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Can mathematics get from other sciences what it got from physics?
Throughout history, physics has been an unparalleled source of '' inspiration'' for discovering/inventing mathematical ideas, which is due to its ability to describe the physical world. But can this connection be made as profoundly with other fields of science like biology?
Can other fields deepen our understanding of mathematics and generate new discoveries/inventions in it? Has this already happened? How so?

What's the meaning of algebraic data type?
I'm reading a book about Haskell, a programming language, and I came across a construct defined "algebraic data type" that looks like
data WeekDay = Mon | Tue | Wed | Thu | Fri | Sat | SunThat simply declares what are the possible values for the type WeekDay.
My question is what is the meaning of algebraic data type (for a mathematician) and how that maps to the programming language construct?

Why should I care about Gauss-Bonnet (and Gaussian curvature)?
The Gauss-Bonnet Theorem (for orientable surfaces without boundary) states that for surface M, with Gaussian curvature at a point K, we have
${\int <!--\; \int \; -->}_{M}KdA=2\mathrm{\pi <!--\; \pi \; -->}\mathrm{\chi <!--\; \chi \; -->}(M).$
Right now, this just says to me that the integral of something I don’t care about is equal to the ratio of a circumference to radius of a (Euclidean) circle times something I do care about. I get why normal curvature is interesting. I get Gaussian curvature is important in the Theorema Egregium and classification of surfaces of constant curvature. In the former, it helps classify surfaces up to isometry and show that developable surfaces are ruled. In the latter, it tells us that Riemannian manifolds with transitive isometry groups locally have a structure from a very small selection. But that’s just Gaussian curvature giving us other things. I don’t understand why I should like it for its own sake, and as such, why I should care about Gauss-Bonnet. I get that it links differential geometry and topology, but maybe there are other ways to do this, and this one feels a little convoluted, and topology is already linked to differential manifolds, and thus Riemannian manifolds, through Poincare-Hopf.

Regression coefficient is zero
I am trying to construct a quadratic regression model for a data as
$Y={\mathrm{\alpha <!--\; \alpha \; -->}}_0+{\mathrm{\alpha <!--\; \alpha \; -->}}_1{X}_1+{\mathrm{\alpha <!--\; \alpha \; -->}}_2{X}_2+{\mathrm{\alpha <!--\; \alpha \; -->}}_3{X}_1{X}_2+{\mathrm{\alpha <!--\; \alpha \; -->}}_4{X}_1^2+{\mathrm{\alpha <!--\; \alpha \; -->}}_5{X}_2^2$
I am getting coefficient of $X_1^2(\to <!--\; \to \; -->{\mathrm{\alpha <!--\; \alpha \; -->}}_4)=0.00000006\approx <!--\; \approx \; -->0.$. How can we interpret this model for $X_1^2$ term? What is the impact of ${\mathrm{\alpha <!--\; \alpha \; -->}}_4=0$ for the model? Adjusted $R^2=0.9988$

1- The science of collecting, organizing, presenting, analyzing, and interpreting data is called ___________.
a. None
b. Regression
c. Time series
d. Statistics

I am studying bioinformatics. I am trying to solve a problem. So we have a gene, whose initial value $x_0$ at time t=0 is $x_0$=1. A perturbation of factor −0.9789812 is applied to it, such that, at time t=10, its value is 0.0210359. The gene is measured at time-points t=[0,1,2...,10]. How can I know the calculate the values at time t=1,2,...,10, given the only information I have is the value at $x_0$ and and the perturbation applied ? I am giving the data here
x = [1.0000000,0.3482754,0.1304151,0.0575881,0.0332433,0.0251052,0.0223848,0.0214755,0.0211715,0.0210698,0.0210359]
But I want to know how would I generate this data. I tried the exponential decay function and the values do not correspond to the ones I have in my data set.
So if the gene has an initial value of 1 and a perturbation of −0.9789812 is applied to it, at the final reading of the gene at time t = 10 is 1−0.9789812 = 0.0210188. The decay is not linear, it seems like it has an exponential curve but I dont know how to fit it.
Note that the only information I have is the initial value of the gene, the perturbation applied. I want to be able to calculate the values of gene at any time given this information.

