sql server – What does Nested Loops join operator has to do with a Scalar?

As far as I understand the docs, the Nested Loops operator is a join operator, i.e., it requires two tables as input.

If that’s correct, then why the Nested Loops operator is used with a Scalar input?

For example, take the following query (from Paul White’s website) and its execution plan.
You can see that the inputs for the Nested Loops operator are (1) a Scalar, and (2) a table (the result of the Index Seek).

I know that a scalar can’t be joined with a table, then what that actually means? What’s actually being joined?

USE AdventureWorks2019;
DECLARE @Like nvarchar(50) = N'D%';
SELECT p.(Name)
FROM   Production.Product AS p
WHERE  p.(Name) LIKE @Like;

enter image description here

BTW, I presume that it’s a pretty basic question, but I couldn’t find a good resource that goes through such basics methodologically, so a recommendation would be much appreciated.

lie groups – Metric of $SO(3)$ : Why is there $frac{1}{8}$ in front of this scalar product?

I’m considering the rotation group, as described by $SO(3)$ (i.e the set of $3 times 3$ orthogonal matrices $R$ of determinant 1, with real components) and $SU(2)$ (the set of $2 times 2$ complex matrices $U$ of determinant 1). The elements of these two groups could be represented like this (the unit matrix is implied implicitely):
begin{align}
R(alpha, n_k) &= 1 cos alpha + sum_{k = 1}^3 lambda_k , n_k sin alpha + n , n^{top} (1 – cos alpha) quad in SO(3), tag{1} \
U(alpha, n_k) &= e^{i vec{sigma} , cdot , vec{n} , alpha / 2} = 1 cos (alpha/2) + i sum_{k = 1}^3 sigma_k , n_k sin (alpha/2) quad in SU(2), tag{2}
end{align}

where $lambda_k$ ($k = 1, 2, 3$) are three $3 times 3$ skew-symetrical matrices that generate the rotations, and $sigma_k$ are the three $2 times 2$ Pauli matrices. The components $n_k$ are three real numbers defining the rotation axis. We could write these components as this:
$$n_k = { sinvartheta cos varphi, : sin vartheta sin varphi, : cos vartheta }. tag{3}$$
Of course, $alpha$ is the rotation angle.

Now, I define a scalar product in $SU(2)$ as a trace:
$$langle , U_1, , U_2 rangle = frac{1}{2} , mathrm{Tr}(U_1^{dagger} , U_2), qquad U_1, , U_2 , in , SU(2).tag{4}$$
Calculating the differential $dU$ gives us a metric on $SU(2)$ (I set $chi equiv alpha / 2$ to simplify things):
$$ds^2 = frac{1}{2} , mathrm{Tr}(dU^{dagger} , dU) = dchi^2 + sin^2 chi , (dvartheta^2 + sin^2 vartheta , dvarphi^2).tag{5}$$
This is a well known metric on $mathcal{S}_3$, the 3-sphere. This is fine, since $SU(2)$ is a compact Lie group, which have the topology of that sphere.

But then I tried to do the same with $SO(3)$ (the calculations are very messy in this case). I got this:
$$frac{1}{8} , mathrm{Tr}(dR^{top} , dR) = dchi^2 + sin^2 chi , (dvartheta^2 + sin^2 vartheta , dvarphi^2).tag{6}$$
The fraction $frac{1}{8}$ on the left part puzzles me. Why this factor, instead of $frac{1}{3}$? This trace of $3 times 3$ matrices doesn’t look like a scalar product. My interpretation of (6) is lacking. How can we define a proper scalar product and Riemanian metric on $SO(3)$ that would show the 3-sphere hidden in it?

linear algebra – Why we can’t subtract scalar product?

We have scalar product property “Distributive over vector addition”: $$(a+b) cdot c=a cdot c space + space b cdot c$$
Now let’s look at this system (we will call it “system-1”):
$$begin{cases} a cdot x leq 1 \ b cdot x leq -1
end{cases}$$

Now if we sum to the second line the first line of the “system-1” we will get (we will call it “system-2”):
$$begin{cases} a cdot x leq 1 \ (a+b) cdot x leq 0
end{cases}$$

So, for the “system-2” we have a solution x=0, but for “system-1” x=0 is not a solution.
Why, if we can subtract and sum lines in the system?

riemannian geometry – Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)

This question is related to a previous one.

Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $partial M$ is mean convex (positive mean curvature).

Question: Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?

magento2 – Where do scalar extension attributes save to?

