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\begin{document}$$\frac{\pi }{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots$$\end{document}π4=1-13+15-17+…
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\begin{document}$$\phi \left( n \right) = n - \mathop \sum \limits_{1\; \le \;l\; \le \;r} \frac{n}{{p_{l} }} + \mathop \sum \limits_{1\; \le \;l\; \le \;j\; \le \;r} \frac{n}{{P_{l} P_{j } }} - \ldots = n \left( {1 - \frac{1}{{P_{1} }}} \right)\left( {1 - \frac{1}{{P_{2} }}} \right) \ldots \left( {1 - \frac{1}{{P_{r} }}} \right)$$\end{document}ϕn=n-∑1≤l≤rnpl+∑1≤l≤j≤rnPlPj-…=n1-1P11-1P2…1-1Pr
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\begin{document}$${\text{e}}^{i\pi } \; = \;\cos \pi + i\sin \pi = \; - 1 + i \cdot 0 = \; - 1$$\end{document}eiπ=cosπ+isinπ=-1+i·0=-1
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\begin{document}$$D = a_{n}^{2n\; - 2} \mathop \prod \limits_{1\; \le \;l\; \le \;j\; \le \;n} \left( {\alpha_{i} - \alpha_{j} } \right)^{2} .$$\end{document}D=an2n-2∏1≤l≤j≤nαi-αj2.
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\begin{document}$$s^{2} = \left( {x^{1} - y^{1} } \right)^{2} + \ldots + \left( {x^{n} - y^{n} } \right)^{2}$$\end{document}s2=x1-y12+…+xn-yn2
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\begin{document}$$\left( {x_{1} \;y_{1} } \right) = \left( {x_{0} + \frac{b}{d}t,\;y_{0} - \frac{a}{d}t} \right),$$\end{document}x1y1=x0+bdt,y0-adt,
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\begin{document}$$\phi = 1 + \frac{1}{{1 \frac{1}{1 + \ldots }}}$$\end{document}ϕ=1+1111+…
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\begin{document}$$\left( {l,\;m,\;n} \right) = \frac{{\left( {\left| {t_{1}^{2} - t_{2}^{2} - t_{3}^{2} } \right|,2\left| {t_{1} \;t_{2} } \right|,2\left| {t_{1} \;t_{3} } \right|} \right)}}{{t_{1}^{2} + t_{2}^{2} + t_{3}^{2} }}$$\end{document}l,m,n=t12-t22-t32,2t1t2,2t1t3t12+t22+t32
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\begin{document}$${}^{N}P_{r} = N \times \left( {N - 1} \right) \times \ldots \left( {N - r + 1} \right)$$\end{document}NPr=N×N-1×…N-r+1
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\begin{document}$$f = a\left( {x + \frac{b}{2a}y} \right)^{2} + \frac{{4ac - b^{2} }}{4a}y^{2} .$$\end{document}f=ax+b2ay2+4ac-b24ay2.
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\begin{document}$$\begin{gathered} \left( {x^{2} y - y^{2} - y^{2} z} \right)dx + \left( {xy^{2} - x^{2} z - z^{2} } \right)dy \\ + \left( {xy^{2} + x^{2} y} \right)dz = 0. \\ \end{gathered}$$\end{document}x2y-y2-y2zdx+xy2-x2z-z2dy+xy2+x2ydz=0.
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\begin{document}$$\int_{{r_{1} }} {} = \;0.$$\end{document}∫r1=0.
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\begin{document}$$^{N} C_{r} = \left( {\begin{array}{*{20}c} N \\ r \\ \end{array} } \right) = \frac{{N \times \left( {N - 1} \right) \ldots \left( {N - r + 1} \right)}}{{r \times \left( {r - 1} \right) \ldots 1}}$$\end{document}NCr=Nr=N×N-1…N-r+1r×r-1…1
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\begin{document}$$\int_{{r_{1} }} {} = p_{1} \int_{{\Delta_{1}^{1} }} {} + p_{2} \int_{{\Delta_{1}^{23} }} {}$$\end{document}∫r1=p1∫Δ11+p2∫Δ123
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\begin{document}$$\left( {\begin{array}{*{20}c} N \\ r \\ \end{array} } \right) = \frac{N!}{{r! \times \left( {N - r} \right)!}}$$\end{document}Nr=N!r!×N-r!
