text:
\section{Introduction}

The problem of constructing many-body wave functions of identical particles is
almost as old as quantum mechanics itself. Soon after Pauli\cite{Pauli25}
articulated his famous principle, Heisenberg\cite{Heisenberg26} and
Dirac\cite{Dirac26} found that it could be mathematically expressed by wave
functions in determinantal form, which came to be known as
Slater\cite{Slater29} determinants. Given that Slater determinants form a
complete basis of Hilbert space, the so-called second-quantized formalism,
which made their manipulation technically feasible even for large systems,
established them as a secure formal foundation for all microscopic
descriptions of many-fermion systems.

While formally universal, the Slater-determinant approach --- as implemented
most characteristically in the time-dependent pertubation
formalism\cite{Abrikosov75,Mahan90} --- has a pattern of success indicating
that it is better adapted to some problems than others. Broadly speaking, it
works when the interacting system can be described in terms of relatively
simple excitations of the non-interacting system. Even an infinity of such
excitations, creating collective modes and new ground states, is within the
purview of the approach, provided that each individual excitation can be
expressed as a small departure from the non-interacting ground state. The
paradigm for such small departures are the well-known particle-hole
excitations of the Fermi sea. Similarly, Cooper pairing is just a resonance in
two-particle head-on collisions.\cite{Mahan90}

On the other hand there are so-called strongly correlated systems, for which
the Slater determinant formulation does not seem to offer advantages by
itself. These are characterized by a number of properties which need not all
be present, but typically include:
\begin{itemize}
 \item a finite number of particles;
 \item localization in space;
 \item strong short-range or unscreened long-range interactions.
\end{itemize}
All three characteristics are present in atoms, nuclei, and small molecules.
The two-dimensional electron gas, found in the inversion layers of
semiconducting heterostructures, has only the third property in the absence of
a magnetic field, but acquires the other two when a strong field is turned on.
Of course, the number of electrons does not suddenly become finite, rather the
localization of trajectories caused by the field means that only a small
number of electrons create a strongly correlated wave function, which is then
repeated throughout space. This number was three in Laughlin's original
description\cite{Laughlin83-1} of the fractional quantum Hall effect (FQHE) at
$1/3$ filling, where the strongly correlated motion in question was their
rotation as the vertices of a ``stiff'' equilateral triangle. Some
contemporary approaches\cite{Hyowon14} to strongly correlated materials share
the same pattern: the local part of the problem is treated by dynamical
mean-field theory\cite{Georges96} (DMFT), while the extension in space is
treated by density-functional theory\cite{Kohn65} (DFT), i.e., by Slater
determinants. A significant difference between the latter two cases is that
all electrons need to be antisymmetrized in the FQHE, while in crystals the
electrons localized within a unit cell need not be antisymmetrized with those
in the other cells. The separation between the local and the itinerant degrees
of freedom in such cases is an important issue beyond the scope of the present
review.

Here, the first tentative steps of a new approach are described, which shows
some promise to be ``native'' for few-particle strongly correlated cases, much
as the Slater determinants natively describe long-wavelength excitations of
extended systems. The present review places this approach in its natural
mathematical context and addresses various physical ramifications in a
qualitative way. Proofs of the main results have appeared
elsewhere.\cite{Sunko16-1} The primary ambition, unfulfilled at present, is to
obtain \emph{realistic} wave functions of few-body systems, without
necessarily improving on existing numerical methods either in terms of speed
or accuracy in the calculation of binding energies. In this sense the
philosophy is converse to that of DFT, which consciously sacrifices realism of
the wave functions in favor of precise calculation of energies.

\section{Mathematical background}

The mathematics invoked in the present work has a long and distinguished
lineage. Combinatorics and the theory of invariants were first brought into
modern form by Leonhard Euler and David Hilbert, respectively. The purpose of
the present section is only to introduce as much background as necessary to
understand the subsequent physical development. The level is chosen to make the
material accessible to any physics student. If the mathematical training of
physicists were devoted half as much to abstract algebra as it is to complex
analysis, the whole section would be unnecessary. The style is informal, with
mathematical rigor left to the references. Although every statement is true as
it stands, the presentation is tendentious in light of the final purpose ---
describing the physical Hilbert space of $N$ particles --- so that it should not
be taken to reflect the attitudes of mathematicians.

