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searching for K-1 508 found (14412 total)

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)}}}

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

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

× ⋯ × ( 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

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

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

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,

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

⁡ ( 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

{\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

∑ 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

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}}

| 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

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 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

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 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

∑ 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

[ 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

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

{\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

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

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

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

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

− 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

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

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

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

) = 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 )

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}}

_{i=1}^{s}b_{i}k_{i},} where k 1 = f ( t n , y n ) , k 2 = f ( t n + c 2 h , y n + ( a 21 k 1 ) h ) , k 3 = f ( t n + c 3 h , y n + ( a 31 k 1 + a 32 k 2 ) h )

⁡ 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

∑ 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

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

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

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

{\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

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 +

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

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

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

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

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

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

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

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 −

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

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

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

{\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

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

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 ]

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 {\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 −

{\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

(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

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}}}

= ∑ 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

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

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

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

{\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

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

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}}}

{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

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 .

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

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

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,

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

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

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

, 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

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

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

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 + .

_{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}\

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

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 ) )

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

{\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

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

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

) 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

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 )

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} ⁠

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

∑ 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 =

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

−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

452 Thermochemistry Heat capacity (C) 70.4 J·mol−1·K−1 Std molar entropy (S⦵298) 115.6 J·mol−1·K−1 Std enthalpy of formation (ΔfH⦵298) −87.5 kJ·mol−1

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

= 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

}})={\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

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

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

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

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}

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

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

( 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

{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

{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

∑ 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}

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

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.

rearranging to k 1 [ E ] [ S ] = ( k − 1 + k 2 ) [ ES ] {\displaystyle k_{1}[{\ce {E}}][{\ce {S}}]=(k_{-1}+k_{2})[{\ce {ES}}]} Dividing by k 1 [ S ] {\displaystyle

by a 200m race; in addition, the men's C-2 500m was replaced by a women's K-1 200m sprint. This was confirmed at an International Canoe Federation board

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

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

_{k}} is rewritten as r k + 1 T A p k = 1 α k r k + 1 T ( r k − r k + 1 ) = − 1 α k r k + 1 T r k + 1 {\displaystyle \mathbf {r} _{k+1}^{\mathsf {T}}\mathbf

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}

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

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 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

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

ICF Canoe Marathon World Championships is an International Canoe Federation competition in canoe marathon in which athletes compete over long distances

= ∑ 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

) + 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

) + 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 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

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

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

(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

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

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

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

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 ) ]

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

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

+ 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

( 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

}{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

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

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})}

− 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

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

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

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

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

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

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

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 ( ∑

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 − 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

= 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 ) ∑

( 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

( 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

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

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

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

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

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

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

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

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

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 =

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

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

= ∏ 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

losing by TKO in the second round. In 2013 he won the WAKO Pro Panamerican K-1 -85 kg/187 lb Championship. In late 2013 he went back to the amateurs to

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 +

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

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 )

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

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

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 +

{\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

\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\

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 … 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

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

K − 1 1 ∏ k = 1 i ln ( k ) ⁡ ( n ) − ρ n ∏ k = 1 K ln ( k ) ⁡ ( n ) + O ( 1 n ) . {\displaystyle \ln a_{n}=-1-\sum _{i=1}^{K-1}{\frac {1}{\prod _{k=1}^{i}\ln

medals which included three golds (1976: K-1 1000 m, 1980: K-1 1000 m, K-4 1000 m) and three bronzes (1976: K-1 500 m, K-4 1000 m; 1980: K-2 500 m). Helm

k − 1 ) ) = p ( y k − 1 | x k − 1 ) p ( x k | x k − 1 ) p ( x k − 1 | y 0 , ⋯ , y k − 2 ) ∫ p ( y k − 1 | x k − 1 ′ ) p ( x k | x k − 1 ′ ) p ( x k −

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 − ◻ →

k = 1 n M | a k − a k + 1 | = ∑ k = 1 n M ( a k − a k + 1 ) = M ∑ k = 1 n ( a k − a k + 1 ) , {\displaystyle \sum _{k=1}^{n}M|a_{k}-a_{k+1}|=\sum _{k

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:

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

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

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 ∞ ( − 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

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

∇ 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}\

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 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

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

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

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

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}

&\ {\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

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

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 ,

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

via F ^ = ∑ k = 1 N ( x ^ k + 1 − F ^ x ^ k ) 2 ∑ k = 1 N x ^ k 2 . {\displaystyle {\widehat {F}}={\frac {\sum _{k=1}^{N}{({\widehat {x}}_{k+1}-{\widehat

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

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

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 ∼

( 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

( 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

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

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

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

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

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

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

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

− 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

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

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

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\

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

\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

− 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

( 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}

{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 ) )

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

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}|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

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

{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

≥ 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

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)

