## 统计代写|随机过程代写stochastic process代考|STATS217

2022年12月29日

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## 统计代写|随机过程代写stochastic process代考|Connection to Poisson-binomial Processes

Rather than directly studying $\eta(z)$, I am interested in applying small perturbations to the summation index $k$ (playing the role of a one-dimensional lattice) in Formula (23). In short, replacing $k$ by $X_k, k=1,2$ and so on, where the $\left(X_k\right)$ ‘s constitute a Poisson-binomial process. This turns $\eta(z)$ into a random function $\eta^{\prime}(z)$ with real and imaginary parts defined respectively by Formulas (20) and (21). The question is this: does the Riemann Hypothesis also apply to the randomized version?

Unfortunately, the answer is negative, unless the scaling factor $s$ in the underlying Poisson-binomial process is very close to zero: see Section 2.3.2. In short, if $s>0, \eta^{\prime}(z)$ – unlike $\eta(z)$ – may have zeroes in the critical strip $0<\sigma<1$, with $\sigma \neq \frac{1}{2}$. A positive answer would have provided some hope and a new path of attack. Another similar attempt, somewhat more promising, is discussed in my article “Deep visualizations to Help Solve Riemann’s Conjecture”, here. Again, $\sigma$ is the real part of $z$. Note that if $s=0$, then $\eta(z)=\eta^{\prime}(z)$.

The videos in Figure 9 (on the left) show the successive partial sums of $\eta^{\prime}(z)$ in the complex plane. The orbits in the video, depending on $s$ and $z=\sigma+i t$, show the chaotic convergence using 10,000 terms in the summation Formulas (20) and (21). If $t$ is large (say $t=10^5$ ), you usually need much more than 10,000 terms to reach the convergence zone. Also, I use a Weibull or Fréchet distribution of parameter $\gamma$, for the underlying Poisson-binomial process: see Formula (37). For standardization purposes discussed in the same section, the intensity is set to $\lambda=\Gamma(1+\gamma)$.

The middle video in Figure 9 shows the convergence path of two orbits (corresponding to two different parameter sets) at the same time, to make comparisons easier. It would be interesting to use a zero of the Riemann zeta function for $z=\sigma+i t$ : for instance, $\sigma=\frac{1}{2}$ and $t \approx 14.134725$. The algorithm to produce the partial sums is in the PB_inference. $\mathrm{Xl}$ s. spreadsheet, in the the Video_Riemann tab. The parameters $\sigma, t, s, \gamma$ are in cells B2: B5 for the first $z$, and $\mathrm{C} 2: \mathrm{C} 5$ for the second one. For more details and source code, see Section $6.7 .1$.

## 统计代写|随机过程代写stochastic process代考|The Story Told by the Videos

The video starts with a chaotic orbit that looks like a Brownian motion. The orbit then gets smoother, jumping from sink to sink until eventually entering a final sink and converging. When $s=0$, the behavior is similar to that pictured in Figure 20. When $s>0$ and $\gamma \neq 0$, the whole orbit looks like a Brownian motion. As $s$ and $\gamma$ get larger, the Brownian motion starts exhibiting a strong clustering structure, with well separated clusters called “sinks”. This is visible in Figure 21. See the discussion accompanying these figures, for additional details about the sinks, and the Brownian versus clustered-Brownian behavior. My question “Is this a Brownian motion?”, posted here on MathOverflow, brings more insights.

The cause of the sinks is well-known, and explained in Exercise 25, for the one-dimensional case. The orbits are very sensitive to small changes in the parameters, especially to tiny moves from the base model $s=0$. Large values of $t$ produce a large number of sinks; the behavior is radically different when $t$ is close to zero. Values of $\sigma$ between $0.1$ and $0.6$ produce similar patterns. Outside this range, the patterns are noticeably different.

The video featuring two orbits has this peculiarity: the orbit on the left, with $s=0$, is non-Brownian; the one on the right with $s=0.05$ and $\gamma=0.005$ is slightly Brownian (barely, because $s$ is still very close to zero, yet the curve is a bit less “curvy”). Despite the tiny difference in $s$, which makes you think that both orbits should converge to close locations, in reality the two orbits move in radically different directions from the very beginning: this is a typical feature of chaotic dynamical systems. In this case, it is caused by choosing a large value for $t\left(t \approx 5.56 \times 10^6\right)$.

The observations (2D points) that generate the orbits, are realizations of a new, very rich class of point processes. Such point processes could have applications in specific contexts (possibly astronomy), as potential modeling tools. Identifying the sinks and counting their number can be done using unsupervised clustering techniques. One might even use the technique described in Section 3.4.3. Finally, the color harmony results from using harmonics, that is, cosine waves with specific periods: see Section $6.7 .1$ for explanations. The next step is to design a black-box algorithm for palette creation, and to automatically generate and add a soundtrack to the video, using related mathematical formulas that produce harmonic sounds. In short, AI-generated art!

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Connection to Poisson-binomial Processes

Fréchet 分布 $\gamma$ ，对于基础泊松二项式过程: 参见公式 (37)。出于同一部分中讨论的标准化目的，强度设置为 $\lambda=\Gamma(1+\gamma)$

## 统计代写|随机过程代写stochastic process代考|The Story Told by the Videos

$s=0$, 是非布朗的；右边的那个 $s=0.05$ 和
$\gamma=0.005$ 有点布朗 (几乎没有，因为 $s$ 仍然非常接近 于零，但曲线不那么”弯曲”)。尽管细微差别 $s$, 这让你 认为两条轨道应该会聚到相近的位置，实际上这两条轨 道从一开始就朝看截然不同的方向运动：这是混沌动力 系统的典型特征。在这种情况下，这是由于为 $t\left(t \approx 5.56 \times 10^6\right)$.

## 有限元方法代写

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## MATLAB代写

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