dg.differential geometry – Proving some identities about the time derivative of the k-th covariant derivatives of scalar curvature under normalized Ricci flow on surfaces

I’m trying to prove the following identities (under the normalized Ricci flow on surfaces, on which $partial_t g = (r-R)g$ holds true, where $r$ denotes the average scalar curvature and has the same sign as the Euler characteristic):

$$
frac{partial}{partial t}left(nabla^{k} Rright)=Delta nabla^{k} R-rleft(nabla^{k} Rright)+sum_{j=0}^{lfloor k / 2rfloor}left(nabla^{j} Rright) otimes_{g}left(nabla^{k-j} Rright)
$$

$$
begin{array}{l}
frac{partial}{partial t}(|nabla^k R |^2) =Deltaleft|nabla^{k} Rright|^{2}-2left|nabla^{k+1} Rright|^{2}-(k+2) rleft|nabla^{k} Rright|^{2}
+left(nabla^{k} Rright) otimes_{g}left(sum_{j=0}^{lfloor k / 2rfloor}left(nabla^{j} Rright) otimes_{g}left(nabla^{k-j} Rright)right)
end{array}
$$

where by $A otimes_g B$ we refer to any tensor field which is a finite linear combination of contractions and metric contractions of the tensor product $A otimes B$. Now, I’ve already proven that the following hold:

$$partial_{t} nabla R=Delta nabla R+frac{3}{2} R nabla R-r nabla R$$
$$
nabla^{n} Delta R-Delta nabla^{n} R=sum_{j=0}^{lfloor n / 2rfloor}left(nabla^{j} Rright) otimes_{g(t)}left(nabla^{n-j} Rright)
$$

$$
begin{array}{c}
nabla^{n} R^{2}=displaystyle{sum_{j=0}^{lfloor n / 2rfloor}left(nabla^{j} Rright) otimes_{g(t)}left(nabla^{n-j} Rright) }\
left(frac{partial}{partial t} Gammaright) otimes_{g(t)}left(nabla^{j} Rright)=(nabla R) otimes_{g(t)}left(nabla^{j} Rright)
end{array}
$$

$$
frac{partial}{partial t}left(nabla_{k_{1}} nabla_{k_{2}} ldots nabla_{k_{n}} Rright)=nabla_{k_{1}}left{partial_t nabla_{k_{2}} ldots nabla_{k_{n}} Rright}-sum_{l=2}^{n}left(partial_{t} Gamma_{k_{1} k_{l}}^{m}right) nabla_{k_{2}} ldots nabla_{k_{l-1}} nabla_{m} ldots nabla_{k_{n}} R
$$

So, to prove the first formula for the evolution of $nabla^k R$, I used recursive applications of this last identity just above, but I’d like someone to check my work. I noticed there would be terms of the form:

begin{align*}
nabla_{k_1} cdots nabla_{k_{n-1}}(partial_t nabla_{k_n} R) &= nabla_{k_1} cdots nabla_{k_{n-1}}(Delta nabla_{k_n}R + frac{3}{2} R nabla_{k_n} R – r nabla_{k_n R}) \
&= nabla_{k_1} cdots nabla_{k_{n-1}} ( nabla_{k_n} Delta R + Sigma + frac{3}{2} R nabla_{k_n} R – r nabla_{k_n R} )\
&=nabla^k Delta R + Sigma – r(nabla^{k} R) \
&=Delta nabla^k R + Sigma – r(nabla^k R)
end{align*}

where by $Sigma$ I’m denoting $displaystyle{sum_{j=0}^{lfloor k / 2rfloor}left(nabla^{j} Rright) otimes_{g}left(nabla^{k-j} Rright)}$ to avoid taking up too much space. The remaining terms are of the form:

$$ begin{aligned}
&left(partial_{t} Gammaright) otimes nabla^{k-1} R=nabla R otimes nabla^{k-1} R=Sigma\
&nabla^r(partial_t Gamma) otimes nabla^{k-r-1} R = nabla^{r+1}R otimes nabla^{k-r-1}R = Sigma end{aligned}
$$

and so we have proved the first identity. But I didn’t manage to prove the second one. We have:

$$begin{aligned}
frac{partial}{partial t}left|nabla^{k} Rright|^{2} &= frac{partial}{partial t}left(g^{i_1 p_1} cdots g^{i_k p_k} nabla_{i_1} cdots nabla_{i_k} R nabla_{p_1} cdots nabla_{p_k} Rright) \
&=(R-r)k |nabla^k R |^2 + 2 langle nabla^k R, partial_t(nabla^k R) rangle
end{aligned}
$$

and since

begin{align*}
2 langle nabla^k R, partial_t(nabla^k R) rangle &= 2 langle nabla^k R, Delta nabla^k R + Sigma – r nabla^k R rangle \
&=2 langle nabla^k R, Delta nabla^k R rangle + 2 langle nabla^k R, Sigma rangle – 2r |nabla^k R|^2
end{align*}

