0707.4470: formula evidence

The rendered column is KaTeX, and KaTeX has no preamble. It cannot use \usepackage or this document’s own macros. Corpus-wide, \bm occurs 327 times, \Perp is defined by no package and spans 10 documents, and 11,088 of 11,624 undefined occurrences are the source document’s own macros. A row that looks wrong here may be correct, and one that looks right here may not compile. The LaTeX report renders through the document’s own preamble and is the surface to judge from.

203 rows

1 row shows a REFINEMENT below, not the OCR reading: 1 verified against the source. Each is marked [refined] beside its identifier. The model is unchanged — the original reading is still what `latex` holds; this report chose the refinement for it. Rows: 0707.4470_FO0175

Inline formulas (first occurrence) (203)

The page, the confidence and the picture of an inline formula are its HOST LINE's --- a formula has none of its own. A line's confidence is not a formula's.

Identifier	Page	Conf.	LaTeX source
0707.4470_FO0001	1	1.000	\nabla \cdot \mathbf{D}=\rho
0707.4470_FO0002	2	1.000	\mathbf{E}=-\nabla \phi
0707.4470_FO0003	3	0.677	\mathbf{B}=\nabla \times \mathbf{A}
0707.4470_FO0004	3	0.677	\mathbf{B}
0707.4470_FO0005	3	1.000	\nabla \cdot \nabla \times=0
0707.4470_FO0006	3	1.000	\nabla \times \nabla=0
0707.4470_FO0007	3	0.959	\mathbf{A} \mapsto \mathbf{A}+\nabla f
0707.4470_FO0008	3	1.000	\nabla \cdot \mathbf{D}-\rho
0707.4470_FO0009	3	1.000	\phi
0707.4470_FO0010	3	1.000	\mathrm{d}^{2}=0
0707.4470_FO0011	4	1.000	F
0707.4470_FO0012	4	1.000	G=* F
0707.4470_FO0013	5	0.979	E
0707.4470_FO0014	5	0.995	B
0707.4470_FO0015	6	1.000	\mathbf{E}=\left(E_{x}, E_{y}, E_{z}\right)
0707.4470_FO0016	6	1.000	\mathbf{B}=\left(B_{x}, B_{y}, B_{z}\right)
0707.4470_FO0017	6	0.995	\mathbf{E}
0707.4470_FO0018	6	0.999	H
0707.4470_FO0019	6	0.999	D
0707.4470_FO0020	6	1.000	\epsilon
0707.4470_FO0021	6	1.000	\mu
0707.4470_FO0022	6	0.968	*_{\epsilon}
0707.4470_FO0023	6	0.968	*_{\mu}
0707.4470_FO0024	6	1.000	\epsilon=\epsilon_{0}
0707.4470_FO0025	6	1.000	\mu=\mu_{0}
0707.4470_FO0026	6	0.531	\epsilon_{0}=\mu_{0}=c=1
0707.4470_FO0027	6	0.925	\rho \mathrm{d} x \wedge \mathrm{~d} y \wedge \mathrm{~d} z
0707.4470_FO0028	6	1.000	J=J_{x} \mathrm{~d} y \wedge \mathrm{~d} z+J_{y} \mathrm{~d} z \wedge \mathrm{~d} x+J_{z} \mathrm{~d} x \wedge \mathrm{~d} y
0707.4470_FO0029	6	1.000	\partial_{t} \rho+\mathrm{d} J=0
0707.4470_FO0030	7	0.992	\rho
0707.4470_FO0031	7	0.992	J
0707.4470_FO0032	7	1.000	\mathscr{J}
0707.4470_FO0033	7	0.986	\mathrm{d} \mathscr{J}=0
0707.4470_FO0034	7	1.000	A
0707.4470_FO0035	7	1.000	F=\mathrm{d} A
0707.4470_FO0036	7	1.000	X
0707.4470_FO0037	7	1.000	\alpha