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

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

( x ) for  k = 1 , ln ⁡ ( ln k − 1 ⁡ ( x ) ) for  k ≥ 2. {\displaystyle \ln _{k}(x)={\begin{cases}\ln(x)&{\text{for }}k=1,\\\ln(\ln _{k-1}(x))&{\text{for

= 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

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

) 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 ] =

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

(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

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}}

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

_{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

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

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

, 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

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

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

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

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

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

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

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

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

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 +

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 − 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

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

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

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

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{

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)

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}

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

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

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}

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}

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

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

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 +

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))

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

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

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

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

= x n _ = x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) ⏞ n  factors = ∏ k = 1 n ( x − k + 1 ) = ∏ k = 0 n − 1 ( x − k ) . {\displaystyle {\begin{aligned}(x)_{n}=x^{\underline

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

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

( 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

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

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 ) =

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

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

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

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

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}}}}

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

) − 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

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

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

(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

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

k ) ( x ) = ( 2 k − 1 ) ! ( k − 1 ) ! 2 f ( x ) ( F ( x ) ( 1 − F ( x ) ) ) k − 1 {\displaystyle f_{X_{(k)}}(x)={\frac {(2k-1)!}{(k-1)!^{2}}}f(x){\Big

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

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

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

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

⟩ | 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

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}}

∑ 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}

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

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

= ( 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

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

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:

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)

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

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

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

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

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 ( ∂

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

{\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}}

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

∂ 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

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

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-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

2000s. He competed in three Summer Olympics, winning a silver medal in the K-1 1000 m event at Athens in 2004. He also finished fourth in the 2008 Olympics

{\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

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

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)

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

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)

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

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

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

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

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

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

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-1 200 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

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

( 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

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

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

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)}

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

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

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

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

= 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

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

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

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

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 )

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

adjusting indices: ( n k ) = 1 ( n + 1 ) B ( n − k + 1 , k + 1 ) . {\displaystyle {\binom {n}{k}}={\frac {1}{(n+1)\,\mathrm {B} (n-k+1,k+1)}}.} Moreover, for

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   [

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 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

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

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 ¬

_{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

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

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

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

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

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

{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

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

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

kayak events. Eight events were held for the women, all in kayak. Women's K-1 1000 m and K-2 1000 m events were added. This was the 28th championships

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

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

) 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

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

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 ¬

{\displaystyle F(a^{k-1}x)} for k = 1 , 2 , … {\displaystyle k=1,2,\dots } , where a {\displaystyle a} is a positive constant, then { X k , k = 1 , 2 , … } {\displaystyle

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

θ = ∑ k = 1 ∞ r k e 2 π i ω k t {\displaystyle \zeta =2\pi t,\quad re^{i\theta }=\sum _{k=1}^{\infty }r_{k}e^{2\pi i\omega _{k}t}} where ω k = 1 n 1 ⋯

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

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

, 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

− 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

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

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

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}}}

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

‖ 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

r ) = { k − 1 sin ⁡ ( r k ) , k > 0 r , k = 0 | k | − 1 sinh ⁡ ( r | k | ) , k < 0. {\displaystyle S_{k}(r)={\begin{cases}{\sqrt {k}}^{\,-1}\sin(r{\sqrt

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 ! (

) 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},}

+ 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

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

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

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

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

⁡ ( ∑ 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

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}}}}

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

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

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

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}

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

coefficients: N ⁡ ( n , k ) = 1 n ( n k ) ( n k − 1 ) {\displaystyle \operatorname {N} (n,k)={\frac {1}{n}}{n \choose k}{n \choose k-1}} The first eight rows

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

of passes p {\displaystyle p} to Ω ( N ∑ k = 1 p h k − 1 / h k ) {\displaystyle \Omega (N\sum _{k=1}^{p}h_{k-1}/h_{k})} where h 0 = N {\displaystyle h_{0}=N}

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

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

{\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

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

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

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

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

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

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

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

{\displaystyle x_{k}} and x k + 1 {\displaystyle x_{k+1}} . q ( p / 4 ) = x k + α ( x k + 1 − x k ) {\displaystyle q(p/4)=x_{k}+\alpha (x_{k+1}-x_{k})} , where k

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

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

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}))\

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}}}}

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}

Gold Time Silver Time Bronze Time K-1 200 m  Piotr Markiewicz (POL)  Sergey Kalesnik (BLR)  Renn Crichlow (CAN) K-1 500 m  Piotr Markiewicz (POL)  Knut

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

{\displaystyle b} . Thus, v k = [ g 1 ( Z k , 1 ) , g 2 ( Z k , 2 ) , … , g B ( Z k , B ) ] {\displaystyle v_{k}=[g_{1}(Z_{k,1}),g_{2}(Z_{k,2}),\ldots ,g_{B}(Z_{k

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

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

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

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:

