fschieber/wit-captions huggingface dataset:
Przykład zastosowania reguły de l’Hospitala dla funkcji 
  
    
      
        
          
            f
            (
            x
            )
          
        
        =
        
          
            sin
            ⁡
            (
            x
            )
          
        
      
    
    {\displaystyle {\color {BurntOrange}f(x)}={\color {BurntOrange}\sin(x)}}
  
 i 
  
    
      
        
          
            g
            (
            x
            )
          
        
        =
        
          
            −
            0
            
              ,
            
            5
            x
          
        
        
          :
        
      
    
    {\displaystyle {\color {Red}g(x)}={\color {Red}-0{,}5x}{:}}
  
 funkcja 
  
    
      
        
          
            h
            (
            x
            )
          
        
        =
        
          
            f
            (
            x
            )
          
        
        
          /
        
        
          
            g
            (
            x
            )
          
        
      
    
    {\displaystyle {\color {Brown}h(x)}={\color {BurntOrange}f(x)}/{\color {Red}g(x)}}
  
 jest nieokreślona w punkcie 
  
    
      
        x
        =
        0
        ,
      
    
    {\displaystyle x=0,}
  
 ale może być kontynuowana jako funkcja ciągła w całym zbiorze 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 z wykorzystaniem definicji 
  
    
      
        
          
            h
            (
            0
            )
          
        
        =
        
          
            
              f
              ′
            
            (
            0
            )
          
        
        
          /
        
        
          
            
              g
              ′
            
            (
            0
            )
          
        
        =
        −
        2.
      
    
    {\displaystyle {\color {Brown}h(0)}={\color {RoyalBlue}f'(0)}/{\color {Blue}g'(0)}=-2.}
Траєкторія польоту при стрибанні з кручі у вводу параболічна, оскільки горизонтальне переміщення є лінійною функцією від часу 
  
    
      
        x
        =
        
          v
          
            x
          
        
        t
      
    
    {\displaystyle x=v_{x}t}
  
, а вертикальне переміщення є квадратичною функцією від часу 
  
    
      
        y
        =
        
          
            
              1
              2
            
          
        
        a
        
          t
          
            2
          
        
        +
        
          v
          
            y
          
        
        t
        +
        h
      
    
    {\displaystyle y={\tfrac {1}{2}}at^{2}+v_{y}t+h}
  
. В результаті, шлях буде задаватися квадратним рівнянням 
  
    
      
        y
        =
        
          
            
              a
              
                2
                
                  v
                  
                    x
                  
                  
                    2
                  
                
              
            
          
        
        
          x
          
            2
          
        
        +
        
          
            
              
                v
                
                  y
                
              
              
                v
                
                  x
                
              
            
          
        
        x
        +
        h
      
    
    {\displaystyle y={\tfrac {a}{2v_{x}^{2}}}x^{2}+{\tfrac {v_{y}}{v_{x}}}x+h}
  
, де 
  
    
      
        
          v
          
            x
          
        
      
    
    {\displaystyle v_{x}}
  
 і 
  
    
      
        
          v
          
            y
          
        
      
    
    {\displaystyle v_{y}}
  
 — горизонтальна і вертикальна компоненти початкової швидкості, a є гравітаційним прискоренням, а h є початковою висотою. Значення a слід задавати від'ємним, оскільки напрям падіння (вниз) є протилежним до вимірювання висоти (вгору).
English: Cliff jumping near the big cave at La Jolla Cove, with La Jolla Shores in the background. Français : Saut depuis une falaise à La Jolla, San Diego, Californie.
Le triangle de Pascal extrait de son manuscrit. Par base Pascal entend une diagonale du triangle. Pascal se sert de la géométrie de son triangle pour en énoncer les propriétés. En termes modernes : les coefficients binomiaux sont définis par la relation 
  
    
      
        
          
            
              (
            
            
              
                n
                +
                1
              
              
                p
                +
                1
              
            
            
              )
            
          
        
        =
        
          
            
              (
            
            
              n
              
                p
                +
                1
              
            
            
              )
            
          
        
        +
        
          
            
              (
            
            
              n
              p
            
            
              )
            
          
        
      
    
    {\displaystyle {\binom {n+1}{p+1}}={\binom {n}{p+1}}+{\binom {n}{p}}}
  
 ; la conséquence XII, celle à propos de laquelle Pascal introduit la récurrence, revient à énoncer que pour tout entier p entre 1 et n, 
  
    
      
