text:
\section{Introduction}
\label{sec:intro}

The quest for continuous models that would govern in some fruitful sense families of discrete processes  is
age-old and continuing to be fascinating~\cite{wanner10}.
Indeed, there is much to be gained if a complicated discrete process can
be adequately approximated by a continuous one in some limit sense:
\begin{itemize}
\item
A simpler structure more amenable to analysis may be considered,
adding to general phenomenological understanding.
\item
Efficient numerical approaches may be constructed for the discrete problem
by going through the continuous one.
\item
Perhaps the more complicated and less elegant discrete structure may not
even have to be considered.
\end{itemize}
A great success story in this regard are the Navier-Stokes equations: for many
fluid flow phenomena \uri{these partial differential equations}
capture the observed flow well enough that we can forget about their derivation while concentrating on understanding their
solutions and on approximating them numerically~\cite{chorin79,chorin}.
At the same time we note, ironically, that the discrete, molecular
description of fluid flow involves a huge number of particles, and that not all fluid phenomena are covered by this continuous model.

A lot of exciting
work that is relevant in our context has been carried out in recent years.
Here is but a partial list of areas and instances of interest:
 \begin{itemize}
\item
{\em Artificial time regularization}.
We return to this below, in Sections~\ref{sec:simple}, \ref{sec:reg}, and~\ref{sec:lsd};
see~\cite{ashudo,zangwill69} for a wider survey.
\item
{\em One-step iterative methods posed as a first order ODE discretization}.
This is a special case of artificial time continuation. Indeed, any iterative algorithm of the form
\[ \xx_{k+1} = \xx_k + \alpha_k \uri{\gb} (\xx_k) \]
which updates a current iterate $\xx_k$ (for a nonnegative iteration counter $k$) using a step size $\alpha_k > 0$
to obtain the next approximation $\xx_{k+1}$
can be viewed as a forward Euler discretization of the ordinary differential equation (ODE)
\[ \frac{d\xx}{dt} = \uri{\gb} (\xx (t)) .\]
This differential equation is obtained from the given discrete process by letting $\alpha_k \rightarrow 0$,
though, while in practice we may well want to keep the step size quite far from zero.
Herein lies a difference between qualitative and quantitative information.
\item
{\em Two-step iterative methods posed as a second order ODE discretization}.
There has been a lot of recent activity in this regard in attempts to better understand accelerated
gradient descent methods for unconstrained optimization~\cite{polyak64,beck2017,su2014,su2016,wibisono2016,betancourt2018,zhang18,dior19}.
We will not dwell further in this article on this mushrooming area of research,
beyond what is exposed in Section~\ref{sec:lsd}.
\uri{We do believe nonetheless that the present paper is relevant in this context.}
\item
{\em Regularizing image processing problems using
\uri{a penalty on the discretized  gradient of an image $u$ in its continuous limit $\int_\Omega  | \nabla u|^s$},
with $s = 2$ or $s=1$.} See, for instance,~\cite{chsh,chpo2016,ashu18}.
We return to this below, in Section~\ref{sec:paradigm}.
\item
{\em Continuation methods for nonlinear equations; homotopy path, etc.}
See, for instance,~\cite{deuflhard04,ashudo} and references therein.
Such methods can be viewed  as artificial time integration, but again we focus on other directions here.
\item
{\em Semiconductor equations}.
See, for instance, the text~\cite{mrs90}. A significant volume of work preceded and followed this book.
In our present context we observe that, unlike the case for fluid flow, more than one continuous process has
been derived and the practical importance of the underlying particle process has not diminished over the years.
\item
{\em Hamilton-Jacobi and mean field in game theory}.
See, for instance,~\cite{ll07,hmc06} and many following articles such as~\cite{gomez13,burger13,gomez14} and others.
Here we do not focus on this large and relatively recent volume of work.

\item
{\em Deep learning}.
The tremendous potential in machine learning and in particular deep learning (DL) techniques has riveted scientists
and engineers in recent years~\cite{Goodfellow-et-al-2016}.
Noting that the connection between consecutive layers in a DL network  resembles a finite difference scheme,
models depending on corresponding limit differential equations and algebraic multigrid methods have been proposed
in order to better design such neural networks.
See, for instance,~\cite{ruha19}.
Interest in two-step optimization algorithms mentioned above is also related to optimization methods in neural networks.

\end{itemize}

With all the promise and excitement of the continuous-limit models, it is also important
to emphasize that care should be taken to ensure that, {\em for the given task},  all important and interesting properties of
the discrete process are captured by the continuous one.
Otherwise, there is the possibility of being restricted to a non-optimal path. Furthermore,
it is important, and occasionally crucial, to distinguish between
{\em qualitative} and {\em quantitative}  relationships between the discrete and the continuous.
Undoubtedly, many researchers have arrived at such a practical conclusion in specific circumstances \uri{(for instance in {\em collective dynamics})},
\uri{with
graduate students who are responsible for the implementation of the relevant algorithms}
often being the first to realize this.

Below we describe several short case studies that aim to highlight various aspects of the issues involved.
We start in Section~\ref{sec:simple} with a simple example which demonstrates that viewing a discrete algorithm as
a discretization of a continuous process may simply be a matter of taste, depending on what one is used to
and feels comfortable with.
Then in Section~\ref{sec:reg} we quickly derive a useful and well-known class of regularization methods for inverse problems
following the artificial time reasoning, so here is a case where the continuous limit offers
a useful and somewhat different perspective.
In Section~\ref{sec:mid}  we then turn the heat up a notch and discuss the potential, if rare, instability of
the classical implicit midpoint method for stiff ODEs. Here we see an instance where, on one hand,
the meaningful continuous limit is not what comes to mind without thought, and on the other hand,
the correct limit leads to a rigorous characterization of unstable scenarios.
In the longer Section~\ref{sec:lsd} we develop a scenario where a discretization of the simplest heat equation with constant
time step \uri{size} exhibits the usual stability limit on that step, arising from a semi-discretization in space;
but  with the freedom of choosing
a variable time step \uri{size} such a stability connection to spatial discretization does not necessarily hold.
We then connect this scenario to the gradient descent method of Section~\ref{sec:simple} and demonstrate
that the lagged steepest descent method~\cite{babo,doeasc} produces such large step sizes.
In Section~\ref{sec:paradigm} we give a quick overview of our past efforts to apply a Tikhonov-type regularization
involving differential equations, as mentioned above, to image and surface processing applications.
These efforts, and many others, have only been partially successful, and we explain why.
Section~\ref{sec:conclusions} seals the paper by offering a few general conclusions.

\uri{In each of our case studies the discrete process has a {\em step size} parameter $h$, and we examine
a limit process for it. If the step size depends on the stage or iteration $k$, denoted $h_k$, then $h = \max_k h_k$.  }

\uri{\paragraph{\bf Remark}

The author's curiosity about the subject of the present work was first aroused before 1980.
We were constructing the general-purpose software package COLSYS~\cite{colsys81}
for boundary value ODE systems~\cite{amr}.
Such discretized nonlinear problems often lead to systems of nonlinear algebraic equations that are difficult to solve,
and we used the damped Newton method with an automatic choice for the damping, or step size, parameter.
The ideal choice for this parameter is $=1$, where a quadratic convergence rate is achieved
for Newton's method.
However, often in applications the Newton direction was  poor, and a rather small step size was required to obtain
decrease is some monitor objective function.
The essential difficulty was that upon moving the iterate by such a small step size the next Newton direction
was also poor, causing a repeated similar difficulty! We ended up providing means in COLSYS to switch to other
methods in such situations. But the lesson relevant  here was that, in practice, letting the step size go to $0$ was not desirable
at all, as this limit process is not ``practically smooth''.
}

\section{Gradient descent stability bound}
\label{sec:simple}

Consider the problem of finding the minimum of a convex $C^1$ function of $n$ variables $f (\xx)$, and denote
\begin{eqnarray}
\xx^* = \arg\min_{\xx \in \Re^n} f(\xx ) . \label{1.1}
\end{eqnarray}
The gradient descent iterative method evaluates at the current iterate $\xx_k$ the gradient $\ensuremath {\vec \nabla} f (\xx_k)$
and sets the next iterate as
\begin{eqnarray}
\xx_{k+1} = \xx_k - \alpha_k \ensuremath {\vec \nabla} f (\xx_k), \label{1.2}
\end{eqnarray}
where $\alpha_k$ is the (positive) step size and $k$ is the iteration counter, $k=0, 1, \ldots $ See, e.g.,~\cite{nw}.

Let us next restrict attention to the quadratic case
\begin{eqnarray}
f(\xx ) = \frac 12 \xx^TA\xx - \bb^T\xx , \label{1.3}
\end{eqnarray}
where $A$ is a given $n \times n$ real symmetric positive definite \uri{(SPD)} matrix and $\bb$ is a given real inhomogeneity vector.
Here $\ensuremath {\vec \nabla} f(\xx) = A\xx - \bb$, and defining (for historical reasons) the residual \uri{$\rr = -\ensuremath {\vec \nabla} f(\xx) = A(\xx^* - \xx)$} we get the iteration
\begin{eqnarray}
\xx_{k+1} = \xx_k + \alpha_k \rr_k. \label{1.4}
\end{eqnarray}
Next we fix the step size $\alpha_k = \alpha$ (called ``learning rate'' in DL parlance) and ask,
what is the upper stability limit on $\alpha$?

There are two methods to answer this simple question:
\begin{enumerate}
\item
Observe that
for the error $\ee_k = \xx_k - \xx^*$ we have the recursion $$\ee_{k+1} = (I - \alpha A)\ee_k ,$$
so $\alpha$ is restricted by  $\| I-\alpha A \|_2 \leq 1$.
Since $A$ can be diagonalized by an orthogonal similarity transformation,
this bound translates to
$1-\alpha \max \lambda_i \geq -1 ,$ where $\lambda_i > 0$ are the eigenvalues of $A$ arranged in decreasing order.
This leads to the stability bound
\begin{eqnarray}
\alpha \leq 2/\lambda_1 .   \label{1.5}
\end{eqnarray}
\item
Another way to see this is by introducing the limit ODE $\frac{d\xx}{dt} =   \rr (\xx(t))$, where $t \geq 0$ is artificial time.
Discretizing this ODE using the forward Euler method~\cite{apbook} clearly gives the gradient descent method  \eqref{1.4},
where the time step size is $h_k = \alpha_k$.
But now, we know the {\em absolute stability} bound for forward Euler! It is the bound~\eqref{1.5}.
\end{enumerate}

\paragraph{\bf Conclusion}
Which method (for arriving at the same result) is better? This depends on one's ``comfort zone'',
or what one is used to.
In fact, the second method
is ``obvious'' if you are used to numerical ODEs; but it is the unnecessarily longer route otherwise.

