ( a k − 1 + a

trial is the first success is Pr ( X = k ) = ( 1 − p ) k − 1 p {\displaystyle \Pr(X=k)=(1-p)^{k-1}p} for k = 1 , 2 , 3 , 4 , … {\displaystyle k=1,2,3

[ H + ] 2 + [ H + ] K 1 + K 1 K 2 = [ H 2 A ] [ H 2 A ] + [ H A − ] + [ A 2 − ] α HA − = [ H + ] K 1 [ H + ] 2 + [ H + ] K 1 + K 1 K 2 = [ HA − ] [ H 2

{\displaystyle p>1} ∑ k = 1 n k p = n p + 1 p + 1 + 1 2 n p + ∑ k = 2 p ( p k ) B k p − k + 1 n p − k + 1 , {\displaystyle \sum _{k=1}^{n}k^{p}={\frac

values of { b k } k = 1 M − 1 {\displaystyle \{b_{k}\}_{k=1}^{M-1}} , { c k } k = 1 M {\displaystyle \{c_{k}\}_{k=1}^{M}} and { y k } k = 1 M {\displaystyle

trapezoids and: G 1 = 1 − ∑ k = 1 n ( X k − X k − 1 ) ( Y k + Y k − 1 ) {\displaystyle G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})} is the resulting

binomial": ( k + r − 1 ) ⋯ ( r ) k ! = ( − 1 ) k ( − r ) ( − r − 1 ) ( − r − 2 ) ⋯ ( − r − k + 1 ) ⏞ k  factors k ! = ( − 1 ) k ( − r − k ) . {\displaystyle

) = p k ( 1 − p ) 1 − k for  k ∈ { 0 , 1 } {\displaystyle f(k;p)=p^{k}(1-p)^{1-k}\quad {\text{for }}k\in \{0,1\}} or as f ( k ; p ) = p k + ( 1 − p )

q: K = 1 2 ( m + n + p ) ( m + n − p ) ( m + n + q ) ( m + n − q ) , {\displaystyle K={\tfrac {1}{2}}{\sqrt {(m+n+p)(m+n-p)(m+n+q)(m+n-q)}},} K = 1 2 p

k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) ⟩ = δ ( k 1 − k 2 ) k 1 2 δ ( k 3 − k 4 ) k 3 2 + δ ( k 1 − k 3 ) k 3 2 δ ( k 2 − k 4 ) k 2 2 + δ ( k 1 − k 4 ) k

k-dimensional polytope that is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k {\displaystyle u_{0},\dots ,u_{k}}

n k ) = n ( n − 1 ) ⋯ ( n − k + 1 ) k ( k − 1 ) ⋯ 1 , {\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},} which can be written

k 1 n E n k 1 | k 1 ( 0 ) ⟩ | n ( 2 ) ⟩ = ( V k 1 k 2 V k 2 n E n k 1 E n k 2 − V n n V k 1 n E n k 1 2 ) | k 1 ( 0 ) ⟩ − 1 2 V n k 1 V k 1 n E k 1 n

− V , {\displaystyle L=T-V,} where T = 1 2 ∑ k = 1 N m k v k 2 {\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}} is the total kinetic energy

− V , {\displaystyle L=T-V,} where T = 1 2 ∑ k = 1 N m k v k 2 {\displaystyle T={\frac {1}{2}}\sum _{k=1}^{N}m_{k}v_{k}^{2}} is the total kinetic energy

⁡ n + ∑ k = 1 n 1 k ) = ∫ 1 ∞ ( − 1 x + 1 ⌊ x ⌋ ) d x . {\displaystyle {\begin{aligned}\gamma &=\lim _{n\to \infty }\left(-\log n+\sum _{k=1}^{n}{\frac

around k = − 1 / 2 {\displaystyle k=-1/2} and even around k = N − 1 / 2. {\displaystyle k=N-1/2.} X k = ∑ n = 0 N − 1 x n cos ⁡ [ π N ( n + 1 2 ) ( k + 1 2

∑ k = 1 n E ⁡ [ ( M k − M k − 1 ) 2 ∣ M 1 , … , M k − 1 ] → 1 {\displaystyle {\frac {1}{n}}\sum _{k=1}^{n}\operatorname {E} \left[\left(M_{k}-M_{k-1}\right)^{2}\mid

_{i}-1}} where { x k } k = 1 k = K {\displaystyle \{x_{k}\}_{k=1}^{k=K}} belong to the standard K − 1 {\displaystyle K-1} simplex, or in other words:

= ⋯ = [ ( k , k ) ] 1 = [ ( 1 , 0 ) ] = [ ( 2 , 1 ) ] = ⋯ = [ ( k + 1 , k ) ] − 1 = [ ( 0 , 1 ) ] = [ ( 1 , 2 ) ] = ⋯ = [ ( k , k + 1 ) ] 2 = [ ( 2

0 n ( n k ) 2 n − k + 1 n − k + 1 B k ( 1 ) = 2 n {\displaystyle -1+\sum _{k=0}^{n}{\binom {n}{k}}{\frac {2^{n-k+1}}{n-k+1}}B_{k}(1)=2^{n}} and − 1 +

infinite series x = b k 10 k + b k − 1 10 k − 1 + ⋯ + b 0 + a 1 10 + a 2 10 2 + ⋯ . {\displaystyle x=b_{k}10^{k}+b_{k-1}10^{k-1}+\cdots +b_{0}+{\frac {a_{1}}{10}}+{\frac

sufficient to prove that k + 1 {\displaystyle k+1} does not divide P {\displaystyle P} . To that end, let k + 1 {\displaystyle k+1} be any prime larger than

k 1 > k 2 > ⋯ > k n > 0 k 1 − s 1 k 2 − s 2 ⋯ k n − s n . {\displaystyle \zeta (s_{1},s_{2},\ldots ,s_{n})=\sum _{k_{1}>k_{2}>\cdots >k_{n}>0}{k_{1

a_{k}=k^{2}} ( a 2 k − 1 ) k = 1 ∞ , a k = k 2 {\displaystyle {(a_{2k-1})}_{k=1}^{\infty },\qquad a_{k}=k^{2}} ( a k ) k = 1 ∞ , a k = ( 2 k − 1 ) 2 {\displaystyle

{\displaystyle k} with a width k 1 / 2 + α {\displaystyle k^{1/2+\alpha }} : P r ( ‖ Z ‖ 2 ∈ [ k − 2 k 1 / 2 + α , k + 2 k 1 / 2 + α + 2 k α ] ) ≥ 1 − e

and one bronze medal. She won three consecutive gold medals in the Women's K‑1 200 metres at the 2012 Summer Olympics, 2016 Summer Olympics and 2020 Summer

0 1 k 1 + k − 1 ( k − 1 + k 1 e − ( k 1 + k − 1 ) t ) + [ P ] 0 k − 1 k 1 + k − 1 ( 1 − e − ( k 1 + k − 1 ) t ) [ P ] = [ A ] 0 k 1 k 1 + k − 1 ( 1 −

1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k , {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}},} is called

is an Olympic gold medallist in the K-1 500m category (2004) and a two-time world champion in K-1 500 (2007) and K-1 1000 (2011), winning four Olympic and

k = p n − k + 1 + p n − k + 2 + ⋯ + p n k = 1 k ∑ i = n − k + 1 n p i {\displaystyle {\begin{aligned}{\textit {SMA}}_{k}&={\frac {p_{n-k+1}+p_{n-k+2}+\cdots

This is the complete list of men's Olympic medalists in canoeing. International Olympic Committee results database

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200 metres (660 ft), 500

k + 1 … g 1 , n 0 1 0 … 0 g 2 , k + 1 … g 2 , n 0 0 1 … 0 g 3 , k + 1 … g 3 , n ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 … 0 … 1 g k , k + 1 … g k , n ] [ m 0 m 1 ⋮ m k − 1 ]

{\displaystyle k} is 1 n − k ⋅ ∏ i = 0 k − 1 n − i − 1 n − i = n − 1 n ⋅ n − 2 n − 1 ⋅ . . . ⋅ n − k n − k − 1 ⋅ 1 n − k = 1 6 ≈ 16.7 % {\displaystyle {\tfrac

p ¯ ( n ) = ∏ k = 1 n − 1 ( 1 − k 365 ) < ∏ k = 1 n − 1 ( e − k 365 ) = e − n ( n − 1 ) 730 . {\displaystyle {\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac

K 1 ′ = [ L ] [ M ] [ L M ] [ LM ] = [ L ] [ M ] K 1 ′ [ L ] + [ LM ] ↽ − − ⇀ [ L 2 M ] K 2 ′ = [ L ] [ LM ] [ L 2 M ] [ L 2 M ] = [ L ] 2 [ M ] K 1 ′

= ∑ k = 1 N Δ S k = ∑ k = 1 N d Q k T → ∫ 0 T 2 d S d T d T = ∫ 0 T 2 C p k T d T {\displaystyle S^{\circ }=\sum _{k=1}^{N}\Delta S_{k}=\sum _{k=1}^{N}{\frac

non-linear differential equations of the second order, I: The equation y" + k(1−y2)y' + y = bλkcos(λt + a), k large". Journal of the London Mathematical

k − 1 + ⋯ + β k − 1 z t + 1 and z ^ t = β 1 z t + 1 + ⋯ + β k − 1 z t + k − 1 . {\displaystyle {\hat {z}}_{t+k}=\beta _{1}z_{t+k-1}+\cdots +\beta _{k-1}z_{t+1}\qquad

k − 1 {\displaystyle k-1} elements of ⁠ V ∗ {\displaystyle V^{*}} ⁠, then define ( ι α w ) ( u 1 , u 2 , … , u k − 1 ) = w ( α , u 1 , u 2 , … , u k −

temperature, which has a unit of joules per kelvin (J⋅K−1) in the International System of Units (or kg⋅m2⋅s−2⋅K−1 in terms of base units). The entropy of a substance

∑ k = 1 n Y k ] = 1 n ∑ i = 1 n ( E ⁡ [ Y i 2 ] − 2 n ( ∑ j ≠ i E ⁡ [ Y i Y j ] + E ⁡ [ Y i 2 ] ) + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E ⁡ [ Y j Y k ] + 1 n 2

as: L p ( x ) = ∑ k = 1 n x k p ∑ k = 1 n x k p − 1 . {\displaystyle L_{p}(\mathbf {x} )={\frac {\sum _{k=1}^{n}x_{k}^{p}}{\sum _{k=1}^{n}x_{k}^{p-1}}}

number: r k r k − 1 ∗ = E s e j ( φ k − φ k − 1 ) + E s e j φ k n k − 1 ∗ + E s e − j φ k − 1 n k + n k n k − 1 ∗ {\displaystyle r_{k}r_{k-1}^{*}=E_{s}e^{j\left(\varphi

(that is, Δ x k = x k − x k − 1 {\displaystyle \Delta x_{k}=x_{k}-x_{k-1}} ), then ∫ a b f ( x ) d x ≈ ∑ k = 1 N f ( x k − 1 ) + f ( x k ) 2 Δ x k . {\displaystyle

can be thought of as measuring the net flux through the boundary of a (k + 1)-parallelotope at each point. The exterior derivative of a differential

integer k is ⌊ log b ⁡ k ⌋ + 1 = ⌈ log b ⁡ ( k + 1 ) ⌉ . {\displaystyle \lfloor \log _{b}{k}\rfloor +1=\lceil \log _{b}{(k+1)}\rceil .} The number of

of time until the occurrence of a fixed number of events. When k = 1 {\displaystyle k=1} , the distribution simplifies to the exponential distribution

{\displaystyle \lambda } , or κ {\displaystyle \kappa } and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low thermal conductivity

it is enough to assume that the ( k − 1 ) {\displaystyle (k{-}1)} -th derivative f ( k − 1 ) {\displaystyle f^{(k-1)}} is differentiable almost everywhere

{\displaystyle \sum _{i=1}^{k}p_{i}=1\,.} Thus, there are only k − 1 {\displaystyle k-1} independent parameters. Variant 1 uses k {\displaystyle k} natural

U ( k ) ( u ) = n ! ( k − 1 ) ! ( n − k ) ! u k − 1 ( 1 − u ) n − k {\displaystyle f_{U_{(k)}}(u)={n! \over (k-1)!(n-k)!}u^{k-1}(1-u)^{n-k}} that is,

A k − 1 A k ) − 1 = A k − 1 A k − 1 − 1 ⋯ A 2 − 1 A 1 − 1 . {\displaystyle (\mathbf {A} _{1}\mathbf {A} _{2}\cdots \mathbf {A} _{k-1}\mathbf {A} _{k})^{-1}=\mathbf

uni-dimensional recurrence: ( n k ) = ( n k − 1 ) ( n − k + 1 ) / k , {\displaystyle {\binom {n}{k}}={\binom {n}{k-1}}(n-k+1)/k,} with the initial value ( n 0

u ( t k ) − u ( t k − 1 ) Δ t = K p e ( t k ) − e ( t k − 1 ) Δ t + K i e ( t k ) + K d e ( t k ) − e ( t k − 1 ) Δ t − e ( t k − 1 ) − e ( t k − 2 )

, 1 k 1 , 1 0 0 ⋯ k 0 , 2 k 1 , 2 k 2 , 2 0 ⋯ k 0 , 3 k 1 , 3 k 2 , 3 k 3 , 3 ⋯ ⋮ ⋮ ⋮ ⋮ ] = [ k 0 , 0 0 0 0 ⋯ k 0 , 1 k 1 , 1 0 0 ⋯ k 0 , 2 k 1 , 2 k

at least in the class C k − 1 {\displaystyle C^{k-1}} since f ′ , f ″ , … , f ( k − 1 ) {\displaystyle f',f'',\dots ,f^{(k-1)}} are continuous on U .

m k = q k m k − 1 + m k − 2 n k = q k n k − 1 + n k − 2 , {\displaystyle {\begin{aligned}m_{k}&=q_{k}m_{k-1}+m_{k-2}\\n_{k}&=q_{k}n_{k-1}+n_{k-2},\end{aligned}}}

k − 1 ∑ i 2 = 0 k − 1 ⋯ ∑ i n = 0 k − 1 Δ h n ( f , x + i 1 h + i 2 h + ⋯ + i n h ) . {\displaystyle \Delta _{kh}^{n}(f,x)=\sum \limits _{i_{1}=0}^{k-1}\sum

X 1 k 1 ⋯ X n k n ] {\displaystyle E[{X_{1}}^{k_{1}}\cdots {X_{n}}^{k_{n}}]} is called a mixed moment of order k {\displaystyle k} (where k = k 1 + .

