From f4458f5c316ed67a5341acc1cf9da99f7ffd4f3e Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Thu, 26 Oct 2023 17:58:48 +0200 Subject: [PATCH] removed vscode folder --- .vscode/ltex.hiddenFalsePositives.en-US.txt | 68 --------------------- 1 file changed, 68 deletions(-) delete mode 100644 .vscode/ltex.hiddenFalsePositives.en-US.txt diff --git a/.vscode/ltex.hiddenFalsePositives.en-US.txt b/.vscode/ltex.hiddenFalsePositives.en-US.txt deleted file mode 100644 index 3d6847b..0000000 --- a/.vscode/ltex.hiddenFalsePositives.en-US.txt +++ /dev/null @@ -1,68 +0,0 @@ -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is constant (in the sense of distributions).\\E$"} -{"rule":"IF_IS","sentence":"^\\QWe say the floating-point representation by truncation of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qfl \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q We say the floating-point representation by rounding of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is: fl \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"MISSING_GENITIVE","sentence":"^\\QEuler method has order of consistency 1, whereas Heun method has order of consistency 2.\\E$"} -{"rule":"HAVE_PART_AGREEMENT","sentence":"^\\QThen, if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for a fixed \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, the convergence of the method has also order \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QFor a method of order \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (see NC:consistencyRK), just start with \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q values \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q of the form: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, and impose: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q There are tables that determine the smallest \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q necessary for a given order \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (see NC:stages-orderRK).\\E$"} -{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\QFor example the Taylor method of order 2 would be: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Note that the Taylor method of order 1 is precisely the NC:euler.\\E$"} -{"rule":"MORFOLOGIK_RULE_EN_US","sentence":"^\\Q[mode=image|tex, width=0.7]Images/euler Explicit Euler method for approximating the ivp \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with different number of steps.\\E$"} -{"rule":"EN_A_VS_AN","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and consider a SVD \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"EN_A_VS_AN","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a SVD of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be such that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThe second equality, follows from NC:eigenvalues_orto,NC:svd-cor.\\E$"} -{"rule":"EXTREME_ADJECTIVES","sentence":"^\\QOtherwise, we say that the method is conditionally absolutely stable The motivation behind this definition of stability is on the stiff equations, which are differential equations for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken extremely small.\\E$"} -{"rule":"NUMEROUS_DIFFERENT","sentence":"^\\QThe shooting method is the process of solving the initial value problem for many different values of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q until one finds the solution \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q that satisfies the desired boundary conditions.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qorder 1 2 3 4 5 6 7 8 \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q 1 2 3 4 6 7 9 11 Number of stages of an explicit RK method needed for a given order of consistency Step-size control for Runge-Kutta methods.\\E$"} -{"rule":"IF_IS","sentence":"^\\QNote that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and suppose that this expectation is finite.\\E$"} -{"rule":"IF_IS","sentence":"^\\QThen: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q We have that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q solves the difference equation: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, whose solution is straightforward (\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q are particular solutions for the case \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, respectively).\\E$"} -{"rule":"IF_IS","sentence":"^\\QThen: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q We have that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q solves the difference equation \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qruin \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qruin \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, whose solution is straightforward.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QTherefore, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q And so the limit has to be \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q (note that the limit does exist because \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is an increasing bounded sequence).\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QNow using SP:2n-n_convinatoria,SP:stirling_polya1 we have: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q The simple random walk on \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is always transient.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QBy SP:thmRec,SP:period_classes we have that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is recurrent and aperiodic.\\E$"} -{"rule":"EN_UNPAIRED_BRACKETS","sentence":"^\\QWe say that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is a stopping time if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q we have: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Intuitively, this condition means that the “decision\" of whether to stop at time \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q must be based only on the information present at time \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, not on any future information.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QIf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is transient, then \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QIf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, then: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a time-homogeneous Markov chain, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a filtration space defined with \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a stopping time.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThat is, if the random walk is in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q are random variables such that: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with probability \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with probability \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q [mode=image|tex,width=0.75]Images/randomWalk A simple random walk of 10000 steps in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"FILE_EXTENSIONS_CASE","sentence":"^\\QAnd this last expression is the joint pdf of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q variables.\\E$"} -{"rule":"IF_IS","sentence":"^\\QThen, for any \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, the following limits exist: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Note that if the limits are finite we have \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"IF_IS","sentence":"^\\QWe define the infinitesimal transition scheme as: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a CTHMC with infinitesimal generator \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and assume that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is finite.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\Q[mode=image|tex,width=0.75]Images/brownianMotion A Brownian motion simulated with 7500 increments.\\E$"} -{"rule":"IF_IS","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be two stochastic processes.