text:
$32 Million Cryptocurrency Scam Ceased by the SEC
SEC has ceased a US$32 million cryptocurrency investment scam created by Steve Chen, who misled hundreds of investors to raise millions of dollars
The US Securities and Exchange Commission (SEC) has ceased a US$32 million cryptocurrency investment scam created by Steve Chen, who misled hundreds of investors to raise millions of dollars, by claiming that the investment firm US Fine Investment Arts, Inc. (USFIA) was a subsidiary of US China Consultation Association, a joint venture between the United States and Chinese governments.
The USFIA (Jason Chen) guaranteed its investors a 64x payout in Gemcoins, in exchange for their investment in the company. Chen personally claimed that USFIA owned large amber mines in the Dominican Republic and Argentina, and that USFIA was set to undergo an initial public offering in the near future.
Foreign investors took interest in the IPO of USFIA and Chen's claims that all investors would be entitled to receive pre-IPO shares in USFIA that would increase in value once the company becomes listed on the public stock market.
Investors were promised a highly profitable payout in Gemcoin, a cryptocurrency created by USFIA as part of the scheme. On its website, USFIA stated that Gemcoin is backed by US$50 billion in assets and has partnered with ApplePay to roll out 2,500 ATMs throughout the United States.
"Gemcoin's value is backed by parent company AFG's 10 Precious Gem mines and Gemcoin are stable as it is backed by its equivalent in precious stones. USFIA has agreements with ApplePay - 2,500 ATM's ready for nationwide distribution for use with our Global VISA Debit Card and joining a ATM network along side Bitcoin," announced USFIA.
Furthermore, Chen and others from USFIA hosted multiple presentations and talks to convince its investors that the company:
Plans to open 3,000 coffee shops in the next three years;
Owned 70% of the amber mines in the Dominican Republic;
founded China Unicorn, an undersecretary to the Secretary of States of the People's Republic of China;
would increase exponentially every two months in value.
The US$32 million raised from the Gemcoin scam was almost immediately wired to Chen's personal accounts in China, which are now frozen due to the complaints of the US SEC. However, the SEC claims that the majority of the funds were already used to purchase luxurious goods such as sports cars, apartments and other properties.
#Investments
ICOs Flow Continues As Regulations Fall Around the World: Expert Blog
SEC Launches Mock ICO to Show Investors Warning Signs of Fraud


CCR Transport Plea - Beauty-FILLED-SHE IS GOING HOME!!! | Cane Corso Rescue, Inc.
Cane Corso Rescue, Inc. would like to thank all our transport angels, volunteers, and friends for supporting Beauty and our mission. Because of YOU this precious baby will be going home this weekend and will start living her happy ever after life. Transport angels please ensure you email me your information for the final run sheet. I have it from most but there a few I am waiting here back from. My email is Cheryl.CCR.Inc@gmail.com. Who needs fireworks when we have 'the village'!
If you cannot help with this transport, please make sure that you use the social share buttons at the bottom of this post in case someone you know may be able to help.
Beauty has received her miracle of a forever family but she needs a few transport angels to deliver her miracle. Can you help deliver the miracle to Beauty and her family? Thank you in advance for considering!
Rescue: Cane Corso Rescue, Inc.
Cane Corso Rescue, Inc. is a 501(c)(3) Non-Profit organization as recognized by the IRS. Your unreimbursed mileage and tolls are a tax-deductible donation to this charity!
Great news our precious Beauty, fondly nicknamed Bee, has found her FUREVER family. We are so elated as not only is this family beyond fabulous but they will be welcoming their second CCR rescue into their homes and hearts. Bee came into rescue as an owner surrender to a local Texas vet who had been providing her routine care. Fortunately, the vet contacted CCR so she would never have to experience the heartbreaking conditions of a shelter or the possibility of being euthanized. Bee has come a long way since in our care, developing into a confident, well- mannered Corso whose stunning beauty is simply breath taking. Bee has been in foster care for several months, she has resided with children, small dogs, and large dogs. She is a healthy, happy girl who is going to give her family a life time of delight. Unfortunately, Bee is in Texas and her family is in Florida so in order for her to receive her miracle we need a quite a few transport angels to get her home! CCR, Bee, and her family would be FUREVER grateful if you would consider being a part of Bee's miraculous journey by driving a leg or two of her transport.
This 3 year old black Beauty is healthy, happy, and ready to receive her miracle. Bee weighs about 100lbs. She is spayed, microchipped, current on vaccinations and preventive care for heartworm, flea and ticks. Bee will travel with a health cert, vaccination records, leash, collar, and food. We are confident she will do well on this long transport as she loves to go for car rides and loves meeting new people. Bee gets along great with other dogs and children but we ask that she not be introduced to any during transport to ensure her level of excitement is kept at a minimum. Bee travels well and since all CCR transports are single dog transports, there is no need to crate her as she will jump right in the back seat and happily go for a ride.
64miles, 1hr, 5min – FILLED – Thank You, Jennifer W!
81miles, 1hr, 12min – FILLED, Thank you-Adrienne W!
106miles, 1hr 39min – FILLED, Thank You -Melanie L!
58miles, 53min – FILLED, Thank you-Robert A!
71miles, 1hr, 2min – FILLED, Thank you-Linda!
59miles, 1hr, 1min – FILLED, Thank you-Emily!
93miles, 1hr, 27min – FILLED, Thank you-Emily!
99miles, 1hr, 35min – Filled, Thank you -Sharon T!
78miles, 1hr, 9min – Filled, Thank you- Corrie!
68miles, 1hr – Filled, Thank you-Anna!
78miles, 1hr, 14min – Filled, Thank you-Anna!
79miles, 1hr, 11min – Filled, Thank you-Anna!
89miles, 1hr, 18min – FILLED! Thank You Marianne M.!
98miles, 1hr, 29min – FILLED! Thank You Marianne M.!
CCR thanks all the wonderful volunteers for their support in getting this lovely girl safely to her fabulous family where she will be loved and cherished forevermore.
If you can help, please reply here in the comments. If you have a link in your comment (includes email addresses) it will be held for moderation. This is to reduce our SPAM count.
The more help needed. ….
Kate R. Can you please email me your information for the run sheet. Cheryl.CCR.Inc@gmail.com. Thank you!
I can possibly do the Baton Rouge to Slidell leg. Let me check with my hubby.
Thank you Kristen for your possible of offer of support. We were able to fill that leg so I hope you and hubby will do something fun. I hope to work with you in the future should CCR have another need in your area.
Hey guys, I would like to help with the Crowley to Baton Rouge leg because I actually already have plans to make that drive from Lafayette that day. Only thing is I would kind of need to be an hour or so earlier if there is any flexibility. My plans are late lunch with the folks. Let me know and if so I would love to help.
Hi Patrick, thanks for offering to help. Any chance you are available to do these legs on Saturday, July 4th?
I can do leg #14 and #15 for my baby girl.
I can do 15. I live 25 minutes from Ocala.
Okay, I can definitely do LEG #3. 🙂 I live in Tomball which is a little North of Houston.
Hi, Melaine…we rescheduled for July 4th weekend, can you still do leg 3? Paws crossed you can as Beauty's family is anxious to get her home.
Hi, Melanie…we got you down for leg 3. I emailed you for your information for the final run sheet. Thank you so much for your support! I look forward to working with you.
Cheryl, sorry for the delay here on this board…my internet has been down all week…but we got in touch…thanks for using other means ;-D So happy I can help get Bee to her new home!!
Where are we at with her???
I can do DFS to Lake City – Legs 11, 12, & 13 – if there arent any other offers – if that helps …..
Hi Anna, We had to reschedule the transport, are you able to do those legs on July 5th?
I removed your phone from the comment as this may pose a privacy risk for you. I however, am not involved in coordinating the transport, but if you are having any website issues, please fill out this form.
Anna. I think I can drive as far as chipley. Help you split up #11. I'm doing #10.
I am in north Florida and do transports. I want to cross post your request but I don't do social media. Do you have an email I can use to send out with the request so people can contact you by email?
Beverly please email me at Cheryl.CCR.Inc@gmail.com so I can send you a run sheet for Beauty's transport. Thank you so much for your support.
Corrie can you please email me at Cheryl.CCR.Inc@gmail.com so I can get your information for the run sheet. Thank you!
can we get the run sheet updated….. bunch of peeps are committing to legs and they still say NEED. Thanks Matt.
I can do Leg #2.
Thank you Adrienne…got you down for leg 2! Only 4 more legs to go!
Hi, Judy…thank you for your offer of support. We were able to fill that leg but would love to work with you in the future should we have another beautiful Corso going home through yur area.
Only 3 to go – thanks Judy.
Only 1 leg to go. Leg 4 is all we need.
I can help with LC to Lafayette. Is she kid friendly to ride with?
Got you down for Leg 5. Emailing for information for the final run sheet. Thank you so much for your support!
Linda can you please email me as your email was returned as undeliverable? Cheryl.CCR.Inc@gmail.com Thank you!
I can do leg 7. I emailed info to Cheryl. I'm in nola and can easily pick her up in Baton Rouge.
Thank you so much for your offer of support. We were able to fill that leg but certainly would love to work with you in the future.
OMG only 2 left to secure… Cato is going to be soooooo happy.
I will do leg 4 Beaumont to Lake Charles.
Oh my GOSH…you FILLED the transport. Beauty is going home! Thank you Robert! I am emailing you now for your information. WE DID IT!
YEAH SO HAPPY FOR CATO GETTING A NEW PLAYMATE and also a GREAT NEW MOM …. MARIANNA IS THE BEST!!!!!
Awesome news! So happy……. so happy for Cato. Thanks Mo for the endorsement – your aren't so bad yourself 🙂 Thanks Cheryl Lutz – you are the best!
Iwill be riding with Marianna…Catos mom..