\subsection{Euler and generating functions}

Euler invented generating functions\cite{Euler51} in 1741, ostensibly to
answer a question from a ``most insightful'' correspondent: in how many ways
can the number $50$ be written as a sum of $7$ distinct natural numbers? Of
course, permutations of the same sum are not counted, so the question may be
rephrased, ``\ldots sum of $7$ \emph{strictly increasing} natural numbers.''
Euler realized that if he could compute the sum
\begin{equation}
\sum_{1\le n_1<n_2<n_3<n_4<n_5<n_6<n_7<\infty}
q^{n_1+n_2+n_3+n_4+n_5+n_6+n_7},
\label{original}
\end{equation}
the solution would appear as the coefficient of $q^{50}$. In terms of
contemporary physics, his task was to compute the degeneracy of the level with
energy $50\,\hbar\omega$ of a one-dimensional harmonic oscillator well, filled
with $7$ fermions, and spectrum given by $\varepsilon_n=n\hbar\omega$, $n\ge
1$, and $q=\exp(-\beta\hbar\omega)$. (The answer is $522$.)

The sum~\eqref{original} is simple enough to be computed directly. By some
clever tricks with indices\cite{Andrews76} one can find that\cite{Euler51}
\be
\sum_{0\le n_1<\ldots<n_N<\infty}q^{n_1+\ldots +n_N}
=q^{N(N-1)/2}\frac{1}{1-q}\frac{1}{1-q^2}\cdots\frac{1}{1-q^N}
\equiv q^{N(N-1)/2}Z_E,
\label{Z1}
\ee
where the origin of the $n_i$ is shifted to zero for later convenience (the
original sum yields the prefactor $q^{N(N+1)/2}$). Interestingly, if one
changes the restrictions on the indices to $\le$ instead of $<$, which means
replacing fermions with bosons, only the prefactor changes to reflect the
different boson ground-state energy. Fermions and bosons are not really
different in one dimension, an observation whose proper generalization to
$d>1$ will become apparent below.

Euler did not compute the sum directly. Instead he presented the result
\eqref{Z1} constructively, as just one particular manipulation of the simpler
generating functions, appearing as terms in the product. He inaugurated the
theory of partitions by extending such manipulations to prove his famous
``pentagonal number theorem,'' a development\cite{Euler51,Andrews76} which
need not concern us here. We shall however need another of his observations:
upon expanding and multiplying the geometric series in the product
\begin{equation}
\frac{1}{1-e_1}\frac{1}{1-e_2}\cdots\frac{1}{1-e_N},
\label{eulergen}
\end{equation}
each distinct monomial $e_1^{n_1}e_2^{n_2}\cdots e_N^{n_N}$ appears exactly
once. While the observation itself may appear obvious enough, it marks a
historical shift towards an abstract use of mathematical symbols, in which
``letters'' are not considered as place-holders for numbers, but as
mathematical objects in their own right. The observation makes sense to one
who is prepared to use a mathematical expression like \eqref{eulergen} as a
sort of symbolic machine to generate other expressions, namely the said
monomials. The statement~\eqref{eulergen} is then a precise prescription to
manipulate symbols, not any numerical value. Here it will be used as a working
part of a bigger symbolic machine, to generate expressions for the shape wave
functions.

\subsection{Hilbert and invariants}

While a particular date may be assigned to the invention of generating
functions, invariants have been present in mathematical thought since the
earliest times. As soon as they were discovered in the 4th century BC, conic
sections were classified according to how the plane cutting the cone was
tilted: too little (ellipse), just enough (parabola), or too much (hyperbola).
The invariant nature of this classification is intuitively obvious to anyone
looking at the intersection from the outside. A significant step forward was
to realize that a small bug living in the plane could also distinguish the
three cases, by fitting the curve to the equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$
and computing the discriminant $B^2-4AC$. This program can be carried out in
two ways. One is to study a small neighborhood of a point on a curve, leading
eventually to differential geometry. The discriminant is then just the Hessian
of the quadratic form, describing the local curvature of the conic.

