number theory – A question in calculating a constant in section square free values of quadratic polynomials

I am studying square free values of quadratic polynomials from class notes and I am struct on a deduction.

Consider this conjecture, I have no problem in understanding it:

enter image description here

Consider this theorem, I have no problem in understanding it.
enter image description here

But I have problem in calculating $c_f$

enter image description here

I don’t understand , how author wrote $rho(p^2) = 1+ (-1/p) $.

Kindly consider giving any hints.

ca.classical analysis and odes – A small question regarding a type of PDE

In doing optimal control of Parabolic PDE’s we often have to solve a problem like that:

$$left{begin{array}{lr} dfrac{partial y}{partial t}-dDelta y(t,x)=f(y(t,x),p(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial p}{partial t}(t,x)+dDelta p(t,x)=g(y(t,x),p(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial y}{partialnu}(t,x)=dfrac{partial p}{partial nu}(t,x)=0 & (t,x)in (0,T)timespartialOmega \ y(0,x)=y_0(x), p(T,x)=p_T(x) & xinOmegaend{array}right.$$

where $Omegasubseteqmathbb{R}^2$ or $mathbb{R}^3$ is a bounded connected set with smooth boundary, and $y,p:(0,T)timesoverline{Omega}to mathbb{R}$ are the primal and the dual state. Here $f,g$ are two smooth functions from $mathbb{R}^2$ to $mathbb{R}$. Also $y_0$ and $p_T$ are two given smooth functions.

How can we deduce the existence and the properties of the solution?

The difficulty is that if we denote $q(t,x)=p(T-t,x)$ (to transform the problem into an INITIAL VALUE PROBLEM) we came across the following system:

$$left{begin{array}{lr} dfrac{partial y}{partial t}-dDelta y(t,x)=f(y(t,x),q(T-t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial q}{partial t}(t,x)-dDelta q(t,x)=g(y(T-t,x),q(t,x)) & (t,x)in (0,T)timesOmega \ dfrac{partial y}{partialnu}(t,x)=dfrac{partial q}{partial nu}(t,x)=0 & (t,x)in (0,T)timespartialOmega \ y(0,x)=y_0(x), q(0,x)=p_T(x) & xinOmegaend{array}right.$$

which looks as a DELAYED PDE. If the terms with $T-t$ would not have appeared then the existence would have been insured by the classical results in A. Pazy – Semigroups of Linear Operators and Applications to Partial Differential Equations

The problem can be implemented numerically via finite differences for example resulting a system of equations, but I wonder if this type of problems has been studied in a theoretical fashion.

I will be thankful for any reference or advice!

Question No.1 – Arithmetic & Algebra

Question No.1 – Arithmetic & Algebra – MathOverflow

A question about the input of special character

When I want to input some special characters, I will use Esc + the pronunciation. Take one for example:
when I want to input theta, I use esc and "th"

But here’s the problem, when I want to insert something in the middle of a piece of code, this method fails:It seems that esc has included the following Cos[x]
How can I solve this problem? or if there exists a more convenient method to input special characters? Thanks for your answer.

measure theory – Question about a proposition in Munkres’s Analysis on Manifolds

I am reading through Munkres’s Analysis on Manifolds, and I get stuck in a proof of the lemma 18.1, that is stated as following:

Lema 18.1 Let $A$ be open in $mathbb{R}^n$; let $g:Ato mathbb{R}^n$ be a function of class $C^1$. If the subset $E$ of $A$ has measure zero in $mathbb{R}^n$, then $g(E)$ has measure zero in $mathbb{R}^n$.

He made out its proof in three steps. The first and second step are mentioned in the third, where he actually prove the theorem. Let me add some pictures of the third step.

enter image description here

enter image description here

(If you need pictures of the other two steps in order to solve the question above, let me know, please)

Note: A $delta$-neighborhood of a set $X$ is the union of all open cubes (in this case) with width $delta>0$ and centered at $xin X.$ The theorem 4.6 in that book states that every compact set $K$ that is contained in an open set $Usubset mathbb{R}^n$ has a $delta$-neighborhood contained in $U$.

So, the problem is here: When he covers the set $E_k$ by countably many cubes $D_i$ with certain properties, he asserts: Because $D_i$ has width less than $delta$, it is contained in $C_{k+1}$.

