text:
\section{Introduction}

\subsection{Statement of the main results}
The goal of this paper is to prove the following results on the log singularity for real analytic functions.
\begin{thm} \label{Thm::W11_intro_1}
Let $f$ be a real analytic germ at the origin in $\R^n$, $n \geq 2$. Suppose the codimension of the zero set of $f$ at $\0$ (denoted $\operatorname{codim}_{\0} (Z_f)$) is at least $2$; then there exists some neighborhood $U$ of $\0$ such that $\log |f| \in W^{1,1} (U)$.
\end{thm}
We can also formulate a global version as follows.
\begin{thm} \label{Thm::W11_intro_2}
Let $f$ be a real analytic function in some neighborhood of a bounded open set $U$ of $\R^n$. Suppose the codimension of the zero set of $f$ in $U$ is at least $2$; then $\log |f| \in W^{1,1}(U)$.
\end{thm}

We remark that the problem is purely local, and \rt{Thm::W11_intro_2} follows from \rt{Thm::W11_intro_1} by a simple local-to-global argument.

We say that $g$ is in $W^{1,1}_\mathrm{loc}(\0)$ (or $L^p_{\mathrm{loc}}(\0)$) at $\0$ if there exists a neighborhood $U$ of $\0$ such that $g \in W^{1,1}(U)$ (or $L^p(U)$), and we say that $g$ has an isolated zero at $\0$ if there exists a neighborhood $U$ of $\0$ such that $Z_f \cap U = \{ \0 \}$, where $Z_f$ denotes the zero set of $f$.
As a special case to \rt{Thm::W11_intro_1}, we conclude that $\log |f| \in W^{1,1}_\mathrm{loc}(\0)$ if $f$ has isolated zero at the origin.

Our results are sharp up to both the codimension of the zero set and the integrability exponent. The result obviously fails if the codimension of the zero set is $1$; take for example $f(x,y) = xy$ defined on $\R^2$, where the zero set is the union of $x$ and $y$ axis. Let $U$ be any neighborhood of the origin, we have
\[
  \int_U \left| \nabla \log |f| \right|^p \, dV = \int_U \left| \frac{\nabla f}{f} \right|^p \, dV \approx
  \int_U \left| \frac{x}{xy} \right|^p \, dV + \int_U \left| \frac{y}{xy} \right|^p \, dV
  = \int_U \frac{1}{|y|^p} \, dV +  \frac{1}{|x|^p} \, dV
\]
which is finite if and only if $p<1$.
On the other hand, for any $\varepsilon>0$ and $2 \leq n-d \leq n$, there exists a real analytic function $f$ with $\operatorname{codim}_\0 Z_f = n-d$, and $\frac{|\nabla f|}{f} \notin L^{1+\varepsilon}_\mathrm{loc} (\0)$; see Example \ref{Ex::sharp}.

For the proof of \rt{Thm::W11_intro_1}, it is not difficult to show that $\log |f| \in L^p_{\mathrm{loc}}$ for any $0 <p< \infty$; the main point is to prove the derivative estimate: $|\nabla \log|f|| = \frac{|\nabla f|}{|f|} \in L^1_{\mathrm{loc}}$. In particular, the $L^1$ integrability reflects the competition between the zeros of the function and its gradient near the singular points where $\nabla f = 0$. It should be noted that in many special cases one can do much better; take the simple example $u = |x|^k$ in $\R^n$. Then $|\nabla u (x)| \approx |x|^{k-1}$, and  $| \nabla \log |u(x)| | = \frac{|\nabla u(x) |}{ |u(x)| } \approx \frac{1}{|x|} \in L^p_\mathrm{loc}$, for any $p<n$.
In fact, we prove the following result which provides a sharp upper bound for the integrability exponent.

\begin{thm} \label{Thm::blow-up_intro}
 Let $f$ be a real analytic function germ at the origin in $\R^n$, $n\geq 2$, with $f(\0) = 0$.  Suppose $\operatorname{codim}_\0 Z_f = n-d \geq 1$ (which is the same as saying $f$ is not identically $0$). Then for each (sufficiently small) neighborhood $\U$ of $\0$,
 \[
   \int_\U \left| \frac{\nabla f}{f} \right|^{n-d} \, dV(x) = \infty.
 \]
 Furthermore, the integrability exponent $n-d$ is sharp in the sense that there exists real analytic function $f$ such that $ \operatorname{codim}_\0 Z_f = n-d$ and $f \in L^{p}_\mathrm{loc}(\0)$, for any $p<n-d$ (See Example \ref{Ex::sharp}.)
\end{thm}

\subsection{Connection with unique continuation problem}
Our original motivation behind  \rt{Thm::W11_intro_1} comes from study of the unique continuation properties of the differential inequality
\eq{Lap_diff_ineq}
  \left| \Delta u \right| \leq V \left| \nabla u \right|, \quad V \in L^p_{\mathrm{loc}}(\0), \; p>0.
\end{equation}
We say that the differential inequality \re{Lap_diff_ineq} satisfies the strong unique continuation property (SUCP), if every solution that vanishes to infinite order at a point vanishes identically. Here a square-integrable function $u$ is said to vanish to infinite order at a point $x_0$ if for all $m \geq 0$,
\[
   \lim_{r \to 0} \frac{1}{r^m} \int_{|x-x_0| < r} |u(x)|^2 \,  dV(x) = 0.
\]
The problem asks for the optimal (minimum) value for $p$ such that \re{Lap_diff_ineq} satisfies the SUCP.

In $\R^2$, the unique continuation properties of \re{Lap_diff_ineq} is closely related to that of the following $\overline\partial$ differential inequality:
\eq{dbar_diff_ineq}
  |\partial{}_{\overline z} w | \leq V |w|, \quad \text{or}
  \quad |\nabla w | \leq V |w|, \quad V \in L^p_\mathrm{loc}(\0).
\end{equation}
Indeed, if \re{dbar_diff_ineq} satisfies the SUCP, then by setting $w = \partial{}_z u $ and using the identity $\Delta u = \partial{}_{\overline z} \partial{}_z u$, we see that \re{Lap_diff_ineq} also satisfies the SUCP (However, the converse is not true, see the preprints \cite{P-Z22} and \cite{F-P18}.)

Regarding the optimal $p$ value, the following conjecture is natural: \re{Lap_diff_ineq} has the SUCP if $V \in L^n_\mathrm{loc}$. To see why, take a function $u$ that vanishes to a finite order $k$ at the origin (so in particular it satisfies the SUCP). Then one would expect $\Delta u$ to vanish to order $k-2$ and $\nabla u$ to vanish to order $k-1$, i.e. $\frac{\Delta u}{|\nabla u|}$ just fails being in  $L^n_{\mathrm{loc}}$. In fact, the function $u(x)= e^{-|x|^{-\varepsilon}}$ for suitable value of $\varepsilon$ shows that SUCP does not hold for $|\Delta u| \leq A|u|$, with $A \in L^p_\mathrm{loc}$, $p<n$.

Now, we can consider the inverse problem to the above unique continuation problem. Namely, we ask what is the optimal (maximum) value of $p$ such that a differential inequality of the form \re{Lap_diff_ineq} and \re{dbar_diff_ineq} with $V \in L^p_\mathrm{loc}(\0)$ hold for all functions satisfying the SUCP property? (such functions are also called quasi-analytic.)

Viewed in this language, our results can now be formulated as follows: $p=1$ is the optimal value for the differential inequality $| \nabla f | / f \in L^p_\mathrm{loc} (\0)$ to hold for all real analytic functions such that $\operatorname{codim}_\0 \geq 2$.
As an immediate follow-up question, we can make the following conjecture:
\begin{quote}
  Suppose $f$ is a real analytic germ at the origin in $\R^n$. Suppose the codimension of the zero set of $f$ is at least $2$. Then $\Delta f/ |\nabla f| \in L^1_\mathrm{loc}(\0)$.
\end{quote}

\subsection{An application to local invariants}
We now give one application of \rt{Thm::W11_intro_1}. First, we recall the Łojasiewicz gradient inequality, which states that given $f$ a real analytic germ at $\0$ such that $f(\0)=0$, there exists some $\beta \in (0,1)$ and some neighborhood $\mathcal{V}$ of $\0$ such that
\begin{equation} \label{L_ineq}
  | \nabla f (x)| \geq c_\beta |f(x)|^\beta, \quad \text{for all $x \in \mathcal{V}$.}
\end{equation}
Here we can assume $\nabla f(\0)= 0$ since otherwise the inequality is trivial.
The \emph{Łojasiewicz exponent of $f$ at $\0$}, denoted by $\beta_0$, is defined to be the infimum of all $\beta$ satisfying \re{L_ineq}. Note that $\beta_0$ itself may not satisfy inequality of the form \re{L_ineq}, as the constant $c_\beta$ could blow up as $\beta \to \beta_0$.

For any real analytic function $f$ with $f(\0)=0$, we define the \emph{singularity exponent of $f$ at $\0$}, denoted by $\alpha_0$, as the supremum of all $\alpha >0$ such that there exists some neighborhood $\mathcal{V}$ of $\0$ with $\int_{\mathcal{V}} |f|^{-\alpha} < \infty$.
By \rt{Thm::W11_intro_1} and inequality \re{L_ineq}, if $\operatorname{codim}_\0 Z_f \geq 2$, then there exists a neighborhood $\U$ of the origin such that
\[
  \infty > \int_{\U} \left| \frac{\nabla f}{ f } \right|
  \geq \int_{\U} \frac{1}{|f|^{1-\beta}}
\]
for any $\beta > \beta_0$. This implies that $(1-\beta) < \alpha_0$.
Letting $\beta \to \beta_0$, we get $1- \beta_0 \leq \alpha_0$, or
\[
\alpha_0 + \beta_0 \geq 1.
\]
We summarize the result in the following corollary:
\begin{cor}
  Let $f$ be a real analytic germ at the origin in $\R^n$ with $f(\0) = 0$. Suppose the codimension of the zero set of $f$ at $\0$ is at least $2$. Let $\alpha_0$ and $\beta_0$ be the singularity exponent and the Łojasiewicz exponent of $f$ at $\0$, respectively. Then
	\[
	\alpha_0 + \beta_0 \geq 1.
	\]
\end{cor}

Here is an outline of the paper. In Section 2, we recall the definition of algebraic, semialgebraic and real analytic sets. We then give two equivalent ways of defining the dimension and codimension of an analytic set.
Lastly we prove some results on removable singularity for Sobolev spaces, which are used later to show that the derivatives in the sense of distribution exist across the singularity set.
Section 3 contains the proof of the main results.  We first prove \rt{Thm::W11_intro_1}. \rt{Thm::W11_intro_2} then follows easily by covering the open set with small balls and applying \rt{Thm::W11_intro_1} for each ball. To prove \rt{Thm::blow-up_intro}, we first show that  $|\nabla f| /f \notin L^n_\mathrm{loc}(\0)$ for any real analytic function $f$ such that $f(\0) =0$ (see \rt{Thm::n_blup}.) For the general case we integrate near a neighboring smooth point of $Z_f$ of dimension $d$ and use \rt{Thm::n_blup}.

The proof of \rt{Thm::W11_intro_1} is organized into four steps. First, we prove the statement when the function is a polynomial in $\R^2$ and the zero set has isolated zero at the origin. We then use induction to show that the statement holds for polynomials with isolated zero in $\R^n$, for all $n\geq 2$. Next, we prove the case for polynomials with zero set of codimension at least $n-2$ by reducing the problem to the case of isolated zero.
Finally, we prove the statement for real analytic functions. The key of the proof is to use the ``finiteness" properties of polynomials and (semi)algebraic sets.

The analogous results for smooth functions fail rather spectacularly. Take $u_\varepsilon = e^{-|x|^{-\varepsilon}}$, $\varepsilon>0$. Then $ \log u_\varepsilon = - |x|^{-\varepsilon}$. It follows that for any $p >0$, there exists some large number $\varepsilon(p)$ such that $\log u_\varepsilon \notin L^p_\mathrm{loc}(\0) $ whenever $\varepsilon > \varepsilon(p)$.

We use $x \lesssim y$ to mean that $x \leq Cy$ where $C$ is a constant independent of $x,y$, and we write $x \approx y$ if $x \lesssim y$ and $y \lesssim x$. For an open subset $\Omega$ of $\R^n$, we denote by $C^\infty_c(\Omega)$ the space of $C^\infty$ function with compact support in $\Omega$.

\begin{ack}
  The authors would like to thank Yifei Pan for many inspiring and helpful discussions over the course of this project, and for showing us the paper of Harvey-Polking \cite{H-P70}.
\end{ack}

\section{Preliminaries}
In this section we review the definitions and concepts used in later proofs.

Let $\mathcal M$ be a connected $n$-dimensional analytic manifold and $\mathcal{U}$ be an open subset of $\mathcal M$. We denote by $ \mathcal{O}^\R_{\mathcal{U}}$ the ring of real analytic functions from $\mathcal{U}$ to $\R$. If $p \in \mathcal M$, we denote by $\mathcal{O}^\R_{\mathcal M,p}$ the set of real analytic germs at $p$.
We denote the set of polynomials in $\R^n$ by $\R[x] = \R[x_1,\dots, x_n]$.
\begin{defn}
 Let $A \subset \mathcal{O}_{\mathcal M}^\R$. We define the \emph{vanishing locus} of $A$, $V(A)$, to be the set of points:
 \[
 V(A):= \{ x \in \mathcal M: f(x) = 0, \: \forall \: f \in A \}.
 \]
\end{defn}
\begin{defn}
 An \emph{algebraic subset of $\R^n$} is a set of the form $V(\mathcal{A})$, where $\mathcal{A} \seq \R[x]$  A subset $X \subset \mathcal M$ is a \emph{(real) analytic subset} of $\mathcal M$ if $X$ is closed in $\mathcal M$ and, for all $x \in X$, there exists an open neighborhood $\mathcal{W} $ of $x$ in $\mathcal M$ and a finite collection $f_1, \cdots, f_j \in \mathcal{O}^{\omega}_{\mathcal{W}}$ such that $\mathcal{W} \cap X = V (f_1, \cdots, f_j)$.
\end{defn}
\begin{defn}
Let $X$ be an analytic subset of $\mathcal M$. A point $p \in X$ is called \emph{smooth, of dimension $d$}, if there exists an open neighborhood $\mathcal{W}$ of $p$ in $\mathcal M$ such that $\mathcal{W} \cap X$ is an analytic sub-manifold of $\mathcal{W}$ of dimension $d$. In other words, there exists $f_{d+1}, \dots, f_n \in \mathcal{O}_{\mathcal{W}}^{\R}$ such that $\mathcal{W} \cap X = V(f_{d+1}, \dots, f_n)$ and  $\nabla f_{d+1}(x), \cdots, \nabla f_n(x) $ are linearly independent at each $x \in \mathcal{W}$,

We denote the set of smooth points of $X$ by $\mathring{X}$, and the set of smooth points of dimension $d$ by $\mathring{X}^{(d)}$.
\end{defn}

\df{Def::dim}
The \emph{dimension} (over $\R$), $\dim X$ of a real analytic set $X \subset \mathcal M$ is the largest $m$ such that $\mathring{X}^{(m)}$ is non-empty. The \emph{dimension of $X$ at a point} $p \in X$, denoted by $\dim_p X$, is the largest $d$ such that $p$ is in the closure of $\mathring{X}^{(d)} $. We say that $X$ is \emph{pure-dimensional} if the dimension of $X$ at each point $p \in X$ is independent of $p$.