        
          
            
              (
            
            
              n
              p
            
            
              )
            
          
        
        =
        
          
            
              n
              −
              p
              +
              1
            
            p
          
        
        
          
            
              (
            
            
              n
              
                p
                −
                1
              
            
            
              )
            
          
        
      
    
    {\displaystyle {\binom {n}{p}}={\frac {n-p+1}{p}}{\binom {n}{p-1}}}
  
, ce que Pascal démontre par récurrence sur n. L'étape de récurrence est exposée sur un cas particulier[3].
Español: Triángulo de Pascal en el escrito original de Pascal
Airy distribution 
  
    
      
        
          A
          
            trans
          
          
            ′
          
        
      
    
    {\displaystyle A_{\text{trans}}^{\prime }}
  
 (solid lines), corresponding to light transmitted through a Fabry-Pérot resonator, calculated for different values of the reflectivities 
  
    
      
        
          R
          
            1
          
        
        =
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{1}=R_{2}}
  
, and comparison with a single Lorentzian line (dashed lines) calculated for the same 
  
    
      
        
          R
          
            1
          
        
        =
        
          R
          
            2
          
        
      
    
    {\displaystyle R_{1}=R_{2}}
  
.[8] At half maximum (black line), with decreasing reflectivities the FWHM linewidth 
  
    
      
        Δ
        
          ν
          
            Airy
          
        
      
    
    {\displaystyle \Delta \nu _{\text{Airy}}}
  
 of the Airy distribution broadens compared to the FWHM linewidth 
  
    
      
        Δ
        
          ν
          
            c
          
        
      
    
    {\displaystyle \Delta \nu _{c}}
  
 of its corresponding Lorentzian line: 
  
    
      
        
          R
          
            1
          
        
        =
        
          R
          
            2
          
        
        =
        0.9
        ,
        0.6
        ,
        0.32
        ,
        0.172
      
    
    {\displaystyle R_{1}=R_{2}=0.9,0.6,0.32,0.172}
  
 results in 
  
    
      
        Δ
        
          ν
          
            Airy
          
        
        
          /
        
        Δ
        
          ν
          
            c
          
        
        =
        1.001
        ,
        1.022
        ,
        1.132
        ,
        1.717
      
    
    {\displaystyle \Delta \nu _{\text{Airy}}/\Delta \nu _{c}=1.001,1.022,1.132,1.717}
  
, respectively.
English: Airy distribution A t r a n s ′ {\displaystyle A'_{\rm {trans}}} (solid lines), corresponding to light transmitted through a Fabry-Pérot resonator, calculated for different values of the reflectivities R 1 = R 2 {\displaystyle R_{1}=R_{2}} , and comparison with a single Lorentzian line (dashed lines) calculated for the same R 1 = R 2 {\displaystyle R_{1}=R_{2}} .[1] At half maximum (black line), with decreasing reflectivities the FWHM linewidth Δ ν A i r y {\displaystyle \Delta \nu _{Airy}} of the Airy distribution broadens compared to the FWHM linewidth Δ ν c {\displaystyle \Delta \nu _{c}} of its corresponding Lorentzian line: R 1 = R 2 = 0.9 , 0.6 , 0.32 , 0.172 {\displaystyle R_{1}=R_{2}=0.9,0.6,0.32,0.172} results in Δ ν A i r y / Δ ν c = 1.001 , 1.022 , 1.132 , 1.717 {\displaystyle \Delta \nu _{Airy}/\Delta \nu _{c}=1.001,1.022,1.132,1.717} , respectively.
Six iterations of a set of states 
  
    
      
        [
        x
        ,
        y
        ]
      
    
    {\displaystyle [x,y]}
  
  passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation 
  
    
      
        
          x
          
            k
            +
            1
          
        
        =
        4
        
          x
          
            k
          
        
        (
        1
        −
        
          x
          
            k
          
        
        )
      
    
    {\displaystyle x_{k+1}=4x_{k}(1-x_{k})}
  
. To expand the state-space of the logistic map into two dimensions,  a second state, 
  
    
      
        y
      
    
    {\displaystyle y}
  
, was created as 
  
    
      
        
          y
          
            k
            +
            1
          
        
        =
        
          x
          
            k
          
        
        +
        
          y
          
            k
          
        
      
