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P values for classification

http://www.arxiv.org/abs/0801.2934
... of classifying $X$ we propose to construct for each $\theta =1,2,...,L$ a p-value $\pi_{\theta}(X,\mathcal{D})$ for the null hypothesis that $Y=\theta$, treating $Y$ temporarily as a fixed parameter. In other words, the point predictor $\widehat{Y}(X,\mathcal{D})$...
Lutz Duembgen, Bernd-Wolfgang Igl, Axel Munk


Arbitrary p Gradient Values

http://www.arxiv.org/abs/1207.4650
...For any prime number p and any positive real number {\alpha}, we construct a finitely generated group {\Gamma} with p-gradient equal to {\alpha}. This construction is used to show that there exist uncountably many pairwise non-commensurable groups th...
Nathaniel Pappas


p adic boundary values

http://www.arxiv.org/abs/math/9901159
...quotients of the dual of a holomorphic representation coming from a p-adic symmetric space, and principal series representations constructed from locally analytic functions on G. We characterize the image of each of our integral transforms as a space...
Peter Schneider, Jeremy Teitelbaum


P values: misunderstood and misused

http://www.arxiv.org/abs/1601.06805
...a coherent picture of what the main criticisms are, and draw together and disambiguate common themes. In particular, we explain how the False Discovery Rate is calculated, and how this differs from a p-value. We also make explicit...
Bertie Vidgen, Taha Yasseri


Test Martingales, Bayes Factors and $p$ Values

http://www.arxiv.org/abs/0912.4269
...st value attained so far by such a martingale, the exaggeration will be limited, and there are systematic ways to eliminate it. The inverse of the exaggerated value at some stopping time can be interpreted as a $p$-value. We give a simple characteriz...
Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, Vladimir Vovk


Combining p values via averaging

http://www.arxiv.org/abs/1212.4966
...This note discusses the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of p-values without making any assumptions about their dependence structure. An old result by Rueschendorf shows that the p-values ...
Vladimir Vovk


P values for high dimensional regression

http://www.arxiv.org/abs/0811.2177
... to reproduce results. Here, we show that inference across multiple random splits can be aggregated, while keeping asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both...
Nicolai Meinshausen, Lukas Meier, Peter Bühlmann


Invariant $P$ values for model checking

http://www.arxiv.org/abs/1001.1886
... clear that there is a better choice for a measure of surprise. This paper is concerned with the definition of appropriate model-based $P$-values for model checking....
Michael Evans, Gun Ho Jang


Exact P values for Network Interference

http://www.arxiv.org/abs/1506.02084
...ng neighboring units matters). Our general approach is to define an artificial experiment, such that the null hypothesis that was not sharp for the original experiment is sharp for the artificial experiment, and such that the randomization analysis f...
Susan Athey, Dean Eckles, Guido Imbens


Irrationality of some p adic L values

http://www.arxiv.org/abs/math/0603277
...We give a proof of the irrationality of the $p$-adic zeta-values $\zeta_p(k)$ for $p=2,3$ and $k=2,3$. Such results were recently obtained by F.Calegari as an application of overconvergent $p$-adic modular forms. In this paper we present an approach ...
F. Beukers



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Expectation value


Expectation value

$\displaystyle{\left\langle{a}\right\rangle}={\left\langle\hat{{a}}\right\rangle}=\int{P}{s}{i}^{\ast}\hat{{a}}{P}{s}{i}{\left.{d}{x}\right.}$

(a) expectation value of a , a operator for a

Time dependence

$\displaystyle\frac{{{d}}}{{{\left.{d}{t}\right.}}}{\left\langle\hat{{a}}\right\rangle}=\frac{{{i}}}{{{h}}}{\left\langle\matrix{\hat{{H}}&\hat{{a}}}\right\rangle}+{\left\langle\frac{{\partial\hat{{a}}}}{{\partial{t}}}\right\rangle}$

t time , h (Planck constant)/(2)

Relation to eigenfunctions

$\displaystyle{\quad\text{if}\quad}\hat{{a}}\psi_{{n}}={a}_{{n}}\psi{\quad\text{and}\quad}{P}{s}{i}=\sum{c}_{{n}}\psi_{{n}}$

n eigen functions of a ,

then (:a:)=sum|c_n|^2 a_n

$\displaystyle{a}{n}{e}{i}\ge{n}{v}{a}{l}{u}{e}{s},{n}{d}{u}{m}{m}{y}\in{d}{e}{x},{c}{n}{p}{r}{o}{b}{a}{b}{i}{l}{i}{t}{y}{a}{m}{p}{l}{i}{t}{u}{d}{e}{s}$