The \emph{codimension $\operatorname{codim}(X)$ of a real analytic set $X \subset \mathcal M$} is defined as $n-d$, where $d =\dim(X)$. The \emph{codimension of $X$ at a point $p \in X$} is defined as $n-d_p$, where $d_p := \dim_p X$.
\end{defn}

We will also need the notion of semi-algebraic sets.
\begin{defn}
  The class of \emph{semialgebraic sets of $\R^n$} is the smallest collection of subsets containing all $\{x \in \R^n: P(x) >0, P \in \R[x] \}$ which is stable under finite intersection, finite union and complement.
\end{defn}
From the above definition, it is clear that every semialgebraic set is the union of finitely many semialgebraic subsets of the form
\[
 \{ x \in \R^n: P(x) = 0 \; \text{and} \; Q_i (x)>0, \: i=1, \dots, \ell; \; P, Q_i \in \R[x]  \}.
\]
In particular, every algebraic set is semialgebraic.
If $X$ is a semialgebraic set,
we can define the notion of smooth points of $X$, the (co)dimension of $X$, and the (co)dimension of $X$ at a point exactly as we did for analytic subsets in \rd{Def::dim}.

The following result is well-known. See for example \cite{St_03} for a direct proof.
\begin{prop} \label{Prop::SA_smooth}
  Let $X \subset \R^n$ be a semialgebraic set. For each $1 \leq k \leq \dim X$, the set of smooth points of $X$ of dimension $k$ is semialgebraic.
\end{prop}

One can also give an (equivalent) definition of dimension and co-dimension by using the concept of stratification for semi-analytic sets (see the classic paper \cite{B-M88}.) Let $X$ be an (semi)analytic subset of $M$; there exists a collection $\{ \mathcal{A}_\alpha \}_\alpha$ of subsets $M$ such that
\begin{itemize}
    \item
    X is the disjoint union of the $\mathcal{A}_\alpha$;
    \item
    Each $A_\alpha$ is an analytic submanifold of $M$ and a connected semianalytic subset of $M$;
    \item
    (``Condition of the frontier") If $\mathcal{A}_\alpha \cap \overline {\mathcal{A}_\beta} \neq \emptyset$, then $\mathcal{A}_\alpha \seq \overline{\mathcal{A}_\beta}$ and $\dim \mathcal{A}_\alpha < \dim \mathcal{A}_\beta$. \item
    $ \{ \mathcal{A}_{\alpha} \}_\alpha$ is locally finite.
\end{itemize}

We can define the \emph{dimension of a (semi)analytic set $X$} by $\dim X = \max_k \dim \mathcal{A}_k$. The definition is independent of the stratification: $\dim X = d$ if and only if $X$ contains an open set homeomorphic to an open ball in $\R^d$, but not an open set homeomorphic to an open ball in $\R^n$, $n >d$. It is also clear that the definition agrees with that of Hausdorff dimension.

Later in the proof, we will need to show that the derivative extends across some singular sets, a phenomenon known as removable singularity. We first state such problem in a slightly more general form. Let $P(x, \Omega)$ be a linear partial differential operator on an open set $\Omega$ in $\R^n$, and let $A$ be a closed subset of $\Omega$. Given a class of distributions on $\mathcal{F} (\Omega)$, we say (following Harvey-Polking \cite{H-P70}) the set $A$ is \emph{removable} for $\mathcal{F}(\Omega)$ if each $f \in \mathcal{F}(\Omega)$ that satisfies $P(x,\Omega)f = 0$ in $\Omega \setminus A$ also satisfies $P(x, \Omega) f = 0 $ in $\Omega$. Questions can then be asked as to what conditions on the coefficients of $P(x,\Omega)$ and on the set $A$ will ensure that $A$ is removable for $\mathcal{F}(\Omega)$. In the remainder of this section, we will prove two results of this type.

We will let $d(x,E)$ to denote the Euclidean distance from the point $x$ to the set $E \subset \R^n$. We define the $\varepsilon$-neighborhood of $E$ by $E_{\varepsilon} := \{ x \in \R^n: d(x, E) <\varepsilon \}$.
The unit ball in $\R^n$ is denoted by $\B$, and the punctured ball $\B \setminus \{ \0 \}$ by $\B_\ast$.
We begin with a result for isolated singularity.
\pr{Prop::rem_sing_1}
Let $\alpha$ be a multi-index with $| \alpha | = m$, for $1 \leq m<n$. Suppose $f \in L^{\frac{n}{n-m}} (\B)$, $g \in L^1 (\B)$, and
  \[
     D^{\alpha} f =g,  \quad (D^{\alpha} = \partial{}_{x_1}^{\alpha_1} \cdots \partial{}_{x_n}^{\alpha_n} )
  \]
 holds in the sense of distributions in $\B_{\ast}$. Then $D^{\alpha} f = g$ holds in the sense of distributions in $\B$.
\end{prop}
\nid \tit{Proof.}
Define a smooth function $\varphi \in C^{\infty}_c (\R^n)$ satisfying
\[
  \varphi (x) =
  \begin{cases}
  0 & \text{if $|x| < 1$};  \\
  1 & \text{if $|x| > 2$}.
  \end{cases}
\]
Let $\varphi_n (x) := \varphi(nx)$. Then $\varphi_n$ satisifes
\[
 \varphi_n (x) =
 \begin{cases}
 0 & \text{if $|x| < \frac{1}{n} $}; \\
 1 & \text{if $|x| > \frac{2}{n}$}.
 \end{cases}
\]
Let $\psi \in C^{\infty}_c (\B)$, and set $\psi_n := \psi \varphi_n$. Since $\psi_n \in C^{\infty}_c (\B_\ast)$, by assumption we have
\eq{ibp}
  \int_{\B} f (D^\alpha \psi_n) = (-1)^{m} \int_{\B} g \, \psi_n.
\end{equation}
Since $\psi_n$ converges to $\psi$ point-wise in $\B$ and $g \in L^1 (\B)$, by the Dominated Convergence theorem the integral on right-hand side converges to
\[
  (-1)^m \int_{\B} g \, \psi.
\]
For the integral on the left-hand side we have
\begin{align} \label{lhs}
 \int_{\B} f (D^\alpha \psi_n)
 &= \int_{\B} f D^\alpha (\psi \varphi_n) \\ \nonumber
 &= \int_{\B} f (D^\alpha \psi) \varphi_n + \sum_{ \substack{|\beta| \geq 1 \\ \beta \leq \alpha}  }  \int_{\B}   f ( D^{\alpha - \beta} \psi)  (D^{\beta} \varphi_n).
\end{align}
Since $f \in L^{\frac{n}{n-m}} (\B) \subset L^1(\B)$, we have
\[
  \int_{\B} f ( D^\alpha \psi) \varphi_n \xrightarrow{\varepsilon \to 0}  \int_{\B} f (D^\alpha \psi).
\]
By the definition of $\varphi_n$, we see that $D^{\beta} \varphi_n$ is compactly supported in $\{x: |x| < \frac{2}{n} \}$, and
\[
\left| D^{\beta} \varphi_n (x) \right|
= n^{|\beta|} \left| \partial{}_y^{\beta} \varphi (nx) \right|
\lesssim n^{|\beta| }.
\]
Denote by $B(x,r)$ the ball centered at $x$ with radius $r$. By Hölder's inequality,
\begin{align*}
  \int_{\B} \left|  f ( D^{\alpha - \beta} \psi)  (D^{\beta} \varphi_n) \right|
 &\leq C \left( \int_{|x|< \frac{2}{n}  } |f|^{\frac{n}{n-|\beta|}} \right)^{\frac{n-|\beta|}{n}}  \left( \int_{|x| < \frac{2}{n}} \left| \partial{}_x^{\beta} \varphi_n \right|^{\frac{n}{|\beta|}} \right)^{\frac{|\beta|}{n}} \\
 &\leq  C \| f \|_{L^{\frac{n}{n- |\beta | }} B(\0, \frac{2}{n}) }  \left( \left( \frac{2}{n} \right)^n n^{|\beta| \cdot \frac{n}{|\beta|}} \right)^{\frac{|\beta|}{n}}   \\
 &\leq C \| f \|_{L^{\frac{n}{n- |\beta | }} B(\0, \frac{2}{n}) } .
\end{align*}
Since $|\beta| \leq |\alpha| = m$, we have $\| f \|_{L^{\frac{n}{n-|\beta|}} \left( B(\0, \frac{2}{n})  \right) } \leq \| f \|_{L^{\frac{n}{n- m }} \left( B(\0, \frac{2}{n})  \right) } $ which converges to $0$ as $n \to \infty$. Putting the results together and letting $n \to \infty$ in \re{ibp} and \re{lhs} we get
\[
  \int_{\B} f (D^\alpha \psi)  = (-1)^m \int_{\B} g  \psi, \quad \psi \in C^{\infty}_c (\B).
\]
Hence $D^\alpha f = g $ in the sense of distribution in $\B$.
\hfill \qed

Next, we prove a removable singularity result for arbitrary set of Hausdorff dimension at most $n-2$.
\begin{prop} \label{Prop::rem_sing_2}
 Let $\Omega$ be an open set in $\R^n$ and let $A \subset \Omega$ be a compact set of dimension at most $n-2$. Suppose $f \in C^\infty (\Omega \setminus A) \cap L^p_\mathrm{loc}(\Omega)$ for some $p>2$, and also $\nabla f \in L^1_\mathrm{loc}(\Omega)$. Then $\nabla f$ is the derivative of $f$ in $\Omega$ in the sense of distributions.
\end{prop}
\begin{proof}
By Harvey-Polking's lemma (\rl{Lem::HP2}), there exists a family of cut-off functions $ \{ \widetilde \chi_\varepsilon \}_{\varepsilon>0} \in C^\infty_c(\Omega)$ such that $0 \leq \widetilde \chi_\varepsilon \leq 1 $, $\widetilde \chi_\varepsilon \equiv 1$ in a neighborhood of $A$, and supp $\widetilde \chi_\varepsilon \seq A_\varepsilon$. Furthermore, the following estimate holds
\begin{equation} \label{chi_ve_grad_est}
  \left| \nabla \widetilde \chi_\varepsilon \right|_{L^{p'}(\Omega)} \leq C_{\alpha,n} \varepsilon^{ l - 1 } \left( \Lambda_{n-lp'}  (A) + \varepsilon \right)^{\frac{1}{p'}}, \quad n-lp' > 0.
\end{equation}
Define $ \chi_\varepsilon := 1 - \widetilde \chi_\varepsilon$, so that $\chi_\varepsilon \equiv 0$ near $A$ and $\chi_\varepsilon \equiv 1$ in $\Omega \setminus A_\varepsilon$. Moreover, $\chi_\varepsilon$ converges to $1$ point-wise in $\Omega \setminus A$ as $\varepsilon \to 0$, and $\chi_\varepsilon$ satisfies estimate \re{chi_ve_grad_est}.
For $\phi \in C^\infty_c(\Omega)$, we can integrate by parts on the domain $\Omega \cap \{ \chi_\varepsilon \geq 0 \}$ to get
\begin{equation} \label{ibp_cutoff}
  \int_{\Omega} (\nabla f) \chi_\varepsilon \phi = - \int_{\Omega} f \nabla (\chi_\varepsilon \phi) = - \int_{\Omega} f (\nabla \chi_\varepsilon) \phi - \int_{\Omega} f \chi_\varepsilon \nabla \phi.
\end{equation}
Since $\nabla f \in L^1_\mathrm{loc}(\Omega)$, by the Dominated Convergence theorem, the integral on the left converges to $\int_{\Omega} (\nabla f) \phi$ as $\varepsilon \to 0$. Similarly, the second integral on the right converges to $\int_{\Omega} f \nabla \phi$. By H\"older's inequality with $\frac{1}{p} + \frac{1}{p'} =1$, we get
\begin{align*}
  \int_{\Omega} \left| f (\nabla \chi_\varepsilon) \phi \right|
  &\lesssim  \| f \|_{L^p(\Omega)} \| \nabla \chi_\varepsilon \|_{L^{p'}(\Omega)}
  \\ &\leq C_{\alpha,n} \varepsilon^{l- 1} \left( \Lambda_{n-lp'} (A) + \varepsilon \right)^{\frac{1}{p'}} \| f \|_{L^p(\Omega)}.
\end{align*}
Since $p'<2$ ($p>2$), we can choose $l>1$ such that $lp'<2$, so $n-lp'> n-2$. By the assumption $\dim A \leq n-2$, we have $\Lambda_{n-lp'}(A)=0$, and therefore
\[
 \int_{\Omega \setminus A} \left| f (\nabla \chi_\varepsilon) \phi \right|
 \leq C_{\alpha,n} \varepsilon^{l-1+\frac{1}{p'}} \| f \|_{L^p(\Omega)}.
\]
In particular the integral converges to $0$ as $\varepsilon \to 0$. Taking the limit in \re{ibp_cutoff} as $\varepsilon \to 0$, we get
\[
  \int_\Omega (\nabla f) \phi = - \int_\Omega f (\nabla \phi),
  \quad \phi \in C^\infty_c(\Omega).
\]
In other words, $\nabla f$ is the derivative of $f$ in $\Omega$ in the sense of distributions.
\end{proof}
\section{Proof of the Main Theorem}
In this section we prove the main results of the paper, \rt{Thm::W11_intro_1} and \rt{Thm::W11_intro_2}.