    
    {\displaystyle y_{k+1}=x_{k}+y_{k}}
  
, if 
  
    
      
        
          x
          
            k
          
        
        +
        
          y
          
            k
          
        
        <
        1
      
    
    {\displaystyle x_{k}+y_{k}<1}
  
 and 
  
    
      
        
          y
          
            k
            +
            1
          
        
        =
        
          x
          
            k
          
        
        +
        
          y
          
            k
          
        
        −
        1
      
    
    {\displaystyle y_{k+1}=x_{k}+y_{k}-1}
  
 otherwise.
English: Six iterations of a set of states $[x,y]$ passed through the logistic map. (a) the blue plot (legend 1) shows the first iterate (initial condition), which essentially forms a circle. Plots in (a), (b) and (c), show the first to the sixth iteration of the circular initial conditions. It can be seen that \emph{mixing} occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogenous and irreversible. The logistic map has equation $x_{k+1} = 4 x_k (1 - x_k )$. In order to expand the state-space of the logistic map into two dimensions, a second state, $y$, was created as $y_{k+1} = x_k + y_k $, if $x_k + y_k <1$ and $y_{k+1} = x_k + y_k -1$ otherwise.
Matching Pennies: Dieses Schaubild stellt die Reaktionskorrespondenzen 
  
    
      
        B
        
          R
          
            1
          
        
        (
        
          q
          
            2
          
        
        )
        :=
        arg
        ⁡
        max
        {
        E
        ⁡
        [
        U
        (
        
          q
          
            1
          
        
        ,
        
          q
          
            2
          
        
        )
        ]
        ∣
        
          q
          
            2
          
        
        ∈
        
          Q
          
            1
          
        
        }
      
    
    {\displaystyle BR_{1}(q_{2}):=\arg \max\{\operatorname {E} [U(q_{1},q_{2})]\mid q_{2}\in Q_{1}\}}
  
 und 
  
    
      
        B
        
          R
          
            2
          
        
        (
        
          q
          
            1
          
        
        )
        :=
        arg
        ⁡
        max
        {
        E
        ⁡
        [
        U
        (
        
          q
          
            1
          
        
        ,
        
          q
          
            2
          
        
        )
        ]
        ∣
        
          q
          
            2
          
        
        ∈
        
          Q
          
            2
          
        
        }
      
    
    {\displaystyle BR_{2}(q_{1}):=\arg \max\{\operatorname {E} [U(q_{1},q_{2})]\mid q_{2}\in Q_{2}\}}
  
 graphisch dar. Die Reaktionskorrespondenzen sind Funktionen, wenn 
  
    
      
        
          q
          
            i
          
        
        ≠
        
          
            
              1
              2
            
          
        
      
    
    {\displaystyle q_{i}\neq {\tfrac {1}{2}}}
  
.[2] Das einzige Nash-Gleichgewicht in gemischten Strategien findet sich im Schnittpunkt B der beiden Beste-Antwort-Korrespondenzen (Die Linien wurden gestrichelt dargestellt, um nicht an ein Hakenkreuz zu erinnern).
English: best response
Rundenfunktion von SHA-224 und SHA-256
   
  
    
      
        Ch
        ⁡
        (
        E
        ,
        F
        ,
        G
        )
        =
        (
        E
        ∧
        F
        )
        ⊕
        (
        ¬
        E
        ∧
        G
        )
      
    
    {\displaystyle \operatorname {Ch} (E,F,G)=(E\land F)\oplus (\neg E\land G)}
  

   
  
    
      
        Ma
        ⁡
        (
        A
        ,
        B
        ,
        C
        )
        =
        (
        A
        ∧
        B
        )
        ⊕
        (
        A
        ∧
        C
        )
        ⊕
        (
        B
        ∧
        C
        )
      
    
    {\displaystyle \operatorname {Ma} (A,B,C)=(A\land B)\oplus (A\land C)\oplus (B\land C)}
  

   
  
    
      
        
          Σ
          
            0
          
        
        (
        A
        )
        =
        (
        A
        
        ⋙
        
        2
        )
        ⊕
        (
        A
        
        ⋙
        
        13
        )
        ⊕
        (
        A
        
        ⋙
        
        22
        )
      
    
    {\displaystyle \Sigma _{0}(A)=(A\!\ggg \!2)\oplus (A\!\ggg \!13)\oplus (A\!\ggg \!22)}
  

   
  
    
      
        
          Σ
          
            1
          
        
        (
        E
        )
        =
        (
        E
        
        ⋙
        
        6
        )
        ⊕
        (
        E
        
        ⋙
        
        11
        )
        ⊕
        (
        E
        
        ⋙
        
        25
        )
      