Ehrenfests theorem

$\displaystyle{m}\frac{{{d}}}{{{\left.{d}{t}\right.}}}{\left\langle{r}\right\rangle}={\left\langle{p}\right\rangle}\ \ \ \ \frac{{{d}}}{{{\left.{d}{t}\right.}}}{\left\langle{p}\right\rangle}=-{\left\langle\nabla{V}\right\rangle}$

m particle mass , r position vector, P momentum , V potential energy

Particle decay


general equation regarding p*p_i

$\displaystyle{p}\cdot{p}_{{i}}={M}{E}_{{i}}\Rightarrow{E}_{{i}}=\frac{{1}}{{M}}{p}\cdot{p}_{{i}}=\frac{{1}}{{M}}{\left({p}_{{1}}\cdot{p}_{{i}}+{p}_{{2}}\cdot{p}_{{i}}\right)}$



the absolute value of the three-momenta

$\displaystyle{\vec{{p}}_{{1}}^{{2}}}={{E}_{{1}}^{{2}}}-{{m}_{{1}}^{{2}}}=\frac{{1}}{{{4}{M}^{{2}}}}{\left({M}^{{4}}-{2}{M}^{{2}}{\left({{m}_{{1}}^{{2}}}+{{m}_{{2}}^{{2}}}\right)}+{\left({{m}_{{1}}^{{2}}}-{{m}_{{2}}^{{2}}}\right)}^{{2}}\right)}={\vec{{p}}_{{2}}^{{2}}}$

the absolute value of the three-momenta,vec p_1^2=E_1^2-m_1^2=1/(4M^2)(M^4-2M^2(m_1^2+m_2^2)+(m_1^2-m_2^2)^2)= vec p_2^2

Tangential Quadrilateral


where p=(L)/(2).

$\displaystyle{L}={a}+{b}+{c}+{d}={2}{\left({a}+{c}\right)}={2}{\left({b}+{d}\right)}$



where p=(L)/(2).

$\displaystyle\alpha+\beta+\gamma+\delta={360}°$



General Quadrilateral


where p=(L)/(2).

$\displaystyle\alpha={120}°$



Special Relativity


Energy Momentum relation, p total momentum in S

$\displaystyle{E}^{{2}}-{p}^{{2}}{c}^{{2}}={E}'^{{2}}-{p}'^{{2}}{c}^{{2}}={{m}_{{0}}^{{2}}}{c}^{{4}}$



Regular Tetrahedron


where p is the perimeter of the cross section.

$\displaystyle{V}=\frac{{{1}}}{{{3}}}{S}_{{B}}{h}=\frac{{{a}^{{3}}}}{{{6}\sqrt{{2}}}}$

Not able to give the formula without the bracket(sin2pi/n)

Bernoulli distribution


the probability distribution of a random variable which takes value 1 with success probability p and value 0 with failure probability

$\displaystyle{f{{\left({k};{p}\right)}}}={\left\lbrace\matrix{{p}&{\quad\text{if}\quad}{k}={1}\\{1}-{p}&{\quad\text{if}\quad}{k}={0}}\right.}$



Generalized extreme value distribution


the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables

$\displaystyle\text{f(x;mu,sigma,xi)=1/sigma t(x)^(xi_1)e^(-t(x))  }\ \text{ for }\ \text{}{x}\in{\left\lbrace{\left({\left[\mu-\frac{\sigma}{\xi},+\infty\right)} {w}{h}{e}{n} \xi\gt{0}\right)},{\left(\mathbb{R}      {w}{h}{e}{n} \xi={0}\right)},{\left({\left(-\infty,\mu-\frac{\sigma}{\xi}\right]} {w}{h}{e}{n} \xi\lt{0}\right)}\right.} $



the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables

$\displaystyle\text{f(x;mu,sigma,xi)=1/sigma t(x)^(xi_1)e^(-t(x))  }\ \text{ for }\ \text{}{x}\in{\left\lbrace{\left({\left[\mu-\frac{\sigma}{\xi},+\infty\right)} {w}{h}{e}{n} \xi\gt{0}\right)},{\left(\mathbb{R}      {w}{h}{e}{n} \xi={0}\right)},{\left({\left(-\infty,\mu-\frac{\sigma}{\xi}\right]} {w}{h}{e}{n} \xi\lt{0}\right)}\right.} $



Kirchhoffs Laws


The sum of the currents values at a node is zero.