\pr{Prop::logLp}
Let $f$ be a real analytic germ at $\0$ in $\R^n$ and suppose $f(\0)=0$; then $ \log |f| \in L^{p}_{\mathrm{loc}}(\0) $, for any $0 < p < \infty$.
\end{prop}
\nid \tit{Proof.}
By Weierstrass Preparation theorem for real analytic functions (\cite[Theorem 6.1.3]{K-P02}, one can find a real analytic coordinate $x = (x_1, \dots, x_n)$ defined in some cube with length $\delta$ centered at $\0$ such that
\eq{Weierstrass_1}
  f (x) = u (x) \left( x_n^m + a_1(x') x_n^{m-1} + \cdots +  a_n (x') \right), \quad x'=(x_1, \dots, x_{n-1}).
\end{equation}
Here $m$ is the vanishing order of $f$ in the $x_n$ direction, $u(\0) \neq 0$ and $a_j$ are real analytic functions in $n-1$ variables with $a_j (\0') = 0$. We can rewrite \re{Weierstrass_1} as
\[
  f(x) = u (x) \prod_{j=1}^{m} ( x_n -  \xi_j (x') ),  \quad x \in [-\delta, \delta]^n,
\]
where the roots $\xi_j (x')$ are either real or complex numbers.
Assuming that $|f|< 1$ in $[-\delta, \delta]^n$, we have
\begin{align*}
\left| \log |f (x)  | \right|
&= \log \frac{1}{|f (x) |}
= \log \frac{1}{|u (x) |} + \sum_{j=1}^m \log \frac{1}{| x_n - \xi_j (x') | },
\quad  x\in [-\delta, \delta]^n.
\end{align*}
Then
\[
  \left| \log |f (x) | \right|^p
  \leq C_p \left( \left( \log \frac{1}{|u (x) |}  \right)^p
  + \sum_{j=1}^m \left( \log \frac{1}{|x_n - \xi_j (x') | } \right)^p  \right),  \quad x\in [-\delta, \delta]^n.
\]
Hence
\[
  \int_{[-\delta, \delta]^n }  \left| \log |f(x)| \right|^p \, dx
 \lesssim \int_{[-\delta, \delta]^n } \left( \log \frac{1}{|u|}  \right)^p \, dx
  + \sum_{j=1}^m \int_{[-\delta, \delta]^n } \left( \log \frac{1}{|x_n - \xi_j (x') | } \right)^p  \, dx.
\]
The first integral is bounded since $u$ does not vanish near $\0$. We estimate each integral in the sum:
\[
   \int_{[-\delta, \delta]^n } \left( \log \frac{1}{|x_n - \xi_j (x') | } \right)^p
   = \int_{[-\delta, \delta]^{n-1} } \left( \int_{[-\delta, \delta]}  \left( \log \frac{1}{|x_n - \xi_j (x') | } \right)^p \, dx_n\right) \, d x'.
\]
The inner integral is bounded by (up to a constant independent of $x'$),
\[
   \int_{ [-\delta, \delta] } \left( \log \frac{1}{|x_n| } \right)^p  \, d x_n.
\]
Since $\log \frac{1}{|x_n|} \leq C_{\varepsilon} |x_n|^{-\varepsilon} $, for any positive $\varepsilon > 0$, we see that for each $0<p<\infty$, the above integral is bounded by some constant $C_p$ which is uniform in $x'$. This shows that $\log |f|$ is locally $L^p$ for any $0<p< \infty$.
\hfill \qed

In what follows we let $S^n$ to denote the unit sphere in $\R^{n+1}$, and we write $d \si (v)$ for the surface area element on $S^n$. For a vector $v \in \R^n$, we denote the hyperplane of its orthogonal complement by $ v^{\perp} = \{ w \in \R^n:  w \cdot v = 0 \}$.

\begin{lemma} \label{Lem::Sn_int_avg}
Let $f$ be a $C^1$ function in the unit ball in $\R^n$. For each $\omega \in S^{n-1}$ and $\rho \in (0, \infty)$, the following holds
\[
   | \nabla_{\omega} f(\rho \omega) |
   \leq C_n \int_{S^{n-2} ( \omega^{\perp}) } | \nabla_{\omega}^{v^{\perp}} f (\rho \omega) | \, d\si (v) .
\]
\end{lemma}
\begin{proof}
We first claim that for any vector $w \in \R^{\ell+1}$,
\begin{equation} \label{Sl_int_avg}
  |w| = c_\ell \int_{S^\ell} \left| w \cdot v \right| \, d \si(v).
\end{equation}
To prove the claim, we estimate the right-hand side of the integral \re{Sl_int_avg}. We use spherical coordinates to parametrize the surface $S^\ell$. Define map $\varphi: [0,\pi]^{\ell-1} \times [0,2\pi] \to S^\ell$ by
\eq{sphere_par}
  \varphi: (\theta_1, \theta_2, \cdots \theta_\ell) \mapsto ( \cos \theta_1, \sin \theta_1 \cos \theta_2, \sin \theta_1 \sin \theta_2 \cos \theta_3, \dots, \left( \prod_{i=1}^{\ell-1} \sin \theta_i \right) \cos \theta_\ell, \prod_{i=1}^{\ell} \sin \theta_i),
\end{equation}
where $\theta_1, \dots, \theta_{\ell-1} \in [0, \pi]$ and $\theta_\ell \in [0, 2\pi]$. The surface area element is given by
\eq{sphere_vol_ele}
  d \si(v) = \sin^{\ell-1} \theta_1 \sin^{\ell-2} \theta_2 \cdots \sin \theta_{\ell-1} \, d \theta_1 \cdots d \theta_\ell.
\end{equation}
Since both sides of \re{Sl_int_avg} are rotation invariant, we can take $w = (|w|, 0, \dots, 0)$. Further, we may assume that the angle between $v$ and $w$ is given by the first coordinate $\theta_1$.
Thus
\begin{align} \label{Sn_int_polar}
 \int_{S^\ell} \left| w \cdot v \right| \, d \si(v)
 &= \int_{\theta_\ell=0}^{2\pi}
 \int_{\theta_{\ell-1}=0}^{\pi}  \cdots \int_{\theta_1 =0}^\pi
 |w| |\cos \theta_1| \sin^{\ell-1} \theta_1 \sin^{\ell-2} \theta_2 \cdots \sin \theta_{\ell-1} \, d \theta.
\end{align}
We have
\begin{align*}
  \int_0^\pi |\cos \theta_1| \sin^{n-1} \theta_1 \, d \theta_1
  &= \int_0^{\pi/2} \cos \theta_1 \sin^{\ell-1} \theta_1 \, d \theta_1 - \int_{\pi/2}^\pi \cos \theta_1 \sin^{\ell-1} \theta_1 \, d \theta_1
  \\ &= 2 \int_0^1 u^{\ell-1}\, du = \frac{2}{\ell},
\end{align*}
and also $\int_0^\pi \sin^k \theta \, d\theta = c_k$, $1 \leq k \leq n-2$. Hence \re{Sl_int_avg} follows from \re{Sn_int_polar}, proving the claim.

Now, we can similarly parametrize the ball of radius $\rho$ using polar coordinates. For fixed $\omega \in S^{n-1}$ and $\rho \in (0,\infty)$, we define the map $\varphi_\rho: [0, \pi]^{n-2} \times [0, 2 \pi] \to S^{n-1}$ by
\[
  \varphi_\rho (\theta):= \rho \varphi(\theta), \quad \varphi_\rho(0) = \rho \omega.
\]
Set $\widetilde f (\theta) := f ( \varphi_\rho(\theta))$ and $\partial{}_\theta \varphi_{\rho}(0) := \left( \partial{}_{\theta_1} \varphi_\rho (0), \dots, \partial{}_{\theta_{n-1}} \varphi_\rho (0) \right) $; then
\begin{equation} \label{ang_der_iden}
  \nabla_\theta \widetilde f (0) = \nabla f (\varphi_\rho(0) ) \cdot \partial{}_\theta \varphi_\rho (0) = \nabla f ( \rho \omega) \cdot \partial{}_\theta \varphi_\rho (0) = \nabla_{\omega} f (\rho \omega).
\end{equation}
where we adopt the notation
\[
  \nabla f ( q) \cdot \partial{}_\theta \varphi_\rho (0)
  := \left( \nabla f ( q) \cdot \partial{}_{\theta_1} \varphi_\rho(0), \dots, \nabla f ( q) \cdot \partial{}_{\theta_{n-1}} \varphi_\rho(0) \right).
\]

Applying \re{Sl_int_avg} to the vector $\nabla_{\theta} \widetilde f (0) \in \R^{n-1}$
and using \re{ang_der_iden}, we have
\begin{align*}
  \left| \nabla_\omega f(\rho \omega) \right|
 = \left| \nabla_\theta \widetilde f (0) \right|
&= c_{n-2} \int_{S^{n-2}} \left| \nabla_\theta \widetilde f (0) \cdot v \right| \, d\si(v)
\\ &\leq C_n \int_{S^{n-2}}  \int_{ \widetilde v \in S^{n-2} \cap v^\perp} \left| \nabla_\theta \widetilde f (0) \cdot \widetilde v \right| \, d \si (\widetilde v) \, d\si(v) \\ &= C_n \int_{S^{n-2}}  \left\{ \int_{ \widetilde v \in S^{n-2} \cap v^\perp} \left| \left( \proj_{v^\perp} \nabla_\theta \widetilde f (0) \right) \cdot \widetilde v \right| \, d \si (\widetilde v) \right\} \, d\si(v).
\end{align*}
Since $S^{n-2} \cap v^\perp = S^{n-3} \subset \R^{n-3}$, by \re{Sl_int_avg} again, the inner integral is equal to $\proj_{v^\perp} \nabla_\theta \widetilde f (0)$ up to a constant factor, which we absorb into $C_n$. Since $\proj_{v^\perp} \nabla_\theta \widetilde f (0) = \proj_{v^\perp} \nabla_\omega f(\rho \omega) = \nabla^{v^\perp}_\omega f (\rho \omega)$, we obtain from above
\begin{align*}
  \left| \nabla_\omega f(\rho \omega) \right|
  &\leq C_n \int_{S^{n-2}} \proj_{v^\perp} \nabla_\theta \widetilde f(0) \, d \si (v)
  \\ &= C_n \int_{S^{n-2}(\omega^\perp)} \left| \nabla^{v^\perp}_{\omega} f(\rho \omega) \right| \, d\si (v).
  \qedhere
\end{align*}
\end{proof}
\begin{lemma} \label{Lem::induct}
 In $\R^n$ with $n \geq 3$, we have
\eq{ind_est}
  \int_{\B^n } | \nabla g | \leq C \int_{S^{n-1}}  \left( \int_{v^{\perp}  \cap \B^n } | \nabla (g |_{v^{\perp}} )  | \, d V_{n-1} \right)  \, d \si (v).
\end{equation}
\end{lemma}
\begin{proof}
For a point $x \in \R^n$, we write it in spherical coordinate $x = \rho \omega$, with $\omega \in S^{n-1}$. Denote the spherical derivative in $\R^n$ by $\nabla_{\omega}$. We have
\[
  | \nabla g (\rho \, \omega ) | \approx | \partial{}_{\rho} g (\rho \omega) | + \frac{1}{\rho} | \nabla_{\omega} g (\rho \omega) |.
\]
Hence
\begin{align} \label{ind_int}
 &\int_{S^{n-1}}  \left( \int_{v^{\perp}  \cap \B^n } | \nabla (g |_{v^{\perp}} )  | \, d V_{n-1} \right)  \, d \si (v) \\ \nonumber
 &\quad \approx \int_{S^{n-1}} \left( \int_{\rho=0}^1 \rho^{n-2}  \left[ \int_{ S^{n-2} (v^{\perp}) }  |\partial{}_{\rho} g (\rho  \omega) |
 + \frac{1}{\rho} |\nabla_{\omega}^{v^\perp} g (\rho \omega) | \, d \si(\omega) \right] \, d \rho \right)  \, d \si(v),
\end{align}
where we set $S^{n-2} (v^{\perp}) = \{w: \| w \| =1, \; w \perp v \}$, and $\nabla_{\omega}^{v^{\perp}}$ denotes the tangential derivative in the $n-1$ dimensional hyperplane $v^{\perp}$. Switching the roles of $v$ and $w$, the above integral is equal to
\[
  \int_{S^{n-1}} \int_{\rho=0}^1 \rho^{n-2} \left( \int_{S^{n-2} (\omega^{\perp}) } |\partial{}_{\rho} g (\rho  \omega) |
  + \frac{1}{\rho} |\nabla_{\omega}^{v^\perp} g (\rho \omega) | \, d \si(v) \right) \, d \rho \,  \, d \si(w).
\]
By \rl{Lem::Sn_int_avg},
\[
   \int_{S^{n-2} (\omega^{\perp} ) } | \nabla^{v^{\perp}}_{\omega} g (\rho \omega) | \, d \si (v)
   \geq c_0 | \nabla_{\omega} g (\rho \omega ) |.
\]
On the other hand, we have $\rho^{n-2} \geq \rho^{n-1}$ for $\rho \in (0,1)$. It follows that the integral \re{ind_int} is bounded below by (up to a positive constant)
\[
  \int_{S^{n-1}}  \int_{\rho=0}^1 \rho^{n-1}  \left( | \partial{}_{\rho} g(\rho \omega) | + \frac{1}{\rho}  \left| \nabla_{\omega} g(\rho \omega)  \right| \right) \,  d \rho \, d \si (w) = \int_{\B^n} |\nabla g | \, dV.
\]
Thus we have proved \re{ind_est}.
\end{proof}

\begin{prop} \label{Prop::der_est_1}
  Let $f$ be polynomial function of degree $N$ in $n$ variables, $n\geq 2$. Suppose $f$ has isolated zero at the origin, that is, there exists some neighborhood $U$ of $\0$ so that $Z_f \cap  U = \{ \0 \}$. Then
 \[
  \int_{U} \left|  \nabla \log |f| \right| < C(N, n),
 \]
 where the constant $C(N,n)$ depends only on the degree $N$ and the dimension $n$.
\end{prop}
\begin{proof}
We prove the proposition in two steps. First, we prove it for $n= 2$, and then we prove the general case by induction on the dimension.