    
    {\displaystyle \Sigma _{1}(E)=(E\!\ggg \!6)\oplus (E\!\ggg \!11)\oplus (E\!\ggg \!25)}
  

 Die Varianten mit 64-Bit-Wörtern nutzen die gleiche Rundenfunktion, nur mit anderen Rotationsweiten für 
  
    
      
        
          Σ
          
            0
          
        
      
    
    {\displaystyle \Sigma _{0}}
  
 und 
  
    
      
        
          Σ
          
            1
          
        
      
    
    {\displaystyle \Sigma _{1}}
Konzentrations-Zeit-Profil einer Reaktion A 
  
    
      
        →
      
    
    {\displaystyle \rightarrow }
  
 P, dessen Ableitung 
  
    
      
        
          
            d
            (
            [
            A
            ]
            
              /
            
            [
            A
            ]
          
          
            0
          
        
        )
        
          /
        
        
          d
        
        t
      
    
    {\displaystyle \mathrm {d([A]/[A]} _{0})/\mathrm {d} t}
  
 und Reaktionsgeschwindigkeit 
  
    
      
        1
        
          /
        
        
          ν
          
            A
          
        
        ⋅
        
          
            d
            (
            [
            A
            ]
            
              /
            
            [
            A
            ]
          
          
            0
          
        
        )
        
          /
        
        
          d
        
        t
      
    
    {\displaystyle 1/\nu _{A}\cdot \mathrm {d([A]/[A]} _{0})/\mathrm {d} t}
  
 als Funktion der Reaktionsdauer t. 
  
    
      
        [
        A
        
          ]
          
            0
          
        
      
    
    {\displaystyle [A]_{0}}
  
 ist die Ausgangskonzentration, [A](t) die Konzentration nach der Reaktionsdauer t und 
  
    
      
        
          ν
          
            A
          
        
        =
        −
        1
      
    
    {\displaystyle \nu _{A}=-1}
  
 die stöchiometrische Zahl von Ausgangsstoff A. 
  
    
      
        
          [
          A
          ]
        
        (
        t
        )
        
          /
        
        
          
            [
            A
            ]
          
          
            0
          
        
      
    
    {\displaystyle \mathrm {[A]} (t)/\mathrm {[A]} _{0}}
  
 ist eine dimensionslose relative Konzentration von A, aufgrund von deren Verwendung die Reaktionsgeschwindigkeit die Dimension einer Frequenz bekommt.
Deutsch: Konzentrations-Zeit-Profil einer Reaktion A --> P, Ableitung desselben und Reaktionsgeschwindigkeit als Funktion der Reaktionsdauer. [A](t) ist die Konzentration eines Ausgangsstoffes A nach der Reaktionsdauer t, [A]0 die Ausgangskonzentration von A vor Reaktionsbeginn und uA = -1 die Stöchiometriezahl von A. [A](t) / [A]0 ist eine dimensionslose relative Konzentration von A.
Latvijas fizioģeogrāfiskā rajonēšana.
I — Piejūras zemiene: 1 — Bārtavas līdzenums, 2 — Piemares līdzenums, 3 — Ventavas līdzenums, 4 — Irves līdzenums, 5 — Engures līdzenums, 6 — Rīgavas līdzenums, 7 — Vidzemes piekrasteII — Kursas zemiene: 1 — Ugāles līdzenums, 2 — Pieventas līdzenumsIII — Rietumkursas augstiene: 1 — Kurmāles pauguraine, 2 — Apriķu līdzenums, 3 — Bandavas pauguraine, 4 — Vārtājas viļņotais līdzenums, 5 — Embūtes pauguraineIV — Ziemeļkursas augstiene: 1 — Dundagas pacēlums, 2 — Vanemas pauguraineV — Austrumkursas augstiene: 1 — Abavas senleja, 2 — Vārmes nolaidenums, 3 — Saldus pauguraine, 4 — Spārnenes viļņotais līdzenums, 5 — Lielauces pauguraineVI — Viduslatvijas zemiene: 1 — Vadakstes līdzenums, 2 — Zemgales līdzenums, 3 — Tīreļu līdzenums, 4 — Upmales paugurlīdzenums, 5 — Taurkalnes līdzenums, 6 — Lejasdaugavas senleja, 7 — Viduslatvijas nolaidenums, 8 — Ropažu līdzenums, 9 — Metsepoles līdzenumsVII — Idumejas augstiene: 1 — Limbažu viļņotais līdzenums, 2 — Augstrozes paugurvalnis, 3 — Gaujas senlejaVIII — Tālavas zemiene: 1 — Burtnieka līdzenums, 2 — Sedas līdzenums, 3 — Trikātas pacēlums, 4 — Trapenes līdzenumsIX — Sakalas augstiene: 1 — Ērģemes pauguraineX — Alūksnes augstiene: 1 — Veclaicenes pauguraine, 2 — Vaidavas pazeminājums, 3 — Malienas pauguraine, 4 — Gulbenes paugurvalnisXI — Vidzemes augstiene: 1 — Aumeisteru paugurvalnis, 2 — Mežoles pauguraine, 3 — Augšgaujas pazeminājums, 4 — Piebalgas pauguraine, 5 — Augšogres pazeminājums, 6 — Vestienas pauguraineXII — Austrumlatvijas zemiene: 1 — Adzeles pacēlums, 2 — Lubāna līdzenums, 3 — Aronas paugurlīdzenums, 4 — Aknīstes nolaidenums, 5 — Jersikas līdzenumsXIII — Augšzemes augstiene: 1 — Sēlijas paugurvalnis, 2 — Ilūkstes pauguraine, 3 — Skrudalienas pauguraineXIV — Mudavas zemiene: 1 — Abrenes nolaidenums, 2 — Zilupes līdzenumsXV — Latgales augstiene: 1 — Burzavas pauguraine, 2 — Rēzeknes pazeminājums, 3 — Maltas pazeminājums, 4 — Rāznavas pauguraine, 5 — Feimaņu pauguraine, 6 — Dagdas pauguraine, 7 — Augšdaugavas pazeminājums
A realization of an exchangeable random graph defined by a graphon. The graphon is shown as a magenta heatmap (lower right). A random graph of size 
  
