## Probability – existence and uniqueness of a stationary measure

The same question was also asked on MSE https://math.stackexchange.com/questions/3327007/existence-and- uniqueness-of-a-stationary-measure.

Recently I asked the following question about MO Attractors in Random Dynamics.

To let $$Delta$$ be the interval $$(-1.1)$$Then we can look at the probability space $$( Delta, mathcal {B} ( delta), nu)$$, from where $$mathcal {B} ( Delta)$$ is the Borel $$sigma$$algebra and $$nu$$ is equal to half the Lebesgue measure.

Then we can equip the room $$Delta ^ { mathbb {N}}: = {( omega_n) _ {n in mathbb {N}}; \ omega_n in Delta \ forall n in mathbb {N} }$$ with the $$sigma$$-Algebra $$mathcal {B} ( Delta ^ { mathbb {N}})$$ (Borel $$sigma$$-algebra of $$Delta ^ { mathbb {N}}$$ induced by the product topology) and the probability measurement $$nu ^ { mathbb {N}}$$ in measurable space$$( Delta ^ { mathbb {N}}, mathcal {B} ( Delta ^ { mathbb {N}}))$$, so that
$$nu ^ { mathbb {N}} left A_1 times A_2 times ldots times A_n times prod_ {i = n + 1} ^ { infty} Delta right) = nu (A_1) cdot ldots cdot nu (A_n).$$

Now let it go $$sigma> 2 / (3 sqrt {3})$$ be a real number and define
$$x _- ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3+ sigma = x,$$
$$x _ + ^ * ( sigma) = text {The unique real root of the polynomial} x ^ 3- sigma = x,$$
that's easy to see $$x _ + ^ * ( sigma) = -x _- ^ * ( sigma)$$,

We can then define the function
$$h: mathbb {N} times Delta ^ mathbb {N} times (x _- ^ * ( sigma), x _ + ^ * ( sigma)) to (x _- ^ * ( sigma), x _ + ^ * ( sigma)),$$
in the following recursive way,

• $$h (0, ( omega_n) _ {n}, x) = x$$. $$forall ( omega_n) _n in mathbb {N}$$ and $$forall x in mathbb {R}$$;
• $$h (i + 1, ( omega_n) _ {n}, x) = sqrt (3) {h (i, ( omega_n) _ {n}, x) + sigma omega_i}.$$

That's how we are for everyone $$x in mathbb {R}$$ and $$( omega_n) _n in Delta ^ mathbb {N}$$Define the following order
$$left {x, sqrt (3) {x + sigma omega_1}, sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2}, sqrt ( 3) { sqrt (3) { sqrt (3) {x + sigma omega_1} + sigma w_2} + sigma w_3}, ldots right }.$$

Now define the following family of Markov kernels
$$P_n (x, A) = nu ^ { mathbb {N}} left ( left {( omega) _ {n in mathbb {N}} in Delta ^ { mathbb { N}}; h (n, ( omega_n) _ {n in mathbb {N}}, x) in A right } right).$$

A probability measure $$mu$$ in the $$((x _- ^ * ( sigma), x _ + ^ * ( sigma)), mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma) ))$$ is a called stationary measure, if

$$mu (A) = int _ {(x _- ^ * ( sigma), x _ + ^ * ( sigma))} P_1 (x, A) text {d} mu (x) ; forall A in mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma))),$$
from where $$mathcal {B} ((x _- ^ * ( sigma), x _ + ^ * ( sigma)))$$ is the Borel $$sigma$$-Algebra. Besides, once $$(x _- ^ * ( sigma), x _ + ^ * ( sigma))$$ it is easy to prove that there is at least one stakionäre measure.

The answer I received to MO suggests that there is only one stationary measure.

Does anyone know if that's true? An indication of such a result is sufficient for my purposes.

