Datasets:
andito
/
mathwriting-google

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\left(a + \sqrt{3} b\right) \left(c + \sqrt{3} d\right) = [(a c + 3 b d) \,\bmod\,q] + \sqrt{3} [(a d + b c) \,\bmod\,q].
(a+\sqrt{3}b)(c+\sqrt{3}d)=[(ac+3bd)modq]+\sqrt{3}[(ad+bc)modq].
synthetic
cfdbf777ce5e01d4
mathwriting-2024/synthetic/cfdbf777ce5e01d4.inkml
P\left( {y,t} \right)
P(y,t)
synthetic
a8fb853f657dbe58
mathwriting-2024/synthetic/a8fb853f657dbe58.inkml
\varphi = -K\Phi(\Phi^{-1}(1-R_1)-\sigma\sqrt{\tau}) + R_2
\varphi=-K\Phi(\Phi^{-1}(1-R_{1})-\sigma\sqrt{\tau})+R_{2}
synthetic
a4f8a5992ca908c0
mathwriting-2024/synthetic/a4f8a5992ca908c0.inkml
h = \left(q + \left\lfloor\frac{13(m+1)}{5}\right\rfloor + K + \left\lfloor\frac{K}{4}\right\rfloor + \left\lfloor\frac{J}{4}\right\rfloor - 2J\right) \bmod 7,
h=(q+\lfloor\frac{13(m+1)}{5}\rfloor+K+\lfloor\frac{K}{4}\rfloor+\lfloor\frac{J}{4}\rfloor-2J)mod7,
synthetic
e0b40c1d3faf7e01
mathwriting-2024/synthetic/e0b40c1d3faf7e01.inkml
P(k) = 2^{-k}.
P(k)=2^{-k}.
synthetic
dcebae6cff257a29
mathwriting-2024/synthetic/dcebae6cff257a29.inkml
S \And \neg S
S\&\neg S
synthetic
c357cddb83d1ca73
mathwriting-2024/synthetic/c357cddb83d1ca73.inkml
\sigma_p^*
\sigma_{p}^{*}
synthetic
3fb82981b4061701
mathwriting-2024/synthetic/3fb82981b4061701.inkml
T_xM\to E_x,
T_{x}M\rightarrow E_{x},
synthetic
9586a42930b53131
mathwriting-2024/synthetic/9586a42930b53131.inkml
y\leftrightarrow (y,y,\ldots,y)
y\leftrightarrow(y,y,...,y)
synthetic
c9bfa6c47f414544
mathwriting-2024/synthetic/c9bfa6c47f414544.inkml
dk_{i,i+1}/df|_{f=f_0}.
dk_{i,i+1}/df|_{f=f_{0}}.
synthetic
99e47ba60d236160
mathwriting-2024/synthetic/99e47ba60d236160.inkml
\gamma = (q Y_{Fo})/(c_v T_o)
\gamma=(qY_{Fo})/(c_{v}T_{o})
synthetic
4f0503e4e0a15642
mathwriting-2024/synthetic/4f0503e4e0a15642.inkml
\mathbf P = \rho_b \mathbf d
P=\rho_{b}d
synthetic
ebf01e6593c7b360
mathwriting-2024/synthetic/ebf01e6593c7b360.inkml
]a,b[
]a,b[
synthetic
59995ec37d7a00c0
mathwriting-2024/synthetic/59995ec37d7a00c0.inkml
\overline{W}_{\dot{\alpha}}
\overline{W}_{\dot{\alpha}}
synthetic
4cf97bbed56f043a
mathwriting-2024/synthetic/4cf97bbed56f043a.inkml
(b^4-d^4-2b^2 d^2)^2
(b^{4}-d^{4}-2b^{2}d^{2})^{2}
synthetic
0c17ed7b1ba7f2ca
mathwriting-2024/synthetic/0c17ed7b1ba7f2ca.inkml
V\, \begin{bmatrix} \frac{d_1}{b_{1,1}}\\ \vdots\\ \frac{d_k}{b_{k,k}}\\ h_{k+1}\\ \vdots\\ h_n \end{bmatrix}\,,
V[\begin{matrix}\frac{d_{1}}{b_{1,1}}\\ \vdots\\ \frac{d_{k}}{b_{k,k}}\\ h_{k+1}\\ \vdots\\ h_{n}\end{matrix}],
synthetic
0cf69befa1b57c32
mathwriting-2024/synthetic/0cf69befa1b57c32.inkml