0707.4470_FO0038	7	1.000	\partial X
0707.4470_FO0039	7	1.000	G=* F=* \mathrm{~d} A
0707.4470_FO0040	7	0.999	\mathrm{d} G=\mathscr{J}
0707.4470_FO0041	8	1.000	\mathrm{d} F=\mathrm{d}^{2} A=0
0707.4470_FO0042	8	0.991	(x, y, z, t)
0707.4470_FO0043	8	1.000	X=\mathbb{R}^{3,1}
0707.4470_FO0044	8	1.000	F=E \wedge \mathrm{~d} t+B
0707.4470_FO0045	8	1.000	G=* F=H \wedge \mathrm{~d} t-D
0707.4470_FO0046	8	1.000	G
0707.4470_FO0047	8	1.000	\alpha, \phi, \gamma
0707.4470_FO0048	9	1.000	\nabla \cdot \mathbf{B}
0707.4470_FO0049	9	1.000	\mathrm{d} A
0707.4470_FO0050	9	1.000	A \mapsto A+\mathrm{d} f
0707.4470_FO0051	9	1.000	\phi=A(\partial / \partial t)=0
0707.4470_FO0052	9	1.000	\mathrm{d}^{2} A=0
0707.4470_FO0053	11	1.000	n
0707.4470_FO0054	11	1.000	K
0707.4470_FO0055	11	1.000	\sigma
0707.4470_FO0056	11	0.999	* K
0707.4470_FO0057	11	0.999	k
0707.4470_FO0058	11	0.364	(n-k)
0707.4470_FO0059	11	0.364	* \sigma
0707.4470_FO0060	11	0.364	(* K
0707.4470_FO0061	11	0.972	(n-1)
0707.4470_FO0062	11	1.000	\alpha^{k}
0707.4470_FO0063	11	1.000	\sigma^{k}
0707.4470_FO0064	11	1.000	\left\langle\alpha^{k}, \sigma^{k}\right\rangle
0707.4470_FO0068	12	1.000	\langle\cdot, \cdot\rangle
0707.4470_FO0069	12	0.994	n-k
0707.4470_FO0070	12	0.884	k+1
0707.4470_FO0071	12	0.884	\mathrm{d} \alpha
0707.4470_FO0072	12	0.604	(k+1)
0707.4470_FO0073	12	1.000	\partial \sigma
0707.4470_FO0074	13	1.000	* \alpha
0707.4470_FO0075	13	1.000	\|\sigma\|
0707.4470_FO0076	13	1.000	\|* \sigma\|
0707.4470_FO0077	13	1.000	\kappa
0707.4470_FO0078	13	0.755	\mathbb{L}^{2}
0707.4470_FO0079	13	0.755	(\cdot, \cdot)
0707.4470_FO0080	13	0.978	\mathrm{CH}(\sigma, * \sigma)
0707.4470_FO0081	13	1.000	\sigma \cup * \sigma
0707.4470_FO0082	13	0.991	\|\mathrm{CH}(\sigma, * \sigma)\|=\binom{n}{k}^{-1}\|\sigma\|\|* \sigma\|
0707.4470_FO0083	13	1.000	\alpha \wedge * \beta
0707.4470_FO0084	13	0.865	\delta
0707.4470_FO0085	13	0.995	\mathrm{d}, *, \delta
0707.4470_FO0086	13	1.000	\sigma_{i}^{k}
0707.4470_FO0087	13	0.803	\alpha_{i}=\left\langle\alpha, \sigma_{i}^{k}\right\rangle
0707.4470_FO0091	14	0.510	\kappa\left(\sigma_{i}^{k}\right) \frac{\left\|* \sigma_{i}^{k}\right\|}{\left\|\sigma_{i}^{k}\right\|}
0707.4470_FO0092	14	1.000	\partial K
0707.4470_FO0093	14	1.000	t_{0}
0707.4470_FO0094	14	1.000	t_{0}, E
0707.4470_FO0095	14	1.000	t_{1}
0707.4470_FO0096	14	1.000	t_{1 / 2}
0707.4470_FO0097	15	0.608	n-1
0707.4470_FO0098	15	0.844	(k-1)
0707.4470_FO0099	15	0.997	*(\partial K)
0707.4470_FO0100	15	1.000	\partial(* K)=*(\partial K)
0707.4470_FO0101	15	1.000	t_{-\epsilon}
0707.4470_FO0102	15	1.000	\epsilon \rightarrow 0
0707.4470_FO0103	16	0.862	k-1
0707.4470_FO0104	16	0.862	\beta
0707.4470_FO0105	16	0.894	* \beta
