以下是维基百科的解释,原地址:https://en.wikipedia.org/wiki/K-function

In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the gamma function.

Formally, the K-function is defined as

It can also be given in closed form as

where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and

Another expression using polygamma function is[1]

Or using balanced generalization of polygamma function:[2]

where A is Glaisher constant.

The K-function is closely related to the gamma function and the Barnes G-function; for natural numbers n, we have

More prosaically, one may write

The first values are

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, … ((sequence A002109 in the OEIS)).

References[edit]

External links[edit]

Weisstein, Eric W. “K-Function”MathWorld.

 

以下是Wolfram的解释,内容来自http://mathworld.wolfram.com/K-Function.html

KFunction

KFunctionReIm

KFunctionContours

此处具体内容请查看原文

 

 

 

请使用手机”扫一扫”x

《贝叶斯方法——概率编程与贝叶斯推断》主要是讲解利用贝叶斯概率方法结合Python编程来求解一些问题。因为我注意到有一本书和研究方向是关于贝叶斯概率方法和量子力学相结合,甚至有量子贝叶斯这种新学科的产生,所以我想看看贝叶斯到底在做什么。

我没有学过概率论,据目前了解,贝叶斯概率是概率方法中的一种,但是作用很大。而且这本书印刷不错,抽空看了几页,文字言简意赅,值得读一读。

   

 

GRTensor 下载地址:https://github.com/grtensor/grtensor

GRTensor 博客:https://hyperspace.uni-frankfurt.de/2016/12/07/grtensoriii-for-maple-has-been-released/

GRTensor wiki介绍:https://github.com/grtensor/grtensor/wiki

GRTensor 官方网站:http://grtensor.phy.queensu.ca/


本段摘选自GRTensor wiki介绍,链接在上面已经给出。

GRTensorIII is provided as Maple package in the  lib/  directory of the project. This lib directory needs to be included in your Maple library path.

This can be accomplished in a Maple session with a command like (changing the path to the directory on your system):

For Windows users (DOUBLE backslashes!!):

If the library is not found the cryptic error message will say something like: “with expects it’s first argument to be of type….”

Alternately, this command can be placed in the Maple init file in the home directory on your computer. See Maple Init File

GRIII can then be loaded via:

And the output will be:

It is then useful to ensure you have a suitable path set for loading and saving metrics. This is controlled by the global variable  grOptionMetricPath . The current option settings are displayed with the command  groptions(); . The grtensor distribution includes a directory with some useful metrics in  metrics/ .

grOptionMetricPath can be configured by assignment in the Maple session or set in your  .mapleinit  file.

GRTensorIII contains a new option variable  grOptionMapletInput  (default  true ). More recent Maple releases have made dialog box input the default mode. The interactive routines in GRTensorIII now use these dialogs by default. If this is not desired, set this option variable to  false . See  ?groptions for a description of all the options for GRTensorIII.

Getting Started

There are multiple approaches to get started with GRTensor:

  • consult the built in help.  ?grtensor  and  ?grt_commands  are a good starting point
  • open one of the sample worksheets in the  worksheets  directory
  • read the PDF docs, typically starting with  grIntro.pdf  in the  docs/  directory.

软件制作界面:

注意,首先设置视频长度(按按帧数设置,比如每秒30帧,共900帧,则一共30秒)

然后按F3开始录制,

在时间轴上将进度条拖动到某一部位,比如300处,再框选局部区域,双击放大,则插入该区域的关键帧。

依此类推,直到末尾。

然后渲染,生成视频,这一步非常耗时。下面的视频一共三十秒,耗时2小时。

不过在这两小时可以做别的事,比如我看完了这本书《Wolfram语言入门》的部分章节,认识到“纯虚函数”的重要性,以及Mathematica中实现流程编程并不方便这一事实。还学到了如何用Mathematica制作声音, 特别有趣!

视频结果:

如何理解四维球体? 十维球体呢?

视频源地址:https://www.youtube.com/watch?v=zwAD6dRSVyI

https://brilliant.org/3b1b尝试基于问题的学习方式! https://www.benbenandblue.com/

特别感谢以下赞助者:http://3b1b.co/high-d-thanks

看看Ben Eater的通道:https://www.youtube.com/user/eaterbc

音乐: 文森特·鲁宾蒂 https://soundcloud.com/vincerubinetti…