\pi\left( \frac{1}{i \pi \omega} + \delta(\omega)\right)
\pi(\frac{1}{i\pi\omega}+\delta(\omega))
synthetic
6c1c2586ed90e61f
mathwriting-2024/synthetic/6c1c2586ed90e61f.inkml
dW = L I \cdot dI
dW=LI\cdot dI
synthetic
55fd4fab42413c47
mathwriting-2024/synthetic/55fd4fab42413c47.inkml
g(0) - g_T(0) = \frac 1 {2 \pi i }\int_C \left( g(z) - g_T(z) \right ) \frac {dz} z = \frac 1 {2 \pi i }\int_C \left( g(z) - g_T(z) \right ) F(z)\frac {dz} z
g(0)-g_{T}(0)=\frac{1}{2\pi i}\int_{C}(g(z)-g_{T}(z))\frac{dz}{z}=\frac{1}{2\pi i}\int_{C}(g(z)-g_{T}(z))F(z)\frac{dz}{z}
synthetic
810c78db1b61ae89
mathwriting-2024/synthetic/810c78db1b61ae89.inkml
B(e_s,e_s)=1
B(e_{s},e_{s})=1
synthetic
b9f7fcf3556af285
mathwriting-2024/synthetic/b9f7fcf3556af285.inkml
\frac{777999225577}{44.38174578\cdot333336668888}
\frac{777999225577}{44.38174578\cdot333336668888}
synthetic
a575ad144bca7a0e
mathwriting-2024/synthetic/a575ad144bca7a0e.inkml
\theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\}^3 \biggr\rangle = \theta_{4}\biggl\langle q\bigl\{\tan\bigl[\tfrac{1}{2}\arctan(1)\bigr]\bigr\} \biggr\rangle \,3^{-1/2} \bigl(\sqrt{3} + \sqrt{2}\,\bigr)^{1/2}
\theta_{4}\langle q\{tan[\frac{1}{2}arctan(1)]\}^{3}\rangle=\theta_{4}\langle q\{tan[\frac{1}{2}arctan(1)]\}\rangle3^{-1/2}(\sqrt{3}+\sqrt{2})^{1/2}
synthetic
5bca8cce98fa576c
mathwriting-2024/synthetic/5bca8cce98fa576c.inkml
f_2(x_0, x_1, x_2)
f_{2}(x_{0},x_{1},x_{2})
synthetic
9f31ed4873cb73a2
mathwriting-2024/synthetic/9f31ed4873cb73a2.inkml
IAS \approx \sqrt{\frac{2 (p_t - p_s)}{\rho(0)}}
IAS\approx\sqrt{\frac{2(p_{t}-p_{s})}{\rho(0)}}
synthetic
4c4d9167134b9e3d
mathwriting-2024/synthetic/4c4d9167134b9e3d.inkml
n'\geq n/2
n^{\prime}\ge n/2
synthetic
a684c870707dbe63
mathwriting-2024/synthetic/a684c870707dbe63.inkml
\forall x \in A_R A \le x
\forall x\in A_{R}A\le x
synthetic
1f17f4813613cdd4
mathwriting-2024/synthetic/1f17f4813613cdd4.inkml
a h
ah
synthetic
ed20cd64f9ca1af5
mathwriting-2024/synthetic/ed20cd64f9ca1af5.inkml
n\in\mathbb P,~a=0,
n\in\mathbb{P},a=0,
synthetic
cdbc93b07dba6270
mathwriting-2024/synthetic/cdbc93b07dba6270.inkml
x(e^{a_1y}-e^{a_2y})-b=xye^{a_1y}-c=0
x(e^{a_{1}y}-e^{a_{2}y})-b=xye^{a_{1}y}-c=0
synthetic
e28c7e09e4b937b0
mathwriting-2024/synthetic/e28c7e09e4b937b0.inkml
\Pi_C\,
\Pi_{C}
synthetic
b40632fee2ba6b95
mathwriting-2024/synthetic/b40632fee2ba6b95.inkml
k \operatorname{cs}
kcs
synthetic
ec77009f79d3aad0
mathwriting-2024/synthetic/ec77009f79d3aad0.inkml
\{(n_1,\dots,n_k)\in \mathbb{N}^k \mid n_1+\cdots+n_k=n\}.\,
\{(n_{1},...,n_{k})\in\mathbb{N}^{k}|n_{1}+\cdot\cdot\cdot+n_{k}=n\}.
