prove: max (w(E), w(E)) is a 1/2 approximation to the value OPT

I don’t know the definition of $1/2$-approximation algorithm, but it looks like what you will need to do is the following argument:

For an enumeration $sigma:Vto{1,2,…,n}$ of the vertices, let, for each input graph $G=(V,E)$, the output of the algorithm be $A_sigma(G)=max(w(overrightarrow{E}),w(overleftarrow{E}))$.

We have that $OPT(G)leq w(E)$.

On the other hand,
$$
begin{align}
A_sigma(G)&=max(w(overrightarrow{E}),w(overleftarrow{E}))\
&geqfrac{w(overrightarrow{E})+w(overleftarrow{E})}{2}\
&= frac{w(E)}{2}
end{align}$$

The last equation is because $overrightarrow{E},overleftarrow{E}$ is a partition of $E$.

Putting these two inequalities together we get $$OPT(G)leq w(E)leq 2A_sigma(G)$$

Therefore $$frac{A_sigma(G)}{OPT(G)}geq frac{1}{2}$$

Approximation in fractional Sobolev space

Assume $Omegasubset Bbb R^d$ is Lipschitz open set. Let $pgeq 1$ and $0<sleq 1/p$.

How to prove that $C_c^infty(Omega)$ is dense in $\$$W^{s,p}(Omega)$.

Recall that,

$$|u|^p_{W^{s,p}(Omega)}= iintlimits_{OmegaOmega}frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$
and

$$W^{s,p}(Omega)= {uin L^p(Omega): |u|^p_{W^{s,p}(Omega)}<infty}$$

equipped with the Banach norm

$$|u|^p_{W^{s,p}(Omega)}= |u|^p_{L^{p}(Omega)}+|u|^p_{W^{s,p}(Omega)}$$

The main difficulty in solving this is to construct a suitable family of cut-off with compact support in $Omega$.

Why isn’t using review counts on Google Maps a good approximation for the popularity of a location?

I read (mirror) this comment that attracted some upvotes stating that:

I don’t believe using (Google Maps) review counts as a proxy for popularity is a good idea for travelers for a multitude of reasons.

Why is using review counts on Google Maps a good approximation for the popularity of a location?

I would have guessed that, approximately, the more people frequent a place (e.g., a restaurant or a market), the more reviews it gets.


The motivations as a traveler to find out which places are most often frequented are well stated in https://www.top-rated.online/ (mirror):

When you travel to a new city, it takes time till you find your new
favorite place or visit what’s best here. You will want to visit the
best places there are, but it is so difficult to find them!

The most reviewed and top rated places are ones that you can trust.
But take everything with a grain of salt.

There are also many hidden gems that are extremely good, but are not
that popular. Also, always keep an eye on worst reviewed places and
tourist traps that you should avoid.


https://infotech.report/news/google-takes-down-fake-reviews-from-maps-platform-using-ai/8665 quotes Kevin Reece, Director of Product Management at Google:

The vast majority of contributions made to Maps are authentic, with policy-violating content seen less than one percent of the time. And we’ll continue to develop new tools and techniques to fight against bad actors. Contributed content is an indispensable part of how we’re making Maps richer and more helpful for everyone.

so it seems policy-violating content wouldn’t have much of an impact.

Approximation of a function by a polynomial (Chebyshev First Kind, Bernstein, etc…) containing only even degrees and constants in a given Range[a,b]

In Mathematica, how can I create a polynomial function containing only even degrees and constants?

That is, I have a function:

$f(x)=frac{pi ^2}{left(frac{pi }{2}-tan ^{-1}(k (x-1))right)^2}$

And I’m looking for a function that generates such a polynomial approximation on arbitrary range with Chebyshev First Kind, Bernstein or another in form:

$p(x) = c_0 + c_1 cdot x^2+ … + c_m cdot x^m$

where $m$ – maximum even degree of polynom.

ClearAll("Global`*")
pars = {k = 1, Subscript((Omega), 0) = 1, (CapitalDelta) = 5}
f = (-ArcTan(k (x - Subscript((Omega), 0))) + Pi/2)/Pi
Plot(f, {x, 0, 5}, PlotRange -> All)
p(x_) = 1/f^2
Plot(1/f^2, {x, 0, 5}, PlotRange -> All)
P = Collect(
   N(InterpolatingPolynomial({{0, 0}, {Subscript((Omega), 0)/2, 
       p(Subscript((Omega), 0)/2)}, {Subscript((Omega), 0), 
       p(Subscript((Omega), 0))}, {(CapitalDelta)/2, 
       p((CapitalDelta)/2)}, {(CapitalDelta), p((CapitalDelta))}}, 
     x)), x) // Simplify
Plot({1/f^2, P}, {x, 0, 2}, PlotRange -> All)

Somewhere on the Internet I read information that in order to get rid of odd degrees you need to include the point {0,0} in the polynomial, which I did with the usual InterpolatingPolynomial command.
Despite this, the odd degrees in the polynomial have been preserved.

at.algebraic topology – relationship between “linear approximation” to immersions and formal immersions

I’m reading these notes

Here, I am regarding $mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M to N$.)