Been reading the doc from Magento and i have been using extension attributes for more complex data structures but i’m basically wanting to add a simple scalar attribute

  <extension_attributes for="MagentoSalesApiDataShipmentTrackInterface">
    <attribute code="track_url" type="string"/>
  </extension_attributes>

And i thought for scalar attributes, i pretty much don’t need to do anything else such as adding plugins, just need to set the value correctly and use repository to save

        $shipment = $this->shipmentRepository->get($shipmentId);

        $track = $this->trackFactory->create()
            ->setTrackNumber('123')
            ->setCarrierCode('abc')
            ->setTitle('super delivery');


        $extensionAttributes = $track->getExtensionAttributes()->setTrackUrl(
            'http://www.trackme.com/abcdef'
        );
        $track->setExtensionAttributes($extensionAttributes);
      

        $shipment->addTrack($track);
        $this->shipmentRepository->save($shipment);

Codes are generated correctly, no errors, but when i try to retrieve it, it doesn’t actually return anything

$track->getExtensionAttributes()->getTrackUrl()

Am i misunderstanding extension attributes then? Does it mean i will always need to add something new to the database, even for scalar types?

sql server – Need Ideas on how to eliminate this scalar funciton or make it faster?

I’m working on pretty lengthy view and one of SQL statements calls a scalar function, which really degrades the performance. The following is the function that is called in a select statement. (sql server 2016)

Really not sure the best ways to modify this process.

Simplified select statement.

Select vendorpartid, 
dbo.ufn_StockUomQuantityToOrder(vpp.priorityLevel, vpp.monthlyUsageRate, vpp.minimumPurchaseUomOrderQuantity, vpp.purchaseUomConversionFactor, vpp.orderFrequencyDays, vpp.boxQuantity) stockUomQuantityToOrder
FROM vendorpartpriority vpp


ALTER FUNCTION (dbo).(ufn_StockUomQuantityToOrder) (
            @priorityLevel decimal(38, 10),
            @monthlyUsageRate decimal(38,10),
            @minimumPurchaseUomOrderQuantity decimal(38, 10),
            @purchaseUomConversion decimal(38, 10),
            @orderFrequencyDays decimal(38,10),
            @boxQuantity int
        )  
        RETURNS int
        WITH SCHEMABINDING
        AS   
        -- Calculate the quantity that needs to be ordered  
        BEGIN  
            DECLARE @quantityToOrderDecimal decimal(38, 10) = NULL;
            DECLARE @quantityToOrderInt int = NULL;
            DECLARE @orderFrequencyMinimumQuantity decimal(38, 10) = NULL;
            DECLARE @minimumStockUomOrderQuantity decimal(38,10) = NULL;
            
            --set the default order quantity
            SELECT @quantityToOrderDecimal = (-1.0 * @monthlyUsageRate * @priorityLevel);
            
            --get the minimum order quantity in stock UOM
            SELECT @minimumStockUomOrderQuantity = (@minimumPurchaseUomOrderQuantity * @purchaseUomConversion);
        
            --calculate the order frequency minimum
            IF(@orderFrequencyDays IS NOT NULL AND @monthlyUsageRate IS NOT NULL)
                SELECT @orderFrequencyMinimumQuantity = (@monthlyUsageRate * @orderFrequencyDays / 30.0);
                    
            --do we need to meet a vendor minimum
            IF (@quantityToOrderDecimal IS NULL OR @quantityToOrderDecimal < @minimumStockUomOrderQuantity)
                SELECT @quantityToOrderDecimal = @minimumStockUomOrderQuantity;
            
            --do we need to meet an order frequency minimum
            IF (@quantityToOrderDecimal IS NULL OR @quantityToOrderDecimal < @orderFrequencyMinimumQuantity)
                SELECT @quantityToOrderDecimal = @orderFrequencyMinimumQuantity;
            
            --convert to the int
            SELECT @quantityToOrderInt = CAST(CEILING(@quantityToOrderDecimal) AS int);
                
            --did we come up with a number that needs to be an increment of boxQuantity
            IF(@quantityToOrderInt IS NOT NULL AND @boxQuantity > 0)
                BEGIN
                    --get the partial box quantity if any
                    DECLARE @partialBox int = @quantityToOrderInt % @boxQuantity;
                    
                    --remove the partial box and add a full one (if we are not of box increments)
                    IF(@partialBox <> 0)
                    BEGIN
                        SELECT @quantityToOrderInt = (@quantityToOrderInt - @partialBox + @boxQuantity);
                    END
                END 
        
            RETURN @quantityToOrderInt;  
        END;
        
GO

2019 isn’t an option now. I have dabbled with the in-line table value, but I’m not quite sure how to get rid of the declares.