synthetic
0273f26fea5342fd
mathwriting-2024/synthetic/0273f26fea5342fd.inkml
\left\langle -2, \{0,1,\mathrm i,1+\mathrm i\}\right\rangle.
\langle-2,\{0,1,i,1+i\}\rangle.
synthetic
9f07cd9c387c502a
mathwriting-2024/synthetic/9f07cd9c387c502a.inkml
X = \sigma_x =\operatorname{NOT} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} ,
X=\sigma_{x}=NOT=[\begin{matrix}0&1\\ 1&0\end{matrix}],
synthetic
662cd3b89f04672d
mathwriting-2024/synthetic/662cd3b89f04672d.inkml
\mathsf{NL} \subseteq \mathsf{SPACE}(\log^2n)
NL\subseteq SPACE(log^{2}n)
synthetic
890433b5fbfd6edc
mathwriting-2024/synthetic/890433b5fbfd6edc.inkml
dX = \mu dt + s dZ
dX=\mu dt+sdZ
synthetic
90a7f9df6679cca5
mathwriting-2024/synthetic/90a7f9df6679cca5.inkml
\text{SWR} = \max \left\{ \frac{R_\text{L}}{\,Z_\text{0}\,} \, , \frac{\,Z_\text{0}\,}{R_\text{L}} \right\}
SWR=max\{\frac{R_{L}}{Z_{0}},\frac{Z_{0}}{R_{L}}\}
synthetic
22eba7bb54a35ee8
mathwriting-2024/synthetic/22eba7bb54a35ee8.inkml
\frac{97.908047487}{222222000004}
\frac{97.908047487}{222222000004}
synthetic
e4d84edce3ee6c27
mathwriting-2024/synthetic/e4d84edce3ee6c27.inkml
\boldsymbol{p} = m\boldsymbol{v}
p=mv
synthetic
0c90dfe27a259c5d
mathwriting-2024/synthetic/0c90dfe27a259c5d.inkml
|\mathbf{B}| = |\mathbf{a}||\mathbf{b}|\sin\theta,
|B|=|a||b|sin\theta,
synthetic
6952b0d5d7dd135b
mathwriting-2024/synthetic/6952b0d5d7dd135b.inkml
w_{j+1}=w_j-\frac{w_j e^{w_j}-z}{e^{w_j}+w_j e^{w_j}}.
w_{j+1}=w_{j}-\frac{w_{j}e^{w_{j}}-z}{e^{w_{j}}+w_{j}e^{w_{j}}}.
synthetic
5de89a80dc7551fc
mathwriting-2024/synthetic/5de89a80dc7551fc.inkml
\hat{\boldsymbol{\tau}} = \hat{\boldsymbol{\tau}}_1 \hat{\boldsymbol{\tau}}_2 \hat{\boldsymbol{\tau}}_3
\hat{\tau}=\hat{\tau}_{1}\hat{\tau}_{2}\hat{\tau}_{3}
synthetic
6eb2ee9f2ee007fb
mathwriting-2024/synthetic/6eb2ee9f2ee007fb.inkml
[0, 3]
[0,3]
synthetic
f948e543b4c0d275
mathwriting-2024/synthetic/f948e543b4c0d275.inkml
\lim_{n \to \infty} x_n = \sqrt S
lim_{n\rightarrow\infty}x_{n}=\sqrt{S}
synthetic
e234591b9ad04775
mathwriting-2024/synthetic/e234591b9ad04775.inkml
\ W_{final}=n*(1-(1-\beta)F_1)*(1-(1-\beta)F_2)...