Without loss of generality we can take $U$ to be the unit ball $\B$ in $\R^2$. Moreover, since $f$ has isolated zero at the origin, we can assume $f >0$ in $\B_{\ast}$. We have
\begin{align} \nonumber
  \int_{\B} | \nabla  \log | f| |
 &\lesssim \int_{\theta =0}^{2 \pi}  \int_{r=0}^1\left( | \partial{}_r (\log f) (r, \theta) | + \frac{1}{r} | \partial{}_{\theta} (\log f) (r, \theta) | \right) r \, dr \, d \theta
 \\ \label{dim2_int}
 &= \int_{\theta =0}^{2 \pi} \int_{r=0}^1  \left| \frac{\partial{}_r f (r, \theta) }{f (r, \theta) } \right| r \, dr \, d \theta
 + \int_{r=0}^1  \int_{\theta =0}^{2 \pi} | \partial{}_\theta (\log f) (r, \theta) | \, d\theta \, d r.
\end{align}
For fixed $\theta \in [0, 2\pi]$, we write $ g_{\theta}(r):= f(r, \theta)$, which is a polynomial in $r$ of degree at most $N$. The coefficients of $g_\theta (r)$ are trigonometric polynomials of $\theta$ of degree at most $N$, so it could have most $2N$ zeros (\cite[p.150]{Po81}.) Therefore $g_\theta$ is not identically zero except possibly for $\theta$ in a finite set $\Gamma$. Take $\theta \notin \Gamma$ and let $m (\theta)$ be the vanishing order of $g_{\theta}$ at $r =0$, so that
\begin{gather*}
 f (r, \theta) = g_{\theta} (r) = c_0 r^{m (\theta) } + O \left( r^{m(\theta)+ 1 } \right),
\\
 \partial{}_r f (r, \theta) = \partial{}_r g_{\theta} (r) = m (\theta) c_0 r^{m(\theta) -1 } + O ( r^{m(\theta)} ).
\end{gather*}
Since the degree of $g_\theta(r)$ is at most $N$, we have $m (\theta) \leq N$ for all $\theta$. Taken together these facts, the first integral in \re{dim2_int} is bounded by
\[
  C_N \int_{\theta =0}^{2 \pi} \int_{r=0}^1 \frac{ m (\theta) r^{m(\theta) - 1}}{r^{m(\theta)} } r \, dr \, d \theta
  \leq C_N \int_{\theta =0}^{2 \pi} \int_{r=0}^1 \, dr \,  d \theta \leq C_N.
\]
For the second integral in \re{dim2_int}, we observe that for fixed $r$, $h_{r} (\theta):= f(r, \theta)$ is a trigonometric polynomial in $\theta$ which has the form
\[
  h_r(\theta) = \sum_{k= -N}^N c_k e^{ik \theta}.
\]
Then $h_r'(\theta) $ has the same form and can have at most $2N$ zeros.
Therefore one can divide the interval $[0, 2 \pi]$ into finitely many subintervals, on each of which $h_r (\theta)$ is monotone, and the number of subintervals is bounded above by $2N$. Taking one such subinterval $[a, b] \subset [0,2\pi]$, we have by monotonicity
\begin{align} \label{thh_int_est}
  \int_{a}^b | \partial{}_{\theta} (\log f) (r, \theta) | \, d \theta
  &= \left| \int_{a}^b \partial{}_{\theta} (\log f) (r, \theta)  \, d \theta \right| \\ \nonumber
  &= \left| \log f(r, b) - \log f(r, a) \right| \\ \nonumber
  &\leq | \log f(r, b) | + | \log f(r,a ) |.
\end{align}
We now use the following fact. Let $g$ be a polynomial of degree N in $\R^2$. Assume that for some open neighborhood $U$ of the origin, $Z_g \cap U = \{ \0 \}$. Then for any compact set $K$ in $U$, there exists some $\alpha = \alpha(N) > 0 $ so that
\eq{Lineq_special}
   |g (x) | \geq c \operatorname{dist} (x, Z_g)^{\alpha},   \quad x \in K.
\end{equation}
Here $\alpha$ can be expressed explicitly in terms of $N$, the degree of $g$. We refer the reader to \cite{J-K11} for an elegant and elementary proof, where it was shown that $\alpha$ can be taken as $N(N-1)$.

Applying inequality \re{Lineq_special} to the function $f$, we obtain for some $\alpha= \alpha(N) >0$,
\[
  | f (r, \theta) | \geq c \, r^{\alpha}.
\]
Therefore from \re{thh_int_est},
\[
  \int_{a}^b | \partial{}_{\theta} (\log f) (r, \theta) | \, d \theta \lesssim 1 + | \log r^{\alpha} |
   \lesssim 1+ \alpha | \log r| .
\]
Since the number of sub-intervals $[a,b]$ is bounded above by $2N$ which is uniformly in $r$, we have
\begin{align*}
  \int_{r=0}^1  \int_{\theta =0}^{2 \pi} | \partial{}_\theta (\log f) (r, \theta) | \, d\theta \, d r
&\leq C(N) \int_{0}^1 (1 + \alpha | \log r | ) \, dr \\
&\leq C(N).
\end{align*}
This completes the proof for $n=2$. We note that the constant depends only on the degree of $f$.

To finish the induction, we note that by the first part of the proof and the induction hypothesis,
\[
   \int_{v^{\perp}  \cap \B^n } | \nabla (g |_{v^{\perp}} ) | \, dV_{n-1} \leq A(N, n),
\]
where $A(N, n)$ is some constant depending only on the degree of $f$ and the dimension. The result then follows by \re{ind_est}.
\end{proof}
We now proceed to the general case $\operatorname{codim}_{\0}(Z_f)= n-d \geq 2$.

\begin{lemma} \label{Lem::Finite}
 Let $f$ be a polynomial in $\R^n$. Suppose the dimension of the zero set of $f$ at $\0$ is $d$; For $l \geq d$, let $\Lambda$ be the set of $l$ dimensional subspace $L$ satisfying the  following property:
\begin{itemize}
\item
$L$ contains a smooth point $p \in Z_f$ of dimension $d$, and there exists a neighborhood $U$ of $p$ such that $U \cap Z_f \seq L$.
\end{itemize}
Then $|\Lambda| < \infty$.
\end{lemma}
\begin{proof}
Let $S$ be the subset of $Z_f$ consisting of all the smooth points of dimension $d$.
We say that a point $p \in S$ satisfies property A if there exists a neighborhood $U$ of $p$ with= $U \cap Z_f \seq L$.

Since $Z_f$ is an algebraic and thus semialgebraic, by \rp{Prop::SA_smooth}, S is a semialgebraic set. By \cite[Theorem 5.21]{B-P-R_03}, the number of connected components of $S$ is finite.  Accordingly, to prove our statement it suffices to show the following: if $L$ contains a point $p$ with property A, then $L$ must contain the entire connected component of $S$ containing $p$, which we denote by $\Gamma_p$.
To this end, we will show that the set of points $p$ satisfying Property A is both open and closed in $S$. It is clear from definition that the set is open. To show it is closed, suppose $p_0 \in S$ is the limit of a sequence of points $p_j$ satisfying property A.
There is a neighborhood $U_0$ of $p_0$, and real analytic functions $\varphi_1, \dots, \varphi_{n-d}$ defined on $U_0$ such that $U_0 \cap Z_f = V(\varphi_1,\dots,\varphi_{n-d})$, and also $\nabla \varphi_1, \dots, \nabla \varphi_{n-d} \in \R^n$ are linearly independent at every point in $U_0$. By rearranging the index if necessary, we can assume that  $[\DD{\varphi_i}{x_j}]_{\substack {1 \leq i \leq d \\ d+1 \leq j \leq n} }$ is non-singular. Write $\varphi = (\varphi_1, \dots, \varphi_{n-d}): \R^n \to \R^{n-d}$ and $p_0 =(a,b) \in \R^d \times \R^{n-d}$. By the implicit function theorem, there exists a (unique) real analytic function $g$ defined on some neighborhood $U_0'$ of $a$ in $\R^d$, such that $\varphi(x, g(x)) =0$ for all $x \in U_0'$. By shrinking $U_0'$ if necessary, we can assume that $(U_0', g(U_0')) \seq U_0$.

Now for each $p_j$ in the sequence, we can find a small neighborhood $U_j$ of $p_j$ such that $U_j \cap Z_f \seq L$. Write $p_j = (a_j, b_j) \in \R^d \times \R^{n-d}$. By the above construction, there exists a neighborhood $U_j' \subset U_0'$ of $a_j$ in $\R^d$ such that $\varphi (x, g(x))=0$ for all $x \in U_j'$; in other words $\{ (x',g(x')): x' \in U_j' \} = (U_j' \times U_j'') \cap Z_f$. Shrinking $U_j'$ if necessary, we can assume that $(U_j', g(U_j')) \subset U_j \cap Z_f \seq L$. Suppose $L$ is given by the equation $h = 0$, where $h$ is a linear function; then $r(x') := h(x', g(x')) \equiv 0$ for all $x' \in U_j'$. In particular, all the derivatives of $r$ vanish at $a_j$. Taking $a_j \to a_0$, we see that $r$ along with all its derivatives vanish at $a_0$. Since $r$ is real analytic, it vanishes identically in some neighborhood $\omega_0$ of $a_0$. We can shrink $\omega_0$ if necessary so that $\omega_0 \subset U_0'$.
Let $\Omega_0$ be any neighborhood of $p_0$ so that its projection onto $\R^d$ is contained in $\omega_0$; then $Z_f \cap \Omega_0 $ is given by $ \{ (x', x'') \in \Omega_0, x''=g(x') \}$. Since $r(x') = h(x',g(x')) \equiv 0$ in $\omega_0$, we have $Z_f \cap \Omega_0 \seq L$, in other words $p_0$ satisfies property A. This shows that the set of all points satisfying property A is closed, finishing the proof.
\end{proof}
\begin{lemma} \label{Lem::proj}
  Let $v_1, \dots, v_l$ be linearly independent vectors in $\R^n$. Let $H$ be an $n-1$ dimensional hyper-plane. For $v \in \R^n$, we denote by $\proj_H v$ the projection of $v$ onto $H$.
Let
\[
  S':= \{ w \in S^{n-1}: \text{ $\proj_{w^\perp} v_1, \dots, \proj_{w^\perp} v_l$ are linearly dependent} \}
\]
and
\[
  \mathcal{L}:= \{ w \in S^{n-1}: w \in \Sp \{ v_1, \dots, v_l \} \} .
\]
Then $S'= \mathcal{L}$.
\end{lemma}
\begin{proof}
First, We observe that both the sets $\{ v \in S^{n-1}: v \in \Sp \{ v_1, \dots, v_l \} \}$ and $S'$ are invariant under Gram-Schmidt process on $\{v_1, \dots, v_l \}$. The first statement is obvious. To see the latter, fix a hyperplane $H$, and suppose for some  $a_1, \dots, a_l$ not all zero,
\[
  a_1 \proj_H v_1 + \cdots + a_l \proj_H v_l = 0.
\]
By rearranging the vectors we can assume $a_l \neq 0$. Applying Gram-Schmidt process to $v_1, \dots, v_l$, we have
\begin{gather*}
  u_1 = v_1 , \quad
  u_2 = v_2 - \frac{v_2 \cdot u_1}{u_1 \cdot u_1} u_1, \quad \dots
  \quad
  u_l = v_l - \sum_{k=1}^{l-1} \frac{v_l \cdot u_k}{u_k \cdot u_k } u_k .
\end{gather*}
We can rewrite the last equation as
$v_l = u_l + \sum_{k=1}^{l-1} \frac{v_l \cdot u_k}{u_k \cdot u_k } u_k$. Consequently, by the linearity of $\proj_H$, we get
\begin{align*}
  \proj_H v_l = \proj_H u_l +  \sum_{k=1}^{l-1} \frac{v_l \cdot u_k}{u_k \cdot u_k } \proj_H u_k.
\end{align*}
Similarly for $1 \leq i \leq l-1$, we can write $\proj_H v_i$ as linear combination of $\proj_H u_i$ with $1 \leq i \leq l-1$. Therefore
\begin{align*}
  0 &= a_1 \proj_H v_1 + \cdots + a_l \proj_H v_l
  \\ &= b_1 \proj_H u_1 + \cdots + b_{l-1} \proj_H u_{l-1} + a_l \proj_H u_l
\end{align*}
for some constants $b_1, \dots, b_{l-1}$. As $a_l \neq 0$ by assumption, we see that $\proj_H u_1, \dots, \proj_H u_l$ are linearly dependent. The same proof in reverse direction shows the converse is true as well, namely, if $\proj_H u_1, \dots, \proj_H u_l$ are linearly dependent, then $\proj_H v_1, \dots, \proj_H v_l$ are linearly independent. This shows that the space $S'$ is invariant under Gram-Schmidt process.

Therefore to prove the lemma, it suffices to prove for the case $v_1, \dots, v_l$ are orthonormal vectors. We extend $v_1, \dots, v_l$ to an  orthonormal basis $B = \{v_1, \dots, v_n \}$ of $\R^n$, and write everything using this basis. In particular, in the new coordinates we have
$v_i = (0, \dots, \underbrace{1}_{i^{th}}, \dots, 0)$. Let $w = (a_1, \dots, a_n)$ in this coordinate, i.e. $w= \sum_{k=1}^n a_k v_k$.

