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Fix an integer <math>t</math>, we count the number of circuits with <math>t</math> gates. We count the number of different ''sequences'' of directed edges that can be added to an empty graph on <math>n</math> v ...

We count the number of different ''sequences'' of directed edges that can be added to an empty graph on <math>n</math> v Multiplying the numbers of choices in all steps, the number of sequences of adding directed edges to an empty graph to form a rooted tree is given b ...

# <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. # <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. ...

# <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. # <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. ...

# <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. # <math>\Delta(k\cdot\tau_{\mathrm{mix}})\le \mathrm{e}^{-k}</math> for any integer <math>k\ge1</math>. ...

:for some integer <math>n\ge 0</math>. State <math>i</math> '''communicates with''' state <ma ...t random sources <math>U_0',U_1',\ldots</math>. So we have two independent sequences: ...

...idable in many [[computation]]s, especially when dividing two numbers in [[integer]] or fixed-point arithmetic; when computing mathematical functions such as * replacing a decimal [[integer]] by an integer with more trailing zeros, e.g. 23,217 people by 23,200 people; or, in gener ...

:for some integer <math>n\ge 0</math>. State <math>x</math> '''communicates with''' state <ma ...t random sources <math>U_0',U_1',\ldots</math>. So we have two independent sequences: ...

:for some integer <math>n\ge 0</math>. State <math>x</math> '''communicates with''' state <ma ...t random sources <math>U_0',U_1',\ldots</math>. So we have two independent sequences: ...

:for some integer <math>n\ge 0</math>. State <math>x</math> '''communicates with''' state <ma ...t random sources <math>U_0',U_1',\ldots</math>. So we have two independent sequences: ...

:for some integer <math>n\ge 0</math>. State <math>i</math> '''communicates with''' state <ma ...t random sources <math>U_0',U_1',\ldots</math>. So we have two independent sequences: ...

...ron goes from energy state (orbit) '''b''' to energy state '''a''' (Again, sequences look backwards, but that is the way they were originally written.): We can see that <math>n</math> must be an integer. This means that the particle can only have special energy values and cann ...