Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 13031 by FilupS last updated on 12/May/17

$$\mathrm{MrW}1$$$$$$$$\mathrm{Going}\mathrm{off}\mathrm{of}\mathrm{Q}12883$$$$$$$$\mathrm{How}\mathrm{many}\mathrm{unique}\mathrm{angles}\mathrm{angles}\mathrm{in}{\mathbb{Z}}^3?$$$$\mathrm{What}\mathrm{about}{\mathbb{Z}}^n?$$

Answered by mrW1 last updated on 12/May/17

$$For1⩽x_i⩽p_{i}with1⩽i⩽nand{x}_i\in {\mathbb{P}}^n$$$$thenumberofuniqueanglesforpoints$$$$P(x_{1,}{x}_2,\cdot \cdot \cdot ,x_i,\cdot \cdot \cdot ,x_n)is$$$$\mathrm{\Phi }=\underset{i=1}{\prod ^n}{p}_i-\underset{x_1=1}{\sum ^{p_1}}\underset{x_2=1}{\sum ^{p_2}}\cdot \cdot \cdot \underset{x_i=1}{\sum ^{p_i}}\cdot \cdot \cdot \underset{x_n=1}{\sum ^{p_n}}sign[GCD(x_1,x_2,\cdot \cdot \cdot ,x_i,\cdot \cdot \cdot ,x_n)-1]$$

Commented by mrW1 last updated on 12/May/17

$$For-r_i⩽x_i⩽p_{i}with1⩽i⩽nand{x}_i\in {\mathbb{Z}}^n$$$$thesolutionis...$$$$inthinking...howtowritetheformula$$

Commented by FilupS last updated on 12/May/17

$$\mathrm{why}\mathrm{is}{x}_i\in {\mathbb{P}}^n?$$

Commented by mrW1 last updated on 12/May/17

$$Iwantedtosolvethissimplecaseat$$$$first:1⩽x_i⩽p_i.$$$$Theresultforthecase-r_i⩽x_i⩽-1is$$$$thesameasfor1⩽x_i⩽r_i.$$$$Itmeanstheproblemfor{x}_i\in {\mathbb{Z}}^{n}will$$$$besolvedasacoupleofproblems$$$$for{x}_i\in {\mathbb{P}}^{n.}$$

Commented by mrW1 last updated on 12/May/17

$$Example:thesolutionfor$$$$-127⩽x⩽128$$$$-128⩽y⩽127$$$$is:$$$$\mathrm{\Phi }=128\times 127-\underset{i=1}{\sum ^{128}}\underset{j=1}{\sum ^{127}}sign(GCD(i,j)-1)$$$$+127\times 127-\underset{i=1}{\sum ^{127}}\underset{j=1}{\sum ^{127}}sign(GCD(i,j)-1)$$$$+127\times 128-\underset{i=1}{\sum ^{127}}\underset{j=1}{\sum ^{128}}sign(GCD(i,j)-1)$$$$+128\times 128-\underset{i=1}{\sum ^{128}}\underset{j=1}{\sum ^{128}}sign(GCD(i,j)-1)$$$$+4$$$$$$$$withline1forquadrant1and$$$$line2forquadrant2and$$$$line3forquadrant3and$$$$line4forquadrant4and$$$$line5forpointsonaxes$$

Commented by FilupS last updated on 12/May/17

$$\mathrm{This}\mathrm{is}\mathrm{all}\mathrm{so}\mathrm{fascinating}\mathrm{and}\mathrm{gives}$$$$\mathrm{me}\mathrm{so}\mathrm{many}\mathrm{more}\mathrm{questions}\mathrm{to}\mathrm{explore}.$$$$$$$$\mathrm{For}\mathrm{example}:$$$$\mathrm{for}u⩽x⩽m\mathrm{and}v⩽y⩽n,x,y\in \mathbb{Z}$$$$\mathrm{What}\mathrm{is}\mathrm{the}\mathrm{average}\mathrm{angle}?$$

Commented by mrW1 last updated on 12/May/17

$${Let}^{\prime }ssay-u⩽x⩽mand-v⩽y⩽n.$$$$$$$$Inquadrant1wehave$$$${\theta }_{ij}={\tan }^{-1}\left(\frac{y}{x}\right)={\tan }^{-1}\left(\frac{j}{i}\right)={\tan }^{-1}\left(\frac{j^{\prime }}{i^{\prime }}\right)$$$$S_1=\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^n}{\theta }_{ij}=\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^n}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$$$$$Inquadrant2wehave$$$${\theta }_{ij}={\tan }^{-1}\left(\frac{y}{x}\right)=\pi -{\tan }^{-1}\left(\frac{j}{i}\right)=\pi -{\tan }^{-1}\left(\frac{j^{\prime }}{i^{\prime }}\right)$$$$S_2=\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^n}{\theta }_{ij}=un\pi -\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^n}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$$$$$Inquadrant3wehave$$$${\theta }_{ij}={\tan }^{-1}\left(\frac{y}{x}\right)=\pi +{\tan }^{-1}\left(\frac{j}{i}\right)=\pi +{\tan }^{-1}\left(\frac{j^{\prime }}{i^{\prime }}\right)$$$$S_3=\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^v}{\theta }_{ij}=uv\pi +\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^v}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$$$$$Inquadrant4wehave$$$${\theta }_{ij}={\tan }^{-1}\left(\frac{y}{x}\right)=2\pi -{\tan }^{-1}\left(\frac{j}{i}\right)=2\pi -{\tan }^{-1}\left(\frac{j^{\prime }}{i^{\prime }}\right)$$$$S_4=\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^v}{\theta }_{ij}=2mv\pi -\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^v}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$$$$$On+xaxis:mpointswithangle0,$$$$On+yaxis:npointswithangle\frac{\pi }{2},$$$$On-xaxis:upointswithangle\pi ,$$$$On-yaxis:vpointswithangle\frac{3\pi }{2},$$$$S_5=m\times 0+n\times \frac{\pi }{2}+u\pi +v\times \frac{3\pi }{2}=\frac{(n+2u+3v)\pi }{2}$$$$$$$$Sumofallangles:$$$$S=S_1+S_2+S_3+S_4+S_5$$$$=\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^n}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$+un\pi -\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^n}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$+uv\pi +\underset{i=1}{\sum ^u}\underset{j=1}{\sum ^v}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$+2mv\pi -\underset{i=1}{\sum ^m}\underset{j=1}{\sum ^v}{\tan }^{-1}\left(\frac{j}{i}\right)$$$$+\frac{(n+2u+3v)\pi }{2}$$$$=\frac{(2un+2uv+4mv+n+2u+3v)\pi }{2}$$$$+[\underset{i=1}{\sum ^m}-\underset{i=1}{\sum ^u}][\underset{j=1}{\sum ^n}{\tan }^{-1}\left(\frac{j}{i}\right)-\underset{j=1}{\sum ^v}{\tan }^{-1}\left(\frac{j}{i}\right)]$$$$$$$$Theaverageangleis$$$$\stackrel{-}{\theta }=\frac{S}{(m+u+1)(n+v+1)}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com