real analysis – Gradient of convex $f$ is $L$-Lipschitz, then how to prove this $f(x^{k+1}) leq f(x) + (x^{k+1}-x)^Tnabla f(x^{k}) + (L/2)|x^{k+1} – x^k|$?

I have been struggling to prove the following inequality given in this paper–please see eq. (34)

$$f(x^{k+1}) leq f(x) + (x^{k+1}-x)^Tnabla f(x^{k}) + frac{L}{2}|x^{k+1} – x^k|$$ for an iterative scheme — ADMM — where $f$ is a convex function whose gradient is $L$-Lipschitz and $k$ is an iteration number.


Although they show some proof but I can barely follow. Please help to clarify this proof. Below I have pasted for your convenience.
begin{align}
f(x) – f(x^{k+1}) – left( x – x^{k+1} right)^T nabla f(x^{k}) &geq ? tag 1\
f(x^{k}) + left( x – x^{k} right)^T nabla f(x^{k})
– f(x^{k+1}) – left( x – x^{k+1} right)^T nabla f(x^{k}) &= ? tag 2\
f(x^{k}) – f(x^{k+1}) – left( x – x^{k+1} right)^T nabla f(x^{k}) &geq frac{-L}{2} | x^{k+1} – x^{k} |^2 tag 3
end{align}

How did they achieve $(1)$ and also $frac{-L}{2} | x^{k+1} – x^{k} |^2 $ ?

From the Lipschitz condition, the bound that can be used as a starting point is
begin{align}
f(y) leq f(x) + left( y – x right)^T nabla f(x) + frac{L}{2} | y – x |_2^2.
end{align}

If I set $y = x^{k+1}$ and $x = x^{k}$, then we have
begin{align}
f(x^{k+1}) leq f(x^{k}) + left( x^{k+1} – x^{k} right)^T nabla f(x^{k}) + frac{L}{2} | x^{k+1} – x^{k} |_2^2.
end{align}

Then, what to do? Can someone please enlighten me and show the steps in between? I would be so grateful to you.

real analysis – Problems with the Riesz Representation Theorem

I am studying Stanislaw Lojasiewicz book – “An introduction to the Theory of Real Functions” and I do not uderstand few things. I hope you’ll help me. Here is what is written:

G is an open set and $Gamma(G)$ is defined as a class of all continous, non-negative functions on compact space such that $varphileq 1$ and cl${x:varphi(x)neq 0}subset G$,

$lambda(G)=sup_{Gamma(G)}I(varphi).$

Now let $G_n$ be a sequence of open sets. Let $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$. Then there exists $varphiinGamma(bigcuplimits_{i=1}^{infty} G_{i})$ such that $L<I(varphi).$

That’s first thing. Why can we define such constant? How can we know it exist? And why existing of constant $L<lambda(bigcuplimits_{i=1}^{infty} G_{i})$ implies fact that $L<I(varphi).$

I will be really thankful for any advices.

Łojasiewicz, Stanisław, An introduction to the theory of real functions. Transl. from the Polish by G. H. Lawden, ed. by A. V. Ferreira, Wiley-Interscience Publication. Chichester (UK) etc.: Wiley. ix, 230 p. textsterling 24.95 (1988). ZBL0653.26001.

Is it possible to notify search engines about non-official restaurant websites competing with the real site?

What can be done about fake websites popping up pretending to be the real one? It is specific to restaurants where there are services provided by food delivery services such as in the UK; Just-eat and a few others. Just-eat is a food portal platform listing thousands of restaurants.

The scenario is, a restaurant has its own website `essex-restaurant.com and uses services from Just-eat and possibly others.

Because Just-eat wants users to go to their platform they make fake websites such as essex-takeaway.com and other platforms make similar such as essex-takeaway.co.uk.

They end up with 3 to 4 websites all competing against each other. This results in the original restaurant website competing with these fake websites to rank.

Is there a way to tell search engines about fake websites, contacting the companies to remove it is not an option as it has been tried without success.

real analysis – Khintchine’s Inequality variant

Let $f_nin L_{infty}((0,1))$ and $(r_n)_n$ be the Radermacher sequence. For each $ninmathbb{N}$ we define the function $g:(0,1)^2tomathbb{R}$ by $g_n(x_1,x_2)=r_n(x_1)f_n(x_2)$. Show that for every $infty>pgeq 1$ there exists a constant $c(p)$ such that: $|sum_{n=1}^kg_n|_{L_p((0,1)^2)}geq c(p)|(sum_{n=1}^k{f_n}^2)^{frac{1}{2}}|_{L_p((0,1))}$.

I can see that this is essentially Khintchine’s Inequality with $f_n$‘s relacing the constants so I suspect that a proof might mimic the original proof but I am not certain how to proceed. Also is there any way to estimate $c(p)$?

real analysis – Point-wise convergence of function

I have been trying out some questions on sequence of functions.In one of those questions,I am supposed to find the point-wise limit of the following sequence of functions defined on ($0,1$) as
$$f_n(x)=begin{cases}
n^2x, text{if 0$le$$x$$le$$frac{1}{n}$}\
-n^2x+2n, text{if $frac{1}{n}$$le$$x$$le$$frac{2}{n}$}\
0, text{if $frac{2}{n}$$le$$x$$le$1}
end{cases}$$

Given this,I am unable to find the limit of f$_n$(x) as n tends to infinity.

Help please!

What's the equivalent of click bait in the real world?

Hello friends,

What’s the equivalent of click bait in the real world?

Convert string to real date time value?

Here is my current input:

"14.05.20
12.37.01"

And yes it seems there is a line break in the cell…

Here is example sheet: https://docs.google.com/spreadsheets/d/14nLk3nVaeknb9T1JS1EIdH5bqkENuj4OgxF0mPCrzmU/edit#gid=0

How could this be turned into an actual date and time?

locking – SQL Server: When is a real shared (S, not IS) lock acquired on a page of a clustered index?

All the explanations I find seem to indicate that, without special hints – which is the case in our software -, shared locks are only acquired for keys, with IS locks at page and object level (and, yes, an S lock on the database).

Lock escalation of S row (key) locks escalates to the table (object) level, so no S page locks can result from this, if I’m correct here.

And foreign key constraint checking also gets (transaction-wide) S locks on keys, if I understand it correctly (see my other question for this).

However, we see lots of S (not IS) page locks in a simple ETL process (doing simple UPDATEs/INSERTs and some DELETEs from a connected server) – where could they come from?

Thanks!
Harald M.

real analysis – minimizing weighted length of closed curves

Let $mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at the origin. Then we define the weighted length of curves in the family as
$$L(gamma):= int_{gamma} frac{2}{1+|z|^2}d|z|,$$where $d|z|$ is the classical length element.

My question is that, is it true that $inf_{gamma in mathcal{A}}L(gamma) =pi$?

I have done some computations for some curves with explicit formulas. For example, if $gamma$ is a unit circle centered at $z=1$, then $gamma in mathcal{A}$ and $L(gamma)=4pi/sqrt{5}$. Also, it seems that when $gamma$ is more and more closed to the line segment $(0,1)$ with multiplicity 2, the weighted length $L(gamma)$ is getting smaller and approaching to $pi$.

Any ideas or comments are really appreciated. Thank you very much for your time.

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