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\begin{document}$$D = \left| {\left( {\gamma_{p}^{i} \cdot \gamma_{n\; - p}^{*j} } \right)} \right| \ne 0.$$\end{document}D=γpi·γn-p∗j≠0.
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\begin{document}$$\left( {\begin{array}{*{20}c} N \\ r \\ \end{array} } \right)\; = \;\left( {\begin{array}{*{20}c} N \\ {N\; - \;r} \\ \end{array} } \right)$$\end{document}Nr=NN-r
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\begin{document}$$\left( {r_{p } \cdot r_{n\; - p}^{*} } \right) = \sum y_{j } \left( {\gamma \gamma_{p} \cdot \gamma_{n\; - p}^{*j} } \right) = 0,$$\end{document}rp·rn-p∗=∑yjγγp·γn-p∗j=0,
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\begin{document}$$P\left( r \right) = \left( {\begin{array}{*{20}c} N \\ r \\ \end{array} } \right)p^{r} \left( {1 - p} \right)^{N - r}$$\end{document}Pr=Nrpr1-pN-r
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\begin{document}$$E_{p} \times E_{q} = - \left( { - E_{p} } \right) \times E_{q} = - E_{p} \times \left( { - E_{q} } \right).$$\end{document}Ep×Eq=--Ep×Eq=-Ep×-Eq.
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\begin{document}$$\mu = 0 \times P\left( 0 \right) + P\left( 1 \right) + 2 \times P\left( 2 \right) \ldots N \times \;P\;(N)$$\end{document}μ=0×P0+P1+2×P2…N×P(N)
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\begin{document}$$R_{k} \left( {K \times K^{\prime};\;\Lambda } \right) = \sum R_{S} \left( {K;\;L} \right)R_{k - s} \left( {K^{\prime};\;L^{\prime}} \right).$$\end{document}RkK×K′;Λ=∑RSK;LRk-sK′;L′.
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\begin{document}$$\sigma^{2} = \left( {0 - \mu } \right)^{2} P\left( 0 \right) + \left( {1 - \mu } \right)^{2} P\left( 1 \right) + \ldots + \left( {N - \mu } \right)^{2} P\left( N \right).$$\end{document}σ2=0-μ2P0+1-μ2P1+…+N-μ2PN.
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\begin{document}$$\frac{du}{{dx}} + bu^{2} = cx^{2}$$\end{document}dudx+bu2=cx2
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\begin{document}$$\sigma^{2} = \mathop \sum \limits_{r = 0}^{N} \left( {r - \mu } \right)^{2} P\left( r \right).$$\end{document}σ2=∑r=0Nr-μ2Pr.
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\begin{document}$$p = p + x\frac{dp}{{dx}} + f\left( p \right)\frac{dp}{{dx}},$$\end{document}p=p+xdpdx+fpdpdx,
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\begin{document}$$\phi \left( n \right) = n\left( {1 - \frac{1}{{p_{1} }}} \right)\left( {1 - \frac{1}{{p_{2} }}} \right) \ldots \left( {1 - \frac{1}{{p_{r} }}} \right).$$\end{document}ϕn=n1-1p11-1p2…1-1pr.
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\begin{document}$$Pdx + Qdy + Rdz = 0$$\end{document}Pdx+Qdy+Rdz=0
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\begin{document}$$\mathop {\lim }\limits_{x \to \infty } \pi \left( x \right)\frac{{\log_{e} x}}{x} = 1.$$\end{document}limx→∞πxlogexx=1.
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\begin{document}$$\frac{dz}{{dx}} = \frac{y}{{\sqrt {\left( {y^{2} - x^{2} } \right)} }}.$$\end{document}dzdx=yy2-x2.
