banach spaces – Integral representation of a bounded bilinear functional

For a given $f in L^{p}((0,1))$, the map $g mapsto B(f,g)$ is a bounded linear functional on $L^{q}((0,1))$. Therefore, (assuming $q in (1,infty)$), there is a $T(f) in L^{q’}((0,1))$ such that
B(f,g) = int_{0}^{1} (Tf)(x) g(x) , dx quad text{if} , , g in L^{q}((0,1)).

Here $q’$ is the dual exponent of $q$ (so that $frac{1}{q} + frac{1}{q’} = 1$), and I have used the well-known result concerning duality in $L^{p}$ spaces (sometimes called Riesz Representation). $T(f)$ is uniquely defined and an exercise shows that the assignment $f mapsto T(f)$ is linear. The boundedness of $B$ implies $T$ is also bounded. (In fact, $|T(f)|_{q’} leq C |f|_{p}$, with $C$ the constant in your question.)

I will have to think about the $q = infty$ case.

linear algebra – Derivative of trace uniformly bounded

I have that $boldsymbol{Omega{(boldsymbol{theta})}}$ is symmetric, positive definite and uniformly bounded where $boldsymbol{theta} in boldsymbol{Theta}$ with $boldsymbol{Theta}$ a compact subset of $R^{n}$ and $n$ is fixed.

I want to show that $frac {partial}{partialboldsymbol{theta’}}trace (
is O(1) uniformly on $boldsymbol{Theta}$, where $boldsymbol{theta_{0}} in boldsymbol{Theta}$ .

For each j=1,2,…,n does it hold that

$frac {partial}{partial{theta_{j}}}trace (
boldsymbol{Omega{(boldsymbol{theta})}}^{-1}boldsymbol{Omega{(boldsymbol{theta_0})}})=trace (frac {partial}{partial{theta_{j}}}(boldsymbol{Omega{(boldsymbol{theta})}}^{-1}boldsymbol{Omega{(boldsymbol{theta_0})}}))=trace (
-boldsymbol{Omega{(boldsymbol{theta})}}^{-1}boldsymbol{Omega{(boldsymbol{theta_0})}} frac {partial{Omega{(boldsymbol{theta})}}}{partial{theta_{j}}}boldsymbol{Omega{(boldsymbol{theta})}}^{-1}boldsymbol{Omega{(boldsymbol{theta_0})}})$

and then since $boldsymbol{Omega{(boldsymbol{theta})}}$ is uniformly bounded, so is its inverse, and its first derivative (or not necessarily and that the first derivative is uniformly bounded needs to be an assumption?) and therefore the whole trace on the right hand side. Subsequently, if the right hand side trace is bounded uniformly for each j, then so is $frac {partial}{partialboldsymbol{theta’}}trace (

domain driven design – Reusing aggregate root key accross bounded contexts?

As the question states, is this bad practice?
I have a User aggregate root in the bounded context of Identity for authenticating the user. In this bounded contexts I have fields for the User related to identification of the User e.g. email, salted pw and so on.

I also have a generic subdomain for handling notifications. In this context a User is a Notificant. In this context, the Notificant has fields for e.g. the number of unread notifications, lastRead etc.

Is it good to reuse the User id in this case, as I know there is a 1-to-1 correspondence between a User and Notificant? Or should I have a field in the Notificant root referencing the User? It feels redundant, because then I have to make a lookup to map between them when I know their relationship is symmetric.

Find a linear bounded automaton that accepts the language $L = { a^{n!} : n geq 0 }$

I need to construct linear bounded automaton for the language $L = { a^{n!} : n geq 0 }$. I know how LBA functions, however, I don’t have a thought how it can check the n! that to in the power of a. I might want to hear a few suggestions, as I am experiencing difficulty in developing the specific LBA for it.

reference request – Derivatives of Measures of Bounded Variation on Intervals

Investigating an abstract Cauchy problem on the space of measures with bounded variation I came up with the following space:

Let $operatorname{BV}(a,b)$ the space of all functions $f:(0, 1) to mathbb C$ with bounded variation, i.e., the supremum of $sum_{i = 1}^n lvert f(x_i) – f(x_{i – 1}) rvert$ over all finite partitions $lbrace x_0, dots, x_nrbrace$, $n in mathbb N$, is finite. Now consider the “Sobolev measure space”
$$mathrm{M}^1(a,b) := lbrace mu in mathrm{M}(a,b) : exists , f in operatorname{BV}(a,b) : mu = f , mathrm d xrbrace. $$
Since each function of bounded variation is differentiable almost everywhere, it is possible, to define the operator
$$A := frac{mathrm d}{mathrm d x}, quad Amu := f’ , mathrm d x, quad D(A) := mathrm{M}^1(a,b). $$
I would like to know if operators in spirit of $A$ and spaces in spirit of $mathrm{M}^1(a,b)$ are already covered in the existing literature.

