sql server – How is data of multi valued attribute say phone number of person relation is stored in table?


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fa.functional analysis – Measurable selection involving measure valued random variable

Let $(Omega, mathcal{F}, mathbb{P})$ be a probability space and let $mathcal{M}(mathbb{R}^d)$ be the space of finite signed measures on $mathbb{R}^d$ endowed with the narrow topology (i.e. the initial topology w.r.t. $C_b(mathbb{R}^d)$, the set of real valued, continuous and bounded functions on $mathbb{R}^d$) and the corresponding Borel $sigma$-algebra. Let $mu: Omega to mathcal{M}(mathbb{R}^d)$ be measurable and let $a in mathbb{R}^d$ be a fixed real number. Let us define the multifunction
$$F : Omega rightrightarrows C_{0,1}(mathbb{R}^d):={ varphi in C_0(mathbb{R}^d) mid |varphi|_{infty} le 1 }$$
(where $C_0(mathbb{R}^d)$ is the Banach space of continuous functions vanishing at infinity with the supremum norm) as
$$F(omega) := left { varphi in C_{0,1}(mathbb{R}^d) mid int_{mathbb{R}^d} varphi text{ d} mu(omega) ge a right }.$$

Can we find a measurable selection $f: Omega to C_{0,1}(mathbb{R}^d)$ of $F$, meaning that $f$ is measurable and $f(omega) in F(omega)$ for every $omega in Omega$?

I tried with the Kuratowski–Ryll-Nardzewski measurable selection theorem but I am not able to prove that ${ omega in Omega mid F(omega) cap U }$ is measurable for every $U subset C_{0,1}(mathbb{R}^d)$ open.

Any hint would be really appreciated!

matrix – The objective function should be scalar valued?

I’m trying to solve the following matrix optimization problem:

Minimize[{Norm[ IdentityMatrix[3] - Inverse[G*G], "Frobenius"], 
  G [Element] Matrices[3, Reals]}, G]

I get the following error:

Minimize::objfs: The objective function {{Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2],Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2],Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2]},{Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2],Sqrt[6 Abs[<<1>>]^2+3 <<1>>^2],Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2]},{Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2],Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2],Sqrt[6 Abs[Times[<<2>>]]^2+3 Abs[Plus[<<2>>]]^2]}} should be scalar-valued.

Obviously the objective function is scalar-valued. What am I doing wrong?

numerical methods – Does there exist an transform that allow me to solve arbitrary linear real valued differential equation by solving upper triangular matrix ODEs?

Linear differential equations can be solved using the matrix exponential. However for upper diagonal matrices we can observer the following simplifications (diagonal matrices commute):

$$
x(t)=exp(begin{bmatrix}
d_1 & u_1 & u_2 \
0 & d_2 & u_3 \
0 & 0 & d_3
end{bmatrix}(t-t_0))x_0 = big(begin{pmatrix}
exp (d_1(t-t_0)) \
exp (d_2(t-t_0)) \
exp (d_3(t-t_0))
end{pmatrix} cdot exp(begin{bmatrix}
0 & u_1 & u_2 \
0 & 0 & u_3 \
0 & 0 & 0
end{bmatrix}(t-t_0))big)x_0
$$

This can be simplified further since the strictly upper diagonal matrix is nil potent:
$$x(t)=Big(begin{pmatrix}
exp (d_1(t-t_0)) \
exp (d_2(t-t_0)) \
exp (d_3(t-t_0))
end{pmatrix} cdot big(begin{bmatrix}
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 1
end{bmatrix}+begin{bmatrix}
0 & u_1 & u_2 \
0 & 0 & u_3 \
0 & 0 & 0
end{bmatrix}(t-t_0)+begin{bmatrix}
0 & 0 & (u_2)^2+u_1u_3 \
0 & 0 & 0 \
0 & 0 & 0
end{bmatrix}(t-t_0)^2big)Big)x_0
$$

$$x(t)=Big(begin{pmatrix}
exp (d_1(t-t_0)) \
exp (d_2(t-t_0)) \
exp (d_3(t-t_0))
end{pmatrix} cdot begin{bmatrix}
1 & u_1(t-t_0) & u_2(t-t_0) + ((u_2)^2+u_1u_3)(t-t_0)^2 \
0 & 1 & u_3(t-t_0) \
0 & 0 & 1
end{bmatrix}Big)x_0
$$

This is extremely simple to solve, in particular if it is know that the degree of nilpotency is much smaller than the size of the matrix. Systems with the same degree of nil potency as this one occur in rigid body dynamics.

Does there exists an transform that transforms an arbitrary real linear ode into (one or more) upper diagonal linear (complex) ODEs whose solutions i can transform back into a solution to the arbitrary linear ODE?