X 1 k 1 ⋯ X n k n ] {\displaystyle E[{X_{1}}^{k_{1}}\cdots {X_{n}}^{k_{n}}]} is called a mixed moment of order k {\displaystyle k} (where k = k 1 + .

that is denoted S = a 1 + a 2 + a 3 + ⋯ = ∑ k = 1 ∞ a k . {\displaystyle S=a_{1}+a_{2}+a_{3}+\cdots =\sum _{k=1}^{\infty }a_{k}.} The nth partial sum Sn

n k − 1 } {\displaystyle \left\{{n \atop k-1}\right\}} since we must partition the remaining n objects into the available ⁠ k − 1 {\displaystyle k-1} ⁠

( k − 1 ) = u k ( k − 2 ) − proj u k − 1 ⁡ ( u k ( k − 2 ) ) , e k = u k ( k − 1 ) ‖ u k ( k − 1 ) ‖ {\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf

of the normalization constraints: α k ′ α k = 1 , k = 1 , … , p {\displaystyle \alpha _{k}'\alpha _{k}=1,k=1,\dots ,p} . As noted above, the results of

n , k ) = 1 2 ( k − 1 ) ( n 2 − ( k − 1 ) n + ( k − 2 ) + ϕ ( n + 1 , k − 1 ) ) = 1 2 ( k − 1 ) ( ( n − 1 ) ( n − ( k − 2 ) ) + ϕ ( n + 1 , k − 1 ) )

{\displaystyle \sigma \colon \mathbb {R} ^{K}\to (0,1)^{K}} , where K > 1 {\displaystyle K>1} , takes a vector z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf

for y is ∑ k = 1 n ( y ) | d d y g k − 1 ( y ) | ⋅ f X ( g k − 1 ( y ) ) , {\displaystyle \sum _{k=1}^{n(y)}\left|{\frac {d}{dy}}g_{k}^{-1}(y)\right|\cdot

This is the complete list of women's Olympic medalists in kayaking. International Olympic Committee results database

k = 1 46 σ ( k ) {\displaystyle \sum _{k=1}^{46}\sigma (k)} 1759 = de Polignac prime 1760 = the number of yards in a mile 1761 = k such that k, k+1 and

G_{2}.} The field K 1 K 2 {\displaystyle K_{1}K_{2}} with Galois group G = Gal ⁡ ( K 1 K 2 / k ) {\displaystyle G=\operatorname {Gal} (K_{1}K_{2}/k)} has

hypergeometric distribution: Pr ( X 1 = k 1 , … , X c = k c ) = ∏ i = 1 c ( K i k i ) ( N n ) {\displaystyle \Pr(X_{1}=k_{1},\ldots ,X_{c}=k_{c})={\frac {\prod

= y k − 1 + x ∗ − x k − 1 x k − x k − 1 ( y k − y k − 1 ) . {\displaystyle y(x_{*})=y_{k-1}+{\frac {x_{*}-x_{k-1}}{x_{k}-x_{k-1}}}(y_{k}-y_{k-1}).} (which

) k − 1 π H k − 1 ( z ) e − z 2 = 2 π d k − 1 d z k − 1 ( e − z 2 ) , k = 1 , 2 , … {\displaystyle \operatorname {erf} ^{(k)}z={\frac {2(-1)^{k-1}}{\sqrt

_{p-1}}-\sum _{k=1}^{p-1}\delta _{\nu _{k}}^{\mu _{p}}\delta _{\nu _{1}\dots \nu _{k-1}\,\nu _{p}\,\nu _{k+1}\dots \nu _{p-1}}^{\mu _{1}\dots \mu _{k-1}\,\mu _{k}\

to pick a ( i k − 1 − ( i 1 − 1 ) ) {\textstyle (i_{k-1}-(i_{1}-1))} -dimensional subspace W k − 1 ′ ⊂ W k ∩ V i k − 1 {\textstyle W_{k-1}'\subset W_{k}\cap

Slalom C-1 men C-2 men K-1 men women Sprint C-1 200 m men C-1 1000 m men C-2 1000 m men K-1 200 m men women K-1 500 m women K-1 1000 m men K-2 200 m men

quaternions, i 2 = j 2 = k 2 = i j k = − 1 {\displaystyle \mathbf {i} ^{2}=\mathbf {j} ^{2}=\mathbf {k} ^{2}=\mathbf {i\,j\,k} =-1} into the stone of Brougham

−273.15 °C, the lowest possible temperature). R has for value 8.314 J/(mol·K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅atm/(mol⋅K). How much gas is present could

Championships with five golds (K-1 1000 m: 2009, 2010, 2013; K-1 5000 m: 2011 and K-4 1000 m: 2011 ) and three silvers (K-1 5000 m: 2010, 2014, 2015). At

C-1 men women K-1 men women Sprint C-1 200 m women C-1 1000 m men C-2 500 m women C-2 1000 m men K-1 200 m men women K-1 500 m women K-1 1000 m men K-2

intermediate can be determined as follows: K 1 = K 1 K − 1 = [ O 2 ] ⋅ [ O ] [ O 3 ] {\displaystyle K_{1}={\frac {K_{1}}{K_{-1}}}={\frac {[{\ce {O2}}]\cdot [{\ce

∑ k = 1 N p k ≥ ∑ k = 1 N q k {\displaystyle \sum _{k=1}^{N}p_{k}\geq \sum _{k=1}^{N}q_{k}} This implies that: − ∑ k = 1 ∞ p k log ⁡ ( p k ) ≤ − ∑ k =

200 m events with one of them being K-1 200 m for the women. The other events for men at 200 m will be C-1, K-1, and K-2. This was confirmed at their

tanker (renamed K.1 after bombing capability removed), 11 converted. Victor BK.1A 3-point in-flight refuelling tanker (renamed K.1A as for K.1), 14 converted

F^{(k+1)}(\xi )} : F ( k + 1 ) ( ξ ) = f ( k + 1 ) ( ξ ) − L ( k + 1 ) ( ξ ) − R ~ ( k + 1 ) ( ξ ) {\displaystyle F^{(k+1)}(\xi )=f^{(k+1)}(\xi )-L^{(k+1)}(\xi

}})={\tfrac {1}{2}}\pi } For example: 4 3 K ( 1 3 ) E ( 1 2 ) + 3 2 E ( 1 3 ) K ( 1 2 ) − 2 K ( 1 3 ) K ( 1 2 ) = 1 2 π {\displaystyle {\tfrac {4}{3}}K({\color

∑ k = 1 p B k k ! ( f ( k − 1 ) ( n ) − f ( k − 1 ) ( m ) ) + R p , {\displaystyle S-I=\sum _{k=1}^{p}{{\frac {B_{k}}{k!}}\left(f^{(k-1)}(n)-f^{(k-1)}(m)\right)}+R_{p}

functions: f n ( y ; θ ) = ∏ k = 1 n f k u n i v a r ( y k ; θ )   . {\displaystyle f_{n}(\mathbf {y} ;\theta )=\prod _{k=1}^{n}\,f_{k}^{\mathsf {univar}}(y_{k};\theta

{2}}} and y k + 1 = ( 1 − f ( y k ) ) / ( 1 + f ( y k ) )   ,   a k + 1 = a k ( 1 + y k + 1 ) 4 − 2 2 k + 3 y k + 1 ( 1 + y k + 1 + y k + 1 2 ) {\displaystyle

standard logistic function, depicted at right, where L = 1 , k = 1 , x 0 = 0 {\displaystyle L=1,k=1,x_{0}=0} , has the equation f ( x ) = 1 1 + e − x {\displaystyle

case above: X k 1 , k 2 , … , k d = X N 1 − k 1 , N 2 − k 2 , … , N d − k d ∗ , {\displaystyle X_{k_{1},k_{2},\dots ,k_{d}}=X_{N_{1}-k_{1},N_{2}-k_{2}

This is a list of medalists from the ICF Canoe Sprint World Championships in men's kayak. Debuted: 1994. Debuted: 1948. Debuted: 1938. Not held: 1948.

mass. The SI unit of molar heat capacity is joule per kelvin per mole, J⋅K−1⋅mol−1. Like the specific heat, the measured molar heat capacity of a substance

differential forms that, given a k-form φ {\displaystyle \varphi } , produces a (k+1)-form d φ . {\displaystyle d\varphi .} This operation extends the differential

{1}{k(k+1)}}=\sum _{k=1}^{\infty }\sum _{n=1}^{k}{\frac {1}{k(k+1)}}} = ∑ k = 1 ∞ k k ( k + 1 ) = ∑ k = 1 ∞ 1 k + 1 = S − 1 {\displaystyle =\sum _{k=1}^{\infty

a subsequence ( a n k ) k = 1 ∞ {\displaystyle \left(a_{n_{k}}\right)_{k=1}^{\infty }} satisfying lim sup n → ∞ | a n k + 1 a n k | ≤ ℓ < r {\displaystyle

functions for k = 1 , . . . , m {\displaystyle k=1,...,m} . Then ‖ ∏ k = 1 m f k ‖ 1 ≥ ∏ k = 1 m ‖ f k ‖ p k . {\displaystyle \left\|\prod _{k=1}^{m}f_{k}\right\|_{1}\geq

{a^{2}+ab+b^{2}}{3}},} m k = ∑ i = 0 k a i b k − i k + 1 . {\displaystyle m_{k}={\frac {\sum _{i=0}^{k}a^{i}b^{k-i}}{k+1}}.} For a random variable following the

}{q}}(2q-\ell )^{2n},} E 2 n = ∑ k = 1 2 n ( − 1 ) k 1 2 k ∑ ℓ = 0 2 k ( − 1 ) ℓ ( 2 k ℓ ) ( k − ℓ ) 2 n . {\displaystyle E_{2n}=\sum _{k=1}^{2n}(-1)^{k}{\frac {1}{2^{k}}}\sum

) + 1 2 ln ⁡ n = n ln ⁡ n − n + 1 + ∑ k = 2 m ( − 1 ) k B k k ( k − 1 ) ( 1 n k − 1 − 1 ) + R m , n , {\displaystyle {\begin{aligned}\ln(n!)-{\tfrac

= ∑ k = 1 K Pr ( Y i = k ) = ∑ k = 1 K 1 Z e β k ⋅ X i = 1 Z ∑ k = 1 K e β k ⋅ X i . {\displaystyle 1=\sum _{k=1}^{K}\Pr(Y_{i}=k)\;=\;\sum _{k=1}^{K}{\frac

obtains the iterative scheme x k + 1 = x k − [ f ″ ( x k ) ] − 1 f ′ ( x k ) , k ≥ 0. {\displaystyle x_{k+1}=x_{k}-[f''(x_{k})]^{-1}f'(x_{k}),\qquad k\geq 0

) ( α − 2 ) ⋯ ( α − k + 1 ) k ! . {\displaystyle {\binom {\alpha }{k}}:={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.} Note that if

1988; K-4 1000 m: 1984), one silver (K-2 1000 m: 1988), and one bronze (K-1 500 m: 1988). His five Olympic medals is the record for the second most Olympic

( x ; k ) = { x k − 1 e − x 2 / 2 2 k / 2 − 1 Γ ( k 2 ) , x ≥ 0 ; 0 , otherwise . {\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma

) = α − β = ( ( 2 k − 1 ) − ( k − 1 ) ) b 2 n + 2 ( ∑ i = 0 n b i ) + ( k − k ) b n + 1 ( ∑ i = 0 n b i ) + ( ( k − 1 ) − ( 2 k − 1 ) ) ( ∑ i = 0 n b

}{q}}(2q-\ell )^{2n},} E 2 n = ∑ k = 1 2 n ( − 1 ) k 1 2 k ∑ ℓ = 0 2 k ( − 1 ) ℓ ( 2 k ℓ ) ( k − ℓ ) 2 n . {\displaystyle E_{2n}=\sum _{k=1}^{2n}(-1)^{k}{\frac {1}{2^{k}}}\sum

1 x ¨ 1 = − ( k 1 + k 2 ) x 1 + k 2 x 2 m 2 x ¨ 2 = k 2 x 1 − ( k 2 + k 3 ) x 2 {\displaystyle {\begin{cases}m_{1}{\ddot {x}}_{1}=-(k_{1

1 ) z Γ ( z + 1 ) ( k − 1 ) ! ∑ r = 1 k − 1 s ( k − 1 , r ) ∑ m = 0 r ( r m ) ( k − 2 ) r − m ζ ( z + 1 − m ) , ℜ ( z ) > k − 1 , k = 3 , 4 , 5 , … {\displaystyle

n = ∑ k = 1 n T k = ∑ k = 1 n k ( k + 1 ) 2 = ∑ k = 1 n ( ∑ i = 1 k i ) {\displaystyle Te_{n}=\sum _{k=1}^{n}T_{k}=\sum _{k=1}^{n}{\frac {k(k+1)}{2}}=\sum

) ( α − 2 ) ⋯ ( α − k + 1 ) k ! . {\displaystyle {\binom {\alpha }{k}}:={\frac {\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}}.} Note that if

as an infimum λ ∗ ( E ) = inf { ∑ k = 1 ∞ ℓ ( I k ) : ( I k ) k ∈ N  is a sequence of open intervals with  E ⊂ ⋃ k = 1 ∞ I k } . {\displaystyle \lambda

k b k + ∑ k = 1 ∞ c k b − k . {\displaystyle (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots )_{b}=\sum _{k=0}^{n}a_{k}b^{k}+\sum _{k=1}^{\infty }c_{k}b^{-k}