We’re then left to prove that:

$$2 langle nabla^k R, Delta nabla^k R rangle = -kR |nabla^k R|^2 + Delta |nabla^k R|^2 – 2 |nabla^{k+1} R|^2$$

but I’ve been stuck on this one for a while. I’d really appreciate some help on this! Thanks in advance.

fourier analysis – Approximate Identities on the Unit Disk and going beyond a power series’ radius of convergence

Let $left{ a_{n}right} _{ngeq0}$ be a bounded sequence of complex numbers, so that the power series $fleft(zright)=sum_{n=0}^{infty}a_{n}z^{n}$ has a radius of convergence $geq1$. Additionally, suppose that $fleft(0right)neq0$, and that $fleft(omegaright)=0$ for some complex number $omega$ with $0<left|omegaright|<1$. Let $nu$ be a positive non-integer real number and let $g_{nu}left(zright)=left(fleft(zright)right)^{1/nu}$. Since $f$ is non-vanishing at $z=0$, $g_{nu}left(zright)$ has a power series expansion: $$g_{nu}left(zright)=sum_{n=0}^{infty}a_{n}left(frac{1}{nu}right)z^{n}$$ with a radius of convergence $r_{nu}inleft(0,1right)$. In particular, thanks to the zero of $f$ at $omega$, we have that $r_{nu}<left|omegaright|$.

Next, let: $$g_{nu,r}left(tright)=left(fleft(re^{2pi it}right)right)^{1/nu}$$

Since $f$ is holomorphic on the open unit disk, $g_{nu,r}in L^{p}left(mathbb{R}/mathbb{Z}right)$ for all $pinleft(1,inftyright)$ and all $rinleft(0,1right)$.

Now, let $left{ K_{r}right} _{r>0}$ be a family of good kernels/an approximate identity on $mathbb{R}/mathbb{Z}$ (ex: $K_{r}left(tright)=P_{r}left(tright)=sum_{ninmathbb{Z}}r^{left|nright|}e^{2npi it}$, the Poisson Kernel), and consider the integral: $$left(K_{rho}*g_{nu,r}right)left(tright)=int_{0}^{1}K_{rho}left(t-xright)g_{nu,r}left(xright)dx$$

Because of the integrability of $g_{nu,r}$ for each $rinleft(0,1right)$, the approximate identity property of the $K_{rho}$s guarantees that $lim_{rhouparrow1}left(K_{rho}*g_{nu,r}right)left(tright)=g_{nu,r}left(tright)$ for almost every $tinmathbb{R}/mathbb{Z}$ (in particular, the limit exists for every $t$ at which $g_{nu,r}$ is continuous). When $r<r_{nu}$, the power series of $g_{nu,r}$ gives us an absolutely convergent Fourier series for $g_{nu,r}$: $$g_{nu,r}left(xright)=sum_{n=0}^{infty}a_{n}left(frac{1}{nu}right)r^{n}e^{2npi ix},textrm{ }forall xinmathbb{R},textrm{ }forallleft|rright|<r_{nu}$$ consequently, we can compute $left(K_{rho}*g_{nu,r}right)left(tright)$ term-by term using this series representation, and hence (writing $eleft(tright)=e^{2npi it}$): $$left|rright|<r_{nu}Rightarrow g_{nu,r}left(tright)=lim_{rhouparrow1}left(K_{rho}*g_{nu,r}right)left(tright)=lim_{rhouparrow1}sum_{n=0}^{infty}a_{n}left(frac{1}{nu}right)r^{n}left(K_{rho}*eright)left(tright)$$ for all $t$ at which $g_{nu,r}$ is continuous. However, when $left|rright|>r_{nu}$, the series for $g_{nu,r}$ fails to converge, and as such, the right-most of the above equalities no longer holds. That is, if we think of the limit: $$lim_{rhouparrow1}sum_{n=0}^{infty}a_{n}left(frac{1}{nu}right)r^{n}left(K_{rho}*eright)left(tright)$$ as a generalization of the abel method of summing divergent series (with respect to the parameter $rho$), while $lim_{rhouparrow1}left(K_{rho}*g_{nu,r}right)left(tright)$ still equals $g_{nu,r}left(tright)$ almost everywhere, this equality does not appear to be realizable as an abel-style summation of a divergent series.

