text:
What a Wild Game - Some Game Videos
I really don't want to spend too much time with words, so instead I'll put up a couple of videos.
Below is Ian Kennedy hitting Yaiel Puig on the head in the 6th inning. It was a dangerous pitch.
As you can see, Ian Kennedy looked disappointed in himself for throwing that pitch. So, I doubt it was a purpose pitch.
On the other hand, Greinke's pitch the following inning that hit Miguel Montero on the numbers was clearly done in retaliation. Both benches were warned, and you would figure that would be the end of it. Below is a video.
But no, this is the 2013 Dodger season. So, naturally Ian Kennedy felt it necessary to bean Greinke. Not on the numbers, mind you, but Kennedy purposefully tried to hit Greinke in the head. The benches clear and punches were thrown. Watch the video below. (check that, click the video below to watch it at MLB.com)
As it happens, the Dodgers pulled out a victory with some timely hitting in their bottom half of the 8th inning.
Wow! This has been some crazy season.
At last count, Puig, Ian Kennedy, Mark McGwire and Kirk Gibson were sent to the showers.
UPDATE: I just thought I'd add that considering Ian Kennedy's (retaliation) pitch was aimed at an opposing players head, I believe he should receive the brunt of any penalties involved. It's one thing to purposefully hit a guy from the numbers down, but when you go for the noggin' then the penalty must be severe.
* Please follow on twitter @ernestreyes *
* Like Dodgers Blue Heaven on facebook *
* Dodgers Blue Heaven home page *
By ernest at Tuesday, June 11, 2013 0 comments
Links Labels: Kirk Gibson, Mark McGwire, Videos, Yasiel Puig, Zach Greinke
Welcome Up to the Blue, Chris Withrow & Alex Castellanos - Some Fantasy Baseball Cards for Them
Don't look now, but a parade of prospects are showing up at Dodger Stadium.
One day after Justin Seller's call up from AAA, he is being sent back down. Furthermore, Scott Van Slyke, who is suffering from left shoulder bursitis, is being placed on the 15-day DL. In their place is minor league outfield prospect Alex Castellanos and pitcher Chris Withrow.
Castellanos came up briefly several times last season, and eventually got into 16 games. Overall, he batted just .174 in 23 at-bats, but did slug 1 home run. We originally attained Castellanos via a trade with the Cardinals involving Rafael Furcal in 2011. Chris Withrow is coming up to the Major Leagues for the second time in his career, although he has yet to actually pitch. He was originally a 1st Round draft pick for the Dodgers in 2007 (20th overall), and was thought to be a potential middle of the order starting pitcher. At one time he was ranked the 48th best prospect in Baseball. Unfortunately, after several years of trying it just didn't pan out. Soon, he was converted into a reliever and spent most of 2012 to today honing that craft. Heck, his numbers in Albuquerque look quite good, considering that place is a pitchers nightmare.
In their honor, I decided to make a fantasy Blue Heaven Baseball card for both of them. See them below. I used photographs available from Jon SooHoo/LA Dodgers.
Click any pic to embiggen.
Via the Dodgers press release:
Withrow, who will wear No. 44, will be making his Major League debut and has gone 4-0 with a 1.93 ERA in 22 relief appearances with the Isotopes. The 24-year-old last pitched on Friday, striking out three of the four batters he faced, and has not allowed a run in his last seven appearances (7.2 IP). Withrow, who was selected by the Dodgers in the first round (20th overall) in the 2007 First-Year Player Draft, has only allowed earned runs in two of his 22 appearances this year.
Castellanos, 26, has a .270 batting average with eight steals, nine home runs and 28 RBI in 43 games with Albuquerque this season. Castellanos went 4-for-23 (.174) with a triple, a homer and three RBI in 16 big league games with the Dodgers last year and tied for the Dodger team lead with both four home runs and 11 RBI in 29 Cactus League games this spring.
Van Slyke is batting .221 with six home runs, five doubles and 11 RBI in 24 games with the Dodgers, while Sellers posted a .188 batting average with a home run, a double and two RBI in 27 contests for the club.
Below are their career stats:
Alex Castellanos
Report Created on Baseball-Reference.com
Chris Withrow
Links Labels: Alex Castellanos, Card Creations, Chris Withrow, Justin Sellers, Rafael Furcal, Scott Van Slyke
Blog Kiosk: 6/11/2013 - Dodgers Links - Puig Mania Just Won't Stop
"Buy me some peanuts and Cracker Jack's, I don't care if I never get back..." Pic via Jon SooHoo/LA Dodgers 2013.
I cannot talk about Brandon League and I refuse to link to any stories about last nights game, lest I be reminded of it.
Via Danny Knobler at CBS Sports, "Stock Watch: Everyone's talking Puig; one says he's Adrian Peterson".
"I always preface this by saying I don't know how his career is going to turn out," said (Matt) Martin, who has since left the Dodgers and taken a job with the Orioles. "But on God-given ability going in, it's like nothing I've ever seen."
Martin kept thinking, and finally he hit on it.
"I said, 'He's Adrian Peterson.' That's how much God-given ability this guy has," Martin said. "That's the comparison I had to give. He's Adrian Peterson."
Ted Berg at For the Win highlights, "Five young MLB stars that could end up the greatest of all time". His list includes Yasiel Puig.
Via Property of Zach, "Riley Breckenridge (Thrice) Starts Baseball-Themed Grindcore Band: Puig Destroyer". Riley's website is here.
Bob Timmerman, briefly taking over for Jon Weisman at Dodger Thoughts, sums up last nights game rather nicely. Woops! I guess I lied.
This is too awesome not to link. At Sons of Steve Garvey they try to talk about something other than Puig. So, Nomo puts up some great Star Wars/Japanese League commercials.
I found this both unfortunate and hilarious at the same time. Peter Mayhew, who played Chewbacca in the original Star Wars films was stopped by the TSA in Denver due to a giant cane (he's 7'3" tall) that he needs to use in order to get around. The cane is shaped like a light saber - I kid you not. Fortunately, TSA came to it's senses. Story and pics via John Aravosis at AmericaBlog.
This is just awesome. The Boston Globe shares pics of the past 50 years of Stanley Cup Championship rings. Wow!
Links Labels: Blog Kiosk, Brandon League, Hockey, Puig Destroyer, Star Wars, Yasiel Puig
There Will be a Puig Sighting at Frank & Son's for a Signing
It's been quite a week for our rookie outfielder. Yasiel Puig has lit the league on fire with his spectacular play on the field and his phenomenal batting at the plate. At this point, he's like the goose that laid the golden egg, and hopefully we don't squander our good fortune.
That being said, Puig has been an absolute monster in the collecting world. His cards are selling for "stupid" money, and as long as he continues to perform at inhuman levels it just might stay there. Of course, the law of averages tell you that that isn't possible. Still, one can dream.
Well, it has just been announced that Yasiel Puig will be making his first paid public signing appearance on Saturday, June 29th for the Dodger Days event at Frank & Sons in the City of Industry. BTW, there will be a whole bunch of Dodgers on hand, so I certainly suggest checking the entire event out.
Usually, I refrain from make this kind of announcement this far in advance of a public signing. I prefer to make once a week post focused on just events happening the immediate weekend, but I figured this was worth breaking that rule.
Puig will only be signing 250 items so there is a bit of urgency if you want to pay to play. On the other hand, the asking price is way beyond most fans pocketbooks. According to their flyer, tickets for small items will be $149.00, large items at $199.00, and premium items at $299.00.
Yeah, you read that right. The current asking prices are at the levels usually reserved for popular Hall of Famers or superstar future Hall of Famers. For comparison, at the National Sports Collectors Convention in Chicago in early August there are only five Baseball players who are priced at or above these levels: Randy Johnson, Chipper Jones, Mike Piazza, Cal Ripken Jr. and Roger Clemens. Another 22 current Hall of Famers are below Puig's prices - some well below.
As you can imagine, there were a couple of complaints on Frank & Sons facebook page, and they responded by noting that it is Puig's management that effectively set the price by asking a huge amount of money for the limited signing. Furthermore, via Sports Collectors Daily, his agents claim that it will be his only autograph show during the season.
Regardless, considering his current hot hand and the current selling prices of his certified autographed cards on eBay, I suspect they will sell out anyway. After all, as some scouts have said, Puig is a once in a generation type of player, and that means he is pure gold in the collecting world.
Links Labels: Auto Op, Yasiel Puig
Dodger Correspondence: Ralph Branca Plays Contract Hardball Against Branch Rickey
Here is another edition of Dodger Correspondence.
This time I share another letter that I found on eBay that was written by former Brooklyn Dodger Ralph Branca to then GM Branch Rickey. As you can see, Branca is seeking more money.
Dear Mr. Rickey,
Enclosed you will find my returned contract. You will also note that it is unsigned. This is to let you know that I am not in accord with you on salary terms for 1945.
Ralph T. Branca
I have no idea what Rickey offered Branca, but I do know that at the time Branca wrote this letter he had yet to win a Major League game. He had just finished up his rookie season in Brooklyn, going 0-2 with a 7.05 ERA in 21 games (20 as a reliever). Clearly, Branca wasn't very effective in the games he pitched. On the other hand, a quick look at his gamelog that season shows that he mostly came in for mop-up duty.
Branca and Rickey eventually came to terms for a reported $3,300 (according to Baseball-Reference). Frankly, I'm not sure if that's right. It seems way too low, or was this the kind of salary a young player would expect to get back then? If so, no wonder he balked at the first offer.
BTW, Ralph Branca pitched much better in 1945. He primarily started for the Dodgers going 5-6 with 15 starts and 1 save. Branca had a ERA of 3.04 and completed 7 games. That's a much better showing for the kid known as Hawk.
(Auction Link)
Links Labels: Branch Rickey, Dodger Correspondence, Ralph Branca
Welcome Up to the Blue, Chris Withrow & Alex Caste...
Blog Kiosk: 6/11/2013 - Dodgers Links - Puig Mania...
There Will be a Puig Sighting at Frank & Son's for...
Dodger Correspondence: Ralph Branca Plays Contract...


text:
\section{Introduction}
Quandles are
non-associative algebraic structures whose study was motivated, in part, by knot theory.  Quandles and racks arose in many different areas of mathematics and appeared in the literature with different names since 1940's (e.g.~\cite{Takasaki}).
In 1982,  Joyce \cite{Joyce} (used the term quandle) and Matveev \cite{Matveev} (who call them distributive groupoids)  introduced independently the notion of a knot quandle $Q(K)$
associated to each oriented knot $K$.
The knot quandle is a complete invariant up to reversed mirror.
Since then quandles have been investigated
for constructing knot and link invariants
(e.g.~\cite{CES, CJKLS, EM}).
Quandles have been also studied
from different point of views such as the study of the set-theoretic Yang-Baxter equation \cite{CES2} and pointed Hopf algebras \cite{AG}.   Topological quandles were introduced in \cite{Rubin} and investigated further in \cite{ChengElhamdadiShekhtman, ClarkSaito, EM}.  Smooth quandles and their cohomology have been introduced by Nosaka~\cite{Nosaka}.

Homology theories of racks \cite{FRS} and quandles \cite{CJKLS} have been defined and investigated algebraically. They have also been used for constructing knot invariants called quandle cocycle invariants.
Continuous cohomology of topological groups  have been studied extensively (see \cite{Stasheff} for an overview), and the problem has been posed how quandle cohomology theories can be extended to topological quandles.

In the present article, a continuous cohomology theory for topological quandles is introduced, and compared to the algebraic theories.
 Extensions of topological quandles are studied with respect to continuous 2-cocycles, and used to show  differences in second cohomology
groups for specific topological quandles,
and to show non-triviality of continuous cohomology groups.
The formula of computing cohomology groups for inverse limits
have been used in various contexts (see  \cite{Wilson} for the case of groups).
We investigate this method in our context of quandles, and
apply it to concrete families of quandles to determine
cohomology groups of their inverse limits.