An alternative to the differential-geometric is the algebraic point of view,
which notes that the fitting needs to evaluate the curve only at a finite
number of points. The coefficients are now obtained by an investigation
limited in a different sense: instead of being continuous in an infinitesimal
neighborhood, it is discrete in a finitely extended neighborhood. The
polynomial is nothing but its list of coefficients, which are numbers in a
certain relationship. The relationship is best presented by keeping the old
notation for a polynomial, but now the variables $x$ and $y$ are just letters,
or notational tags, to keep track of the roles played by the coefficients,
e.g. when calculating the discriminant. Relabeling the tags by a Euclidean
transformation of the pair $(x,y)$ changes the coefficients without changing
the discriminant: it is said to be an \emph{invariant} of the quadratic form
under action of the Euclidean group in the plane. In this ``classical''
program, usually associated with Cayley, the main concern was how to find all
the invariants of a particular group action in any given case.

If Euler had opened the study of generating functions, Hilbert was widely
credited with closing the classical study of invariants, by proving a
structural result which relegated the computation of particular invariants to
applications. His attitude can be introduced in a succession of steps. First,
there is a mapping between the coefficients and roots of a polynomial, namely
if the roots are $t_i$, $i=1,\ldots N$, the coefficients are $(-1)^ke_k$,
where $e_k$ are the elementary symmetric functions,
\begin{align}
e_1&=t_1+\ldots+t_N=\sum_it_i,\nonumber\\
e_2&=t_1t_2+t_1t_3+\ldots+t_{N-1}t_N=\sum_{i<j}t_it_j,\nonumber\\
&\vdots\nonumber\\
e_N&=t_1t_2\cdots t_N.\label{ek}
\end{align}
These are just the Vi\`ete formulas, first encountered in high-school algebra.
All functions of the coefficients, including the invariants such as the
discriminant, are thus necessarily symmetric functions of the roots. Taking
the roots as the new variables, one parametrizes all the polynomials at once.
Instead of the old dummy variables $x,y$ being used as handles for group
action in the geometric ``outer'' space over which the polynomials were
defined, attention shifts to the abstract ``inner'' space of the polynomials
themselves, as parametrized by their roots. This crucial change of focus is
almost imperceptible in the notation, because of course the roots may be
visualized as points in the very same geometric space in which the original
polynomial was being evaluated. In the mathematical literature, it is not
uncommon to treat the $e_i$ themselves as free parameters, given that they
parametrize the polynomials equally well. As will appear below, in physics the
difference between the two must remain marked, because roots are related to
coefficients like one-body to many-body wave functions.

Taking the discriminant of a quadratic $x^2-e_1x+e_2$ as an example,
\begin{equation}
e_1^2-4e_2=(t_1-t_2)^2,
\end{equation}
one sees immediately that it is an invariant of the Euclidean group, which by
definition preserves distances. In fact the discriminant may in general be
defined as the square of the Vandermonde determinant\cite{Aitken39} $\Delta$,
\begin{equation}
\Delta(t_1,\ldots,t_N)\equiv
\left|\begin{matrix}
t_1^{N-1}&\cdots&t_N^{N-1}\\
\vdots&&\vdots\\
t_1&\cdots&t_N\\
1&\cdots&1
\end{matrix}\right|=
\prod_{1\le i<j\le N} (t_i-t_j).
\label{vandermonde}
\end{equation}
One immediate consequence of this point of view is that one has generated a
great many rather obvious invariants: because all functions of the
coefficients are symmetric in the roots, they are all invariant under the full
group of permutations of the $N$ roots, the symmetric group $S_N$. These may
be understood as invariants with respect to renaming the roots. They can all
be generated by Euler's expression~\eqref{eulergen}. To be explicit, the
monomials generated by~\eqref{eulergen} are a basis in the space of
invariants: any invariant of the symmetric group can be expressed as a linear
combination of these monomials, i.e.\ a polynomial in the $e_i$,
$i=1,\ldots,N$.