All that being said, I have two questions:

(I) Let $rinleft(r_{nu},1right)$, so that $fleft(zright)$‘s power series about $z=0$ converges for $left|zright|=r$, but $g_{nu}left(zright)$‘s power series about $z=0$ diverges for all $left|zright|=r$. Under these hypotheses, as discussed above, the limit $lim_{rhouparrow1}left(K_{rho}*g_{nu,r}right)left(tright)$ exists for almost every $t$. Is there a way to express the value of the limit (when it converges to $g_{nu,r}$) in terms of power series coefficents of $g_{nu,r}$ (maybe expanded about some point other than $0$), or something similar? Here, case I’m most interested in is when $K_{rho}=P_{rho}$, the Poisson Kernel. So understanding that particular situation is paramount, though information about the more general case would also be desirable.

(II) The Parseval-Gutzmer formula establishes the identity: $$int_{0}^{1}left|sum_{n=0}^{infty}c_{n}left(re^{2npi it}right)^{n}right|^{2}dt=sum_{n=0}^{infty}left|c_{n}right|^{2}r^{2n}$$ for all $rgeq0$ which are less than the radius of convergence of the power series $sum_{n=0}^{infty}c_{n}z^{n}$. For our function $fleft(zright)$, this gives: $$int_{0}^{1}left|fleft(re^{2pi it}right)right|^{2/nu}dt=sum_{n=0}^{infty}left|a_{n}left(frac{1}{nu}right)right|^{2}r^{2n},textrm{ }forall rinleft(0,r_{nu}right)$$ where, recall, $r_{nu}$ is the radius of convergence of the power series of $f^{1/nu}left(zright)$ about $z=0$. Since $tmapsto f^{1/nu}left(re^{2pi it}right)$ is bounded for all $rinleft(0,1right)$, however, the left-hand side of this identity exists even for $rinleft(r_{nu},1right)$. Is there a way to “correct”/alter the right-hand side of the identity so as to extend it to all $rinleft(0,1right)$? And if so, how would one do this?

co.combinatorics – Combinatorics and geometry underlying a refined Pascal matrix/Newton identities

The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for transforming between complete homogeneous symmetric polynomials/functions and elementary symmetric polynomials/functions. Certain Koszul duals are related to this.

The algebraic combinatorics of the complementary reciprocal of a Taylor series/e.g.f. is governed by the antipode/refined Euler characteristic classes of the permutahedra or, equivalently, by surjective mappings, so I have an indirect geometric combinatorial interpretation of ‘scaled’ versions of the Newton identities, but I’m looking for more direct interpretations.

What combinatoric/geometric structures are enumerated by the integer coefficients of these partition polynomials for conversion of an o.g.f. into a reciprocal o.g.f.?

fa.functional analysis – Doubt from a paper on approximate identities

I am currently reading this paper on approximate identities of Ternary Banach algebras. Assume that $(A, (.,.,.))$ is a ternary Banach algebra. A bounded net $(e_{alpha}, f_{alpha})$ is said to be left-bounded approximate identity for $A$ if $lim_{alpha}(e_{alpha}, f_{alpha},a)=a$ for all $a in A$. Can someone clarify my following doubts from the same paper:

What is the definition of bounded net? Also at many places in the same paper for instance in Theorem $2.2$, the product $e_{alpha}f_{alpha}$ is used. What is the meaning of product in ternary Algebra?

All these doubts are not explained in paper. Thank you very much in advance!

Get identities of authors of wordpress.com websites?

How can I found out the real identities of authors of wordpress.com websites?

For example, there is a wordpress.com website which seems to be involved in terrorist activities:

https://staatsterrorlive.wordpress.com

I have tried with several domain tools but got only the data of wordpress.com itself.

domain-driven design: can entity identities be written in a DDD world?

When designing entities, I instinctively put their identities as read-only properties.

class MyEntity
{
    MyEntity(Id id) { Id = id; }

    Id { get; }
}

But I wonder, can identity be a property of writing?
Should we consider the controller / pointer itself to be part of the identity or not?

I did not work with Entity framework, but it seems that they expose identities as properties of writing.
Do you see any problem with this fact?

Hyperledger tissue nodeOU not activated. Identities cannot be distinguished

When I try to approve a chaincode package, a timeout error appears.
Peer records show the following warnings. Although I have the config.yaml file in the MSP directory with NodeOUs enabled for true, it says NodeOus not enabled. Can anyone help me solve this problem? I use Hyperledger Fabric 2.0 binaries.
identity 0 does not satisfy principal: The identity is not a [PEER] under this MSP [org1MSP]: NodeOUs not activated. Cannot tell apart identities.