The following is the organization of the paper.  Section~\ref{Prel} gives the preliminary background on quandles.  In Section~\ref{secCohomology}, a continuous cohomology is introduced for topological quandles, and    the  first cohomology groups of topological Alexander quandles is investigated.
Furthermore
 extensions of  quandles by continuous 2-cocycles is studied  in the topological context.
 Section~\ref{InvLim} discusses the notion of inverse and direct limits of quandles and their cohomology groups, with examples presented.
Further examples of non-trivial cohomology groups are presented in Appendix~\ref{sec:nontriv},  and
the results that continuous isomorphism classes differ from algebraic isomorphism classes for
topological quandles are presented in
Appendix~\ref{ContinIsom},
that were obtained by W.~Edwin Clark.

\section{Preliminary} \label{Prel}

In this section we review basic definitions and properties of quandles.
A rack is a set $X$, together with a binary operation $ *: X \times X \longrightarrow X$ satisfying the following axioms:
\begin{itemize}
\item[(i)] for all $x,y\in X$, there exists a unique $z\in X$ such that  $z * x = y$;
\item[(ii)] for all $x,y$ and $z$ in $X$: $(x*y)*z = (x*z)*(y*z)$.
\end{itemize}
Property (ii) is commonly referred to as right self-distributivity.
Property (i)
states that the map $R_x$, defined by right $*$-multiplication by the element $x\in X$, is a bijection of $X$ onto itself.
A {\it quandle}  is an idempotent rack:
\begin{itemize}
\item[(iii)]
for all $x\in X$, $x*x = x$.
\end{itemize}

 A {\it quandle homomorphism} between two quandles $X, Y$ is
 a map $f: X \rightarrow Y$ such that $f(x*_X y)=f(x) *_Y f(y) $, where
 $*_X$ and $*_Y$
 denote
 the quandle operations of $X$ and $Y$, respectively.
   The subgroup of ${\rm Sym}(X)$ generated by the permutations ${ R}_x$, $x \in X$, is
called the {\it {\rm inner} automorphism group} of $X$,  and is
denoted by ${\rm Inn}(X)$.
A quandle is {\it indecomposable} (we use this term instead of {\it connected} to avoid confusion with topological meaning) if ${\rm Inn}(X)$ acts transitively on $X$.

A topological space $X$ with a continuous binary operation $(x,y) \mapsto x*y$ satisfying right self-distributivity and such that $R_x$ is a homeomorphism for all $x \in X$ is called a {\it topological rack}.   If moreover, the operation $*$ is idempotent then $X$ is called a {\it topological quandle}.
 Any rack $(X,*)$ can be regarded as a topological rack if $X$ is endowed with the discrete topology.

\begin{ex}
Typical examples of quandles arise as topological quandles when considered with topological structures as follows.

 Any topological group $G$ becomes a topological quandle with operation $*$ given by conjugation: $x*y = y^{-1} x y$. This quandle is
 denoted ${\rm Conj}(G)$, the conjugation quandle associated to $G$.

For a topological group $G$ and a continuous automorphism $f: G \rightarrow G$,
$x*y=f(xy^{-1})y$ for $x, y \in G$ defines a topological quandle structure on $G$. This is called a {\it generalized Alexander quandle}
and is denoted by $(G, f)$. If $G$ is abelian it is called an Alexander quandle.

In particular,
for any $T \in {\rm GL}(n, \mathbb{R})$,
$\mathbb{R}^{n}$ can be given a topological quandle structure by defining $x*y = T x + (I - T)y$, for all $x,y \in \mathbb{R}^{n}$,
where $I$ denotes the identity matrix.
\end{ex}

\begin{ex}\label{ex:Rubin}
 Consider the $n$-dimensional sphere $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$. The operation $x*y = 2(x\cdot y)y - x$, for all $x,y \in \mathbb{S}^{n}$ endows the sphere with a topological quandle structure, where
 $x\cdot y$ denotes the standard inner product in $\mathbb{R}^{n+1}$.
Also, this operation induces a topological quandle structure on the real projective space $\mathbb{RP}^{n}$ giving a topological quandle homomorphism $S^n \rightarrow  \mathbb{RP}^{n}$.  Knot colorings by these quandles have been studied in \cite{Rubin}.

Similarly
for a topological abelian group $G$,
a topological quandle structure on $G$ is defined by the rule $x*y = 2y - x$. These
are a generalization of what is commonly referred to as Takasaki quandles \cite{Takasaki}.
\end{ex}

For more informations about quandles, discrete and topological, the interested reader can consult \cite{Joyce, Matveev, CJKLS, EN, EM, Rubin}.

\section{Continuous Cohomology of Topological Quandles} \label{secCohomology}

In this section we
define a {\it continuous cohomology theory} for topological quandles.
A similar theory for  smooth quandles was also defined and studied by Nosaka \cite{Nosaka} independently.
Rack and quandle homology theories have been defined and studied in several different contexts and have been generalized
(\cite{AG,CES,CJKLS}, for example).
In this section we follow \cite{CES} and in Section~\ref{sec:AG} we discuss
a generalization of the quandle cohomology defined in \cite{AG} for topological quandles.

Let $X$ be a topological quandle.
Let  $A$ be a topological abelian group, $T: A \rightarrow A$ be a continuous automorphism, and $A$ is also considered with the generalized Alexander quandle structure  $(A, T)$.
Consider the following abelian groups:
\[ \Gamma^n(X,A) = \{ f: X^n \rightarrow A\ |\ f\  \text{is continuous}, \ f(x_1, \cdots, x_n)=0 \; \text{if} \;  x_i=x_{i+1} \; \text{for some i} \ \}, \]
where $X^n$ is given the product topology induced by $X$  and the sum in $\Gamma^n$ is induced by pointwise addition in $A$.
Define  the following maps $\Gamma^n(X,A) \longrightarrow \Gamma^{n+1}(X,A)$, $n \in \mathbb{N}$:
\begin{eqnarray*}
& &  \delta^i_0 f(x_1, \ldots , x_{n+1}) = f(x_1, \ldots , \hat{x_i} , \ldots , x_{n+1}); \\
& & \delta^i_1 f(x_1, \ldots , x_{n+1}) = f(x_1*x_i, \ldots ,x_{i-1}*x_{i},x_{i+1},\ldots , x_{n+1}).
\end{eqnarray*}
We  define the chain complex:
\[\cdots \rightarrow \Gamma^n(X,A) \xrightarrow{\delta} \Gamma^{n+1}(X,A)\rightarrow\cdots\]
by setting:
\[ \delta^n =
\sum_{i=1}^{n+1}(-1)^i [ T \delta^i_0 - \delta^i_1 ] . \]
We  define the $n^{th}$ continuous cohomology group of $X$ with coefficients in $A$ by
\[ H^n_{\rm TC}(X,A) = \frac{ {\rm ker} (\delta^n)}{ {\rm im} (\delta^{n-1})},\]
assuming that the map $\delta^0$ is defined to be the canonical inclusion of the trivial group into $\Gamma^1(X,A)$, i.e. $H^1_{TC}(X,A) = \Gamma^1(X,A)$.

When $T=1$, the groups $H^n_{TC}(X,A)$ are called (untwisted)
 continuous quandle cohomology groups and will be denoted $H^n_{C}(X,A)$.
Furthermore, in Section~\ref{sec:AG}, a generalization of the twisted cohomology theory using quandle modules defined in \cite{AG} is defined and denoted by $H^n_{\rm GC}$.

Any topological quandle can be viewed as a discrete quandle, by forgetting the topological structure; it is clear that the chain complex above is a subcomplex of the usual (discrete) chain complex associated to $X$.
The original  and twisted quandle cohomology groups were denoted by $H_{\rm Q}$ and $H_{\rm T}$.
Our notation is summarized as follows.

\begin{center}
\begin{tabular}{llllll}
$H_{\rm Q}$ & : & Original (untwisted) & \quad
$H_{\rm T}$ & : & Original twisted \\
$H_{\rm C}$ & : & Continuous (untwisted) & \quad
$H_{\rm TC}$ & : & Continuous twisted \\
$H_{\rm GC}$ & : & Continuous generalized (quandle module)
\end{tabular}
\end{center}

\begin{ex} \label{ex:12}
Let $X$ be a topological quandle and $(A,T)$ be a topological Alexander quandle. Then
a continuous map $\eta: X \rightarrow A$ is a continuous 1-cocycle
if it satisfies
$T[\eta(y) - \eta (x) ] - [\eta (y) - \eta(x*y) ] =0$, that is, $\eta$ is a continuous quandle homomorphism,
$\eta(x*y)=T \eta (x) + (1-T) \eta (y)$.
If, in particular,  $T=1$ and $A$ is considered as a topological abelian group with trivial quandle structure, then the 1-cocycle condition is
$\eta (x*y) = \eta(x)$.

	A continuous map $\phi: X^2 \rightarrow A$ is a  2-cocycle if and only if it satisfies the condition:
	\[
	T\phi(x_1,x_2) + \phi(x_1*x_2,x_3) = \\
	T\phi(x_1,x_3) + (1-T)\phi(x_2,x_3) +  \phi(x_1*x_3,x_2*x_3)
	\]
    and $\phi(x,x)=0$.
	These considerations appear in \cite{CES} except the requirement of continuity.
\end{ex}

\subsection{Continuous First Cohomology}

Let $X$ be a topological quandle and $(A, T)$ be a topological Alexander quandle.
As mentioned in Example~\ref{ex:12}, it follows from the definition that $H^1_{TC}(X,A)$ is the group of
continuous quandle homomorphisms $\eta: X \rightarrow A$.

As a special case of $T=1$, where $A$ is regarded as a topological abelian group with the trivial quandle structure,
we have the following.
\begin{pro}
Let $X$ and $A$ be as above. If $X$ is indecomposable, then  the first cohomology group $H^1_{C}(X,A)$ is isomorphic  to $A$.
\end{pro}

\begin{proof}
The proof is similar to the original case, and our purpose here to include the proof is to observe that  the continuity does not play a role in the argument below as the 1-cocycle condition implies functions being constant.

Let $x, x'\in X$ be arbitrary elements of $X$ and let $f : X \rightarrow A$ be a $1$-cocycle. By indecomposability  of $X$ there exist $y_1, \ldots , y_n\in X$ such that $(\cdots (x*^{\epsilon_1}y_1)*^{\epsilon_2}\cdots *^{\epsilon_n}y_n )= x'$, such that $\epsilon_i=\pm 1$, where $*^{-1}$ is defined by $x*^{-1}y=z$ if
$z*y=x$.
Recall that in this case the 1-cocycle condition is $f(x*y)=f(x)$, which also implies $f(x*^{-1} y)=f(x)$.
Therefore
\[f(x') = f(\cdots (x*^{\epsilon_1}y_1)*^{\epsilon_2} \cdots *^{\epsilon_n} y_n ) = f(\cdots (x*^{\epsilon_1}y_1)*^{\epsilon_2} \cdots *^{\epsilon_{n-1}}y_{n-1} )\]
where the second equality follows from the $1$-cocycle condition for $f$.
Inductively
it follows that $f$ is a constant map.
On the other hand, any constant map satisfies the cocycle condition and is continuous, hence it is in $H^1_C(X,A)=Z^1_C(X, A)$.
Hence there is a bijective correspondence between $H^1_C(X,A)$ and $A$
that
respects the group structures as the group operation of cocycles is pointwise, and
the lemma follows.
\end{proof}

\begin{pro}
Let $X=({\mathbb R}^n, S)$ and $A=({\mathbb R}^m, T)$
be  indecomposable Alexander quandles, where $S, T$ are continuous additive automorphisms.
Then $H^1_{TC} (X, A)$ is isomorphic to
\[
\{ \ F + a : {\mathbb R}^n  \rightarrow {\mathbb R}^m \ | \ a \in A, \ F \ \text{is linear,}\  FS=TF \ \} .
\]
\end{pro}

\begin{proof}
Since $X$ and $A$ are indecomposable, we have $I-S$ and $I-T$ invertible.
Let  $G \in H^1_{TC} (X, A)$.
Then $G$ is a continuous quandle homomorphism $G: X \rightarrow A$.
Then for all $a \in A$, $G+a \in H^1_{TC} (X, A)$.
For any $G \in H^1_{TC} (X, A)$, there is $a \in A$ such that $(G+a)(0)=0$.
By Lemma~\ref{lem:similar}, $F=G+a$ is linear, and $FS=TF$.
Hence the result follows.
\end{proof}

\begin{pro}
Let $X=({\mathbb R}^n, T)$
be  an indecomposable Alexander quandle.
Then $H^1_{\rm TC}(X, X) \not\cong H^1_{\rm T}(X, X)$.
\end{pro}

\begin{proof}
By Proposition~\ref{prop:noniso}, there are quandle isomorphisms that are not continuous.
Hence we have that $H^1_{\rm TC}(X, X)$ is a proper subgroup of $H^1_{\rm T}(X, X)$.
\end{proof}

\subsection{Continuous Second Cohomology and Extensions}

We  define the concept of extension of a quandle, in the topological context, following \cite{CENS, CES}, see also \cite{EM}.