0707.4470_FO0106	16	0.991	F, G
0707.4470_FO0107	16	0.989	\mathbb{R}^{3,1}
0707.4470_FO0108	16	1.000	p
0707.4470_FO0109	16	1.000	\Delta x, \Delta y, \Delta z, \Delta t
0707.4470_FO0110	17	0.925	E_{x} \Delta x \Delta t
0707.4470_FO0111	17	1.000	x t
0707.4470_FO0112	17	1.000	E_{y} \Delta y \Delta t
0707.4470_FO0113	17	1.000	y t
0707.4470_FO0114	17	1.000	E_{z} \Delta z \Delta t
0707.4470_FO0115	17	1.000	z t
0707.4470_FO0116	17	1.000	B_{x} \Delta y \Delta z
0707.4470_FO0117	17	1.000	y z
0707.4470_FO0118	17	1.000	B_{y} \Delta z \Delta x
0707.4470_FO0119	17	1.000	x z
0707.4470_FO0120	17	1.000	B_{z} \Delta x \Delta y
0707.4470_FO0121	17	0.998	x y
0707.4470_FO0122	17	0.930	\left[x_{k}, x_{k+1}\right] \times
0707.4470_FO0123	17	1.000	\left[y_{l}, y_{l+1}\right] \times\left[z_{m}, z_{m+1}\right] \times\left[t_{n}, t_{n+1}\right]
0707.4470_FO0124	17	0.965	\left.F\right\|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}
0707.4470_FO0125	17	1.000	\left[x_{k}, x_{k+1}\right] \times\left\{y_{l}\right\} \times\left\{z_{m}\right\} \times\left[t_{n}, t_{n+1}\right]
0707.4470_FO0126	18	1.000	\mathrm{d} F
0707.4470_FO0127	20	1.000	\mathbf{D}=\epsilon \mathbf{E}, \mathbf{B}=\mu \mathbf{H}
0707.4470_FO0128	21	1.000	\kappa=1
0707.4470_FO0129	21	1.000	\kappa=-1
0707.4470_FO0130	21	1.000	E \Delta t
0707.4470_FO0131	21	1.000	H \Delta t
0707.4470_FO0132	21	1.000	B^{n}
0707.4470_FO0133	21	1.000	H^{n}
0707.4470_FO0134	21	1.000	E^{n+1 / 2}
0707.4470_FO0135	21	1.000	D^{n+1 / 2}
0707.4470_FO0136	21	1.000	\mathrm{d}_{1}
0707.4470_FO0137	21	1.000	\mathrm{d}_{1}^{T}
0707.4470_FO0138	21	1.000	\mathrm{d} F=0
0707.4470_FO0139	23	1.000	\Delta t_{\sigma}=t_{\sigma}^{n+1}-t_{\sigma}^{n}
0707.4470_FO0140	23	1.000	\Delta t
0707.4470_FO0141	23	1.000	\Theta_{\sigma} \cap \Theta_{\sigma^{\prime}}=\left\{t_{0}\right\}
0707.4470_FO0142	23	1.000	\sigma \neq \sigma^{\prime}
0707.4470_FO0143	23	1.000	e
0707.4470_FO0144	23	1.000	t_{e}^{k+1 / 2}=\left(t_{e}^{k+1}+t_{e}^{k}\right) / 2
0707.4470_FO0145	23	1.000	\Theta_{e} \supset \Theta_{\sigma}
0707.4470_FO0146	23	1.000	e \subset \sigma
0707.4470_FO0147	24	1.000	\nVdash_{\left\{t_{\sigma}^{n}=t_{e}^{m}\right\}}
0707.4470_FO0148	24	1.000	t_{\sigma}^{n}=t_{e}^{m}
0707.4470_FO0149	24	1.000	t_{\sigma}^{n+1}
0707.4470_FO0150	24	1.000	B_{\sigma}^{n+1}
0707.4470_FO0151	24	1.000	H_{\sigma}^{n+1}=*_{\mu}^{-1} B_{\sigma}^{n+1}
0707.4470_FO0152	24	1.000	D_{e}^{m+3 / 2}
0707.4470_FO0153	24	1.000	E_{e}^{m+3 / 2}=*_{\epsilon}^{-1} D_{e}^{m+3 / 2}
0707.4470_FO0154	24	0.902	\mathbf{x}
0707.4470_FO0155	24	1.000	\dot{\mathbf{x}}
0707.4470_FO0156	25	0.748	A_{e} \leftarrow A_{e}^{0}, E_{e} \leftarrow E_{e}^{1 / 2}, \tau_{e} \leftarrow t_{0} / /