W_{final}=n*(1-(1-\beta)F_{1})*(1-(1-\beta)F_{2})...
synthetic
95ec6d6c51584e26
mathwriting-2024/synthetic/95ec6d6c51584e26.inkml
P^{2m}(R)=E[\Pr[\sigma(x)\in R]]\leq \max_{x\in X^{2m}}(\Pr[\sigma(x)\in R]),
P^{2m}(R)=E[Pr[\sigma(x)\in R]]\le max_{x\in X^{2m}}(Pr[\sigma(x)\in R]),
synthetic
5f09529f8ab6b937
mathwriting-2024/synthetic/5f09529f8ab6b937.inkml
\mathcal{FL^-}
FL^{-}
synthetic
7bb510a88a863100
mathwriting-2024/synthetic/7bb510a88a863100.inkml
S, T \subseteq Y,
S,T\subseteq Y,
synthetic
9fca1c9a11fb1a85
mathwriting-2024/synthetic/9fca1c9a11fb1a85.inkml
H_A \otimes H_B
H_{A}\otimes H_{B}
synthetic
99ac1454f4a60d4f
mathwriting-2024/synthetic/99ac1454f4a60d4f.inkml
f(k,r)\le C^k
f(k,r)\le C^{k}
synthetic
20661fead373fe6e
mathwriting-2024/synthetic/20661fead373fe6e.inkml
g(-t)
g(-t)
synthetic
64500930b87cff61
mathwriting-2024/synthetic/64500930b87cff61.inkml
\tfrac12+\Omega(1/\sqrt\Delta)
\frac{1}{2}+\Omega(1/\sqrt{\Delta})
synthetic
b2383ef54f58c6e7
mathwriting-2024/synthetic/b2383ef54f58c6e7.inkml
(A f)(x) = -i \frac{\mathrm{d}}{\mathrm{d}x} f(x) \,,
(Af)(x)=-i\frac{d}{dx}f(x),
synthetic
ec3c942306675b5a
mathwriting-2024/synthetic/ec3c942306675b5a.inkml
\cos(3\theta_c)=\sin\left(\left(3\theta_s - \frac{\pi}{2}\right) + \frac{\pi}{2} \right)
cos(3\theta_{c})=sin((3\theta_{s}-\frac{\pi}{2})+\frac{\pi}{2})
synthetic
bc57314355d8cfca
mathwriting-2024/synthetic/bc57314355d8cfca.inkml
a_i,\ a_{i+1}
a_{i},a_{i+1}
synthetic
8dde15ec88ec3623
mathwriting-2024/synthetic/8dde15ec88ec3623.inkml
\underline{\mathbf{\hat{T}}}: \mathcal{V} \rightarrow \mathcal{V}
\underline{\hat{T}}:V\rightarrow V
synthetic
0fd7a293390fa228
mathwriting-2024/synthetic/0fd7a293390fa228.inkml
\Delta\lambda = \lambda^2\frac{\delta D}{2D\Delta D}.
\Delta\lambda=\lambda^{2}\frac{\delta D}{2D\Delta D}.
synthetic
51ccff9f473295d8
mathwriting-2024/synthetic/51ccff9f473295d8.inkml
E \approx K E_0
E\approx KE_{0}
synthetic
01c1f901328ecb2e
mathwriting-2024/synthetic/01c1f901328ecb2e.inkml
(\nabla^2 + k^2)\mathbf{B} = 0,\, \mathbf{E} = -\frac{i}{k} \nabla \times \mathbf{B}.
(\nabla^{2}+k^{2})B=0,E=-\frac{i}{k}\nabla\times B.