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, the trajectories are almost surely non-decreasing and have jumps of size at most \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"FINAL_ADVERB_COMMA","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q in \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q The Brownian trajectories pass through every point \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q infinitely many times almost surely.\\E$"} -{"rule":"SENTENCE_WHITESPACE","sentence":"^\\QRn.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QIf \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, then the center of the osculating circle at \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q has coordinates \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q given by: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q [mode=image|tex,width=0.8]Images/oscu-circle Osculating circle of a cycloid at a certain point Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open interval, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a curve and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an arc-length parametrization of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q of class \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and suppose that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"IF_IS","sentence":"^\\QThen: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q for some \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\Q[mode=image|tex,width=]Images/involute-evolute Construction of the evolute and involute of a curve Curvature of plane curves.\\E$"} -{"rule":"EN_A_VS_AN","sentence":"^\\QRecall that an Euclidean motion is a function that preserves the distance, that is, if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is an Euclidean motion, then \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"EN_A_VS_AN","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an Euclidean motion.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q we have: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open interval, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a curve, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an arc-length parametrization of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q of class \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, where \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\QSO\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is arc-length parametrized and the TNB frame of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q And the curvature and torsion of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q are: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Fundamental theorem of curves.\\E$"} -{"rule":"EN_A_VS_AN","sentence":"^\\QMoreover, if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is another curve arc-length parametrized by \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q satisfying these restrictions, then there exists an Euclidean motion that carries \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q into \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\Q[mode=image|tex,width=0.45]Images/theorem_immersions Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open set, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a submersion at \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be the projection map into the first coordinate.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\Q[mode=image|tex,width=0.45]Images/theorem_submersions Submanifolds of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qif \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q such that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q then this root is at most double.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open bounded connected set such that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is of class \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QThen: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qg \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a compact oriented surface without boundary.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a compact oriented surface without boundary and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a vector field tangent to \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q with isolated singularities whose indexes are \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QThen: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a compact oriented surface without boundary and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a vector field tangent to \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a compact oriented surface with boundary, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open neighbourhood of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ADVERB_OR_HYPHENATED_ADJECTIVE","sentence":"^\\QLet \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a compact oriented surface, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be an open neighbourhood of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QWe denote by \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the set of all differentiable vector fields on \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q that are tangent to \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QWe say that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is tangent to \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qa hyperbolic point if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qa plane point if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qa parabolic point if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q but \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qwith initial conditions \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"SENTENCE_WHITESPACE","sentence":"^\\QR3.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QWe denote by \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q the set of all differentiable vector fields defined on \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, given \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q we have that: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q where \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q by RFA:measureB,RFA:measureC.\\E$"} -{"rule":"COMMA_COMPOUND_SENTENCE","sentence":"^\\QGiven a function \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, we say that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q exists and it is finite if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is integrable on \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"SENTENCE_WHITESPACE","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"SENTENCE_WHITESPACE","sentence":"^\\Q\\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Qa.e.\\E$"} -{"rule":"TO_TWO","sentence":"^\\QNow extend this to positive simple functions and the to positive measurable functions.\\E$"} -{"rule":"COMMA_PARENTHESIS_WHITESPACE","sentence":"^\\QThe equality is held if: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q [mode=image|tex,width=0.7]Images/young We say that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q are Hölder conjugates if \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"SENTENCE_WHITESPACE","sentence":"^\\QLp. spaces.\\E$"} -{"rule":"COMMA_COMPOUND_SENTENCE","sentence":"^\\QWe will denote \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q and we will say that \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is the Fourier series of \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q is a continuous square-integrable martingale with: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Let \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q be a Brownian motion.\\E$"} -{"rule":"ENGLISH_WORD_REPEAT_BEGINNING_RULE","sentence":"^\\QThen, for \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q: \\E(?:Dummy|Ina|Jimmy-)[0-9]+$"}