## Projections in CAR Algebra – MathOverflow

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## Add new field in MongoDB projections

I have a Mongo collection (store details) like below …

{
_I would: "",
storeName: "store1",
Items: [
{
itemName: "mongo",
itemPrice: 20,
itemAvailablity: 100
},
{
itemName: "apples",
itemPrice: 50,
itemAvailablity: 70
}
]
},
{
_I would: "",
storeName: "store2",
Items: [
{
itemName: "banana",
itemPrice: 10,
itemAvailablity: 30
},
{
itemName: "apple",
itemPrice: 45,
itemAvailablity: 90
}
]
},
{
_I would: "",
storeName: "store3",
Items: [
{
itemName: "apple",
itemPrice: 10,
itemAvailablity: 30
},
{
itemName: "mongo",
itemPrice: 30,
itemAvailablity: 50
}
]
}


From the above data, I would like to receive certain item details along with storeName.

If I want to receive "Mongo" details from all stores, my expected issue is as follows

[
{
itemName: "mongo",
itemPrice: 20,
itemAvailablity: 100,
storeName: "store1"
},
{
itemName: "mongo",
itemPrice: 30,
itemAvailablity: 50,
storeName: "store3"
}
]


I'm trying different Mongo aggregation queries, but I did not get the expected output

Can somebody help me with it

Thank you

## fa.functional analysis – Strong differentiability of spectral projections

To let $$H$$ to be a Hilbert room and $$W$$ Be a dense subspace equipped with another norm that turns it into a Hilbert space. To let $$(A (t)) _ {t in[0,T]}$$ to be a family of operators $$B (B, H)$$ (Restricted operators of $$W$$ to $$H$$), which deal with discrete spectrum as unrestricted operators in $$H$$ with domain $$W$$, Accept that $$A (t)$$ is differentiable depending on $$t$$ in strong topology $$B (B, H)$$ and this 0 is not in the spectrum of $$A (t)$$ for each $$t$$, Does this mean that the positive spectral projection of $$A (T)$$i.e. $$chi_ {[0infty)}(A(t))[0infty)}(A(t))[0infty)}(A(t))[0infty)}(A(t))$$is differentiable in $$t$$ in terms of strong topology $$B (H, H)$$? Does anyone know a reference where such a statement could be found?

## ag.algebraic geometry – correct projections

To let $$D subset mathbb {C} ^ k$$ be your preferred complex domain.
Suppose we get a correct holomorphic map $$f colon D to mathbb {C} ^ {k + 2}$$,

Let us take $$k + 1$$ generic linear functions $$l_i colon mathbb {C} ^ {k + 2} an mathbb {C}$$, $$i = 1, lpoints, k + 1$$ and make a linear map $$L colon mathbb {C} ^ {k + 2} an mathbb {C} ^ {k + 1}$$ by $$L: = (l_1, ldots, l_ {k + 1})$$,

Is it true that $$L circ f$$ is also good? How can you show that?

I understand that mathstackexchange is an appropriate place for that
a question. I just want to leave it here.

## Functional Analysis – Projections in the tensor product of Neumann algebras

This question seems to be fundamental, but I have already asked an expert who does not know the answer, so I would like to post here.

To let $$M$$ and $$N$$ be from Neumann algebras and let $$M bar { otimes} N$$ be it from Neumann algebra tensor product.

Question: Can any projection in $$M bar { otimes} N$$ as expressed
the supremum (join) of projections of the form $$p otimes q$$, from where $$p$$
and $$q$$ are projections in $$M$$ and $$N$$, respectively?

## fa.functional analysis – Number of projections – Can one explain the magic?

I have included a calculation that calculates the Chern number of the first Landau stage (the result is $$-1$$) and the complete work can be found here
Click me. I have trouble understanding
what happened here.

The projection is given by the integral core

$$Pi (x, y) = frac {qB} {h} e ^ {- (qB / 4 hbar) (xy) ^ 2-i (qB / 2 hbar) x Wedge y}.$$

The authors calculate a "derivative" of this expression and get an integral expression for the Chern character, claiming it is the same $$-1$$, It does not seem to be a very sophisticated calculation, but it's hard to understand, so I'd like to ask here, as it's very likely I'm missing something.

## Mesage theory – When does measurability of projections imply measurability in product $sigma$ algebra?

To let $$(X, mathcal {X})$$ and $$(Y, mathcal {Y})$$ be and let measurable spaces $$mathcal {Z}$$ be the product $$sigma$$Algebra on $$X times Y$$, Accept $$A subset X times Y$$ is such that his projections are up $$X$$ and $$Y$$ are measurable w.r.t. their respective $$sigma$$-algebras.

Question: Under what natural assumptions
$$(X, mathcal {X})$$ and $$(Y, mathcal {Y})$$
can we close that
$$A in mathcal {Z}$$?

For example, does $$mathcal {X} = 2 ^ X$$ suffice? What if we add the condition $$Y = {0,1 } ^ X$$, and $$mathcal {Y}$$ is the cylindrical one $$sigma$$-Algebra?

## postgresql – Postgres – Using an alias for subquery projections in a "FROM clause" on a "WHERE clause" does not work

For example, if you have a table to store users of your application, call these users with the two fields "ID" and "E-mail."

Select the ID and send it by e-mail
from (
Select ID, send e-mail
of users
t
where electronic_mail = & # 39; & # 39;


Postgres complain that

ERROR: column "electronic_mail" does not exist

Disclaimer: This example is deliberately simple and stupid, just to indicate the problem that arises. You can imagine that I have to think through a json column. I think an array of complex elements to get a single scalar value of each. (I can share some code if you want to see it).

This is quite strange for me, because I really do not understand what the complication would be, probably something is not aware of me. As far as I work with Postgres, the alias columns can be easily used in a where clause.

## banach blanks – If the ranges of all f.d base projections in $X$ are isometric to subspaces of $Y$, does it follow that $X$ is isometric for a subspace of $Y$?

To let $$X$$ be a Banach room with a shudder base and $$Y$$ a Banach room. To let $$P_N$$ denote the coordinate projections relative to the base of $$X$$, and let $$X_N$$ denote their ranges. specifically, $${x_n } _ {n = 1} ^ { infty}$$ is a base in $$X$$ and
$$P_N (x) = sum_ {n = 1} ^ Na_nx_n, quad hbox {with} quad x = sum_ {n = 1} ^ { infty} a_nx_n$$
Accept this for everyone $$N$$the finally dimensional space $$X_N$$ isometric to a subspace of $$Y$$, Does it follow that? $$X$$ isometric to a subspace of $$Y$$?

If $$Y = L_p$$ With $$1 leq p leq 2$$ then the answer is affirmative, but the general case escapes me.