text:
\section{Introduction}
Radiometric age dating has been popular for decades in groundwater hydrology and for more than a century in other disciplines such as geology, archeology, and so on. It is one of the recent advances in hydrogeology to clarify the definitions of the kinematic, mean, and radiometric ages of groundwater. In general, there is an increasing consensus that the apparent radiometric age, which is the most measurable quantity, cannot be considered to represent the mean age, which may be the most representative for the average residence time of water molecules in the subsurface (\cite{Bethke2002},~\cite{Bethke2008},~\cite{Sanford2011}). Discrepancies between
ages obtained by means of different approaches have been frequently reported, and in the majority of the cases purely advective ages are compared to ages definitions that include advection and dispersion processes (\cite{Castro2005},~\cite{LaBolle2006},~\cite{Bethke2008}). Concentration-based ages are simply apparent ages that do not a priori relate to the mean age of a water sample (\cite{Sanford2011}). Radiometric ages are classically derived from the interpretation of chemical/isotopic data that ignores transport characteristics although isotope transport undergoes dispersion/diffusion, decay and to some extend reaction processes in the subsurface. The kinematic age of groundwater is defined as the time for which a groundwater molecule may travel along a one-dimensional stream line from the time of recharge (birth). Thus, this kinematic age is calculated based only on the advective component of the migration of water molecules (piston flow) without considering the mixing processes among water molecules which enter the system at different times in the past (different birthdates). Since the kinematic age does not consider mixing, it cannot be directly translated from any chemical signature in the real world where various types of mixing always exist. Irrespective of the importance of the kinematic age it should be noted that it is a function of only the groundwater velocity which is one of the most fundamental physical parameters to describe the groundwater flow system. When the ages are known at two different locations, hydrogeologists may hope to be able to calculate the average velocity of groundwater dividing the distance by the difference in age along this distance.\\
A radiometric age is a measurable quantity for a given water sample, based on the change in the activity of decaying solutes. When the initial activity at birth and the rate of the decay (change) are known, the radiometric age can be calculated from the current activity at given locations. This assumes that the water sample under consideration is in a closed system, i.e. a system that does not exchange any material with its surrounding. Radiometric age thus differs from the mean age of groundwater, which is defined as the first temporal moment of the age probability density function for given samples, describing how age distributes as a result of advection and mixing processes. Age mass is a useful extensive (additive) property, defined as the product of age and mass for a given mass (e.g. see~\cite{Goode1996},~\cite{Varni1998},~\cite{Ginn1999}). Mathematically an intensive property or the age probability density, defined as the age mass divided by the total mass, is a popular concept to derive the equation that describes the mixture of different ages of groundwater.\\
\cite{Bethke2008} clarify the difference between the sample apparent (radiometric) age and the mean age of groundwater, the mathematical description of which has been well established by~\cite{Goode1996},~\cite{Varni1998},~\cite{Ginn1999},~\cite{Cornaton2006}, and~\cite{Ginn2009} among others. Although it seems that there are many different versions of mathematical equations to describe the mean age, age probability density, and age mass, conceptual difference is minor and consists in the assumed steady or transient flow conditions or different dependent variables as an integral moment or probability density (\cite{Ginn2007},~\cite{Cornaton2007}).\\
The recent finding that the radiometric age may not be representative for the mean residence time or the velocity of groundwater can imply that the only way to analyze the mean residence time is to consider applications of the mean age equation. This may lead to questioning the relevance of the radiometric ages frequently estimated by isotope hydrologists and for what purpose the radiometric age is measured. Such an age may not be the most relevant data to be used for calibrating a groundwater flow field in a numerical model (for example by solving inverse problems constrained by radiometric age data and traditional hydrological data). As indicated by~\cite{Bethke2008}, time is ripe for developing a new paradigm that would account for the recent findings.\\
In this paper, we first review mathematical equations to describe the transport of the kinematic age, mean age, and decaying solute and illustrate the differences among them. The relationships between a measurable quantity and more intrinsic quantities are then analyzed. A discussion on what can be interpreted from measurable quantities and under which conditions is finally made.

\section{Mathematical Description of Kinematic, Radiometric and Mean Ages}
This Section is dedicated to the definition of the kinematic, radiometric and mean ages, and to the comparison between each age property. One-dimensional and two-dimensional analyses are made in order to reveal the bias induced by the use of apparent ages for calculating apparent flow velocities.

\subsection{One-Dimensional Analysis}
Considering a simple one-dimensional flow system where groundwater recharges at the origin ($x = 0$) and the flow system is homogeneous in a steady state regime ($v(x,t)=v$), the kinematic age ($A_k$) is straightforward to calculate from its definition:
\begin{equation}
A_k = \frac{x}{v}
\label{eq:kinematicAge}
\end{equation}
Steady-state transport of a decaying solute can be described using a simple advection-dispersion-reaction equation as:
\begin{equation}
- v \frac{{d C(x)}}{{d x}} + D \frac{{d^2 C(x)}}{{d x^2 }} - \lambda C(x) = 0
\label{eq:1DADE}
\end{equation}
where $D$ is the dispersion coefficient and $\lambda$ is the constant for the first order decay to satisfy $C(t) = C(t = 0)e^{- \lambda t}$. Since the boundary concentration at origin is assumed to be known, $C(x = 0) = C_0$, the solution of Eq.~(\ref{eq:1DADE}) is
\begin{equation}
C(x) = C_0 \exp \left ( \frac{x v (1-\beta)}{2 D} \right )
\label{eq:concSol}
\end{equation}
where
\begin{equation}\label{eq:betadef}
\beta = \sqrt{1 + \frac{4 \lambda D}{v^2}}
\end{equation}
and where use has also been made of the condition $- \left.D \frac{\partial C(x)}{\partial x} \right|_{x = \infty } = 0$. The apparent age ($A_a$) is a direct translation of the initial concentration and the rate of change in the first order kinetic decay and is classically given as
\begin{equation}
A_a(x) = - \frac{1}{\lambda} \ln \left( \frac{C}{C_0} \right) = \frac{x v}{2\lambda D}\left( \beta - 1 \right)
\label{eq:apparentAge}
\end{equation}
It is obvious from Eq.~(\ref{eq:apparentAge}) that the apparent age is influenced by the mixing ($D$) and the solute decay rate ($\lambda$) as well as by the flow velocity ($v$) and the location ($x$). Eqs.~(\ref{eq:kinematicAge}) and~(\ref{eq:apparentAge}) also indicates that apparent age $A_a$ can only relate to kinematic age for the asymptotic situation, i.e. $A_a = A_k$ when $\frac{v^2}{\lambda D} \rightarrow \infty$. \\

The equation describing mean age ($A_m$) is well established (e.g. see~\cite{Goode1996},~\cite{Varni1998}) and a simplified version for the given one-dimensional domain can be written as
\begin{equation}
- v \frac{d A_m(x)}{dx} + D\frac{d^2 A_m(x)}{dx^2} + 1 = 0
\label{eq:meanAgeEqn}
\end{equation}
When the birth for groundwater is assumed at the inlet and a boundary condition for the mean age is given as $A_m(x = 0) = 0$, the mean groundwater age is same as the kinematic age:
\begin{equation}
A_m(x) = \frac{x}{v}
\label{eq:meanAgeSol}
\end{equation}
for which the condition $- \left.D \frac{\partial A_m(x)}{\partial x} \right|_{x = \infty } = 0$ has also been used. Equations~(\ref{eq:kinematicAge}) and~(\ref{eq:meanAgeSol}) imply that if the flow system is one-dimensional or if there is only one velocity component, the mean age which is same as the kinematic age is only a function of a flow velocity and of the location in the flow system, and thus it is straightforward to translate the mean age to describe the flow velocity. In other words, when one knows the groundwater mean ages ($A_m(x_1)$ and $A_m(x_2)$) at two different locations ($x_1$ and $x_2$) in a one-dimensional flow system, the distance between two locations divided by the difference in mean age can provide an average velocity between two points:
\begin{equation}
v = \frac{x_2 - x_1}{A_m(x_2) - A_m(x_1)}
\label{eq:veloApproxFromMA}
\end{equation}
When the apparent radiometric ages ($A_a(x_1)$ and $A_a(x_2)$) are known, the average velocity between two locations can also be calculated from~(\ref{eq:apparentAge}) such that
\begin{equation}
v = v_a - \frac{\lambda D}{v_a}
\label{eq:veloApproxFromAA}
\end{equation}
where the apparent velocity $v_a$ is defined as $[x_2  - x_1 ]/[A_a (x_2 ) - A_a (x_1 )]$, by analogy with Eq.~(\ref{eq:veloApproxFromMA}). Eq.~(\ref{eq:veloApproxFromAA}) indicates that the groundwater velocity can be calculated using the apparent velocity when the decaying property of the solute ($\lambda$) is known and the mixing processes ($D$) are clearly defined. It is also implied from~(\ref{eq:veloApproxFromAA}) that the apparent velocity generally overestimates the actual velocity and that the difference between them becomes greater when $v_a$ is smaller and the product $\lambda D$ is larger. When the dispersion coefficient is defined as $D = D_{diff} + \alpha_l v$, Eq.~(\ref{eq:veloApproxFromAA}) can be re-arranged to give
\begin{equation}
v = \frac{v_a (1 - \lambda D_{diff} /v_a^2)}{1 + \lambda \alpha_l / v_a}
\label{eq:veloApproxFromAA2}
\end{equation}
where $D_{diff}$ is the effective diffusion coefficient in the porous medium and $\alpha_l$ is the coefficient of longitudinal dispersivity. Defining the dimensionless variable $\Pi_a = \frac{v_a^2}{\lambda D}$ (defined as Jenkins number in~\cite{Mouchet2008}), Eq.~(\ref{eq:veloApproxFromAA}) takes the form:
\begin{equation}
v = v_a \left (1 - \frac{1}{\Pi_a} \right )
\label{eq:veloApproxFromAAadim}
\end{equation}
The variable $\Pi_a$ is similar to the P\'{e}clet number ($Pe$) if one considers a characteristic distance $L = \frac{v_a}{\lambda}$ as being the distance that tracer molecules can travel if they move at the average velocity $v_a$ during time $\lambda^{-1}$ ($Pe = \frac{v_a L}{D} = \frac{v_a^2}{\lambda D} = \Pi_a$). \\