Remark: A concept closely related to my questing is the so called Skohorod differentiability that can be found for instance in Bogachev, Vladimir I., Differentiable measures and the Malliavin calculus, ZBL1247.28001. However, the definition of this kind of differentiability is then defined only for measures on $mathbb R$ via the duality to $C_b(mathbb R)$. It is then proven that a measure on $mathbb R$ is Skohorod differentiable if and only if it has a density of bounded variation, which was my main motiviation to define the space above. But I cannot believe that I am the first to come up with the pretty simple idea to consider this space.

complexity theory – Is the While programming language with bounded number of variables Turing complete?

In our lecture it was proven that WHILE-Programs are turing-complete. In short a WHILE-Program only allows the following:

Variables             x1,x2,...  (initially all 0 and can only be positive)
Constants c           0,1,2,3,...
Assignment            xi := c
Addition/Subtraction  xi := xj + c, xi := xj - c
while-loops           while(xi != 0) { ... }

Now we restrict the Language to k-VARIABLE-WHILE meaning that only a bounded number of variables can be used.
Is 100-VARIABLE-WHILE turing-complete? I think it isn’t but I am not quite sure which function I can construct that needs more than 100 variables.

Find area bounded by four curves

I need to find area bounded by four curves:
y=x2, y=2x2, x+y=1, x+y=2. How do I do that?

fa.functional analysis – Duality of finite signed measures and bounded continuous functions

Let $E$ be a metric space, $C_b(E)$ denote the space of bounded continuous functions $Etomathbb R$ (equipped with the supremum norm), $mathcal M(E)$ denote the space of finite signed measures on the Borel $sigma$-algebra $mathcal B(E)$ (equipped with the total variation norm) and $$langle f,murangle:=int f:{rm d}mu;;;text{for }(f,mu)in C_b(E)timesmathcal M(E).$$

I’m searching for a reference of a functional analytic proof showing that a linear functional $varphi$ on $C_b(E)$ is continuous if and only if $$existsmuinmathcal M(E):varphi=langle;cdot;,murangletag1.$$

All similar results (e.g. in Bogachev’s Measure Theory) I’ve found are either treating way more general settings or establish the result in a way where I got the feeling that the arguments can be significantly simplified once one is aware of certain basic results on locally convex topologies arising from duality pairings.

In general, if $X,Y$ are $mathbb R$-vector spaces, $langle;cdot;,;cdot;rangle$ is a duality pairing between $X$ and $Y$ and $sigma(X,Y)$ denotes the topology on $X$ generated by $$p_y(x):=|langle x,yrangle|;;;text{for }xin X$$ for $yin Y$, we know that for $varphiin X^ast$

  1. $varphi$ is $sigma(X,Y)$-continuous;
  2. $exists kinmathbb N:exists y_1,ldots,y_kin Y:exists cge0:|varphi|le cdisplaystylemax_{1le ile k}p_{y_i}$;
  3. $exists yin Y:varphi=langle;cdot;,yrangle$

are equivalent.

Maybe we need to impose further assumptions (e.g. restrict ourselves to the subspace $mathcal R(E)$ of Radon measures in $mathcal M(E)$), but I think we should be able to find a proof for the desired claim using the aforementioned equivalence.

Maybe a result similar to Bogachev, but in the present simpler setting, can be established: It holds $(1)$ for a Radon measure $mu$ if and only if $varphi$ satisfies $$forallvarepsilon>0:exists Ksubseteq Etext{ compact }:forall fin C_b(E):left.fright|_K=0Rightarrow|varphi(f)|levarepsilonleft|fright|_inftytag2.$$

reference request – Diameter of finite and bounded martingales

I am interested in the solution of the following problem: for a fixed $n$, find

$$sup mathbb{E}max_{1le i<jle n}|M_i-M_j|^p,$$
where supremum is taken over all the martingales
$$M_1, M_2, dots, M_n,$$
such that $ M_nle 1 $ a.e. I am mostly concerned with the case of $ p $ equal to $ 1 $ or $ 2$.

This problem looks as if it must have been solved a long time ago. Have You ever encountered it? I know that there are similar articles for continues time case (with cadlag assumptions) but I would like to find a discrete version as stated above.

I would be very grateful if someone would point me towards an appropriate book or paper.

real analysis – Bounded functions and inclusion on the $L^{p} $ spaces

Let $f in L^p(X) $ be a bounded function, show that $forall q geq p, ; f in L^q(X)$

I’m aware that this result holds for the case where $mu(X)<infty$ but I can’t see how to prove this for when the measure of the whole space is not necessarily finite. I’ve tried using Holder’s inequality but that didn’t seem to help much.