I’m not 100% sure on the nulls. My gut says I shouldn’t be passing any values that are nulls, but looks like input table has quite a few in different columns.

Right now I’m trying to make some glaring performance changes until we get time to dismantle the whole process.

statistics – Derivative of term that involving vectors, transposes, and matrices, with respect to a scalar

I have to differentiate the following expression with respect to $k$ and equate it to $0$,
$$L=frac{n}{2}mathrm{tr}left(Sigma^{-1}left{S+left({overline{X}}-kmu_0right)left({overline{X}}-kmu_0right)’right}right)$$
where, $n,Sigma_{ptimes p},{mu_0}_{(ptimes1)},{overline{X}}_{(ptimes1)},S_{(ptimes p)}$ are known.

The final expression is given as
$$nmu_0Sigma^{-1}left({overline{X}}-kmu_0right)=0.$$

I am not very familiar with differentiation when matrices and vectors are involved. Could I get a basic/general explanation to how to final expression is obtained?

physics – The Weyl Scalar: does anyone here knows a code to calculate it?

From pure differential geometry we have an important tensor called Weyl Tensor. Its components are given by:

$$C_{ijkl} = R_{ijkl} – frac{2}{n-2}(g_{ik}R_{jl} + g_{jl}R_{ik} – g_{il}R_{jk} – g_{jk}R_{il} ) – frac{2}{(n-1)(n-2)}R(g_{il}g_{jk} – g_{ik}g_{jl}).tag{1}$$

I would like to calculate via a mathematica code the scalar given by:

$$W = C_{ijkl}C^{ijkl} tag{2}$$

Does anyone knows a code for this one calculation $(2)$ (given, of course, a metric tensor matrix)?

python – How to prevent scalar multiplication by odd numbers?

I have a script that does scalar multiplication of points on an Elliptic Curve.

class Point(object):
    def __init__(self, _x, _y, _order = None): self.x, self.y, self.order = _x, _y, _order
    def calc(self, top, bottom, other_x):
        l = (top * inverse_mod(bottom)) % p
        x3 = (l * l - self.x - other_x) % p
        return Point(x3, (l * (self.x - x3) - self.y) % p)
    def double(self):
        if self == INFINITY: return INFINITY
        return self.calc(3 * self.x * self.x, 2 * self.y, self.x)
    def __add__(self, other):
        if other == INFINITY: return self
        if self == INFINITY: return other
        if self.x == other.x:
            if (self.y + other.y) % p == 0: return INFINITY
            return self.double()
        return self.calc(other.y - self.y, other.x - self.x, other.x)
    def __mul__(self, e):
        if self.order: e %= self.order
        if e == 0 or self == INFINITY: return INFINITY
        result, q = INFINITY, self
        while e:
            if e&1: result += q
            e, q = e >> 1, q.double()
        return result
    def __str__(self):
        if self == INFINITY: return "infinity"
        return "%x %x" % (self.x, self.y)
def inverse_mod(a):
    if a < 0 or a >= p: a = a % p
    c, d, uc, vc, ud, vd = a, p, 1, 0, 0, 1
    while c:
        q, c, d = divmod(d, c) + (c,)
        uc, vc, ud, vd = ud - q*uc, vd - q*vc, uc, vc
    if ud > 0: return ud
    return ud + p


p, INFINITY = 377, Point(None, None)
g = Point(18, 26)
EvenNumber = 2
 
result = '  Scalar:    %xn   Point: %s' % (EvenNumber, g*EvenNumber)
f = open('Result.txt', 'a')
f.write(result)
f.close()

I cannot understand in more detail the code itself. I need to change the algorithm of this script so that points on the plane of the curve are added only if an even number is specified.

I.e:

EvenNumber = 2
EvenNumber = 4
EvenNumber = 6
EvenNumber = 8
EvenNumber = 10
EvenNumber = 12
...
...
...
etc

Is it possible to do so to completely eliminate the odd numbers in scalar multiplication?

differential equations – Nonlinear PDE – Coefficients do not evaluate to a numerical scalar

I am trying to solve a nonlinear differential equation but I get an error message regarding the coefficients. Please help ..

n = 1/2;
cof = ((1/4 (Inactive(D)(u(r, θ), r))^2 + 
   1/4/r^2 (Inactive(D)(u(r, θ), θ))^2)/2)^((n - 1)/
2);
NDSolveValue({1/r D(cof  r  Inactive(D)(u(r, θ), r), r) + 
1/r D(cof  1/r  Inactive(D)(
    u(r, θ), θ), θ) == -1, 
  DirichletCondition(u(r, θ) == 0, True)}, 
 u(r, θ), {r, 0, 1}, {θ, 0, Pi})