Finally, the mean age of a radioactive tracer ($A_{m,\lambda}$) can be obtained by calculating the first two temporal moments of the corresponding age concentration density function (pdf). The age concentration pdf $g_\lambda(x,\tau)$ of a radioactive tracer (where $\tau$ denotes the variable age) is solution of the advection-dispersion-reaction equation
\begin{equation}\label{eq:tracerpdfEqn}
\frac{\partial g_\lambda(x,\tau)}{\partial \tau} = - v \frac{\partial g_\lambda(x,\tau)}{\partial
x} + D \frac{\partial^2 g_\lambda(x,\tau)}{\partial x^2} - \lambda g_\lambda(x,\tau)
\end{equation}
with the boundary conditions $g_\lambda(0,\tau) = \delta(\tau)$ (zero age flux concentration) and $- \left.D \frac{\partial g_\lambda(x,\tau)}{\partial x} \right|_{x = \infty } = 0$:
\begin{equation}\label{eq:tracerpdfSol}
g_\lambda(x,\tau) = \frac{x}{\sqrt{4 \pi D} \tau^{3/2}} \exp \left ( -\frac{( x - v \tau )^2}{4 D \tau} - \lambda \tau \right ) = \exp \left ( -\lambda \tau \right ) g_0(x,\tau)
\end{equation}
For instance, the zero-order temporal moment of $g_\lambda(x,\tau)$ gives the mass of radioactive tracer in the system:
\begin{equation}\label{eq:componentM0}
m_{0,\lambda}(x) = \int_0^\infty {g_\lambda(x,\tau) d\tau} = \exp \left ( \frac{x v (1-\beta)}{2 D} \right )
\end{equation}
in which $\beta$ has been defined in Eq.~(\ref{eq:betadef}). The first-order temporal moment of $g_\lambda(x,\tau)$ gives the mean age concentration of the radioactive tracer:
\begin{equation}\label{eq:componentM1}
m_{1,\lambda}(x) = \int_0^\infty {\tau g_\lambda(x,\tau) d\tau} = \frac{x}{\beta v} m_{0,\lambda}(x)
\end{equation}
The mean age of the radioactive tracer is then obtained as
\begin{equation}\label{eq:componentMA}
A_{m,\lambda}(x) = \frac{m_{1,\lambda}(x)}{m_{0,\lambda}(x)} = \frac{x}{\beta v} = \frac{A_m(x)}{\beta}
\end{equation}
Note that $m_{0,\lambda}(x)$ satisfies the tracer mass equation~(\ref{eq:1DADE}) using the boundary conditions $m_{0,\lambda}(0) = 1$ (unit mass input) and $- \left.D \frac{\partial m_{0,\lambda}(x)}{\partial x} \right|_{x = \infty } = 0$, and that $m_{1,\lambda}(x)$ satisfies the tracer mean age concentration equation
\begin{equation}\label{eq:componentM1Eqn}
- v \frac{\partial m_{1,\lambda}(x)}{\partial
x} + D \frac{\partial^2 m_{1,\lambda}(x)}{\partial x^2} - \lambda m_{1,\lambda}(x) + m_{0,\lambda}(x) = 0
\end{equation}
with the boundary conditions $m_{1,\lambda}(0) = 0$ (zero mean age concentration) and $- \left.D \frac{\partial m_{1,\lambda}(x)}{\partial x} \right|_{x = \infty } = 0$.\\

Dimensionless variables can be used to describe the relations between the apparent age, mean tracer age and mean water age, and between the apparent velocity and the actual velocity. The ratio of the apparent and mean ages $r_A = A_a /A_m$ and the ratio of the apparent and mean velocities $r_v = v_a / v$ can be defined using the dimensionless variables $\Pi = v^2 /\lambda D$, $\Pi_a = v_a^2 /\lambda D$, $\Pi_{a,diff} = v_a^2 /\lambda D_{diff}$, and $\Pi_{a,disp} = v_a /\lambda \alpha _l$. The following relations are derived from~(\ref{eq:apparentAge}),~(\ref{eq:veloApproxFromAA}), and~(\ref{eq:veloApproxFromAA2}) (see Fig.~\ref{fig:fig1}):
\begin{subequations}\label{eq:ratios}
\begin{equation}\label{eq:ratios1}
r_A = \frac{\Pi }{2}\left( {\sqrt {1 + \frac{4}{\Pi}} - 1} \right)
\end{equation}
\begin{equation}\label{eq:ratios2}
r_v = \frac{{\Pi_a }}{{\Pi_a - 1}} = \frac{1}{2}\left( {1 + \sqrt {1 + \frac{4}{\Pi}}} \right) = \frac{{1 + 1/\Pi_{a,disp} }}{{1 - 1/\Pi_{a,diff}}}
\end{equation}
\end{subequations}
\begin{figure}[h]
  \begin{center}
  \includegraphics[width=\textwidth]{fig1.jpg}
  \end{center}
  \caption{\small The ratios of (a) apparent and mean ages, and (b) apparent and mean velocities as functions of the dimensionless variables $\Pi = v^2 /\lambda D$ and $\Pi_a = v_a^2 /\lambda D$.}
 \label{fig:fig1}
\end{figure}
A first fundamental consideration arising from the analysis of Eqs.~(\ref{eq:componentMA}),~(\ref{eq:meanAgeSol}) and~(\ref{eq:ratios1}) is that the inequality $A_{m,\lambda} \leq A_a \leq A_m$ always holds, as already shown by~\cite{Deleersnijder2001}. \cite{Delhez2003} also demonstrated that this inequality is valid for more general contexts. Another interesting aspect is that the age and velocity ratios in Eqs.~(\ref{eq:ratios1}) and~(\ref{eq:ratios2}) are independent on the $x$-coordinate. Eq.~(\ref{eq:ratios1}) and Fig.~\ref{fig:fig1}a indicate that the apparent age generally underestimates the mean age and that the ratio between two ages becomes larger as the decay and dispersion dominate the advective transport and vice versa. Eq.~(\ref{eq:ratios1}) also indicates that the age ratio $r_A$ does not depend on the spatial coordinate $x$. When the parameter $\Pi$ decreases (i.e. $\lambda D$ increases), then the bias increases, and vice-versa. Thus apparent ages are reliable only if $\Pi$ is large enough, or consequently only if the product $\lambda D$ is small enough. In Eq.~(\ref{eq:ratios2}) and Fig.~\ref{fig:fig1}b, the apparent velocity is similar to the actual velocity when the transport regime is advection-dominated ($\Pi > 1$), and the actual velocity can be calculated from the apparent velocity when $\Pi_a > 1$. When the dispersion dominates the transport ($\Pi_a \leq 1$), it may not be possible to estimate the actual velocity from the apparent velocity.