( x ; k ) = { x k − 1 e − x 2 / 2 2 k / 2 − 1 Γ ( k 2 ) , x ≥ 0 ; 0 , otherwise . {\displaystyle f(x;k)={\begin{cases}{\dfrac {x^{k-1}e^{-x^{2}/2}}{2^{k/2-1}\Gamma

A k + 1 {\displaystyle A_{k+1}} and morphism A k → A k + 1 {\displaystyle A_{k}\to A_{k+1}} such that A k − 1 → A k → A k + 1 {\displaystyle A_{k-1}\to

(1948: K-1 1000 m, K-1 10000 m; 1952: K-1 1000 m, 1956: K-1 1000 m, K-1 10000 m; 1960: K-2 1000 m), one silver (1952: K-1 10000 m), and one bronze (K-1 1000

⁠ k 1 , k 2 ∈ K {\displaystyle k_{1},k_{2}\in K} ⁠ and ⁠ v 1 , v 2 ∈ V {\displaystyle v_{1},v_{2}\in V} ⁠, one has ( k 1 + v 1 ) ( k 2 + v 2 ) = k 1 k

collections { E k } k = 1 ∞ {\displaystyle \{E_{k}\}_{k=1}^{\infty }} of pairwise disjoint sets in Σ, μ ( ⋃ k = 1 ∞ E k ) = ∑ k = 1 ∞ μ ( E k ) . {\displaystyle

≈ ∑ k = 1 N 1 2 ( f ( x k − 1 ) + f ( x k ) ) Δ x , {\displaystyle \int _{a}^{b}f(x)\,dx\approx \sum _{k=1}^{N}{\tfrac {1}{2}}{\bigl (}f(x_{k-1})+f(x_{k}){\bigr

result. A cos ⁡ ( ω 1 t + k 1 ) cos ⁡ ( ω 2 t + k 2 ) = A 2 cos ⁡ [ ( ω 1 t + k 1 ) + ( ω 2 t + k 2 ) ] + A 2 cos ⁡ [ ( ω 1 t + k 1 ) − ( ω 2 t + k 2 ) ]

+ k − 1 k ) = ( n + k − 1 ) ! k ! ( n − 1 ) ! = n ( n + 1 ) ( n + 2 ) ⋯ ( n + k − 1 ) k ! , {\displaystyle \left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose

the identity ∑ k = 1 ∞ − 1 ln ⁡ ( 1 − p ) p k k = 1. {\displaystyle \sum _{k=1}^{\infty }{\frac {-1}{\ln(1-p)}}\;{\frac {p^{k}}{k}}=1.} This leads directly

π ∑ n = k ∞ ( − 1 ) n ( n k ) ( a − 1 ) n − k ∫ 0 2 π d θ = ∑ n = k ∞ ( − 1 ) n ( n k ) ( a − 1 ) n − k = ( − 1 ) k ∑ m = 0 ∞ ( m + k k ) ( 1 − a ) m

( k ) = { 1 / 2 if  k = − 1 , 1 / 2 if  k = + 1 , 0 otherwise. {\displaystyle f(k)=\left\{{\begin{matrix}1/2&{\text{if }}k=-1,\\1/2&{\text{if }}k=+1,\\0&{\text{otherwise

1 − ⋯ − g k − 1 ∼ g k {\displaystyle f-g_{1}-\cdots -g_{k-1}\sim g_{k}} takes its full meaning if g k + 1 = o ( g k ) {\displaystyle g_{k+1}=o(g_{k})}

indices a k − 1 + ∑ i = 0 j − 1 | Q i k | , … , a k − 1 + ( ∑ i = 0 j | Q i k | ) − 1 {\textstyle a_{k-1}+\sum _{i=0}^{j-1}|Q_{i}^{k}|,\dots ,a_{k-1}+\left(\sum

silvers (K-1 1000 m and K-2 500 m: both 1982), and a bronze (K-1 1000 m: 1983). He won both of his Olympic gold medals at Los Angeles in 1984, in the K-1 1000

− 2 x k − ∑ k = 1 ∞ a k − 1 x k + 2 ∑ k = 0 ∞ a k x k . {\displaystyle 1=\sum _{k=2}^{\infty }a_{k-2}x^{k}-\sum _{k=1}^{\infty }a_{k-1}x^{k}+2\sum _{k=0}^{\infty

length k {\displaystyle k} is: w a ( k ) = a 2 k ( 1 − a ) k − 1 . {\displaystyle w_{a}(k)=a^{2}k(1-a)^{k-1}{\text{.}}} In this equation, k is the number

E 2 n + F 2 n = ∏ k = 1 n ( E 2 + 2 E F cos ( 2 k − 1 ) π 2 n + F 2 ) = ∏ k = 1 n ( E 2 − 2 E F cos ( 2 k − 1 ) π 2 n + F 2 ) {\displaystyle

function of p 1 , … , p k − 1 {\displaystyle p_{1},\ldots ,p_{k-1}} is [ ∏ i = 1 k − 1 B ( a i , b i ) ] − 1 p k b k − 1 − 1 ∏ i = 1 k − 1 [ p i a i − 1 ( ∑

Women's C–2 500 m results Women's K-1 200 m results Women's K-1 500 m results Women's K-1 1000 m results Women's K-1 5000 m results Women's K-2 200 m results

= 2 k + 1 + ∑ i = 1 n ( 2 k + 1 ) b 2 i + ( 2 k + 1 ) b 2 i − 1 = 2 k + 1 + ∑ i = 1 2 n ( 2 k + 1 ) b i = ∑ i = 0 2 n ( 2 k + 1 ) b i = ( 2 k + 1 ) ∑

authors define the range as 0 ≤ k < 1 {\displaystyle 0\leq k<1} , but some define it as − 1 < k < 1 {\displaystyle -1<k<1\,} . Allowing negative values

K − 1 {\displaystyle \alpha \approx {\frac {0.038}{T_{m}}}-7.0\cdot 10^{-6}\,\mathrm {K} ^{-1}} In the table below, the range for α is from 10−7 K−1 for

following formula w ( n , g ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 w ( n − k 1 − k 2 , g − 2 ) = ∑ k 1 = 0 n ∑ k 2 = 0 n − k 1 ⋯ ∑ k g = 0 n − ∑ j = 1 g − 1 k

t h ( i , j , k − 1 ) , {\displaystyle \mathrm {min} {\Big (}\mathrm {shortestPath} (i,j,k-1),} s h o r t e s t P a t h ( i , k , k − 1 ) + s h o r t

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

0 , … , k − 1 [ a , b ] = [ t 0 , t 1 ) ∪ [ t 1 , t 2 ) ∪ ⋯ ∪ [ t k − 2 , t k − 1 ) ∪ [ t k − 1 , t k ) ∪ [ t k ] a = t 0 ≤ t 1 ≤ ⋯ ≤ t k − 1 ≤ t k =

(plural), placed before the numerals followed by "/-". Amounts less than K. 1/- are typically denoted with the number of pyas following "-/" (e.g. -/50

winningest medalist in men's kayak with his 21st career medal, earning that in a K-1 200 m silver. Rauhe eclipsed that record he had tied at the previous championships

= ∏ k = 1 ∞ ( 1 + x k ) = ∏ k = 1 ∞ 1 1 − x 2 k − 1 . {\displaystyle \sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty

( D f 2 ∘ f 3 . . n ) ⋯ ( D f n − 1 ∘ f n . . n ) D f n = ∏ k = 1 n [ D f k ∘ f ( k + 1 ) . . n ] {\displaystyle {\begin{aligned}Df_{1\,.\,.\,n}&=(Df_{1}\circ

x , t ) = cos ⁡ ( k 1 x − ω 1 t ) + cos ⁡ ( k 2 x − ω 2 t ) = 2 cos ⁡ ( ( k 2 − k 1 ) x − ( ω 2 − ω 1 ) t 2 ) cos ⁡ ( ( k 2 + k 1 ) x − ( ω 2 + ω 1 )

n_{1}+\cdots +n_{k}=n\}.\,} Its number of elements is ( n + k − 1 k − 1 ) . {\displaystyle {n+k-1 \choose k-1}.} In matrix notation, E ⁡ ( X ) = n p , {\displaystyle

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

the interval [ k τ , ( k + 1 ) τ ] {\displaystyle [k\tau ,(k+1)\tau ]} as y k = 1 τ ∫ k τ ( k + 1 ) τ x ˙ ( t ) d t = x ( ( k + 1 ) τ ) − x ( k τ ) τ {\displaystyle

season to obtain k 1 {\displaystyle k_{1}} , k 0 {\displaystyle k_{0}} , and k − 1 {\displaystyle k_{-1}} . Values of k 1 {\displaystyle k_{1}} , k 0 {\displaystyle

∑ k = 1 ∞ 1 k ( 2 k − 1 ) = lim n → ∞ 2 ∑ k = 1 n ( 1 2 k − 1 − 1 2 k ) = lim n → ∞ 2 ∑ k = 1 n ( 1 2 k − 1 + 1 2 k − 1 k ) = 2 lim n → ∞ ( ∑ k = 1 2

k + 1 f ( x ; c , k , λ ) = c k λ ( x λ ) c − 1 [ 1 + ( x λ ) c ] − k − 1 {\displaystyle {\begin{aligned}f(x;c,k)&=ck{\frac {x^{c-1}}{(1+x^{c})^{k+1}}}\\[6pt]f(x;c

k + 1 ) n k {\displaystyle \zeta (k+1)n^{k}} : σ k ( n ) = ζ ( k + 1 ) n k [ 1 + ( − 1 ) n 2 k + 1 + 2 cos ⁡ 2 π n 3 3 k + 1 + 2 cos ⁡ π n 2 4 k + 1 +

x , t ) = cos ⁡ ( k 1 x − ω 1 t ) + cos ⁡ ( k 2 x − ω 2 t ) = 2 cos ⁡ ( ( k 2 − k 1 ) x − ( ω 2 − ω 1 ) t 2 ) cos ⁡ ( ( k 2 + k 1 ) x − ( ω 2 + ω 1 )

k + 1 ) n k {\displaystyle \zeta (k+1)n^{k}} : σ k ( n ) = ζ ( k + 1 ) n k [ 1 + ( − 1 ) n 2 k + 1 + 2 cos ⁡ 2 π n 3 3 k + 1 + 2 cos ⁡ π n 2 4 k + 1 +

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

any k − 1 are known, the remaining one is uniquely determined. Basically, one can say, there are only k − 1 freely determined binn counts, thus k − 1 degrees

\varepsilon } . We have | ∑ i = k + 1 s ∫ Γ i A d x + B d y | ≤ 1 2 ε ∑ i = k + 1 s Λ i . {\displaystyle \left\vert \sum _{i=k+1}^{s}\int _{\Gamma _{i}}A\,dx+B\

, k 1 … k n ⟩ = a i † ( k 1 ) ⋯ a i † ( k n ) | i , 0 ⟩ } , {\displaystyle {\mathcal {H}}_{\rm {i}}=\operatorname {span} \{\left|\mathrm {i} ,k_{1}\ldots

fourteen golds (K-1 500 m: 2013, 2014, 2018; K-2 200 m: 2011; K-2 500 m: 2009, 2010, 2015, 2018; K-4 500 m: 2009, 2011, 2013, 2014, 2018 and K-1 200 m relay:

k − 1 r = k [ 1 + ( k − 1 ) r 2 + ⋯ ] {\displaystyle \lambda _{k}={\frac {p_{k}}{p}}={\frac {r_{k}}{r}}={\frac {(1+r)^{k}-1}{r}}=k\left[1+{\frac {(k-1)r}{2}}+\cdots

{\displaystyle r=0} , go back to step 3. Calculate s = k − 1 ( z + r d A ) mod n {\displaystyle s=k^{-1}(z+rd_{A})\,{\bmod {\,}}n} . If s = 0 {\displaystyle

described with k1, k−1 and k2, which are the rate constants of their corresponding intermediate reaction steps: L n M − L ⇌ + L , k − 1 − L , k 1 L n M − ◻ →

numbers k 1 , k 2 , … , k u {\displaystyle k_{1},k_{2},\dots ,k_{u}} are powers of (not necessarily distinct) primes, or k 1 {\displaystyle k_{1}} divides

contain at least k + 1 objects. For arbitrary n and m, this generalizes to k + 1 = ⌊ ( n − 1 ) / m ⌋ + 1 = ⌈ n / m ⌉ {\displaystyle k+1=\lfloor (n-1)/m\rfloor

variable is developed as V 1 / k 1 V 2 / k 2 ∼ F k 1 , k 2 ′ ( λ ) {\displaystyle {\frac {V_{1}/k_{1}}{V_{2}/k_{2}}}\sim F'_{k_{1},k_{2}}(\lambda )} If J ∼

season to obtain k 1 {\displaystyle k_{1}} , k 0 {\displaystyle k_{0}} , and k − 1 {\displaystyle k_{-1}} . Values of k 1 {\displaystyle k_{1}} , k 0 {\displaystyle

virial is − 1 2 ∑ k = 1 N F k ⋅ r k = 1 2 ∑ k = 1 N ∑ j < k d V j k d r j k r j k = 1 2 ∑ k = 1 N ∑ j < k n α r j k n − 1 r j k = 1 2 ∑ k = 1 N ∑ j < k n V