0707.4470_FO0157	25	1.000	\tau_{\sigma} \leftarrow t_{0}
0707.4470_FO0158	25	0.999	t_{\sigma}^{1}
0707.4470_FO0159	25	0.851	Q . \mathrm{push}\left(t_{\sigma}^{1}, \sigma\right) / /
0707.4470_FO0160	25	0.873	(t, \sigma) \leftarrow Q . \operatorname{pop}() / /
0707.4470_FO0161	25	0.873	t
0707.4470_FO0162	25	1.000	A_{e} \leftarrow A_{e}-E_{e}\left(t-\tau_{e}\right) / /
0707.4470_FO0163	25	0.570	t<
0707.4470_FO0164	25	1.000	B_{\sigma} \leftarrow \mathrm{d}_{1} A_{e}
0707.4470_FO0165	25	1.000	H_{\sigma} \leftarrow *_{\mu} B_{\sigma}
0707.4470_FO0166	25	0.998	D_{e} \leftarrow *_{\epsilon} E_{e}
0707.4470_FO0167	25	1.000	D_{e} \leftarrow D_{e}+\mathrm{d}_{1}(e, \sigma) H_{\sigma}\left(t-\tau_{\sigma}\right)
0707.4470_FO0168	25	1.000	E_{e} \leftarrow *_{\epsilon} D_{e}
0707.4470_FO0169	25	0.999	\tau_{\sigma} \leftarrow t / /
0707.4470_FO0170	25	0.997	t_{\sigma}^{\text {next }}
0707.4470_FO0171	25	0.438	Q . \mathrm{push}\left(t_{\sigma}^{\text {next }}, \sigma\right) / /
0707.4470_FO0172	25	1.000	x
0707.4470_FO0173	25	1.000	y
0707.4470_FO0174	27	0.998	3+1
0707.4470_FO0175 [refined: source]	29	0.919	\mathscr{J}
0707.4470_FO0176	29	1.000	\mathrm{d} * \mathrm{~d} A=\mathscr{J}
0707.4470_FO0178	29	1.000	\theta_{\mathscr{L}}
0707.4470_FO0179	29	1.000	(n+1)
0707.4470_FO0180	29	0.976	(n+2)
0707.4470_FO0181	29	0.976	\omega_{\mathscr{L}}
0707.4470_FO0182	29	1.000	\omega_{\mathscr{L}}=-\mathbf{d} \theta_{\mathscr{L}}
0707.4470_FO0183	29	1.000	\mathbf{d}^{2}=0
0707.4470_FO0184	29	1.000	n=0
0707.4470_FO0185	29	1.000	\omega_{L}
0707.4470_FO0186	30	1.000	K \subset \mathscr{K}
0707.4470_FO0187	30	1.000	S_{d}
0707.4470_FO0188	30	1.000	\theta_{\mathscr{L}_{d}}
0707.4470_FO0189	30	1.000	\omega_{\mathscr{L}_{d}}=-\mathbf{d} \theta_{\mathscr{L}_{d}}
0707.4470_FO0190	30	1.000	\mathbf{d}^{2} S_{d}=0
0707.4470_FO0191	30	1.000	\alpha, \beta
0707.4470_FO0195	31	1.000	\rho=\nabla \cdot \mathbf{D}
0707.4470_FO0196	31	1.000	f
0707.4470_FO0197	31	1.000	A_{t}=0
0707.4470_FO0198	31	0.749	A_{x}, A_{y}, A_{z}
0707.4470_FO0199	31	0.749	\mathbf{A}
0707.4470_FO0200	31	0.880	A=\mathbf{A}^{b}
0707.4470_FO0201	31	1.000	\mathrm{d}_{t}
0707.4470_FO0202	32	0.934	\mathrm{d}_{s}
0707.4470_FO0203	32	0.934	\mathrm{d}=\mathrm{d}_{t}+\mathrm{d}_{s}
0707.4470_FO0204	32	0.934	\mathrm{d} t
0707.4470_FO0205	32	0.934	\mathrm{d}_{s} A
0707.4470_FO0206	32	1.000	\mathrm{d}_{t} A
0707.4470_FO0207	32	0.999	\mathrm{d}_{s} D=\rho
0707.4470_FO0208	32	1.000	t_{f}
0707.4470_FO0209	32	1.000	\Sigma
0707.4470_FO0210	32	1.000	\alpha=\mathrm{d}_{s} f
0707.4470_FO0211	33	1.000	\mathrm{d}_{s} D-\rho
0707.4470_FO0212	33	0.986	\nabla \cdot \mathbf{D}
0707.4470_FO0213	36	1.000	\mathbb{R}^{3}