$\displaystyle\Sigma{I}_{{i}}={0}$



Von Mises–Fisher distribution


a probability distribution on the (p-1)-dimensional sphere in p-dimensional R-vectorial space, applied, for example, to model the interaction of electric dipoles in an electric field

$\displaystyle{{f}_{{p}}{\left(\vec{{x}};\vec{\mu},\kappa\right)}}={C}_{{p}}{\left(\kappa\right)}{\exp{{\left(\kappa\vec{\mu}^{{T}}\vec{{x}}\right)}}} \ \text{ for the random p-dimensional unit vector }\ \vec{{x}},$



Negative multinomial distribution


a generalization of the negative binomial distribution (NB(r, p)) to more than two outcomes

$\displaystyle{f{{\left({k}_{{1}},\ldots{k}_{{m}};{k}_{{0}},\vec{{p}}\right)}}}=\Gamma{\left({\sum_{{{i}={0}}}^{{m}}}{k}_{{i}}\right)}\frac{{{{p}_{{0}}^{{{k}_{{0}}}}}}}{{\Gamma{\left({k}_{{0}}\right)}}}{\prod_{{{i}={1}}}^{{m}}}{\left(\frac{{{p}_{{i}}^{{{k}_{{i}}}}}}{{{k}_{{i}}!}}\right)}\ \text{ for }\ {k}_{{i}}\in\mathbb{N},{1}\le{i}\le{m},\text{where }\ {k}_{{0}}\in\mathbb{N},\text{the number of failures before the experiment is stopped, }\ {\quad\text{and}\quad}\vec{{p}}\in\mathbb{R}^{{m}}\text{, the m-vector of success probabilities.}$



Fréchet distribution


a special case of the generalized extreme value distribution

$\displaystyle{f{{\left({x};\alpha,{s},{m}\right)}}}=\frac{\alpha}{{s}}{\left(\frac{{{x}-{m}}}{{s}}\right)}^{{-{1}-\alpha}}{e}^{{-{\left(\frac{{{x}-{m}}}{{s}}\right)}^{{-{\alpha}}}}} \ \text{ for }\ {x}\gt{m}$



Binomial distribution


the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p

$\displaystyle{f{{\left({k};{n},{p}\right)}}}={\left(\matrix{{n}\\{k}}\right)}{p}^{{k}}{\left({1}-{p}\right)}^{{{n}-{k}}}$



Degenerate distribution


the probability distribution of a random variable which only takes a single value

$\displaystyle{f{{\left({k}_{{0}}\right)}}}=\delta{\left({x}-{k}_{{0}}\right)}$



Truncated normal distribution


the probability distribution of a normally distributed random variable whose value is either bounded below or above (or both)

$\displaystyle{f{{\left({x};\mu,\sigma,{a},{b}\right)}}}=\frac{{\phi{\left(\frac{{{x}-\mu}}{\sigma}\right)}}}{{\Phi{\left(\frac{{{b}-\mu}}{\sigma}\right)}-\Phi{\left(\frac{{{a}-\mu}}{\sigma}\right)}}}\ \text{ for }\ {a}\le{x}\le{b}\text{, where }\ \phi\ \text{ is the probability density function of the standard normal distribution and }\ \Phi\ \text{ is its cumulative distribution function.}$



Folded normal distribution


a probability distribution related to the normal distribution, of which the probability mass to the left of the x = 0 is "folded" over by taking the absolute value

$\displaystyle\text{the random variable Y = |X| has a folded normal distribution: }\ {{f}_{{Y}}{\left({x};\mu,\sigma\right)}}=\frac{{1}}{{\sigma\sqrt{{{2}\pi}}}}{e}^{{-\frac{{\left({x}-\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}}+ \frac{{1}}{{\sigma\sqrt{{{2}\pi}}}}{e}^{{-\frac{{\left({x}+\mu\right)}^{{2}}}{{{2}\sigma^{{2}}}}}} \ \text{ for }\ {x}\ge{0}$



Matrix normal distribution


a generalization of the multivariate normal distribution to matrix-valued random variables

$\displaystyle{{f}_{{{n}\times{p}}}{\left({X};{M},{U},{V}\right)}}=\frac{{\exp{{\left(-\frac{{1}}{{2}}{t}{r}{\left[{V}^{{-{{1}}}}{\left({X}-{M}\right)}^{{T}}{U}^{{-{{1}}}}{\left({X}-{M}\right)}\right]}\right)}}}}{{{\left({2}\pi\right)}^{{{n}{p}{/}{2}}}{\left|{V}\right|}^{{{n}{/}{2}}}{\left|{U}\right|}^{{{p}{/}{2}}}}} \ \text{ for }\ {X}\in\mathbb{R}^{{{n}\times{p}}}\text{, where }\ {M}\in\mathbb{R}^{{{n}\times{p}}}\text{, }\ {U}\in\mathbb{R}^{{{n}\times{n}}}\text{, positive-definite, and }\ {V} \in\mathbb{R}^{{{p}\times{p}}},\ \text{ positive-definite.}$



Uniform distribution (discrete)


a symmetric probability distribution whereby a finite number of values are equally likely to be observed

$\displaystyle{f{{\left({k};{n}\right)}}}=\frac{{1}}{{n}}$





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