    
      
        n
      
    
    {\displaystyle n}
  
 is generated by independently assigning to each vertex 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        n
        }
      
    
    {\displaystyle k\in \{1,\dotsc ,n\}}
  
 a latent random variable     
  
    
      
        
          U
          
            k
          
        
        ∼
        
          U
        
        (
        0
        ,
        1
        )
      
    
    {\displaystyle U_{k}\sim \mathrm {U} (0,1)}
  
 (values along vertical axis) and     including each edge 
  
    
      
        (
        k
        ,
        l
        )
      
    
    {\displaystyle (k,l)}
  
 independently with probability 
  
    
      
        f
        (
        
          U
          
            k
          
        
        ,
        
          U
          
            l
          
        
        )
      
    
    {\displaystyle f(U_{k},U_{l})}
  
.     For example, edge 
  
    
      
        (
        3
        ,
        5
        )
      
    
    {\displaystyle (3,5)}
  
 (green, dotted) is present with probability     
  
    
      
        f
        (
        0.72
        ,
        0.9
        )
      
    
    {\displaystyle f(0.72,0.9)}
  
; the green boxes in the right square represent the     values of 
  
    
      
        (
        
          u
          
            3
          
        
        ,
        
          u
          
            5
          
        
        )
      
    
    {\displaystyle (u_{3},u_{5})}
  
 and 
  
    
      
        (
        
          u
          
            5
          
        
        ,
        
          u
          
            3
          
        
        )
      
    
    {\displaystyle (u_{5},u_{3})}
  
. The upper left     panel shows the graph realization as an adjacency matrix.
English: The generative model for an exchangeable random graph defined by a graphon. The graphon is shown as a magenta heatmap. A random graph is generated by assigning each vertex k {\displaystyle k} a U [ 0 , 1 ] {\displaystyle U[0,1]} random variable u k {\displaystyle u_{k}} and including each edge ( k , l ) {\displaystyle (k,l)} independently with probability W ( U k , U l ) {\displaystyle W(U_{k},U_{l})} . For example, edge ( 3 , 5 ) {\displaystyle (3,5)} (green, dotted) is present with probability W ( 0.72 , 0.9 ) {\displaystyle W(0.72,0.9)} ; the green boxes in the right square represent the values of ( u 3 , u 5 ) {\displaystyle (u_{3},u_{5})} and ( u 5 , u 3 ) {\displaystyle (u_{5},u_{3})} . The upper left panel shows the graph realization as an adjacency matrix. This figure was originally made for (and is included in) https://arxiv.org/abs/1512.03099. It is intended for inclusion in graphon