DOKUMEN123.COM

I am a Ph.D. of Mathematics from Israel. See User:MathKnight.

Gallery

Oxalis pes-caprae, Israel
Anemone coronaria, Israel
IDF D9 די-9 דובי
Saltwater Crocodile (Crocodylus prorsus)

Crocodylus niloticus (order Crocodilia)
Lightsaber - The coolest weapon ever
Oxalis pes-caprae
Nile Crocodile (Crocodylus niloticus)

Vibeke Stene - Gothic angel of music
Simone Simons - Gothic beauty
Coccinella septempunctata מושית השבע
Protaetia cuprea ignicollis נחושתית הקוצים
Shofar - Jewish ritual horn

Epitonium scalare (Precious Wentletrap)
Lupinus pilosus
Gothic architecture
Oxalis pes-caprae, חמציץ נטוי, Israel

Cologne cathedral, Gothic style
Triceratops - skulls
Colorful Anemone coronaria in Dalia, Israel
Tower of London

Ulm Minster
Jerusalem, Walls and Tower of David, Israel
מגדל דוד וחומות ירושלים, ירושלים, ישראל
Castel Nuovo, Naples, Italy
Clermont-Ferrand Cathedral, Rayonnant Gothic style

Coracias garrulus - כחל מצוי
Ulm Minster
YAMAM ימ"מ - Israel's Counter-Terrorism Unit
Flags of Israel in Jerusalem

Useful mathematics

Classification Theory
- Binary classifier
- Sensitivity and Specificity
- Receiver operating characteristic (ROC)
Basic Probability and Statistics
- Probability $\left(\Omega ,{\mathcal {A}},\mathbb {P} \right)$
- Statistics
- Expected value $\mathbb {E} (X)=\sum _{i}x_{i}\cdot \mathbb {P} (X=x_{i})=\int x\,\mathrm {d} \mathbb {P} (X=x)=\int xf(x)\mathrm {d} x$
- Variance and Standard deviation $\sigma ^{2}=\operatorname {Var} (X)=\operatorname {cov} (X,X)=\mathbb {E} {\big [}(X-\mathbb {E} [X])^{2}{\big ]}=\mathbb {E} [X^{2}]-(\mathbb {E} [X])^{2}$
- Covariance $\sigma _{X,Y}^{2}=\operatorname {cov} (X,Y)=\mathbb {E} {\big [}(X-\mathbb {E} [X])(Y-\mathbb {E} [Y]){\big ]}=\mathbb {E} [XY]-\mathbb {E} [X]\mathbb {E} [Y]$
- Covariance matrix $\Sigma =\mathbb {E} \left[(\mathbf {X} -\mathbb {E} [\mathbf {X} ])(\mathbf {X} -\mathbb {E} [\mathbf {X} ])^{\rm {T}}\right]$ , if $\mathbf {X} =(X_{i})_{i=1}^{n}$ then $\Sigma _{i,j}=\operatorname {cov} (X_{i},X_{j})$
- Conditional probability $\mathbb {P} (A\mid B)={\frac {\mathbb {P} (A\cap B)}{\mathbb {P} (B)}}$
- Bayes' theorem $\mathbb {P} (A\mid B)={\frac {\mathbb {P} (B\mid A)\,\mathbb {P} (A)}{\mathbb {P} (B)}}$
- Independence (probability theory) $A\perp \!\!\!\perp B\iff \mathbb {P} (A\cap B)=\mathbb {P} (A)\mathbb {P} (B)\iff \mathbb {P} (A)=\mathbb {P} (A\mid B)\iff \mathbb {P} (B)=\mathbb {P} (B\mid A)$
- Conditional expectation
- Normal distribution (Gaussian) $X\sim {\mathcal {N}}(\mu ,\sigma ^{2})$ means $\mathrm {d} \mathbb {P} (x)=f(x\mid \mu ,\sigma ^{2})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left(-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}\right)$
Statistics
- Statistics
- Statistical hypothesis testing
- p-value
- Linear regression
- Spearman's rank correlation coefficient
- Likelihood function ${\mathcal {L}}(\theta \mid x)=p_{\theta }(x)=\mathbb {P} _{\theta }(X=x)$
- Maximum likelihood estimation (MLE) ${\hat {\theta }}=\operatorname {arg\,\max } \{{\mathcal {L}}(\theta \mid x)\ \mid \ \theta \in \Theta \}$
Signal Processing
- Signal processing
- Time series
- Fourier analysis
- Fourier transform and Laplace transform
- Discrete-time Fourier transform (DTFT) $X_{2\pi }(\omega )=\sum _{n=-\infty }^{\infty }x[n]\,e^{-i\omega n}\ \implies \ X_{1/T}(f)=X_{2\pi }(2\pi fT)\ {\stackrel {\mathrm {def} }{=}}\sum _{n=-\infty }^{\infty }\underbrace {T\cdot x(nT)} _{x[n]}\ e^{-i2\pi fTn}\;{\stackrel {\mathrm {Poisson\;f.} }{=}}\;\sum _{k=-\infty }^{\infty }X\left(f-k/T\right)$
- Convolution $(f\ast g)(\tau )=\int _{-\infty }^{\infty }f(t)g(\tau -t)\mathrm {d} t=\int _{-\infty }^{\infty }f(\tau -t)g(t)\mathrm {d} t$
- Cross-correlation $(f\star g)(\tau )=\int _{-\infty }^{\infty }f^{*}(t)g(t+\tau )\mathrm {d} t=\int _{-\infty }^{\infty }f(t-\tau )g(t)\mathrm {d} t$
- Auto-correlation
- Autocovariance: for $(X_{t})_{t\in \mathbb {N} }$ : $C_{XX}(t,s)=\operatorname {cov} (X_{t},X_{s})=\mathbb {E} [X_{t}X_{s}]-\mathbb {E} [X_{t}]\mathbb {E} [X_{s}]$
- Low-pass filter
- Recursive least squares filter (RLS)
- Kalman filter (excellent introduction here)
- Autoregressive–moving-average model (ARMA) = Autoregressive model (AR) and Moving-average model (MA)
- Savitzky-Golay filter
- Window function
Differential equations
- Partial differential equation (PDE)
- Finite element method (FEM)
Mathematical programming and Numerical analysis