Assume we are given a quandle $X$ and an Alexander quandle
 $(A,T)$.
For a $2$-cocycle $\psi \in Z^2_{T}(X, A)$ (so that $\; \psi(x,x)=0 $ for all $x \in X$), a quandle structure is defined on $X \times A$ by
\[ (x,a)*(y,b) = (x*y, a*b+ \psi (x,y))\]
 for all $x,y \in X$ and $a,b \in A$, as in  \cite{CENS}.
 The resulting quandle is denoted by
 $X \times_{\psi} A$ and called an {\it extension} of $X$ by $A$.
 The projection $\pi: X \times_{\psi} A \rightarrow X$ is a quandle homomorphism.

For a topological quandle $X$ and a topological Alexander quandle $(A, T)$, we observe the following.
Assume $\psi \in Z^2_{TC}(X, A)$, a continuous 2-cocycle.
Then $X \times_{\psi} A$ has the product topology and  the quandle operation on $X \times_{\psi} A$
is continuous, therefore $X \times_{\psi} A$ is a topological quandle.
Furthermore, the projection $\pi: X \times_{\psi} A \rightarrow X$ is continuous, hence
$\pi$ is a topological quandle morphism.
Although we call $X \times_{\psi} A$  the extension of
a topological quandle, we require continuity as described above.
For topological spaces, it appears desirable to consider fiber bundles instead of product spaces. However, for the purpose of investigating differences of cohomology groups between continuous and algebraic, and for non-triviality of continuous cohomology groups, in this article we focus on the product cases. See also Section~\ref{sec:principal} that is related to this issue.

 We
 define morphisms in the class of
 extensions of $X$ by the abelian group $A$ and, consequently, define an equivalence relation corresponding to the isomorphism classes. The class of  extensions of $X$ by $A$ can be viewed therefore as a category.
 Consider two topological  extensions $X \times_{\psi} A$ and $X \times_{\phi} A$
 where $\psi$ and $\phi$ are two $2$-cocycles. A morphism $X \times_{\psi} A  \rightarrow X \times_{\phi} A$ of  extensions of $X$ by $A$ is a morphism of topological quandles $f: X\times_{\psi}  A \rightarrow X\times_{\phi} A$  making the following diagram commute:
 \[
 \begin{tikzcd}
 X\times_\psi A \arrow [r, "f"]\arrow[d] &  X\times_\phi A\arrow[d] \\
 X \arrow[r, equal] & X
 \end{tikzcd}
 \]
  In particular, if $f$ is an isomorphism of topological quandles with the property of making the above diagram commute, it will be called an {\it isomorphism of topological  extensions}.
  Two extensions are {\it equivalent} if there is an isomorphism $f$ as above.
 We now prove the following result, analogous to the classification of the second cohomology group for group cohomology and the corresponding result for discrete quandles, as in \cite{CENS, CES}.

 \begin{pro}
 There is a bijective correspondence between equivalence classes of topological abelian extensions of $X$ by $A$ and the second cohomology group $H_{TC}^2(X,A)$ of $X$ with coefficients in $A$.
 \end{pro}

\begin{proof}
Although computations below are similar to those in \cite{CES}, we examine   topological aspects of the argument.
 Assume $ X\times_\psi A $ and $ X\times_\phi A $ are two topological  extension of $X$ with $\psi$ and $\phi$ cohomologus $2$-cocycles (i.e. they differ by a coboundary). Consider the map $f: X\times A \rightarrow X\times A$, $ (x,a) \mapsto (x, a + g(x))$, where $g : X \rightarrow A$ is such that $\delta g = \psi - \phi$.
 Since $g \in Z^1_{TC}(X,A)$, $g$ is continuous, and so is $f$.
 We have
 \[ f((x,a)*(y,b)) = f(x*y, a*b +\phi(x,y)) = (x*y, a*b  + \phi(x,y) + g(x*y)).\]
 On the other hand we have
 \[ f(x,a)*f(y,b) = (x, a+ g(x))*(y, b + g(y)) = (x*y, a*b+ g(x)*g(y) + \psi(x,y)).\]
  These two terms are equal since $\phi = \psi + \delta g$, hence $f$ is an isomorphism of quandles.
 Since it is also a homeomorphism and clearly makes the required diagram commute, we get that $X\times_\psi A $ and $X\times_\phi A $ are equivalent.

 Conversely, assume $X\times_\psi A $ and $X\times_\phi A $ are equivalent. Say $f: X\times A \rightarrow X\times A$ is an isomorphism of topological  extensions. Since, by definition, both $\pi (x,a)=x$ and $\pi (f(x,a))=x$,
 the map $f$ is determined by its second component. Using the group structure of $A$ we can also write $f$ as
 $f(x,a)=(x, a+g(x))$
 for some map $g : X \rightarrow A$.  Now the continuity of $f$ implies the  continuity of $g$.
Since $f$ is a morphism of quandles we get, for all $x,y \in X$ and all $a,b\in A$,
\begin{eqnarray*}
\lefteqn{
  (x*y, a*b + g(x*y) + \psi(x,y)) = f((x,a)*(y,b)) }\\
 &=&
f(x,a)*f(y,b) = (x*y, a *b + g(x)*g(y) + \phi(x,y)).
\end{eqnarray*}
Equating  the second component, we find that $\psi$ and $\phi$ differ by $\delta g$,  i.e. they are representatives of the same cohomology class, since $g$ is continuous.
\end{proof}

\begin{lem}\label{lem:Kronecker}
Let $X$ be a topological quandle, and $(A, T)$ be a topological Alexander quandle.
Let $\alpha \in Z_n^{\rm T}(X, A)$ be an $n$-cycle (in the usual sense of discrete homology), and $\phi \in Z^n_{\rm TC}(X, A)$ be a continuous $n$-cocycle.
If $\phi(\alpha )\neq 0$, then $[\phi] \neq 0 \in H^n_{\rm TC}(X, A)$.
\end{lem}

\begin{proof}
The standard argument applies as follows in continuous case.
Suppose $\phi$ is a coboundary in $H^2_C(X,A)$. Then, by definition, there exists some continuous $f: X \rightarrow A$ such that $\delta f = \phi$ which, in particular, means that $[\phi] = 0$ in $H^2(X,A)$ (discrete cohomology).
As a consequence, applying the Kronecker pairing we get:
\[ \phi(\alpha ) = \bra{[\phi]}\ket{[\alpha]} =
 \bra{[\delta f ]}\ket{[\alpha]} =  \bra{[ f ]}\ket{[\partial \alpha]}  = 0\]
 since $\alpha \in Z_n^{\rm T}(X, A)$.
This contradicts the assumption.
\end{proof}

	Let $(G,  +)$ be a topological abelian group.
 Consider $G^m$ with the quandle structure given by the rule:
\[
(a_1, \ldots , a_m) * (b_1, \ldots , b_m) = (a_1, a_2 + b_1 - a_1, \ldots , a_m + b_{m-1} - a_{m-1}).
\]
By direct computation we see that the operation just defined respects the defining axioms for a quandle structure and it is continuous, hence define a topological quandle structure on $G^{m}$.
This construction is motivated from \cite{CENS}.

\begin{pro}
Let $(G, +)$ be a topological abelian group, $ x \neq 0$,  and
$(G^m, *)$ be the topological quandle defined as above.
Then $H^2_{\rm C}( G^{m}, G ) \neq 0$.
\end{pro}

\begin{proof}
Consider the following $2$-cycle (in the usual sense of discrete homology):
\[
\alpha = (0, \ldots , 0) \times (x, 0, \ldots , 0) + (0, x , 0, \ldots , 0)\times (-x, x, 0, \ldots , 1) \ ,
\]
where $\times$ has been used to better indicate that $\alpha$ is an element of $G^{m}\times G^{m}$.
By direct computation using the boundary map, it follows that $\alpha$ is indeed a $2$-cycle.
Consider also the $2$-cocycle:
\[
\phi : G^{m}\times G^{m} \longrightarrow G \]
defined by
\[ \phi ( \, (a_1, \ldots , a_m)\times(b_1, \ldots , b_m) \, ) = b_m - a_m \ .
 \]
Again by direct computation using the coboundary map it can be shown that $\phi$ is a cocycle.
Applying $\phi$ to $\alpha$ we get $\phi (\alpha) = x \neq 0$. Hence
by Lemma~\ref{lem:Kronecker} $\phi$ is not null-cohomologus, and we obtain $H^2_{\rm C}( G^{m}, G )\neq 0$.
\end{proof}

Note that a constant map $\phi$ in $\Gamma^n(X,A)$
is the zero map for $n>1$.

\begin{pro}\label{prop:linear}
Let $X=({\mathbb R}^n, S)$ and $A=({\mathbb R}^m, T)$ be  Alexander quandles, where $S \in {\rm GL}_n(\mathbb{R})$ and
 $T \in {\rm GL}_m(\mathbb{R})$, respectively.
Then $H^2_{\rm TC} ( X, A ) \neq 0$ if one of the following conditions hold.
\begin{itemize}
\item[{\rm (i)}]
There is a nonzero vector $v$ and $m \times n$ matrix $C$ such that
$  S^2-S+1 =  T^2 - T +1  =0 $ and $ C - TCS \neq  0$.
\item[{\rm (ii)}]
$T=-I $ and $S$ has eigenvalue $-1$.
\end{itemize}
\end{pro}

\begin{proof}
	Consider the matrix $M:=\begin{pmatrix}
	S & 0 \\
	C & T
	\end{pmatrix} \in {\rm GL}_{m+n}(\mathbb{R})$.
	Then $X \times A$
	can be considered as an Alexander quandle with operation
	$$\begin{pmatrix}
		x_1\\
		a_1
	\end{pmatrix} *\begin{pmatrix}
	x_2\\
	a_2
	\end{pmatrix}:= M \begin{pmatrix}
	x_1 \\
	a_1
	\end{pmatrix} +(I_{m+n} -M)\begin{pmatrix}
	x_2 \\
	a_2
	\end{pmatrix} . $$
	 A direct computation shows that $\begin{pmatrix}
	x_1 \\
	a_1
	\end{pmatrix} *\begin{pmatrix}
	x_2 \\
	a_2
	\end{pmatrix}= \begin{pmatrix}
	x_1*x_2\\
	a_1*a_2 +\phi(x_1,x_2)
	\end{pmatrix} $ and thus the quandle $X \times A$ is a topological twisted extension of the topological quandle $X$ with the continuous $2$-cocycle $\phi: \mathbb{R}^n \times  \mathbb{R}^n \rightarrow \mathbb{R}^m$ given explicitly by the formula
	\[
	\phi(x_1,x_2)=C(x_1-x_2).
	\]
We find a 2-cycle $w \in Z_2^{\rm T}(X,A)$ such that $\phi(w)\neq 0$ under each of the conditions (i) and (ii), then
Lemma~\ref{lem:Kronecker} implies $H^2_{\rm TC} ( X, A ) \neq 0$.
Let $w=(u_0, v_0) + T(u_1, v_1)$. This is an element of the free module over $A$ generated by pairs of elements of
$X$  as in \cite{CES}.
Then one computes
\begin{eqnarray*}
\partial w &=&
[T (u_0) + (I-T) (v_0) - (S u_1 + (I-S) v_0) ) ] \\
& & + \quad T [ T (u_1) + (I-T) (v_1) - (S u_1 + (I-S) v_1) ) \\
&=&
[ ( v_0)  -  (S u_1 + (I-S) v_0)  ] \\
& & +
T [ (u_0) - ( v_0 ) + ( v_1 ) - (S u_1 + (I-S) v_1) ]
+
T^2 [ (u_1) - ( v_1) ]
. \quad (*)
\end{eqnarray*}
where $(v_1)$ is a 1-chain, and not an element of $X$.