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\begin{document}$$\mathop \sum \limits_{d|n} \phi \left( d \right) = n,$$\end{document}∑d|nϕd=n,
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\begin{document}$$\frac{dz}{{dx}} = \frac{y}{x + z}.$$\end{document}dzdx=yx+z.
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\begin{document}$$\frac{df}{{dx}} + \frac{df}{{da}}\frac{da}{{dx}} + \frac{df}{{db}}\frac{db}{{dx}} = \frac{d\aleph }{{dx}},$$\end{document}dfdx+dfdadadx+dfdbdbdx=dℵdx,
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\begin{document}$$Z = \frac{{\overline{x} - \mu_{n} }}{{\frac{\sigma }{\sqrt n }}}$$\end{document}Z=x¯-μnσn
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\begin{document}$$\frac{d\phi }{{dy_{1} }}dy_{1} + \frac{d\phi }{{dy_{2} }}dy_{s} = 0$$\end{document}dϕdy1dy1+dϕdy2dys=0
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\begin{document}$$v = \left\{ {\frac{{\left\{ {\frac{{s_{1}^{2} }}{{n_{1} }} + \frac{{s_{2}^{2} }}{{n_{2} }}} \right\}^{2} }}{{\frac{{s_{1}^{4} }}{{n_{1}^{2} \left( {n_{1} - 1} \right)}} + \frac{{s_{2}^{4} }}{{n_{2}^{2} \left( {n_{2} - 1} \right)}}}}} \right\}$$\end{document}v=s12n1+s22n22s14n12n1-1+s24n22n2-1
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\begin{document}$$m\left( {\overline{AB} \cup \overline{CD} } \right) = m\left( {\overline{AB} } \right) + m\left( {\overline{CD} } \right) - m\left( {\overline{AB} \cap \overline{CD} } \right).$$\end{document}mAB¯∪CD¯=mAB¯+mCD¯-mAB¯∩CD¯.
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\begin{document}$$b = \frac{{\sum x_{i} y_{i} - \frac{1}{n}x_{i} \sum y_{i} }}{{\sum x_{i}^{2} - \frac{1}{n}\left( {\sum x_{i} } \right)^{2} }}$$\end{document}b=∑xiyi-1nxi∑yi∑xi2-1n∑xi2
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\begin{document}$$V = lwh$$\end{document}V=lwh
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\begin{document}$$r = \frac{{\sum \left( {x_{i} - \overline{x}} \right)\left( {y_{i} - \overline{y}} \right)}}{{\left[ {\sum \left( {x_{i} - \overline{x}} \right)^{2} \sum \left( {y_{i} - \overline{y}} \right)^{2} } \right]^{\frac{1}{2}} }}.$$\end{document}r=∑xi-x¯yi-y¯∑xi-x¯2∑yi-y¯212.
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\begin{document}$$a^{2} + b^{2} = c^{2}$$\end{document}a2+b2=c2
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\begin{document}$$C = 1 + \frac{1}{{3\left( {K + 1} \right)}}\left\{ {\sum \frac{1}{{\left( {n_{j} - 1} \right)}} - \frac{1}{{\sum \left( {n_{j} - 1} \right)}}} \right\}$$\end{document}C=1+13K+1∑1nj-1-1∑nj-1
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\begin{document}$$\frac{AB}{{A^{\prime}B^{\prime}}} = \frac{BC}{{B^{\prime}C^{\prime}}} = \frac{CA}{{C^{\prime}A^{\prime}}}$$\end{document}ABA′B′=BCB′C′=CAC′A′
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\begin{document}$$X^{2} = \mathop \sum \limits_{i = 1}^{K} \frac{{\left( {O_{i} - E_{i} } \right)^{2} }}{{E_{i} }}$$\end{document}X2=∑i=1KOi-Ei2Ei
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\begin{document}$$\Delta \left( \frac{a}{b} \right) \approx \frac{b\Delta a - a\Delta b}{{b^{2} }}.$$\end{document}Δab≈bΔa-aΔbb2.