) ( 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

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

{\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

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

× 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

has the Maclaurin series: γ = 1 1 − β 2 = ∑ n = 0 ∞ β 2 n ∏ k = 1 n ( 2 k − 1 2 k ) = 1 + 1 2 β 2 + 3 8 β 4 + 5 16 β 6 + 35 128 β 8 + 63 256 β 10 + ⋯

K + 1 ) k + 1 − 1 {\displaystyle {\hat {N}}_{C}={\frac {(n+1)(K+1)}{k+1}}-1} The example (n, K, k) = (10, 15, 5) gives N ^ C = ( n + 1 ) ( K + 1 ) k +

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

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

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

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

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

) 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 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

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

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

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

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

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

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

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

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

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

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

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

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

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

traditionally written with a single separator: r = b k b k − 1 ⋯ b 0 . a 1 a 2 ⋯ {\displaystyle r=b_{k}b_{k-1}\cdots b_{0}.a_{1}a_{2}\cdots } Here . is the decimal

shows that E ⁡ [ S n 2 k + 1 ] = 0 ; E ⁡ [ S n 2 k ] = ( n k ) ( 2 k ) ! 2 k n k = n ( n − 1 ) ⋯ ( n − k + 1 ) n k ( 2 k − 1 ) ! ! {\displaystyle

− 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

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

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

while for odd n it is n ! ! = ∏ k = 1 n + 1 2 ( 2 k − 1 ) = n ( n − 2 ) ( n − 4 ) ⋯ 3 ⋅ 1 . {\displaystyle n!!=\prod _{k=1}^{\frac {n+1}{2}}(2k-1)=n(n-2)(n-4)\cdots

{\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

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

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

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

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

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

in Beijing, China from August 8 to August 24, 2008. At the men's slalom K-1 event, Benjamin Boukpeti won a bronze medal in canoeing, the first Olympic

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}}}}

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 ,

, 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

probabilities. The associated random walk is W n = ∑ k = 1 n X k {\displaystyle W_{n}=\sum _{k=1}^{n}X_{k}} so that, for each n, Wn is drawn from a shifted

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

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

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|

states that π 4 = 1 − 1 3 + 1 5 − 1 7 + 1 9 − ⋯ = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 , {\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac

= ‖ 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

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}})}

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

function δ ( k 1 , k 2 ) {\displaystyle \delta (k_{1},k_{2})} that defines an abstract notion of the distance between the keys k 1 {\displaystyle k_{1}} and

= 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

momenta" Πk defined as the Fourier transforms of the pk: Q k = 1 N ∑ l e i k a l x l Π k = 1 N ∑ l e − i k a l p l . {\displaystyle {\begin{aligned}Q_{k}&={\frac

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)

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}

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

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

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}

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 x} : x 2 k ( k + 1 ) f k + 2 ( x ) = f k + 1 ( x ) − f k ( x ) . {\displaystyle {\frac {x^{2}}{k(k+1)}}f_{k+2}(x)=f_{k+1}(x)-f_{k}(x).} Proof:

golds (K-1 200 m: 2006, K-2 200 m: 2003, K-2 1000 m: 2003, K-4 200 m: 2002, 2006; K-4 500 m: 2006, K-4 1000 m: 2005, 2006, 2007), six silvers (K-1 4 × 200

⋅ k 1 {\displaystyle n\cdot {\frac {k}{1}}} n ⋅ k 1 {\displaystyle n\cdot {\frac {k}{1}}} 2 1 {\displaystyle 2^{1}} No No d=1 for in-place, k / 1 {\displaystyle

… , 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

with Alan Thompson, Ian Ferguson and Paul MacDonald. Bramwell also won a K-1 10000 m bronze at the 1985 ICF Canoe Sprint World Championships in Mechelen

with six golds (K-1 500 m: 1973, 1975; K-1 1000 m: 1973, 1974; K-1 4 x 500 m: 1971, K-2 10000 m: 1973), two silvers (K-1 500 m: 1973, K-1 4 x 500 m: 1973)

1♣ opening promises 17 or more HCP, with step answers showing controls (K=1 and A=2 controls) or HCP. 1♦ being negative showing 0-5 HCP and 1♥ showing

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

mtbf ( c 1 ; … ; c n ) = ( ∑ k = 1 n 1 mtbf ( c k ) ) − 1 , {\displaystyle {\text{mtbf}}(c_{1};\dots ;c_{n})=\left(\sum _{k=1}^{n}{\frac

meaning "Pressure is inversely proportional to Volume", or P V = k 1 {\displaystyle PV=k_{1}} , or P 1 V 1 = P 2 V 2 {\displaystyle P_{1}V_{1}=P_{2}V_{2}}

= 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

he won three medals with a gold (1992: K-4 1000 m) and two bronzes (1988: K-1 1000 m, K-4 1000 m). He was born in Berlin. Wohllebe also won eighteen medals