## postgresql – Ensure uniqueness of values ​​in bigint arrays created after merging two bigint arrays

What is the most efficient way to get the uniqueness of the values ​​in `bigint` Array created by merging 2 others `bigint` Arrays?
For example, this operation `select ARRAY[1,2] || ARRAY[2, 3]` should give as a result `1,2,3`, I checked the extension `intarray` and see, it does not work with `bigint`,

## Probability – Uniqueness of the martingale problem for the Levy operator

Consider the following Levy-type operator:
$$L_t varphi (x) = int_ {R ^ d} big ( varphi (x + z) – varphi (x) -1_ {| z | leq 1} z cdot nabla varphi (x) big) kappa (x, z) nu (dz), quad forall varphi in C_c ^ 2 (R ^ d),$$
from where $$nu$$ is a symmetric delivery measure, $$kappa ( cdot, z) in C ^ infty (R ^ d)$$ (smooth in $$x$$but maybe degenerate and unlimited) and for everyone $$x$$.
$$int_ {R ^ d} (| z | ^ 2 wedge1) kappa (x, z) nu (dz) < infty.$$
Then the martingale problem is for $$L$$ has a unique solution? I think the conclusion should be correct, because the coefficient is smooth, but I can not find a reference. Thanks for your help.

## Algorithms – Latest hash functions to test their speed and uniqueness

I'm a tenth grader and need to do a research project. I'm doing a scientifically fair experiment on "The Impact of Different Cryptographic Hash Functions on Decryption Times and Singularity". To test this, I wanted to know what software / platform I should use to test these different hash functions, and what are the latest hash functions I should use. I only need 3-5 functions and my control is SHA-256 because it is the most used one. Which other newer hash functions should I use and what makes them special?

Here's a link to a good amount of data I found, but they're pretty outdated and I wanted to know if there are newer functions: Which hashing algorithm works best for uniqueness and speed?

This can be a scientifically fair experiment as I test 200,000 words with some hash functions. I basically test this. I'm sorry if you got confused. The output is what I will insert into my data table.

Many thanks

## SQL Server – Index Uniqueness overhead

I have had a constant debate with various developers in my office about the cost of an index and whether the uniqueness is beneficial or costly (probably both). At the heart of the problem are our competing resources.

background

I have previously read a discussion in which a `Unique` Index is no additional cost to maintain since a `Insert` The operation implicitly checks where it fits into the B-tree and, if a duplicate is found in a non-unique index, appends a unique to the end of the key but otherwise inserts it directly. In this sequence of events a `Unique` Index has no additional costs.

My colleague fights this statement by saying so `Unique` is enforced as a second operation after finding the new position in the B-tree and is therefore more expensive to maintain than a non-unique index.

In the worst case, I've seen tables with an identity column (inherently unique), which is the table's clustering key, but explicitly stated to be ambiguous. On the other hand, my obsession with uniqueness is the worst, and all indices are created as unique. If it is not possible to define an explicitly unique relationship to an index, I append the table's PK to the end of the index to make sure the index is unique. Uniqueness is guaranteed.

question

The uniqueness has extra costs for the back end of one `Insert` compared to the cost of maintaining a non-unique index? Secondly, what is wrong about appending the primary key of a table to the end of an index to ensure uniqueness?

Example table definition

``````Create the #test_index table
(
id int not null identity (1, 1),
dt datetime not null default (current_timestamp),
val varchar (100) not null,
is_deleted bit not null default (0),
Primary key not grouped (ID descending),
uniquely grouped (dt desc, id desc)
);

Create an index
[nonunique_nonclustered_example]
on #test_index
(is deleted)
include
(Val);

Create a unique index
[unique_nonclustered_example]
on #test_index
(is_deleted, dt desc, id desc)
include
(Val);
``````

## Uniqueness of the gamma function with the Euler reflection formula

Is the gamma function unique? $$f (x)$$ so that
$$f (x) f (1-x) = dfrac { pi} { sin pi x}?$$
I looked up the Bohr Mollerup theorem, although I wonder if it is possible to uniquely characterize the gamma function with less than 3 conditions.

## real analysis – uniqueness for autonomous ODE with discontinuous coefficient

Consider the following scalar ODE: for all initial data $$x (0) =: x_0 in (a, b) subset mathbb {R} _ +$$.
$$begin {equation} dot {x} (t) = begin {cases} 1-x, & text {if} xc end {cases} \ end {equation}$$
from where $$c in (a, b)$$, We search for $$textbf {Caratheodorys Solution}$$ for all $$t geq 0$$,

