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searching for K-1 503 found (14413 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)}}}

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

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,

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

{\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)+(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 )

∑ 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

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

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}}

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

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

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

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

[ 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

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

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

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

) = 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 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

_{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 )

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

⁡ 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 > 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

_{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:

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 +

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

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

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

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 −

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

{\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

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

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

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 … 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 ]

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

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

{\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

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

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

(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

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 {\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 −

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

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 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

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

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

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

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

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

, 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

{\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

{\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,

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}}}

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

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

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

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

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

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

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

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} ⁠

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 + .

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

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

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

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

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 =

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

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

_{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}\

) 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

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

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

= 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 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

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

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

∑ 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}

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

−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

}})={\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

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

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

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

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.

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

{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

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

{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}} 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

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}

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

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

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

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

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

= ∑ 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

{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

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

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

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

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

( 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

(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 / 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

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 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

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 ( ∑

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

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

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

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 ) ]

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})}

⁠ 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

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

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

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}

) = 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 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

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 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

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

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

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

= 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 ) ∑

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

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

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

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 =

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 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

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

∑ 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

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

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

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 ∞ ( 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

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 −

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

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

( 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 ) 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 +

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 ∞ ( 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

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

\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\

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

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

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:

( 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

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

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

{\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

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 )

− 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

, 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

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 − ◻ →

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

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

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

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

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 ∼

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

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

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

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

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}

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

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)

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

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

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

( 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

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

( 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

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

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 ∞ ( − 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

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 ,

Schüttpelz. The next year, Idem switched to the K-1 events. At the 1988 Olympics in Seoul, she finished ninth in the K-1 500 m and fifth with the West German quartet

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)}}

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

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

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

∇ 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}\

^{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

&\ {\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

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

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

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

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

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

_{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

\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

&\ {\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

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

{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

, 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

( 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}

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

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}}

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

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

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

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

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

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

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

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

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

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

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{

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 +

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

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)

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 = 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}

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

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

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\

= 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

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

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

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

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

= 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

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

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

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

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

∑ 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}

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

) 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 ] =

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

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

= 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

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

_{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 ) ( 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

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 +

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)

(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

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

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-1 200 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

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

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  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

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

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

{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 ) )

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

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

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

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}

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

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 ) =

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))

= 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

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

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

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

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

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

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

≥ 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)  Anton Ryakhov (UZB)  Roman Petrukanecas (LTU) K-1 500 m  Nathan Baggaley (AUS)  Petar Merkov (BUL)

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 ( 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

( 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

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

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)

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

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:

( 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     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

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}}}}

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

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

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 , 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

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

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

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

∂ 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

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

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

) 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

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

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

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

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

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

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

⟩ | 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

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

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

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

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

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

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}}

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

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)

_{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

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

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

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

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

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

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

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

{\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}}

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

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

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

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

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

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

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 , 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 )

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

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

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 ⋯

{\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 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

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 . {\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},}

‖ 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

{\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

{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

+ 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

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

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

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

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

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

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}}}

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

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

, 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

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

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

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

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 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

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

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 ,

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}

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

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)}

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

⁡ ( ∑ 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

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

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

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

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

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

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

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

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

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

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   [

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

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

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

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

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

{\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

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

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

{\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

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

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

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

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

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

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

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

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

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

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

× 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

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

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}))\

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)

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

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

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

{\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

= 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

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 ,

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

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}

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 ! (

) 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

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 +

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

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}}}}

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

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

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

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

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

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

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)

= 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

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

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

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

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

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:

, 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

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−, 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

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

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

+ ‖ 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

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}}}}

− 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

= ‖ 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

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

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

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\

− 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

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

(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

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

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|

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

+ 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

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}

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

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

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, 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

10000 m: 1983), four silvers (K-1 4 x 500 m: 1974, K-2 500 m: 1977, K-2 10000 m: 1979, K-4 1000 m: 1974) and a bronze (K-1 10000 m: 1982). Динамо. Энциклопедия

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

{\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

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

five golds (K-1 4 x 500 m: 1948, 1950, 1954; K-2 500 m: 1950, K-2 1000 m: 1950) and one silver (K-1 500 m: 1948). Note that the K-1 500 m, K-1 4 x 500 m

k 1 j y 2 j = y 1 j − δ 1 j q 1 = k 1 j − δ 1 j q 1 y 3 j = k 1 j − δ 1 j q 1 − δ 2 j q 2 ⋮ y i j = k 1 j − ∑ r = 1 i − 1 δ r j q r = { k 1 j j ≥ i k

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

The Pentax K-1 is the first production Pentax full-frame digital SLR camera. As the flagship model of the Pentax K-mount system, it includes several new

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

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

| n k 1 , n k 2 . . . ⟩ = ( b k 1 † ) n k 1 n k 1 ! ( b k 2 † ) n k 2 n k 2 ! . . . | 0 k 1 , 0 k 2 , . . . ⟩ {\displaystyle |n_{\mathbf {k} _{1}},n_{\mathbf

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;

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

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

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

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 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

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

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

) ( 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

{\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