Identifier

Page

Conf.

LaTeX source

Rendered

Image

0707.4470_FO0001

1.000

\nabla \cdot \mathbf{D}=\rho

0707.4470_FO0002

1.000

\mathbf{E}=-\nabla \phi

0707.4470_FO0003

0.677

\mathbf{B}=\nabla \times \mathbf{A}

0707.4470_FO0004

0.677

\mathbf{B}

0707.4470_FO0005

1.000

\nabla \cdot \nabla \times=0

0707.4470_FO0006

1.000

\nabla \times \nabla=0

0707.4470_FO0007

0.959

\mathbf{A} \mapsto \mathbf{A}+\nabla f

0707.4470_FO0008

1.000

\nabla \cdot \mathbf{D}-\rho

0707.4470_FO0009

1.000

\phi

0707.4470_FO0010

1.000

\mathrm{d}^{2}=0

0707.4470_FO0011

1.000

0707.4470_FO0012

1.000

G=* F

0707.4470_FO0013

0.979

0707.4470_FO0014

0.995

0707.4470_FO0015

1.000

\mathbf{E}=\left(E_{x}, E_{y}, E_{z}\right)

0707.4470_FO0016

1.000

\mathbf{B}=\left(B_{x}, B_{y}, B_{z}\right)

0707.4470_FO0017

0.995

\mathbf{E}

0707.4470_FO0018

0.999

0707.4470_FO0019

0.999

0707.4470_FO0020

1.000

\epsilon

0707.4470_FO0021

1.000

\mu

0707.4470_FO0022

0.968

*_{\epsilon}

0707.4470_FO0023

0.968

*_{\mu}

0707.4470_FO0024

1.000

\epsilon=\epsilon_{0}

0707.4470_FO0025

1.000

\mu=\mu_{0}

0707.4470_FO0026

0.531

\epsilon_{0}=\mu_{0}=c=1

0707.4470_FO0027

0.925

\rho \mathrm{d} x \wedge \mathrm{~d} y \wedge \mathrm{~d} z

0707.4470_FO0028

1.000

J=J_{x} \mathrm{~d} y \wedge \mathrm{~d} z+J_{y} \mathrm{~d} z \wedge \mathrm{~d} x+J_{z} \mathrm{~d} x \wedge \mathrm{~d} y

0707.4470_FO0029

1.000

\partial_{t} \rho+\mathrm{d} J=0

0707.4470_FO0030

0.992

\rho

0707.4470_FO0031

0.992

0707.4470_FO0032

1.000

\mathscr{J}

0707.4470_FO0033

0.986

\mathrm{d} \mathscr{J}=0

0707.4470_FO0034

1.000

0707.4470_FO0035

1.000

F=\mathrm{d} A

0707.4470_FO0036

1.000

0707.4470_FO0037

1.000

\alpha

0707.4470_FO0038

1.000

\partial X

0707.4470_FO0039

1.000

G=* F=* \mathrm{~d} A

0707.4470_FO0040

0.999

\mathrm{d} G=\mathscr{J}

0707.4470_FO0041

1.000

\mathrm{d} F=\mathrm{d}^{2} A=0

0707.4470_FO0042

0.991

(x, y, z, t)

