8

$\mathcal L(a-1,\{a,a\})$ is the hypersurface of the determinant $0$ matrices in the rank $a \times a$. I'm afraid that this hypersurface is not a topological manifold, and hence also not a smooth one. To see this, it is sufficient to look in a small neighborhood of a rank $a-2$ matrix. Were the hypersurface a manifold, that space would have to be ...


6

Try this reference: O:H-O Bond Anomalous Relaxation Resolving Mpemba Paradox, by Xi Zhang Yongli Huang, Zengsheng Ma and Chang Q Sun http://arxiv.org/abs/1310.6514 P.S. I see you have already found this reference. Some useful information about Mpemba effect can be found here http://math.ucr.edu/home/baez/physics/General/hot_water.html By the way it seems ...


5

Some buzzwords that should lead to some non-textbook examples: In the fields of uncertainty quantification, statistical inverse problems or Bayesian inference one wants, for example, compute conditional expectations for posterior distributions. The domain of integration has as many dimensions as the the quantity of interest has degrees of freedom, and this ...


4

There is a method, provided the times can be scaled to be integers not too large. Consider that the times are $t_1,\ldots,t_{40}$. The problem can be described like this: for some given $N$, how many of the $\binom{40}{20}$ sums of 20 of the times are $\ge N$? We can assume $1\le N\le T = t_1+\cdots+t_{40}$. Now define $$F(x,y)=\prod_{j=1}^{40}\, (1 + xy^{...


4

For small $n$ Monte Carlo integration is not needed. For $n$ up to 100 see Kolmogorov-Smirnov Tests when Parameters are Estimated with Applications to Tests of Exponentiality and Tests on Spacings (table 3). These exact results were used to construct the Graphs for Use with the Lilliefors Test for Normal and Exponential Distributions. There are also ...


4

The convergence of the discretized version $\max_{x \in \{0, \ldots, N\}} |W(x/N)|$ of $M:=\max_{0 \le t \le 1} |W(t)|$ to $M$ will be very slow -- at the rate of $1/\sqrt N$, according to Korolyuk 1961 and Nagaev 1970 (Korolyuk 1961 apparently exists only in Russian, but can be rather easily read using e.g. Google Translate), and then you will have to ...


4

Joris, So we assume we know only the marginals $p(x_i)$ and the probabilities that $p(x_i=x_j)$. In terms of the physics' "spin" notation, $s_i=\pm 1$, this means that we know $\left<s_i \right> \equiv \sum_{s_i} s_i p(s_i)$ and $\left<s_i s_j \right> \equiv \sum_{s_i s_j} s_i s_j p(s_i,s_j)$. There are many different joint probability ...


4

Since this question is still open, I take the liberty of pointing to a recent survey of the status of the Mpemba effect, Pathological Water Science -- Four Examples and What They Have in Common, which draws the following conclusion: If confounding factors (such as evaporation, dissolved gases, mixing by convective currents, inefficient thermal contacts) are ...


3

I tried Brendan's method on some data: $t_1 \ldots t_{40}$ are a sample of size $40$ from the exponential distribution with parameter $1$. I want to bound the probability $p$ that $\sum_{i \in S} t_i \ge 29$, where $S$ is a random subset of size $20$. An upper bound is the probability that $\sum_{i \in S} x_i \ge 29000$ where $x_i = \lceil 1000 t_i \rceil$, ...


3

Efficient Simulation of the Wishart model (2009) shows how to simulate the fractional non-central Wishart distribution for all $\nu>p-1$. See section 6.1.2.b for $\nu\geq p+1$ and page 41 and following for $p-1<\nu<p+1$.


3

Try: X. Zhang, Y. Huang, Z. Ma, Y. Zhou, J. Zhou, W. Zheng, Q. Jiang, and C.Q. Sun, Hydrogen-bond memory and water-skin supersolidity resolving the Mpemba paradox. PCCP, 2014. 16(42): 22995-23002. X. Zhang, Y. Huang, Z. Ma, Y. Zhou, W. Zheng, J. Zhou, and C.Q. Sun, A common supersolid skin covering both water and ice. PCCP, 2014. 16(42): 22987-22994.


3

For instance, if you want to simulate an AR1 process in MATLAB, you could proceed as follows: alpha = 0.8; % Value smaller than 1 for the process to be stationary sigma = 1.3; M = 1e3; X = zeros(M, 1); X(1) = randn; % Initialize for k = 2:M X(k) = alpha*X(k-1) + randn*sigma; end This generates M points in time of the model: $$X_k = \alpha X_{k-1} + \...


2

What you are doing above looks like forward Euler to me, which is very bad for Hamiltonian systems since it tends to continually add energy. (Backward Euler tends to continually remove energy.) The simplest sound method is the Verlet integrator, which is 2nd order and symplectic. You can read about it in http://books.google.ca/books/about/...


2

Here are a few papers that discuss high-dimensional Monte Carlo integrals, together with quotes from Math Reviews. MR2719643 (2011i:65038) Griebel, Michael; Holtz, Markus; Dimension-wise integration of high-dimensional functions with applications to finance, J. Complexity 26 (2010), no. 5, 455ā€“489. "In addition to error bounds, the authors also ...


2

If $B_j$, $j = 1 \ldots \infty$, are independent Bernoulli(1/2) random variables, then $X = 2 \sum_{j=1}^\infty 3^{-j} B_j$ has a Cantor distribution. You could let $U$ be uniform on $[0,1]$ and take $B_j$ to be the $j$'th base-2 digit of $U$ after the "decimal" point.


2

Any real normal matrix $M$ can be written as $M=O\,\mathrm{diag}(B_1,\ldots,B_\ell)\,O^t$ where $O$ is orthogonal and where the blocks $B_j$ are either $1\times 1$ real numbers or $2\times2$ matrices of the form: $$ \left[\begin{matrix} a & b \\- b& a\end{matrix}\right],\qquad a\in\mathbb R,\qquad b>0. $$ This provides a way to sample a real $n\...


1

Your question is a bit similar to a hanging post here How can we simulate from a geometric mixture? To cite @whuber's comment under the SE post, Without additional assumptions, this seems unlikely. ...Suppose that associated with each $f_i$ is an interval $I_i$ on which $f_iā‰¤1$ and $Pr_i(I_i)>1āˆ’\epsilon$, outside of which $0<f_i<\epsilon$, ...


1

Below some references regarding distributional properties of Wiener chaoses The book, Gaussian Hilbert spaces, by S. Janson, is a standard reference to start with. In particular, you might want to read Chapters 2, 5 and 6. Chapter 6 contains general tail bound. For precise moment and tails estimates, you can have a look at this: https://projecteuclid.org/...


1

The method above to converge to the uniform distribution is in the right spirit to generalize the Galton board. A simple regular repeating pattern. The following is much less aesthetic (and my crude illustration makes it worse) but does show how to get an exact uniform distribution, at least with $2^k$ cells.


1

How come CFL condition only depends on the equation , while the Condition number is dependent on the approximation operator as well? The CFL condition is a restriction on the relation between the time and space steps in order for convergence to be possible. It does depend on what you call "the approximation operator", since the numerical domain of ...


1

(1) You might look at simulations of the Game of Life, e.g., at the Wikipedia article: There is also a method, applicable to other cellular automata too, for implementation of the Game of Life using arbitrary asynchronous updates whilst still exactly emulating the behaviour of the synchronous game. (2) Following Qiaochu's suggestion, you might look at ...


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