(i) Set $u_1=v_0$ and $v_1=S u_0  + (I- S) v_0$. Then the expression  $(*)$ vanishes if
$$(S^2 - S + 1) (u_0 - v_0)=(T^2 - T + 1)(u_0 - v_0) =0 . $$
Hence the assumption $  S^2-S+1  = 0= T^2 - T +1  $ implies  $\partial w=0$.

One computes $\phi(w)= ( C - TCS) (u_0 - v_0)$.
One can choose  nonzero vectors $u_0$ and $v_0$ satisfying this condition under the assumption  $C - TCS \neq 0$.
Hence
$\phi(w)= ( C - TCS) (u_0 - v_0) \neq 0 $ follows from the assumption.
From Lemma~\ref{lem:Kronecker} we obtain that $\phi$ is not null cohomologous.

(ii) Set  $T=-I$. Further, set $v_0=0$, $u_1= -u_0$, and $v_1=0$. Then $(*)$ vanishes if there exists
$u_0$ such that $Su_0=-u_0$, and this condition is satisfied by the assumption.
Then one computes
$\phi(w)= C(u_0 - 0) + C( - u_0 - 0) = C( 2 u_0)$. By choosing $C$ and $u_0$ such that $C u_0 \neq 0$,
one obtains a non-trivial 2-cocycle $\phi$.
\end{proof}

\begin{ex}
	Let $n=4$, $m=2$, $S=T \oplus T$, and $T=\begin{pmatrix}
	0 & -1\\
	1 & 1
	\end{pmatrix}$.
	Then $S^2-S+1=0=T^2-T+1$.
	Let $C=(I, I)$ where $I$ is $2\times 2$ identity matrix. Then $C-TCS= (I-S^2 , I-S^2 )$
	is not the zero matrix,
	and conditions in (i) in  Proposition~\ref{prop:linear} are satisfied, and we obtain  $H^2_{\rm TC} ( X, A ) \neq 0$
	for this specific example.
\end{ex}

\begin{rmk}
	The conditions of Proposition~\ref{prop:linear} (i) can be extended to those for higher degree alternating polynomials.
	Since the idea of proof is the same but the computations are lengthy, we delay its statement and proof to Appendix~\ref{sec:nontriv}.
\end{rmk}

\subsection{Continuous Cohomology with Quandle Modules}\label{sec:AG}

The goal of this subsection is to introduce  a topological version of the cohomology theory generalized in
\cite{AG}  and exhibit explicit examples with non-trivial continuous cohomology.
Before defining the generalized cohomology theory of topological quandles we
present a few
preliminary concepts.

 Let $X$ be a topological quandle.  Recall~\cite{AG, EM} that an $X$-module is a triple $(A,\eta,\tau)$ where $A$ is a topological abelian group, $\eta$ is a family of continuous group automorphisms $\eta_{x,y}:A\rightarrow  A$  and $\tau$  is a family of continuous group morphisms $\tau_{x,y}:A \rightarrow  A$
 such that

\begin{enumerate}
\item
$\eta_{x * y,z}\; \eta_{x,y} =  \eta_{x * z,y * z}\; \eta_{x,z}$,
\item
$ \eta_{x * y,z}\; \tau_{x,y} =  \tau_{x * z,y * z}\; \eta_{y,z}$,
\item
$\tau_{x * y,z}= \eta_{x * z,y * z} \tau_{x,z}+
 \tau_{x * z,y * z}\tau_{y,z}$, \quad and

  \item
   $\tau_{x,x} + \eta_{x,x} ={\rm id}_{ A}$.
\end{enumerate}

We consider the abelian groups
 $\Gamma^n(X, A)$, $\delta^i_0$ and $\delta^i_1$  as defined above in Section~\ref{secCohomology}.  Define the differentials by the following formula
  \begin{eqnarray*}
 \delta^n & : =&\sum_{i=2}^{n+1}(-1)^i\left(\eta_{[x_1,\cdots,\widehat{x}_i,\cdots, x_{n+1}],[x_i,\cdots, x_{n+1}]}\delta^i_0 - \delta^i_{  1}\right) \\
  & & + \ \tau_{ [x_2, x_3, \ldots, x_{n+1}], [x_1, x_3, \ldots, x_{n+1}]}\delta^1_0
 \end{eqnarray*}
 where
 $$[x_1, x_2, x_3, \ldots, x_{n}] = (( \cdots ( x_1 * x_2) * x_3 ) \cdots ) * x_n . $$
  The resulting cohomology groups  are denoted by
 $H^n_{GC}(X, A)$.

 As in the discrete case, it is easy to see that we obtain a cochain complex \[\cdots \rightarrow \Gamma^n(X,A) \xrightarrow{\delta} \Gamma^{n+1}(X,A)\rightarrow\cdots\]
  Following \cite{AG}, if $X$ is a topological quandle and  $(A, \eta, \tau)$ is a topological quandle module, we can define a topological quandle structure on $X \times A$ by:
\[ (x,a)*(y,b) = (x*y, \eta_{x,y}(a) + \tau_{x,y}(b) + \kappa_{x,y}),\]
 for all $x,y \in X$ and $a,b \in A$, where $X \times A$ is given the product topology.
This formula defines a topological quandle structure if and only if
$\kappa_{x,y}$ is a $2$-cocycle of this cohomology theory.

  We have that the
  projection onto the
  first factor
   is
   an epimorphism of topological quandles.

Let $G$ be the subgroup of ${\rm GL}(2n, \mathbb{R})$ for a positive integer $n$, consisting of block matrices of the form
$$G= \left\{ \
\left. E= \left(
\begin{array}{cc} S & O \\ C & T \end{array}
\right) \
  \right| \
S, T  \in {\rm GL}(n, \mathbb{R} ), C \in {\rm M}(n, \mathbb{R} ) \
\right\} , $$
where $O$ denotes the zero matrix,
and consider $X=G  \times  \mathbb{R}^n $
with quandle operation
$$
( E_0, x_0) * ( E_1,   x_1)=
( E_1 E_0 E_1^{-1},   S_1 x_0 + (I - S_1) x_1 )  ,
$$
where
$E_i = \left(
\begin{array}{cc} S_i & O \\ C_i & T_i \end{array}
\right)
$
for $i=0,1$.
Let $A=\R^{n}$ and consider endomorphisms of $A$ defined by
$\eta_{ (E_0, x_0), (E_1, x_1) } (a) = T_1 a $ and  $\tau_{ (E_0, x_0), (E_1, x_1) } (a) = (I - T_1) a $.
It is checked by computation that these define a $X$-module structure on $A$.

\begin{thm}\label{thm:AG}
$H^2_{GC}(X, A) \neq 0$.
\end{thm}

\begin{proof}
The quandle operation on $X \times A= G \times \mathbb{R}^{2n} $ defined by
$$
( E_0, u_0) * ( E_1,   u_1)=
( E_1 E_0 E_1^{-1},   E_1 u_0 + (I - E_1) u_1 )  ,
$$
where $u_i=(x_i, a_i)$ and $a_i \in A$ for $i=0,1$ are computed as operation on $X \times A$ as
\begin{eqnarray*}
\lefteqn{[ \   (E_0, x_0), a_0 \ ] * [ \ ( E_1,  x_1) , a_1 \ ] }\\
& = &
[ \  E_1 E_0 E_1^{-1},   S_1 x_0 + (I - S_1) x_1,  T_1a_0 + (I - T_1) a_1  +  C_1 (x_0 - x_1) ) ] .
\end{eqnarray*}
Let $p: X \times A \rightarrow X$ by $p(\ [ (E, x) ,a) ] \ ) = (E, x)$.
Then we find that $p$ defines the extension of $X$ by the $X$-module $A$,
with the 2-cocycle
$\kappa_{( E_0, x_0), ( E_1, x_1)}=  C_1 (x_0 - x_1)$.

We show that $\kappa$ is not a coboundary.
Let $E= \left(
\begin{array}{cc} -I  & O \\ C & -I  \end{array}
\right) $, and
 $w= [ (E, x), (E, 0) ] - [ ( E, -x), (E, 0) ]$ be a 2-chain.
 Since
 $\partial( \, ( x, y) \, ) = \eta_{x, y} (x ) + \tau_{ x, y}(y) - (x*y)$,
one computes
  \begin{eqnarray*}
\partial (w) & = &  (-I) ( E,  x)  + (2I) ( E, 0 ) - (E, (-I)x  +(2I) 0  ) \\
& - & [   (-I) ( E,  - x)  + (2I) ( E, 0 ) - (E, (-I)(-x)  +(2I) 0  ) \quad = \quad 0.
 \end{eqnarray*}
 Hence $w$ is a 2-cycle.
One also computes
\begin{eqnarray*}
 \kappa_{( E, x) , (  E, 0) } - \kappa_{  ( E, -x) , (E, 0) }
&=&
 C_1(x - 0 ) - C_1(-x - 0 )  \\
&=&
 C_1(2x )
\end{eqnarray*}
and by choosing $x, C_1$ such that $C_1x \neq 0$, we obtain that $\kappa$ is not a coboundary
by the argument similar to Lemma~\ref{lem:Kronecker}.
\end{proof}

The construction of $X\times A$ in Theorem~\ref{thm:AG} is similar to the notion of $G$-family of quandles defined in \cite{IIJO}.

\subsection{Principal Bundles as Quandle Extensions}
\label{sec:principal}

In this section we describe  quandle extensions of topological quandles
that are not defined by continuous 2-cocycles but can be described by principal bundles in a natural manner with discontinuous 2-cocycles.
As a consequence we observe the difference between continuous and algebraic cohomology theories.

Let $A$ be a topological abelian group and $p: E \rightarrow X$ be a principal $A$-bundle;
a fiber bundle with  a fiber preserving right action of $A$ on $E$ that acts freely and transitively.

\begin{df}[cf.~\cite{Eis-cover}]
{\rm
Let $E,X$ be connected topological quandles and  $A$ be a topological abelian
group.
A {\it principal  (abelian)
quandle extension by $A$} is a continuous surjective quandle homomorphism
$p: E \rightarrow X$ that is a principal $A$-bundle
such that
for all $x, y \in X$ and $a \in A$, the following conditions hold:
(i)  $(x*y)\cdot a= (x \cdot a)* (y \cdot a)$,
(ii) $ (x \cdot a)*y= (x * y) \cdot a$.
}
\end{df}

The quandle homomorphism in Example~\ref{ex:Rubin} is a  principal  abelian quandle extension by $\Z_2$.

\begin{lem}
Let $A$ be a topological abelian
group and
$p: E \rightarrow X$ be a  principal abelian quandle extension by $A$.
Let $s: X \rightarrow E$ be a set-theoretic section; $p \circ s={\rm id}_X$.
Then for all $x, y \in X$, there exists a unique element $a \in A$ such that
$s(x)*s(y)=s(x*y)\cdot a$.
\end{lem}

\begin{proof}
Since $p$ is a quandle homomorphism, we have
$$p( s(x)*s(y) ) = (p s)(x)* ( ps)(y)= x*y= (ps)(x*y). $$

Since $A$ acts  freely and transitively, there is a unique $a$ such that $s(x)*s(y)=s(x*y)\cdot a$.
\end{proof}

In the preceding lemma, the unique element $a$ is determined by $x, y \in X$, so that we denote it by $a=\phi(x,y)$.
Then  we obtain a function $\phi: X \times X \rightarrow A$.

\begin{lem} \label{lem:princ2cocy}
Let $\phi: X \times X \rightarrow A$ be  defined as above.
Then $\phi$ is a quandle (abelian) 2-cocycle.