Why this is true? I mean, if each cube $D_i$ is centered at some point lying at $C_k$ it is clearly true, but we don’t know if this happens. I tried to give a proof that we can assume that each $D_i$ can be choosen in a way that is centered in $C_k$ but I couldn’t prove that.

Can you help me to justify that assertion on the book? Thanks in advance.

sharepoint online – Question for Site Collections [Beginner]

I’m beginner in SharePoint. I’m working in SharePoint 2013.

I’m confused about site collections.

When we are getting site collection, we can do it like this e.g.:

SPSite site = new SPSite("localhost:32000");

Doesn’t this mean that we get more Site Collections than one? If I understand, localhost:32000 in this case is url of web application. And I know web application can have more site collections.

And when we want to create web site to web collection, shouldn’t we specify for which site collection are we making web site?

So am I wrong, can anyone explain please this?

soft question – What’s the point of differential geometry?

I’ve been self studying differential geometry for a little while now (4-6 months). I am learning from Lee’s Introduction to Smooth Manifolds, and I just don’t quite get the point of the subject. Why do we study the constructions that we study, such as differential forms, submanifolds, vector bundles, etc. ? What is the goal of differential geometry, i.e., what is the motivation for studying it?

Edit: As Will Sawin suggested, I will describe things I find motivating. I find beautiful structures, unsolved problems, and a general goal of what we are trying to do in the subject to be motivating. For instance, the motivation in topology (point-set) is to find topological invariants of a given space. This leads us to connectedness, compactness, the fundamental group, etc. In this example, we have a general rule for what we are trying to accomplish. What is this such rule in differential geometry?

Support Hosting Question | Web Hosting Talk


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(Let’s put aside all the other things like speed, price, performance etc)

Regards.

arithmetic geometry – A specific Diophantine equation related to the congruent number question

Let $n$ be an odd square free natural number. J.B. Tunnel in his 1983 paper, showed that a number $n$ is congruent, if and only if the number of triples of integers satisfying $2x^2+y^2+8z^2=n$ is equal to twice the number of triples of integers satisfying $2x^2+y^2+32z^2=n$. This is by assuming the BSD conjecture, but still we do not know an efficient (polynomial time) algorithm to determine whether a number is congruent or not, from the theorem stated above.

I am trying to move with this problem a little. A simple observation tells us that, if $(alpha,beta,gamma)$ satisfies $2x^2+y^2+32z^2=n$, then $(alpha,beta,2gamma)$ and $(alpha,beta,-2gamma)$ satisfies $2x^2+y^2+8z^2=n$. So, like this we deduce that if $2x^2+y^2+8z^2=n$ has twice integral solution than $2x^2+y^2+32z^2=n$, then $2x^2+y^2+8z^2=n$ cannot have its integral solution with $z$ odd.

So, now the problem reduces to for what values of $n$, we will have an integral solution of $2x^2+y^2+8(2z+1)^2=n$. If it has a solution then $n$ is not congruent, otherwise it is. Now, I do not know how to proceed any further. As, the equation is not homogenous, one cannot directly invoke Hasse Minkowski’s local global principle, so trying to solve over $p$-adics is not an option. However, if one fails to find solution over $mathbb{Q}_p$ for any $p$ for a particular $n$, then $n$ is congruent. By this, I was able to prove that the numbers $nequiv 5$ mod $8$ and $nequiv 7$ mod $8$ are always congruent, as this type of numbers were failing to give any solution mod $8$ and hence, there was no $mathbb{Q}_2$ solutions. But this will never say whether a number is not congruent.

I do not have any idea to proceed with the problem. Again, the diophantine problem is for what $n$, does $2x^2+y^2+8(2z+1)^2=n$ has integral solutions. Any suggestions or directions to move will be really helpful.

ag.algebraic geometry – A question on effective divisors

Let $X$ be a projective variety with two morphisms $f:Xrightarrow Y$ and $g:Xrightarrow Z$. Assume that $Pic(X) = f^{*}Pic(Y)oplus g^{*}Pic(Z)$. Then if $D$ is a divisor on $X$ we can write $D = f^{*}D_Y + g^{*}D_Z$, where $D_Y,D_Z$ are divisors on $Y$ and $Z$ respectively.

If $D$ is effective are then $D_Y$ and $D_Z$ effective as well?

This holds for instance when $X = Y times Z$ is a product and $f,g$ are the projections onto the factors.

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