We now show that $S' = \mathcal{L} $. We will prove that $\mathcal{L} \seq S'$ and $S' \seq \mathcal{L}$. Suppose $w \in \mathcal{L}$; then $w = \{a_1, \cdots, a_l, 0, \dots, 0 \}$, where $ \sum_{k=1}^l a_k^2 = 1$ and the last $n-l$ coordinates are $0$. Without loss of generality we can assume $a_l \neq 0$.
The projection of $v^i$ onto the hyperplane $w^\perp$ is
\eq{vi_proj}
  \proj_{w^\perp} v_i = v_i - \frac{v_i \cdot w}{w \cdot w} w = v_i - a_i w, \quad 1 \leq i \leq l.
\end{equation}
We claim that $\sum_{i=1}^{l-1} \left( -\frac{a_i}{a_l} \right) \proj_{w^\perp} v_i = \proj_{w^\perp} v_l $.
Indeed,
\begin{align*}
 \sum_{i=1}^{l-1} \left( -\frac{a_i}{a_l} \right) \proj_{w^\perp} v_i
 &= \sum_{i=1}^{l-1} \left( -\frac{a_i}{a_l} \right) (v_i - a_i w)
 \\ &= - \sum_{i=1}^{l-1} \left( \frac{a_i}{a_l} \right)  v_i + \sum_{i=1}^{l-1} a_i^2 \frac{w}{a_l}
 \\ &= - \sum_{i=1}^{l-1} \left( \frac{a_i}{a_l} \right)  v_i + (1- a_l^2) \frac{w}{a_l}
 \\ &= \frac{1}{a_l} \left( w- \sum_{i=1}^{l-1} a_i v_i \right) - a_l w
 \\ &= v_l - a_l w = \proj_{w^\perp} v_l.
\end{align*}
It follows that $w \in S'$. Next we show $S' \seq \mathcal{L}$. For this we will show that if $w \notin \mathcal{L}$, then $w \notin S'$. Suppose for some constants $c_1, \dots, c_l$,
\[
  c_1 \proj_{w^\perp} v_1 + \cdots + c_l  \proj_{w^\perp} v_l = 0.
\]
Using $\proj_{w^\perp} v_i = v_i - a_i w $ (by \re{vi_proj}), we have
\[
  c_1 (v_1 -a_1 w) + \cdots + c_l (v_l - a_l w) = 0,
\]
or
\[
  c_1 v_1 + \cdots + c_l v_l - (c_1 a_1 + \cdots + c_l a_l ) w = 0.
\]
Since $w \notin \mathcal{L} = \Sp \{ v_1, \dots, v_l \} $, we must have $c_1 a_1 + \cdots + c_l a_l = 0$. Hence $c_1 v_1 + \cdots + c_l v_l = 0$. Since by assumption $v_1, \cdots, v_l$ are linearly independent, we have $c_1 = \cdots = c_l = 0$, which implies that $\proj_{w^\perp} v_1 , \cdots, \proj_{w^\perp} v_l $ are linearly independent. By definition this means $w \notin S'$. The proof is now complete.
\end{proof}

\begin{lemma} \label{Lem::Dichotomy}
Let $f$ be a real analytic germ at $\0$ in $\R^n$. Suppose $\operatorname{codim}_\0(Z_f) = n-d \leq n-1$. Let $H$ be an $n-1$ dimensional hyperplane through the origin and let $\widetilde f := f|_H$ be the restriction of $f$ on $H$. Then $\operatorname{codim}_\0 Z_{\widetilde f} < n-d$ if and only if for every (sufficiently small) neighborhood $U_0$ of the origin, there exists a smooth point $p$ of $Z_f$ dimension $d$ with $p \in H$, and a neighborhood $U_p$ of $p$ in $U_0$ such that $U_p \cap Z_f \seq H$.
\end{lemma}
\begin{proof}
The `` if " part is obvious. If $p$ satisfies the given condition stated in the proposition, then there exists some neighborhood $\widetilde U_p$ of $p$ in $H$ such that $\widetilde U_p \cap Z_f$ is a $d$ dimensional submanifold of $H$, so $p$ is a smooth point of $\widetilde f$ of dimension $d$. By assumption, there exists a sequence of such points $p$ converging to $\0$, so $\dim_\0 Z_{\widetilde f}=d$, and $\operatorname{codim}_\0 Z_{\widetilde f} = n-1-d < n-d$.

For the ``only if" part, we will prove the contrapositive statement. Suppose for some neighborhood $U_\0$ of the origin, there does not exist a smooth point $p$ with the given property, we want to show that $\operatorname{codim}_\0 Z_{\widetilde f} \geq n-d$.

Under this assumption there are two possibilities. Either a) there is no point of $Z_f$ of dimension $d$ in $\widetilde U_0 = U_0 \cap H$, or b)there exists a point $p$ of $Z_{\widetilde f}$ of dimension $d$ in $\widetilde U_\0 = U_\0 \cap H$, such that the tangent space $T_p (Z_f)$ does not lie in $H$.
If case (a) occurs, then every point of in $\widetilde U_0$ has dimension less than $d$, so $\operatorname{codim}_\0 Z_{\widetilde f} \geq n-1 - (d-1) = n-d$.

If case (b) occurs, there exists a neighborhood $\omega$ of $p$ such that $\omega \cap Z_f = V(h_1, \dots, h_{n-d})$, where $h_i$ are real analytic functions and $\nabla h_i$ are linearly independent everywhere in $\omega$. Let $\widetilde \omega := \omega \cap H$ and $\widetilde h_i := h_i|H$. By shrinking $\widetilde \omega$ we can assume $\widetilde w \subset \widetilde U_0$.
We claim the vectors $\nabla \widetilde h_i \in \R^{n-1}$ are linearly independent at each point in $\widetilde \omega$. Assume for the moment the validity of the claim.  Since $\widetilde \omega \cap Z_{\widetilde{f}} = V(\widetilde h_1, \dots, \widetilde h_{n-d})$, this means $\widetilde f$ has codimension $n-d$ at $p$. Since this is true for all smooth points $p \in Z_{\widetilde f} \cap \widetilde U_\0 $ of dimension $d$, it follows that $\operatorname{codim}_\0 \widetilde f \geq n-d$.

It remains to prove the claim. It suffices to show that $\nabla \widetilde h_i $ are linearly independent at $p$, since the rank of the Jacobian matrix $[\DD{\widetilde h_i}{x_j}]$ cannot decrease locally near $p$. Seeking contradiction,
suppose that the vectors $\nabla \widetilde h_i(p) \in \R^{n-1}$ are linearly dependent. Since $\nabla \widetilde h_i(p) = \proj_H \nabla h_i (p)$ and $H=v^\perp$, by \rl{Lem::proj} we must have $v \in \Sp \{ \nabla h_i (p) \}_{i=1}^{n-d}$. On the other hand, $\nabla h_i(p)$ are perpendicular to $T_p (Z_f)$, the tangent plane of $Z_f$ at $p$. Thus $v \perp T_p (Z_f)$, or $T_p(Z_f) \seq H$, which contradicts the assumption. Thus the claim holds and the proof is now complete.
\end{proof}

\begin{prop} \label{Prop::der_est_2}
 Let $f$ be a polynomial of degree $N$ in $\R^n$, $n \geq 2$. Suppose the codimension of the zero set of $f$ at $\0$ is $n-d \geq 2$; then
 $ \nabla \log |f|$ is $L^1_{\mathrm{loc}}(\0)$. That is, there exists a neighborhood $U$ of $\0$ and some constant $C(N,n)$ such that
 \eq{der_est_2}
    \int_U \left| \nabla \log|f| \right| < C(N,n).
 \end{equation}
\end{prop}
\begin{proof}
By using stratification, there exists some small neighborhood $\U$ of $\0$ such that $Z_f \cap \U$ is a finite disjoint union of $\mathcal{A}_\ell$, which are strata with maximum dimension $d$. Furthermore, $\0$ is in the closure of at least one of the $d$-dimensional strata, which we denote by $\mathcal{A}_0$.

Fix $m=n-d \geq 2$, we will do induction on the ambient dimension.
If $U \subset \R^m$, then the assumption that $\operatorname{codim}_\0 Z_f = m$ implies that $f$ is non-vanishing or has an isolated zero at the origin (Here we need to use the fact that $Z_f$ is algebraic and therefore consists of finitely many connected components.) Consequently, \re{der_est_2} holds by \rp{Prop::der_est_1}.
In general, suppose we have proved the statement for $U \subset \R^k$ with $k \geq m$, and $\operatorname{codim}_\0 Z_f \geq m$ in $U$, we want to show that it also holds for $ V\subset \R^{k+1}$ and $\operatorname{codim}_\0 Z_f \geq m$ in $V$. If $\operatorname{codim}_\0 Z_f=k+1$, then the result is given by \re{Prop::der_est_1}. If $\operatorname{codim}_\0 Z_f < k+1$, we apply \re{ind_est} with $g = \log|f|$:
\begin{equation} \label{n-d_induc}
\int_{\B^{k+1}} | \nabla (\log | f | ) |
  \leq C \int_{S^k}  \left( \int_{v^{\perp}  \cap \B^k } | \nabla ( \log |(f|_{v^{\perp}})| ) | \, d V_{n-1} \right)  \, d \si (v) .
\end{equation}

In view of \rl{Lem::Finite} and \rl{Lem::Dichotomy} , $\operatorname{codim}_\0 Z_{f|_{v^\perp}} \geq \operatorname{codim}_\0 Z_f \geq m$ for all $v \in S^k$ except a finite set. By the induction hypothesis, the inner integral is bounded by a constant $C(k,n)$ depending only on $k$ and $N$, so the whole integral is bounded by $C(k,n)$. This completes the induction and the proof.
\end{proof}

Combining the above results, we can now prove \rt{Thm::W11_intro_1}.
\th{Thm::W11}
  Let $f$ be a real analytic germ at the origin in $\R^n$, $n\geq 2$. Suppose the codimension of the zero set of $f$ at $\0$ is at least $n-2$; then $\log |f| \in W^{1,1}_{\mathrm{loc}} (\0)$.
\eth
\begin{proof}
If $f$ is a polynomial, then the result follows immediately from \rp{Prop::rem_sing_2}, \rp{Prop::logLp} and \rp{Prop::der_est_2}. What remains to show is that \rp{Prop::der_est_2} still holds if $f$ is real analytic. Expanding $f$ locally into convergent power series, we can write $f = f_1 + f_2$, where $f_1$ is a polynomial and $|f_2| < \frac{1}{2} |f|$. Write
\[
  f = f_1 \frac{f}{f_1}.
\]
Set $u = f/f_1$. Then $u$ does not vanish near the origin. Indeed, we have
\[
  | u | = \left| \frac{f}{f_1} \right| = \left| \frac{f}{f-f_2} \right|
  \geq \frac{|f|}{|f| + |f_2|} \geq \frac{|f| }{|f| + \frac{1}{2} |f| } = \frac{2}{3}.
\]
Now
\[
  \left| \frac{\nabla f }{f} \right| = \left| \frac{\nabla (f_1 u) }{f_1 u}  \right|
  \leq \left| \frac{\nabla u}{u} \right| + \left| \frac{\nabla f_1 }{f_1} \right|.
\]
The first term on the right-hand side is bounded by some constant near $\0$ and thus in $L^1_\mathrm{loc}(\0)$; the second term is in $L^1_\mathrm{loc}(\0)$ by \rp{Prop::der_est_2}. This completes the proof.
\end{proof}

\begin{thm} \label{Thm::W11_2}
 Let $f$ be a real analytic function in some neighborhood of a bounded open set $U$ of $\R^n$. Suppose the codimension of the zero set of $f$ in $U$ is at least $2$; then $\log |f| \in W^{1,1}(U)$.
\end{thm}
\begin{proof}
The assumption implies that the dimension of $Z_f$ at each point is at least 2.
 We can cover $\overline U$ by finitely many balls. Furthermore, we can shrink the balls so that
 \begin{enumerate}
     \item
   For any given ball $B(p,r)$ with center $p$ and $f(p) \neq 0$, we have that $\overline{B(p,r)}  \cap Z_f = \emptyset$.
    \item
   For any given ball $B(p,r)$ with center $p$ and $f(p) = 0$, $\log |f| \in W^{1,1}(B(p,r))$.
 \end{enumerate}
 where (2) holds because of \rp{Thm::W11}. The theorem then follows easily.
\end{proof}

In order to prove \rt{Thm::blow-up_intro}, we
first show that the integral blows up when the exponent is the dimension of space.
\th{Thm::n_blup}
 Let $f$ be a real analytic germ at the origin in $\R^n$ with $f(\0)= 0$. Then for every (sufficiently small) neighborhood $\mathcal{U}$ of $\0$
\[
 \int_{\mathcal{U}} \left| \frac{ \nabla f }{f} \right|^n \, d V(x)= \infty.
\]
\eth
\begin{proof}
We prove by contradiction. Without loss of generality we can assume $f$ is real analytic in $\mathcal{U} = B_\varepsilon(\0)$, a ball of radius $\varepsilon$ centered at $\0$. Suppose
\eq{qtfinite}
   \int_{\mathcal{U}} \left| \frac{ \nabla f }{f} \right|^n \, dV(x) < \infty.
\end{equation}
Define
\[
  A(x) =
  \begin{cases}
  \left| \frac{\nabla f (x)}{f (x)}  \right|,  & f (x) \neq 0,  \\
  0, & f (x)=0.
  \end{cases}
\]
Then $A \in L^n (B_{\varepsilon} (\0))$. Using spherical coordinate $ x = \rho  \omega$, $\omega \in S^{n-1}$, we have
\eq{Vdint}
  \int_{B_{\varepsilon} (\0)} A^n(x) \, d V(x) = \int_{S^{n-1}} \int_0^{\varepsilon}  A^n( \rho \omega) \rho^{n-1} \, d \rho \, d \si (\omega),
\end{equation}
where we denote by $d \si (\omega)$ the area element on the $n-1$ dimensional unit sphere $S^{n-1} \subset \R^n$.
By \re{qtfinite}, the integral in \re{Vdint} is finite. Hence Fubini's theorem implies that, for almost all $\omega_0 \in S^{n-1}$,
\[
   \int_{0}^{\varepsilon} A^n (\rho \omega_0) \rho^{\rho -1} \, d \rho < \infty,
\]
or
\eq{vrld}
A (\rho \omega_0) \rho^{\frac{n-1}{n}} \in L^n (0, \varepsilon).
\end{equation}
Fix $\omega_0 \in S^{n-1}$ such that \re{vrld} holds and also $f(\rho \omega_0)$ is not identically $0$ for $\rho \in (0,\varepsilon)$. Define $\varphi_{\omega_0} (\rho) := f(\rho \omega_0)$, with $\varphi_{\omega_0} (0) = 0$. By choosing $\varepsilon$ small, we have $\varphi_{\omega_0}(\rho) \neq 0 $ if $\rho \in (0,\varepsilon) \setminus \{ 0 \}$ ($\varphi$ is a one variable real analytic function and thus have isolated zero at $0$.) Taking derivative,
\[
  |\varphi'(\rho) | = \left| \nabla f (\rho \omega_0) \cdot \omega_0 \right|  \leq  \left| \nabla f (\rho \omega_0) \right|.
\]
By definition of $A$, we have for $\rho \neq 0$,
\begin{align*}
|\varphi'(\rho) | &\leq | \nabla f (\rho \omega_0) |  \\
&= A (\rho \omega_0) | f ( \rho \omega_0) |   \\
&= A (\rho \omega_0) |\varphi( \rho) |   \\
 &= A (\rho \omega_0) \rho^{\frac{n-1}{n}}  |\varphi(\rho) |  \rho^{\frac{1-n}{n}}.
\end{align*}
Hence
\[
   \rho^{\frac{n-1}{n}} \frac{|\varphi'( \rho) |}{|\varphi ( \rho) | } \leq  A (\rho \omega_0)   \rho^{\frac{n-1}{n}}, \quad \rho \neq 0.
\]
Since $\varphi (\rho)$ is real analytic and is not identically $0$, it has finite order vanishing, i.e. there exists $N$ such that
\begin{gather*}
  \varphi (\rho) = \rho^N + O(\rho^{N+1}), \\
  \varphi'(\rho) = N \rho^{N-1} + O(\rho^N).
\end{gather*}
and $\frac{\varphi'(\rho)}{\varphi(\rho)} \approx \frac{1}{\rho}$ for $\rho$ small.
It follows that
\begin{align*}
 A(\rho \omega_0)  \rho^{\frac{n-1}{n}}  \geq \rho^{\frac{n-1}{n}} \left| \frac{\varphi'(\rho)}{\varphi(\rho)} \right|
 \approx \rho^{\frac{n-1}{n}} \rho^{-1}
 = \rho^{-\frac{1}{n}} \notin L^n(0, \varepsilon),
\end{align*}
which contradicts with \re{vrld}.
\end{proof}