I’ve gathered from context that there is a connection between formal immersions and the sheaf of sections $mathcal{F}(U):=Gamma(V_n(TU) times_{GL_n} mathrm{Imm}(mathbb R^n,N))$ (defined for an arbitrary open $mathbb R^n to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don’t know why, although I assume it agrees in the sense of functor calculus.

My questions are the following: is the topological sheaf of formal immersions isomorphic to $mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the “obvious” maps $mathrm{Imm}(U,N) to mathrm{Imm}^f(U,N),,,, f mapsto (df,f)$?

approximation – uncapacitated facility location problem using local search

I’m studying about UFLP using the book The Design of Approximation Algorithms Ch 9 starting page 233 (there is an electronic free edition), I ran into some unclear steps in the book and need some help with it.

In few words the UFLF deals with finding a subset of facilities from a given set of potential facility locations to meet the demands of all the customers such that the sum of the opening cost for each of the opened facilities and the service cost (or connection cost) is minimized

We can do the following local steps on current solution
1. We can open one additional facility an “add” move
2. We can close one facility that is currently open a “delete” move
3. We can do both of these simultaneously a “swap” move

Let’s assume we have optimal solution and let S* be it’s open
enter image description here

What I need help with :

enter image description here

Proof from the book page 235 :
enter image description here

Not clear what is “open the additional facility i*”
isn’t i* in S* already open ? where does this facility open ? in S ? if so why its marked with * isn’t it for the optimal solution ?

I also need help with :

proof from page 237

enter image description here

enter image description here
enter image description here

If the function returns the nearest facility in S, what is the meaning of choosing i’ ? isn’t it just one facility that is the closest in S ? I mean isn’t R of size 1 ? (line 2)

There is a typo and it’s not lemma 9.3 but lemma 9.1 in the second par, my second question here is also similar to the first question what is i* here ?

Any clarification will be happily welcome, thank you

approximation – How to find x’s for representing a specific type of integral using a quadrature formula?

I have this integral which I want to approx with quadrature formula for some fixed $n$:
$$
intlimits_0^{+infty} x e^{-x} f(x) dx approx sumlimits_{k = 0}^n A_k f(x_k)
$$

I found info about how to find $A_k$:
$$
A_k = dfrac{n!Gamma(alpha + n + 1)}{x_kleft(L_n’^{(alpha)}(x_k)right)^2},
$$

when $Gamma(x)$ – gamma-function and $L_n(x)$ – Laguerre polynom.

But how to find $x_k$?

Approximation force of tensor product splines

To let $ Omega $ be a rectangle in 2-D, and $ f in W ^ s_p ( Omega) $. I'm looking for a result on how good a tensor product spline quasi interpolation is $ Qf $ approximately $ f $. In fact, I need to know how good the partial derivatives of $ Qf $ about that of $ f $.
This should be a standard result. And I was hoping to find it in Schumaker's book.

In section 12.3 of Schumaker (3rd edition) the approximation result, which seems to follow an article by Dahmen-Devore-Scherer from 1980, only gives the approximation force of $ Qf $ to $ f $ (As in the original paper by Dahmen-Devore-Scherer.) No mention is made of how this result can be extended to the derivatives.

I then looked up an earlier (1975) paper by Lyche-Schumaker ("Local Spline Approximation Methods, J. of Approximation Theory, 1975) and it seems to contain the desired result. The picture below shows the result of this paper. With the heavy ones Notations are impossible to understand what it says from what is shown below, but if you only read the first line, the result never seems to be the most pervasive $ p = 2 $ Case. I wonder if there is a typo there.

Does anyone know where to find a good report on this result?

(I solve a variation problem and have to bind it $ L ^ 4 $ Errors in the first derivatives, i.e. $ | d Q f – d f | _ {L ^ 4 ( Omega)} $, to the $ f in W ^ {2} _2 $. So the level of generality of Lyche-Schumaker's result is good if I can only use it.)

Enter the image description here