synthetic
c9e0be5155262518
mathwriting-2024/synthetic/c9e0be5155262518.inkml
1_\mathcal{D}
1_{D}
synthetic
de96a2ec644a9bbb
mathwriting-2024/synthetic/de96a2ec644a9bbb.inkml
=\tfrac{ (x+x^2+x^3+...+x^n) + x(x+x^2+...+x^{n-1}) + ... + (x^n) } {1 + x + x^2 + x^3 + ...}=
=\frac{(x+x^{2}+x^{3}+...+x^{n})+x(x+x^{2}+...+x^{n-1})+...+(x^{n})}{1+x+x^{2}+x^{3}+...}=
synthetic
e3b276aa99655ca9
mathwriting-2024/synthetic/e3b276aa99655ca9.inkml
\mathbf{n} \cdot \nabla\psi
n\cdot\nabla\psi
synthetic
d8e6cd364c070508
mathwriting-2024/synthetic/d8e6cd364c070508.inkml
T= (2\pi)^4\frac{constant^3}{G^2 \left(M_1 + M_2\right)^2}
T=(2\pi)^{4}\frac{constant^{3}}{G^{2}(M_{1}+M_{2})^{2}}
synthetic
2b66e829fe5cb928
mathwriting-2024/synthetic/2b66e829fe5cb928.inkml
\int_{-\infty}^{\infty}f(x)\,dx = \int_{-1}^{1} f\left(\frac{t}{1-t^2}\right)\frac{1+t^2}{(1-t^2)^2}\,dt,
\int_{-\infty}^{\infty}f(x)dx=\int_{-1}^{1}f(\frac{t}{1-t^{2}})\frac{1+t^{2}}{(1-t^{2})^{2}}dt,
synthetic
ecd503b77dee7bf8
mathwriting-2024/synthetic/ecd503b77dee7bf8.inkml
\int_{-\infty}^\infty \mathit{He}_m(x) \mathit{He}_n(x)\, e^{-\frac{x^2}{2}} \,dx = \sqrt{2 \pi}\, n!\, \delta_{nm},
\int_{-\infty}^{\infty}He_{m}(x)He_{n}(x)e^{-\frac{x^{2}}{2}}dx=\sqrt{2\pi}n!\delta_{nm},
synthetic
40d9e39280c281c3
mathwriting-2024/synthetic/40d9e39280c281c3.inkml
X = \cup_i \psi_i(X_i),
X=\cup_{i}\psi_{i}(X_{i}),
synthetic
25602195264de5bb
mathwriting-2024/synthetic/25602195264de5bb.inkml
w^0_{,1111} + 2\,w^0_{,1212} + w^0_{,2222} = \cfrac{q}{D}
w_{,1111}^{0}+2w_{,1212}^{0}+w_{,2222}^{0}=\frac{q}{D}
synthetic
5ac15a71a085db35
mathwriting-2024/synthetic/5ac15a71a085db35.inkml
U_1 \in [10V,11V]
U_{1}\in[10V,11V]
synthetic
6568b38c1640b9c4
mathwriting-2024/synthetic/6568b38c1640b9c4.inkml
\sum_{i=1}^m a_{ij} x_i + e_j t_j \ge g_j,
\sum_{i=1}^{m}a_{ij}x_{i}+e_{j}t_{j}\ge g_{j},
synthetic
b3b0392fb33e3126
mathwriting-2024/synthetic/b3b0392fb33e3126.inkml
lastBlock \leftarrow 0
lastBlock\leftarrow0
synthetic
a8d93d11d7b4c625
mathwriting-2024/synthetic/a8d93d11d7b4c625.inkml
\big(K = K_{\text{max}} = K_{\text{Ic}}\big)
(K=K_{max}=K_{Ic})
synthetic
601c38c88e987986
mathwriting-2024/synthetic/601c38c88e987986.inkml
t_1 t_2 = 1
t_{1}t_{2}=1
synthetic
e675b97c5e722dc7
mathwriting-2024/synthetic/e675b97c5e722dc7.inkml
= 1.0200\ldots-1
=1.0200...-1
synthetic
f980ffc5f542e591
mathwriting-2024/synthetic/f980ffc5f542e591.inkml
\operatorname{cl}_D(A),
cl_{D}(A),
synthetic
4b141fda507d5131
mathwriting-2024/synthetic/4b141fda507d5131.inkml
-2\pi n
-2\pi n
synthetic
4f2abc86c8f29d0d
mathwriting-2024/synthetic/4f2abc86c8f29d0d.inkml
\frac{3[\gamma+\ln(2\pi f_H\tau)]-\ln 2}{4\pi^2}