\end{lem}

\begin{proof}
We perform the following computations analogous to those in \cite{CENS} and \cite{Eis-cover}:
\begin{eqnarray*}
 (s (x) * s(y) ) * s(z)
 &=& [\, s( x*y ) \cdot \phi(x, y) \, ] * s(z) \\
 &=& [\, s(x*y) * s(z) \, ] \cdot \phi (x,y) \\
 &=&  s( (x*y)*z )
  \cdot [ \phi (x*y, z) \phi (x,y) ] , \\
  (s (x) * s(z) ) * ( s(y) * s(z) )
  &=&  [\, s( x*z ) \cdot \phi(x, z) \, ] *  [\, s( y*z) \cdot \phi(y, z) \, ] \\
   &=& ( [\, s( x*z ) \cdot \phi(x, z)\phi(y, z)^{-1}  \, ] *  s( y*z) ) \cdot \phi(y, z)  \\
      &=& ( s( x*z )  *  s( y*z) ) \cdot [  ( \phi(x, z)\phi(y, z)^{-1}  ) \phi(y, z)  ] \\
   &=& s( ( x*z )  * ( y*z) ) \cdot [  (\phi(x*z, y*z)  \phi(x, z)  ] ,
 \end{eqnarray*}
 and $s(x)*s(x)=s(x*x)\cdot \phi(x,x)$ gives $\phi(x,x)=0$.
Hence $\phi$ satisfies the 2-cocycle condition.
\end{proof}

Nosaka pointed out to us that the argument works also for non-abelian groups $A$.
See \cite{AG} for non-abelian 2-cocycles.

\begin{ex}\label{ex:proj}
{\rm
Consider $p: S^2 \rightarrow \mathbb{RP}^2$ in Example~\ref{ex:Rubin}.
Let
$$P_+:= \{ (x,y,z) \in S^2 : z >0 \ {\rm or} \
z=0, y>0 \ {\rm or} \ y=z=0, x>0 \}$$
 and $P_-:= S^2 \setminus P_+$.
Let $s: \mathbb{RP}^2 \rightarrow S^2$ be a set-theoretic section defined by
$s( [x]) =x$ where $x\in P_+$.
Then the map $\phi$ of the preceding lemma provides a non-zero 2-cocycle.
For example, $\phi( [1,0, 0] , [0,1,0]) = 1 \in \Z_2$.
In this case, as a set $S^2$ is regarded as $\mathbb{RP}^2 \times \mathbb{Z}_2$.
}
\end{ex}

\begin{rmk}
{\rm
Let  $p: E \rightarrow X$ be a  principal abelian quandle extension by $A$,
and fix a set-theoretic section $s: X \rightarrow E$.
For any given  $u \in E$, let $x=p(u)$, then there is a unique $a=a_s(u)$ such that $u=s(x)\cdot a$.
Similarly for $v \in E$ let $y=p(v)$ and $v=s(y) \cdot b$.
Then one computes
$$ u*v= ( s(x) \cdot a ) * ( s(y) \cdot b ) = [ (  s(x) \cdot (a b^{-1} ) ) * s(y) ] \cdot b = [ s(x) * s(y) ] \cdot a
=  s(x*y)  \cdot  ( a \/ \phi(x, y) ) . $$
Note that this equality  $ ( s(x) \cdot a ) * ( s(y) \cdot b ) =  s(x*y)  \cdot  ( a \, \phi(x, y) )$
compares to the equality
$ (x, a) * (y, b)= (x*y, a + \phi(x, y) ) $ for $E=X \times_\phi A$ in the case $T=1$.
}
\end{rmk}

\begin{lem} \label{lem:zero}
Let $p: E \rightarrow X$ be a  principal abelian quandle extension by $A$.
If $A$ is discrete and $\phi$ is a continuous 2-cocycle, then $\phi(x,y)=0$ for all $x, y \in X$.
\end{lem}

\begin{proof}
From the assumptions $\phi$ is a constant map. From the quandle condition $\phi(x,x)=0$ for all $x \in X$,
it follows that $\phi(x,y)=0$ for all $x, y \in X$.
\end{proof}

Note that in this case  it follows that $s$ is a quandle homomorphism.

For  $p: S^2 \rightarrow \mathbb{RP}^2$ in Example~\ref{ex:Rubin} we have the following proposition.

\begin{pro}
$H^2_{\rm Q}(\mathbb{RP}^2, \Z_2) \neq 0$, yet   $H^2_{C} (\mathbb{RP}^2 , \Z_2)=0$.
\end{pro}

\begin{proof}
Let $\phi$ be the  quandle 2-cocycle constructed in Example~\ref{ex:proj}.
By Lemma~\ref{lem:princ2cocy} and Example~\ref{ex:proj}, $\phi$ is a (discrete) quandle $2$-cocycle,
that yields a non-trivial extension, and therefore, $\phi$ is non-trivial in $H^2_{\rm Q}(\mathbb{RP}^2, \Z_2) $.

By Lemma~\ref{lem:zero}, any continuous 2-cocycle gives rise to the trivial extension, and hence
$H^2_{C} (\mathbb{RP}^2 , \Z_2)=0$.
\end{proof}

\section{Inverse Limits of Quandles and their Cohomologies}\label{InvLim}

In this section we apply a method of computing cohomology of inverse limits to determine continuous cohomology of quandles.

\subsection{Basic Construction of Inverse and Direct Limits}

Suppose we are given a projective system of quandles $(X_n, \psi_n)_{n\in\mathbb{N}}$:
\[
X_1 \stackrel{\psi_1}{\longleftarrow} X_2 \stackrel{\psi_2}{\longleftarrow}  \cdots \longleftarrow X_n \stackrel{\psi_n}{\longleftarrow} \cdots
\]
where each $\psi_n$ is a quandle morphism.
We
define the inverse limit of the projective system, $\varprojlim X_n$, as the subset of $\prod_{n\in\mathbb{N}} X_n$ of sequences $(x_0, x_1, \ldots , x_n, \ldots)$ satisfying $\psi_n(x_{n+1}) = x_{n}$ for all $n\geq 1$.
We
give $\varprojlim X_n$ the $*$ operation induced componentwise by the operations of the $X_n$. This construction together with the canonical projection maps $\varprojlim X_n \rightarrow X_i$, for each $i \in \mathbb{N}$, satisfies the usual universal property for an inverse limit of a projective system indexed by the natural numbers, see Remark~\ref{rem:invsys} below.

The same construction can be defined
 for a projective system of topological quandles, where each morphism is now required to be continuous, and $\varprojlim X_n$ is endowed with the initial topology with respect to the projection maps.
 The initial topology is the coarsest topology that makes projections $p_i : \varprojlim X_n \rightarrow X_i$ continuous, and in our case,
 it is the same topology as the subspace topology of the product space.

A particular case is to start with a projective system of discrete (possibly finite) quandles. The topology on $\varprojlim X_n$ will be the subspace topology of the product $\prod_{n\in\mathbb{N}} X_n$, where the latter space has the product topology of discrete spaces.
Thus taking the inverse limit provides a method to build topological quandles from discrete quandles.
The following example is of central importance.

\begin{ex}
Fix a prime $p\in \mathbb{N}$. Put $X_n = \mathbb{Z}/p^n\mathbb{Z}$ together with the standard dihedral quandle operation $x*y=2y-x$. There are canonical projections $\psi_n : X_{n+1} \rightarrow X_{n}	$ obtained by reducing $mod \ p^{n}$ a representative of a class modulo $p^{n+1}$. These maps are
ring homomorphisms and, as a consequence, quandle morphisms, since the quandle structure on $X_n$ is obtained from the ring operations.
By definition, the inverse limit of this projective system is the ring of p\,-adic integers $\mathbb{Z}_p$ and it inherits a topological quandle structure from the dihedral quandles $X_n$.
To be precise, the same quandle operation would be obtained on $\mathbb{Z}_p$ defining the Alexander quandle structure with $T = -1$, via the ring operations on $\mathbb{Z}_p$. That is,
$( \varprojlim X_n, *)$ is isomorphic to $( \mathbb{Z}_p , -1)$.
\end{ex}

\begin{rmk}\label{rem:invsys}
More generally, if we start with a directed set $I$ and a projective system of (topological) quandles $(X_i, \psi_{i,j})$, where the morphisms $\psi_{i,j}$ satisfy the usual compatibility relations, we can define $\varprojlim X_n = \{ x\in\prod_{n\in I} X_n\ | \ \psi_{i,j}\pi_j(x) = \pi_i(x) \ $for all$ \ i,j \ $ with$ \  j \geq i  \}$  where $\pi_i$ is the canonical projection onto the $i^{th}$ factor.
They satisfy the universal property depicted below.

\[	\begin{tikzcd}
		& Y\arrow[d,dashed,"\exists!"]\arrow[ddr] \arrow[ddl]& \\
		& \varprojlim X_n\arrow[dr]\arrow[dl] &     \\
X_j\arrow[rr,"\psi_{i, j}"]	&         & X_i
			\end{tikzcd}\]

Then we can  endow it again with the $*$ operation induced pointwise by the quandle operations in each $X_i$ and get an inverse limit for the projective system we started with.
 If we start with topological quandles, the topology of $\varprojlim X_i$ will be again the initial topology with respect to the projections.
By definition it follows that the inverse limit of (topological) quandles, is the usual inverse limit in the category of topological spaces, equipped with a continuous binary operation that turns it into a topological quandle.
\end{rmk}

We
recall the following results from point-set topology, proofs of which  can be found in \cite{Wilson}:
\begin{lem}
	Let $(X_i, \psi_{i,j}) $be a projective system of topological spaces. Then the following results hold:
	\begin{itemize}
	\item[{\rm (a)}] If each $X_i$ is compact and Hausdorff, so is $\varprojlim X_i$;
	\item[{\rm (b)}]  If each $X_i$ is totally disconnected, so is $\varprojlim X_i$;
	\item[{\rm (c)}]  If each $X_i$ is a nonempty compact Hausdorff, then $\varprojlim X_i$ is nonempty.
	\end{itemize}
\end{lem}

\begin{lem}\label{lem:cptH}
	Let $(X_i, \psi_{i,j})$ be a projective system of nonempty compact Hausdorff spaces and let $Y$ be a discrete space.
	Then any continuous map $f : \varprojlim X_i\rightarrow Y $ factors through $X_i$ for some $i \in I$.
\end{lem}

Observe also that taking inverse limits in the category of topological spaces commutes with the operation of taking cartesian products, $ (\varprojlim X_i)^{\bullet} = \varprojlim (X_i^{\bullet})$.

A similar construction applies to
 an inductive system of quandles
\[
X_1 \xrightarrow{\phi_1} X_2 \xrightarrow{\phi_2} \cdots \xrightarrow{\phi_{k-1}} X_k \xrightarrow{\phi_k} \cdots
\]
to obtain a quandle structure on the direct limit
$\varinjlim X_k$. In this case $\phi_k$ are continuous quandle morphisms.
The direct limit is endowed with  the final topology (or inductive topology),  the finest topology which makes the  functions $X_k \rightarrow \varinjlim X $ continuous.
In particular, if $X_k$ are discrete, then $\varinjlim X_k$
is discrete.