We are now ready to prove \rt{Thm::blow-up_intro}, which gives a refinement of the above result based on information about codimension of the zero set.
\begin{thm}
  Let $f$ be a real analytic germ at the origin in $\R^n$ with $f(\0) = 0$. Suppose $\operatorname{codim}_\0 (Z_f) = n-d$. Then for every (sufficiently small) neighborhood $\mathcal{U}$ of $\0$,
  \[
     \int_{\mathcal{U}} \left| \frac{\nabla f}{ f} \right|^{n-d} = \infty.
  \]
\end{thm}
\begin{proof}
By definition, for any $\varepsilon >0$ there exists a smooth point $x_0$ of $Z_f$ in $\mathcal{U} $ such that $\dim_{x_0} (Z_f) = d$. Then one can find a coordinate system $(y',y'')$, with $y'=(y_1, \cdots, y_d)$, $y''=(y_{d+1}, \cdots y_n)$, in a small neighborhood $\mathcal{W} $ of $x_0$ such that $\mathcal{W} \subset \mathcal{U}$ and
\[
  Z_f \cap \mathcal{W} = \{ y_{d+1} = \cdots = y_n = 0 \}.
\]
Here, $y = ( y_1, \cdots, y_d )$ is a local coordinate chart of $Z_f $ in $\mathcal{W}$. Assume for the moment that $\int_{\mathcal{W}} \left| \frac{\nabla f}{f} \right|^{n-d} < \infty$. Let $f_{y'} (y'') = f (y', y'')$ be the restriction of $f$ onto the $y'$-plane.
For each fixed $y'$, the function $f_{y'}$ satisfies $f_{y'}(\0) = 0$. Thus by Fubini's theorem,
\[
  \infty > \int_{\mathcal{W}} \left| \frac{\nabla_y f (y)}{f(y)} \right|^{n-d} \, dy
  =  \int \int \left| \frac{\nabla f (y', y'')}{f(y', y'')} \right|^{n-d} \, dy'' \, dy'
  \geq \int \int \left| \frac{(\nabla f_{y'}) (y'') }{f_{y'} (y'')}\right|^{n-d} \, dy'' \, dy'
\]
Now the integral on the right-hand side is infinity by \rt{Thm::n_blup}, which is contradiction. Therefore we conclude that $\int_{\mathcal{W}} \left| \frac{\nabla f}{f} \right|^{n-d} = \infty$ and
\[
    \int_{\mathcal{U}} \left| \frac{\nabla f}{f} \right|^{n-d} > \int_{\mathcal{W}} \left| \frac{\nabla f}{f} \right|^{n-d} = \infty.
\]
\end{proof}

The following example shows that, the integrability exponents in \rt{Thm::W11_intro_1} and  \rt{Thm::blow-up_intro} are optimal.

\ex{Ex::sharp}
Let $f$ be a function on $\R^n$, $n \geq 2$, of the form
 \[
    f(x) = x_1^{2 r_1} + x_2^{2 r_2} + \cdots + x_k^{2 r_k},  \quad k \geq 2,
 \]
 where $r_k$ are integers satisfying $1 \leq r_1 \leq r_2 \dots \leq r_k$.
 Then for any bounded neighborhood $U$ of $\0$, the following holds
\eq{exam_int_cvg}
   \int_U \left| \frac{\nabla f}{f}\right|^{\gamma} < \infty
\end{equation}
if and only if
\[
  \gamma <  1 + \frac{r_1}{r_2} + \cdots + \frac{r_1}{r_k}.
\]
If $k=1$, i.e. $f(x) = x_1^r$, then \re{exam_int_cvg} holds if and only if $\gamma < 1$.
\end{exmp}
\begin{rem}
 By choosing suitable values for $r_1, \cdots, r_k$, we see that for every $\varepsilon>0$, there exists a real analytic function $f$ such that $f \notin L^{1+\varepsilon}(\0)$. This shows that $L^1$ integrability is the best possible in \rt{Thm::W11_intro_1}. On the other hand, for $g(x) = x_1^2 + x_2^2 + \cdots + x_k^2$, by the above computation we have $|\nabla g| /g \in L^p_\mathrm{loc}(\0)$, for any $p <k$ with $k= \operatorname{codim}_{\0}(Z_f)$, but $g \notin L^k_\mathrm{loc}(\0)$. This proves the optimality of the exponent in \rt{Thm::blow-up_intro}.
\end{rem}
\nid \textit{Proof of Example \ref{Ex::sharp}.}
We have, up to a nonzero constant,
\[
\left| \frac{ \nabla f (x) }{f (x)} \right| = \frac{\sqrt{|x_1|^{2 (2r_1- 1)} +  \dots + |x_n|^{2 (2 r_k- 1)} }}{|x_1|^{2 r_1} + \dots + |x_n|^{2 r_k}}.
\]
Set $u_1 = |x_1|^{r_1}, \dots, u_k = |x_k|^{r_k}$. Then $|x_1| = u_1^{\frac{1}{r_1}}, \dots, |x_k| = u_k^{\frac{1}{r_k}}$, and
\[
\left| \frac{ \nabla f (x) }{f (x)} \right| = \frac{ \sqrt{u_1^{4 - \frac{2}{r_1}} + \dots + u_k^{4- \frac{2}{r_k}} } }{u_1^2 + \dots + u_k^2}.
\]
Writing $y= (y',y'')$ where $y'= (y_1, \dots, y_k)$ and $y''=(y_{k+1}, \dots, y_n)$, the integral becomes
\eq{eheadcoc1}
\int_U \left| \frac{\nabla f (x) }{f(x)} \right|^{\gamma} \, dx
= \int_{U''} \int_{U'_{y''}} \frac{ \left( y_1^{4 - \frac{2}{r_1}} + \dots + y_k^{4- \frac{2}{r_k}} \right)^{\frac{\gamma}{2}} } { \left( y_1^2 + \dots + y_k^2 \right)^{\gamma} } \frac{y_1^{\frac{1}{r_1} -1} \dots y_k^{\frac{1}{r_k} -1} }{r_1 \dots r_k} \, dy' \, dy''.
\end{equation}
where we denote by $U''$ the projection of $U$ onto the $y''$ space and $U'_{y''} := \{ y' \in \R^n: (y',y'') \in U \}$.
Using spherical coordinate $y = \rho  \omega$ with $\rho \in (0,\infty)$ and $\omega = (\omega_1, \dots, \omega_k) \in S^{k-1}$, we get
\begin{align*}
\frac{ \left( y_1^{4 - \frac{2}{r_1}} + \dots + y_k^{4- \frac{2}{r_k}} \right)^{\frac{\gamma}{2}} } { \left( y_1^2 + \dots + y_k^2 \right)^{\gamma} }
&= \frac{ \left( (\rho \omega_1)^{4 - \frac{2}{r_1} } + \dots + (\rho \omega_k)^{4 - \frac{2}{r_k} }  \right)^{\frac{\gamma}{2}} }{\rho^{2 \gamma}} \\
&= \frac{ \left( \rho^{4 - \frac{2}{r_1}} \left(  \omega_1^{4- \frac{2}{r_1}}  + \rho^{\alpha_2} \omega_2^{4- \frac{2}{r_2}}   \dots + \rho^{\alpha_k} \omega_k^{4 -  \frac{2}{r_k}}  \right) \right)^{\frac{\gamma}{2}} }{\rho^{2 \gamma}}  \\
&= \rho^{- \frac{\gamma}{r_1}} \left( \omega_1^{4 - \frac{2}{r_1}} + \rho^{\alpha_2} \omega_2^{4 - \frac{2}{r_2}}  \dots + \rho^{\alpha_k} \omega_k^{4 - \frac{2}{r_k}} \right)^{\frac{\gamma}{2}},
\end{align*}
where we set
\[
\alpha_2 = \frac{2}{r_1} - \frac{2}{r_2}, \quad \dots, \quad \alpha_k = \frac{2}{r_1} - \frac{2}{r_k}.
\]
Hence up to a positive constant the inner integral in \re{eheadcoc1} is bounded by
\eq{eheadcoc2}
\int_{\rho =0}^{1}  \int_{S^{k-1}} \frac{\left( \omega_1^{4 - \frac{2}{r_1}} + \rho^{\alpha_2} \omega_2^{4 - \frac{2}{r_2}}  \dots + \rho^{\alpha_k} \omega_k^{4 - \frac{2}{r_k}} \right)^{\frac{\gamma}{2}}  }{ \omega_1^{1- \frac{1}{r_1}} \dots \omega_k^{1- \frac{1}{r_k}}  }   \,d\si (\omega) \, \left( \rho^{- \frac{\gamma}{r_1}}  \rho^{\frac{1}{r_1} -1} \dots \rho^{\frac{1}{r_k} -1} \rho^{k-1} \right) \, d \rho,
\end{equation}
where $d \si(\omega)$ denotes the area element on $S^{k-1}$.
The integral over the sphere is bounded up to a constant by
\eq{sphere_int}
  \int_{S^{k-1}} \frac{\omega_1^{2 \gamma - \frac{\gamma}{r_1}} + \rho^{\frac{\alpha_2 \gamma}{2}} \omega_2^{2 \gamma-  \frac{\gamma}{r_2}} + \dots + \rho^{\frac{\alpha_k \gamma}{2}} \omega_k^{ 2 \gamma - \frac{\gamma}{r_k} }  }{ \omega_1^{1- \frac{1}{r_1}} \dots \omega_k^{1- \frac{1}{r_k}} }  \, d \si(\omega).
\end{equation}
In view of the sphere parametrization formula  \re{sphere_par} and \re{sphere_vol_ele},
the integral \re{sphere_int} is bounded for any $\gamma>0$. Hence the integral \re{eheadcoc2} is bounded if and only if
\begin{align*}
-1 &< - \frac{\gamma}{r_1} + \frac{1}{r_1} - 1 + \dots + \frac{1}{r_k} - 1 + k - 1 \\
&= - \frac{\gamma}{r_1} + \frac{1}{r_1} + \dots + \frac{1}{r_k} -1,
\end{align*}
or
\[
\gamma < 1 + \frac{r_1}{r_2} + \dots + \frac{r_1}{r_k}.  \qquad \qed
\]


I've been living in the land of Dorothy and her spinning world today. As the outside rang with whistles of the wind and tornado sirens, I stayed warm inside listening to the new Taylor Swift album while editing photos.
For as much drama as there may be in the world of Ms. Swift, I still appreciate her lyrics. The lyric in the photo above is from a song titled Innocent. It's a nice little reminder to me that God loves me no matter the mistakes I have made.


text:
Lee Lozano: Private Book 6
$25.00 Member Price: $22.50
This is the sixth volume in Karma's 11-volume facsimile printing of Lee Lozano's "Private Book" project. A prolific writer and documenter of both her art and her relationships, the public and private, the painter Lee Lozano (1930–99) kept a series of personal journals from 1968 to 1970 while living in New York's SoHo neighborhood. In 1972 she rigorously edited these books, thus completing the project. Eleven of these private books survive, containing notes on her work, detailed interactions with artist friends and commentary on the alienations of gender politics, as well as philosophical queries into art's role in society and humorous asides from daily life.
Published by Karma, 2019
Spiral-Bound softcover
Dimensions: 5 x 3 inches (12.7 x 7.62 cm)
Edition of 3,000 copies
200 Words to Help You Talk about Art
200 Words to Help You Talk about Sexuality & Gender
30,000 Years of Art, Mini Edition
50 Feminist Art Manifestos
52 Artists: A Feminist Milestone
6 Spices, 60 Dishes: Indian Receipes That Are Simple, Fresh, and Big on Taste
This is the sixth volume in Karma's 11-volume facsimile printing of Lee Lozano's "Private Book" project. A prolific writer and documenter of both her art and her relationships, the public and private, the painter Lee Lozano (1930–99) kept a series of personal journals from 1968 to 1970 while living in New York's SoHo neighborhood. In 1972 she rigorously edited these books, thus completing the project. Eleven of these private books survive, containing notes on her work, detailed interactions...

redpajama_set_name: RedPajamaCommonCrawl


text:
Dame Kelly Holmes keeps on running
Lydia Slater
Saturday April 28 2012, 1.01am, The Times
Dame Kelly Holmes: "I'm bloated and unfit, and not as defined as I was. But you never like your body, do you?"
TIMES PHOTOGRAPHER, JON ENOCH
The trainers at Matt Roberts's City gym have seen too many scantily-clad A-listers to be easily impressed. But there's a palpable thrill in the room when Dame Kelly Holmes walks in. The Olympic athlete is a vision in clingy black leggings and a black and fuchsia vest from her own sportswear range which, with fitness equipment in co-ordinating colours, is now sold in Tesco.
Although to everyone in the gym she embodies physical perfection, she seems to perceive only degeneration. She says that she's "bloated" and "really unfit". "I used to train three times a day, six times a week, and once on the seventh day. Now, I'm lucky if I train four times a week. But keeping fit is vital, especially as you get