In the next step, obtaining ``interesting'' invariants becomes a program of
reduction of the complete monomial basis obtained above. This program cannot
miss any invariants, because \emph{any} finite group operating on the $N$
points $t_i$ can be understood as a subgroup of the permutation group, a
result due to Cayley. For example, some invariants under root reflection
$t_i\to-t_i$ are generated from the coefficients of even index $e_{2i}$, and
others from squares of coefficients of odd index, $e_{2i+1}^2$. The
corresponding reduced generating expression~\eqref{eulergen} is obtained
simply by squaring all the $e_i$ with odd indices appearing in the
denominators. However, this expression misses ``mixed'' reflection-symmetric
terms, such as $e_1e_3$. A moment's thought suffices to realize that the
generating function of all invariants of (say) cubic polynomials
$x^3-e_1x^2+e_2x-e_3$ under root reflection is just
\begin{equation}
\frac{1}{2}\left(\frac{1}{1-e_1}\frac{1}{1-e_2}\frac{1}{1-e_3}+
\frac{1}{1+e_1}\frac{1}{1-e_2}\frac{1}{1+e_3}\right)
=\frac{1+e_1e_3}{(1-e_1^2)(1-e_2)(1-e_3^2)},
\label{e123}
\end{equation}
because it lists all and only monomials which are unchanged by the change of
sign. In other words, any invariant under root reflection takes the form
\begin{equation}
R(e_1,e_2,e_3)=P(e_1^2,e_2,e_3^2)\cdot 1+Q(e_1^2,e_2,e_3^2)\cdot e_1e_3,
\label{PQ}
\end{equation}
where $P$ and $Q$ are arbitrary polynomials in three variables.

An expression like~\eqref{e123} or~\eqref{PQ} is the most general answer one
can give to a problem of finding invariants. Hilbert's conclusive contribution
was to realize not only that such expressions were always possible, but in
addition that the numerator in Eq.~\eqref{e123}, or the sum in Eq.~\eqref{PQ},
will be \emph{finite}, i.e.\ not an infinite series, for a very large class of
symmetry groups,\cite{Derksen02} so large that physicists need not be
concerned with the exceptions.

\subsection{The algebra of invariants}

The structure of Hilbert's insight is visible in Eq.~\eqref{PQ}: in some
sense, there are only two interesting invariants in the problem, namely the
set $\{1,e_1e_3\}$, listed in the numerator of Eq.~\eqref{e123}. By contrast,
$P$ and $Q$ are arbitrary, once one has identified their allowed variables.
This separation follows the natural reasoning above, which quickly identified
$e_{2i}$ and $e_{2i+1}^2$ as obvious invariants under root reflection, but was
hesitant how to include $e_1e_3$ properly. Terminologically, the monomials
appearing in the geometric series (or in $P$ and $Q$) are called
\emph{primary}, while those appearing in the polynomial in the numerator of
Eq.~\eqref{e123} are called \emph{secondary} invariants.\cite{Derksen02}

The form of the expression~\eqref{PQ} indicates that the secondary invariants
act as a basis, spanning the whole space of invariants. This insight is
formalized by noting that polynomials are a vector space over the real (or
complex) numbers, so that ordinary multiplication of polynomials is vector
multiplication in that space, therefore polynomials naturally form an algebra.
The basis vectors intuited above are just the generators of that algebra. Any
invariant $p$ may be written
\begin{equation}
p=\sum_{i=1}^Dh_ig_i,
\label{palg}
\end{equation}
where the $h_i$ are arbitrary polynomials in the primary invariants and $g_i$
are secondary invariants, or generators of the algebra of invariants. The sum
is always finite, by Hilbert, for all classical and finite groups acting on
polynomials over the complex field.\cite{Derksen02} It is said that such an
algebra is \emph{finitely generated}, where $D$ is its dimension [number of
its generators, $D=2$ in the example~\eqref{PQ}].

