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Operation ResearchQuestion and Answers: Page 4

Question Number 1448 Answers: 1 Comments: 2

x=(x_1 ,x_2 ),y=(y_1 ,y_2 ) η:[0,1)^4 →[0,1] η(x,y):=med[(((1−x_1 )^y_1 +(1−y_1 )^x_1 )/2),(((1−x_2 )^y_2 +(1−y_2 )^x_2 )/2)] med(x,y):=((min(x,y)+max(x,y))/2) η(x,y)=^? η(y,x) η(x,y)=0⇔^? x=y η(x,z)≤^? η(x,y)+η(y,z)

$$\boldsymbol{{x}}=\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} \right),\boldsymbol{{y}}=\left({y}_{\mathrm{1}} ,{y}_{\mathrm{2}} \right) \\ $$$$\eta:\left[\mathrm{0},\mathrm{1}\right)^{\mathrm{4}} \rightarrow\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\eta\left(\boldsymbol{{x}},\boldsymbol{{y}}\right):=\mathrm{med}\left[\frac{\left(\mathrm{1}−{x}_{\mathrm{1}} \right)^{{y}_{\mathrm{1}} } +\left(\mathrm{1}−{y}_{\mathrm{1}} \right)^{{x}_{\mathrm{1}} } }{\mathrm{2}},\frac{\left(\mathrm{1}−{x}_{\mathrm{2}} \right)^{{y}_{\mathrm{2}} } +\left(\mathrm{1}−{y}_{\mathrm{2}} \right)^{{x}_{\mathrm{2}} } }{\mathrm{2}}\right] \\ $$$$\mathrm{med}\left({x},{y}\right):=\frac{\mathrm{min}\left({x},{y}\right)+\mathrm{max}\left({x},{y}\right)}{\mathrm{2}} \\ $$$$\eta\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)\overset{?} {=}\eta\left(\boldsymbol{{y}},\boldsymbol{{x}}\right) \\ $$$$\eta\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)=\mathrm{0}\overset{?} {\Leftrightarrow}\boldsymbol{{x}}=\boldsymbol{{y}} \\ $$$$\eta\left(\boldsymbol{{x}},\boldsymbol{{z}}\right)\overset{?} {\leqslant}\eta\left(\boldsymbol{{x}},\boldsymbol{{y}}\right)+\eta\left(\boldsymbol{{y}},\boldsymbol{{z}}\right) \\ $$

Question Number 1351 Answers: 0 Comments: 2

W{f(x)}(t)=∫_0 ^(1/t) f(x)ln(xt)dx,t>0 W{f(x)+g(x)}(t)=^? W{f(x)}(t)+W{g(x)}(t) W{cf(x)}(t)=^? cW{f(x)}(t) W{1}(t)=? W{x}(t)=? W{x^n }(t)=?,n∈N W{f′(x)}(t)=?

$$\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}/{t}} {\int}}{f}\left({x}\right)\mathrm{ln}\left({xt}\right){dx},{t}>\mathrm{0} \\ $$$$\mathcal{W}\left\{{f}\left({x}\right)+{g}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right)+\mathcal{W}\left\{{g}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{{cf}\left({x}\right)\right\}\left({t}\right)\overset{?} {=}{c}\mathcal{W}\left\{{f}\left({x}\right)\right\}\left({t}\right) \\ $$$$\mathcal{W}\left\{\mathrm{1}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}\right\}\left({t}\right)=? \\ $$$$\mathcal{W}\left\{{x}^{{n}} \right\}\left({t}\right)=?,{n}\in\mathbb{N} \\ $$$$\mathcal{W}\left\{{f}'\left({x}\right)\right\}\left({t}\right)=? \\ $$

Question Number 1055 Answers: 0 Comments: 0

f:[0,1]→R g:[0,1]×N→R g_n (x)=f[g_(n−1) (x)] g_0 (x)=x A{f}(n)=∫_0 ^1 f(t)g_n (t)dt A{f+g}=^? A{f}+A{g} A{kf}=^? kA{f}

$$\:{f}:\left[\mathrm{0},\mathrm{1}\right]\rightarrow\mathbb{R} \\ $$$${g}:\left[\mathrm{0},\mathrm{1}\right]×\mathbb{N}\rightarrow\mathbb{R} \\ $$$${g}_{{n}} \left({x}\right)={f}\left[{g}_{{n}−\mathrm{1}} \left({x}\right)\right] \\ $$$${g}_{\mathrm{0}} \left({x}\right)={x} \\ $$$$\mathscr{A}\left\{{f}\right\}\left({n}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}{f}\left({t}\right){g}_{{n}} \left({t}\right){dt} \\ $$$$\mathscr{A}\left\{{f}+{g}\right\}\overset{?} {=}\mathscr{A}\left\{{f}\right\}+\mathscr{A}\left\{{g}\right\} \\ $$$$\mathscr{A}\left\{{kf}\right\}\overset{?} {=}{k}\mathscr{A}\left\{{f}\right\} \\ $$

Question Number 1034 Answers: 1 Comments: 0

x=i+f,i∈N,f∈[0,1) x^2 =(i+f)^2 =i^2 +2if+f^2 x∈R_+ i+2f=2⇒x^2 =2i+f^2 2if=0⇒x^2 =i^2 +f^2

$${x}={i}+{f},{i}\in\mathbb{N},{f}\in\left[\mathrm{0},\mathrm{1}\right) \\ $$$${x}^{\mathrm{2}} =\left({i}+{f}\right)^{\mathrm{2}} ={i}^{\mathrm{2}} +\mathrm{2}{if}+{f}^{\mathrm{2}} \\ $$$${x}\in\mathbb{R}_{+} \\ $$$${i}+\mathrm{2}{f}=\mathrm{2}\Rightarrow{x}^{\mathrm{2}} =\mathrm{2}{i}+{f}^{\mathrm{2}} \\ $$$$\mathrm{2}{if}=\mathrm{0}\Rightarrow{x}^{\mathrm{2}} ={i}^{\mathrm{2}} +{f}^{\mathrm{2}} \\ $$

Question Number 746 Answers: 1 Comments: 1

lets ⊞:(R^+ )^2 →R^+ defined by x⊞y=(√(⌊x⌋⌈x⌉))+y 1. x⊞y=^? y⊞x 2.x⊞(y⊞z)=^? (x⊞y)⊞z 3.∃e,∀x∈R^+ ,x⊞e=x ? 4.∃e,∀x∈R^+ ,e⊞x=x ?

$${lets}\:\boxplus:\left(\mathbb{R}^{+} \right)^{\mathrm{2}} \rightarrow\mathbb{R}^{+} \\ $$$${defined}\:{by}\:{x}\boxplus{y}=\sqrt{\lfloor{x}\rfloor\lceil{x}\rceil}+{y} \\ $$$$\mathrm{1}.\:{x}\boxplus{y}\overset{?} {=}{y}\boxplus{x} \\ $$$$\mathrm{2}.{x}\boxplus\left({y}\boxplus{z}\right)\overset{?} {=}\left({x}\boxplus{y}\right)\boxplus{z} \\ $$$$\mathrm{3}.\exists{e},\forall{x}\in\mathbb{R}^{+} ,{x}\boxplus{e}={x}\:? \\ $$$$\mathrm{4}.\exists{e},\forall{x}\in\mathbb{R}^{+} ,{e}\boxplus{x}={x}\:? \\ $$

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