\subsection{Coupled One-Dimensional Solutions for Two-Dimensional Analysis of Aquifer-Aquitard Systems}
A simple two-dimensional domain can be considered when mobile and immobile (or a lot less mobile) waters co-exist in a groundwater system. Such two-dimensional analyses have been widely adapted for a fracture-matrix system (\cite{Cornaton2008}) and also in an aquifer-aquitard system (\cite{Bethke2002,Bethke2008},~\cite{Ginn2009}). In these flow systems, the flow in a mobile region is essentially one-dimensional and the velocity is large compared to that in an immobile area so that it is safe to assume that the velocity in the immobile region is negligible. When an aquitard (or a matrix block) has a dimension of $2L$ perpendicular to the flow direction $x$ between two aquifers of a dimension $2l$ (or two fractures of an aperture value of $2l$), transport of the mean age or decaying solute can be described using an advection-dispersion-reaction equation in the aquifer and the diffusion-reaction equation in the aquitard (see Figure~\ref{fig:fig2}).\\

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=0.6\textwidth]{fig2.jpg}
  \end{center}
  \caption{\small Illustration of the aquifer-aquitard system used to derive two-dimensional analytical solutions.}
 \label{fig:fig2}
\end{figure}

The kinematic age in the mobile aquifer ($A_{k,m}$) is, by definition (see Eq.~(\ref{eq:kinematicAge})), a direct translation of the velocity in the mobile region ($v_x$):
\begin{equation}
A_{k,m} = \frac{x}{v_x}
\label{eq:mobilekinematicAge}
\end{equation}
Since the velocity in the immobile region is assumed to be negligible, the kinematic age in the immobile aquitard ($A_{k,im}$) is infinite for all $x$ and $z$, where $z$ represents a direction perpendicular to the flow direction along the aquifer.\\

Steady-state transport of a decaying solute in the aquifer is described by the following one-dimensional advection-dispersion-reaction equation such that
\begin{equation}
- v_x \frac{d C_m(x)}{dx} + D_x \frac{d^2 C_m(x)}{dx^2} - \lambda C_m(x) + \left. {\frac{n_{im} D_z }{n_m l}\frac{d C_{im}(z|x)}{dz}} \right|_{z = 0}  = 0
\label{eq:mobileADE}
\end{equation}
where $n_m$ and $n_{im}$ represent the porosities in the mobile and immobile regions, respectively. Steady-state transport in the aquitard can also be described by a one-dimensional diffusion-reaction equation:
\begin{equation}
D_z \frac{d^2 C_{im} (z|x)}{dz^2} - \lambda C_{im}(z|x) = 0
\label{eq:immobileADE}
\end{equation}
When the concentrations in the interface between mobile and immobile regions are assumed to be identical ($C_{im} (z|x) = C_m (x)$) and the mass flux in the middle of the aquitard is negligible ($\left. { - D_z \frac{d C_{im}(z|x)}{dz}} \right|_{z = L}  = 0$), then the solution of Eq.~(\ref{eq:immobileADE}) is given as
\begin{equation}
C_{im}(z|x) = C_m(x)\frac{{e^{\sqrt {{\textstyle{\lambda \over {D_z}}}} (z - L)} + e^{- \sqrt {{\textstyle{\lambda \over {D_z}}}} (z - L)}}}{{e^{\sqrt {{\textstyle{\lambda \over {D_z}}}} L}  + e^{- \sqrt {{\textstyle{\lambda \over {D_z}}}}L}}}
\label{eq:immobileSol}
\end{equation}
Using Eq.~(\ref{eq:immobileSol}), Eq.~(\ref{eq:mobileADE}) can be re-written as
\begin{equation}
- v_x \frac{d C_m(x)}{dx} + D_x \frac{d^2 C_m(x)}{dx^2} - (\lambda + \gamma) C_m(x) = 0
\label{eq:mobileADE2}
\end{equation}
where
\begin{equation}
\gamma = \frac{{n_{im} \sqrt {\lambda D_z }}}{{n_m l}}\frac{{e^{\sqrt {{\textstyle{\lambda \over {D_z }}}} L} - e^{- \sqrt {{\textstyle{\lambda \over {D_z }}}} L} }}{{e^{\sqrt {{\textstyle{\lambda \over {D_z }}}} L} + e^{- \sqrt {{\textstyle{\lambda \over {D_z}}}} L} }}
\label{eq:gamma}
\end{equation}
The solution of Eq.~(\ref{eq:mobileADE2}) is similar to Eq.~(\ref{eq:concSol}) where $\lambda$ is replaced by $(\lambda + \gamma)$:
\begin{equation}
C_m(x) = C_m(0) \exp \left[ {\frac{x v_x}{2 D_x}\left( {1 - \sqrt {1 + \frac{{4(\lambda + \gamma)D_x}}{{v_x^2}}}} \right)} \right]
\label{eq:mobileConc}
\end{equation}
and the apparent age in the aquifer ($A_{a,m}$) is also similar to Eq.~(\ref{eq:apparentAge}) such that
\begin{equation}
A_{a,m}(x) = - \frac{1}{\lambda} \ln \left( {\frac{C_m(x)}{C_m(0)}} \right) = \frac{x v_x}{2\lambda D_x}\left( {\sqrt {1 + \frac{4(\lambda + \gamma)D_x}{v_x^2}} - 1} \right)
\label{eq:mobileAA}
\end{equation}
The apparent age in the aquitard ($A_{a,im}$) can also be calculated using~(\ref{eq:immobileSol}) and~(\ref{eq:mobileConc}):
\begin{equation}
A_{a,im}(x,z) = - \frac{1}{\lambda} \ln \left( {\frac{C_{im}(z|x)}{C_m(0)}} \right) = A_{a,m}(x) + \frac{1}{\lambda} \ln \left[ {\frac{{e^{\sqrt {{\textstyle{\lambda \over {D_z}}}} L} + e^{- \sqrt {{\textstyle{\lambda \over {D_z}}}}L}}}{{e^{\sqrt {{\textstyle{\lambda \over {D_z}}}} (z - L)} + e^{- \sqrt {{\textstyle{\lambda  \over {D_z}}}} (z - L)}}}} \right]
\label{eq:immobileAA}
\end{equation}
The results~(\ref{eq:mobileAA}) and~(\ref{eq:immobileAA}) indicate that the apparent age in the aquifer is determined by the decay rate of the solute and the mixing processes in both the aquifer and aquitard, and that the apparent age in the aquitard is same as that in the aquifer at the interface between the two domains and has a maximum value at the center of the aquitard when $z = L$. The maximum of the apparent age is found at $z = L$:
\begin{equation}
\max [A_{a,im}(x,z)] = A_{a,im}(x,z = L) = A_{a,m} + \frac{1}{\lambda} \ln \left[ {\frac{{e^{\sqrt {{\textstyle{\lambda \over {D_z}}}} L} + e^{- \sqrt {{\textstyle{\lambda \over {D_z}}}} L}}}{2}} \right]
\label{eq:MaximmobileAA}
\end{equation}
The maximum age increases as the aquitard becomes thicker (larger $L$), the decaying rate of the solute is higher (larger $\lambda$), and the effective diffusion coefficient ($D_z$) is smaller. Fig.~\ref{fig:fig3} illustrates the distributions of the concentration of a decaying solute and the apparent age in such a two-dimensional system when $l = 5$m, $L = 10$m, $v_x = 10$m/yr, $\lambda = \ln 2/100{\rm{yr}}$, $D_x = \alpha_l v_x  = 100{\rm{m}}^{\rm{2}} {\rm{/yr}}$, $D_z = 0.004{\rm{m}}^{\rm{2}} {\rm{/yr}}$, and $n_m = n_{im} = 0.1$.\\

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=\textwidth]{fig3.jpg}
  \end{center}
  \caption{\small Illustration of the distributions of (a) relative concentration of a decaying solute and (b) apparent age in a two-dimensional aquifer-aquitard system.}
 \label{fig:fig3}
\end{figure}

Mean groundwater age in the aquifer ($A_{m,m}$) and in the aquitard ($A_{m,im}$) can be described using equations similar to~(\ref{eq:meanAgeEqn}),~(\ref{eq:mobileADE}) and~(\ref{eq:immobileADE}):
\begin{equation}
- v_x \frac{d A_{m,m}(x)}{dx} + D_x \frac{d^2 A_{m,m}(x)}{dx^2} + \left. {\frac{n_{im} D_z}{n_m l}\frac{d A_{m,im}(z|x)}{dz}} \right|_{z = 0} + 1 = 0
\label{eq:mobileMAEq}
\end{equation}
\begin{equation}
D_z \frac{d^2 A_{m,im}(z|x)}{dz^2} + 1 = 0
\label{eq:immobileMAEq}
\end{equation}
By applying the following boundary conditions:
\begin{subequations}\label{eq:mobimmobMABCs}
\begin{equation}
A_{m,im} (z = 0|x) = A_{m,m} (x)
\label{eq:mobimmobMABC1}
\end{equation}
\begin{equation}
\left. {- D_z \frac{d A_{m,im}(z|x)}{dz}} \right|_{z = L}  = 0
\label{eq:mobimmobMABC2}
\end{equation}
\begin{equation}
A_{m,m}(x = 0) = 0
\label{eq:mobimmobMABC3}
\end{equation}
\end{subequations}
The solutions of~(\ref{eq:mobileMAEq}) and~(\ref{eq:immobileMAEq}) are given as
\begin{subequations}\label{eq:mobimmobMASol}
\begin{equation}
A_{m,m}(x) = \frac{x}{v_x} \left( 1 + \frac{n_{im} L}{n_m l} \right)
\label{eq:mobimmobMASol1}
\end{equation}
\begin{equation}
A_{m,im}(z|x) = \frac{1}{2 D_z}z(2L - z) + A_{m,m}(x)
\label{eq:mobimmobMASol2}
\end{equation}
\end{subequations}
Eq.~(\ref{eq:mobimmobMASol1}) corresponds to the result of~\cite{Bethke2002}, the so-called groundwater age paradox, which is actually a simple direct consequence of the hypotheses of steady-state conditions, constant porosity and no diffusion in the direction of aquifer flow in the aquitard (\cite{Ginn2009}). Clearly, the mean age in the aquifer is a function of the ratio of the water volumes of the two systems ($\frac{n_{im} L}{n_m l}$) to the kinematic age. Eq.~(\ref{eq:mobimmobMASol2}) indicates that the age in the aquitard is a travel time by pure diffusion from the aquifer to the aquitard midpoint. It is parabolic with its maximum located at the midpoint ($\max [A_{m,im} ] = A_{m,m} + L^2 /2D_z$, when $z = L$). Fig.~\ref{fig:fig4} illustrates the distributions of the groundwater mean age for the same groundwater flow system in Fig.~\ref{fig:fig3}. Fig.~\ref{fig:fig5} illustrates the distributions of the kinematic, apparent, and mean ages along the flow direction in the mobile region and the apparent and mean ages across the immobile region (at $x = 1000$m) of the two-dimensional system shown in Figures~\ref{fig:fig3} and~\ref{fig:fig4}. It is clear from Fig.~\ref{fig:fig5} that the apparent age overestimates the kinematic age but underestimates the groundwater mean age in the aquifer. Groundwater mean age in the aquitard is significantly higher than the apparent age although both ages are influenced by the mixing processes in the aquitard. This is because the apparent age is influenced by the decay rate of the solute. From Eq.~(\ref{eq:mobileAA}), although the relationship between apparent age and mean age is more complex than in the one-dimensional system, it is unique and thus the apparent age can be used to estimate the mean age for a given distance from the source, groundwater velocity, the rate of decay for the solute, and the thickness and diffusive properties of the immobile region surrounding the mobile region.