For the existence:

i): If $$x (0) in (a, c)$$: I found a solution
$$x (t) = 1- (1-x_0) e ^ {- t}$$ to the $$t < tilde {t}$$ and $$x (t) = c$$ to the $$t geq tilde {t}$$ from where $$tilde {t}$$ is the time $$x (t)$$ reached $$c$$ and notice that $$x (t)$$ is strictly increasing $$(a, c)$$;

ii): If $$x (0) = 0$$is a solution $$X (t) = 0$$ for all $$t geq 0$$;

iii): if $$x (0) in (c, b)$$is a solution
$$x (t) = x_0e ^ {- t}$$ to the $$t leq has {t}$$ and $$x (t) = c$$ for all $$t geq has {t}$$, from where $$has {t}$$ is the time $$x (t)$$ reachese $$c$$, Note that $$x (t)$$ is strictly decreasing while it is in $$(c, b)$$,

My main concern, however, is the $$textbf {uniqueness}$$ the Caratheodory solution for this ODE. Are these solutions unique? If not, can we impose mild conditions to restore uniqueness?

## Number Theory – Strong uniqueness of Euler's Totient function

To let $$f: mathbb N to mathbb C$$ be an arithmetic function. Define $$varphi_f (n)$$ according to the following formula:

$$varphi_f (n) = sum _ { substack {k leq n \ (k, n) = 1}} f (k).$$

In other words, $$varphi_f (n)$$ is the sum of $$f$$ about totatives of $$n$$, For example when $$f = delta_1 (n)$$ then $$varphi_f (n) = 1$$, if $$f = 1$$ then $$varphi_f (n) = varphi (n) –$$ Euler's Totientenfunktion. Accept that $$f$$ is completely multiplicative. Investigation of the first values ​​of $$f$$ (up to $$about 40$$) shows that if $$varphi_f$$ is only multiplicative if $$f = 1$$ or $$f = delta_1$$,

Obviously, the direct analysis of about 40 cases is not the most revealing way to prove this kind of assertion. This leads us to a more general question. Let's call an arithmetic function $$g$$ finally multiplicative if there is a multiplicative function $$G$$ so that $$g (n) = G (n)$$ to the $$n$$ big enough. Is it true that if $$f$$ is completely multiplicative and $$varphi_f$$ Finally, it is either multiplicative $$f = delta_1$$ or $$f = 1$$?

## ordinary differential equations – on the proof of the uniqueness / existence theorem

I try to show that $$| Underline {x} (t) – Underline {y} (t) | leq left | Underline {x} (t_0) – Underline {y} (t_0) right | + int_ {t_0} ^ {t} left | underline {f} ( underline {x}, s) – underline {f} ( underline {y}, s) right | ds,$$
from where $$underline {x} (t) = underline {x} (t_0) + int_ {t_0} ^ {t} underline {f} ( underline {x}, s) ds, underline { y} (t) = underline {y} (t_0) + int_ {t_0} ^ {t} underline {f} ( underline {y}, s) ds.$$ This is part of my proof for the uniqueness / existence proposition.

This is my previous work.

begin {align} | underlined {x} (t) – underlined {y} (t) | & leq | underlined {x} (t) | + left | – underlined {y} (t) right | \ & = left | underline {x} (t_0) + int_ {t_0} ^ {t} underline {f} ( underline {x}, s) ds right | + left | – underline {y} (t_0) – int_ {t_0} ^ {t} underline {f} ( underline {y}, s) ds right | \ & leq | underline {x} (t_0) | + left | int_ {t_0} ^ {t} underline {f} ( underline {x}, s) ds right | + left | – underline {y} (t_0) right | + left | – int_ {t_0} ^ {t} underline {f} ( underline {y}, s) ds right | \ end
I do not understand how to combine the terms as desired.

## Rotations – Anchoring matrices in the linear equation system for uniqueness

This book "Articulated Motion and Deformable Objects: 7th International Conference" (1) describes how a unique (non-zero) solution to this homogeneous system of linear equations arises $$MR = [… -I … R_{ij} …][… R_i … R_j]^ T = 0$$ for relative rotation decoding (where $$j = 1.2.3$$). This equation is derived from the relationships between relative rotations and absolute rotations $$R_ {i, j} = R_iR_j$$ reformulated to $$R_ {i, j} R_i – R_j = 0$$,

Thus, it can be solved by anchoring one or more rotation matrices in the least square sense $$R_i$$ Guarantee uniqueness: $$M ^ TM + lambda I_k = M ^ TR_i + lambda ^ 2 cdot R_k$$However, I do not understand where the parameter is $$lambda$$ comes out and how to get it and what is meant by "solution in the least square sense".

Any help is greatly appreciated!