\section{Regularizing ill-posed problems}
\label{sec:reg}

In our second case study, let us consider for simplicity the linear problem
\begin{eqnarray}
A\xx = \bb , \label{2.1}
\end{eqnarray}
where $A$ and $\bb$ are as described in Section~\ref{sec:simple}, but now $A$ is also large, sparse and ill-conditioned.
\uri{In terms of the SVD~\cite{agbook}
\begin{subequations}
\begin{eqnarray}
A = U \Sigma U^T , \label{new1a}
\end{eqnarray}
where $U$ is an orthogonal matrix and $\Sigma$ is diagonal, containing the singular values $s_i$ in decreasing
order on its main diagonal,
the solution is strictly given by $\xx = U\yy$, where the components of the vector $\yy$ are
\begin{eqnarray}
y_i = s_i^{-1} (U^T\bb)_i  \quad i = 1, 2, \ldots , n.    \label{new1b}
\end{eqnarray}
But if $s_{i_0} \ge 0$ is extremely small (representing a perturbation of $0$ due to noise, say) for some $i_0 \leq n$, then we wish to avoid \eqref{new1b}
for all $i \geq i_0$.
The solution process then has to be regularized somehow.
\label{new1}
\end{subequations}}

The {\em truncated SVD} is one possible method, \uri{where we set $y_i = 0$ for all $i \geq i_0$.}
But the SVD transformation involves large, full matrices,
and we want to take advantage of the sparsity of $A$.
An appropriate Tikhonov regularization is possible, see, e.g., \cite{ehn1}; \uri{we will not describe this further here.}

An alternative to Tikhonov's method is to consider the artificial time formulation from Section~\ref{sec:simple}, \uri{written as}
\begin{eqnarray}
\frac  {d\xx}{dt} = \bb - A\xx . \label{2.2}
\end{eqnarray}
This well-known method has been called {\em exponential filtering} in~\cite{carezh,care}.
For the artificial time ODE, at time {$t$} the effect of the reciprocal singular value  {$s_i^{-1}$} is replaced by
{$\omega(s_i^2) s_i^{-1}$}, where
\begin{eqnarray}
\omega (s) = 1 - e^{-ts} . \label{2.2}
\end{eqnarray}

Thus, the effect of small singular values gets dampened while
large ones remain almost intact for an appropriate  {\em finite time} instance $t$.
This artificial finite time serves as the regularization parameter~\cite{ashudo},
\uri{gauging the amount by which inverse singular values are dampened}.
Next, we can use {forward Euler}, or \uri{some} other time discretization hopefully with \uri{large step sizes $h_k$}, to roughly integrate the ODE.

The resulting filter function, as it turns out, behaves quite well and not so differently from the Tikhonov filter
with regularization weight $\beta$ given by
\begin{eqnarray}
\omega_{\rm T}  (s) = \frac{s}{s+\beta} . \label{2.3}
\end{eqnarray}
For the choice of artificial time $t_\beta=\frac 1{2\beta}$ the curves in Figure~\ref{fig1} are quite similar.

\begin{figure}[htb]
  \begin{center}
    \includegraphics[scale=0.45]{omega.pdf}
    \end{center}
\caption{Exponential filter  $\omega (s) = 1 - e^{-ts}$
and Tikhonov filter $\omega_{\rm T}  (s) = \frac {s}{s + \beta} $
for $t\beta = 1/2$. \label{fig1}}

\end{figure}

Some relevant questions, discussed in~\cite{ashudo}, are:
\begin{enumerate}
\item
Using an explicit discretization method, can we take large steps even if the ODE is stiff?
\item
Is the resulting regularization effective?
(Note that we can view it  as a {\em continuation method}: we want to get to {$t=t_\beta$} quickly, without approximating the ODE trajectory accurately.)
\end{enumerate}

\paragraph{\bf Conclusion}
We will not dwell on this further, \uri{since this case study is documented in a similar context elsewhere.}
But \uri{let us}  observe that here the artificial ODE yields a practically useful regularizer!
At the same time, the task of approximating the ODE~\eqref{2.2} should not be taken
too zealously: in fact, using a higher order method such as explicit two-stage Runge-Kutta~\cite{apbook}
does not prove more useful than forward Euler for the given purpose.
\uri{Moreover, this entire regularization approach does have competition: for instance, our own favourite
has often
been a subspace regularization method, applying just a few conjugate gradient iterations to a singular linearized system~\cite{doas1,rodoas}.}

\section{Stability of the midpoint method}
\label{sec:mid}

In our third case study we start with a given ODE system (so the independent variable $t$ is not artificial)
and consider another, implicit discretization.
Let us denote $\dot u = \frac{du}{dt}$.
For the ODE system
\begin{eqnarray}
\dot \uu = \ff (t,\uu ), \quad 0 \leq t \leq T, \label{1}
\end{eqnarray}
the well-known implicit midpoint method reads
\begin{eqnarray}
\frac {\uu_{k+1} - \uu_k }h = \ff \left(\frac{t_{k+1}+t_k}2, \frac{\uu_{k+1}+\uu_k}2 \right) . \label{2}
\end{eqnarray}
Here  $h = h_k$  is a step  size from time $t_k$ (where for an initial value problem $\uu_k$ is known) to time $t_{k+1}$ (where $\uu_{k+1}$ is  unknown).

This method is implicit, second order accurate, A-stable, AN-stable (meaning it is stable for $f = \lambda(t)u$ for any $\Re (\lambda) \leq 0$),
 algebraically stable, symplectic for Hamiltonian systems, symmetric \uri{(reversible)}, and it conserves quadratic invariants.
 Collocation at Gaussian points generalizes midpoint  to higher order methods and ODE systems~\cite{amr}.
 The method has been applied  in the context of boundary value ODEs~\cite{amr}, geometric integration~\cite{hlw}
 and stiff initial value ODEs (IVODEs).
 We continue with the latter.

 Despite all its good properties (and perhaps because of some of them), the implicit midpoint method is not always stable for well-posed, stiff IVODE systems!
 Our task is next to analyze this situation.
 To recall, a stiff ODE typically has more than one time scale, and we aim to employ a time step $h$ that is
 commensurate with the slow scale but not with the fast one~\cite{hw,bawi,kale}.
 In such a constellation, though, {\em the continuous limit of the discrete structure} \eqref{2} {\em \uri{is}  not the original
 problem~\eqref{1}}. This is so, because to get from \eqref{2} to \eqref{1} we have to let $h \rightarrow 0$,
 and once we do that the problem is no longer stiff in the sense that the fast scale is eventually also resolved by the small step size.
 Instead, we have to resort to numerical singular perturbation techniques.

 To study this further, consider the linear non-autonomous  ODE system
 \begin{eqnarray}
 \dot  \uu =  \omega A(t) \uu, \label{3}
 \end{eqnarray}
 where $\omega$ is a parameter that is allowed to grow large  (so we must consider $\omega \gg 1$),
  and $\| A \| = {\Oh} (1)$ and nonsingular. It is possible to generalize \eqref{3} to the case where $A = A(t; \omega )$.
  In \eqref{3} we are essentially studying a linearized version of~\eqref{1} that governs the propagation of an error.
 Assume that the IVODE for~\eqref{3} is well-posed (or ``stable'').
 The midpoint method reads
 \begin{eqnarray}
\frac {\uu_{k+1} - \uu_k }h =  \omega A\left(\frac{t_{k+1}+t_k}2 \right) \frac{\uu_{k+1}+\uu_k}2 , \label{4}
\end{eqnarray}
but we cannot conclude much when
\begin{eqnarray}
\gamma = \frac 14 \omega h^2 \label{5}
\end{eqnarray}
 is fixed and not small,  because we can't take the limit of $h \rightarrow 0$ without also changing $\omega$
 and thus the ODE~\eqref{3},  as explained above.

Specifically,  what we are looking at instead is the limit process of $\left( h \rightarrow 0, \ \omega \rightarrow \infty \right)$, such that $\gamma$ stays constant
(e.g., $\gamma = 1$).

We next introduce an old but not necessarily well-known  trick~\cite{kreiss72}.
For analysis purposes, let us
define for each integer $k$
\begin{eqnarray}
\vv_k = (-1)^k \uu_k . \label{6}
\end{eqnarray}
Obviously, \uri{the} boundedness properties of \uri{the sequences} $\{\vv_k\}$ and $\{\uu_k\}$ are the same: $|\uu_k| = |\vv_k| ~\forall k$.

Substituting \eqref{6} in \eqref{4} and cancelling out $(-1)^k$ we obtain
 \begin{eqnarray}
\frac {\vv_{k+1} + \vv_k }h = \omega A\left(\frac{t_{k+1}+t_k}2 \right) \frac{\vv_{k+1}-\vv_k}2 . \nonumber
\end{eqnarray}
Multiplying throughout by $2/h$ \uri{then yields} an approximation for $\dot \vv$ at the right hand side.
Multiplying further by $(\omega A)^{-1}$ we can rewrite this as
 \begin{eqnarray}
\frac {\vv_{k+1} - \vv_k }h = \frac 1{\gamma} A^{-1}\left(\frac{t_{k+1}+t_k}2 \right) \frac{\vv_{k+1}+\vv_k}2 . \label{7}
\end{eqnarray}
Now, for \eqref{7} with $\gamma$ fixed we can finally take the limit $h \rightarrow 0$, obtaining the {\em ghost ODE}
\begin{eqnarray}
\dot \vv = \gamma^{-1}  A^{-1}(t)  \vv . \label{8}
\end{eqnarray}
{\em The stability of the midpoint method therefore depends on the stability of the ghost ODE~\eqref{8},
not on that of the given ODE \eqref{3}}.

