multivariable calculus – approximating unbounded integral in higher dimension

In multivariable calculus, I usually encounter this: if $f: mathbb{R}^n rightarrow mathbb{R}$, we can define $int_{mathbb{R^n}} f(x) dx = lim_{R rightarrow infty} int_{B(0,R)} f(x) dx$ if this limit exists where $B(0,R)$ is the ball centered at $0$ of radius $R$. However, for the case $n=1$, this is not true: consider the function $f(x)=x$. We know that $int_{mathbb{R}} x dx$ is undefined, but $lim_{R rightarrow infty} int_{B(0,R)} x dx = 0$. Can anyone explain to me why?

fa.functional analysis – Are there references to functional variants of the Unbounded Knapsack Problem?

Looking for a version of the following problem, extended to solutions in $ell^{infty}(mathbb{N})$

Unbounded Knapsack Problem

$ max_{x_1,…,x_n} sum_{i=1}^n v_ix_i$

$text{ subject to }$

$sum_{i=1}^n w_ix_i leq W$

$w_i,v_i in mathbb{R}^+ ;;forall i$

$x_i in mathbb{N}$

What I’d like to examine is the extended version

“Functional” Unbounded Knapsack Problem

$ max_{x_1,…,x_n} langle v,x rangle$

$text{ subject to }$

$langle w,x rangle leq W$

$w,v in ell^{infty}:w_i,v_i > 0;forall i in mathbb{N}$

$x in mathbb{N}^{infty}$

I’m having a hard time finding any existing literature on this problem (or its continuous cousin defined on $L^{infty}$).

More generally, is there something akin to Calculus of Variations (except we’re optimizing functionals defined on $ell^{infty}$). Is there a variant of the Euler-Lagrange equation for functionals in sequence spaces?

set theory – Unbounded set in $V[G]$ has an unbounded subset in $V$?

This is a repost of the same question on math.SE, which received several comments but no answers/comments on the first question.

Suppose $kappa$ is a cardinal preserved in the generic extension $V(G)$. Let $Y subseteq kappa$ be an unbounded set in $V(G)$. Does there always exist an $X in V$ such that $X subseteq Y$ and $X$ is unbounded?

This question comes from Section 18 of James Cummings’ Singular Cardinal Arithmetic. In this context, he forced with the Levy collapses $operatorname{Col}(omega,delta^{+omega}) times operatorname{Col}(delta^{+omega+2},<kappa)$, and fixed “$X subseteq gamma$ unbounded of order type $delta_V^{+omega+1}$…”. In the next line, he then says that “However, it is easy to see that there is $Y in V$ with $Y subseteq X$ unbounded”.

This observation is certainly not easy for me, so my questions are:

  1. Why do such a $Y$ exist?
  2. Is the existence of such a $Y$ limited to forcing with Levy collapse?

Any help is appreciated.

Note: Andreas Blass provided a counterexample to the second question, which is a Mathias real.

functional analysis – May I ask if positive definite second order elliptic operator over an unbounded domain can give rise to compact semigroup?

Let $A$ denote some positive definite second order elliptic operator which is defined over $L^2(Omega)$ with domain $D(A) = H^{2m}(Omega) cap H^{m}_{0}(Omega)$. Here $Omega$ is a bounded domain in $R^m$. Assume that the coefficient functions of $A$ are nice enough, then it can be shown that the semigroup generated by $A$ is compact.

Now what if $Omega$ is $(0, infty)$? Is it possible that $A$ can still generate compact semigroup, if replacing the function spaces in the bounded case by some weighted function spaces? If so, are there any references please?

Thank you!

reference request – Invariant on C*-algebras-number of closed unbounded derivation it admitted

In working of the unbounded derivation of C*-algebras. I observed the following: For topological manifold $M$, the number of closed, linear independent, unbounded derivation it admitted on $C(M)$ is exactly the dimension of $M$.

Of course this is true for smooth manifold. But I found that it may holds for arbitrary manifold. I try to google it but seems like no positive results. I would like to know if my result is known and well-studied. Thank you in advance.

fa.functional analysis – Regularity of solution of a Fokker-Planck PDE with unbounded drift

Let $A$ be a positive definite symmetric matrix and $bin C^1(mathbb R^d!times!(0,infty);mathbb R^d) ,cap, C(mathbb R^d!times!(0,infty);mathbb R^d)$. Consider the parabolic PDE
$$ partial_t rho(x,t) ,=, nabla_{!x}cdotbig(,A;nabla_{!x}rho(x,t),+,b(x,t);rho(x,t),big)$$
for $xinmathbb R^d$, $t>0$ and initial condition $rho(x,0)=nu(x)$ continuous such that $nugeq0$ and $int_{mathbb R^d}nu(x),d x=1,$.

Assume, e.g., that $b,,text{D}_x b$ have polynomial growth in $x$. Can I conclude that the solution $rhoin C^1(mathbb R^d!times!(0,infty))cap, C(mathbb R^d!times!(0,infty))$ ?

I cannot find results about the regularity in classical sense of solutions of these parabolic PDEs. Bogachev, Krylov, Rochner, Schaposhnikov in their book “Fokker-Planck-Kolmogorov equations” study the measure solutions of these (and much more general) parabolic PDEs, but I couldn’t find such a result about regularity in classical sense.

dg.differential geometry – Area-minimising hypersurface with unbounded area growth

Let $T$ be an $n$-dimensional area-minimising hypersurface in $mathbf{R}^{n+1}$. If $T$ has bounded area growth in the sense that there is a constant $C > 0$ so that $mathcal{H}^n(T cap B_R) leq C R^n$ for all $R > 0$, then there are rigidity theorems for $T$. For example, when $n leq 6$ then the work of Simons (1) implies that $T$ must be an $n$-dimensional plane. (In larger dimensions there are singular area-minimising hypercones.)

Question. Is there an example of an area-minimising hypersurface with unbounded growth? Could such an example exist in low dimensions, when $n leq 6$? What about $n = 2$?

(1) James Simons. Minimal varieties in Riemannian manifolds. Annals of Mathematics, Second Series, Vol. 88, No. 1 (1968), pp. 62-105.

reference request – Can the subdifferential become unbounded at interior points?

Consider $f: mathbb{R}^n to overline{mathbb{R}}$ a lower-semicontinuous, proper, closed and convex. My question is, can the subdifferential of $partial f$ be unbounded in the interior of $text{dom}(f)$ (all points where $f(x) < infty$?

I was pretty convinced that it can only be unbounded on the boundary and not the interior until I found the following counter-example (unfortunately I’ve lost track of where I encountered it, possibly in some online notes by Dimitri Bertsekas):

Suppose $f(x) = |x|^2, x = (x_1, x_2, x_3)$ is defined on an affine hyperplane of $mathbb{R}^3$, then the subdifferential necessarily includes a ray that is perpendular to that hyperplane at any point in the interior of the domain, hence the subdifferential of this function at any point (interior or not) is unbounded.

Can someone please verify or provide guidance on this matter?

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swift – Unbounded Memory Growth IOS App (CFString) – How to debug?

I have an IOS app that displays images and text in a UITableView. When I scroll up and down I get this unbounded memory growth. I have used Allocations to attempt to find out where this memory growth comes from. Please see below. From what I can see, CFString takes up a huge amount of memory. It doesn’t appear to be from my code(based on StackTrace of CFString) as seen in Figure 2.
Please can someone advise? I’m really stuck.

Figure 1
I am looking at CFString and the persistent memory is huge.

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Figure 2
Clicking on CFString:

enter image description here