0707.4470_FO0043

1.000

X=\mathbb{R}^{3,1}

0707.4470_FO0044

1.000

F=E \wedge \mathrm{~d} t+B

0707.4470_FO0045

1.000

G=* F=H \wedge \mathrm{~d} t-D

0707.4470_FO0046

1.000

0707.4470_FO0047

1.000

\alpha, \phi, \gamma

0707.4470_FO0048

1.000

\nabla \cdot \mathbf{B}

0707.4470_FO0049

1.000

\mathrm{d} A

0707.4470_FO0050

1.000

A \mapsto A+\mathrm{d} f

0707.4470_FO0051

1.000

\phi=A(\partial / \partial t)=0

0707.4470_FO0052

1.000

\mathrm{d}^{2} A=0

0707.4470_FO0053

1.000

0707.4470_FO0054

1.000

0707.4470_FO0055

1.000

\sigma

0707.4470_FO0056

0.999

* K

0707.4470_FO0057

0.999

0707.4470_FO0058

0.364

(n-k)

0707.4470_FO0059

0.364

* \sigma

0707.4470_FO0060

0.364

(* K

0707.4470_FO0061

0.972

(n-1)

0707.4470_FO0062

1.000

\alpha^{k}

0707.4470_FO0063

1.000

\sigma^{k}

0707.4470_FO0064

1.000

\left\langle\alpha^{k}, \sigma^{k}\right\rangle

0707.4470_FO0068

1.000

\langle\cdot, \cdot\rangle

0707.4470_FO0069

0.994

n-k

0707.4470_FO0070

0.884

k+1

0707.4470_FO0071

0.884

\mathrm{d} \alpha

0707.4470_FO0072

0.604

(k+1)

0707.4470_FO0073

1.000

\partial \sigma

0707.4470_FO0074

1.000

* \alpha

0707.4470_FO0075

1.000

|\sigma|

0707.4470_FO0076

1.000

|* \sigma|

0707.4470_FO0077

1.000

\kappa

0707.4470_FO0078

0.755

\mathbb{L}^{2}

0707.4470_FO0079

0.755

(\cdot, \cdot)

0707.4470_FO0080

0.978

\mathrm{CH}(\sigma, * \sigma)

0707.4470_FO0081

1.000

\sigma \cup * \sigma

0707.4470_FO0082

0.991

|\mathrm{CH}(\sigma, * \sigma)|=\binom{n}{k}^{-1}|\sigma||* \sigma|

0707.4470_FO0083

1.000

\alpha \wedge * \beta

0707.4470_FO0084

0.865

\delta

0707.4470_FO0085

0.995

\mathrm{d}, *, \delta

0707.4470_FO0086

1.000

\sigma_{i}^{k}

0707.4470_FO0087

0.803

\alpha_{i}=\left\langle\alpha, \sigma_{i}^{k}\right\rangle

0707.4470_FO0091

0.510

\kappa\left(\sigma_{i}^{k}\right) \frac{\left|* \sigma_{i}^{k}\right|}{\left|\sigma_{i}^{k}\right|}

0707.4470_FO0092

1.000

\partial K

0707.4470_FO0093

1.000

t_{0}

0707.4470_FO0094

1.000

t_{0}, E

0707.4470_FO0095

1.000

t_{1}

0707.4470_FO0096

1.000

t_{1 / 2}

0707.4470_FO0097

0.608

n-1

0707.4470_FO0098

0.844

(k-1)

0707.4470_FO0099

0.997

*(\partial K)

0707.4470_FO0100

1.000

\partial(* K)=*(\partial K)