\subsection{Cohomology of Inverse Limits}
The purpose of this section is to apply the tools from the continuous cohomology groups of the inverse limit of a projective system to finite discrete quandles $(X_n, \psi_n)_{n\in\mathbb{N}}$, to obtain the cohomology groups of $\varprojlim X_n$.
The tools used here depend only on the functoriality of twisted cohomology of quandles, the universal property of colimits and the factorization property of continuous maps between an inverse limit and a discrete space.
Suppose we are given a projective system $(X_n, \psi_n)$ of discrete quandles:
\[
X_1 \xleftarrow{\psi_1} X_2 \xleftarrow{\psi_2}\cdots \xleftarrow{\psi_{n-1}}X_n \xleftarrow{\psi_n} \cdots
\]
We also consider an inductive system of Alexander quandles $(A_k,\phi_k)$, where $(A_k, T_k)$ are topological Alexander quandles ($A_k$ is a topological abelian groups and $T_k$ are continuous automorphisms of $A_k$):
\[
A_1 \xrightarrow{\phi_1} A_2 \xrightarrow{\phi_2} \cdots \xrightarrow{\phi_{k-1}} A_k \xrightarrow{\phi_k} \cdots
\]
From the data above we can construct an inductive system of cohomology groups.
Below we suppress subscripts of cochain groups for simplicity.
Define cochain maps $C^\bullet(X_n, A_n) \longrightarrow C^\bullet(X_{n+1}, A_{n+1})$ by the following diagram:
\[
\begin{tikzcd}
C^\bullet(X_n, A_n) \arrow[r,"\psi_n^*"]\arrow[dr,dashed,"\tau_n"] & C^\bullet(X_{n+1}, A_{n})\arrow[d,"\phi_n \circ -"]\\
 & C^\bullet(X_{n+1}, A_{n+1})
\end{tikzcd}
\]
where the vertical map is the change of coefficients induced by $\phi_n$ and the horizontal map is the dual map of the projection $\psi_n$.
We obtain consequently an inductive system in the category of groups.
Consider now the following diagram:
\[\begin{tikzcd}
& \vdots \arrow[d,"\delta^{\bullet-1}"]& \vdots \arrow[d,"\delta^{\bullet-1}"]&\\
\cdots \arrow[r] & C^\bullet(X_n,A_n) \arrow[r,"\tau_n"]\arrow[d,"\delta^\bullet"] & C^\bullet(X_{n+1},A_{n+1}) \arrow[r]\arrow[d,"\delta^\bullet"] & \cdots \\
\cdots \arrow[r]& C^{\bullet+1}(X_n,A_n) \arrow[r,"\tau_n"]\arrow[d,"\delta^{\bullet+1}"]& C^{\bullet+1}(X_{n+1},A_{n+1}) \arrow[r]\arrow[d,"\delta^{\bullet+1}"] & \cdots \\
&  \vdots & \vdots&
\end{tikzcd}\]
where $\delta$ indicates the cohomology differentials and the maps $\tau$ are defined above.
Since $\psi^*_n$ and $\phi_n\circ -$ commute with differentials, the diagram is commutative, so that each $\tau_n$ induces a map on cohomology (which we will still denote by $\tau_n$).
Thus we obtain the inductive systems of cohomology groups:
\[H^\bullet_T(X_0,A_0) \xrightarrow{\tau_0}  H^\bullet_T(X_1,A_1) \xrightarrow{\tau_1} \cdots \xrightarrow{\tau_{n-1}}  H^\bullet_T(X_n,A_n) \xrightarrow{\tau_n} \cdots  \]
from which we derive their inductive limits: $\varinjlim H^\bullet_T(X_n,A_n)$.

\begin{sloppypar}
Our next goal is to relate $H^\bullet(X_n,A_n)$ to $H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l)$. More precisely we will show that $\varinjlim H^\bullet(X_n,A_n)$ is isomorphic to the continuous cohomology group $H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l)$ when each $A_l$ is a discrete group and the topologies on $\varprojlim X_k$ and $ \varinjlim A_l$ are the projective and inductive limit topologies.
\end{sloppypar}

Let us construct first a morphism from $\varinjlim H^\bullet_T(X_n,A_n)$ to $H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l)$ using the universal property of direct limits.
Consider the following diagram, corresponding to any representative $f$ of a class $[f]\in H^\bullet_T(X_n,A_n)$:
\[\begin{tikzcd}
(\varprojlim X_k)^\bullet \arrow[r,dashed]\arrow[d,"\pi^\bullet_n"] & \varinjlim A_l \\
X_n^\bullet \arrow[r,"f"] & A_n \arrow[u,"\iota_n"]
\end{tikzcd}\]
where $\pi^\bullet_n$ is the canonical projection $(\varprojlim X_k)^\bullet \rightarrow X^\bullet_n$ and $\iota_n : A_n \rightarrow \varinjlim A_l$ is the natural morphism of $A_n$ into the direct limit.
Observe that if $\iota_n f \pi_n^\bullet$ is
as above, then it factors through $X^\bullet_n$
by definition and, in particular, any preimage of a subset in $\varinjlim A_l$ is a basis element of the topology of $\varprojlim X_k^\bullet$ since each $X_k$ is a discrete topological space and $\varprojlim X_k$ is endowed with the projective limit topology. Since $\varinjlim A_l$ is discrete, being a direct limit of discrete spaces; it follows then that $\iota_n f \pi_n^\bullet$ is continuous. Also, the correspondence $\{f:X_n^\bullet \rightarrow A_n\} \stackrel{\iota_n f \pi_n^\bullet}{\longrightarrow} \{ (\varprojlim X_k )^\bullet \rightarrow\varinjlim A_l \} $ respects equivalence classes, so it induces a well defined map $\sigma_n:H^\bullet_T(X_n,A_n)\rightarrow H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l)$.
We obtain
therefore the following diagram:
\[
\begin{tikzcd}
\cdots \arrow[r] & H^\bullet_T(X_{n-1},A_{n-1} ) \arrow[r] \arrow[dr,"\sigma_{n-1}"]& H^\bullet_T(X_n,A_n)\arrow[r]\arrow[d,"\sigma_n"] & H^\bullet_T(X_{n+1}, A_{n+1}) \arrow[r]\arrow[dl,"\sigma_{n+1}"] & \cdots\\
& & H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l) & &
\end{tikzcd}
\]
which can be seen to be commutative by a direct computation.
 By the universal property of colimits we obtain a unique morphism $\varinjlim H^\bullet_T(X_n,A_n)\xrightarrow{\Psi}H^\bullet_{TC}(\varprojlim X_k, \varinjlim A_l)$.
 In fact, $\Psi$ is an isomorphism.
 We include a proof to illustrate the computations that will be given in corollaries below for specific quandles.

\begin{pro}
     Suppose $(X_n,\psi_n)$ is a projective system of discrete quandles and $(A_m,\phi_m)$ is a direct system of discrete abelian groups. Then the map $\Psi$ defined above is an isomorphism.

\end{pro}
\begin{proof}
\begin{sloppypar}
Suppose $\alpha \in \varinjlim H^\bullet_T(X_n,A_n)$ is mapped to zero by $\Psi$. By construction of $\Psi$, it means that there exists some $i \in \mathbb{N}$ such that $[\iota_i f \pi_i^\bullet]$ is the zero class in $H_{TC}^\bullet(\varprojlim X_k, \varinjlim A_l)$, where $f$ is a representative of a class $[f] \in H^\bullet_T(X_i,A_i)$, $\pi_i^\bullet$ is the projection on the $i^{th}$ factor	and $\iota_i$ is the natural map $A_i \rightarrow \varinjlim A_l$. So $\iota_i f \pi_i^\bullet$ is a coboundary of some continuous $g : (\varprojlim X_k)^{\bullet-1} \rightarrow \varinjlim A_l$ which  factors through some $X^{\bullet-1}_j$, $j\in \mathbb{N}$. Choosing $t\in \mathbb{N}$ large enough it follows that $[\iota_t f \pi_t^\bullet] = 0$ in $H^\bullet_T(X_t,A_t)$.  Since $[\iota_i f \pi_i^\bullet]\sim[\iota_t f \pi_t^\bullet]$ in $\varinjlim H^\bullet_T(X_n,A_n)$ it follows that $\alpha$ is the zero class.
\end{sloppypar}

Suppose now we are given a class $[\beta] \in H_{TC}^\bullet(\varprojlim X_k, \varinjlim A_l)$. Since $\beta$ is continuous and $\varinjlim A_l$ is discrete, it factors through some $X_i^\bullet$. Therefore its image in $\varinjlim A_l$ is finite and it will be contained in some $A_j$. Choosing $t\in \mathbb{N}$ large enough, we get that $[\beta]$ is the image $T\overline{[h]}$, for some $[h] \in H^\bullet_T(X_t,A_t)$, where the bar symbol indicates that we are considering a representative class in $ \varinjlim H^\bullet_T(X_n,A_n)$.
	\end{proof}

    See \cite{Wilson} for an analogous statement for group cohomology.

Observe that the construction of the morphism $\Psi$ and the proof above are still valid if we consider topological compact Hausdorff quandles $X_n$ and replace each $H^\bullet_T(X_n,A_n)$ by their continuous counterparts, in virtue of Lemma~\ref{lem:cptH}.

	Fix an odd prime $p\in\mathbb{Z}$ and choose $u\in\mathbb{Z}$ such that $(u, p)=1$. Then multiplications by $u$ and by $1-u$ define automorphisms of
    $\mathbb{Z}/p^n\mathbb{Z}$, for any $n\in\mathbb{N}$.
    Thus we obtain an Alexander quandle structure on $\mathbb{Z}/p^n\mathbb{Z}$, which will be denoted by $X^u_{n}$.
	Also recall that there are natural maps $ \mathbb{Z}/p^n\mathbb{Z} \longrightarrow \mathbb{Z}/p^{n+1}\mathbb{Z}$ given by multiplication by $p$, which commute with the action by $\mathbb{Z}$ and consequently a direct system, whose direct limit is the Pr\"ufer group $\mathbb{Z}(p^\infty)$.
    Thus we have an  Alexander quandle
    $(\mathbb{Z}(p^\infty), u)$, denoted also by  $\mathbb{Z}(p^\infty)_u$.