The exercise to design a stable IVODE problem~\eqref{3} such that the IVODE for~\eqref{8}
is unstable is solved in~\cite{a2} for the limit case $1/\omega = 0$. It can be extended to a large but finite $\omega$.
Then the computed solution using the implicit midpoint method applied to a well-posed problem can be made to blow up;
see~\cite{a2}.\footnote{\uri{Further, although it is not strictly a matter of stability, we urge the
interested reader to  see Example~2.1 in~\cite{ar2},
where the analysis trick \eqref{6} is used to show that the symplectic midpoint method could produce consistently wrong results
for highly oscillatory non-autonomous Hamiltonian systems.}}

The above analysis can be summarized \uri{as follows}:
\begin{theorem}
Assume that the given IVODE for~\eqref{3} is well-posed, and
consider the midpoint discretization~\eqref{4} with $\uu_0 = \uu (0)$ for $k = 0, 1, \ldots, N-1$ and $Nh = T$.

Then this method is stable along the ray $\left( h \rightarrow 0,~\omega \rightarrow \infty \right)$ such that $\gamma$ defined in~\eqref{5}
is held fixed, iff
the IVODE for the ghost ODE~\eqref{8} is well-posed.
\end{theorem}

If $A$ is a constant matrix, then the stability of the midpoint method is guaranteed. But in the more general case it is not.
Fortunately, such midpoint instability is rather rare in practice, and this is important especially for boundary value ODEs,
where general-purpose codes typically implement symmetric schemes.\\

\paragraph{\bf Conclusion}
In this case study we have learned two lessons. The first is that just because a differential equation looks like
a plausible continuous limit does not mean that it is the correct continuous limit for a discrete process
in the context of a given task.
The second lesson is that with some extra care and ingenuity it may be possible to use a continuous limit to prove
interesting theoretical results that are of practical importance.

\section{Forward Euler for the heat equation and chaotic descent}
\label{sec:lsd}

In this case study we obtain some unexpected and perhaps counter-intuitive results
by considering time-stepping with highly variable step sizes.

Consider the simple heat equation on a unit square in space and time
\begin{eqnarray}
 \uri{\frac{\partial u}{\partial t} \equiv }  u_t = \Delta u + b, \quad 0 < x,y < 1, \; t \geq 0 \label{5.1}
 \end{eqnarray}
 with homogeneous boundary conditions (BC) and initial condition $u(0,x,y) = u_0 (x,y)$ \uri{that satisfies the BC}.
 Here $\Delta u = u_{xx} + u_{yy}$ is the Laplacian, and $b > 0$ is a constant for simplicity.
 Integrating in time to steady state, we obtain
 the model Poisson problem via a {\em continuation method} in the time variable
\begin{eqnarray}
 -\Delta u = b, \quad 0 < x,y < 1
 \label{5.2}
 \end{eqnarray}
subject to the same BC.\footnote{\uri{A time-honoured {\em continuation method} for finding an approximte solution for a tough nonlinear
elliptic PDE $\psi(u) = 0$ is to numerically integrate the parabolic PDE $u_t = \psi(u)$, starting from an initial guess $u_0$,
until a suitable error tolerance is satisfied. For our purpose here we consider the simplest linear version of this.}}

 A forward Euler discretization of~\eqref{5.1} in time gives the recursion
 \begin{eqnarray}
 u_{k+1} = u_k + h_k \Delta u_k + h_kb , \quad k = 0, 1, \ldots  . \label{11}
 \end{eqnarray}
This is clearly unstable for any series of positive time \uri{step sizes} $h_k$.
The {\em unconditionally unstable} semi-discretization \eqref{11} is then our continuous model.

 Next, we discretize the partial differential equation (PDE)~\eqref{5.1}  in space first, with constant spacing $\xi = 1/\sqrt{n}$ in both $x$ and $y$ directions.
 Reshaping the 2D array of unknowns at spatial mesh points into a vector $\uu$ of length $n$ we obtain an ODE system
 \begin{eqnarray}
 \frac{d\uu}{dt} = -A \uu +\bb, \label{11.5}
 \end{eqnarray}
where the $n \times n$ matrix $A$ is SPD. Using  straightforward centred differences for the Laplacian,
$A$ has $5$ \uri{nonconsecutive} nonzero diagonals
(see, e.g., Section~7.1 of~\cite{agbook}).

Our discrete model is obtained upon applying the same forward Euler method as for the continuous model
to \eqref{11.5}, obtaining
 \begin{eqnarray}
 \uu_{k+1} = \uu_k + h_k (\bb - A\uu_k)  . \label{12}
 \end{eqnarray}
This recusion is the same as \eqref{1.4} in Section~\ref{sec:simple}.
Evaluating eigenvalues, the stability \uri{restriction} \eqref{1.5} translates here to the condition
\begin{eqnarray}
h \leq \frac 14 \xi^2 , \quad {\rm where~} h = \max_k h_k . \label{5.5}
\end{eqnarray}
So, when we let $\xi \rightarrow 0$
the stability region for $h$ shrinks like $\xi^2$.
The unconditional instability as above is effectively reached in the continuous limit for \eqref{11}.
For a uniform step size, then, the continuous model governs the discrete process well.

But next we ask,  is this shrinking stability limit on the maximum step size necessary even if we allow $h_k$ to vary?
The answer turns out to be negative!
Indeed, consider a sequence of several small step sizes $h_k$ effectively reducing the amplitude of high eigenvalue modes of the residual,
followed by a large step reducing low mode amplitudes while inflaming the high modes to some extent.
The latter are reduced again using further small steps.
The largest step size $h$ depends on the low modes
rather than the high modes, and as
such it can be independent of $\xi$.
This possibility is not available for the (semi-) continuous limit scheme, only the fully discrete one!

\uri{\paragraph{\bf Remark}
Before proceeding to see an actual method which produces large step sizes, let us note that it is $h$ and
not $\xi$ which is in our focus of interest here. Indeed, the entire PDE setup can be replaced by a time-dependent linear ODE
system discretized as in~\eqref{12},
where the constant SPD matrix $A$ has eigenvalues that grow larger and larger
and the question is whether $h$ must then tend to zero for convergence to steady state.}

\subsection{Faster gradient descent}
\label{sec:faster}

To actually obtain such a sequence of time steps as described above, we use the interpretation of
the forward Euler discretization as that of gradient descent, as in Section~\ref{sec:simple}.
Since we are switching from numerical PDEs to optimization,
let us change notation slightly to conform to different standards by letting  {$\xx \leftarrow \uu, \ \alpha_k \leftarrow h_k$}.
Then our forward Euler  for the discretized heat equation becomes
{gradient descent} for the quadratic objective function~\eqref{1.3}
\[ \xx_{k+1} = \xx_k + \alpha_k \rr_k, \quad \rr_k = \bb - A\xx_k  .\]
We next concentrate on choosing the step size $\alpha_k = h_k$.

The steepest descent (SD) choice is
\begin{eqnarray}
\alpha_k^{SD} = \frac{\rr_k^T\rr_k}{\rr_k^TA\rr_k} \equiv \frac {(\rr_k, \rr_k)}{(\rr_k, A\rr_k)}  . \label{5.6}
\end{eqnarray}
This is the ``greedy choice'', obtained upon performing exact line search. It is known to perform well for the first
few steps but to become slower later on~\cite{akaike}. Asymptotically \uri{SD performs as well}
as the best
constant step choice, for which the bound~\eqref{5.5} holds.

The lagged steepest descent (LSD) step size~\cite{babo} uses the same expression evaluated at the previous iterate:
 \begin{eqnarray}
\alpha_k^{LSD} = \frac{\rr_{k-1}^T\rr_{k-1}}{\rr_{k-1}^TA\rr_{k-1}}  =  \frac {(\rr_{k-1}, \rr_{k-1})}{(\rr_{k-1}, A\rr_{k-1})}  . \label{5.7}
\end{eqnarray}
This then is a two-step method, using information about both $\xx_k$ and $\xx_{k-1}$ to define $\xx_{k+1}$.\footnote{In~\cite{doeasc}
we found experimentally that the overall best  faster gradient descent methods are the two-step ones.}\\

\paragraph{\bf Example}
Returning to our heat equation with $b = 1$, we apply the scheme~\eqref{12} with $h_k = \alpha_k^{LSD}$ to advance to steady state,
stopping when $\| \rr_k \| \leq 10^{-6} \| \rr_0 \|\ .$
The maximum  step size $h$ is recorded in Table~\ref{tab1} for a decreasing sequence of spatial steps $\xi$.
\begin{table}[htb]
\begin{center}
\begin{tabular}{|l|cccc|}
\hline
$\xi$ & $2^{-5}$ & $2^{-6}$ & $2^{-7}$ & $2^{-8}$    \\
\hline
$h$ & .05 & .039  & .043 & .035 \\
\hline
\end{tabular}
\caption{Maximum time step $h$ for the heat-to-Poisson \uri{process}
as a function of spatial step $\xi$.
\label{tab1}}
\end{center}
\end{table}
\uri{Evidently, the maximum step size $h$ does not shrink as a function of $\xi$ in this example.}

Continuing our discussion of the LSD method, it is obvious from Table~\ref{tab1} that the maximum step size $h$ does not
satisfy the bound~\eqref{5.5}.
In other words, some step sizes $h_k=\alpha_k$ disobey the fixed-step stability limit which relates to the continuous PDE.
This causes a chaotic effect in the resulting method \uri{which nonetheless converges}~\cite{doeasc}.

There are many other faster gradient descent variants; our favourite among those is the one where
the SD step~\eqref{5.6} is simply updated only at every second iteration~\cite{rasv}.
In experiments its efficiency is comparable to that of LSD.