Condition for existence of an orthonormal matrix whose column space is orthogonal to the column space of another matrix
While I was reading a statistics paper, I came across one statement that I don't understand (I just have basic linear algebra knowledge).
Assume (in the context of regressions), we have a $n\times <!--\; \times \; -->p$ data matrix X, assuming that X is invertible and n>p. The paper states
"$U\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->p}$ is an orthonormal matrix whose column space is orthogonal to that of X s.t. $U^{T}X=0$": such matrix exists if $n\ge <!--\; \ge \; -->2p$. I don't understand where the last statement comes from.
I know that the nullspace of X has dimension n−rank(X)=n−p in full rank case and U is the orthonormal basis of the null space of X. But I don't get the link why U only exists, if $n\ge <!--\; \ge \; -->p+rank(X)$, i.e. $n\ge <!--\; \ge \; -->2p$.

How to show the interpretability of NMF by a small qualitative example on a toy data?
In some paper, such as, Nonnegative Matrix Factorization: A Comprehensive Review, I see the interpretability of Nonnegative matrix factorization (NMF). However, I don't know the means of this. How to show the interpretability of NMF by a small qualitative example on a toy data?
In addition, what is interpretability? Especially in Non-negative Matrix Factorization, how to understand interpretability?

Roadmap for learning Topological Data Analysis?I'm a math major who has recently graduated and I will be starting full time work in 'data analysis'.Having finished with decent marks and still being incredibly interested in mathematics, I was thinking of pursuing graduate study/research at some point in the future. I was reading up about possible areas of study for this when I came across topological data analysis, which (as I understand it) is an application of algebraic topology to data analysis.Given my situation, I was intrigued by the concept and I would like to do some self study so I can have a working understanding of the subject. I have only done basic undergraduate abstract algebra, analysis and point set topology, and I am currently reading Munkres' Topology (Chapter 9 onwards). How do I get from where I am now to understanding the theory behind TDA and being able to apply it?My knowledge on further mathematics is far from extensive and I would appreciate any advice on links/texts which I could use to learn the relevant material.

A computer A generated some data that is sent to computers B, C and D in a described manner. Find probabilty of computer B receiving correct data.
A computer A generated some data that can only be either 0 or 1. The data was sent over local network to computer B, which sends it to computer C, which consequently sends it to computer D. Taking the noises into consideration, the probability of the data getting transferred from one computer to another unchanged is just 35%. If it is known that the data that was generated by computer A got correctly to computer D, find the probability that computer B received the correct data.
Any tips on how should I start? What's the idea here and how can I interpret the known fact that computer D received the correct data?

What are your ideas about the topic of Gathering and Cleaning the Data?
Why do you think this topic is important before you proceed with interpreting the results of a certain set of data?

I have an experiment that relies on measuring temperature as a function of a given condition. I change the condition, allow the temperature to stabilize as much as possible, then take 3 measurements in a row, which are all slightly different, as the values tend to oscillate.
For example, I might have a set of readings that looks like this:
X Temperature
10.1 80.3
10.8 82.1
10.3 78.9
20.4 100.2
20.0 101.1
20.2 101.0
30.1 139.1
30.0 140.2
30.0 138.2
Which I will combine to look like this:
uniqueX Xmean Xstd Tmean Tstd
_______ ______ ________ ______ _______
10 10.2 0.1 80.433 1.6042
20 20.2 0.2 100.77 0.49329
30 30.033 0.057735 139.17 1.0017
% Matlab code for computation
xNom = round(X);
[uniqueX,~,subs] = unique(xNom);
Tmean = accumarray(subs, T, [], @mean);
Tstd = accumarray(subs, T, [], @std);
Xmean = accumarray(subs, X, [], @mean);
Xstd = accumarray(subs, X, [], @std);
tb = table(uniqueX, Xmean, Xstd, Tmean, Tstd);
I'm wondering: what is the best way to plot such data, i.e. when three data points like this are combined to represent one point, what is the best value to use for the error bars on a plot of T vs X, for example in this case?

Derivation of formula for mean of a dataset after adding a new data point
I was reading an article that stated that if we know the mean ${\stackrel{¯<!--\; ¯\; -->}{x}}_{prev}$ of a dataset with n datapoints, and that if we add a new data point $x_k$ to the dataset, then the new mean ${\stackrel{¯<!--\; ¯\; -->}{x}}_{new}$ can be expressed by the formula:
${\stackrel{¯<!--\; ¯\; -->}{x}}_{new}=\frac{1}{n+1}\underset{i=1}{\overset{n+1}{\sum <!--\; \sum \; -->}}{x}_i=\frac{1}{n+1}(x_k+\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i)$
It is not obvious to me how this expression is true.
This is my attempt at deriving the expression:
${\stackrel{¯<!--\; ¯\; -->}{x}}_{new}=\frac{1}{n+1}\underset{i=1}{\overset{n+1}{\sum <!--\; \sum \; -->}}{x}_i=\frac{1}{n+1}(x_k+\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i)$
Note that $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{x}_i=n{\stackrel{¯<!--\; ¯\; -->}{x}}_{prev}$
${\stackrel{¯<!--\; ¯\; -->}{x}}_{new}=\frac{1}{n+1}(x_k+n{\stackrel{¯<!--\; ¯\; -->}{x}}_{prev})$
but is does not seem like this is getting me anywhere.

