real analysis – Prove the existence of local minimum of a multi-variable absolute value function.

I have the following function
$$H(mathbf{x}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}|+cdots+a_{n}|x_{n}-A_{n}|
+b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}|+cdots+a_{n}|x_{n}-B_{n}|
+c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+cdots+c_{n}|x_{n}-x_{n-1}|+c_{n+1}|x_{n+1}-x_{n}|$$

where all coefficients are positive.

For two dimensional case, I can plot $$H(x_{1},x_{2}) = a_{1}|x_{1}-A_{1}|+a_{2}|x_{2}-A_{2}|
+b_{1}|x_{1}-B_{1}|+b_{2}|x_{2}-B_{2}|
+c_{1}|x_{1}-x_{0}|+c_{2}|x_{2}-x_{1}|+c_{3}|x_{3}-x_{2}|$$

and it indeed has only one minimum,

enter image description here

However, I cannot prove it in a more general case. Does anyone once had encountered such problems?

real analysis – Can you give me an example of such family of functions in $L^{p}(mathbb{R})$?

I’m prepearing for the functional analysis test and I have a doubt on $L^{p}$ spaces. Kolmogorov criterion characterizes precompactness in $L^{p}(mathbb{R}^{d})$ for any $pneq +infty$, and says the following:

$Hsubseteq L^{p}(mathbb{R}^{d})$ is precompact if and only if:

  1. $H$ is bounded

  2. $limlimits_{Rrightarrow 0}hspace{2mm}suplimits_{fin H}int_{|x|geq R}|f(x)|^{p}dx=0$

  3. $limlimits_{|a|rightarrow0}hspace{2mm}suplimits_{fin H}|T_{a}f-f|_{p}=0$

Our prefessors told us that usually this criterion fails on the third point, but so far I did a lot of exercises and it always worked (up to mistakes). Since it can’t always be like that, as any arbitrary collection of functions can’t be precompact, I’m asking:

Can you give me an example where such criterion fails? Is there a general idea one should check to see “at first glance” when this criterion indeed fails?

real analysis – if $lim _{xrightarrow infty}{f'(x)}=0$ then does $lim_{x rightarrow infty}{f(x)}$ exist in the broad sense

Let $f$ be a differentiable a function in $mathbb{R}$, and let $lim _{xrightarrow infty}{f'(x)}=0$

Does $lim_{x rightarrow infty}{f(x)}$ exist in the broad sense?

I’m really lost here. This exercise is from a section on MVT, and intuitively it seems to be correct, but I can’t seem to find a lead. If someone could just give me a hint that would be great.

So far my best shot has been using Heine’s definition of the limit, but no dice.

real analysis – Norm of a smooth function is smooth?

My lecture notes seem to use this, although I can’t shake the feeling that it’s wrong. Let $phi: (a,b) to mathbb{R}^n$ for $n ge 2$ be smooth (infinitely differentiable) satisfying $|phi ‘(t)| neq 0$ for all $t in (a,b)$. Then, $|phi'(t)|$ is also a smooth function.

Does anyone have a nice proof for this? (My first ‘counterexample’ was $log$, but it doesn’t satisfy the condition $|phi ‘(t)| neq 0$.) Also, is $|phi(t)|$ a smooth function too?

Although this is from a course in elementary differential geometry, my question is essentially a calculus question…

real analysis – Convergence of the Power Method

Given a vector $x^{(0)}$ with $left|x^{(0)}right|_{2}=1$, I wish to show that the power method :
$$
x^{(k)}=frac{A x^{(k-1)}}{left|A x^{(k-1)}right|_{2}}, quad mu^{(k)}=left(x^{(k)}right)^{T} A x^{(k)}
$$

generates the pair $left(x^{(k)}, mu^{(k)}right)$ such that $displaystyle x^{(k)}=frac{A^{k} x^{(0)}}{left|A^{k} x^{(0)}right|_{2}}$ for any matrix $A$.

My attempt by induction:

Base Case: For $k=1, displaystyle x^{(1)}=frac{A x^{(0)}}{left|A x^{(0)}right|_{2}}$ by definition.

For the inductive step, it seems trivial but I can’t seem to figure it out, any help is much appreciated!

real analysis – formulate a mathematical solution to answer a multiple choice question

If I have a multiple choice question where $int_ {e} ^ {2e} frac {x} {ln(x)} dx $ is equal to:

a) 6.55…

b) 7.28…

c) 7.93…

d) 7.96…

e) 8.10…

f) 10.10…

the … means that there is integer number.

Find a method based on the ZF axiom to find the right answer?

I have already found a method but with the definition of a non-mathematical Z axiom.

who can solve any MCQ of this kind, poropser by geniuses of math for geniuses in math.

conversion – How to convert a real number to a string only in the original length?

The following function seems to do what you want

realToString[x_Real] := StringReplace[ToString[x,InputForm],
   StringExpression[a__~~"`"~~__~~"*^"~~b__]->a~~"*^"~~b];

For example try

FullForm[realToString@
     2.9525730420542015686180548`15.954589770191005*^1798]

which returns

"2.9525730420542015686180548*^1798"

If you want to limit the number of decimal digits then try

realToString[x_Real, nd_Integer:18] := Module[{a, b, ab=
  First[List@@StringReplace[ToString[x, InputForm],
  StringExpression[a__~~"`"~~__~~"*^"~~b__] -> {a ,b}]]},
  {a, b}=ab; StringTake[a, Min[StringLength@a, nd+1]]~~"*^"~~b];

For an example of this try

FullForm[realToString[
 2.9525730420542015686180548`15.954589770191005*^1798, 4]]

which returns

"2.952*^1798"

There are other ways to get Mathematica to do what you want,
but this approach is more user programmable in my opinion.

lighting – Why real time lightning fails to show shadows on terrain in Unity?

Flicking shadows on terrain can often be fixed by adjusting the “Bias” and “Normal bias” of the light source. However, adjusting those settings does have side effects, so you should pay attention to how the settings affect your shadows in different areas (especially on high-detail objects close to the camera). In the worst-case scenario, you might have to use tricks, like rendering nearby objects with one camera/directional light, and distant objects with a different camera/directional light, to get close and distant objects to both look right.

Other possible causes of flickering shadows:

  • The camera’s near clipping plane is set too low (it should usually be at least 0.3 and may need to be set higher if the far clipping plane is very large).
  • The camera is very far away from the world center (more than 100,000 units if I remember correctly).

Baked lighting doesn’t cast real-time shadows (e.g. character shadows) because it’s baked. Baking your lighting essentially renders the shadows into a texture that is applied on top of the meshes. For traditional baked lighting, there’s very little performance cost but it’s completely static. However, you can combine real-time and baked lighting using mixed lighting.

Baked and real-time lighting are important concepts, so if you weren’t already familiar with the differences between them, you should take some time to thoroughly read the Unity documentation on the different lighting modes.