Quantum Field Theory: completing the “A Bridge between Mathematicians and Physicists” series

I decided to read the series “A Bridge between Mathematicians and Physicists” written by Eberhard Zeidler. But when I read the preface of the first book I realized that at first this series should be composed of six volumes, namely:

  1. Quantum Field Theory I: Basics in Mathematics and Physics
  2. Quantum Field Theory II: Quantum Electrodynamics
  3. Quantum Field Theory III: Gauge Theory
  4. Quantum Field Theory IV: Quantum Mathematics
  5. Quantum Field Theory V: The Physics of the Standard Model
  6. Quantum Field Theory VI: Quantum Gravity and String Theory

I have only the first three books. Searching for the last three I discovered that Eberhard Zeidler died and therefore the last three volumes will never be published.

For this reason I ask: could someone please indicate me books that cover the subject of the last three books and that focus a lot on the mathematical part?

Thank you for your attention.

Pads Approximants of Power Series With Natural Boundaries

Consider a power series $ \sum_{n=0}^{\infty}c_{n}z^{n}$ for which $ c_{n}\in\left\{ 0,1\right\}$ for all $ n$ . One can write this as: $ $ \varsigma_{V}\left(z\right)\overset{\textrm{def}}{=}\sum_{v\in V}z^{v}$ $ for some set $ V$ of positive integers. I call this the “set-series” of $ V$ . There is a beautiful theorem due to Otto Szëgo which, for the case of set-series, shows that $ \varsigma_{V}\left(z\right)$ is either a rational function whose poles are simple and located at roots of unity, or that $ \varsigma_{V}\left(z\right)$ is a transcendental function with the unit circle ($ \partial\mathbb{D}$ ) as a natural boundary.

Natural boundaries generally occur as the result of singularities clustering arbitrarily close to one another. My intuition tells me that in the case where $ \varsigma_{V}\left(z\right)$ has a natural boundary (example: $ V=\left\{ 2^{n}:n\geq0\right\}$ , $ V=\left\{ n^{2}:n\geq0\right\}$ , etc), the clustering singularities in question are simple poles.

I figure a good way to try to see this would be via Padé approximants. The “rigorous” statement of my intution would then be something along the lines of: for an appropriately chosen sequence of Padé approximants $ \left\{ P_{n}\left(z\right)\right\} _{n\geq1}$ of $ \varsigma_{V}\left(z\right)$ (where $ \varsigma_{V}\left(z\right)$ has a natural boundary on $ \partial\mathbb{D})$ , for every $ \epsilon>0$ and every $ \xi\in\partial\mathbb{D}$ , there is an $ N_{\epsilon,\xi}\geq1$ so that, for all $ n\geq N_{\epsilon,\xi}$ , any pole $ s$ of $ P_{n}\left(z\right)$ satisfying $ \left|s-\xi\right|<\epsilon$ is necessarily simple.

With the literature on Padé Approximants appears to be quite extensive (while the literature on natural boundaries appears to be comparatively paltry), I was wondering if anyone knew of anything about this question, or something similar. Insight and/or references would be most appreciated.

Lazy streams and infinite series

I just started Unix System Programming with Standard ML and starting on page 22 Shipman begins to explain a pure functional way of avoiding the constant state changes of typing at a keyboard:

A lazy stream is an infinite list of values that is computed lazily. The stream of keystrokes that a user presses on the keyboard can be represented as an infinite (or arbitrarily long) list of all of the keystrokes that the user is ever going to press. You get the next keystroke by taking the head element of the stream and saving the remainder. Since the head is only computed lazily, on demand, this will result in a call to the operating system to get the next keystroke. What’s special about the lazy stream approach is that you can treat the entire stream as a value and pass it around in your program. You can pass it as an argument to a function or save the stream in a data structure. You can write functions that operate on the stream as a whole. You can write a word scanner as a function, toWords, that transforms a stream of characters into a stream of words. A program to obtain the next 100 words from the standard input might look something like apply show (take 100 (toWords stdIn)) where stdIn is the input stream and take n is a function that returns the first n elements in a list. The show function is applied to each word to print it out. The program is just the simple composition of the functions. Lazy evaluation ensures that this program is equivalent to the set of nested loops that you would write for this in an [imperative] program. This is programming at a much higher level than you typically get with [imperative] languages.

I take it he doesn’t mean just a list of the 26 letters of the alphabet, rather, something more like every possible word and combination thereof, i.e., similar to the theoretical possibility of finding yours and everyone’s birthday if you look hard enough in the irrational never-repeating infinite stream of $ \pi$ places. That is, somewhere in the infinite series of $ \pi$ decimal places is your birthday mmddyyyy-style. Is this in fact what is meant here? Also, is this in any way related to infinite recursion?

In a similar vein, I once was told that no functional language is truly pure, and he gave the example of a function that “goes out” and gets the exact time from an atomic clock. Is this the same sort of issue, where, e.g., an infinite list of possible times is available for a list operation?

Fourier series representation of piecewise function

$ $ {Expand} \; f(x)= \begin{cases} 2{A\over L}x & 0\leq x\leq {L\over 2} \ \ 2{A\over L}\left(L-x\right) & {L\over 2}\leq x\leq L \end{cases} $ $

I have determined $ A_0$ (but omitted) to be $ A_0=A$ . For $ \ A_n \ $ and $ \ B_n$ , I’m rather confused where to start. For $ A_n$ : $ $ A_n={2\over L}\left[2{A\over L}\int_{0}^{L\over 2}x\cos\left({2n\pi x}\over L\right)dx+2{A\over L}\left(L\int_{L\over 2}^L\cos\left({2n\pi x}\over L\right)dx-\int_{L\over 2}^Lx\cos\left({2n\pi x}\over L\right)dx\right)\right]$ $ And a similar case to $ B_n$ , I’m not quite sure if I’m on the right path.