View student performance by various student groups, including gender, race/ethnicity, family income (based on eligibility for the National School Lunch Program), parental education, students with disabilities (SD), and English language learners (ELL).
Move the cursor over data points to see more detail.
Click "Chart Options" below the figures to view different types of charts or change the format and layout of the existing chart.
When viewing average score figures as a bar chart, use the drop down menu to show data for each group. When viewing average score figures as a line chart, click on a group within the legend to show detailed information about its performance.
When viewing the achievement level or percentile figures, use the drop down menu to show data for each group.
Compared to 2009, the average reading score was higher in 2011 in Charlotte.
Beginning in 2009, results for charter schools are excluded from the TUDA results if they are not included in the school district's Adequate Yearly Progress (AYP) report to the U.S. Department of Education.


text:
Do it Best Names New Vice President of Ecommerce
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Practice these 5 Skills to Make Employee Training Easier
84 Lumber Announces 1st Team HQ Award Winners
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IHA Announces GIA Student Design Competition Winners
February 7, 2022 February 10, 2022 0 Comments 2022, association, competition, design, housewares, international, student, winners
ROSEMONT, IL – Industrial design students from Western Washington University (WWU) in Bellingham have captured four of six top prizes in the International Housewares Association's 29th annual Global Innovation Awards (gia) Student Design Competition. This year's winners include products that help reduce food and e-waste; a self-watering, upside-down herb grower; a portable immersion boiler for tea and other beverages; and a potty that reminds caregivers to "bathroom" their children, while making potty time fun.
The competition challenges college students to redesign a current housewares product to meet the needs of the future or to create a concept for a new product. Winning projects are selected for their innovation, understanding of production and marketing principles, and quality of entry materials.
The winning students and their products will be on display during The Inspired Home Show, IHA's global home + housewares marketplace, March 5-7 at Chicago's McCormick Place, the first in-person gathering of the industry's premier marketplace since 2019. View the winners and their products on the GIA student design webpage
Madeline Gerst, a WWU junior from Seattle, Wash., captured first place and $3,500 for her WAVE UV-C Breadbox. Joining her are two second- and three third-place winners who will share $7,000 in prize money. The top six finishers will receive an all-expense paid trip to Chicago for The Inspired Home Show. The three winning schools will receive grants totaling $3,000.
Gerst said, "I am elated and honored to have this opportunity to share my work. Winning this competition opens doors to exciting new career opportunities for my future. I hope to make valuable connections with fellow students, designers, companies and innovation leaders at the Show in March. I would like to thank my professors, peers and friends whose support throughout the design process contributed to the success of this project."
"Madeline's winning entry is thoughtful, purposeful, user-centered design at its best," said competition judge Chris Cunningham, IDSA, Cunningham Design, Chicago. "The WAVE UV-C Breadbox reinvigorates a stagnant market/category to meet today's user needs with a clever application of new technology to refresh a historically low-tech solution. Madeleine also makes excellent use of prototyping and design-thinking methodologies to determine the technology component and final solution. Her product concept is incredibly on trend now with the rise in home bread baking and I can see a manufacturer partnering with her and bringing it to market."
Globally, almost 900,000 tons of bread, or around 24 million slices each day, are wasted annually, mostly due to an inherently short shelf life and poor storage conditions.Gerst's winning product concept, WAVE, uses germicidal UV-C light to sterilize pathogens on the surface of bread, significantly extending its shelf life and eliminating the need for harsh chemical preservatives and single-use plastic packaging.
IHA received entries from 26 schools in the U.S., Canada, Austria, Mexico and Romania. Vicki Matranga, IHA's design programs coordinator and manager of the Student Design Competition, said, "IHA's program has become known as the gold standard for college-level competitions. Many U.S. professors – and an increasing number internationally – assign the program annually to industrial design students because it is a real-world exercise and every entry receives feedback from two industry professionals. Students must identify user needs and opportunity spaces in the marketplace, research competitive available products, test models with users and consider production issues."
Second and Third Place Winners
Second-place awards of $2,300 each went to: Rose Kirby, a WWU junior from Seattle, for her Osmo Evaporative Refrigerator; and Nikolaus Potapow, a graduate of FH Joanneum University of Applied Sciences in Graz, Austria, for his RUFUS User Relationship for Extending Appliance Life.
Osmo is a countertop evaporative refrigerator that uses only the power of water to keep fresh foods cool and healthy. Most fruits and vegetables should be stored in a cool and humid environment. The fridge is often too dry, the counter too warm. The first-in, last-out design of conventional refrigerators invites waste, while Osmo helps keep foods visible and healthy.
Rufus is a rotary-drive food mixer with several blender attachments that provides a deeper user-product connection. It engages the user to assemble, maintain and repair the mixer to ultimately lengthen the product's life span and reduce e-waste.
Third-place awards of $1,200 each went to: WWU junior Elliot Quasha from Seattle, for the Tē Immersion Boiler; WWU junior Carson Porter-Keese from Kent, Wash., the Gro Herb Planter and Western Michigan University junior Isabella Waite from Morley, Mich., for Pip the Potty Pal.
Tē is a portable alternative to an electric kettle that allows the user to heat water for tea or reheat beverages with a portable immersion boiler. Pip the Potty Pal solves two main problems: children not wanting to take breaks from playing to go potty, and parents, nannies or childcare workers forgetting to remind children to use the bathroom. This product helps adults keep track of their child's bathroom schedule and makes potty breaks fun for children. Gro is a self-supporting planter system that can efficiently grow herbs upside down to save counter space for kitchen activities. In addition, Gro waters itself by absorbing and using moisture from the atmosphere.
Five Honorable Mentions Awarded
Five projects merited an honorable mention and their designers received $250 cash award:
Team of Sam Eisenberg, Mary Arevalo, Amberly Dawson, Sean Ostrowski and Kasey Connelly, Virginia Polytechnic Institute and State University juniors, Mosaic Smart Puzzle
Emily Goergen, Iowa State University senior, Namu Nebulizer
Mohammad Shadab Naveed, Arizona State University junior, Stoven Camp Stove
Rinona Metaj and Jaclyn Jamieson, Carleton University seniors, Singing Stovetop Kettle
Yuanqing Li, Cleveland Institute of Art senior, Mosha Litter Box
Judges Include Seven Previous Winners
This year's winners were chosen by a respected jury of 14, including designers at consumer and industrial product companies, design consultants, educators and seven former winners. Altogether, judges spent many volunteer hours reviewing the submissions, which consist of written materials, sketches, engineering drawings and photos.
Along with Cunningham, the jury included: Brian Bock, industrial designer, Hamilton Beach Brands, Glen Allen, Va.; 2015 winner Evan Cincotta, head of design, Susteau, New York City; Earle Cramer, principal industrial designer, Newell Brands, Kalamazoo, Mich.; 1995 winner Robert Giacolo, new product development program manager, Transcendia, Franklin Park, Ill..; Carly Hagins, IDSA, associate professor of product design, Western Michigan University, Kalamazoo; 2002 winner Lisa Heim-Yoo, design lead, Nu-Way Industries, Inc., Des Plaines, Ill.; 2010 winner Teddy Lu, creative director, VEO, Chicago; 2018 winner Anastasia Miller, senior industrial designer, Doris Dev, Brooklyn, N.Y.; 1996 winner Jennifer Nemec, Ideation Studio, Chicago; Audra Norvilas, senior design manager, Kimberly-Clark, Chicago; David Richter-O'Connell, IDSA, assistant professor, industrial design, University of Wisconsin-Stout, Menomonie; 2018 winner Brandon Rodriguez, industrial designer, Smart Design, Brooklyn, N.Y.; and Kimberlee Wilkens, director, BDes in industrial design, and assistant professor, industrial design, University of Illinois at Chicago.
29 Years of Exceptional Student Design
This is the 29th year that winning design students have been honored by IHA at its annual trade show, expanding awareness of careers in industrial design among students and highlighting the impact of design on the global housewares industry. Since the competition began in 1993, more than 6,200 college students have entered, and each has come away with an educational experience that the design profession recognizes as unique.
"Winning the IHA Student Competition was a turning point in my career," said 2018 winner Anastasia Miller, one of this year's judges. "I remember being equally shocked and elated when Vicki called to inform me that my project partner and I had won the competition. From there, it was a whirlwind of interviews, meeting up with past winners and flying to Chicago to exhibit our design. During the Show I met future colleagues, friends and professional advisors. As a judge this past year, it was exciting to see students thinking holistically about the design process, and to welcome a new batch of winners into the IHA community."
The Inspired Home Show, IHA's global home + housewares marketplace, will be held March 5-7 at Chicago's McCormick Place Complex. For more information about the 2022 Show and to register for a Show badge, please visit TheInspiredHomeShow.com/register. Before picking up their Show badge at McCormick Place, all exhibitors and attendees will be required to demonstrate proof of vaccination OR produce a negative Covid-19 test that had been administered in the previous 72 hours. Information on what precautions are being taken to ensure the health and safety of the industry is available on our Covid Readiness webpage.
About IHA
The International Housewares Association is the 84-year-old voice of the housewares industry. The not-for-profit, full-service association sponsors the world's premier exposition of products for the home, The Inspired Home Show, IHA's Global Home + Housewares Market (TheInspiredHomeShow.com), and offers its 1,400 member companies a wide range of services, including industry and government advocacy; export assistance; industry and consumer trends through the IHA Market Watch Report; executive management peer groups; a B2B community at Housewares.org; group buying discounts on business solutions services; direct-to-consumer engagement through TheInspiredHome.com; and an independent news and information platform through HomePageNews.com.
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text:
Oxyura vantetsi Worthy, 2005:263
New Zealand stiff-tailed duck, Van Tets' duck (proposed)
Last record: Holocene
North Island, New Zealand
Holotype: NMNZ S.001081
Type locality: Poukawa XII.2, North Island, New Zealand
Paratypes:
NMNZS.3370 (right humerus)
NMNZ S.3527 (right humerus)
NMNZ S.5915 (left humerus)
NMNZ S.12165 (left humerus)
NMNZ S.13730 (right humerus)
Above: New Zealand Stiff-tailed Duck. Oxyura vantetsi. From the series: Extinct Birds of New Zealand., 2005, Masterton, by Paul Martinson. Purchased 2006. © Te Papa. CC BY-NC-ND 4.0. Te Papa (2006-0010-1/30)
Worthy, Trevor H. (2005). A new species of Oxyura (Aves: Anatidae) from the New Zealand Holocene. Longmann Symposium, Memoirs of the Queensland Museum 51(1): 259-275. [automatic download]
Horn, P. L. (1983). Subfossil avian remains from Poukawa, Hawkes Bay, and the first record of Oxyura australis (Blue-billed Duck) from New Zealand. Journal of the Royal Society of New Zealand 13: 67-78.
Hugh Robertson, John Dowding, Graeme Elliott, Rod Hitchmough, Colin Miskelly, Colin O'Donnell, Ralph Powlesland, Paul Sagar, Paul Scofield, Graeme Taylor. (2013). Conservation status of New Zealand birds, 2012. New Zealand Threat Classification Series 4. 22 pp.
Hume, Julian Pender and Walters, Michael. (2012). Extinct Birds. London: T & AD Poyser. 544 pp.
Robertson, H. A., Baird, K. A., Elliott, G. P., Hitchmough, R. A., McArthur, N. J., Makan, T. D., Miskelly, Colin M., O'Donnell, C. F. J., Sagar, P. M., Scofield, R. P., Taylor, G. A. and Michel, P. (2021). Conservation status of birds in Aotearoa New Zealand, 2021. New Zealand Threat Classification Series 36. Department of Conservation, Wellington. 43 pp.
Tennyson, Alan J. D. and Bartle, J. A. (2008). Catalogue of type specimens of birds in the Museum of New Zealand Te Papa Tongarewa. Tuhinga 19: 185-207.
Tennyson, Alan J. D. and Martinson, P. (2006). Extinct Birds of New Zealand. Te Papa Press, Wellington. [p. 48]
Worthy, Trevor H. (2004). The Holocene fossil waterfowl fauna of Lake Poukawa, North Island, New Zealand. Tuhinga 15: 77-120.
<< Back to the Anseriformes (Waterfowl) database


