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Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now. To shift it into the third dimension I need to somehow combine the Bézier curves. Given two curves in each two dimensions, how can I combine them to build a surface? I thought about something like a curve of a curve. Or maybe I could multiply them somehow. How can I combine them to cause the wanted behavior? Is there a common approach? In the image you can see the input, on the left, and the output, on the right, of the algorithm that works in a two dimensional space. For the real application there is a three dimensional input. The algorithm relays only on the adjacent tiles. Given these it will define the control points of the mentioned Bézier cur...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Taken from P. Suppes "Introduction to logic" pp16 Starting from the sentence "If $P$ tautologically implies $Q$, then.." Question: Is it true that, at least within propositional logic, we are able to prove only tautologies or do I mis-understand something? If yes, is propositional logic special in some way? Is the notion of tautology meaningful only in propositional logic? logic propositional-calculus share | cite | improve this question edited Nov 29 '17 at 18:53 Mauro ALLEGRANZA 60.8k 4 46 105 asked Nov 29 '17 at 16:32 Alvin Lepik 2,448 9 20 add a comment  |  up vote 0 down vote favorite Taken from P. Suppes "Introduction to logic" pp16 Starting from the sentence "If $P$ tautologically implies $Q$, then.." Question: Is it true that, at least within propositional logic, we are able to prove only tautologi...

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $(W_s)_s geq 0$ be a Wiener process and $tau$ be a random variable with an exponential distribution with parameter $lambda$. Suppose that $W$ and $tau$ are independent. In this question, we see that the distribution of the stopped Wiener process $W_tau$ corresponds to a Laplace distribution with scale parameter $fracsigmasqrt2lambda$ where $sigma$ is the instantaneous variance of the Wiener process. Suppose now, that the variance $sigma^2$ also follows a stochastic process such that we have: $$dS = sigma SdW_s \ dsigma^2 = alphasigma^2dt + xisigma^2dW_sigma$$ where $alpha$ and $xi$ are independent of $S$ and $dW_s$ and $dW_sigma$ are independent Wiener processes. My aim is to derive the distribution of $log S_tau$. First, if $barV_T$ denotes the mean variance over some time interval $[0,T]$ defined by $$barV_T = frac1T intlimits_0^Tsigma^2(t)dt,$$ it is easy to show (Lemma of Îto) that $$...