B k + 1 ( x k + 1 − x k ) = ∇ f ( x k + 1 ) − ∇ f ( x k ) . {\displaystyle B_{k+1}(\mathbf {x} _{k+1}-\mathbf {x} _{k})=\nabla f(\mathbf {x} _{k+1})-\nabla

the fixed numerical value of the Boltzmann constant k to 1.380649×10−23 J⋅K−1, (J = kg⋅m2⋅s−2), given the definition of the kilogram, the metre, and the

seven silvers (K-1 1000 m: 2010, K-2 500 m: 2013, K-4 200 m: 2007, 2009, K-4 500 m: 1997, 1998, 2007), and three bronzes (K-1 500 m: 2001, K-1 1000 m: 1999

k + 1 {\displaystyle s_{k+1}} and t k + 1 {\displaystyle t_{k+1}} are coprime. The relation a s k + 1 + b t k + 1 = 0 {\displaystyle as_{k+1}+bt_{k+1}=0}

d 1 − k ζ ( k ) − 1 = ζ ( k − 1 ) ζ ( k ) . {\displaystyle \mathrm {E} (k)=\sum _{d=1}^{\infty }d^{1-k}\zeta (k)^{-1}={\frac {\zeta (k-1)}{\zeta (k)}}

k = 1 ∞ ( − 1 ) k − 1 e − 2 k 2 x 2 = 2 π x ∑ k = 1 ∞ e − ( 2 k − 1 ) 2 π 2 / ( 8 x 2 ) , {\displaystyle \operatorname {Pr} (K\leq x)=1-2\sum _{k=1}^{\infty

its k nearest neighbors (k is a positive integer, typically small). If k = 1, then the object is simply assigned to the class of that single nearest

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

with a ⁠Z/1 − K⁠ impedance between the first node and ground and a ⁠KZ/K − 1⁠ impedance between the second node and ground. Since impedance varies inversely

− 1 k ) ( N − r − k K − k ) ( N K ) = 1 ( N K ) ∑ k = 0 K ( k + r − 1 k ) ( N − r − k K − k ) = 1 ( N K ) ( N K ) = 1 , {\displaystyle \sum _{k=0}^{K}\Pr(X=k)=\sum

i k + 1 , … , i n d x i k + 1 ∧ ⋯ ∧ d x i n . {\displaystyle {\star }\alpha ={\frac {1}{(n-k)!}}({\star }\alpha )_{i_{k+1},\dots ,i_{n}}dx^{i_{k+1}}\wedge

( n , k ) = ∑ i = 0 k ( − 1 ) i ( n + 1 i ) ( k + 1 − i ) n . {\displaystyle A(n,k)=\sum _{i=0}^{k}(-1)^{i}{\binom {n+1}{i}}(k+1-i)^{n}.} For fixed n

x k + 1 − x k x k − x k − 1 | log ⁡ | x k − x k − 1 x k − 1 − x k − 2 | . {\displaystyle q\approx {\frac {\log \left|\displaystyle {\frac {x_{k+1

September 2021. "Men's K-1 200" (PDF). Retrieved 19 September 2021. "Men's K-1 500" (PDF). Retrieved 19 September 2021. "Men's K-1 1000" (PDF). Retrieved

k ) ( 1 − x − 2 ) k = n 2 ∑ k = 0 ⌊ n 2 ⌋ ( − 1 ) k ( n − k − 1 ) ! k ! ( n − 2 k ) !   ( 2 x ) n − 2 k  for  n > 0 = n ∑ k = 0 n ( − 2 ) k ( n + k −

following construction: p e = ∑ k p k 12 ^ = ind. ∑ k p k 1 ^ p k 2 ^ = ∑ k n k 1 N n k 2 N = 1 N 2 ∑ k n k 1 n k 2 {\displaystyle p_{e}=\sum _{k}{\widehat {p_{k12}}}{\overset

This is a list of medalists from the ICF Canoe Sprint World Championships in women's kayak. Debuted: 1994. Debuted: 1938. Not held: 1948. Resumed: 1950

c g = lim k 1 → k 2 ω 1 − ω 2 k 1 − k 2 = lim k 1 → k 2 Ω ( k 1 ) − Ω ( k 2 ) k 1 − k 2 = d Ω ( k ) d k . {\displaystyle c_{g}=\lim _{k_{1}\,\to \,k_{2}}{\frac

A rocket attack was carried out on the K-1 Air Base in Kirkuk province in Iraq on 27 December 2019. The air base was one of many Iraqi military bases

Time K-1 200 m  Caroline Brunet (CAN)  Josefa Idem (ITA)  Rita Kőbán (HUN) K-1 500 m  Caroline Brunet (CAN)  Josefa Idem (ITA)  Rita Kőbán (HUN) K-1 1000

Men's C–1 200 m Relay Results Men's K–1 200 m Results Men's K–1 500 m Results Men's K–1 1000 m Results Men's K–1 5000 m Results Men's K–2 200 m Results

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

Event Gold Silver Bronze K-1 200 m  Ronald Rauhe (GER)  Oleksiy Slivinskiy (UKR)  Anton Ryakhov (UZB) K-1 500 m  Ákos Verecki (HUN)  Anton Ryakhov (UZB)

∇ f ( x ) = ∑ k = 1 M λ k ∇ g k ( x ) ⟺ ∇ f ( x ) − ∑ k = 1 M λ k ∇ g k ( x ) = 0   . {\displaystyle \nabla f(\mathbf {x} )=\sum _{k=1}^{M}\lambda _{k}\

( 2 k + 1 ) . {\displaystyle (2^{r}+1)\mid (2^{k}+1).} Because 1 < 2 r + 1 < 2 k + 1 {\displaystyle 1<2^{r}+1<2^{k}+1} , it follows that 2 k + 1 {\displaystyle

&\ {\mbox{ if }}\ k<1\ {\mbox{ or }}\ N<k~.\end{cases}}} where HN is a normalization constant: The Nth harmonic number: H N ≡ ∑ k = 1 N   1   k   . {\displaystyle

Fu Factory K-1 World GP K-1 World GP 2006 K-1 Premium 2005 Dynamite!! K-1 World GP 2005 K-1 World Max 2005 K-1 Premium 2004 Dynamite!! K-1 World Grand

^{k}|n_{1}+\cdots +n_{k}=n\}.\,} Its number of elements is ( n + k − 1 k − 1 ) . {\displaystyle {n+k-1 \choose k-1}.} In matrix notation, E ⁡ ( X ) = n p , {\displaystyle

− τ ) B u ( τ ) d τ x [ k + 1 ] = e A ( k + 1 ) T x ( 0 ) + ∫ 0 ( k + 1 ) T e A [ ( k + 1 ) T − τ ] B u ( τ ) d τ x [ k + 1 ] = e A T [ e A k T x ( 0

context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. Some theories further generalize

2-spinors. In three dimensions, the Levi-Civita symbol is defined by: ε i j k = { + 1 if  ( i , j , k )  is  ( 1 , 2 , 3 ) , ( 2 , 3 , 1 ) ,  or  ( 3 , 1 ,

1972: K-1 1000 m) and one silver (1968: K-1 1000 m). Shaparenko also won 13 medals at the ICF Canoe Sprint World Championships with seven golds (K-1 1000

(K-1 200 m: 1994, 1995; K-1 500 m: 1995, K-2 200 m: 1994, K-4 200 m: 1994, 1998, 1999; K-4 500 m: 1986, 1999), ten silvers (K-1 500 m: 1991, 1994; K-1

adopted at the 26th CGPM in late 2018, with a value of kB = 1.380649×10−23 J⋅K−1. In fundamental physics, the mapping E=kBT which converts between the characteristic

Time K-1 200 m  Caroline Brunet (CAN)  Josefa Idem (ITA)  Rita Kőbán (HUN) K-1 500 m  Caroline Brunet (CAN)  Josefa Idem (ITA)  Rita Kőbán (HUN) K-1 1000

Event Gold Silver Bronze K-1 200 m  Ronald Rauhe (GER)  Oleksiy Slivinskiy (UKR)  Anton Ryakhov (UZB) K-1 500 m  Ákos Verecki (HUN)  Anton Ryakhov (UZB)

positive, we set ∫ ( ∑ k a k 1 S k ) d μ = ∑ k a k ∫ 1 S k d μ = ∑ k a k μ ( S k ) {\displaystyle \int \left(\sum _{k}a_{k}1_{S_{k}}\right)\,d\mu =\sum

C} , and v = v k + P − p k p k + 1 − p k ( v k + 1 − v k ) . {\displaystyle v=v_{k}+{\frac {P-p_{k}}{p_{k+1}-p_{k}}}(v_{k+1}-v_{k}).} The 50% weighted percentile

( z − z 0 ) k + c k + 1 ( z − z 0 ) k + 1 + ⋯ + c n ( z − z 0 ) n . {\displaystyle p(z)=a+c_{k}(z-z_{0})^{k}+c_{k+1}(z-z_{0})^{k+1}+\cdots +c_{n}(z-z_{0})^{n}

_{t=1}^{m}k_{t}=n,} and ( n k 1 , k 2 , … , k m ) = n ! k 1 ! k 2 ! ⋯ k m ! {\displaystyle {n \choose k_{1},k_{2},\ldots ,k_{m}}={\frac {n!}{k_{1}!\,k_{2}!\cdots

partial sum of the series is the triangular number ∑ k = 1 n k = n ( n + 1 ) 2 , {\displaystyle \sum _{k=1}^{n}k={\frac {n(n+1)}{2}},} which increases without

satisfies the recurrence formula p k p k − 1 = a + b k , k = 1 , 2 , 3 , … {\displaystyle {\frac {p_{k}}{p_{k-1}}}=a+{\frac {b}{k}},\qquad k=1,2,3,\dots

\deg H_{k}<\deg G.} Setting F k − 1 = F C + Q , {\displaystyle F_{k-1}=FC+Q,} one gets F G k = H k G k + F k − 1 G k − 1 , {\displaystyle {\frac {F}{G^{k}}}={\frac

log ⁡ log ⁡ n ) k − 1 ( k − 1 ) ! , {\displaystyle \pi _{k}(n)\sim \left({\frac {n}{\log n}}\right){\frac {(\log \log n)^{k-1}}{(k-1)!}},} a result of

years older he won two medals – gold in the K-4 500 m and silver in the K-1 500 m, an unprecedented achievement for a fifteen-year-old. After winning

formula (also implemented in the PyAstronomy package for Python): T = 4600 K ( 1 0.92 ( B - V ) + 1.7 + 1 0.92 ( B - V ) + 0.62 ) . {\displaystyle T=4600\

, k − 1 ) = k ( k + 1 ) L ( n , k + 1 ) + 2 k L ( n , k ) + L ( n , k − 1 ) {\displaystyle {\begin{aligned}L(n+1,k)&=(n+k)L(n,k)+L(n,k-1)\\&=k(k+1)L(n

an infinite sum of sinusoidal waves: x ( t ) = 4 π ∑ k = 1 ∞ sin ⁡ ( 2 π ( 2 k − 1 ) f t ) 2 k − 1 = 4 π ( sin ⁡ ( ω t ) + 1 3 sin ⁡ ( 3 ω t ) + 1 5 sin

k + 1 ) a n , k + 1 k = 0 , a n , k − 1 − ( k + 1 ) a n , k + 1 k > 0 , {\displaystyle a_{n+1,k}={\begin{cases}-(k+1)a_{n,k+1}&k=0,\\a_{n,k-1}-(k+1)a_{n

one has scalar multiplication on the field: k 1 ⊗ k 2 = k 1 k 2 . {\displaystyle k_{1}\otimes k_{2}=k_{1}k_{2}.} The most difficult to verify is the compatibility

{M} _{o1n}^{(1)}(k_{1},\mathbf {r} )\right),} H 1 = − k 1 ω μ 1 ∑ n = 1 ∞ E n ( d n M e 1 n ( 1 ) ( k 1 , r ) + i c n N o 1 n ( 1 ) ( k 1 , r ) ) . {\displaystyle

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

= k 1 + k 2 2 {\displaystyle A={\frac {k_{1}+k_{2}}{2}}} and B = k 1 − k 2 2 . {\displaystyle B={\frac {k_{1}-k_{2}}{2}}.} This function average, f3

C-2 1000m men C-4 500m men women Kayak events K-1 200m men women K-1 500m men women K-1 1000m men women K-1 5000m men women K-2 200m women K-2 500m men

A^{k}} because A k v = A k − 1 A v = λ A k − 1 v = ⋯ = λ k v . {\displaystyle A^{k}{\textbf {v}}=A^{k-1}A{\textbf {v}}=\lambda A^{k-1}{\textbf {v}}=\dots =\lambda

finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution

x k + 1 = 2 x k x 1 − x k − 1 , y k + 1 = 2 y k x 1 − y k − 1 . {\displaystyle \displaystyle {\begin{aligned}x_{k+1}&=2x_{k}x_{1}-x_{k-1},\\y_{k+1}&=2y_{k}x_{1}-y_{k-1}

are known for having very low coefficients of thermal expansion (≈3 × 10−6 K−1 at 20 °C), making them more resistant to thermal shock than any other common

x k } = p k    for    k = 1 ,   2 ,   …   {\displaystyle \operatorname {Pr} \{\ X=x_{k}\}=p_{k}\qquad ~{\mbox{ for }}~\quad k=1,\ 2,\ \ldots ~} then the

with eight golds (K-1 1000 m: 1985, K-1 10000 m: 1986, K-2 500 m: 1987, K-4 1000 m: 1986, 1987, 1989, 1990, 1991), four silvers (K-1 1000 m: 1986, 1987

at the final kinetic expression: [ A ] = k -1 [ A ] 0 k 1 + k -1 + k 1 [ A ] 0 k 1 + k -1 exp ⁡ ( − k 1 + k -1 ) t {\displaystyle [A]={\frac

mathematical notation for summation: ∑ k = 1 n k 3 = ( ∑ k = 1 n k ) 2 . {\displaystyle \sum _{k=1}^{n}k^{3}=\left(\sum _{k=1}^{n}k\right)^{2}.} This identity

12 n k ( k + 1 ) ∑ j = 1 k ( r ¯ ⋅ j − k + 1 2 ) 2 {\displaystyle Q={\frac {12n}{k(k+1)}}\sum _{j=1}^{k}\left({\bar {r}}_{\cdot j}-{\frac {k+1}{2}}\right)^{2}}

is an integer, f ( k ; ρ ) = ρ ρ ! ( k − 1 ) ! ( k + ρ ) ! . {\displaystyle f(k;\rho )={\frac {\rho \,\rho !\,(k-1)!}{(k+\rho )!}}.} The parameter ρ {\displaystyle