\frac{3[\gamma+ln(2\pi f_{H}\tau)]-ln2}{4\pi^{2}}
synthetic
dd911d9eeb2ce8cf
mathwriting-2024/synthetic/dd911d9eeb2ce8cf.inkml
\overline{v_n^2} = \text{v}_\text{rms}^2
\overline{v_{n}^{2}}=v_{rms}^{2}
synthetic
317c3a6062c244bb
mathwriting-2024/synthetic/317c3a6062c244bb.inkml
\alpha : A \longrightarrow FA
\alpha:A\longrightarrow FA
synthetic
71d1c0379582f09d
mathwriting-2024/synthetic/71d1c0379582f09d.inkml
s^2 = \frac{n}{n-1} s_n^2
s^{2}=\frac{n}{n-1}s_{n}^{2}
synthetic
6708f4a4265efe9e
mathwriting-2024/synthetic/6708f4a4265efe9e.inkml
\Delta^*
\Delta^{*}
synthetic
bcf1c918e3b5897d
mathwriting-2024/synthetic/bcf1c918e3b5897d.inkml
T_2 = \frac{1}{4} \sum_{i,j} \sum_{a,b} t_{ab}^{ij} \hat{a}^a \hat{a}^b \hat{a}_j \hat{a}_i,
T_{2}=\frac{1}{4}\sum_{i,j}\sum_{a,b}t_{ab}^{ij}\hat{a}^{a}\hat{a}^{b}\hat{a}_{j}\hat{a}_{i},
synthetic
0d4b52496e1c75b6
mathwriting-2024/synthetic/0d4b52496e1c75b6.inkml
4 \times 2r = 8r
4\times2r=8r
synthetic
0a50a3a3a6cca752
mathwriting-2024/synthetic/0a50a3a3a6cca752.inkml
f(z):=g\pi Hz
f(z):=g\pi Hz
synthetic
17506ce485696232
mathwriting-2024/synthetic/17506ce485696232.inkml
\lambda_{1}g_{1}(x)=\cdots = \lambda_{m}g_{m}(x) = 0
\lambda_{1}g_{1}(x)=\cdot\cdot\cdot=\lambda_{m}g_{m}(x)=0
synthetic
2af34e0ad0a3429d
mathwriting-2024/synthetic/2af34e0ad0a3429d.inkml
I_d \equiv \int_1^\infty e^{- {\gamma}/{\theta}} (\gamma^2-1)^{{d}/{2}} \mathrm{d}\gamma.
I_{d}\equiv\int_{1}^{\infty}e^{-\gamma/\theta}(\gamma^{2}-1)^{d/2}d\gamma.
synthetic
6427e170a0eb3d8c
mathwriting-2024/synthetic/6427e170a0eb3d8c.inkml
\frac{94.5}{75}
\frac{94.5}{75}
synthetic
246e82d0e68ead8b
mathwriting-2024/synthetic/246e82d0e68ead8b.inkml
C = \frac{q}{V},
C=\frac{q}{V},
synthetic
cd95914936dbf6a9
mathwriting-2024/synthetic/cd95914936dbf6a9.inkml
\left|\frac{e(S, T)}{e(G)} - \alpha \beta\right| \leq \frac{\lambda}{\sqrt{d_Ld_R}} \sqrt{\alpha \beta (1 - \alpha) (1 - \beta)} \leq \frac{\lambda}{\sqrt{d_Ld_R}} \sqrt{\alpha \beta}\,.
|\frac{e(S,T)}{e(G)}-\alpha\beta|\le\frac{\lambda}{\sqrt{d_{L}d_{R}}}\sqrt{\alpha\beta(1-\alpha)(1-\beta)}\le\frac{\lambda}{\sqrt{d_{L}d_{R}}}\sqrt{\alpha\beta}.
synthetic
cfd12dae86ccfa0f
mathwriting-2024/synthetic/cfd12dae86ccfa0f.inkml
\frac{112222222222}{880000011666}
\frac{112222222222}{880000011666}
synthetic
a2c387a890009e28
mathwriting-2024/synthetic/a2c387a890009e28.inkml
(\Pi_{\text{ACCEPT}},\Pi_{\text{REJECT}})
(\Pi_{ACCEPT},\Pi_{REJECT})
synthetic
bf06ab49e1d4e3b6
mathwriting-2024/synthetic/bf06ab49e1d4e3b6.inkml
\kappa_r^{F_n} = \frac{n \kappa_r}{\sigma^r n^{r/2}} = \frac{\lambda_r}{n^{r/2 - 1}} \quad \mathrm{where} \quad \lambda_r = \frac{\kappa_r}{\sigma^r}.