Figure~\ref{fig:stepsizes} displays the step sizes observed in the LSD iteration.
Notice how much larger the largest step sizes are from the maximum constant step size.
 \begin{figure}[htb]
 \begin{center}
 \leavevmode
 \includegraphics[scale=0.17]{lsd_stepsizes.png}

\caption{Gradient descent with step sizes by~\eqref{5.7} for
the discretized heat equation
with $n = 63^2$.
The calculated step sizes are displayed vs iteration counter $k$.
The stability limit for a constant step size gives the straight blue line.  \label{fig:stepsizes}}
 \end{center}
\end{figure}

The actual residual norm $\| \rr_k \|_2$ and objective function error $f(\xx_k) - f (\xx^*)$ with $f$
given by~\eqref{1.3} are given in Figures~\ref{fig:lsd_rerr} and ~\ref{fig:lsd_ferr}, respectively.
 \begin{figure}[htb]
 \begin{center}
 \leavevmode
 \includegraphics[scale=0.17]{lsd_min_rerr.png}
\caption{Convergence behaviour of LSD for the model Poisson problem  with $n = 63^2$.
The errors $\| \rr_k \|$
are displayed as a function of iteration counter $k$.
\label{fig:lsd_rerr}}
 \end{center}
\end{figure}
 \begin{figure}[htb]
 \begin{center}
 \leavevmode
 \includegraphics[scale=0.17]{lsd_min_ferr.png}
\caption{Convergence behaviour of LSD for the model Poisson problem  with $n = 63^2$.
The errors $f( \xx_k) - f(\xx^*)$
are displayed as a function of iteration counter $k$.
\label{fig:lsd_ferr}}
 \end{center}
\end{figure}
In these figures we observe a rather non-monotone convergence, and yet the overall convergence rate
does not appear to be slow.

As mentioned in Section~\ref{sec:intro}, there has been a lot of recent interest in  two-step optimization methods
that accelerate gradient descent.
We will not get into related ODE formulations, and instead mention the most popular of these.
The celebrated Nesterov's method~\cite{nesterov1983method}
for the problem~\eqref{1.1}, i.e., the unconstrained minimization of a function $f(\xx)$,  is given by
\begin{subequations}
\begin{eqnarray}
\yy_{k+1} &=& \xx_k + \beta_k (\xx_k - \xx_{k-1}), \label{9a5a}\\
\xx_{k+1} &=& \yy_{k+1} - \alpha_k \ensuremath {\vec \nabla} f (\yy_{k+1})  \label{9a5b}.
\end{eqnarray}
Note that, unlike LSD, the gradient is not evaluated at $\xx_k$, but rather at its modification $\yy_{k+1}$.
Hence, the search direction is not the residual $\rr_k$.
\label{9a5}
\end{subequations}

It can be shown that under suitable conditions this method satisfies
\begin{eqnarray}
f(\xx_k) - f(\xx^*) = \Oh \left( k^{-2}\right) . \label{5.8}
\end{eqnarray}
Furthermore, there is an oracle that says that such a rate is optimal.
Note that SD and constant-step gradient descent
do not achieve \eqref{5.8}~\cite{nesterov18}.
It is therefore interesting to compare the method~\eqref{9a5} to the faster gradient descent LSD for the
case where $f$ is convex quadratic as in~\eqref{1.3}.

However, the non-monotonic convergence displayed in Figures~\ref{fig:lsd_rerr} and~\ref{fig:lsd_ferr} makes such
a comparison difficult.
Fortunately, these errors can be ``monotonized'' by ditching the heat equation interpretation
and evaluating, following the $k$th iteration,
\begin{subequations}
\begin{eqnarray}
\ell_k = \arg\min_{1 \leq l \leq k} \| \ensuremath {\vec \nabla} f (\xx_l)\| . \label{ella}
\end{eqnarray}
 Note that storage and CPU time can be saved by evaluating this at each step $k > 0$ as
 \begin{eqnarray}
\ell_k = \begin{cases} k & {\rm if~}  \| \ensuremath {\vec \nabla} f (\xx_k)\| \leq \| \ensuremath {\vec \nabla} f(\xx_{\ell_{k-1}}) \| \cr
             \ell_{k-1} & {\rm otherwise} \end{cases}  . \label{ellb}
\end{eqnarray}
We can then report $\xx_{\ell_k}$ at any real time, if required, as the ``best'' approximate solution at the end of iteration $k$ (although
we don't use it for the evaluation of the next iteration), because this represents  what we know at the end
of step $k$ better than $\xx_k$ does, in general. So to assess the iterative process we may record also
\begin{eqnarray}
{\rm Egrad}_k = \| \ensuremath {\vec \nabla} f(\xx_{\ell_k}) \| , \quad {\rm and~~~}   {\rm  Ef}_k =  f(\xx_{\ell_k}) - f(\xx^*) ,
\label{ellc}
\end{eqnarray}
the latter still involving a generally unknown quantity.
\label{ell}
\end{subequations}

The important point to remember is that all we need to report in practice is the computed solution upon termination.
So, if our
termination criterion is $\| \ensuremath {\vec \nabla} f (\xx_k) \| \leq {\tt tol}$ for some error tolerance {\tt tol}, then upon
achieving this we stop.

For the quadratic optimization problem we use the notation
\begin{eqnarray}
{\rm Er}_k = \| \rr_{\ell_k} \| .
\label{errs}
\end{eqnarray}
These residual error measures are by definition monotonically non-increasing,  and calculating them at the end of each iteration
is straightforward.
Furthermore, stopping the iteration at the first $k$ such that $\| \rr_k \| \leq {\tt tol} \| \bb \|$ is the same as
using  ${\rm Er}_k \leq {\tt tol} \| \bb \|$ for termination.

For the method~\eqref{9a5} there is the question of determining appropriate step sizes $\beta_k$ and $\alpha_k$.
If we keep both step sizes constant, then the iteration is stationary and standard techniques (involving the largest
and perhaps also smallest eigenvalues of the Hessian) apply.
For the calculations in Figures~\ref{fig:lsd_nesr} and~\ref{fig:lsd_nesf} we fixed the momentum parameter $\beta = 0.95$
\uri{(after some trial-and-error)}
and calculated $\alpha_k$ by the SD formula~\eqref{5.6} with $\rr_k$ replaced by $\bb - A\yy_{k+1}$.

 \begin{figure}[htb]
 \begin{center}
 \leavevmode
 \includegraphics[scale=0.17]{lsd_rerr.png}
\caption{Convergence behaviour of monotonized LSD (LSDm) as well as SD, conjugate gradient (CG)
and Nesterov's (Nes) for the model Poisson problem  with $n = 63^2$.
The errors $\| \rr_k \|$
are displayed as a function of iteration counter $k$.
\label{fig:lsd_nesr}}
 \end{center}
\end{figure}
 \begin{figure}[htb]
 \begin{center}
 \leavevmode
 \includegraphics[scale=0.17]{lsd_ferr.png}
\caption{Convergence behaviour of monotonized LSD (LSDm) as well as SD, conjugate gradient (CG)
and Nesterov's (Nes) for the model Poisson problem  with $n = 63^2$.
The errors $f( \xx_k) - f(\xx^*)$
are displayed as a function of iteration counter $k$.
\label{fig:lsd_nesf}}
 \end{center}
\end{figure}

In the present convex quadratic optimization context, all four methods depicted in these figures are Krylov-subspace methods
generating iterates  in the same subspace at each $k$. So, the superiority of CG is a foregone conclusion,
as it is optimal on each such subspace (see, e.g.,~\cite{greenbaum}).
More interestingly, note the closeness of the LSD method and the accelerated gradient descent method~\eqref{9a5},
both being far better than SD and not much worse than CG. This suggests that the monotonized lagged steepest descent method
also satisfies the ``optimal'' estimate~\eqref{5.8}.\\

\paragraph{\bf Conclusion}
This longer case study, which involves some
new work, demonstrates a situation where the continuous
differential equation limit~\eqref{11} misses out on some interesting and unusual action that occurs at the discrete process level.

\section{Gradient-based penalties in image processing}
\label{sec:paradigm}

In this case study we consider image and surface processing problems, where the given problem is entirely
discrete.
Instances include image and surface denoising, deblurring
and inpainting, data consolidation, morphing, and much more. We can often view this as an instance of data
fitting: the given data $d$ relates to an image or object surface (e.g., it can be a point cloud),
the required operation such as denoising is described by a {\em forward operator} $\phi$,
and we seek another image or surface $u$ in \uri{2D or 3D, respectively,} such that $\phi(u)$ fits the data $d$ to a desired extent.
Often, however,  a differential term is added in an attempt to regularize and control the desired process,
and the resulting differential problem is then subsequently discretized in order  to compute a solution.

\uri{Note that we are deliberately using the notation $u$ both for a discrete image/object and for this image/object interpolated in such
a way that its gradient in two or three space variables, denoted $\nabla u$, may be considered and differentiated. It is then
discretized (or restricted) back to the space where the data $d$ lives.
The step size $h$ is a parameter of such a discretization, say the distance between neighbouring image data locations.}
Here we examine the utility of \uri{such a} continuous model, \uri{which the computer never ``sees''}.

To be concrete, let us consider the problem
\uri{\begin{eqnarray}
\min_u f(u), \quad {\rm where~} f(u) = \| \phi (u) - d \|_2^2 + \beta \int_\Omega | \nabla u |^s , \label{41}
\end{eqnarray}}%
which we will call {\em the paradigm}, for short.
Here $\Omega$ is the domain of the image or the 3D object and $\beta > 0$ is \uri{a fixed, noise-dependent}
regularization weight as in Figure~\ref{fig1}.
Indeed,~\eqref{41} is in the typical form of Tikhonov regularization for an inverse problem~\cite{ta,ehn1,kaso}.
However, we stress that our concern here is with problems where $\phi$ is not a discretization of a continuous structure,
and the only source of differential terms, whether explicit or inverted, is the regularization term.
As such there is some cause for minor mathematical discomfort, because the leading (or, highest order) term of the
PDE in the boundary value problem that arises in
the variational form  comes from the regularization rather than the data fitting term in~\eqref{41}.