The obvious idea of splitting up the matrix in an strictly upper, strictly lower and an diagonal part doesn’t work as the strictly lower part doesn’t commute with (diagonal+strictly upper). I’m not very familiar with matrix transforms or transforms of differential equations.

nt.number theory – Integer valued polynomials and polynomials with integer coefficients

It is well known that the subring $S$ of integer valued polynomials ${mathbb Q}(x)$ is generated by the binomial functions $P_n={x choose n}$. One can ask a dual question: how to characterize the polynomial functions ${mathbb Z} to {mathbb Z}$ which come from an element in ${mathbb Z}(x)$.
I understand one can write down derivative as a (terminating) series of difference derivatives and thus express each coefficients in terms of values of the polynomial but does this (or another) procedure lead to a neat answer?

There is a necessary condition for a polynomial function $f:{mathbb Z} to {mathbb Z}$ to come from an element in ${mathbb Z}(x)$, namely for every $n$ the residue of $f(x) mod n$ depends only on the residue of $x mod n$. This necessary condition is not sufficient but am interested in the subring of elements in $S$
satisfying that necessary condition. Is there a nice set of generators and/or a basis?

plotting – If the level set of real valued function f(x,y,z) is a curve then will ContourPlot3D fail to plot points?

I have the function $f(x,y,z)=frac{x^2}{2}+x y+frac{y^2}{2}+z^2$. The solution set to the equation $f(x,y,z)=0$ is the line parametrized as $(t,-t,0)$. When I use ContourPlot3D on f=0 I get an empty graph. I begin to suspect that ContourPlot3D only works for level sets that are surfaces. Is this correct?

Auto-sorting in google sheets and excel by custom valued sorting

I want my data to be auto sorted by 2 columns every-time I append or change values. Also sorting is not just A to Z or Ascending/Descending but Custom valued sorting as example shown below:

1- High
2- Normal
3- Low

Can anyone help the soonest?

Is every field the residue field of a discretely valued field of characteristic 0?

Let $k$ be a field of positive characteristic $p$. Is there necessarily a discrete valuation ring of characteristic $0$ with maximal ideal $(p)$ and residue field isomorphic to $k$?

multivariable calculus – Newton method for matrix valued function.

We know the Newton method for $x in mathbb{R^n}$ is of the form
$$ x^{t+1} = x^t – (nabla^2 f(x^t))^{-1} nabla f(x^t))$$
How can we define similar update relation when we have a function which is matrix valued?

Let $f:R^{mtimes n} rightarrow R$ be a twice differentiable function. Then The gradient itself is a real $m times n$ matrix and the “hessian” is a tensor. How do I generalize newton method to such case? Can someone suggest some references in which such methods are discussed?

pr.probability – Characterizing ‘very homogeneous’ finitely valued stochastic processes

Fix a positive integer $n$. Let $X = {X_i}_{i in mathbb{N}}$ be a discrete time stochastic process such that each $X_i$ is a ${0,dots,n-1}$-valued random variable. Suppose that the joint probability distributions of any finite sequence of $X_i$‘s only depends on the order of their indices, or to be more precise suppose that $X$ satisfies the following:

  • For any $kin mathbb{N}$, any two increasing sequences of indices $i_0<i_1 < cdots i_{k-1}$ and $j_0 < j_1<cdots<j_{k-1}$, and any function $f : {0,dots,k-1} to {0,dots,n-1}$, $$mathbb{P}(X_{i_0} = f(0) wedge X_{i_1} = f(1) wedge cdots wedge X_{i_{k-1}} = f(k-1)) = mathbb{P}(X_{j_0} = f(0) wedge X_{j_1} = f(1) wedge cdots wedge X_{j_{k-1}} = f(k-1)).$$

Call such a stochastic process ‘strongly homogeneous.’ I’m trying to understand what the set of strongly homogeneous stochastic processes looks like. This is my approach so far:

The set of ${0,dots,n-1}$-valued discrete time stochastic processes can be understood as the set of Borel probability measures on the (compact) space $A = {0,dots,n-1}^{mathbb{N}}$. This is a subset of Banach space of (regular Borel) signed measures on $A$. Let $S$ be the set of such measures corresponding to a strongly homogeneous stochastic process. It’s not hard to check that $S$ is a convex, weak* closed set, and therefore that the Krein-Milman theorem applies to it. This gives us that every element of $S$ is in the weak* closure of the convex hull of the set of extreme points of $S$ (where a point is extreme if it is not the convex combination of any other elements of $S$). This leads to my precise question.

Question: What are the extreme points of $S$?

Note that the extreme points of the set of all probability measures on $A$ is precisely the set of Dirac measures on $A$, but a similar statement here is not sufficient. For instance, if $n=2$, then the only strongly homogeneous Dirac measures are those concentrated on the constant $0$ sequence or the constant $1$ sequence, but convex combinations of these do not have the measure corresponding to a sequence of i.i.d. fair coin flips in their weak* closure.

My suspicion is that the measures corresponding to i.i.d. sequences are the extreme points, but I haven’t proved either that they are all extreme points or that all extreme points are of that form. (Proving that they are all extreme points should be easy, however.)