0707.4470_FO0101

1.000

t_{-\epsilon}

0707.4470_FO0102

1.000

\epsilon \rightarrow 0

0707.4470_FO0103

0.862

k-1

0707.4470_FO0104

0.862

\beta

0707.4470_FO0105

0.894

* \beta

0707.4470_FO0106

0.991

F, G

0707.4470_FO0107

0.989

\mathbb{R}^{3,1}

0707.4470_FO0108

1.000

0707.4470_FO0109

1.000

\Delta x, \Delta y, \Delta z, \Delta t

0707.4470_FO0110

0.925

E_{x} \Delta x \Delta t

0707.4470_FO0111

1.000

x t

0707.4470_FO0112

1.000

E_{y} \Delta y \Delta t

0707.4470_FO0113

1.000

y t

0707.4470_FO0114

1.000

E_{z} \Delta z \Delta t

0707.4470_FO0115

1.000

z t

0707.4470_FO0116

1.000

B_{x} \Delta y \Delta z

0707.4470_FO0117

1.000

y z

0707.4470_FO0118

1.000

B_{y} \Delta z \Delta x

0707.4470_FO0119

1.000

x z

0707.4470_FO0120

1.000

B_{z} \Delta x \Delta y

0707.4470_FO0121

0.998

x y

0707.4470_FO0122

0.930

\left[x_{k}, x_{k+1}\right] \times

0707.4470_FO0123

1.000

\left[y_{l}, y_{l+1}\right] \times\left[z_{m}, z_{m+1}\right] \times\left[t_{n}, t_{n+1}\right]

0707.4470_FO0124

0.965

\left.F\right|_{k+\frac{1}{2}, l, m} ^{n+\frac{1}{2}}

0707.4470_FO0125

1.000

\left[x_{k}, x_{k+1}\right] \times\left\{y_{l}\right\} \times\left\{z_{m}\right\} \times\left[t_{n}, t_{n+1}\right]

0707.4470_FO0126

1.000

\mathrm{d} F

0707.4470_FO0127

1.000

\mathbf{D}=\epsilon \mathbf{E}, \mathbf{B}=\mu \mathbf{H}

0707.4470_FO0128

1.000

\kappa=1

0707.4470_FO0129

1.000

\kappa=-1

0707.4470_FO0130

1.000

E \Delta t

0707.4470_FO0131

1.000

H \Delta t

0707.4470_FO0132

1.000

B^{n}

0707.4470_FO0133

1.000

H^{n}

0707.4470_FO0134

1.000

E^{n+1 / 2}

0707.4470_FO0135

1.000

D^{n+1 / 2}

0707.4470_FO0136

1.000

\mathrm{d}_{1}

0707.4470_FO0137

1.000

\mathrm{d}_{1}^{T}

0707.4470_FO0138

1.000

\mathrm{d} F=0

0707.4470_FO0139

1.000

\Delta t_{\sigma}=t_{\sigma}^{n+1}-t_{\sigma}^{n}

0707.4470_FO0140

1.000

\Delta t

0707.4470_FO0141

1.000

\Theta_{\sigma} \cap \Theta_{\sigma^{\prime}}=\left\{t_{0}\right\}

0707.4470_FO0142

1.000

\sigma \neq \sigma^{\prime}

0707.4470_FO0143

1.000

0707.4470_FO0144

1.000

t_{e}^{k+1 / 2}=\left(t_{e}^{k+1}+t_{e}^{k}\right) / 2

0707.4470_FO0145

1.000

\Theta_{e} \supset \Theta_{\sigma}

0707.4470_FO0146

1.000

e \subset \sigma

0707.4470_FO0147

1.000

\nVdash_{\left\{t_{\sigma}^{n}=t_{e}^{m}\right\}}

0707.4470_FO0148

1.000

t_{\sigma}^{n}=t_{e}^{m}

0707.4470_FO0149

1.000

t_{\sigma}^{n+1}

0707.4470_FO0150

1.000

B_{\sigma}^{n+1}

0707.4470_FO0151

1.000

H_{\sigma}^{n+1}=*_{\mu}^{-1} B_{\sigma}^{n+1}

0707.4470_FO0152

1.000

D_{e}^{m+3 / 2}

0707.4470_FO0153

1.000

E_{e}^{m+3 / 2}=*_{\epsilon}^{-1} D_{e}^{m+3 / 2}

0707.4470_FO0154

0.902

\mathbf{x}

0707.4470_FO0155

1.000

\dot{\mathbf{x}}

0707.4470_FO0156

0.748

A_{e} \leftarrow A_{e}^{0}, E_{e} \leftarrow E_{e}^{1 / 2}, \tau_{e} \leftarrow t_{0} / /