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=0.90\textwidth]{fig4.jpg}
  \end{center}
  \caption{\small Illustration of the groundwater mean age distribution in a two-dimensional aquifer-aquitard system.}
 \label{fig:fig4}
\end{figure}

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=\textwidth]{fig5.jpg}
  \end{center}
  \caption{\small Illustration of the kinematic, apparent, and groundwater mean age distributions (a) along the flow direction in the mobile region and (b) across the immobile region ($x = 1000$m) of the two-dimensional system illustrated in Figures~(\ref{fig:fig3}) and~(\ref{fig:fig4}).}
 \label{fig:fig5}
\end{figure}

\subsection{Two-Dimensional Analysis}
Effects of transverse mixing can be analyzed by considering a two-dimensional system where the flow velocity can be approximated by a one-dimensional uniform velocity ($v_x$) but where recharge is limited at the origin ($x = 0$, $y = 0$). The kinematic age is approximated along the streamline from the source ($y = 0$) using the location and the one-dimensional velocity:
\begin{equation}
A_k(x, y = 0) \approx \frac{x}{v_x}
\label{eq:kinematicAge2D}
\end{equation}

The steady-state transport of a decaying solute is described by
\begin{equation}
- v_x \frac{\partial C(x,y)}{\partial x} + \nabla \cdot {\bf{D}} \cdot \nabla C(x,y) - \lambda C(x,y) = 0
\label{eq:solute2DADE}
\end{equation}
where the two-dimensional operator is defined as $\nabla  \equiv (\partial /\partial x,\partial /\partial y)$. When the dispersion coefficient ${\bf{D}} \equiv (D_{xx}, D_{xy}; D_{yx}, D_{yy})$ is such that $D_{xx} = \alpha_l v_x + D_{diff}$, $D_{xy} = D_{yx} = 0$, and $D_{yy} = \alpha_t v_x + D_{diff}$, Eq.~(\ref{eq:solute2DADE}) simplifies in
\begin{equation}
- v_x \frac{\partial C(x,y)}{\partial x} + D_{xx} \frac{\partial^2 C(x,y)}{\partial x^2} + D_{yy} \frac{\partial^2 C(x,y)}{\partial y^2} - \lambda C(x,y) = 0
\label{eq:solute2DADE2}
\end{equation}
For the given boundary condition of continuous mass injection with the rate of $Q C_0$ ($\textrm{M} \textrm{T}^{-1}$) at the origin, where $Q$ is the inflow rate and $C_0$ the flux concentration, the solution of Eq.~(\ref{eq:solute2DADE2}) is given by
\begin{equation}
C(x,y) = \frac{Q C_0}{2\pi n\sqrt {D_{xx} D_{yy}}}\exp \left( {\frac{v_x x}{2D_{xx}}} \right) K_0 \left( {\sqrt {\left( {\frac{v_x^2}{4D_{xx}} + \lambda } \right)\left( {\frac{x^2}{D_{xx}} + \frac{y^2}{D_{yy}}} \right)} } \right)
\label{eq:solute2Dsol}
\end{equation}
where $n$ is the porosity and $K_0$ is the modified Bessel function of second kind and zero order. When an asymptotic form of $K_0(p)$ ($\approx \sqrt {\pi / 2 p}e^{- p}$ for large $p$) is introduced in Eq.~(\ref{eq:solute2Dsol}) (e.g. see~\cite{Wilson1978}), then it reads:
\begin{equation}
C(x,y) = \frac{Q C_0}{n \sqrt {8\pi D_{xx} D_{yy} r/B}}\exp \left( \frac{x - r}B \right)
\label{eq:solute2Dsol2}
\end{equation}
where
\begin{equation}
B = \frac{2D_{xx}}{v_x} \quad\quad \textrm{and} \quad\quad r = \sqrt {\left( 1 + \frac{2B\lambda}{v_x} \right)\left( x^2 + \frac{D_{xx}}{D_{yy}}y^2 \right)}
\label{eq:Bandrdef}
\end{equation}
If the apparent age is defined from the ratio in concentrations of decaying and conservative solutes ($\lambda = 0$), then it is given as
\begin{subequations}
\begin{equation}
A_a(x,y) = - \frac{1}{\lambda} \ln \left( {\frac{C}{{C(\lambda = 0)}}} \right) = \frac{1}{\lambda}\left( {\frac{1}{4} \ln \kappa - \frac{r}{B}\left( {\frac{{1 - \sqrt \kappa}}{{\sqrt \kappa}}} \right)} \right)
\label{eq:apparentage2D}
\end{equation}
\begin{equation}
A_a(x,y = 0) = \frac{1}{\lambda}\left( {\frac{1}{4}\ln \kappa - \frac{x}{B}\left( {1 - \sqrt \kappa} \right)} \right)
\label{eq:apparentage2Dyzero}
\end{equation}
\end{subequations}
with
\begin{equation}
\kappa = 1 + \frac{{2B\lambda }}{{v_x}}
\label{eq:kappadef}
\end{equation}
An asymptotic solution for $A_m(x,y)$ is obtained when solving the mean age equation
\begin{equation}
- v_x \frac{\partial A_m(x,y)}{\partial x} + D_{xx} \frac{\partial^2 A_m(x,y)}{\partial x^2} + D_{yy} \frac{\partial^2 A_m(x,y)}{\partial y^2} + 1 = 0
\label{eq:MAeqn2D}
\end{equation}
Using the zero mean age boundary condition $A_m(x = 0, y = 0) = 0$, this solution is:
\begin{subequations}
\begin{equation}
A_m(x,y) = \frac{1}{v_x} \left( {\sqrt {x^2 + \frac{D_{xx}}{D_{yy}}y^2} + \frac{D_{xx}}{v_x}} \right)
\label{eq:MAsol2D}
\end{equation}
\begin{equation}
A_m(x,y = 0) = \frac{x}{v_x} + \frac{D_{xx}}{v_x^2}
\label{eq:MAsol2Dyzero}
\end{equation}
\end{subequations}
Eqs.~(\ref{eq:apparentage2D}) and~(\ref{eq:MAsol2D}) show a dependency of age with transverse dispersion $D_{yy}$. The ratio of apparent age and mean age is more complex than in the one-dimensional system but it still can be computed and potentially used to assess the dispersive properties of the aquifer. Eqs.~(\ref{eq:apparentage2Dyzero}) and~(\ref{eq:MAsol2Dyzero}) indicate that the apparent and mean ages are not influenced by the transverse dispersion along the streamline from the source. The apparent and mean ages, compared to the kinematic age, yield the following relation:
\begin{subequations}
\begin{equation}
r_{A_{mk}}(x,y = 0) = \frac{A_m}{A_k} = 1 + \frac{D_{xx}}{x v_x}
\label{eq:ratioA2D1}
\end{equation}
\begin{equation}
r_{A_{ak}}(x,y = 0) = \frac{A_a}{A_k} = \frac{v_x}{4\lambda x}\ln \kappa + \frac{v_x^2}{2\lambda D_{xx}}\left( {\sqrt \kappa - 1} \right)
\label{eq:ratioA2D2}
\end{equation}
\end{subequations}
In Eq.~(\ref{eq:ratioA2D1}), it is clear that ratio between the kinematic and mean ages becomes unity for large $x$ and $v_x$, and small $D_{xx}$. When $D_{xx} \approx \alpha_l v_x$, $r_{A_{mk}} \approx 1 + \alpha_l / x$ and thus the mean and kinematic ages becomes similar as the location is farther from the source compared to the longitudinal dispersivity. In Eq.~(\ref{eq:ratioA2D2}), it is also clear that $r_{A_{ak}} \approx \frac{v_x^2}{2\lambda D_{xx}}\left( {\sqrt \kappa - 1} \right)$ for large $x$, in which case the one-dimensional solution~(\ref{eq:ratios1}) is recovered. As a result, it can be concluded that the effect of dimensionality on age bias is minor if the velocity field is uniform.