1 ) ( 3 k + 1 ) − k ( 3 k + 1 ) = ( k + 1 ) ( 3 k + 1 ) ( 3 k + 1 ) − k ( 3 k + 1 ) = k ( 3 k + 1 ) ( 3 k + 1 ) + ( 3 k + 1 ) ( 3 k + 1 ) − k ( 3 k +

noticing that m j {\displaystyle m_{j}} has to be zero for j > n − k + 1 {\displaystyle j>n-k+1} leads to a somewhat simpler formula expressed in terms of partial

is thus given by (see binomial coefficient): 2 k + 1 ( n k + 1 ) {\displaystyle 2^{k+1}{n \choose {k+1}}} The extended f-vector for an n-orthoplex can

) k / ( 1 − τ ) {\textstyle dw/d\tau =-f'(k)k/(1-\tau )} , we have that: d w d x = − f ′ ( k ) k 1 − τ f ′ ( k ) 1 − τ [ τ f ′ ( k ) f ″ ( k ) + k ] =

2F(k) − 1 is the cdf of a half-logistic distribution. Specifically, G ( k ) = 1 − e − k 1 + e − k  for  k ≥ 0. {\displaystyle G(k)={\frac {1-e^{-k}}{1+e^{-k}}}{\text{

digits: A = a k a k − 1 . . . a 1 {\displaystyle A={a_{k}}{a_{k-1}}...{a_{1}}} B = b k b k − 1 . . . b 1 {\displaystyle B={b_{k}}{b_{k-1}}...{b_{1}}} such

are obtained by successive approximation: β j k + 1 = β j k + Δ β j , {\displaystyle {\beta _{j}}^{k+1}={\beta _{j}}^{k}+\Delta \beta _{j},} where a superscript

{R} ^{3}} , with the only "non-injectivity" being K ( 0 ) = K ( 1 ) {\displaystyle K(0)=K(1)} . Topologists consider knots and other entanglements such

expressed as [citation needed] ρ ( τ ) = ∑ k = 1 p a k y k − | τ | , {\displaystyle \rho (\tau )=\sum _{k=1}^{p}a_{k}y_{k}^{-|\tau |},} where y k {\displaystyle

sawtooth ( t ) = − 2 a π ∑ k = 1 ∞ ( − 1 ) k sin ⁡ ( 2 π k f t ) k {\displaystyle x_{\text{sawtooth}}(t)=-{\frac {2a}{\pi }}\sum _{k=1}^{\infty }{(-1)}^{k}{\frac

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200, 500, 1000 or 5000 metres

Event Gold Silver Bronze K-1 200 m  Carlos Pérez (ESP) 35.289  Tomasz Mendelski (POL) 35.697  Anton Ryakhov (RUS) 35.733 K-1 500 m  Nathan Baggaley (AUS)

in the Montreal program though four 500 m events for men were (C-1, C-2, K-1, and K-2). 1976 Summer Olympics official report Volume 3. pp. 167–85. (in

well-known identity: n 2 = ∑ k = 1 n ( 2 k − 1 ) , {\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1),} we obtain the following derivation: ∑ k = 1 n k 3 = 1 + 8 + 27 +

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200, 500, 1000 or 5000 metres

well-known identity: n 2 = ∑ k = 1 n ( 2 k − 1 ) , {\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1),} we obtain the following derivation: ∑ k = 1 n k 3 = 1 + 8 + 27 +

final, giving Mexico their first-ever world championship gold medal. The K-1 1000 m, the Blue Riband event, was won by Markus Oscarsson of Sweden. Germany's

defined for positive integers k ≥ 1 {\displaystyle k\geq 1} , and its probability mass function is given by P ( x = k ) = 1 ζ ( s ) k − s , {\displaystyle

(1988: K-1 500 m, K-4 1000 m) and two silver (1992: K-1 500 m, K-4 1000 m). He also won fourteen medals at the world championships: six golds (K-1 1000

and truncate: ⌈ total votes k + 1 ⌉ ≈ ⌊ total votes k + 1 + 1 ⌋ {\displaystyle \left\lceil {\frac {\text{total votes}}{k+1}}\right\rceil \approx \left\lfloor

given by x d = x c + x u − x c 2 K 1 r u 2 ( 1 − 1 − 4 K 1 r u 2 ) y d = y c + y u − y c 2 K 1 r u 2 ( 1 − 1 − 4 K 1 r u 2 ) , {\displaystyle {\begin{aligned}x_{\mathrm

1 ) ( 3 k + 1 ) − k ( 3 k + 1 ) = ( k + 1 ) ( 3 k + 1 ) ( 3 k + 1 ) − k ( 3 k + 1 ) = k ( 3 k + 1 ) ( 3 k + 1 ) + ( 3 k + 1 ) ( 3 k + 1 ) − k ( 3 k +

X_{0}=i)>0\}} The state is periodic if k > 1 {\displaystyle k>1} ; otherwise k = 1 {\displaystyle k=1} and the state is aperiodic. A state i is said

of unity, as ( z k ) n = z k n = ( z n ) k = 1 k = 1. {\displaystyle (z^{k})^{n}=z^{kn}=(z^{n})^{k}=1^{k}=1.} This is also true for negative exponents

gold in the K-1 500 m and bronze in the K-1 1000 m events. He also won two medals at the ICF Canoe Sprint World Championships with a gold (K-1 5000 m: 2010)

_{2}\left(1-{\frac {1}{(1+k)^{2}}}\right)~.} Let x = 1 k 1 + 1 k 2 + ⋯ {\displaystyle x={\cfrac {1}{k_{1}+{\cfrac {1}{k_{2}+\cdots }}}}} be the continued fraction

{1}{n}}\sum _{i=1}^{n}d_{i}^{2m}} , then S n ( 2 m ) = ( S n ( 2 ) ) m + ∑ k = 1 ⌊ m / 2 ⌋ ( m 2 k ) ( 2 k k ) R 2 k ( S n ( 2 ) − R 2 ) k ( S n ( 2 ) )

absolutely: ∑ k = 1 ∞ | ( a k − a k − 1 ) z k | = ∑ k = 1 ∞ | a k − a k − 1 | ⋅ | z | k ≤ ∑ k = 1 ∞ ( a k − 1 − a k ) {\displaystyle \sum _{k=1}^{\infty

distribution function is F ( k ) = 1 − τ T k 1 , {\displaystyle F(k)=1-{\boldsymbol {\tau }}{T}^{k}\mathbf {1} ,} for k = 1 , 2 , . . . {\displaystyle k=1

k = 1 d − 1 ( 2 η − 2 + d − k ) ( d − k ) ∏ k = 1 d − 1 [ B ( η + ( d − k − 1 ) / 2 , η + ( d − k − 1 ) / 2 ) ] d − k {\displaystyle C=2^{\sum _{k=1}^{d-1}(2\eta

K-2 races were ruled by the German paddlers. However, Katalin Kovács won K-1 1000 m and 500 m, and gained two silvers in the K-4 events, while former

(K-1 200 m: 1997, 1998, 1999, 2003; K-1 500 m: 1997, 1998, 1999; K-1 1000 m: 1997, 1999; K-4 200 m: 1995), seven silvers (K-1 200 m: 1995, 2002; K-1 500

won 26 medals at the ICF Canoe Sprint World Championships with 20 golds (K-1 200 m: 2007, 2009, 2010; K-2 200 m: 2005, 2006, 2009, 2010; K-2 500 m: 2005

Phytomenadione, also known as vitamin K1, phylloquinone, or phytonadione, is a vitamin found in food and used as a dietary supplement. It is on the World

( x ) − ( − 1 ) n d n f ( x ) d x n ) d x = e x ∑ k = 1 n ( − 1 ) k − 1 d k − 1 f ( x ) d x k − 1 + C {\displaystyle \int {e^{x}\left(f\left(x\right

∑ k = 1 n ( − 1 ) k − 1 ( k − 1 ) ! B n , k ( μ 1 ′ , … , μ n − k + 1 ′ ) , {\displaystyle \kappa _{n}=\sum _{k=1}^{n}(-1)^{k-1}(k-1)!B_{n,k}(\mu '_{1}

≥ 0 and ℓ ≥ 2 we have x k ( 1 − x ) ℓ = ( 1 − 2 x + x 2 ) x k ( 1 − x ) ℓ − 2 = ( 1 + x 2 ) x k ( 1 − x ) ℓ − 2 − 2 x k + 1 ( 1 − x ) ℓ − 2 . {\displaystyle

d t k + a k − 1 d k − 1 x d t k − 1 + ⋯ + a 1 d x d t + a 0 x = 0. {\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots

k 1 , n ) = 1 1 ≤ k 1 , k 2 , … , k s ≤ n gcd ( k 1 − 1 , k 2 , … , k s , n ) = φ ( n ) σ s − 1 ( n ) . {\displaystyle \sum _{\stackrel {1\leq k_{1}

defined as M x k + 1 = N x k + b , k ≥ 0 , {\displaystyle M\mathbf {x} ^{k+1}=N\mathbf {x} ^{k}+b,\quad k\geq 0,} or, equivalently, x k + 1 = x k + M − 1

the following: SFD b ⁡ ( n ) = ∑ i = 0 k − 1 d i ! . {\displaystyle \operatorname {SFD} _{b}(n)=\sum _{i=0}^{k-1}d_{i}!.} where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle

approximate Hessian B k + 1 {\displaystyle B_{k+1}} , or directly its inverse H k + 1 = B k + 1 − 1 {\displaystyle H_{k+1}=B_{k+1}^{-1}} using the Sherman–Morrison

i.e. d d x x k = k x k − 1 . {\displaystyle {\frac {d}{dx}}x^{k}=kx^{k-1}.} When n = k + 1 {\displaystyle n=k+1} , d d x x k + 1 = d d x ( x k ⋅ x ) =

) − k 1 ( 1 − θ i t ) − k 2 = ( 1 − θ i t ) − ( k 1 + k 2 ) . {\displaystyle \varphi _{X+Y}(t)=\varphi _{X}(t)\varphi _{Y}(t)=(1-\theta it)^{-k_{1}}(1-\theta

= λ k − 1 p k − 1 ( t ) + μ k + 1 p k + 1 ( t ) − ( λ k + μ k ) p k ( t ) , k = 1 , 2 , … , {\displaystyle p_{k}^{\prime }(t)=\lambda _{k-1}p_{k-1}(t)+\mu

Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men women K-2 1000 m men K-4 500 m women K-4

success. Its probability mass function is p X ( k ) = ( 1 − p ) k − 1 p {\textstyle p_{X}(k)=(1-p)^{k-1}p} . An example is tossing a coin until the first

equations: d N 1 d t = r 1 N 1 K 1 − N 1 − α N 2 K 1 {\displaystyle {\frac {dN_{1}}{dt}}=r_{1}N_{1}{\frac {K_{1}-N_{1}-\alpha N_{2}}{K_{1}}}\,} Island biogeography:

men's 500 m events (C-1, C-2, K-1 and K-2) at the Summer Olympics with the 200 m events (men's C-1, K-1 and K-2; and women's K-1) for the upcoming Summer Olympics

for k = 1:n r = sqrt(L(k, k)^2 + x(k)^2); c = r / L(k, k); s = x(k) / L(k, k); L(k, k) = r; if k < n L((k+1):n, k) = (L((k+1):n, k) + s * x((k+1):n))

m bronze. In the K-1 1000m however, he could only finish 17th. At the 2002 European under-23 Championships in Zagreb, he won the K-1 1000 m gold medal

total welfare is then W = ∑ k = 1 n V k ( D k ) − ∑ k = 1 n C k ( S k ) {\displaystyle W=\sum _{k=1}^{n}V_{k}(D_{k})-\sum _{k=1}^{n}C_{k}(S_{k})} The economic

R_{i}} with r k , k R i − r i , k R k r k − 1 , k − 1 . {\textstyle {\frac {r_{k,k}R_{i}-r_{i,k}R_{k}}{r_{k-1,k-1}}}.} This produces a row echelon form

i = k + 1 n ( ∂ i a ) d ξ i ∧ d ξ 2 ∧ ⋯ ∧ d ξ k , d α = ∑ i = k + 1 n ( ∂ i a ) d ξ i ∧ d ξ 1 ∧ d ξ 2 ∧ ⋯ ∧ d ξ k , ι X ( d α ) = − ∑ i = k + 1 n ( ∂

K-1 200 m  Caroline Brunet (CAN)  Josefa Idem (ITA)  Éva Dónusz (HUN) K-1 500 m  Caroline Brunet (CAN)  Katrin Borchert (AUS)  Josefa Idem (ITA) K-1 1000

defined by the following formula. X = F 0 k 1 ( 1 − r 2 ) 2 + ( 2 ζ r ) 2 . {\displaystyle X={F_{0} \over k}{1 \over {\sqrt {(1-r^{2})^{2}+(2\zeta r)^{2}}}}

i 2 1 ≤ i ≤ k 1 k + 1 ≤ i ≤ n − k 0 otherwise {\displaystyle \sigma _{i}={\begin{cases}{\sqrt {1+\gamma _{i}^{2}}}&1\leq i\leq k\\1&k+1\leq i\leq

⟩ | 2 = | ∑ k = 1 n u k v ¯ k | 2 ≤ ⟨ u , u ⟩ ⟨ v , v ⟩ = ( ∑ k = 1 n u k u ¯ k ) ( ∑ k = 1 n v k v ¯ k ) = ∑ j = 1 n | u j | 2 ∑ k = 1 n | v k | 2

function, or connection function is K k c c ( x , z ) = ∑ k 1 , … , k d , ∑ j = 1 d k j = k k ! k 1 ! ⋯ k d ! ( 1 d ) k ∏ j = 1 d 1 ⌈ 2 k j x j ⌉ = ⌈ 2 k

= ( k     k + 1 ) ⋅ ( k + 1     k + 2 ) ⋯ ( l − 1     l ) ⋅ ( l − 2     l − 1 ) ⋯ ( k     k + 1 ) . {\displaystyle (k~~l)=(k~~k+1)\cdot (k+1~~k+2)\cdots