0707.4470_FO0157

1.000

\tau_{\sigma} \leftarrow t_{0}

0707.4470_FO0158

0.999

t_{\sigma}^{1}

0707.4470_FO0159

0.851

Q . \mathrm{push}\left(t_{\sigma}^{1}, \sigma\right) / /

0707.4470_FO0160

0.873

(t, \sigma) \leftarrow Q . \operatorname{pop}() / /

0707.4470_FO0161

0.873

0707.4470_FO0162

1.000

A_{e} \leftarrow A_{e}-E_{e}\left(t-\tau_{e}\right) / /

0707.4470_FO0163

0.570

0707.4470_FO0164

1.000

B_{\sigma} \leftarrow \mathrm{d}_{1} A_{e}

0707.4470_FO0165

1.000

H_{\sigma} \leftarrow *_{\mu} B_{\sigma}

0707.4470_FO0166

0.998

D_{e} \leftarrow *_{\epsilon} E_{e}

0707.4470_FO0167

1.000

D_{e} \leftarrow D_{e}+\mathrm{d}_{1}(e, \sigma) H_{\sigma}\left(t-\tau_{\sigma}\right)

0707.4470_FO0168

1.000

E_{e} \leftarrow *_{\epsilon} D_{e}

0707.4470_FO0169

0.999

\tau_{\sigma} \leftarrow t / /

0707.4470_FO0170

0.997

t_{\sigma}^{\text {next }}

0707.4470_FO0171

0.438

Q . \mathrm{push}\left(t_{\sigma}^{\text {next }}, \sigma\right) / /

0707.4470_FO0172

1.000

0707.4470_FO0173

1.000

0707.4470_FO0174

0.998

3+1

0707.4470_FO0175 [refined: source]

0.919

\mathscr{J}

0707.4470_FO0176

1.000

\mathrm{d} * \mathrm{~d} A=\mathscr{J}

0707.4470_FO0178

1.000

\theta_{\mathscr{L}}

0707.4470_FO0179

1.000

(n+1)

0707.4470_FO0180

0.976

(n+2)

0707.4470_FO0181

0.976

\omega_{\mathscr{L}}

0707.4470_FO0182

1.000

\omega_{\mathscr{L}}=-\mathbf{d} \theta_{\mathscr{L}}

0707.4470_FO0183

1.000

\mathbf{d}^{2}=0

0707.4470_FO0184

1.000

n=0

0707.4470_FO0185

1.000

\omega_{L}

0707.4470_FO0186

1.000

K \subset \mathscr{K}

0707.4470_FO0187

1.000

S_{d}

0707.4470_FO0188

1.000

\theta_{\mathscr{L}_{d}}

0707.4470_FO0189

1.000

\omega_{\mathscr{L}_{d}}=-\mathbf{d} \theta_{\mathscr{L}_{d}}

0707.4470_FO0190

1.000

\mathbf{d}^{2} S_{d}=0

0707.4470_FO0191

1.000

\alpha, \beta

0707.4470_FO0195

1.000

\rho=\nabla \cdot \mathbf{D}

0707.4470_FO0196

1.000

0707.4470_FO0197

1.000

A_{t}=0

0707.4470_FO0198

0.749

A_{x}, A_{y}, A_{z}

0707.4470_FO0199

0.749

\mathbf{A}

0707.4470_FO0200

0.880

A=\mathbf{A}^{b}

0707.4470_FO0201

1.000

\mathrm{d}_{t}

0707.4470_FO0202

0.934

\mathrm{d}_{s}

0707.4470_FO0203

0.934

\mathrm{d}=\mathrm{d}_{t}+\mathrm{d}_{s}

0707.4470_FO0204

0.934

\mathrm{d} t

0707.4470_FO0205

0.934

\mathrm{d}_{s} A

0707.4470_FO0206

1.000

\mathrm{d}_{t} A

0707.4470_FO0207

0.999

\mathrm{d}_{s} D=\rho

0707.4470_FO0208

1.000

t_{f}

0707.4470_FO0209

1.000

\Sigma

0707.4470_FO0210

1.000

\alpha=\mathrm{d}_{s} f

0707.4470_FO0211

1.000

\mathrm{d}_{s} D-\rho

0707.4470_FO0212

0.986

\nabla \cdot \mathbf{D}

0707.4470_FO0213

1.000

\mathbb{R}^{3}