\section{Implications for Tritium Age Dating}
Transient transport of isotopic signatures has been widely used to measure the age of groundwater. One of the most popular radioisotopes in dating modern groundwater is the tritium which was produced by the atmospheric testing of thermonuclear bombs between 1951 and 1980 (activity peaked at 1963, e.g. see~\cite{Maloszewski1982},~\cite{Zuber2005},~\cite{IAEAUNESCO2001},~\cite{Morgenstern2010}). There can be various different approaches to dating groundwater using tritium as explained by~\cite{Clark1997} and they may be categorized as: (\emph{i}) identifying the spatial location of concentration (or activity) peak, (\emph{ii}) calculating the time for decay from a known input level to the measured level, (\emph{iii}) analyzing concentration time series at a specific point. \cite{Clark1997} indicated that the tritium age dating could be most appropriate when the influence of the mixing and dispersion is minimal for isotope activities. It is interesting to note that the spatial location of the tritium activity peak in the subsurface is influenced only by the advection and thus it is related only to the kinematic age, while the attenuation and decay can be strongly influenced by the mixing or other dilution processes. Both of these measures, however, could be misleading if they are used to analyze the mean residence time or mean age of the sample.\\

Transport of tritium in a one-dimensional domain can be described by the following advection-dispersion-reaction equation:
\begin{equation}
\frac{\partial C(x,t)}{\partial t} =  - v\frac{\partial C(x,t)}{\partial x} + D\frac{\partial ^2 C(x,t)}{\partial x^2} - \lambda C(x,t)
\label{eq:TritiumADE}
\end{equation}
For the simplistic analysis, a steady flow field and an initial Gaussian slug input of tritium is assumed at the origin:
\begin{equation}
C(x, t = 0) = \frac{1}{\sqrt {4 \pi D t_0}}\exp \left( - \frac{x^2}{4 D t_0} - \lambda t_0 \right)
\label{eq:GaussianSlugTritium}
\end{equation}
The variable $t_0$ denotes the initial time at which the concentration defined in Eq.~(\ref{eq:GaussianSlugTritium}) is taken as the initial concentration. The solution of Eq.~(\ref{eq:TritiumADE}) is given by
\begin{equation}
C(x,t) = \frac{1}{\sqrt {4 \pi D(t + t_0)}}\exp \left( {- \frac{(x - vt)^2}{4 D (t + t_0)} - \lambda (t + t_0)} \right)
\label{eq:TritiumSol}
\end{equation}
In Eq.~(\ref{eq:TritiumSol}), the position of the peak is located at $x = v t$ with the peak value being
\begin{equation}
C(x = v t,t) = \frac{e^{- \lambda (t + t_0)}}{\sqrt {4 \pi D(t + t_0)}}
\label{eq:TritiumPeak}
\end{equation}
If the location of the peak ($x = x_{peak}$) is known at given time $t = t_{peak}$, groundwater velocity is calculated as $v = x_{peak} / t_{peak}$  and the kinematic age of groundwater at $x = x_{peak}$ is given by $A_k (x = x_{peak} ) = t_{peak}$, which is same as the mean age of groundwater in a one-dimensional flow system ($A_k = A_m$). It is noted again that this is based only on the fact that the peak migrates with the rate of groundwater velocity and is not related to the decay of the solute.\\

If the age is calculated from the change in peak concentration over the time period $t_{peak}$ (also known as time lag method), the apparent age of groundwater is given by
\begin{align}\label{eq:weakform32}
A_a(x = x_{peak}) = {} & - \frac{1}{\lambda}\ln \left( \frac{C}{C_0} \right) =
- \frac{1}{\lambda} \ln \left( {\frac{\sqrt {t_0} e^{- \lambda t_{peak}}}{\sqrt {t_{peak} + t_0}}} \right) \\
 & = t_{peak} + \frac{1}{\lambda} \ln \left( \sqrt {1 + \frac{t_{peak}}{t_0}} \right) \nonumber
\label{eq:AA_tritium}
\end{align}
Because the apparent age calculation is based on the rate of decrease in concentration values, additional dilution by the dispersion or mixing can make it greater than the kinematic age. It is clear that apparent age increases as the time for the chemical measure increases.\\

In a hypothetical one-dimensional flow field, groundwater velocity ($v$) is assumed to be 100 m/yr with the dispersion coefficient of 1000m. For the sake of simplicity, activity at time zero is assumed to be Gaussian with the peak at $x = 0$ ($t_0  = 1{\rm{yr}}$). Fig.~\ref{fig:fig6}a shows the spatial distribution of tritium activity at $t =$~0, 10, and 20 years and Fig.~\ref{fig:fig6}b shows the yearly time series at $x = 1000$m. The spatial distribution of tritium activity at $t = 10$years indicates that it peaks at $x = 1000$m and thus the kinematic age or the mean age can be calculated as 10 years. This is valid also for the conservative tracer. If the simple rate of decay is used to calculate the apparent age at $t = 10$years, then $A_a(x = x_{peak} ) \approx 31.5$. Using the ratio between non-decaying and decaying solutes, one obtains $A_a(x = x_{peak}) = - (1/\lambda) \ln \left( C/C_{\lambda = 0} \right) \approx 11.0$years.\\

\begin{figure}[h]
  \begin{center}
  \includegraphics[width=\textwidth]{fig6.jpg}
  \end{center}
  \caption{\small (a): Spatial distribution of tritium activity; (b): Temporal distribution of tritium activity, with indication on the criterion $p(t)$ of~\cite{Fritz1991}. Solid lines: decaying tracer; Dashed line: conservative tracer.}
 \label{fig:fig6}
\end{figure}

\cite{Fritz1991} estimated groundwater mean residence time using tritium activities measured from the samples collected at the same location in different years. They suggested to use the criterion $p(t) = C(t) \exp (- \lambda t)/C(t + \Delta t)$ to determine whether the peak passed the location or not. If $p(t) > 1$, then the peak has passed the observation point, whereas if $p(t) < 1$, then the observation point is sampling the leading edge. From Fig.~\ref{fig:fig6}b, $p(t \approx 8.8 \, {\rm{years}}) = 1$ and so the age is determined to be 8.8 years at $x = 1000$ m. The method by~\cite{Fritz1991} is derived from the decay equation. If the observation point is sampling the leading edge, then $dC(t)/dt > - \lambda C(t)$ and for the tail, $dC(t)/dt < - \lambda C(t)$ since the advection and dispersion increases the concentration before the peak passes the location while concentration decreases after the peak. Although this is a semi-quantitative approach to determine the relative location of the peak in the flow system, it closely approximates the kinematic. \cite{Delhez2008} however show how such apparent ages systematically underestimate the mean age.\\

It is implied from the above analyses that the isotopic age dating could provide an approximate mean age or kinematic age when the flow system is homogeneous and uniform (or approximated as one-dimensional), when the mixing is minimal, and if the spatial and temporal location of the peak is used to measure the age. However, the calculated age is based on the radioactivity (radiometric age) and may not be representative for the mean residence time of the sample or groundwater velocity at given locations.

\section{Numerical Example}
\cite{Schwartz2010} analyzed the $^{14}\textrm{C}$ apparent age at Yucca Mountain in Nevada, in a steady flow system and also in a historic transient since the last glacial maximum, and indicated that accounting for the historic transients in recharge in the flow system makes a significant difference in apparent age distribution at present compared to that simulated using the present-day steady flow system. In this study, the mean age simulated using Eq.~(\ref{eq:apparentAge}) is compared to the radiometric age translated from the results of the simulation using Eq.~(\ref{eq:1DADE}) in the same steady flow system used by~\cite{Schwartz2010} (Fig.~\ref{fig:fig7}). Details for the domain, initial and boundary conditions, and all the parameters used for the simulations are referred to~\cite{Schwartz2010}: in the two-dimensional vertical flow domain (approximately 20 km long and 2 km deep) consisting of different volcanic aquifers and aquitards, water table location is 500 m below the ground surface or deeper, and recharge ranges between 0.05 to 10 mm/year.\\

\begin{figure}[!t]
  \begin{center}
  \includegraphics[width=0.8\textwidth]{fig7.jpg}
  \end{center}
  \caption{\small (a) Mean and (b) radiometric ages simulated using Eqs.~(\ref{eq:apparentAge}) and~(\ref{eq:1DADE}) in Yucca Mountain, Nevada. (c) Relative difference. Steady flow field is assumed for both simulations and the black solid lines indicate the location of the water table.}
  \label{fig:fig7}
\end{figure}

Fig.~\ref{fig:fig7} shows the distributions of the simulated mean and radiometric ages in the steady flow system. From the results in Fig.~\ref{fig:fig7}, it is clear that the simulated ages are similar in pattern (if the water is older in terms of radiometric age, then the mean residence time is greater based on the mean age) but the radiometric age underestimates the mean residence time, especially when the water becomes older, agreeing with the analysis in the previous sections. In the unsaturated zone above the water table where transport is mostly advection-dominated, the two age distributions are almost identical except for the low permeable-low recharge region where the Caldera Barrier exists (\cite{Schwartz2010}). In the saturated zone, the two age distributions are relatively distinct since the mixing of waters of different origins and the dispersion processes play more important roles.