	\begin{cor}
		$H^1_{TC}(\varprojlim X^u_n,\mathbb{Z}(p^\infty)_u) \cong  \mathbb{Z}(p^\infty)_u \times \mathbb{Z}(p^\infty)_u$
		for any  $u\in\mathbb{Z}$ such that $(u, p)=1$.
	\end{cor}
\begin{proof}
	As in Section 4.1 it is possible to show that the first twisted (discrete) cohomology group $H_T^1(X^u_n, \mathbb{Z}/p^n\mathbb{Z})$ is  the abelian group of affine maps $\{f_{\alpha,\beta}:\mathbb{Z}/p^n\mathbb{Z}\rightarrow\mathbb{Z}/p^n\mathbb{Z}\mid f(x) = \alpha x + \beta,\ \alpha,\beta\in \mathbb{Z}/p^n\mathbb{Z}\}$ which can be seen to be isomorphic to $\mathbb{Z}/p^n\mathbb{Z}\times \mathbb{Z}/p^n\mathbb{Z}$ for all $n\in\mathbb{N}$, via the isomorphism $f_{\alpha,\beta}\mapsto\alpha\times\beta$.
	Using the definition of $\sigma_n:H^1_T(X^u_n,\mathbb{Z}/p^n\mathbb{Z}) \rightarrow H^1_T(X^u_{n+1},\mathbb{Z}/p^{n+1}\mathbb{Z})$ we have that $(\sigma_nf) (x) = (pf\pi_n) [x] = p\alpha [x] + p\beta$, from which we obtain the direct limit
	\[
	\mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z} \rightarrow \cdots \rightarrow \mathbb{Z}/p^n\mathbb{Z}\times\mathbb{Z}/p^n\mathbb{Z} \rightarrow \cdots
	\]
	where each map is just multiplication by $p$ on each coordinate. \\
	It follows that $\varinjlim H^1_T(X^u_n,\mathbb{Z}/p^n\mathbb{Z}) = \mathbb{Z}(p^\infty)_u\times\mathbb{Z}(p^\infty)_u$ and the result follows.
\end{proof}

\begin{cor} $H_C^3(\varprojlim R_{p^n}, \mathbb{Z}/p\mathbb{Z}) = 0$, where $R_{p^n}$ denotes the dihedral quandle on $p^n$ elements and the cohomology group is meant to be untwisted.
	\end{cor}
\begin{proof}
\begin{sloppypar}
	For a given odd prime, it has been computed by Mochizuki \cite{Mo}, that $H^3(R_{p^n}, \mathbb{Z}/p\mathbb{Z}) = \mathbb{Z}/p\mathbb{Z}$, where $R_{p^n}$. Directly from the proof in \cite{Mo}  it also follows that the map $H^3(R_{p^n}, \mathbb{Z}/p\mathbb{Z}) \rightarrow H^3(R_{p^{n+1}}, \mathbb{Z}/p\mathbb{Z})$ induced by the canonical projection $ R_{p^{n+1}} \rightarrow R_{p^n}$ is the trivial map. We obtain the inductive system:
    \end{sloppypar}
	\[
	\mathbb{Z}/p\mathbb{Z} \xrightarrow{0} \mathbb{Z}/p\mathbb{Z} \xrightarrow{0} \cdots \xrightarrow{0} \mathbb{Z}/p\mathbb{Z} \xrightarrow{0} \cdots
	\]
	whose direct limit is the trivial group.
\end{proof}

redpajama_set_name: RedPajamaArXiv


text:
Q: odbc connection for PL/SQL I'm using PL/sql developer to create table.i used conncetion codings like
Class.forName("oracle.jdbc.driver.OracleDriver");
con = DriverManager.getConnection( "jdbc:oracle:thin:@localhost:1521:test")
ps = con.createStatement();

SQL_Str="Select * from dhanu_demo where uname like ('" + UName +"')"
rs=ps.executeQuery(SQL_Str);

rs.next();
dbusername=rs.getString(1).toString();
dbpassword=rs.getString(2).toString();

now i want to create dsn name through
controlpanal->performance&maintannence->datasource->system domain->?
i dunno which driver i need to select here. can anyone tell me. how to connect my jsf page to PL/SQL database??

A: For your way of connecting there is no need to create dsn. because it uses the type 4 driver from oracle. Moreover to interact with database in jsf
Check out the following link http://www.mkyong.com/jsf2/jsf-2-0-jdbc-integration-example/

redpajama_set_name: RedPajamaStackExchange


text:
Hackers backed by foreign states attacked 12,000 Google users
Government-supported hackers tried to steal 12,000 Google credentials in 2019
Google's Threat Analysis Group (TAG) revealed in a report that they distributed more than 12,000 warnings to users in nearly 150 countries that include the United States of America. The recipients were notified that they have been subject of state-sponsored phishing attempts. The foreign government-supported hackers have tried to obtain the Google login credentials of thousands of people using "credential phishing emails." Those attacks generally have one goal – to hijack the victim's Gmail account by tricking users into thinking that they are on a Google website, and their password needs to be changed.
The emails usually contain links that forward the potential victims to a spoofed website – a webpage that has the appearance of the real one. The fake website encourages the victims to update their login details to increase security. However, instead of changing passwords, people end up sharing their login details and unintentionally end up giving up access to their accounts. Very often, victims do not even know that they are being hacked as the hacker does not use the hacked account to cause immediate trouble but monitors it until they decide the best way to strike.
Possible strikes would include:
Finding a way to access your internet banking or even bitcoin wallet
Looking for materials that could be used for blackmailing
Stealing trade secrets
A first step towards overtaking your identity
Nowadays hackers find it particularly easy to commit fraud after they hijack an account. Email inboxes are often filled with personal information such as passport scans, birth certificates, etc. Google's Threat Analysis group also revealed in the report that they are actively battling groups of hackers backed by foreign governments who are working on spreading disinformation.
TAG said that there are certain groups of people who are considered high-risk. Such userbase consists of journalists, human rights activists, and people involved in shaping the political landscape in their home countries. TAG strongly encourages people with such backgrounds to always have additional layers of security on all their devices.
Knowing that government-sponsored hackers have targeted thousands of people might sound scary, but the report revealed that the number hadn't changed much when compared with the data from the last few years. The attacks have been going on for years! According to the information presented in the report, Google stated that the number of alerts sent by them this year was similar to the number they observed in the same period of 2018 and 2017.
Having the most reliable protection is crucial not only for high-risk people wanting to maintain the highest level of security possible but also for everyone not ready to become a victim of a cybercrime.
Working Abroad: What Countries Have the Fastest Internet?
The GDPR in 2019: the year of the million euro fine
Porn: You know when you can see it, but does a computer?
How to limit work-from-home distractions?
Cyber criminals targeting Olympics
Xavier Mertens: "Cryptojacking is one of the most brilliant attacks I've seen"
What can organizations learn from military cyberdefense?
Robert Proctor says:
This is not a political comment. What else came out today is that the origin of these attempts have come from several eastern countries including Turkey, Russia and the Ukraine. As this would be a potential threat to our democracy, I wonder what our government is doing with this news.

redpajama_set_name: RedPajamaCommonCrawl


text:
Big Box Retail
City Investment
Søstrene Grene opens its world's largest store in Berlin
11.02.2022 Berlin | Deal announcement
Danish lifestyle chain Søstrene Grene is opening a new store at Schloßstrasse 30 in Berlin's Steglitz district in April 2022. The previous tenants at the location were Tivola Shoes and, for a transitional period, an outlet store. Lührmann advised on the lease and acted as intermediary.
The new Søstrene-Grene store on Berlin's second largest shopping street has almost 750 square metres of retail space on two levels. There, Søstrene Grene (translated: the Grene sisters) will in future present its wide range of home accessories, kitchen items, DIY supplies, party goods, school and office supplies and toys. With its store concept 'Retail for the Senses', the new branch provides customers with a whole new shopping experience, which Søstrene Grene itself describes as follows: 'The new concept offers the unmistakable Søstrene Grene feeling for all the senses – a feeling of joy, relaxation, 'hygge' (coziness), enchantment and constant change'.
'We are very pleased to have gained Søstrene Grene as a very desirable new tenant for the store at Schloßstraße 30', explains Christoph Schulz of Lührmann Berlin. The new store in a striking corner location is by far the largest Søstrene Grene store worldwide.
In recent years, the Danish lifestyle brand Søstrene Grene, founded in 1973 in Aarhus, has firmly established itself in Germany: there are now more than 50 stores across the country, making Germany Søstrene Grene's second-largest market outside Denmark. In 2022, the company plans to open more; by the middle of the year, Søstrene Grene aims to have a total of 60 stores in Germany.
With over 240 stores in 15 countries, Søstrene Grene has become a household name throughout Europe. The stores are extremely popular with customers – thanks to their charming winding aisles and their range of Scandinavian design.
More about Søstrene Grene: www.sostrenegrene.com/de
We connect potential – as Germany's real estate specialists for prime locations. There is a perfect location for every brand, every tenant and every project. Lührmann Germany, headed by Managing Directors Achim Weitkamp, Mario Hampel and Michael Backhove, operates from six locations. Berlin, Düsseldorf, Frankfurt, Hamburg, Munich and Osnabrück form the optimal network to consolidate our expertise in the business areas of Retail, Office, Hotel & Hospitality, City Investment and Big Box Retail for you. This enables us to reach all locations in German metropolitan regions within a very short time. And our excellent knowledge of properties, people and markets, combined with a passion for the perfect deal, provide the conditions for a perfect match. And this has been the case since 1992.
These include essential cookies that are necessary for the operation of the site, as well as others that are used only for anonymous statistical purposes, for comfort settings or to display personalized content. You can decide for yourself which categories you want to allow. Please note that based on your settings, not all functions of the website may be available.
Choose cookies
Your cookie preferences have been saved.
To load this element, it is required to consent to the following cookie category: {category}.
Load {category} cookies


text:
Triple S Elects New Board Members
Triple S announced its newly elected board of directors and executive committee at the buying group's annual stockholders meeting earlier this month.
Elected board members for the coming year are:
Aaron Buchko, Bradley University
Michael Chiappe, California Janitorial Supply
Jim Chittom, Jr., Athens Janitor Supply Co., Inc.
Kenneth R. Crutcher, Kenway Distributors, Inc.
James Egan, Egan Supply Co.
Butch Ellis, Lansing Sanitary Supply, Inc.
Bruce Janvey, I. Janvey & Sons, Inc.
Dutch Owens, Gem Supply Co.
Morten Riegg, GORM, Inc.
Michael J. Tighe, Industrial Cleaning Products, Inc.
John B. Treat, Treat's Solutions, Inc.
Additionally, the board elected Owens as chairman of the board; Chittom, Jr., as secretary; Forrest T. Thompson IV of Triple S as treasurer; and Alan E. Sadler of Triple S as clerk.
About Triple S
Based in Billerica, MA, Triple S is a member-owned national distribution services and logistics company that provides facility maintenance solutions to the education, health care, commercial, retail, industrial, and government markets through a network of independent member-dealers and regional distribution centers. For more information, visit www.triple-s.com.


However, let's check concrete examples of these "ancient behemoths". Summary Weight is huge factor in heavyweight boxing. Whether it's MMA or kickboxing or Kung-Fu or wrestling: Boxing like other contact sports is divided into weight..
If your work history has fewer than 35 years, the missing years will be counted as zeros. For married couples, the available options for filing strategies grow. Most people file before they can receive their..
of Liver Diseases position paper on acute liver failure 2011. Moving out to California, he co-founded Intel Corporation in 1968. Bernal W, Wang Y, Maggs J,. "The Significance of the Frontier in American History".
American Dream, come True. The American Dream, scholarship helps the dream of higher education become a reality. This scholarship picks up where other aid and scholarships leave off, ensuring that bright, qualified students have the means to continue their education. At its best, the American Dream can be both moving and inspiring, but it is also problematic, not least because it is largely false.
Stravitz RT, Kramer AH, Davern T, et al, for the Acute Liver Failure Study Group. We will all be sacrificed. Upon arriving in America, Yang knew how Religions Influence Gender Identity just one word of English: "shoe." But in spite of the language barrier, Yang excelled in school and attended Stanford, graduating in 1990. Rosenberg, Spreading the American Dream : American Economic and Cultural Expansion (1982). . 26 Carlin pointed to "the big wealthy business interests that control things and make all the important decisions" as having a greater influence than an individual's choice.