There is a huge volume of literature on such methods; see, for instance,~\cite{sapiro,chsh,of,chpo2016}, to name but a few.
The parameter $s$ typically equals $2$, representing significant smoothing \uri{(diffusion)},
or $1$  for {\em total variation} (TV),
\uri{which typically yields piecewise smoothing and is also referred to as {\em anisotropic diffusion}}.
Other values such as $0 < s < 1$  are not our concern here.
The Euler-Lagrange equations for \eqref{41} give a boundary value PDE, and the problem actually solved is a discretization of such a PDE.
 The gradient in \eqref{41} can be replaced by a higher order differential operator, which is again not the focus here.
 If $s = 1$, then some \uri{further} regularization of the regularizer is required, and in~\cite{ahh} we proposed a method for controlling
 a Huber switch between $s=1$ and $s=2$ for this purpose.

 There are lots of situations where the paradigm is the thing to consider!
 \uri{Indeed, it introduces additional information which the discrete $\phi (u)$ as such does not have
 that relates the values of the $u$-unknowns to their neighbouring values. This in turn
 potentially enables a significantly simpler optimization process for $f(u)$.}

 However, already around the turn of the 21st century researchers
 found experimentally that the differential term in~\eqref{41} can introduce side effects that pollute the \uri{visual} quality of the obtained result.
 A ``social bifurcation'' \uri{has} formed, where researchers and practitioners in computer graphics and image processing
\uri{have}  turned more towards purely data driven methods, while mathematicians and numerical analysts \uri{have} continued to explore the paradigm and other such methods
 due to their more coherent structure that in turn enables development of better and more complete theory.

 Around the year 2005, armed with the code from~\cite{ahh}, we started exploring ways to introduce
 the paradigm for various image processing applications in computer graphics. Here the visual quality of the results is what matters,
 and no theory can compensate for lack in \uri{this} regard.
 We considered surface triangle mesh denoising in applications that involve intrinsic texture \uri{and

 sharp corners~\cite{huas,huas2}.} Together with computer graphics and vision experts we then considered  consolidation of unorganized point clouds
 to allow quality surface reconstruction~\cite{hlzac}, point-resampling in the presence of edges~\cite{hwgcaz},
 and image tele-registration and structure-driven completion~\cite{hyglcac2013}.
 Unfortunately, while working on these projects our version of the paradigm was consistently beaten by other, discrete approaches,
 and there is no continuous model left in the final, \uri{published} version of any of
 \uri{\cite{huas,huas2,hlzac,hwgcaz,hyglcac2013}!}

 Let us add a few more details regarding the research just mentioned.
 \begin{itemize}
 \item A noisy triangle mesh representing a surface has noisy
 triangle vertices, which are vectors in 3D. Several researchers have considered forms of
 anisotropic Laplace-Beltramy PDE for denoising surface triangle meshes.
 But in \cite{huas} we learned (not before failing to extend the image denoising method of~\cite{tnv} to surfaces)
  that the most effective way to denoise such a vertex is to find the normal direction to the surface
 there and denoise in that specific direction. To find such a normal, the standard technique is to perform a principal component analysis (PCA)
 on a group of neighboring vertices. This process, described in detail in~\cite{huas}, is necessarily geometrically {\em local}, in marked
 difference to the {\em global} nature of an expression such as the penalty integral in~\eqref{41}.
 The obtained visual results are better, the complexity of the algorithm is linear in the number of unknowns(!), and
 the corresponding {\sc Matlab} code has less than 50 lines. We feel that, unfortunately,
 one just can't practically do all that with anisotropic Laplace-Beltramy.
 \item
 In~\cite{hwgcaz} there is a construction of normals for a consolidated point cloud that represents a surface with an edge.
 Our algorithm is shown to perform better than  applying an $\ell_1$ regularization to the crude PCA normals
 (which is related to TV).
 \item
 The tele-registration algorithm in~\cite{hyglcac2013} involves a step to find salient curves in an image.
 This is a task for TV regularization, but the algorithm of~\cite{ffls2008} proved to be more robust.
 Further, towards the end of the image completion procedure there
 the algorithm and software of~\cite{barnes2009,darabi2012}, which do not employ any \uri{direct} version of the paradigm,
 were used for  inpainting.
 \end{itemize}

 In~\cite{ashu18} we have summarized the pros and cons of the continuous paradigm~\eqref{41} vs discrete processes of
 data-driven reconstruction.

 \paragraph{\bf Conclusion}
 The essential  advantage in using the paradigm is that one can often obtain a more solid theoretical backing to algorithms
 \uri{that are occasionally more clearly derived based on solid principles}
 from this global and generic point of view. The essential disadvantage, however, is that this approach leads to algorithms that
 could be outperformed by more brute force techniques,  as described for the examples above.
 This is so especially if (i) the forward operator $\phi$ is simple, and (ii) the data $d$ is of high \uri{visual} quality.

\section{Conclusions}
\label{sec:conclusions}

Recent years have seen a welcome development where knowledge and expertise  that have been gained over decades
in modelling and computationally solving differential equations
find use in the context of discrete problem areas such as optimization, \uri{game theory}, graphics and image processing.
The author, who has dealt with the numerical solution of differential equations most of his adult life
while being a member of a computer science department, finds special reasons to rejoice.
At the same time, we advocate balance and controlled euphoria.
It is crucial to examine with an open mind the utility of such approaches and specific methods,
from perspectives that include applicability,  relative advantage, and
extensions in the context of the task at hand.

We have demonstrated in a sequence of case studies that employing differential equations in this way can
be at times very useful, at times limiting, and at times just a matter of comfort.
One example of such ``comfort'', not mentioned hitherto, is the use of global properties of dynamical systems, such as
conservation laws, \uri{symplecticity} and reversibility, at the discrete level without requiring
\uri{extreme} closeness of discrete and continuous model solutions.
We advocate to become involved in this fascinating research avenue, as we ourselves are, while
exercising practical caution.

\uri{We have deliberately avoided any significant discussion of stochastic optimization and stochastic differential equation methods
and models, as well as randomized algorithms.
Our point is not to ignore these exciting areas, but rather to emphasize
that significant issues arise more basically even in their absence.}

\subsection*{Acknowledgment}
I thank Drs Jelena Diakonikolas, Arieh Iserles, Alexander Madureira, Fred Roosta, Lars Ruthotto, Mark Schmidt
and Jorge Zubelli for fruitful discussions.
During several preparation stages of this paper I enjoyed the hospitality of IMPA, Rio de Janeiro.

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
PUBLISHED: March 10, 2007 at 5:31 a.m. | UPDATED: August 15, 2016 at 7:56 p.m.
To place a reunion notice, fax information (school name, class year, date of reunion, place, contact name and phone number) to 925-943-8362; e-mail kbennett1@cctimes.com; or mail to "Reunions," Contra Costa Newspapers, P.O. Box 8099, Walnut Creek, CA 94596. Announcement forms may be obtained by calling 925-943-8235.
Alameda High School, class of 1987 — is planning a reunion for the last weekend of July. www.alamedahighclassof87.com.
Alhambra High School, class of 1987 — Planning a reunion for Nov. 3 at the Crowne Plaza in Concord. Contact Tiffany (Mitchell) Colvin at tippy44@hotmail.com, 772-240-6616 or www.myspace.com/alhambra_hs_reunion_1987.
Acalanes High School, classes of 1960-62 — Planning a multiclass reunion for Aug. 4 at Diablo Creek Golf Club. Sharon (Jones) Fraetis, acalanes1962@yahoo.com.
Acalanes High School, class of 1967 — Planning a reunion for Sept. 29 at Scott's Restaurant in Walnut Creek. Linda Ehrich, 925-253-4654; acalanesreunion67@gmail.com.
Clayton Valley High School, class of 1987 — Planning a reunion for Oct. 13 at the Hilton Concord. www.greatreunions.com.
College Park High School, class of 1982 — Planning a cruise on a Hornblower's Yacht for its reunion July 28. Deadline April 1. Janine Davis or Sue (Brown) Firman, 925-679-9858 or cphs1982reunion@aol.com.
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Fremont High School, class of 1967 — Planning a reunion for Aug. 25 at Golden Gate Fields in Albany. www.greatreunions.com.
Liberty High School, classes of 1905-1956 — Planning a reunion for 1 p.m. May 19 at the Flor do Oakley Hall, 520 Second St., Oakley. Linda Gregory, 925-634-4877; Alta Ackerman, 925-634-1906.
Liberty High School, classes of 1978-1990 — Planning a multiclass reunion for Aug. 4 at the Crowne Plaza Hotel in Concord. $87. Class members and teachers may contact Linda at libertyhimultiru@aol.com. Registration forms available at www.libertyreunions.talkspot.com.
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Home » Suspicions about what happened on the wedding day of Mary of Denmark are confirmed
Suspicions about what happened on the wedding day of Mary of Denmark are confirmed
The people of Denmark and supporters of European royalty were very excited in 2004 that Crown Prince Frederik would marry Crown Princess Mary , an Australian girl he met in a bar during the 2000 Sydney Olympics. Pre-wedding events included a boat race, a rock and roll concert, receptions and two tiara events. But what was Maria wearing?
The gown for Mary Donaldson, now known as Princess Mary of Denmark, was the brainchild of Danish designer Uffe Frank, who used ivory duchess satin with a sheen of mother-of-pearl, fully lined with silk organza. The scoop neckline barely touched the beautiful bride's shoulders before descending to sleeves that wrapped around her arm like a lily.
Regrettably, dear Mary of Denmark's mother, Henrietta Clark Donaldson, was unable to attend her wedding day as she tragically died on November 20, 1997. So, the new European royal had her mother's wedding ring sewn close to her heart between the folds of fabric of her wedding dress. Mary's bouquet, which included eucalyptus trees to honor her Australian roots, was brought to Scotland by an aunt to rest on the grave of Henrietta and her grandparents after the wedding.
Mary of Denmark's wedding day was bittersweet for the bride
The most unique feature of the dress was Mary's skirt, where satin panels fanned out from her hip to reveal eight yards of traditional lace. Irish lace was another sentimental touch , this time in honor of Frederik's family. It was part of the original wedding gift to Crown Princess Margaret of Sweden (Frederik's great-grandmother) which included the famous family veil.
Mary wore Crown Princess Margaret 's veil , inherited from the family by Queen Ingrid. Crown Princess Mary of Denmark is the only sister-in-law to date to wear the veil . She anchored him with her wedding tiara, a gift from the Queen and Prince Henrik. Mary also wore new brilliant-cut diamond earrings set in platinum with South Sea Island pearls.