\section{Conclusions}
The presented one-dimensional and two-dimensional analyses reveal that the differences between mean age (as obtained from the direct physical modelling of mean age fate) and apparent age (as obtained from isotope concentrations) can be very important, and that the competition between decay and dispersion coefficient is a major factor that can explain such differences. More precisely, the larger the product $\lambda D$, the larger the difference between mean and apparent ages. It is also shown that apparent age systematically overestimates kinematic age and underestimates mean age. This statement is verified in the context of aquifer-aquitard systems, for which coupled one-dimensional and two-dimensional solutions also show that mean age in the aquitard is significantly higher than the apparent age although both ages are influenced by the mixing processes in the aquitard. A correction to the average apparent velocity $v_a$ that is obtained from two apparent age dates at two distinct locations is also provided. This correction is a direct function of the dimensionless variable $\frac{v_a^2}{\lambda D}$, and indicates that the average velocity may be calculated from the apparent velocity only when $v_a > \sqrt{\lambda D}$. The analytical analysis is concluded by the study of the effects of transverse mixing on apparent and mean ages. The results indicate a relatively minor effect of dimensionality for uniform flow fields, and that apparent and mean ages are not influenced by transverse dispersion along the streamline from the source.\\
The differences in apparent age and mean age observed with simple system configurations can be expected to be much larger in more realistic and complex situations, e.g. when several flow systems tend to mix, with the consequence that dispersion of ages is not the result of mechanical dispersion only, but also a consequence of the travel
times repartition within each flow system (being itself a consequence of the shape and
length of flow paths, thus of the geological structure of the aquifer and of the boundary
conditions). A numerical analysis of age distribution at Yucca Mountain in Nevada is presented to illustrate the magnitude of these age differences one can expect in realistic situations.\\
Our results suggest that the use of the classical piston-flow model (and by extension of the exponential model
and others) tends to be irrelevant for age dating and particularly for the interpretations that are frequently carried out to derive average flow velocities. Age definitions based on the radioactivity of isotopes may not be representative of the mean residence time of the sample or for the groundwater velocity at given locations, and may not be suitable in many cases for the calibration of hydrogeological models.

\section*{Acknowledgements}
Eric Deleersnijder is a Research Associate with the Belgian Fund for Scientific Research (F.R.S.-FNRS). His contribution to the present study was achieved in the framework of the Interuniversity Attraction Pole TIMOTHY (www.climate.be/timothy), which is funded by the Belgian Science Policy office (www.belspo.be) under contract IAP6.13.

\bibliographystyle{elsarticle-harv}
\newpage

redpajama_set_name: RedPajamaArXiv


text:
Q: ng-class expression not resolving I just started to try out Angular last day. I'm trying to use bootstrap tabs feature. I want to add class 'active' dynamically to the active li, depending on which tab user clicks. Below is my HTML snippet.

{{currentTab}}

Events

Surveys

Forums

What I am doing is, when user clicks on any link, I'm caling setTab() method with the tab name and changing the value if 'currentTab'.As you can see, I'm printing the value of 'currentTab' on top of the page, and it is printing as 'Events' first time. And when user clicks on different tabs, I'm able to change the value of currentTab (the value printed on page is changing). But the only thing is, 'active' class is not added to the selected tab. For each li, I check whether currentTab value is matching with specific tab, by calling isTab() method. I have put some logs in the isTab() method, but no logs are getting printed.
Below is my module and controller definition.
var app = angular.module('myApp',[]);
app.controller('TabController',['$scope','$http', function($scope, $http) {
$scope.currentTab = 'Events';

$scope.setTab = function(tabName)  {
    console.log('in setTab '+tabName);      // this is getting called
    $scope.currentTab = tabName;
};

$scope.isTab = function(tabName)  {
    console.log('in isTab '+tabName);                  // this is never printed!
    return $scope.currentTab === tabName;
    //return true;
};

}]);

A: This is because of controller as syntax. If you use this one, you should define all view model data, including funcitons, in this:
this.isTab = function(tabName)  {
    console.log('in isTab '+tabName);
    return this.currentTab === tabName;
};

It will work.

redpajama_set_name: RedPajamaStackExchange


The Xiaomi Mi Mix has been turning quite a few heads, with its near-bezelless design. The handset definitely stands out amid an ever-growing crowd of "slab" design smartphones. However, even with its reduced footprint, a 6.4-inch panel is still beyond the comfort zone for many users.
Don't get us wrong, the Mi Mix has been enjoying amazing popularity and will likely continue to do so in the foreseeable future. However, to broaden the new design's reach and appeal even more, Xiaomi is allegedly already working on a 5.5-inch variant.
The Mi Mix Nano has already featured in a few leaks and today brings yet another one. Industry sources now claim that the phone's release is just around the corner - the first week of December, to be exact. There is also a brand new live photo of the device, which seems to reaffirm that it will feature the same eye-catching design as its bigger sibling.
Specs should be quite similar as well and include a Snapdragon 821 SoC and 4GB of RAM. The FullHD resolution will likely be preserved as well, which translates to a higher pixel density on the smaller panel. Just like its bigger sibling, the Mi Mix Nano should launch running Android 6 Marshmallow, with a Nougat update expected to hit some time in Q1 2017.
Kinda ridiculous, isn't it? The smartphone industry is so stupid.


IDC says that there will be drastic growth worldwide in mobile workers, and that UC will be one of the key drivers. By 2013, the firm says, there will be almost 1.2 billion mobile workers wandering around the planet. The bottom line is pretty clear: UC and mobility are gradually merging.
The most interesting finding of IDC's "Worldwide Mobile Worker Population 2009-2013 Forecast" is that there will be 1.2 billion mobile workers roaming the earth by 2013. That is more than one-third of the world's work force and is, of course, a startling finding.
�the most significant gains will be in the emerging economies of Asia/Pacific, where a strong economic recovery and new interest in unified communications will drive healthy growth in all aspects of mobility spending.
It's fair to assume that IDC thinks that UC will be a key part of that growth everywhere, not just Asia-Pac. The report makes a number of other claims. The specificity is overstated, as it is in most analysts' reports. Guessing what the future brings in general is dangerous enough; assigning specific numbers and percentages is a bit silly.
But they are worthy of note for the general trend lines. IDC concludes that the United States now has and will continue to have the highest concentration of mobile workers. This is a bit of a surprise because the United States traditionally hasn't been the cutting-edge nation in mobility.
The study also says that Asia/Pacific (excluding Japan) has the most mobile workers and that Western Europe will pass the United States in total number by the end of the term of the study. Finally, Japan will have about the same percentage of mobile workers as the United States in 2013. This total – in the 75 percent range – is about as high as the concentration will ever get.
For IDC's predictions to be realized, mobile device vendors and other players in the ecosystem must make moves to bring functionality of various sorts to smartphones. This is happening. Within the past week, for instance, Skype and Verizon Wireless announced an exclusive -- and, to some, highly intriguing -- arrangement. Another (and more subtle) example is D2 Technologies' upgrade of its mCUE line to enhance IP capabilities offered to devices using the Android mobile operating system.


text:
page_title: Understanding Docker
page_description: Docker explained in depth
page_keywords: docker, introduction, documentation, about, technology, understanding

# Understanding Docker
**What is Docker?**

Docker is an open platform for developing, shipping, and running applications.
Docker is designed to deliver your applications faster. With Docker you can
separate your applications from your infrastructure AND treat your
infrastructure like a managed application. Docker helps you ship code faster,
test faster, deploy faster, and shorten the cycle between writing code and
running code.

Docker does this by combining a lightweight container virtualization platform
with workflows and tooling that help you manage and deploy your applications.

At its core, Docker provides a way to run almost any application securely
isolated in a container. The isolation and security allow you to run many
containers simultaneously on your host. The lightweight nature of containers,
which run without the extra load of a hypervisor, means you can get more out of
your hardware.

Surrounding the container virtualization are tooling and a platform which can
help you in several ways:

* getting your applications (and supporting components) into Docker containers
* distributing and shipping those containers to your teams for further development
and testing
* deploying those applications to your production environment,
 whether it be in a local data center or the Cloud.

## What can I use Docker for?

*Faster delivery of your applications*

Docker is perfect for helping you with the development lifecycle. Docker
allows your developers to develop on local containers that contain your
applications and services. It can then integrate into a continuous integration and
deployment workflow.

For example, your developers write code locally and share their development stack via
Docker with their colleagues. When they are ready, they push their code and the
stack they are developing onto a test environment and execute any required
tests. From the testing environment, you can then push the Docker images into
production and deploy your code.

*Deploying and scaling more easily*

Docker's container-based platform allows for highly portable workloads. Docker
containers can run on a developer's local host, on physical or virtual machines
in a data center, or in the Cloud.

Docker's portability and lightweight nature also make dynamically managing
workloads easy. You can use Docker to quickly scale up or tear down applications
and services. Docker's speed means that scaling can be near real time.

*Achieving higher density and running more workloads**

Docker is lightweight and fast. It provides a viable, cost-effective alternative
to hypervisor-based virtual machines. This is especially useful in high density
environments: for example, building your own Cloud or Platform-as-a-Service. But
it is also useful for small and medium deployments where you want to get more
out of the resources you have.

## What are the major Docker components?
Docker has two major components:

* Docker: the open source container virtualization platform.
* [Docker Hub](https://hub.docker.com): our Software-as-a-Service
  platform for sharing and managing Docker containers.

**Note:** Docker is licensed under the open source Apache 2.0 license.

## What is Docker's architecture?
Docker uses a client-server architecture. The Docker *client* talks to the
Docker *daemon*, which does the heavy lifting of building, running, and
distributing your Docker containers. Both the Docker client and the daemon *can*
run on the same system, or you can connect a Docker client to a remote Docker
daemon. The Docker client and service communicate via sockets or through a
RESTful API.

![Docker Architecture Diagram](/article-img/architecture.svg)

### The Docker daemon
As shown in the diagram above, the Docker daemon runs on a host machine. The
user does not directly interact with the daemon, but instead through the Docker
client.

### The Docker client
The Docker client, in the form of the `docker` binary, is the primary user
interface to Docker. It accepts commands from the user and communicates back and
forth with a Docker daemon.