What does an increase in the mean, but the median is the same signify?
I have been collecting some data from a project (check if something is affected by the environment it is in) using some sensors. I collected some data before making a change in the environment and then after. My hypothesis is that it should be affected by the environment (the number must decrease).
To test this, I did some calculations (in Excel) and noticed that there is a small increase in the mean, but the median remains the same. I don't know how to interpret this (I only have some basic knowledge about statistics). Can I simply assume that there was indeed an increase (and that my hypothesis is false), or is there something more to it? After searching online, I found something about called "hypothesis testing of means" that seems to be applicable here, but I have to idea how to do this (tried watching videos about it, but didn't understand how to use it in my case). I calculated the variance and the standard deviation if it helps. Thanks.
Here is the result of my data:
Sample size: 60
BEFORE:
mean: 3.35
median: 3.5
variance: 0.25
standard dev.: 0.50
AFTER:
mean: 3.51
median: 3.5
variance: 0.20
standard dev.: 0.45

I was reading a tutorial written on Linear Regression by Avi Kak. There is a part about geometric interpretation of linear regression on pg.19.
The optimum solution for β~ that minimizes the cost function C($\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$) in Eq. (14) possesses the following geometrical interpretation: Focusing on the equation $\sim <!--\; \sim \; -->$y = X$\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$, the measured vector $\sim <!--\; \sim \; -->$y on the left resides in a large N dimensional space. On the other hand, as we vary $\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$ in our search for the best possible solution, the space spanned by the product X$\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$ will be a (p+1)-dimensional subspace (a hyperplane, really) in the N dimensional space in which $\sim <!--\; \sim \; -->$y resides. The question now is: which point in the hyperplane spanned by X$\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$ is the best approximation to the point $\sim <!--\; \sim \; -->$y which is outside the hyperplane. For any selected value for $\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$, the “error” vector $\sim <!--\; \sim \; -->$y − X$\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$ will go from the tip of the vector X$\mathrm{\beta <!--\; \beta \; -->}\sim <!--\; \sim \; -->$ to the tip of the $\sim <!--\; \sim \; -->$y vector. Minimization of the cost function C in Eq. (14) amounts to minimizing the norm of this difference vector.
I could not understand how to relate N-dimensional space and (p+1)-dimensional subspace. B vector defines a (p+1) dimensional subspace but I could not understand why N dimensional space contains the (p+1) subspace. As I understand in (p+1) each dimension means features but in N dimensional space each dimension means a data point. I'm a lot confused about the idea. Are there any other resource that explains the idea in a much more detail? or Could anyone explains the idea how these spaces relate?

Interpreting Coin Toss Data. Biased or Not?
We have run an experiment in which some good chap has sat down and flipped a coin 100 times. At the end of the 100 flips he has tallied 40 Heads and 60 Tails. Now this seems like something is up with the coin. The question is whether or not this coin is biased.
I have already determined that the mean p=1/2 and the standard deviation is 5. If we take the mean as a random variable of a normal distribution about the mean, then the experimental results we obtained are 2sigma from the mean. My first question is what does it mean if the results are outside one sigma?
Next I proceeded to find the probability that the p-value of the getting a heads was instead 4/10. I used the function 100C40 *p^40 *(1-p)^60 and integrated this (dp) from 0.35 to 0.45. My result was 0.006. Now does this mean that the probability of getting a p is between 0.35 and 0.45 is 0.006 ? But I feel in this method I should be comparing this p against something.
I suppose my problem really lies in interpreting the results and their meaning.