The big hardware stores and supermarkets clearly know zip about gardening because they haven't yet cottoned on to this plant now harvest later concept. I went to buy some cool season seedlings on the weekend, and surprise, surprise, the vegie selection was in full blown spring and early summer planting mode. I went home empty handed and I'm kind of glad – I'll now have to sow seed and be a bit more patient than I would have otherwise. This is a good outcome, especially with our contrary weather. By the time the seeds are up and ready for transplanting, things might be a bit more settled.
In subtropical areas that don't get frost, you can pretty much plant whatever fruit trees you like during February. If you've had flooding, wait for your soil to dry so it's no longer waterlogged. If not then late summer and autumn is a better planting time than spring for many trees as wet season rains generally do a good job of watering new plants and getting them off to a good start. Try citrus, guava, lychee, avocado, and mango, among others.


Turn the RF gain full CCW, AF gain to where you can hear. Note that there is almost no hum/noise in the AM position (even with AF gain full CW). Now switch to CW-SSB. Note the rise in hum!
My first thought was the BFO but pulling V20 has no effect. Also, no cathode-fil shorts anywhere. Furthermore since I have found this in all -A4's it appears to be a design deficiency. I do realize the "condx" under which one hears this hum is not "normal" so I can understand how it could go undetected. My best guess at this point is some sort of pickup in the wiring harness and especially the long lead from the output of the BFO to V11.
Anyone else experience this or have any ideas?
The frequency of the hum may be a clue, if 60 hz its probably from heater current or possibly from the bias rectifier. If 120 hz from the power supply. The AM/SSB switch changes three things; the input to the audio amplifer; the configuration of the noise limiter; and it also grounds the screen of the BFO in AM. See if pulling the noise limiter tube makes any difference. Also pull the product detector. You may be able to trace the hum with a scope. If you have encountered this hum in several receivers it may be a hum loop due to a bad ground. That may be hard to find but the usual voodoo of working all the screws that may have grounds made through them is at least worth trying. It may not be a design problem so much as something that shows up with age. I had a mysterious hum in my R-388 that turned out to be a poor ground return for a filament. The return was made through the frame of the headphone jack which was not making good contact due to the paint on the panel. The hum was there all the time even when the rectifier was pulled.


Bob Kelly (Lead Vocal, Guitar and Sax Solos). Vocal group Neil Wood, Johnny Biggerstaff, and Bill Byrd. Various Musicians from 1950 recording sessions in Dallas and Fort Worth. Clifford Herrings Recording Company, Sellers Studio and Top Ten Recording Studio (owned by Bob Kelly).
Joe Turner, Clovers, Drifters, Chuck Berry, and early 1950 rockers.
1954 to 1959 Original Recordings of songs written and performed by Bob "Git it" Kelly. In 1958,Gene Vincent recorded "Git it" and "Somebody Help Me" written by Kelly. Now the original Kelly versions are on this CD. Also, Mac Curtis recorded "What You Want" in 1957 and Bob Kelly's version is here on MP3 as well as the CD. All of the other songs from this CD are Classic's from this great Rockabilly Legend and some are released for the first time. Check out "Chalypso" and reference to Steverino, from the old Steve Allen late night TV show. "Mmm Love" has a Whistling Solo, thats different.
Bob Kelly Fender Strat and Ovation Electric Acoustic Guitar, Fender Bass, Martin Alto and Conn Tenor Sax, Keyboards.
Bob Kelly Stage Songs Libra Records 1101, also 1954/1959 Rockabilly Original Records Libra Records 1202 thru Hep Cat Record Distributors.
Gene Vincent "Lost Dallas Sessions" with Bob Kelly versions of "Git it" and "Somebody Help Me".
Source: http://www.mp3.com - listen to mp3 samples from Bob's new CD.


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text:
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Anouck Girard
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3049 FXB
Michigan Engineering | College Administration, 1221 Beal Avenue, Ann Arbor, MI 48109-2102
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He has extensive commercial experience with a wide range of software packages, and has worked in a variety of application domains (including finance, law, recruitment, tourism, construction and mining).
Kaz has been managing projects and supervising teams successfully for over 10 years. He has experience in all aspects of small business operations – including recruitment, sales, marketing, risk management, and business development.
Kaz holds a Diploma of Information Technology (TAFE NSW), a Bachelor of Computer Science degree (University of Newcastle), and a Master of Information Technology degree (Queensland University of Technology). He has also been appointed as a university lecturer and tutor, and is currently on the teaching staff at Queensland University of Technology.
Robert Bray has over 15 years of commercial experience in IT. His depth of knowledge across all aspects of corporate IT services, and his ability to find creative solutions to complex problems, have enabled him to excel in a multitude of Japha projects over the past decade.
His enthusiasm for learning means that Robert is always up-to-date on the newest technologies and how to utilise them effectively.
Robert holds a Bachelor of Information Technology (Network Systems) degree from Queensland University of Technology.
Aaron's logical approach to systems design - combined with his ability to learn new technologies quickly, and his innate capacity for "seeing the bigger picture" - have made him a valued member of the Japha team.
As with all Japha team members, Toni is extremely hard-working and committed to our clients and the success of their projects. His experience with a wide range of technologies has proved extremely useful on a range of software projects.
Toni holds a Diploma of Information Technology and a Bachelor of Information Technology (Computer Science) degree, both from Queensland University of Technology. He also has several overseas qualifications in web development.
Jonny Hall has experience in several areas of IT, including software development, web development, IT support and user training.
Jonny's enthusiasm for building software is evident in the projects he works on. He is also an active member of QUT Code Network, and a regular participant in hackathon events.
Jonny is currently completing a Bachelor of Information Technology (Computer Science) degree at Queensland University of Technology.
Tyler Chan has brought his skills in animation and graphic design to a variety of industries in Australia and Hong Kong over the past 7 years. His portfolio spans many areas of work - including a feature documentary, educational software, website design, several animation projects, and mobile games.
Tyler holds a Bachelor of Animation (First Class Honours) degree from the Queensland College of Art, Griffith University.


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Paula Abdul cancels Kansas City show due to injury
"Due to a temporary injury Paula Abdul is unable to perform this evening," the post from Live Nation said. "We wish Paula a speedy recovery."
Updated: 10:35 AM CDT Jun 12, 2017
(Photo by Al Wagner/Invision/AP)
SOURCE: (Photo by Al Wagner/Invision/AP)
Paula Abdul will not be performing tonight at Sprint Center due to an injury, concert promoter Live Nation announced in a social media post."Due to a temporary injury Paula Abdul is unable to perform this evening," the post from Live Nation said. "We wish Paula a speedy recovery." Live Nation said if you decide not to attend tonight's concert due to this, you are asked to visit the box office immediately.Boyz II Men and New Kids on the Block! are still scheduled to perform this evening.Abdul cancelled a show in Denver over this past weekend as well."Thank you to all my fans for your constant love & support," Abdul Tweeted. "I treasure each & every one of you & I'm most grateful for your understanding..."For free news updates on the go, download the KMBC 9 News app
Paula Abdul will not be performing tonight at Sprint Center due to an injury, concert promoter Live Nation announced in a social media post.
Live Nation said if you decide not to attend tonight's concert due to this, you are asked to visit the box office immediately.
Boyz II Men and New Kids on the Block! are still scheduled to perform this evening.
Abdul cancelled a show in Denver over this past weekend as well.
"Thank you to all my fans for your constant love & support," Abdul Tweeted. "I treasure each & every one of you & I'm most grateful for your understanding..."
For free news updates on the go, download the KMBC 9 News app