\kappa_{r}^{F_{n}}=\frac{n\kappa_{r}}{\sigma^{r}n^{r/2}}=\frac{\lambda_{r}}{n^{r/2-1}}where\lambda_{r}=\frac{\kappa_{r}}{\sigma^{r}}.
synthetic
335bdf52bdf62522
mathwriting-2024/synthetic/335bdf52bdf62522.inkml
\left[(x_1-\bar{x})^2 + (x_2-\bar{x})^2\right] /(n-1) = (1+1)/1 = 2
[(x_{1}-\overline{x})^{2}+(x_{2}-\overline{x})^{2}]/(n-1)=(1+1)/1=2
synthetic
29daa0b85a2bb210
mathwriting-2024/synthetic/29daa0b85a2bb210.inkml
\sqrt{1 + 1 + 1 + 1 + 1 +1} = \sqrt{6} = 2.4495,
\sqrt{1+1+1+1+1+1}=\sqrt{6}=2.4495,
synthetic
1a8a19d2f1b6d203
mathwriting-2024/synthetic/1a8a19d2f1b6d203.inkml
\sin(\theta) \approx \pi - \theta
sin(\theta)\approx\pi-\theta
synthetic
a61048dd6e551766
mathwriting-2024/synthetic/a61048dd6e551766.inkml
=(\vec{N} \cdot (\{ 0.133; \; 0.65; \; 0 \}))^3=(0\cdot 0.133 + 1\cdot 0.65 + 0 )^3=0.65^3=0.274625.
=(\vec{N}\cdot(\{0.133;0.65;0\}))^{3}=(0\cdot0.133+1\cdot0.65+0)^{3}=0.65^{3}=0.274625.
synthetic
78995fb29db53e68
mathwriting-2024/synthetic/78995fb29db53e68.inkml
K=\left\{p(\cdot):p(\cdot)=\sum_{i=1}^n a_i f_i(\cdot ; \theta_i), a_i>0, \sum_{i=1}^n a_i=1, f_i(\cdot ; \theta_i)\in J\ \forall i,n\right\}
K=\{p(\cdot):p(\cdot)=\sum_{i=1}^{n}a_{i}f_{i}(\cdot;\theta_{i}),a_{i}>0,\sum_{i=1}^{n}a_{i}=1,f_{i}(\cdot;\theta_{i})\in J\forall i,n\}
synthetic
fd7440602b67038c
mathwriting-2024/synthetic/fd7440602b67038c.inkml
x^{\prime}=\sqrt{s}x,\quad y^{\prime}=y,\quad z^{\prime}=z,\quad t'=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,\quad s=1-\frac{u^{2}}{v^{2}}
x^{\prime}=\sqrt{s}x,y^{\prime}=y,z^{\prime}=z,t^{\prime}=\frac{t}{\sqrt{s}}-\sqrt{s}\frac{u}{v^{2}}x,s=1-\frac{u^{2}}{v^{2}}
synthetic
16bbb3b490bc5723
mathwriting-2024/synthetic/16bbb3b490bc5723.inkml
\frac{daf(x)b}{d x} = a\ \frac{df(x)}{d x}\ b
\frac{daf(x)b}{dx}=a\frac{df(x)}{dx}b
synthetic
8f8d2c3925ab2d59
mathwriting-2024/synthetic/8f8d2c3925ab2d59.inkml
e=\frac{0.34812}{1+e^{(1403.79\text{ K}-T)/540.196\text{ K}}}
e=\frac{0.34812}{1+e^{(1403.79K-T)/540.196K}}
synthetic
046174841447fe27
mathwriting-2024/synthetic/046174841447fe27.inkml
\frac{\frac{\frac{111888833111}{666999991144}}{888885554449*\frac{446666600555}{24.27842}}}{a}
\frac{\frac{\frac{111888833111}{666999991144}}{888885554449*\frac{446666600555}{24.27842}}}{a}
synthetic
24f5e8a7701c996b
mathwriting-2024/synthetic/24f5e8a7701c996b.inkml