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\begin{document}$$x^{2} = \mathop \sum \limits_{i = 1}^{p} \frac{{\left( {n_{ij} - n_{i.} n_{.j} /N} \right)^{2} }}{{n_{i.} n_{.j} /N}}$$\end{document}x2=∑i=1pnij-ni.n.j/N2ni.n.j/N
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\begin{document}$$\left\{ {A,B,C,\left. D \right\}} \right. \cap \left\{ {C,D,\left. E \right\}} \right. = \left\{ {C,\left. D \right\}} \right..$$\end{document}A,B,C,D∩C,D,E=C,D.
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\begin{document}$$Z = \frac{{\beta_{2} - \beta_{2} }}{{\sqrt {\sum \left( {\frac{1}{{f_{i} }}} \right)} }}$$\end{document}Z=β2-β2∑1fi
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\begin{document}$$\left( {x + \frac{q}{p}} \right)\left( {x^{\prime} + \frac{q}{p}} \right) = \frac{{q^{2} }}{{p^{2} }} - \frac{s}{p}.$$\end{document}x+qpx′+qp=q2p2-sp.
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\begin{document}$$x \wedge y = y \wedge x$$\end{document}x∧y=y∧x
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\begin{document}$$V\left( {AD,BC} \right) = \left( {z_{1} z_{2} ,\;z_{3} z_{4} } \right).$$\end{document}VAD,BC=z1z2,z3z4.
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\begin{document}$$\left( {x \wedge \left( {y \vee z} \right)} \right) = \left( {x \wedge y} \right) \vee \left( {x \wedge z} \right)$$\end{document}x∧y∨z=x∧y∨x∧z
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\begin{document}$$\frac{AB}{{AC}} \cdot \frac{DC}{{DB}} = \left| k \right|,$$\end{document}ABAC·DCDB=k,
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\begin{document}$$\left( {x \wedge x^{\prime}} \right) = 0$$\end{document}x∧x′=0
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| 44 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {x + \sqrt {x^{2} - bc } } \right)\left( {y + \sqrt {y^{2} - ca} } \right)\left( {z + \sqrt {z^{2} - ab} } \right) = abc$$\end{document}x+x2-bcy+y2-caz+z2-ab=abc
| 60 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {x \vee 0} \right) = x$$\end{document}x∨0=x
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| 45 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2xyz - ax^{2} - by^{2} - cz^{2} + abc = 0.$$\end{document}2xyz-ax2-by2-cz2+abc=0.
| 61 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\left( {\left\{ q \right\}} \right) = h\left( {\bigcap\limits_{q \in I} I \cap \bigcap\limits_{q \notin I} {\left( {I^{\prime}} \right)} } \right)$$\end{document}hq=h⋂q∈II∩⋂q∉II′
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| 46 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{ayz}{{\left( {y - z} \right)^{2} }} = \frac{bzy}{{\left( {z - x} \right)^{2} }} = \frac{cxy}{{\left( {x - y} \right)^{2} }}$$\end{document}ayzy-z2=bzyz-x2=cxyx-y2
| 62 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0 \wedge x = \left( {x \wedge x^{\prime}} \right) \wedge x$$\end{document}0∧x=x∧x′∧x
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| 47 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S = - \;a_{0}^{2} \phi \left( {a_{1} } \right)\phi \left( {a_{2} } \right).$$\end{document}S=-a02ϕa1ϕa2.
| 63 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{array}{*{20}c} {T\frac{e}{N}} \\ {var} \\ \end{array} = I_{P} = \begin{array}{*{20}c} {T\frac{e}{N}} \\ {proper} \\ \end{array}$$\end{document}TeNvar=IP=TeNproper
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| 48 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\log z = 2\left\{ {\frac{z - 1}{{z + 1}} + \frac{1}{3}\left( {\frac{z - 1}{{z + 1}}} \right)^{2} + \ldots } \right\}$$\end{document}logz=2z-1z+1+13z-1z+12+…
| 64 |
\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{array}{*{20}c} {T_{{\overline{N}}} } \\ {\text{var}} \\ \end{array} = T_{{\overline{N}}} .$$\end{document}TN¯var=TN¯.
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Fig. 1