Let's do this fabulous song next week. This is the hit by Roberta Flack. Listen to those guitars, Dave. Wanna sing it Maria?
Good night last night. Well done everyone.
Next week, I've asked Maria if she can come. If she can, then she will give me some songs she wants to try out and we'll do those. I'll mail you all.
Attached is Autumn Leaves in Ddm. You'll love it, I promise. Where else can you practise your Dbm with others listening? I'm not going to bring Bb and Eb flat copies, trusting you can do it yourselves from this attachment. But if you get into trouble with it, let me know.
We might have another go at Afro Blue, with other folks hitting things rather than stroking them, just to get the right Latin feel. We need a percussion intro really.
We can also do the following, as practise for Terry's session coming up on October 20th. It is best if we know the material already then we can get further with him.
Let's do Feel like making Love. Geoff says we have to sort out how to get into it, and we could do it with a funky sound. Good song for that.
Maria could you plan to do this one if you come?
You folks have got the very first look at this, a brand new Google map showing the main venues for jazz from Chichester to Dover. Can't get all the occasional and tiny ones in, obviously.
Man Jazz Venues in Sussex, East and West.
For the moment the workshop is full, until someone drops out. there are about 10 of us each week.
Personnel are Andrew Batchelar, alto; Geoff Valenti trumpet; Dave Moore, guitar; Maria Ball, vocal; Vic Crowley, bass; Bob Savary, drums; John Winkler, keyboard.
We have honestly tried to make this one of the really good short videos you can find of a jazz band.
The brief to Callum Robertston was to make a video packed with movement. The sound is being heard only on a computer after all. The sound is a backing track.
Bands are notoriously difficult to film. There are amplifiers, music stands, mics, cables. They look a mess at a distance and the players are separated from each other. Ugh!
You can do it very well if you spend a lot of money on staging it, but we have only peanuts to spend.
Callum decided get the movement by using a lot of very short sequences, 1sec 2 secs with rapid cutting. He used close ups, reveals, he also added soft focus and slow motion to some sequences. It produces a huge variety of look. But he also showed one shot of the whole band at the end, and he got the front line smiling and enjoying themselves.
The band plays a range of different styles, so we started with a full on jazz song by Bill Evans, called Funkaleroo, then we switched the sound track to our Maria singing a cool soft jazz version of a standard "It Could Happen to you"
We were worried about the rough jump, but I think the tension caused by this brings attention back to the video.
In this blog We will produce playing tips for jazz piano, and jazz guitar together with jazz saxophone. We will cover jazz chords, jazz guitar chords, and we will deal with jazz scales. We will cover jazz songs. This site is all about jazz improvisation. you can sign up directly to this blog site as a FOLLOWER, bottom rh side panel, you'll get all the posts.
Oscar Petersen practised for an average of 5 hours a day. He said, "You need to do enough practice to get your technique to the point where you can express your ideas. It depends how heavy those ideas are."
Later in life he halved his practice routine. Sensible chap.
Most professionals I know do practice, and practice much more than amateurs. Quiet Paul Desmond seems to be in awe of Charlie Parker in this interview.
WHAT WE'LL DO NEXT SUNDAY. THIS WILL KEEP YOU OUT OF MISCHIEF.
Autumn Leaves in Gm (lets call it Bb maj), We'll do it in Dmaj (Bbm). This is part of our doing songs in unusual keys. Just 30 minutes on it.
Lets try out three things led by Dave's Guitar. The Use me riff for Lullaby of Birdland So What also the Superstition riff for Love for Sale. Dave can you just run through a mediaeval version of Greensleeves for us.
You are trying it out I know.
CAN YOU FOLKS PLEASE PREPARE A PAD WITH THE FOLLOWING SONGS WHICH WE DID LAST YEAR. ADD NEW ONES AS WE GO.
We will select songs from this list for working on. Sometimes the key may have varied from the standard because of the vocal need. Add Autumn Leaves Gm please. We'll use it as a practice piece for more difficult key.
Love for sale Eb (not F, no vocals) Gtr using Superstition riff.
During the season we will select from these songs we have not done before. I have in mind building a song list from the 70's for the new jazz show next year.
I'll circulate them on BIAB files when I get them all. One or two we have done years ago. I've also got in mind some of these for vocals.
We have a problem. We have three newcomers trying us out for the first time. Music sheets for them. Geoff has kindly arranged to have three copies of BIAB software for them. In a couple of weeks' time we'll have a 30 minute session about how to use BIAB.
We need charts for Eb and Bb instruments, and one is a Bb bari part. We probably have some music in the Bb and Eb boxes at present. If we select some other songs to do then, they can share existing charts.
I'll bring the charts for these.
THE WAY YOU LOOK TO-NIGHT. LISTEN TO THE EXCHANGES AT THE END.
I copied the Blues in Time CD for everyone, without explaining why. I've got spare copies for the newcomers.
These two do and their backing section do nearly everything that we plan to do in the workshop. Have a listen. They are listening closely to each other while they play. For example, on track 4 Mulligan plays a little quote in his solo. Not to be outdone Desmond then plays a quote in his. They are just having fun with each other. Behind Desmond's solo is a soft descending harmony which is Mulligans trademark in all his arrangements. Its soft, quiet and cool. On other tracks listen to the way they copy each other exchanging 4s, sometimes they use a counter point harmony.
Desmond's solos are full of little playing patterns transposed, Mulligan too plays patterns but with a more flowing line. We'll have a go at this.
WE DID IT ONCE. BRILLIANTLY.
Andrew and Geoff exchanged 4's, and they really played off each other, they copied, they drove, they played fast and slow, loud and soft but they had a huge amount of fun playing for each other. It made them laugh together, it made the audience laugh, and it was totally musical.
For me it was the epitome of what a band should be about. It was a team game with the two forwards exchanging passes in the box.
That is part of what we are going to strive for in the new workshops.
judgement, no criticism, just fun.
THESE ARE SOME OF THE THINGS WE'LL WORK ON.
FRONT LINE LEADERS TO INVENT VARIETY, melodies/melody plus solo/sharing/ensemble/ backing/breaks/ 8's 4's 2's hand overs, harmonies, jamming, codas, intros all sorts of stuff.
BAND PLAYING ON THURSDAY WEEK FOR WOMEN'S INSTITUTE. SHOULD BE REALLY NICE.
This is what some of our country's top professionals have said about the following issue.
Interesting topic, most music that I've come into contact with has genre specific vocabulary, that's to say that each style of music has its own peticular rhythmic, harmonic and melodic structures, licks and cliches.
it's very important as a contemporary musician to be aware of what these are so the layman and scholars alike can identify what kind of music they are listening to.
As a drummer I'm happy to say that my instrument (the drum kit) was being developed at the same time as jazz music was one of the most popular music forms, so you could say that jazz music is inherent in everything that is played on the drums up to this day, but you would be wrong as various technical and stylistic advances have been made since the 1970s ( ie linear patterns, hybrid rudiments) from where jazz music's major developments have been through mixing it with rock, world and other musical forms.
So being creative??? Or improvising within a recognised musical framework or instant composition. Like any part of music the more you practise improvising the better you will become, but soloing isn't just wiggling your fingers or noodling. It should be based on phrasing, rhythmic and melodic development. I personally only use practised patterns or licks as platforms to start with or to refocus a solo if I feel that it has become too abstract. This of course is down to personal taste which gives my solos the stamp of my own musical identity.
So this leads to my question, which is: how creative can you be if you are restricting your solo to fit within a certain genre? I personally feel that you should express your full experience and artistry in all scenarios, even if it means playing some Lydian dominant quintuplet arpeggios on a gypsy jazz gig. If you can hear it play it, but don't try and crowbar the latest lick you've just been shedding.
So in conclusion, learn the history and the vocabulary of the music (and your instrument), listen to how the greats do it, don't be afraid to experiment, don't be a jazz snob. John Coltrane was the best at being himself and you will never be a better John Coltrane than he was. Enjoy yourself.
Hi John, thanks for your question! Well I believe that if you actually use a creative approach to your practice sessions thinking of as many creative, fun and exciting ways to work on a concept then this will come out in your playing. So basically taking a concept or idea, working on it to get it under your fingers & in your ear then the next step practicing it in as many crazy ways as you can come up with. This is what I do with my students at Sussex Jazz Guitar School. The blog attached describes the Creative practice process we will be exploring in our next class 28/9. All the best..
September 28th class will be taking a jazz standard tune and tackling it using a multitude of approaches, expanding on concepts from previous classes plus new ones. We will be developing Creative and fun approaches to practice specific material because if we practice at home in a fun, creative way.
.Paul also wrote: "Thanks John, on this other blog here there's a section "exploring concepts" which explains it a bit. If you practise creatively you'll play creatively.
said this in response to the creative question and sent in this brilliant clip of "Elvis" discussing atonality. You must look at it.
"John I really like it - and I like the idea of making a few provocative statements to get the juices flowing! I must find the Elvis extract for you - it is about this!!
"Also I think it is always worth emphasising how life-changing this music can be, and how that is reinforced when we get together and exchange knowledge and enthusiasm for artists and recordings that we know and love... listening is the key!!"
THE ORIGINAL POST THEY ARE RESPONDING TO.
Oooh! Hot topic. I can feel the coals about to be dropped on my head.
Who is this guy who can't play, doesn't know anything much, to put forward a view on creativity in this highly specialised field?
Well, no more than the average viewer in the Tate Gallery who likes/dislikes a painting even though they can't draw.
Personal comments these, but they relate directly to our workshop and what we plan to do. These are not your views maybe, there as many views on this topic as there are players. Start heating up the coals now. Comments welcome.
Jazz musicians are improvisers by nature. We constantly explore the unexplored, experiment, and try new things. That is what we do, once we have our technique established and even before then. From the start we'll be encouraged to do our own thing.
But look at the notes below. You notice how many of these great innovators did their best work with others. They feed off their rhytm section, they feed off each other. Without their favourite playing partners they were often a bit diminished.
Scientific studies, carried out in laboratories show that when jazz musicians improvise, their brains turn off areas linked to self-censoring and inhibition, and turn on those areas that let self-expression flow.
To develop creativity you take big problems and then break them into small tasks.
It takes a lot of courage to be willing to sound awful from time to time in order to develop. Everyone is scared of making mistakes. If not, we are not trying hard enough.
THELONIUS MONK Known to his wife as Melodius Hunk.
Who were the creative jazz people of my generation? Thelonius Monk - a legend in his time for his unusual harmonies and soloing style, his chord voicings, his timing, his original songs, copied by everyone in his day. Self-styled founder of Bebop. Some top jazz players would not play with him. I can take him in small bites, but I don't want a full evening of him. He hasn't been copied. Notice that. Genuinely creatively people leave a legacy, if it doesn't last it is more likely to be a novelty. Thelonius Monk was hospitalized twice for what was probably bipolar disease. Notice how some disability can lead you into original directions.
THE MJQ, Modern Jazz Quartet.
They made a genuinely, long lasting impression on the music world as a whole. They were probably the foundation of much of the more melodic contemporary jazz style of to-day. They made jazz acceptable to classical musicians who recognised some of their baroque influences. I could listen to them all night. John Lewis plays the simplest of solos, spare, and open, a perfect counter to the powerful Milt Jackson on the vibraharp, with Percy Heath the driving force on the bass.
Mulligan and Baker in their pianoless quartet. The pianist forgot to turn up one night. They found that this enabled them to free off from the harmony more easily. For me they were really original in the way they harmonised behind each other. Sometimes they did not even discuss the songs before they played them but they achieved wonders just playing off each other Their quartet only last for 9 months but they left a long lasting impression on the jazz world and helped to set up the West Coast movement.
Their original work in using different time signatures is what they are known for, but I loved their Jazz at the College gigs from their early days. Desmond's fluent patterns, Brubeck playing chords. Interesting point here. He had to play chords at this time because he had an accident that stopped his fingers from soloing. Wonderfully original and musical were these tracks. Later he dropped the chord playing, when his fingers got better. Pity. Notice how some disability can lead you into original directions.
Going back a bit now, but his trademark was the heavy block chords. Self taught, he sounded like a one man orchestra. Lovely sounds. He couldn't read music. Notice how some disability can lead you into original directions.
A lot of argument here. He was put down by jazz musicians for being a Hotel pianist. So much of a hotel pianist he outsold nearly all the others with his records, with very sweet harmonies. This is George Shearing speaking, "Your intent, then, is to speak to your audience in a language you know, to try to communicate in a way that will bring to them as good a feeling as you have yourself" Notice his concentration upon the audience result. Lesson there for some players.
He was blind. Notice how some disability can lead you into original directions.
BILL EVANS and his trio. He changed the style of piano playing more than any other. Introspective, deeply thoughtful, amazing range of different ideas which all fitted together somehow. He developed rootless chords and relied hevily on chords tones. First man to rely heavily on the bass player so heavily.
CHARLIE PARKER. He might not have started bebop but he and others pushed it to its limits. His trademark was unique phrasing and complex melodic lines. This, for me, is jazz musician's music. I can understand its brilliance, the technique but it does not sit easy on my ear. That's just me. I don't like Tracey Emin, but I do like David Hockney.
The atonal movement which followed Bebop, wrecked jazz. It was not creative, it did not have a long lasting effect it was nothing more than a novelty. For my ear it sounded awful. A disastrous novelty.
Research into jazz creativity, using brain scans.
The workshop will be opening again on Sunday September 15th. For the moment it is full, until someone drops out.