I'm reading about Bayesian data analysis by Gelman et al. and I'm having big trouble interpreting the following part in the book (note, the rat tumor rate $\mathrm{\theta <!--\; \theta \; -->}$ in the following text has:$\mathrm{\theta <!--\; \theta \; -->}\sim <!--\; \sim \; -->Beta(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})$
Choosing a standard parameterization and setting up a ‘noninformative’ hyperprior dis- tribution.
Because we have no immediately available information about the distribution of tumor rates in populations of rats, we seek a relatively diffuse hyperprior distribution for $(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})$. Before assigning a hyperprior distribution, we reparameterize in terms of $\text{logit}(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}})=\log <!--\; \; -->(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\beta <!--\; \beta \; -->}})$ and $\log <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->})$, which are the logit of the mean and the logarithm of the ‘sample size’ in the beta population distribution for $\theta$. It would seem reasonable to assign independent hyperprior distributions to the prior mean and ‘sample size,’ and we use the logistic and logarithmic transformations to put each on $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->})$ scale. Unfortunately, a uniform prior density on these newly transformed parameters yields an improper posterior density, with an infinite integral in the limit $(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->})\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$, and so this particular prior density cannot be used here.
In a problem such as this with a reasonably large amount of data, it is possible to set up a ‘noninformative’ hyperprior density that is dominated by the likelihood and yields a proper posterior distribution. One reasonable choice of diffuse hyperprior density is uniform on $(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}},(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-<!--\; -\; -->1/2})$, which when multiplied by the appropriate Jacobian yields the following densities on the original scale,
$p(\mathrm{\alpha <!--\; \alpha \; -->},\mathrm{\beta <!--\; \beta \; -->})\propto <!--\; \propto \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-5/2},$
and on the natural transformed scale:
$p(\log <!--\; \; -->\left(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\beta <!--\; \beta \; -->}}\right),\log <!--\; \; -->(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}))\propto <!--\; \propto \; -->\mathrm{\alpha <!--\; \alpha \; -->}\mathrm{\beta <!--\; \beta \; -->}(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-5/2}.$
My problem is especially the bolded parts in the text.
Question (1): What does the author explicitly mean by: "is uniform on $(\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}},(\mathrm{\alpha <!--\; \alpha \; -->}+\mathrm{\beta <!--\; \beta \; -->}{)}^{-<!--\; -\; -->1/2})$
Question (2): What is the appropriate Jacobian?
Question (3): How does the author arrive into the original and transformed scale priors?
To me the book hides many details under the hood and makes understanding difficult for a beginner on the subject due to seemingly ambiguous text.
P.S. if you need more information, or me to clarify my questions please let me know.

Survivor function of a variable that has discrete and continuous components
I'm currently reading The Statistical Analysis of Failure Time Data by Kalbfleisch and Prentice and had trouble at arriving at the expression for the survivor function of a random variable T having both discrete and continuous components. The setup is the following:
Let T be a random variable on $[0,\mathrm{\infty <!--\; \infty \; -->})$ with survivor function F(t)=P(T>t). Then
if T is absolutely continuous with density f, then the hazard function $\mathrm{\lambda <!--\; \lambda \; -->}$ can be defined as
$\mathrm{\lambda <!--\; \lambda \; -->}(t):=\underset{h\to <!--\; \to \; -->0^{+}}{lim}\frac{P(t\le <!--\; \le \; -->T<t+h\mid <!--\; \mid \; -->T\ge <!--\; \ge \; -->t)}{h}=\frac{f(t)}{F(t)}$
for $t\ge <!--\; \ge \; -->0$, and hence we have
$F(t)=\exp <!--\; \; -->(-<!--\; -\; -->{\int <!--\; \int \; -->}_0^t\mathrm{\lambda <!--\; \lambda \; -->}(s)\mathrm{d}s),t\ge <!--\; \ge \; -->0.$
if T is discrete taking on the values $0\le <!--\; \le \; -->a_1<a_2<\cdots <!--\; \cdots \; -->$, then we define the hazard at $a_i$ as
${\mathrm{\lambda <!--\; \lambda \; -->}}_i=P(T=a_i\mid <!--\; \mid \; -->T\ge <!--\; \ge \; -->a_i),i=1,2,\dots <!--\; \dots \; -->.$
Then we can show that
$F(t)=\underset{j\mid <!--\; \mid \; -->a_j\le <!--\; \le \; -->t}{\prod <!--\; \prod \; -->}(1-<!--\; -\; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_j),t\ge <!--\; \ge \; -->0.$
These expressions for the survivor functions I am ok with. Now they write the following:
More generally, the distribution of T may have both discrete and continuous components. In this case, the hazard function can be defined to have the continuous component ${\mathrm{\lambda <!--\; \lambda \; -->}}_c(t)$ and discrete components ${\mathrm{\lambda <!--\; \lambda \; -->}}_1,{\mathrm{\lambda <!--\; \lambda \; -->}}_2,\dots <!--\; \dots \; -->$ at the discrete times $a_1<a_2<\cdots <!--\; \cdots \; -->$
The overall survivor function can then be written
$\begin{array}{}\text{(1)}& F(t)=\exp <!--\; \; -->(-<!--\; -\; -->{\int <!--\; \int \; -->}_0^t{\mathrm{\lambda <!--\; \lambda \; -->}}_c(s)\mathrm{d}s)\underset{j\mid <!--\; \mid \; -->a_j\le <!--\; \le \; -->t}{\prod <!--\; \prod \; -->}(1-<!--\; -\; -->{\mathrm{\lambda <!--\; \lambda \; -->}}_j).\end{array}$
That T has both discrete and continuous components means that the distribution of T is of the form
$P_T(\mathrm{d}x)=f_c(x)\mathrm{\lambda <!--\; \lambda \; -->}(\mathrm{d}x)+\underset{j=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{b}_j{\mathrm{\delta <!--\; \delta \; -->}}_{a_j}(\mathrm{d}x)$
or equivalently
$P(T\in <!--\; \in \; -->A)={\int <!--\; \int \; -->}_A{f}_c(x)\mathrm{d}x+\underset{j\mid <!--\; \mid \; -->a_j\in <!--\; \in \; -->A}{\sum <!--\; \sum \; -->}{b}_j$
for some sequence $a_1<a_2<\cdots <!--\; \cdots \; -->$ and $b_i\in <!--\; \in \; -->(0,1)$ and some non-negative measurable function $f_c$ with ${\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{f}_c\mathrm{d}\mathrm{\lambda <!--\; \lambda \; -->}+\underset{i=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{b}_j=1$. If we define
${\mathrm{\lambda <!--\; \lambda \; -->}}_c(t)=\frac{f_c(t)}{P(T\ge <!--\; \ge \; -->t)}=\frac{f_c(t)}{F(t)},t\ne <!--\; \ne \; -->a_i,$
and
${\mathrm{\lambda <!--\; \lambda \; -->}}_i=P(T=a_i\mid <!--\; \mid \; -->T\ge <!--\; \ge \; -->a_i),$
then how do I show (and is it even true) that the survivor function of T is given by (1)?