text:
45th Anniversary of Women's Ordination: Talk from The Rev. Canon Judy Rois
by Christ Church Cathedral, Vancouver
Sun, 28 November 2021
1:00 PM – 2:30 PM PST
690 Burrard St
Vancouver, BC V6C 2L1
Join Christ Church Cathedral to celebrate the 45th anniversary of the ordination of women with the Rev. Canon Judy Rois.
Christ Church Cathedral is pleased to welcome the Rev. Judy Rois as part of the Cathedral's celebration of the 45th anniversary of the ordination of women priests in the Anglican Church of Canada.
On Sunday November 28 the Rev. Rois will be the guest preacher at the Cathedral's 10:30am Choral Eucharist.
Following the service, at 1pm she will give a talk exploring the key advancements in Canadian society and politics that advanced the role of women. Her talk will also explore how women have contributed to the mission and ministry of the Church.
The 10:30am service takes place in person and is also live streamed at thecathedral.ca/live
The talk following the service will take place in-person and will be live streamed at thecathedral.ca/live. Register here to attend the talk in person.
About the Rev. Judy Rois
Ordained in 1985 The Rev. Judy Rois was the first female Vicar of St. James' Cathedral in Toronto. Born and raised in Calgary, Alberta, Judy moved to Toronto as a teenager. She earned a Master of Divinity degree from Wycliffe College in Toronto and a Master of Theology degree from Trinity College, Toronto. She was ordained in 1985 in the Diocese of Toronto and served three parishes as Rector and was the first female Vicar of St. James' Cathedral in Toronto. After earning a Doctor of Ministry degree in Homiletics in Chicago, Judy was appointed the George Snell Lecturer in Homiletics at Trinity College in Toronto and lecturer at Queen's College at Memorial University in St. John's, Newfoundland. After 26 years in full-time parish ministry, Judy was named Executive Director of the Anglican Foundation of Canada. Over the course of 11 years, Judy travelled to every province and territory in Canada as the Foundation's emissary, and she oversaw over 6 million dollars in grants to support ministry and mission in Canada.
She is the author of a number of academic papers, one of which received international attention at the University of Oxford Roundtable in the UK. She has also written two books of prayers for children, a book on generosity for young people, and more recently, prayers during a pandemic.
If you plan to attend the 1pm talk in-person:
-Masks must be work at all time while you are in the building
-Vaccine passports are required to attend the talk.
-If you are feeling at all unwell on the day of this event, you are asked to stay home and take part via the live stream.
Vancouver Seminars
Vancouver Spirituality Seminars
#theology
#vancouver
#anglican
#christ_church_cathedral
#judy_rois
Organizer Christ Church Cathedral, Vancouver
Organizer of 45th Anniversary of Women's Ordination: Talk from The Rev. Canon Judy Rois
We envision a Cathedral which is radically inclusive, restlessly inquisitive, intentional, and inspired. This is a place that is called to lead and to serve, in communion and in community, with healing, reconciliation, and love, living out Christian mission. Together, we look to the power of Christ for the transformation of each person, this city, and the world.
45th Anniversary of Women's Ordination: Talk from The Rev. Canon Judy Rois at Christ Church Cathedral
690 Burrard St, Vancouver, BC V6C 2L1, Canada
Christ Church Cathedral, Vancouver
Returning Whence We Came: Discerning God's Will for Us
Room 308 - St. Mark's College, Vancouver
#Spirituality #Seminar
The Impact of Technology on Relationships: Should We Worry? Carol Breuss
St. Mark's College at UBC, Vancouver
Fri, 24 Jun 7:00 pm
Worship Central Conference 2022
The Centre For Performing Arts, Vancouver
Fri, 27 May 5:30 pm
Impact Conference
Chandos Pattison Auditorium, Surrey
Fri, 11 Feb 7:00 pm
Momentum 2022
Surrey Pentecostal Assembly, Surrey
Sat, 7 May 9:00 am
Coffee Date
Broadway Church, Vancouver
Sat, 29 Jan 11:00 am
Family Constellations Workshop
1718 Venables Street, Vancouver
#Spirituality #Class
Church Leadership Worship Experience 2022 | reGROUP Lite | Vancouver
Harvest City Church, Vancouver


Help an organization understand the value of data-driven decision-making and ensure your analytics recommendations are implemented.
Due to rapid innovation in data collection and processing technologies, every organization is looking for professionals who can deliver business intelligence through analytics. Columbia University's Master of Science in Applied Analytics prepares managers and team leaders to make data work through instruction in data science, frameworks, and visualization — as well as leadership, management, and communication.
The program has four study options: part-time and full-time study primarily on Columbia's campus in New York City; part-time entirely online via Columbia's flexible and interactive networked learning platform; and part-time through a combination of online coursework and multi-day intensive study in San Francisco.
Recognize potential challenges to and opportunities for an organization's strategic direction and identify the corresponding analytics problems.
Identify analytical tools and methods that are appropriate to address different types of business problems.
Source, store, and model the internal and external data needed to answer questions and produce value.
Implement change based on data-driven decisions.
Evaluate and communicate the impact of applied analytics.
The program is ideal for individuals looking to enhance decision-making through data as managers or team leaders within a wide range of functions, including marketing, sales, operations, or finance, with a mandate to enhance decision-making through data.
Following successful completion of the program, international degree-holders may be eligible for F1 Optional Practical Training (OPT) and the opportunity to work in the U.S. for up to one year. Because this is a STEM-eligible program, international degree-holders may be eligible for an F1 STEM OPT Extension, which allows F1 students to apply for an additional 24 months of OPT.


The AWS SDK for .NET has a utility class, Amazon.CloudFront.AmazonCloudFrontCookieSigner, for creating signed cookies to access private content served using Amazon CloudFront. This blog contains details on usage of this utility class along with sample code.
Specifying AmazonCloudFrontCookieSigner.Protocols.Https as the protocol parameter creates a cookie with incorrect policy; the policy contains a resource restriction of "http*://" instead of "https://" .
CloudFront distributions configured to serve HTTP and HTTPS requests are affected by this issue, unless "Viewer Protocol Policy" is configured as HTTPS. In this case, CloudFront will block attempts to access content over HTTP.
Update your dependency to the latest version of the SDK. The fix contains a change to the AmazonCloudFrontCookieSigner.Protocols enum's underlying values (a breaking change) and requires a recompilation of the consuming application. The assembly version of the SDK package has been updated for this fix. There are no other breaking API changes in this version.


text:
Q: VBA Excel wait for shell with ansys to finish I have written a macro to run Ansys and if Ansys is closed the macro should run on.
My Problem here is that I run Ansys with some scripts so the code for running it is
retval = shell("C:\...\runwb2 -X -R C:\...\Script.py -F C:\...\Workbenchproject.wbpj", vbNormalFocus)

This works perfectly fine until I want to wait for it to finish
Using
Dim wsh As Object
Set wsh = VBA.CreateObject("WScript.Shell")
cmd = "C:\...\runwb2 -X -R C:\...\Script.py -F C:\...\Workbenchproject.wbpj"
retval = wsh.Run(cmd, 1,True)

gives me a runtime error -2147024894 (0x80070002)
If it is possible I don't want to use something big like ShellAndWait I found earlier, just something simple
Thanks in advance

A: To pause for a five seconds....
'// Start Code Snippet

Application.Wait TimeSerial(Hour(Now()), Minute(Now()), Second(Now()) + 5)

'// End Code Snippet

However, per your code you are setting WaitOnReturn to TRUE, which SHOULD be doing what you want it to do... so I don't know why you'd be getting an error. let me know if this helps.. thanks, Brian

A: If have found my problem.
The path to run ANSYS Workbench was the default installation path with the folder "ANSYS Inc", the space is not a problem if I use shell() but if I use the WScript.Shell the space produces the runtime error. I had to change the Path so there is no more space and now it works fine

redpajama_set_name: RedPajamaStackExchange


Internet of Things investment will support Lebanon's IT market growing to LBP 820 billion by 2019, and will transform daily lives and businesses, industry experts announced during the recent SmartEx Lebanon exhibition.
Lebanon's nationwide digital transformation, guided by the Lebanon 2020 Digital Telecom Vision, which is seeing USD 600 million invested in fixed and 5G mobile broadband, is laying the foundation for the Internet of Things era. As a result, BMI Research shows Lebanon's IT market will grow at 10 percent CAGR to LBP 820 billion in 2019.
By connecting people, objects, and devices to the cloud and networks, Lebanon's organizations can analyze data in real-time for new business insights. Energy, agriculture, and healthcare are the three industry verticals that have tremendous potential to transform Lebanon's economy, society, and environment, argues global digital transformation enabler SAP.
"Lebanon has the opportunity to adopt the Internet of Things era as the next Industrial Revolution, with an inter-connected, smart, and communicating world that will transform organizations and daily lives," said Sameer Areff, Chief Operating Officer, SAP Middle East South, on the sidelines of his keynote "Future of the Internet of Things". "Smart solar grids can power Lebanon's energy future, connected farms can help growers to predict crop yields, customer demand, and sign smart leasing contracts, and digital patient records can personalize and enhance patient treatment."
Across the wider Middle East and North Africa, more than 75 percent of organizations ranked the Internet of Things as an investment priority in 2017, according to the recent SAP and Oxford Economics Digital Transformation Executive Study.
In Lebanon, SAP's channel partner ecosystem is playing a key role in implementing these Internet of Things projects. Organizations have the opportunity to integrate emerging innovations such as artificial intelligence, machine learning, and blockchain.
Supporting the Lebanese workforce of the future, SAP inspires digital skills development in youth with its University Alliances partners: the American University of Beirut and Lebanese American University.
As Silver Sponsor of SmartEx, SAP's visitors can experience Internet of Things innovations running on the SAP Leonardo digital innovation system, SAP S/4HANA real-time business suite, and SAP HANA in-memory platform.


Hanoi authorities have decided to "urgently" build a new water pipeline that serves more than 70,000 families after the existing pipe broke early the same day -- the ninth time in two years.
"This is the ninth time the pipe broke down and nobody can assert that it won't break again," said Nguyen Quoc Hung, deputy chairman of the capital's municipal People's Committee, the city government.
"We can't let tens of thousands of families continue to suffer from water shortages due to pipe breakdowns. We can't let residents depend on such a troubled pipe," Hung said at a meeting on Saturday afternoon.
The VND1.5-trillion (US$70.3 million) pipeline began bringing clean water from the Da River Water Factory in the neighboring province of Hoa Binh to families in six Hanoi districts in 2008.
The pipeline, 47 kilometers long, broke for the first time in February 2012. Breakdowns occurred more often this year, with three ruptures reported in April, one in June, and two this month (July 10 and 12).
Nguyen Van Ton, general director of the Vinaconex Corporation, which manages the pipeline, said the problem was fixed on Sunday.
Poor-quality construction materials have been blamed for the continuous breakdowns, according to inspectors from the construction ministry.
Hung ordered the city's construction department to urgently build a new pipeline that stretches 28 kilometers from Hoa Lac Road to Beltway No. 3.
He made this decision after the existing pipe experienced repeated ruptures within this section.
"Counting from today (July 12), it shouldn't take more than 70 days to complete a new, high-quality pipe," said Hung.
The new pipe, which will parallel to the existing one, is expected to supply 100,000 cubic meters per day when it comes into operation. This will help the existing pipe reduce its capacity to 200,000 cubic meters per day.


text:
Rok jedna (orig. Year One) je americká filmová komedie z roku 2009 režiséra Harolda Ramise a producenta Judda Apatowa v hlavních rolích s Jackem Blackem a Michaelem Cerou.

Děj
Zed je lovec a Oh je sběrač. Poté, co se dozvědí, že Zed snědl jablko ze stromu poznání dobra a zla, je šaman a Zarlak vyženou pryč od kmene. Když Zed omylem podpálí vesnici, Oh se rozhodne jít s ním, aby objevil vše, co svět nabízí. Po cestě potkají Kaina a Ábela. Kain zabije Ábela a řekne Zedovi a Ohovi, že musí utéct s ním, jinak by je obvinili z Ábelovy vraždy.

Následně Zed a Oh zjistí, že dívky z jejich kmene, se kterými chtějí "ulehnout" (Maya a Eema), byly chyceny a prodány do otroctví. Pokusí se je vykoupit na svobodu, ale Kain místo toho prodá Zeda a Oha také do otroctví. Když se dostanou se svým majitelem a dalšími otroky pocházejícími rovněž z jejich kmene do vesnice, zaútočí na ni Sodomité a otroky unesou. Zed a Oh ale utečou do pouště, v noci chtějí osvobodit Mayu a Eemu.

V noci ale usnou a ráno zjistí, že jsou Sodomité pryč. Vydají se je hledat a na jedné hoře narazí na Abraháma, který se zrovna chystá obětovat svého syna Izáka. Zed ho zastaví, tvrdí, že ho tam vyslal Bůh. Abrahám je vezme s sebou do své hebrejské vesnice a vypráví jim o městech Sodoma a Gomora.

Zed a Oh se rozhodnou jít do Sodomy poté, co se Abrahám rozhodne je obřezat. Po příchodu do Sodomy jsou zatčeni. Kain, nyní sodomský voják, je zachrání od uvěznění, když je nazve svými bratry. Kain se jim omluví za to, že je prodal do otroctví a nabídne jim jídlo. Během prohlídky města jim Kain nabídne, aby se stali také vojáky. Krátce na to si ve městě všimnou princezny Innany. Ta se postí, protože se cítí provinile za to, že většina města hladoví. Večer je Zed pozván na párty na rozhovor s Innanou.

Uvnitř paláce Zed uvidí Mayu a Eemu, jak pracují jako otrokyně. Oh musí následovat zženštilého velekněze. Zed se setká s Innanou a ta ho žádá, aby vstoupil do Nejsvětější svatyně a řekl jí, jaké to je. Princezna si myslí, že Zed je "vyvolený". Ve svatyni Zed narazí na Oha, který se tam schovává před veleknězem. Začnou se hádat a následně jsou uvězněni za to, že do svatyně vkročili. Jsou odsouzeni k ukamenování k smrti, ale Zed obyvatele města přesvědčí, aby byli soucitní, a tak jsou místo toho odsouzeni k těžké práci, dokud nezemřou. Král pak oznámí, že obětuje bohům svou nevlastní dceru (Innanu) a dvě další panny (Mayu a Eemu).

Zed přeruší obřad a tvrdí, že je "vyvolený". Začne povstání. Oh zachrání Eemu. Do města přijede Abrahám s Hebrejci, aby svrhli krále. Oh a Eema spolu mají v paláci sex, což znamená, že Eema již nemůže být obětovaná. Potom vylezou ven, aby pomohli Zedovi v boji proti vojákům (včetně Kaina). Dav zabije všechny vůdce a prohlásí Zeda jako "vyvoleného" novým vládcem. Zed to odmítne a předá vládu Innaně. Místo toho se z něj stane cestovatel, na cesty se vydá s Mayou. Oh se stane vůdcem vesnice, kde začal celý příběh. Zed a Oh se rozloučí a vydají se každý na svou vlastní cestu.

Obsazení

Ohlas
Rok jedna sklidil negativní reakce kritiky. Server Rotten Tomatoes dává filmu na základě 162 hodnocení kritiků skóre 14%. Server Metacritic hodnotí film 34 body ze 100 na základě 28 recenzí. Uživatelé ČSFD snímek hodnotí průměrně 53%.

Během úvodního víkendu ve Spojených státech snímek utržil necelých 20 milionů USD a stal se tak čtvrtým nejúspěšnějším filmem víkendu po snímcích Návrh, který měl premiéru ve stejný den, Pařba ve Vegas a Vzhůru do oblak. Celkově ve Spojených státech Rok jedna utržil více než 43 milionů USD a v zahraničí téměř 17 milionů USD.

Odkazy

Reference

Externí odkazy

Americké filmové komedie
Americké dobrodružné filmy
Filmy studia Columbia Pictures
Filmy z roku 2009
Filmy Harolda Ramise

redpajama_set_name: RedPajamaWikipedia


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