Event Gold Silver Bronze K-1 200 m  Ronald Rauhe (GER) 35.798  Vince Fehérvári (AUS) 36.411  Anton Ryakhov (UZB) 36.417 K-1 500 m  Nathan Baggaley (AUS)

x = ∑ k = 1 ∞ a k ⋅ 10 − k {\displaystyle x=\sum _{k=1}^{\infty }a_{k}\cdot 10^{-k}} and y = ∑ k = 1 ∞ b k ⋅ 10 − k {\displaystyle y=\sum _{k=1}^{\infty

including general relativity. Let x1, ..., xn be variables, denote for k ≥ 1 by pk(x1, ..., xn) the k-th power sum: p k ( x 1 , … , x n ) = ∑ i = 1 n

including general relativity. Let x1, ..., xn be variables, denote for k ≥ 1 by pk(x1, ..., xn) the k-th power sum: p k ( x 1 , … , x n ) = ∑ i = 1 n

f ( n ) = L − a max ( − 1 , min ( k − 3 , 9 − k , 1 ) ) {\displaystyle f(n)=L-a\max(-1,\min(k-3,9-k,1))} where k , n ∈ R ≥ 0 {\displaystyle k,n\in \mathbb

where k 1 = γ / J 1 {\displaystyle k_{1}=\gamma /J_{1}} , k 2 = γ / J 2 {\displaystyle k_{2}=\gamma /J_{2}} , k = k 1 + k 2 {\displaystyle k=k_{1}+k_{2}}

Slalom C-1 men C-2 men K-1 men women Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men

variable β K + 1 {\displaystyle \beta _{K+1}} with a newly introduced last column of X being unity i.e., X i ( K + 1 ) = 1 {\displaystyle X_{i(K+1)}=1} for

∂ k 1 + ⋯ + k n f ( ζ 1 , ζ 2 , … , ζ n ) ∂ z 1 k 1 ⋯ ∂ z n k n | ≤ M k 1 ! ⋯ k n ! r 1 k 1 ⋯ r n k n {\displaystyle \left|{\frac {\partial ^{k_{1}+\cdots

would be to use two keys ( K 1 , K 2 ) {\displaystyle (K1,K2)} instead of one, and encrypt each block twice: E K 2 ( E K 1 ( plaintext ) ) {\displaystyle

1 1 φ ( y ) cos ⁡ ( 2 k + 1 ) π y 2 d y = ∫ − 1 1 ( a cos ⁡ π y 2 cos ⁡ ( 2 k + 1 ) π y 2 + a ′ cos ⁡ 3 π y 2 cos ⁡ ( 2 k + 1 ) π y 2 + ⋯ ) d y {\displaystyle

medals at the ICF Canoe Sprint World Championships with four golds (K-1 500 m: 1997, K-1 1000 m: 1997, K-4 500 m: 1997, K-4 1000 m: 1999), five silvers (K-2

( k − 1 ) cos ⁡ θ + r cos ⁡ ( ( k − 1 ) θ ) y ( θ ) = r ( k − 1 ) sin ⁡ θ − r sin ⁡ ( ( k − 1 ) θ ) {\displaystyle {\begin{aligned}&x(\theta )=r(k-1)\cos

Event Gold Silver Bronze K-1 200 m  Ronald Rauhe (GER)  Anton Ryakhov (UZB)  Roman Petrukanecas (LTU) K-1 500 m  Nathan Baggaley (AUS)  Petar Merkov (BUL)

k ( x ) ≤ f k + 1 ( x ) {\displaystyle (1-\varepsilon )s(x)\leq f_{k}(x)\leq f_{k+1}(x)} so x ∈ B k + 1 s , ε . {\displaystyle x\in B_{k+1}^{s,\varepsilon

arithmetico-geometric series has the form S n = ∑ k = 1 n A k G k = ∑ k = 1 n ( a + ( k − 1 ) d ) b r k − 1 = b ∑ k = 0 n − 1 ( a + k d ) r k = a b + ( a

{\displaystyle D} as polynomials D = d k − 1 x k − 1 + d k − 2 x k − 2 + . . . + d 1 x + d 0 {\displaystyle \mathbf {D} =d_{k-1}x^{k-1}+d_{k-2}x^{k-2}+...+d_{1}x+d_{0}}

1 k 0 0 0 1 k 0 0 0 2 k ] [ 1 0 1 1 2 0 0 1 − 1 ] − 1 = [ 2 − 2 k − 1 + 2 k 2 − 2 k + 1 0 1 0 − 1 + 2 k 1 − 2 k − 1 + 2 k + 1 ] . {\displaystyle

had the engine of the Model 11 and the airframe of the Model 22. The Ohka K-1 was an unpowered trainer version with water ballast instead of warhead and

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

) d ( a , b ) − ∑ k = 1 5 d ( a , k ) − ∑ k = 1 5 d ( b , k ) {\displaystyle Q_{1}(a,b)=(n-2)d(a,b)-\sum _{k=1}^{5}d(a,k)-\sum _{k=1}^{5}d(b,k)} = ( 5

obtain k 1 [ CH 3 CHO ] − 2 k 4 [ CH 3 ] 2 = 0 {\displaystyle k_{1}{\ce {[CH3CHO]}}-2k_{4}{\ce {[CH3]}}^{2}=0} so that ( 4 ) . . . [ CH 3 ] = k 1 2 k 4

S k − L | ≤ | S k − S k + 1 | = a k + 1 . {\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!} Suppose we are given

K-1 5000 m 2014 Moscow K-1 5000 m 2018 Montemor-o-Velho K-1 5000 m 2019 Szeged K-4 500 m 2021 Copenhagen K-2 500 m 2011 Szeged K-1 5000 m 2011 Szeged K-4

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200, 500, 1000 or 5000 metres

are calculated using Newton's divided differences method. Given a set of k + 1 data points ( x 0 , y 0 ) , … , ( x j , y j ) , … , ( x k , y k ) {\displaystyle

expansion for the logarithm: ln ⁡ ( 1 + x ) = − ∑ k = 1 ∞ ( − 1 ) k k x k {\displaystyle \ln(1+x)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\,x^{k}} the logarithmic

Men's C–1 200 m Relay Results Men's K–1 200 m Results Men's K–1 500 m Results Men's K–1 1000 m Results Men's K–1 5000 m Results Men's K–2 200 m Results

k − 1 d i b i {\displaystyle n=\sum _{i=0}^{k-1}d_{i}b^{i}} and therefore F b ( n ) = ∏ i = 0 k − 1 d i = d k − 1 ∏ i = 0 k − 2 d i < d k − 1 b k − 1

2010. Classed as K-1 A before 2015. First contested in 2010. Classed as K-1 TA before 2015. First contested in 2010. Classed as K-1 LTA before 2015. Was

{\displaystyle k\in \mathbb {N} } . Then F(k + 1) = f(F(k)) = f(G(k)) = G(k + 1). Hence F(k) = G(k) implies F(k + 1) = G(k + 1). By induction, F(n) = G(n) for all

Championships with four golds (K-1 1000 m: 1987, K-1 10000 m: 1985, 1987, 1991), a silver (K-1 10000 m: 1990), and a bronze (K-1 1000 m: 1991). He received

1 K n i ( Y ¯ i ⋅ − Y ¯ ) 2 / ( K − 1 ) {\displaystyle \sum _{i=1}^{K}n_{i}({\bar {Y}}_{i\cdot }-{\bar {Y}})^{2}/(K-1)} where Y ¯ i ⋅ {\displaystyle {\bar

conductivity m⋅kg⋅s−3⋅K−1 kelvin per watt K/W thermal resistance m−2⋅kg−1⋅s3⋅K reciprocal kelvin K−1 thermal expansion coefficient K−1 kelvin per metre K/m

Summer Olympics in the K-4 1000 m event. He won the silver medal in the K-1 1000 metres event at the 2016 Summer Olympics in Rio de Janeiro. He was also

k = 1 , 2 , … } {\displaystyle \{X_{k},k=1,2,\dots \}} , if they are independent and the cdf of X k {\displaystyle X_{k}} is given by F ( a k − 1 x

with four golds (K-1 4 x 500 m: 1963, K-2 500 m: 1963, K-2 1000 m: 1963, K-4 1000 m: 1966), a silver (K-4 1000 m: 1963), and a bronze (K-1 4 x 500 m: 1966)

e k + 1 p k + 1 2 e k + 1 … p n 2 e n = p 1 e 1 … p n e n p 1 … p k . {\displaystyle {\sqrt {p_{1}^{2e_{1}+1}\cdots p_{k}^{2e_{k}+1}p_{k+1}^{2e_{k+1}}\dots

the K-1 200 m event at the 2012 Summer Olympics in London, a gold medal (with Cristian Toro) in the K-2 200 m event and a bronze medal in the K-1 200

k − 1 ∑ n = 1 2 k − 1 cos ⁡ ( n − 1 / 2 2 k − 1 x ) , ∀ k ≥ 1 , {\displaystyle \prod _{n=1}^{k}\cos \left({\frac {x}{2^{n}}}\right)={\frac {1}{2^{k-1}}}\sum

in five Summer Olympics, he won five medals which included three golds (K-1 500 m: 1996, K-2 1000 m: 1996, 2000), one silver (K-2 1000 m: 2004), and

if we define the sequence pk by p n k = ( n + k − 1 k − 1 ) {\displaystyle p_{n}^{k}={n+k-1 \choose k-1}} then the Cesàro sum Ck is defined by Ck(s) =

_{k=1}^{n}{\left(-1\right)^{k-1}{\frac {d^{k-1}f\left(x\right)}{dx^{k-1}}}}+C} ∫ e − x ( f ( x ) − d n f ( x ) d x n ) d x = − e − x ∑ k = 1 n d k − 1 f

bronze (K-1 500 m: 2008). Wagner-Augustin also won 26 medals at the ICF Canoe Sprint World Championships with ten golds (K-1 1000 m: 2005, K-1 4 × 200

Dostál Canoeing Men's K-1 1000 m  Silver Jaroslav Kulhavý Cycling Men's cross-country  Bronze Jiří Prskavec Canoeing Men's slalom K-1  Bronze Ondřej Synek

case of p = 1 , k = 1 {\displaystyle p=1,k=1} yields the harmonic series, which diverges. The case of p = 2 , k = 1 {\displaystyle p=2,k=1} is the Basel

Germany 1980 Moscow K-1 500 m 1988 Seoul K-2 500 m 1988 Seoul K-4 500 m 1988 Seoul K-1 500 m Representing  Germany 1992 Barcelona K-1 500 m 1996 Atlanta

x , t ) = cos ⁡ ( k 1 x − ω 1 t ) + cos ⁡ ( k 2 x − ω 2 t ) = 2 cos ⁡ ( ( k 2 − k 1 ) x − ( ω 2 − ω 1 ) t 2 ) cos ⁡ ( ( k 2 + k 1 ) x − ( ω 2 + ω 1 )

k 1 {\displaystyle k_{1}} and k 2 {\displaystyle k_{2}} dimensional marginal distributions X {\displaystyle X} and Y {\displaystyle Y} , such that k 1

m in 2011), a silver (K-1 4 x 200 m, 2014) and two bronze (K-1 4 x 200 m: 2009, K-1 500 m 2013). Hybois also competed in the K-1 500 m event at the 2008

Laplacian operator, defined by Δ = ∇ 2 = ∑ k = 1 n ∂ 2 ∂ x k 2 . {\displaystyle \Delta =\nabla ^{2}=\sum _{k=1}^{n}{\frac {\partial ^{2}}{\partial x_{k}^{2}}}

obtain k 1 [ CH 3 CHO ] − 2 k 4 [ CH 3 ] 2 = 0 {\displaystyle k_{1}{\ce {[CH3CHO]}}-2k_{4}{\ce {[CH3]}}^{2}=0} so that ( 4 ) . . . [ CH 3 ] = k 1 2 k 4

energy K by an additive constant, which gives K 0 − K 1 = E ( 1 1 − v 2 c 2 − 1 ) {\displaystyle K_{0}-K_{1}=E\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)}

{N} } to be the following: F b ( n ) = ∑ i = 0 k − 1 d i k . {\displaystyle F_{b}(n)=\sum _{i=0}^{k-1}d_{i}^{k}.} where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle

The 1st Army (German: 1. Armee / Armeeoberkommando 1 / A.O.K. 1) was an army level command of the German Army in World War I. It was formed on mobilization

we get: A = ∑ k = 1 K α β C k ¬ n + V β {\displaystyle A=\sum _{k=1}^{K}{\frac {\alpha \beta }{C_{k}^{\neg n}+V\beta }}} B = ∑ k = 1 K C k d β C k ¬

for k {\displaystyle k} : for k = 1 {\displaystyle k=1} one gets the identity mapping, for k = − 1 {\displaystyle k=-1} one gets the reflection at the

Sprint World Championships with three golds (K-1 4×500 m: 1948, 1950; K-4 1000 m: 1948) and a silver (K-1 500 m: 1950). Lennart Klingström. Sports-reference

be defined by S n ( x ) x := ∏ k = 1 n ( 1 − x 2 k 2 ⋅ π 2 ) {\displaystyle {\frac {S_{n}(x)}{x}}:=\prod \limits _{k=1}^{n}\left(1-{\frac {x^{2}}{k^{2}\cdot

Yael; Chazan, Michael; Arnold, M.; Aumaître, G.; Bourlès, D.; Keddadouche, K. (1 May 2021). "Magnetostratigraphy and cosmogenic dating of Wonderwerk Cave:

x_{2}-3~\end{bmatrix}}_{k}\end{aligned}}} and X k + 1   =   X k − C k + 1 {\displaystyle X_{k+1}~=~X_{k}-C_{k+1}} The iterations should be repeated until   [

finished eighth in the K-2 1000 m event, but was eliminated in the heats of the K-1 500 m event. He competed again in the K-2 1000 m at the 2008 Summer Olympics

Men's K-1 1000m  Gold Rüdiger Helm Bernd Olbricht Harald Marg Bernd Duvigneau Canoeing Men's K-4 1000m  Gold Birgit Fischer Canoeing Women's K-1 500m  Gold