text:
GeodSoft
Using the Member Database to Control Web Site Security
Association web site security automated by using codes from the member database to control access to member only areas. Automated export and online registration take staff out of routine authorization maintenance. Highly granular access control to resources based on member type and section / special interest group memberships.
Automated Maintenance
Member Authentication
NT Users and OS Security
Member Data Export
Application Level Security
Adaptive Scripts
Batch Updates
Most web server software provides for public and private areas. IIS on NT and Apache on most platforms provide for tie-ins with operating system security and using OS usernames and passwords. Beyond that, providing granular security becomes a matter for custom programing. While ATLA NET was on the legal publisher's host, we were limited to public and private areas. All registered members had access to all private areas.
When ATLA took control of its own site, the two major functional areas that it gained control over were the list server set up and site security. Hosting our own site let us take site security in directions that few sites other than intranets have gone with highly granular security and selective access to limited areas. Intranets, however, tend to get their security definitions from the underlying OS. Network administrators maintain the user databases or directories against which the web server software authenticates users. Since this has been a core part of network administration since the inception of LANs there is no special additional overhead other than perhaps adding a few web specific user groups.
Where the number of users, in this case members, is enormous compared to the number of support staff, any administrative procedure that requires staff involvement to set up or control member access to a web site quickly becomes a major administrative burden and in the long run probably can not be supported. ATLA had already taken the first steps towards automating web user administration while still on the legal publishers site.
Though the Netscape server authentication file provided only for public and private areas and no differentiation within the private areas, this file was maintained via an online registration process where ATLA members created their own user names and passwords. To register a member had to provide their last name, their ATLA member number and their bar year or birth date. We examined all the data in the member database and considered what we could have the member enter, that they would know or have easy access to, that the public could not easily get and that would match unambiguously against the actual data stored in our member database.
Lawyer's last names are public information but neither ATLA's member ID or a lawyer's birthday or bar year are easily obtained by persons not associated with the member. When we considered what data to verify members against we always tried to keep in mind that our primary concern was to protect access to the system so that non members could not get into and use the online member services as members. Our traditional computer security concerns were relatively minor compared to this. Simply put, there is very little data of value to anyone except members and their legal opponents on the web site. Most of the information is derived from other sources and is well backed up. The site is not an e-commerce site so a denial of service attack or even the complete loss of the system to crackers would be more of a (big) nuisance than a catastrophe.
Since ATLA does not do and is not likely ever to do background checks on its members, anyone who really wants access to the system merely needs to fraudulently complete a membership application claiming they are a lawyer who has just started to practice law. This puts the price of illicit admission at about $50. When you consider that, it's pointless to go to the lengths that many sites need to protect themselves. Until early 1999 when we created a public member directory that gave away one of the key pieces of information necessary to enter the site as a member, we felt that our security was appropriate to our needs.
On the old system the user assigned themselves a username and a password and when they completed registration, the username and encrypted password were saved in a file, used by the Netscape server, to grant on/off access to the private areas of the site. The new system changed this so that the registration process actually created a new NT user record from the username and password provided by the member registrant. In addition, codes in the member record were used to add the member to as many NT user groups as necessary, one NT group per each code. Thus member record codes translated directly to membership in NT user groups which were in turn, granted access to various resources on the system.
The preceding statement is not strictly true. In ATLA's membership system codes that may grant access to resources on the web site come from at least four different sets of codes in the member record. Theoretically, at any time new areas of the site could be added that correspond to an entirely different set of codes in the member record. There is a single primary member type, potentially a multiplicity of other member types, committee codes and a catch all category code.
The primary member type correlates most importantly with voting status and dues level. The other member types were for ATLA Sections which are extra fee specialty groups focused on specific areas of the law. Some resources are available only to voting members and some benefits go only with a particular dues level such as the higher paying types getting larger, more comprehensive, directory listings.
Many of ATLA's access rules are not so simple however but rather based on certain combinations of codes. When looking at the resources associated with Sections some were simple and depended only on membership in the Section but others depended on being a voting member and a member of any one or more sections. Still others required being a member of the specific section and others both a voting member and a member of the specific section. The very highest level of primary membership overrode most of these and provided access to nearly all the section related resources, regardless of whether the member was a member of the section or of any section. Finally some sections had special access rules that allowed types of members who were barred from accessing comparable resources in other Sections, to use certain resources within the specific Section area.
Clearly, trying to embed complex logic like this in various routines, or even a centralized security module on the web site that is not directly tied to the member database and has no knowledge of the member database structure, would likely cause problems. The solution was to translate various combinations of codes into a series of simple codes in one place. The data export module that prepared data from the membership database for the web site performed this function. This export program, used procedural logic to create the various relationships between different sets of codes and to explicitly account for the unique cases.
The first exports, were initially developed in 1995 when ATLA's site was remotely hosted, ATLA had no continuous high speed connectivity to the Internet, and VPNs and real time ODBC back end database to web site connectivity were things of the future. In very resource limited environments such as most associations there is a strong tendency to make minor modifications to existing infrastructure. If such changes can achieve the functionality that is needed immediately, there is little inclination to review core infrastructure to determine how well it's likely to meet long terms needs. Further, "long term" needs in an association environment are much more volatile, being subject to rapid change based on current leadership, than long term needs in a traditional profit oriented business.
For a number of services that are provided by ATLA's web site, the export model works quite well. The procedural logic in the export routine is able to translate complex business rules and multiple codes stored in different locations in the member record into single codes that grant or deny access to specific resources on the web site.
For the protection of static resources, some codes passed to the web site are turned into NT user groups. There is a table that defines which member codes become which NT user groups. For dynamic content provided via scripts, as mentioned previously, a limitation in NT security required the development of application level security via a function that all protected scripts had to call.
Ultimately, regardless of the functionality provided by NT and IIS, such a module would have been needed to provide controlled access to specific resources to which the scripts gave access. As an example, ATLA Online includes a significant number of list servers which are available to various subsets of ATLA's members. The same basic scripts provide the font ends for all lists. The security module needs to determine which member is running the script and whether or not they are allowed to access the list they are trying to get to.
The security module is called at the beginning of each private script and determines if the user has access to the script. The security module is passed the logical name of the function to be protected. If the user has access, control is returned to the calling function, otherwise the security module terminates the program and displays an HTML page that indicates the user is not authorized to use the function they are trying to access.
Though the security model is largely table driven, there is more to it than this. The function names are hierarchical in nature. The security module always attempts to determine access based on the most specific function name provided but if there is no table entry that matches the current function name, that name is reduced to the next higher level and another check performed. As long as a match is found at some level, the remote users access codes are checked against those allowed for the function being tested and if an exact match is found, access is allowed.
The multi level naming structure allows broad classes of security access to be defined but specific security can be placed on any function, any point within a function or any resource accessed by a function, as necessary. This allows for highly flexible, but low maintenance application security, as specific table entries do not need to be created for every possible function point to be protected. Safety is provided at two levels: once a match is found in the security tables, the user must have access at that level. No further checking is done if they do not. Access is denied. Unless at least one match is found in the security table, no access is allowed to any user. This prevents careless security setup from unintentionally opening a protected function to unauthorized users.
Further, any protected function may define numerous points within its own logic that can have different function names and thus different levels of access for various members. This allows scripts to dynamically adjust the content they display depending on who is running them. Thus, a link to a resource the user does not have access to is not displayed at all rather than being displayed and inviting the user to click on it only to be met with a password prompt or access denied message. Alternatively, if the member might become eligible to use the resource they might be shown instructions on using the resource or a description of the resource, which in turn may dynamically adjust to indicate why the user is seeing instructions and not the actual resource and what the user must to gain access to the actual resource.
Where a script provides access to and protects a resource, such as a Section membership related benefit, the scripts pass the section name (or a shortened version) between themselves which causes the appropriate resource to be presented to the user. The function names that are passed to the security function are dynamically created before calling the security function and where variable resources such as section related benefits are involved, the section name becomes part of the function name that is passed. By systematic use of the multilevel naming structure, highly flexible security can be provided with a minimum of administrative maintenance. Only a limited number of security definition tables need to be maintained. All user specific setup is handled automatically by the system.
So far everything that has been described covers, the initial user account creation and setup and dynamic access to interactive web functions. One important piece is missing. In any membership organization, over time, some significant number of members will lapse, upgrade or otherwise change their membership status. The web site has to dynamically accommodate this without either staff or member intervention. In an environment where the integration with the member database is handled via data export, as long as this function is performed automatically at the end of each business day, for most practical purposes, the web member database is kept synchronized with the central member database.
Updating the web member data is all that is necessary to control access to the dynamic site content. Dynamic content is accessed through scripts that call the central security module, each time a script is run. The security module in turn checks the current member data. This leaves at least two other areas which must be dealt with separately. Unless the NT user record which is created during the online registration is removed or disabled a lapsed member would continue to have access to static site content. Also any application, such as list servers, that have their own application security are not protected from changed member status just because the web member data has been updated. List server integration and security will be discussed at length in the next section on list server integration.
In general, what is needed is a batch function that examines all users created via the online registration process which then determines these users current access. The batch process cannot process all NT users as there are likely to be users created by normal administrative processes and not via the online registration. Specifically administrators, server operators and web content developers are created outside the registration process and have access to the web site quite different than that provided members via the online registration process. It's worth noting that the registration process operates as the anonymous user which is defined as an NT account operator. An account operator can create and delete users but only users with no administrator or operator privileges. This is believed to be safer and more secure than the alternative of storing an administrator password somewhere on the server that the registration process would use to change its user context to an administrator account context.
Also the batch process cannot process all member records because most members have never registered and do not have matching NT users. The online registration process maintains a file that correlates NT user names with ATLA member IDs. It's that file that the batch process uses and for each entry in it, finds the corresponding member record and NT user record. As long as the member remains active and eligible to use the site the NT user record is kept. If the member lapses, the NT user record is removed and an email composed notifying the former user that their access to the web site has been disabled and explaining why and what to do to regain access. As long as the member remains eligible, then all NT user groups are compared against the codes in the member record and the table of member codes and corresponding NT user groups. The NT user is added to or removed from NT user groups based on the current codes in their member record. This allows the NT OS to grant or deny access to all static web pages based on current membership status.
<P The Transition <
^ Large Project Contents ^
> List Server Integration N>
Top of Page - Site Map
Copyright © 2000 - 2014 by George Shaffer. This material may be distributed only subject to the terms and conditions set forth in http://GeodSoft.com/terms.htm (or http://GeodSoft.com/cgi-bin/terms.pl). These terms are subject to change. Distribution is subject to the current terms, or at the choice of the distributor, those in an earlier, digitally signed electronic copy of http://GeodSoft.com/terms.htm (or cgi-bin/terms.pl) from the time of the distribution. Distribution of substantively modified versions of GeodSoft content is prohibited without the explicit written permission of George Shaffer. Distribution of the work or derivatives of the work, in whole or in part, for commercial purposes is prohibited unless prior written permission is obtained from George Shaffer. Distribution in accordance with these terms, for unrestricted and uncompensated public access, non profit, or internal company use is allowed.
Large Project >
security.htm
<P The Transition
N> List Server Integration
About GeodSoft
- Large Web Project
- - Historical Background
- - Standardization
- - Selection of NT
- - List Server Selection
- - Transition to NT
- - Security and Members
- - Integration Issues
- - List Server Issues
- - Document Sharing
- - Electronic Contracts
- - Performance Surprise
- Designing GeodSoft
- Building GeodSoft
- Good Passwords
- 10 Security Steps
- Intrusion Detection
- Secure Shell (SSH)
- Harden OpenBSD
- Dual Boot Open Source
- Style Sheets (CSS)
- Time Synchronization
- Product Packages
- Software Licenses
- Open Source Limits
- Server Comparison
- Bogus PHP Bug
- Corel Linux
Book: Assn. Webs
- Network & Web Basics
- Assn. Computer Security
- Publishing Myth
Password Evaluator
Crack Time Calculator
Copyright © 2000-2014, George Shaffer. Terms and Conditions of Use.


Drivers around the Ashburton District are being urged to take care on the roads following a number of wet weather events that have caused widespread potholes and washouts.
Ashburton District Council Service Delivery Group Manager Neil McCann says contractors are working very hard to repair the roads, however there is a lot of work to get to and it will take time.
"After the heavy rainfall event on 21 July, contractors quickly began repairing the roads, however after the rainfall we've had in recent days, some of the work they completed has been undone and will need to be carried out again. On top of this, we are expecting more rainfall which will likely do further damage.
"Water is a road's worst enemy. It seeps into minute cracks and erodes the quality of the bitumen. As pressure such as vehicles driving on the road continues to push the water into the cracks, the road begins deteriorating. That's when we end up with potholes and washouts.
"We have a big list of roads to attend to, but we can't get to all of it at once. In the interim, I urge all drivers to be cautious and slow down, especially on roads they aren't familiar with. It's also important to keep a safe distance from vehicles in front of you so you have time to spot potholes and avoid hitting them. Please be assured that we are doing our best to repair the roads. We really appreciate your patience."