How to apply INAR model to a simple time series model in Python

As a course project for Time Series Analysis, I used ARIMA for a very simple model – (Analyzing number of deaths in each episode of game of thrones and forecasting the number of deaths in the final episode), thus there wasn’t as much data for this. I have been asked to redo it using INAR model. I was wondering if that could be achieved using ARIMA and applying zero for the MA and lag part, which would just give it AR. But I’m confused as to how to make it Integer valued. I need this done in Python and was wondering if there is a model out there for this.

This is the code I used for forecasting but I’m sure INAR is more than this.

#Prediction     model = ARIMA(df, order=(15, 0, 0))     model_fit = model.fit(disp=False)     prediction = model_fit.forecast()[0]     print(prediction) 

What graphics cards can I upgrade to if I have an HP Slimline Desktop PC 270-p0xx series?

I am looking to upgrade my computer’s graphics card, because I currently am running on Integrated Graphics. It would be good to know what I could upgrade to, and know if I need to get a stronger fan, and power supply. Here are my current specifications:

Processor: Intel Core i7-7700T

RAM: 12 GB

Graphics: Intel HD Graphics 630

I do not know my current power supply wattage, so any help would be appreciated. Thanks!

Fourier series Parseval equality with partial sums

Let $ f \in \mathcal{R}_\left[-\pi,\pi\right]$ be function with period $ 2\pi$ . We denote n-th partial Fourier series sum of function $ f$ with $ S_n(x)$ . Prove that: $ $ \int_{-\pi}^{\pi}\left(f(x)-S_n(x) \right)^2dx = \pi\sum_{k=n+1}^{\infty}(a_k^2+b_k^2) $ $

I know that Parseval equality(is it called identity?) is true: $ $ \frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)dx = \frac{a_0^2}{2} + \sum_{k=1}^{\infty}(a_k^2+b_k^2) $ $

I tried the following (without much loss of generality I have assumed that $ a_0 = 0$ ) to obtain Parseval equality back: $ $ \int_{-\pi}^{\pi}\left(f(x)-S_n(x) \right)^2dx = \pi\sum_{k=n+1}^{\infty}(a_k^2+b_k^2) $ $ $ $ \frac{1}{\pi}\left(\int_{-\pi}^{\pi}f(x)^2dx -2 \int_{-\pi}^{\pi}f(x)S_n(x)dx + \int_{-\pi}^{\pi}S_n(x)^2dx \right)= \sum_{k=n+1}^{\infty}(a_k^2+b_k^2) $ $ Now, we have: $ $ \int_{-\pi}^{\pi}f(x)S_n(x)dx = \int_{-\pi}^{\pi}f(x)\left[\frac{a_0}{2} + \sum_{k=1}^n\left(a_kcos(kx) + b_kcos(kx)\right)\right]dx = $ $ $ $ = \sum_{k=1}^n\int_{-\pi}^{\pi}f(x)a_kcos(kx)dx+ \sum_{k=1}^n\int_{-\pi}^{\pi}f(x)b_kcos(kx)dx = $ $ $ $ = \sum_{k=1}^na_k\int_{-\pi}^{\pi}f(x)cos(kx)dx+ \sum_{k=1}^nb_k\int_{-\pi}^{\pi}f(x)cos(kx)dx = $ $ $ $ = \sum_{k=1}^na_ka_k+ \sum_{k=1}^nb_kb_k = $ $ $ $ = \sum_{k=1}^na_k^2+ \sum_{k=1}^nb_{k}^2 = \sum_{k=1}^na_k^2+b_k^2 $ $

So the main equation is now: $ $ \frac{1}{\pi}\left(\int_{-\pi}^{\pi}f(x)^2dx -2\sum_{k=1}^na_k^2+b_k^2 + \int_{-\pi}^{\pi}S_n(x)^2dx \right)= \sum_{k=n+1}^{\infty}(a_k^2+b_k^2) $ $

I feel like I got closer, because I have the $ a_k^2 + b_k^2$ sums from 1 to n and from n+1 to infinity. Is that correct line of reasoning? How can I proceed from here?

2 Dimensional Array for storing a series of different data types in C#

I want to create an array with different datatype of columns for storing a series of data in C# assume we have 1000 rows of data so we could not create a class and instantiate it 1000 times what i whant is exactly like an excel worksheet , like the picture Connecting to a data base is a very slow proccess.

how can i do this? enter image description here

Ideal in ring of power series

Let $ K$ be a field of characteristic $ p$ and $ A_n \colon= K[[X_1,\ldots,X_n]]$ be a $ n$ -variable formal power series ring over $ K$ such that $ n, p \geq 3$ .

Consider the ideal $ I$ defined by \begin{equation*} I \colon= (X_1^{p},X_2^{p^2},X_3^{p^3},\ldots,X_n^{p^n}). \end{equation*} Suppose that $ \alpha, \beta \in A_n$ be two different prime elements such that $ \alpha \notin I$ and $ \beta \notin I$ .

Q. Does it always hold that the multiplication $ \alpha \beta \notin I$ ?