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.