bric"],"L-ST-C":"[]-[]-[US]","MSP":"org1MSP","OU":["peer"]} isn't eligible for channel twoorgschannel : implicit policy evaluation failed - 0 sub-policies were satisfied,
but this policy requires 1 of the 'Readers' sub-policies to be satisfied```

co.combinatorics – For human tests of two novel combinatorial identities

by $ n = 0,1,2, ldots $let's define the polynomial
$$ S_n (x): = sum_ {k = 0} ^ n binom {x / 2} k binom {(x-1) / 2} k binom {- (x + 1) / 2} { nk} binom {- (x + 2) / 2} {nk}. $$
Such polynomials occur in some series to $ 1 / pi $ discovered by me in 2011, see Conjecture 4 of my article List of conjecture series for powers of $ pi $ and other constants

In 2011, I found the following two novel identities for polynomials $ S_n (x) $:
$$ S_n (x) = binom {-1/2} n sum_ {k = 0} ^ n (-1) ^ k binom {x} k ^ 2 binom {-1-x} {nk} tag {1} $$
Y
$$ begin {aligned} & sum_ {k = 0} ^ n binom nk (-1) ^ kS_k (x)
\ = & sum_ {k = 0} ^ n binom {x / 2} k binom {- (x + 1) / 2} k binom {(x-1) / 2} {nk} binom {- (x + 2) / 2} {nk}. end {aligned} tag {2} $$

Note that $ (1) $ implies symmetric identity
$$ sum_ {k = 0} ^ n (-1) ^ k binom xk ^ 2 binom {-1-x} {nk} = sum_ {k = 0} ^ n (-1) ^ k binom {-1-x} k ^ 2 binom x {nk}, $$
that was demonstrated without a computer in my work Supercongruences involving dual sequences, Finite Fields Appl. 46 (2017), 179-216.

By the Zeilberger algorithm, yes $ u_n $ is the left side or the right side of $ (1) $ Then we have the recurrence relationship:
begin {align} 4 (n + 2) ^ 3u_ {n + 2} = & 2 (2n + 3) (2n ^ 2 + 6n + x ^ 2 + x + 5) u_ {n + 1}
\ & – (n + 1) (2n + 1) (2n + 3) u_n. end {align}

Similarly, yes $ v_n $ denotes the left side or the right side of $ (2) $, then we have the recurrence
begin {align} 4 (n + 2) ^ 3v_ {n + 2} = & 2 (2n + 3) (2n ^ 2 + 6n-x ^ 2-x + 5) v_ {n + 1}
\ & – (n + 1) (2n-2x + 1) (2n + 2x + 3) v_n. end {align}

Thus $ (1) $ Y $ (2) $ Have tests through a computer.

Question. How to provide human proof of identities. $ (1) $ Y $ (2) $?

Your comments are welcome!

differential geometry: how do you reduce an equation that involves Trig identities in Mathematica?

I am trying to automatically generate the Christoffel symbol in Mathematica. I am starting with the formulas:

X(r_, theta_) := r*Cos(theta); 
Y(r_, theta_) := r*Sin(theta); 
R(x_, y_) := Sqrt(x^2 + y^2); 
Theta(x_, y_) := ArcTan(x, y);

Now, without trying to analyze the equations, I am trying to implement this in Mathematica. So far I have:

D(D(X(r, theta), r), r)*D(R(x, y), x) + D(D(Y(r, theta), r), r)*D(R(x, y), y)
D(D(X(r, theta), r), theta)*D(R(x, y), x) + D(D(Y(r, theta), r), theta)*D(R(x, y), y)

What results in this:

0
(y Cos(theta))/Sqrt(x^2 + y^2) - (x Sin(theta))/Sqrt(x^2 + y^2)

Now I know that I can reduce the second result using the fact that $ y = r space Sin (theta) $ Y $ x = r space Cos (theta) $ Y $ r = sqrt {x ^ 2 + y ^ 2} $, but I want Mathematica to do the reduction for me. How do I tell Mathematica to reduce this equation (hint: the result should be 0)?

complex geometry: Kähler identities in the holomorphic vector pack

I am trying to understand this article. On page 307 section 2:

I have a Kähler collector $ M $ shaped like Kähler $ omega $, we define the Lefschetz operator: $ L ( alpha) = omega wedge alpha $.

Here in the article we have a locally free sheaf $ F $, then there is a metric in $ F $. How do we define a metric in $ F $ Or did you mean a vector package and a locally free property means that the section bundle of this vector package is locally free? It also defines the $ bar { partial} $ Laplacian $ Delta $, so I guess the vector pack is also holomorphic.

It also states that $ (L, Delta) = -i theta $ where $ theta $ It is the curvature of the metric connection. I tried to search the literature, but I could only find the identities of Kähler in the variety itself. I don't know how to prove this relationship. There was no specific metric connection. Does that mean that the shape of curvature only depends on the metric and not on the metric connection?