{v} _{k},} if k > 1 , {\displaystyle k>1,} and v 1 = 0 {\displaystyle \mathbf {v} _{1}=\mathbf {0} } if k = 1. {\displaystyle k=1.} Thus, a set of

airship) (ZNN-G) (ZGT) H class blimp (observation) J class blimp (trainer) K-1 blimp (experimental) K class blimp (anti-submarine patrol) (ZNP-K) (ZSG-2

‖ P ‖ → 0 ∑ k = 1 n ( X t k − X t k − 1 ) 2 {\displaystyle [X]_{t}=\lim _{\Vert P\Vert \rightarrow 0}\sum _{k=1}^{n}(X_{t_{k}}-X_{t_{k-1}})^{2}} where

) k + 1 . {\displaystyle F_{b}(n)=k(b-1)^{k+1}.} This suggests that k ( b − 1 ) k + 1 ≥ n ≥ b k − 1 , {\displaystyle k(b-1)^{k+1}\geq n\geq b^{k-1},}

for South Korea and for Japan at the 2002 Asian Games. He is the former K-1 HERO's Light Heavyweight Grand Prix Tournament Champion. Before his professional

are calculated using Newton's divided differences method. Given a set of k + 1 data points ( x 0 , y 0 ) , … , ( x j , y j ) , … , ( x k , y k ) {\displaystyle

+ k 1 ) 3 = ( x 2 + k 1 2 ) ( x + k 1 ) = x 3 + k 1 2 x + x 2 k 1 + k 1 3 , {\displaystyle c_{1}=(x+k_{1})^{3}=(x^{2}+k_{1}^{2})(x+k_{1})=x^{3}+k_{1

square ( t ) = 4 π ∑ k = 1 ∞ sin ⁡ ( ( 2 k − 1 ) t ) 2 k − 1 . {\displaystyle f_{\text{square}}(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\sin {\big

form α k y n + k + α k − 1 y n + k − 1 + ⋯ + α 0 y n = h [ β k f ( t n + k , y n + k ) + β k − 1 f ( t n + k − 1 , y n + k − 1 ) + ⋯ + β 0 f ( t n ,

was redefined so that the Boltzmann constant is exactly 1.380649×10−23 J⋅K−1, and the triple point of water became an experimentally measured constant

K-1 5000 m 2014 Moscow K-1 5000 m 2018 Montemor-o-Velho K-1 5000 m 2019 Szeged K-4 500 m 2021 Copenhagen K-2 500 m 2011 Szeged K-1 5000 m 2011 Szeged K-4

formula states that ρ ( A ) = lim k → ∞ ‖ A k ‖ 1 / k {\displaystyle \rho (A)=\lim _{k\to \infty }\|A^{k}\|^{1/k}} . Both of these results are shown below

consider the put/call parity equation at two different strike prices K 1 {\displaystyle K_{1}} and K 2 {\displaystyle K_{2}} . The stock price S will disappear

u d u = ∫ 0 1 [ ∑ k = 1 n ( n k ) ( − u ) k − 1 ] d u = ∑ k = 1 n ( n k ) ∫ 0 1 ( − u ) k − 1 d u = ∑ k = 1 n ( n k ) ( − 1 ) k − 1 k . {\displaystyle

k = 1 n ( − 1 ) k + 1 ( ∑ 1 ⩽ i 1 < ⋯ < i k ⩽ n | A i 1 ∩ ⋯ ∩ A i k | ) {\displaystyle \left|\bigcup _{i=1}^{n}A_{i}\right|=\sum _{k=1}^{n}(-1)^{k+1}\left(\sum

isomorphic to the eight-element subset { 1 , i , j , k , − 1 , − i , − j , − k } {\displaystyle \{1,i,j,k,-1,-i,-j,-k\}} of the quaternions under multiplication

and men's and women's K-1) resumed at the 2010 championships.) and the 200 m events for C-1, C-2, and C-4 (men only), and K-1, K-2, and K-4 (both men

in five Summer Olympics, he won five medals which included three golds (K-1 500 m: 1996, K-2 1000 m: 1996, 2000), one silver (K-2 1000 m: 2004), and

Harbor, operated off the United States West Coast, and took part in Operation K-1, the Aleutian Islands campaign, and the Guadalcanal campaign, including the

⁡ ( ∑ k = 1 ∞ α k λ ( z k − 1 ) ) , ( | z | ≤ 1 ) {\displaystyle P_{Y}(z)=\sum \limits _{i=0}^{\infty }P(Y=i)z^{i}=\exp \left(\sum \limits _{k=1}^{\infty

should be within 4 parts in 10,000, or 0.04%, corresponding to k = 1.0004 {\displaystyle k=1.0004} . Since sec ⁡ φ {\displaystyle \sec \varphi } attains

recursion formula a k + 1 a k = cos ⁡ ( k γ ) sin ⁡ ( ( k + 1 ) γ ) {\displaystyle {\frac {a_{k+1}}{a_{k}}}={\frac {\cos(k\gamma )}{\sin((k+1)\gamma )}}} and

, k − 1 ( x ) + t i + k + 1 − x t i + k + 1 − t i + 1 B i + 1 , k − 1 ( x ) . {\displaystyle B_{i,k}(x):={\frac {x-t_{i}}{t_{i+k}-t_{i}}}B_{i,k-1}(x)+{\frac

kickboxer. A four-time K-1 World Champion, Hoost is considered to be one of the greatest kickboxers of all time. Debuting in 1993 at the K-1 World Grand Prix

than or equal to n. Then π 2 ( n ) = ∑ k = 1 π ( n ) [ π ( n p k ) − k + 1 ] {\displaystyle \pi _{2}(n)=\sum _{k=1}^{\pi \left({\sqrt {n}}\right)}\left[\pi

96 kg (212 lb) Sport Country  Germany Sport Sprint kayak Event(s) K-1 200 m, K-1 500 m, K-1 1000 m, K-2 200 m, K-4 500 m, K-4 1000 m Club Kanu Club Dresden

base b {\displaystyle b} as the k-digit number d k − 1 d k − 2 . . . d 1 d 0 {\displaystyle d_{k-1}d_{k-2}...d_{1}d_{0}} where each digit d i {\displaystyle

K 1 0 1 Paulenka K 0 0 0 Paveisininkai K 7 3 4 Pertakas K 5 3 2 Purviniai K 3 2 1 Semoškai K 26 13 13 Simaniškė K 1 0 1 Sirguškė K 6 4 2 Stalai K 1 1

power: ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 + k 2 + ⋯ + k m = n k 1 , k 2 , ⋯ , k m ≥ 0 ( n k 1 , k 2 , … , k m ) x 1 k 1 ⋅ x 2 k 2 ⋯ x m k m {\displaystyle

Farkas József Hunics C-2 10000 m — 55:15.6 Lajos Kiss K-1 1000 m 4:35.8 2 Q 4:16.2 Ferenc Hatlaczky K-1 10000 m — 47:53.3 Imre Vágyóczky Zoltán Szigeti K-2

gold medal in Women's K-4 500 metres, and a silver medal in the Women's K-1 500 metres At the 2010 ICF Canoe Sprint World Championships in Poznań, she

c n − m = − 1 m ∑ k = 1 m c n − m + k tr ⁡ ( A k )     ( 1 ≤ m ≤ n )   . {\displaystyle c_{n}=1;~~~c_{n-m}=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\operatorname

kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u,t)} which (roughly speaking) yields an inverse transform: f ( t ) = ∫ u 1 u 2 ( T f ) ( u ) K − 1 ( u ,

recursion formula a k + 1 a k = cos ⁡ ( k γ ) sin ⁡ ( ( k + 1 ) γ ) {\displaystyle {\frac {a_{k+1}}{a_{k}}}={\frac {\cos(k\gamma )}{\sin((k+1)\gamma )}}} and

Farkas József Hunics C-2 10000 m — 55:15.6 Lajos Kiss K-1 1000 m 4:35.8 2 Q 4:16.2 Ferenc Hatlaczky K-1 10000 m — 47:53.3 Imre Vágyóczky Zoltán Szigeti K-2

r ) = π k k ! r 2 k , V 2 k + 1 ( r ) = 2 k + 1 π k ( 2 k + 1 ) ! ! r 2 k + 1 = 2 ( k ! ) ( 4 π ) k ( 2 k + 1 ) ! r 2 k + 1 . {\displaystyle

{\displaystyle x_{\varnothing },} the root of the tree, followed, for k = 1 , … , n , {\displaystyle k=1,\ldots ,n,} by a family of s k {\displaystyle s^{k}} 2 vectors

identically distributed U(0, 1) random variables: X = ∑ k = 1 n U k . {\displaystyle X=\sum _{k=1}^{n}U_{k}.} The probability density function (pdf) for

π ∑ k = 0 n − 1 ( n − k − 1 ) ! k ! ( z 2 4 ) k + 2 π J n ( z ) ln ⁡ z 2 − ( z 2 ) n π ∑ k = 0 ∞ ( ψ ( k + 1 ) + ψ ( n + k + 1 ) ) ( − z 2 4 ) k k ! (

become P k | k − 1 = F k − 1 P k − 1 | k − 1 F k − 1 T + L k − 1 Q k − 1 L k − 1 T {\displaystyle {\boldsymbol {P}}_{k|k-1}={{\boldsymbol {F}}_{k-1}}{{\boldsymbol

K 1 0 1 Paulenka K 0 0 0 Paveisininkai K 7 3 4 Pertakas K 5 3 2 Purviniai K 3 2 1 Semoškai K 26 13 13 Simaniškė K 1 0 1 Sirguškė K 6 4 2 Stalai K 1 1

equation: J k ∗ ( x n − k ) = min u n − k { f ^ ( x n − k , u n − k ) + J k − 1 ∗ ( g ^ ( x n − k , u n − k ) ) } {\displaystyle J_{k}^{\ast }\left(\mathbf

r 2 + ∑ k = 1 n − 1 r 2 ( ∏ m = 1 k − 1 sin 2 ⁡ ( φ m ) ) d φ k 2 {\displaystyle ds^{2}=dr^{2}+\sum _{k=1}^{n-1}r^{2}\left(\prod _{m=1}^{k-1}\sin ^{2}\left(\varphi

variables on the unit interval, Uk: X = 1 n ∑ k = 1 n U k . {\displaystyle X={\frac {1}{n}}\sum _{k=1}^{n}U_{k}.} The equation defining the probability

kickboxer. A four-time K-1 World Champion, Hoost is considered to be one of the greatest kickboxers of all time. Debuting in 1993 at the K-1 World Grand Prix

power: ( x 1 + x 2 + ⋯ + x m ) n = ∑ k 1 + k 2 + ⋯ + k m = n k 1 , k 2 , ⋯ , k m ≥ 0 ( n k 1 , k 2 , … , k m ) x 1 k 1 ⋅ x 2 k 2 ⋯ x m k m {\displaystyle

below (with constant factor k 1 {\displaystyle k_{1}} ) ∃ k 1 > 0 ∃ k 2 > 0 ∃ n 0 ∀ n > n 0 : {\displaystyle \exists k_{1}>0\,\exists k_{2}>0\,\exists

considering the expression ∑ k = 1 n f ( γ ( t k ) ) [ γ ( t k ) − γ ( t k − 1 ) ] = ∑ k = 1 n f ( γ k ) Δ γ k . {\displaystyle \sum _{k=1}^{n}f(\gamma (t_{k}))\

equation ‖ ∑ k = 1 n v k ‖ 2 = ∑ k = 1 n ‖ v k ‖ 2 {\displaystyle {\biggl \|}\sum _{k=1}^{n}\mathbf {v} _{k}{\biggr \|}^{2}=\sum _{k=1}^{n}\|\mathbf

integers) is equivalent to the statement that we have ∏ p ∈ P k 1 1 − 1 p = ∑ n ∈ N k 1 n , {\displaystyle \prod _{p\in P_{k}}{\frac {1}{1-{\frac {1}{p}}}}=\sum

2 ∑ k = 1 N ( m k r ˙ k ( q , q ˙ ) ⋅ r ˙ k ( q , q ˙ ) ) {\displaystyle T({\boldsymbol {q}},{\boldsymbol {\dot {q}}})={\frac {1}{2}}\sum _{k=1}^{N}{\biggl

Helm Canoeing men's K-1 1000 m  Gold Joachim Mattern Bernd Olbricht Canoeing men's K-2 500 m  Gold Carola Zirzow Canoeing women's K-1 500 m  Gold Klaus-Jürgen

K + 1 ) / 2 ⌉ {\displaystyle \lceil (K+1)/2\rceil } items new node has ⌊ ( K + 1 ) / 2 ⌋ {\displaystyle \lfloor (K+1)/2\rfloor } items Copy ⌈ ( K + 1

proved that if c − 2 ∑ k = 1 n ‖ x k ‖ 2 ≤ Ave ± ⁡ ‖ ∑ k = 1 n ± x k ‖ 2 ≤ c 2 ∑ k = 1 n ‖ x k ‖ 2 {\displaystyle c^{-2}\sum _{k=1}^{n}\|x_{k}\|^{2}\leq

A−, so [ H + ] 2 = K 1 K 2 {\displaystyle [{\ce {H+}}]^{2}=K_{1}K_{2}} Therefore, taking cologarithms, the pH is given by p I = p K 1 + p K 2 2 {\displaystyle

a 3-cycle : ( k     k + 1 ) ⋅ ( k + 1     k + 2 ) = ( k     k + 2     k + 1 ) {\displaystyle (k~~k+1)\cdot (k+1~~k+2)=(k~~k+2~~k+1)} . Therefore S n +

× A k − 1 × A k + 1 × … , × A K → 2 A k {\displaystyle f_{k}:{\mathcal {A}}_{1}\times \ldots \times {\mathcal {A}}_{k-1}\times {\mathcal {A}}_{k+1}\times

{\displaystyle \mathbf {u} ,\mathbf {v} } with dimensions ( k 1 , k 2 , … , k m ) {\displaystyle (k_{1},k_{2},\dots ,k_{m})} and ( l 1 , l 2 , … , l n ) {\displaystyle