### Inside Docker
To understand Docker's internals, you need to know about three components:

* Docker images.
* Docker registries.
* Docker containers.

#### Docker images

A Docker image is a read-only template. For example, an image could contain an Ubuntu
operating system with Apache and your web application installed. Images are used to create
Docker containers. Docker provides a simple way to build new images or update existing
images, or you can download Docker images that other people have already created.
Docker images are the **build** component of Docker.

#### Docker Registries
Docker registries hold images. These are public or private stores from which you upload
or download images. The public Docker registry is called
[Docker Hub](http://index.docker.io). It provides a huge collection of existing
images for your use. These can be images you create yourself or you
can use images that others have previously created. Docker registries are the
**distribution** component of Docker.

####Docker containers
Docker containers are similar to a directory. A Docker container holds everything that
is needed for an application to run. Each container is created from a Docker
image. Docker containers can be run, started, stopped, moved, and deleted. Each
container is an isolated and secure application platform. Docker containers are the
 **run** component of Docker.

##So how does Docker work?
So far, we've learned that:

1. You can build Docker images that hold your applications.
2. You can create Docker containers from those Docker images to run your
   applications.
3. You can share those Docker images via
   [Docker Hub](https://hub.docker.com) or your own registry.

Let's look at how these elements combine together to make Docker work.

### How does a Docker Image work?
We've already seen that Docker images are read-only templates from which Docker
containers are launched. Each image consists of a series of layers. Docker
makes use of [union file systems](http://en.wikipedia.org/wiki/UnionFS) to
combine these layers into a single image. Union file systems allow files and
directories of separate file systems, known as branches, to be transparently
overlaid, forming a single coherent file system.

One of the reasons Docker is so lightweight is because of these layers. When you
change a Docker image—for example, update an application to a new version— a new layer
gets built. Thus, rather than replacing the whole image or entirely
rebuilding, as you may do with a virtual machine, only that layer is added or
updated. Now you don't need to distribute a whole new image, just the update,
making distributing Docker images faster and simpler.

Every image starts from a base image, for example `ubuntu`, a base Ubuntu image,
or `fedora`, a base Fedora image. You can also use images of your own as the
basis for a new image, for example if you have a base Apache image you could use
this as the base of all your web application images.

> **Note:** Docker usually gets these base images from
> [Docker Hub](https://index.docker.io).
>
Docker images are then built from these base images using a simple, descriptive
set of steps we call *instructions*. Each instruction creates a new layer in our
image. Instructions include actions like:

* Run a command.
* Add a file or directory.
* Create an environment variable.
* What process to run when launching a container from this image.

These instructions are stored in a file called a `Dockerfile`. Docker reads this
`Dockerfile` when you request a build of an image, executes the instructions, and
returns a final image.

### How does a Docker registry work?
The Docker registry is the store for your Docker images. Once you build a Docker
image you can *push* it to a public registry [Docker Hub](https://index.docker.io) or to
your own registry running behind your firewall.

Using the Docker client, you can search for already published images and then
pull them down to your Docker host to build containers from them.

[Docker Hub](https://index.docker.io) provides both public and private storage
for images. Public storage is searchable and can be downloaded by anyone.
Private storage is excluded from search results and only you and your users can
pull images down and use them to build containers. You can [sign up for a storage plan
here](https://index.docker.io/plans).

### How does a container work?
A container consists of an operating system, user-added files, and meta-data. As
we've seen, each container is built from an image. That image tells Docker
what the container holds, what process to run when the container is launched, and
a variety of other configuration data. The Docker image is read-only. When
Docker runs a container from an image, it adds a read-write layer on top of the
image (using a union file system as we saw earlier) in which your application can
then run.

### What happens when you run a container?
Either by using the `docker` binary or via the API, the Docker client tells the Docker
daemon to run a container.

    $ docker run -i -t ubuntu /bin/bash

Let's break down this command. The Docker client is launched using the `docker`
binary with the `run` option telling it to launch a new container. The bare
minimum the Docker client needs to tell the Docker daemon to run the container
is:

* What Docker image to build the container from, here `ubuntu`, a base Ubuntu
image;
* The command you want to run inside the container when it is launched,
here `/bin/bash`, to start the Bash shell inside the new container.

So what happens under the hood when we run this command?

In order, Docker does the following:

- **Pulls the `ubuntu` image:** Docker checks for the presence of the `ubuntu`
image and, if it doesn't exist locally on the host, then Docker downloads it from
[Docker Hub](https://index.docker.io). If the image already exists, then Docker
uses it for the new container.
- **Creates a new container:** Once Docker has the image, it uses it to create a
container.
- **Allocates a filesystem and mounts a read-write _layer_:** The container is created in
the file system and a read-write layer is added to the image.
- **Allocates a network / bridge interface:** Creates a network interface that allows the
Docker container to talk to the local host.
- **Sets up an IP address:** Finds and attaches an available IP address from a pool.
- **Executes a process that you specify:** Runs your application, and;
- **Captures and provides application output:** Connects and logs standard input, outputs
and errors for you to see how your application is running.

You now have a running container! From here you can manage your container, interact with
your application and then, when finished, stop and remove your container.

## The underlying technology
Docker is written in Go and makes use of several Linux kernel features to
deliver the functionality we've seen.

### Namespaces
Docker takes advantage of a technology called `namespaces` to provide the
isolated workspace we call the *container*.  When you run a container, Docker
creates a set of *namespaces* for that container.

This provides a layer of isolation: each aspect of a container runs in its own
namespace and does not have access outside it.

Some of the namespaces that Docker uses are:

 - **The `pid` namespace:** Used for process isolation (PID: Process ID).
 - **The `net` namespace:** Used for managing network interfaces (NET:
 Networking).
 - **The `ipc` namespace:** Used for managing access to IPC
 resources (IPC: InterProcess Communication).
 - **The `mnt` namespace:** Used for managing mount-points (MNT: Mount).
 - **The `uts` namespace:** Used for isolating kernel and version identifiers. (UTS: Unix
Timesharing System).

### Control groups
Docker also makes use of another technology called `cgroups` or control groups.
A key to running applications in isolation is to have them only use the
resources you want. This ensures containers are good multi-tenant citizens on a
host. Control groups allow Docker to share available hardware resources to
containers and, if required, set up limits and constraints. For example,
limiting the memory available to a specific container.

### Union file systems
Union file systems, or UnionFS, are file systems that operate by creating layers,
making them very lightweight and fast. Docker uses union file systems to provide
the building blocks for containers. Docker can make use of several union file system variants
including: AUFS, btrfs, vfs, and DeviceMapper.

### Container format
Docker combines these components into a wrapper we call a container format. The
default container format is called `libcontainer`. Docker also supports
traditional Linux containers using [LXC](https://linuxcontainers.org/). In the
future, Docker may support other container formats, for example, by integrating with
BSD Jails or Solaris Zones.

## Next steps
### Installing Docker
Visit the [installation section](/installation/#installation).

### The Docker User Guide
[Learn Docker in depth](/userguide/).

redpajama_set_name: RedPajamaGithub


text:
package io.sidecar.util;

import com.google.common.collect.ImmutableList;
import com.google.common.collect.ImmutableSet;
import com.google.common.collect.Lists;
import com.google.common.collect.Sets;
import org.testng.annotations.Test;

import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Set;

import static org.testng.Assert.assertEquals;

public class CollectionUtilsTest {

    @Test
    public void testFilterNullsList() {
        assertEquals(CollectionUtils.filterNulls(Lists.newArrayList("foo", null, "bar")),
                Lists.newArrayList("foo", "bar"));
    }

    @Test
    public void testFilterNullsSet() {
        assertEquals(CollectionUtils.filterNulls(Sets.newHashSet("foo", null, "bar")),
                Sets.newHashSet("foo", "bar"));
    }

    @Test
    public void testFilterNullsListParamIsNull() {
        assertEquals(CollectionUtils.filterNulls((List)null), ImmutableList.of());
    }

    @Test
    public void testFilterNullsSetParamIsNull() {
        assertEquals(CollectionUtils.filterNulls((Set)null), ImmutableSet.of());
    }

}


The Chevrolet Corvair was one of those cars that got a bad rap before it could even get started. It was a completely unique design that America just wasn't ready for it. People just assumed that the engine should be in front and were just really weirded out by a rear engine car. The Corvair had a lot going for it. It had really nice curves and really appealed to the younger generation. The kids really got into the Corvair and everybody wanted one. It was sporty and different looking and that made it somewhat rebellious and every young kid wants to be able to do something that the older folks didn't do. But some good stories have a bad ending and the Chevrolet Corvair has just that. Some people said that the rear engine design made it hard to handle on the streets. They said that the Corvair was dangerous and once the Corvair got the bad reputation, there was no turning back. A good start ended far too early. Now looking back the Corvair wasn't as bad as everyone thought and if given a little time the Corvair could have been made into a classic but alas it wasn't. Now owners of the Corvair have reveled in its uniqueness and have breathed life into an old classic. But like so many classic cars before it the stock radio was not up to snuff. Now you want your Corvair to be better than it ever has right? So you need to get over to Vintage Car Radio and checkout the line of Ken Harrison radios. Ken Harrison radios are built with exacting standards and are built to ensure the highest quality which means a long life in your Corvair. So get to Vintage Car Radio and get yourself a Ken Harrison radio.