Springfield Guest House is a large Victorian house built in 1874. It boasts a period dining room, separate lounge and all of the rooms are beautifully and elegantly furnished.
The house is set in its own private grounds and includes a summer house as well as being walking distance from Haworth - Home of the Bronte Sisters. The period dining room offers traditional home cooked Yorkshire food and a full English breakfast which sets one up for the day.
There is off road parking and central heating throughout. We are within easy reach of Bradford, Skipton and Calderdale.
As well as being land of the Bronte Sisters The Worth Valley Railway nearby was used in the film "The Railway Children."


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Born in Kenya, of Rwandan parents, Shad was raised in London, Ontario. His debut album When This Is Over (2005) was self-made, financed with the $17,500 he won from 91.5 The Beat's Rhythm of the Future talent competition during his time as an undergraduate student at Wilfrid Laurier University. The album was recognized for Shad's self-deprecating lyrics, which are atypical of the current rap scene and focusing on social causes—for example, the track "I'll Never Understand" examines the Rwandan genocide and includes poetry written by his mother, Bernadette Kabango.
Recorded by Dan Cantor / Notable Productions


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Bioinformatics Analysis of Transcriptomics Data Reveals That SRSF1 Is a Novel Molecular Brake for the T Cell Activation Program and Controls Key Cytokine Signaling Genes Implicated in Systemic Lupus Erythematosus
Rhea Bhargava1, Michelle S. Lee 2, Takayuki Katsuyama 3 and Vaishali R. Moulton 3, 1Beth Israel Deaconess Medical Center Harvard Medical School, Boston, MA, 2Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, 3Beth Israel Deaconess Medical Center, Harvard Medical School, Boston, MA
Keywords: autoimmunity and Gene Expression, Bioinformatics, SLE, T cells
Session Title: T Cell Biology & Targets in Autoimmune & Inflammatory Disease Poster
Background/Purpose: T cells from systemic lupus erythematosus (SLE) patients exhibit a hyperactive phenotype with defects in homeostasis, signaling and cytokine production. By discovery approaches, we previously identified the serine arginine-rich splicing factor 1 (SRSF1) in human T cells and showed that it controls genes involved in signaling and cytokine production. We further showed that SRSF1 expression is decreased in SLE T cells, and associates with worse disease. We recently found that mice with conditional knockout of Srsf1 in T cells (Srsf1-T cell ko) exhibit T cell hyperactivity, systemic autoimmunity and lupus nephritis. However, little is known about the molecular targets controlled by SRSF1. Our goal is to identify the genes and pathways controlled by SRSF1 in mice and by comparative analysis evaluate if these are implicated in SLE.
Methods: Effector CD4 T (Teff) cells were generated by stimulating naïve CD4 T cells from spleens of Srsf1-T cell-ko or control WT (n=3) mice for 72h with CD3/CD28 antibodies. RNA from CD4 Teffs was subjected to RNA-sequencing (R-seq). Data was analyzed for differentially expressed (DE) genes, gene set enrichment, Kyoto encyclopedia of genes and genomes (KEGG) and gene ontology (GO) pathways. Gene expression omnibus (GEO)2R was used to identify a publicly available dataset (GDS4888) from CD4 T cells in patients with active SLE (n=6, SLEDAI range 6-22, ANA and dsDNA positive) and 4 healthy controls. Comparative analysis of mouse (Srsf1-T cell ko) and human (SLE) gene array data was performed using Metascape to identify overlapping gene signatures in SLE patients controlled by SRSF1.
Results: R-seq analysis from CD4 Teff cells of Srsf1-T cell ko mice revealed 608 significant DE genes compared to WT mice at the 2-fold cutoff with p< 0.05. In the human SLE CD4 T cells transcriptomics analysis, SRSF1 was confirmed to be significantly downregulated in active SLE patients compared to healthy controls. 290 genes were significantly (p< 0.05) changed between patients with active SLE and healthy controls. The top pathways represented in the DE genes in ko mice were cell cycle, Th1 and Th2 differentiation, Th17 differentiation and cytokine-cytokine receptor interaction. Overall the CD4 Teffs showed an elevated T cell activation gene signature. Pathway analysis of the 290 DE genes in SLE patients identified interferon signaling, cytokine production, cytokine receptor interaction, cell migration and lysosomal clearance pathways. Overlapping genes between human and mouse transcriptomics data were analyzed. Specifically, we found 11 genes (CCR1, RHOG, ELL2, IFI16, IFIT3, OAS2, ZER1, PRKD2, RGS3, SAT1, SOCS1) to be significantly altered in active SLE patients, which were regulated by SRSF1 as confirmed by our mouse R-seq analysis.
Conclusion: SRSF1 controls genes involved in T cell homeostasis, activation, cytokine regulation/signaling and differentiation, which are altered in patients with active SLE. Therefore SRSF1 is an important regulator of T cell function and its deficiency may lead to the hyperactive T cell phenotype in active SLE patients. Targeting SRSF1 to correct the aberrant T cell phenotype may lead to novel therapeutics particularly in SLE patients with decreased SRSF1.
Disclosure: R. Bhargava, None; M. Lee, None; T. Katsuyama, None; V. Moulton, None.
Bhargava R, Lee M, Katsuyama T, Moulton V. Bioinformatics Analysis of Transcriptomics Data Reveals That SRSF1 Is a Novel Molecular Brake for the T Cell Activation Program and Controls Key Cytokine Signaling Genes Implicated in Systemic Lupus Erythematosus [abstract]. Arthritis Rheumatol. 2019; 71 (suppl 10). https://acrabstracts.org/abstract/bioinformatics-analysis-of-transcriptomics-data-reveals-that-srsf1-is-a-novel-molecular-brake-for-the-t-cell-activation-program-and-controls-key-cytokine-signaling-genes-implicated-in-systemic-lupus-e/. Accessed January 19, 2022.
ACR Meeting Abstracts - https://acrabstracts.org/abstract/bioinformatics-analysis-of-transcriptomics-data-reveals-that-srsf1-is-a-novel-molecular-brake-for-the-t-cell-activation-program-and-controls-key-cytokine-signaling-genes-implicated-in-systemic-lupus-e/


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21st January 2021 Going Postal Arts, Media, Television 0
Fiona Bruce presents another hour of BBC programming from Northern Ireland.
On the panel are: Brandon Lewis MP, secretary of state for Northern Ireland, Conservative, Louise Haigh MP, shadow secretary of state for Northern Ireland, Labour, Arlene Foster MLA, first minister of Northern Ireland and leader of the Democratic Unionist Party, Michelle O'Neill MLA, deputy first minister of Northern Ireland and cice president of Sinn Fein, and Anand Menon, professor of European Politics and Foreign Affairs at King's College London and director of UK in a Changing Europe.
Anand Menon
Arlene Foster
Louise Haigh
Michelle O'Neill
The Question Time That Never Was
24th October 2019 Going Postal Arts, Media, Television 0
Graphic EJ – Updated by Sam. Fiona Bruce presents an hour of topical debate from Leicester. On the panel are: Norman Lamont, former chancellor of the exchequer 1990-93, and a Conservative peer since 1998; Richard [more…]
25th June 2021 Always Worth Saying Arts, Literature, Media, Review, Television 0
Question Time 24th June 2021 Panel: Robert Buckland (Conservative) Jon Ashworth (Labour) Caroline Lucas (Green) Kate Andrews (The Spectator) Victor Adebowale (NHS Confederation) Venue: London Question one. Is it not double standards to allow UEFA [more…]
2nd April 2020 Going Postal Arts, Media, Television 0
Graphic EJ – Updated by Sam. Fiona Bruce presents an hour of topical debate from Rugby. On the panel are: Matt Hancock MP, health secretary, Conservative; Yvette Cooper MP, chair of the commons' home affairs [more…]
Crimean War, Part Three
In 1954, as the centenary of the end of the Crimean War approached, my uncle John Alldridge produced a series of articles for the Manchester Evening News which vividly described that campaign. In this article, [more...]
The Unseen Path – Part Thirty Five
"Hi Lena, how are you?" She had been a couple of minutes late and, due to constant demands from her work tablet for her attention, hadn't been able to prepare herself for what could be [more...]
Postcard From Kyrgyzstan
5th January 2022 0
There's something I've been meaning to tell Swiss Bob but haven't got round to. Perhaps you could oblige? Next time you bump into him, drop into the conversation that, after two years sitting around on [more...]
Graphic EJ – Updated by Sam. Fiona Bruce presents an hour of topical debate from Thetford. On the panel: Nicky Morgan MP, former education secretary, Conservative; Anneliese Dodds MP, Labour's shadow treasury minister and a [more...]
Question Time 28th May 2020 Panel: Helen Whately (Conservative) Ian Murray (Labour) John Swinney (Scottish National Party) Layla McCay (NHS Confederation) Alex Massie (Journalist) Venue: Scotland Proprietor of the Coniston & Windemere Surgical Appliance & [more...]