This is data I am thinking about after reading sections 1,2,3 of chapter 2 on schemes from Hartshorne's Algebraic Geometry.
Basically, I know very little and I'm very uncomfortable with schemes.
Let X be a scheme.
We know that every point is in some open affine $U_i\cong <!--\; \cong \; -->\mathrm{Spec}<!--\; \; -->(A_i)$. So, we can cover X be open affines $U_i\cong <!--\; \cong \; -->\mathrm{Spec}<!--\; \; -->(A_i)$.
Now, we can intersect any open subset of X with the cover of open affines.
(1) Does this mean that any open subset of X be can covered (abusing notation) by basic open subsets $D(f_{i_j})\subset <!--\; \subset \; -->\mathrm{Spec}<!--\; \; -->(A_i)$? Therefore, any point in X is in some (abusing notation) $D(f_{i_j})\cong <!--\; \cong \; -->\mathrm{Spec}<!--\; \; -->(A_{i_{f_{i_j}}})$?
An exercise shows that any open subset is a scheme via the induced scheme structure.
(2) Does this mean that any cover of X will give us a cover by open affines? For example, take any open subset U. Then U is a scheme via the induced scheme structure. So, we can cover U via open affines. Since U is open, then these open affines are also open affines of X?
(3) If $p\in <!--\; \in \; -->X$ is in some open affine $U\cong <!--\; \cong \; -->\mathrm{Spec}<!--\; \; -->(A)$, can we also keep finding smaller and smaller open affines containing p? How do these smaller and smaller open affines relate to U and X? How do the rings relate to each other&

Properties of Finite Differences
I have seen this statement in different variants, but could not find a proof:
If the n-th order differences of equally spaced data are non-zero and constant then the data can be represented by a polynomial.
I interpret this statement more formally as follows:
Let $f:<!--\; :\; -->\mathbb{Z}\to <!--\; \to \; -->\mathbb{Z}$be any function such that the finite differences $D_r$, defined by
$D_{r+1}(x)=D_r(x+1)-<!--\; -\; -->D_r(x)$ for $r\ge <!--\; \ge \; -->0$ (where $D_0:=f$ by convention)
are such that $D_r$ is constant and non-zero for any $r\ge <!--\; \ge \; -->r_0$. Then f is a polynomial in x (with Q coefficients) and degree $r_0$.
Question: Is this correct as I stated it? How does one prove that such f must be a polynomial?