This gives ∑ k = 0 n C k = 1 + ∑ k = 1 n ( n k ) M k − 1 . {\displaystyle \sum _{k=0}^{n}C_{k}=1+\sum _{k=1}^{n}{\binom {n}{k}}M_{k-1}.} The generating function

2008 featured Dream and K-1 fights Dynamite!! 2009 featured fights by Dream, Sengoku and K-1 Dynamite!! 2010 featured Dream and K-1 fights. Fight For Japan:

= 1 4 ∑ k = 1 ∞ ( − 1 ) k − 1 ( k − 1 ) ! 3 ( 56 k 2 − 32 k + 5 ) ( 2 k − 1 ) 2 ( 3 k ) ! . {\displaystyle \zeta (3)={\frac {1}{4}}\sum _{k=1}^{\infty

500 m for Germany) and two bronzes (both Australia, 1996: K-2 500 m, 2000: K-1 500 m). During her career, she has represented four countries: East Germany

respectively: d N 1 d t = r 1 N 1 K 1 − N 1 − α N 2 K 1 {\displaystyle {dN_{1} \over dt}=r_{1}N_{1}{K_{1}-N_{1}-\alpha N_{2} \over K_{1}}} d N 2 d t = r 2 N 2 K

C ( u − k , 1 ) Δ y k − 1 + C ( u − k + 1 , 2 ) Δ 2 y k − 2 + C ( u − k + 2 , 3 ) Δ 3 y k − 3 + ⋯ = y k + ( u − k ) Δ y k − 1 + ( u − k + 1 ) ( u − k

that for all k > 1, σ ( n ) n 1 + ε ≥ σ ( k ) k 1 + ε {\displaystyle {\frac {\sigma (n)}{n^{1+\varepsilon }}}\geq {\frac {\sigma (k)}{k^{1+\varepsilon

− T 1 ) k − 1 = k R T 1 k − 1 [ ( P 2 P 1 ) ( k − 1 ) / k − 1 ] {\displaystyle W_{comp,in}={\frac {kR(T_{2}-T_{1})}{k-1}}={\frac {kRT_{1}}{k-1}}\left[\left({\frac

particle, 0.239 kcal·mol−1, or 83.6 cm−1. At room temperature (25 °C, or 298.15 K) 1 kJ·mol−1 is approximately equal to 0.4034  k B T {\displaystyle k_{B}T}

) n − k − 1 d z = ∮ γ ∑ n = − ∞ ∞ b n ( z − c ) n − k − 1 d z . {\displaystyle \oint _{\gamma }\,\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n-k-1}\,dz=\oint

The EMER K-1, sometimes known as the EMER K1 or EMERK-1, was a prototype bullpup assault rifle made in Myanmar by the Electrical Mechanical and Engineering

Sprint C-1 500 m men C-1 1000 m men C-2 500 m men C-2 1000 m men K-1 500 m men women K-1 1000 m men K-2 500 m men women K-2 1000 m men K-4 500 m women K-4

ICF Canoe Sprint World Championships with eight golds (K-1 500 m: 2005, 2011, K-2 200 m: 2007, K-1 4 × 200 m: 2009, 2010, 2011, K-2 500 m: 2007, K-4 200

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

linear with the velocity we may write: − ( k − 1 ) i j μ φ u j − ∂ i p = 0 , {\displaystyle -\left(k^{-1}\right)_{ij}\mu \varphi u_{j}-\partial _{i}p=0\

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200 metres (660 ft), 500

golds (K-1 10000 m: 1981, 1983; K-2 1000 m: 1979, K-4 10000 m: 1975), three silvers (K-1 10000 m: 1979, 1982; K-2 1000 m: 1978), and three bronzes (K-1 1000

+ ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)} for some K > 1. {\displaystyle K>1.} A quasi-seminorm on a vector space X {\displaystyle X} is a

K + 1 ) / 2 ⌉ {\displaystyle \lceil (K+1)/2\rceil } items new node has ⌊ ( K + 1 ) / 2 ⌋ {\displaystyle \lfloor (K+1)/2\rfloor } items Copy ⌈ ( K + 1

or four people (4). For each of the specific canoes or kayaks, such as a K-1 (kayak single), the competition distances can be 200 metres (660 ft), 500

m k = ( 1 − c ) p k + 1 − p k − 1 x k + 1 − x k − 1 {\displaystyle {\boldsymbol {m}}_{k}=(1-c){\frac {{\boldsymbol {p}}_{k+1}-{\boldsymbol {p}}_{k-1}}{x_{k+1}-x_{k-1}}}}

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

, p k − 1 {\displaystyle 1,p,p^{2},\dots ,p^{k-1}} which sum to p k − 1 p − 1 {\displaystyle {\frac {p^{k}-1}{p-1}}} , which is at most p k − 1 {\displaystyle

formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If

= k 1 [ tBuBr ] − k − 1 [ tBu + ] [ Br − ] − k 2 [ tBu + ] [ H 2 O ] {\displaystyle {\frac {d[{\text{tBu}}^{+}]}{dt}}=0=k_{1}[{\text{tBuBr}}]-k_{-1

correlation. The test statistic is: Q = n ( n + 2 ) ∑ k = 1 h ρ ^ k 2 n − k {\displaystyle Q=n(n+2)\sum _{k=1}^{h}{\frac {{\hat {\rho }}_{k}^{2}}{n-k}}} where

2 ∑ n = 0 ∞ ∏ k = 1 n 2 k z 2 ( 2 k + 1 ) ( 1 + z 2 ) . {\displaystyle \arctan(z)={\frac {z}{1+z^{2}}}\sum _{n=0}^{\infty }\prod _{k=1}^{n}{\frac

− sgn ⁡ ( a k + 1 , k k ) ∑ j = k + 1 n ( a j k k ) 2 r = 1 2 ( α 2 − a k + 1 , k k α ) v 1 k = v 2 k = ⋯ = v k k = 0 v k + 1 k = a k + 1 , k k − α 2

respectively: d N 1 d t = r 1 N 1 K 1 − N 1 − α N 2 K 1 {\displaystyle {dN_{1} \over dt}=r_{1}N_{1}{K_{1}-N_{1}-\alpha N_{2} \over K_{1}}} d N 2 d t = r 2 N 2 K

The SI unit for heat capacity of an object is joule per kelvin (J/K or J⋅K−1). Since an increment of temperature of one degree Celsius is the same as

he entered: K-1 500 m, K-2 500 m and K-2 1000 m. He also won twelve medals at the ICF Canoe Sprint World Championships, with nine golds (K-1 500 m: 1979

series in s: Li s ⁡ ( z ) = ∑ k = 1 ∞ z k k s = z + z 2 2 s + z 3 3 s + ⋯ {\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}=z+{z^{2}

the computation of the product ∏ k = 1 n f k = ∏ k = 1 A f k f k . {\displaystyle \prod _{k=1}^{n}f_{k}=\prod _{k=1}^{A}f_{k}^{f_{k}}.} Applying log2

) ( a ∣ b ) ⋯ ( a ∣ b ) ⏟ k − 1  times . {\displaystyle (a\mid b)^{*}a\underbrace {(a\mid b)(a\mid b)\cdots (a\mid b)} _{k-1{\text{ times}}}.\,} On the

(born 20 June 1997) is a Hungarian sprint canoeist. He competed in the men's K-1 1000 metres event at the 2016 Summer Olympics. He won the same event in the

k, the sample size, is N ^ = k + 1 k m − 1 = m + m k − 1 {\displaystyle {\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1} . This can be seen as a very

have 0 ≤ ∑ k = 1 n ( a k + | a k | ) ≤ ∑ k = 1 n 2 | a k | . {\displaystyle 0\leq \sum _{k=1}^{n}(a_{k}+\left|a_{k}\right|)\leq \sum _{k=1}^{n}2\left|a_{k}\right|

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

k = 1 ∞ β k δ x k ( x ) {\displaystyle f(x)=\sum _{k=1}^{\infty }\beta _{k}\delta _{x_{k}}(x)} , where { x k } k = 1 ∞ {\displaystyle \{x_{k}\}_{k=1}^{\infty

is at most k − 1 2 . {\displaystyle {\frac {k-1}{2}}.} At least one of the prime powers dividing N exceeds 1062. N < 2 ( 4 k + 1 − 2 k + 1 ) {\displaystyle

approximated as m 1 = M 1 + 2.5 ⋅ K 1 log 10 ⁡ ( d B S d 0 ) + 5 log 10 ⁡ ( d B O d 0 ) {\displaystyle m_{1}=M_{1}+2.5\cdot K_{1}\log _{10}{\left({\frac

500 m for Germany) and two bronzes (both Australia, 1996: K-2 500 m, 2000: K-1 500 m). During her career, she has represented four countries: East Germany

v o l ( F ) = 1 k ! ∏ 1 ≤ m < n ≤ k + 1 ( t i n − t i m ) . {\displaystyle \mathrm {nvol} (F)={\frac {1}{k!}}\prod _{1\leq m<n\leq k+1}{(t_{i_{n}}-t_{i_{m}})}

linear with the velocity we may write: − ( k − 1 ) i j μ φ u j − ∂ i p = 0 , {\displaystyle -\left(k^{-1}\right)_{ij}\mu \varphi u_{j}-\partial _{i}p=0\

{\displaystyle \phi } is a Π k 1 {\displaystyle \Pi _{k}^{1}} formula, and similar, Π k + 1 1 {\displaystyle \Pi _{k+1}^{1}} has the form ∀ R 0 … ∀ R

= ‖ U k − 1 ‖ 1 ‖ U k − 1 ‖ ∞ ‖ U k ‖ 1 ‖ U k ‖ ∞ 4 {\displaystyle \gamma _{k}={\sqrt[{4}]{\frac {\left\|U_{k}^{-1}\right\|_{1}\left\|U_{k}^{-1}\right\|_{\infty

Championships with four golds (K-1 1000 m: 1975 (tied with Poland's Grzegorz Śledziewski, K-1 10000 m: 1974, 1975, 1977) and two bronzes (K-1 1000 m: 1974, 1977)

the particle accelerations squared, S = 1 2 ∑ k = 1 N m k a k 2 , {\displaystyle S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,} where the

k − 1 H 2 k − 1 H 2 k − 1 − H 2 k − 1 ] = H 2 ⊗ H 2 k − 1 , {\displaystyle H_{2^{k}}={\begin{bmatrix}H_{2^{k-1}}&H_{2^{k-1}}\\H_{2^{k-1}}&-H_{2^{k-1

root of x k + 1 = 2 k + 1 ( x − 1 ) , β k = 2 − α k k + 1 − k α k {\displaystyle x^{k+1}=2^{k+1}(x-1),\beta _{k}={\frac {2-\alpha _{k}}{k+1-k\alpha _{k}}}}

given by E ⁡ ( X k ) = ( − 1 ) k + 1 k ! α T − ( k + 1 ) s . {\displaystyle \operatorname {E} (X^{k})=(-1)^{k+1}k!\mathbf {\alpha } T^{-(k+1)}\mathbf {s}

with three golds (K-1 1000 m: 1971, 1975 (tied with Italy's Oreste Perri); K-2 500 m: 1974), five silvers (K-1 500 m: 1970, 1977; K-1 1000 m: 1973, 1974;

operator, P j k = 1 2 ( σ → j ⋅ σ → k + 1 )   . {\displaystyle P_{jk}={\frac {1}{2}}\,\left({\vec {\sigma }}_{j}\cdot {\vec {\sigma }}_{k}+1\right)~.} Its

1 > χ α , k − 1 2 {\displaystyle T_{1}>\chi _{\alpha ,k-1}^{2}} where Χα,k − 12 is the α-quantile of the chi-squared distribution with k − 1 degrees of

k 2 ( E N C k 1 ( P ) ) P = D E C k 1 ( D E C k 2 ( C ) ) {\displaystyle {\begin{aligned}C&={\mathit {ENC}}_{k_{2}}({\mathit {ENC}}_{k_{1}}(P))\\P&={\mathit

Canoeing: 2012 – K-1 200 m 2016 – K-1 200 m 2020 – K-1 200 m 2020 – K-2 500 m 2020 – K-1 500 m 2024 – K-4 500 m 2024 – K-2 500 m 2024 – K-1 500 m Four Ian

+ ‖ y ‖ ) {\displaystyle \|x+y\|\leq K(\|x\|+\|y\|)} for some K > 1. {\displaystyle K>1.} A quasi-seminorm on a vector space X {\displaystyle X} is a

{F(1-p)p}}}}} MGF 1 + ∑ k = 1 ∞ ( ∏ r = 0 k − 1 α + r 1 − F F + r ) t k k ! {\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{{\frac

= p r ( 1 − p ) − r Γ ( r ) k ! ( 1 − p ) k + r ∫ 0 ∞ λ k + r − 1 e − λ d λ ⏟ = Γ ( r + k ) = Γ ( r + k ) Γ ( r ) k ! ( 1 − p ) k p r {\displaystyle

+ r − 1 k ) ( 1 − q ) k q r ⋅ q α − 1 ( 1 − q ) β − 1 B ( α , β ) d q = 1 B ( α , β ) ( k + r − 1 k ) ∫ 0 1 q α + r − 1 ( 1 − q ) β + k − 1 d q {\displaystyle

Operation K (K作戦, Kē-Sakusen) was a Japanese naval operation in World War II, intended as reconnaissance of Pearl Harbor and disruption of repair and salvage

… , q k − 1 , q k + 1 , … , q N ; p 1 , p 2 , … , p k − 1 , p k + 1 , … , p N ; ψ ; t ) . {\displaystyle H=H(q_{1},q_{2},\ldots ,q_{k-1},q_{k+1},\ldots

a stable bound state of n particles with masses { m k } k = 1 n {\displaystyle \{m_{k}\}_{k=1}^{n}} corresponds to a pole in the S-matrix with a center-of-mass

rectangle, its made by obtaining a cylinder of radius a + ( k + 1 ) Δ x {\displaystyle a+(k+1)\Delta x} with height f ( a + k Δ x ) {\displaystyle f(a+k\Delta