An additional structure can be inferred from Eq.~\eqref{palg}. Let $g_i'$,
$i=1,\ldots,M$, be some linear combinations of the generators $g_i$. Then
polynomials $q$ of the form
\begin{equation}
q=\sum_{i=1}^Mh_ig_i',
\label{ideal}
\end{equation}
with $h_i$ the same as before, form an \emph{ideal} in the space of invariants
$p$ in Eq.~\eqref{palg}. This proposition is easily checked from the
definition. Ideals are subsets closed under addition,
$q+q'=\sum(h_i+h_i')g_i'$, and absorptive under multiplication,
$h'q=\sum(h'h_i)g_i'$. An important question now is whether the $q$'s form a
proper subset of the $p$'s. Even if $M<D$, one cannot be sure that the
generators $g_i$ were really the minimal set needed to span the space of
$p$'s, because a clever choice of the $g_i'$ could in principle transfer some
of the burden to the coefficients $h_i$. In other words, because of the
flexibility of polynomial multiplication, the dimension of the algebra of
invariants can be difficult to divine in practice, leading to a large body of
research devoted to basis-reduction algorithms.\cite{Derksen02} Sometimes,
however, one can argue \emph{a fortiori} that the most efficient
representation has been achieved. Such is our running example~\eqref{PQ},
noting that the term $e_1e_3$ is even under root reflection but cannot appear
in $P$ or $Q$.

\subsection{Grading and Hilbert series}

Instead of explicitly generating all polynomial invariants, one may ask how
many of a given degree exist. Clearly each elementary symmetric function $e_k$
is of degree $k$. Hence the replacement $e_k\to q^k$ in the
example~\eqref{e123} gives the answer: the number of independent invariants of
degree $m$ is the coefficient of $q^m$ in the expression
\begin{equation}
\frac{1+q^4}{(1-q^2)^2(1-q^6)}.
\label{hilbex}
\end{equation}
Such a series is called a Hilbert (sometimes\cite{Stanley78} Poincar\'e)
series. The classification of polynomials by degree is a particular example of
a \emph{grading}, which is any decomposition of an algebra (or ring) in a
direct sum of subspaces $M_i$, labeled by natural numbers $i$, which has the
property $M_iM_j\subseteq M_{i+j}$.

The Hilbert series is an important practical tool for finding invariants,
because computing it is in principle much easier than finding a closed
invariant-generating function like Eq.~\eqref{e123}. Sometimes it gives
sufficient hints of the structure of the space of invariants to enable the
design of concrete algorithms for their construction. As we shall see now,
that is the case with many-body Hilbert space.

\section{Main results}

Having established the mathematical language, all the main results concerning
many-body Hilbert space can be stated transparently. The proofs appear
elsewhere.\cite{Sunko16-1}

\subsection{Sum over states}

The natural grading for wave functions is the number of one-body nodes. In
order to make the node-numbers independent in the various coordinates, only
bases separable in Cartesian coordinates are considered. Then the most
efficient way to represent one-body wave functions is as pure powers of some
formal variables, with the exponent denoting the grade, e.g., $t^lu^mv^n$ in
three dimensions. In one dimension, the ground-state Slater determinant then
becomes just the Vandermonde determinant~\eqref{vandermonde}. Any $N$-body
wave function can be readily recovered from this representation:
\begin{equation}
t_i^lu_j^mv_k^n\to H_l(x_i)e^{-x_i^2/2}
H_m(y_j)e^{-y_j^2/2}H_n(z_k)e^{-z_k^2/2}
\end{equation}
for the 3D harmonic oscillator, say, with $i,j,k=1,\ldots,N$. In this mapping,
the one-body functions must not be individually normalized. Only the final
superpositions are normalized before concrete calculations.

The Hilbert series for this grading is just the physical partition function of
the harmonic oscillator, because the energy is linear in the number of nodes
in that case. Thus all oscillator counting results become completely general.
The price to pay is, of course, that wave functions classified by grade are
not necessarily the eigenfunctions of any Hamiltonian, as soon as one invokes
some other one-body realization than the oscillator one, say $\cos mx$ in
place of $H_m(x)e^{-x^2/2}$ above. However, that is a very small price,
because one does not hope anyway that the initial choice of basis will solve
any non-trivial problem. Replacing $e^{-\beta\hbar\omega}$ by the letter $q$,
the Hilbert series for $N$ identical particles in $d$ dimensions
reads\cite{Sunko16-1}
\begin{equation}
Z_d(N,q)=Z_E(N,q)^dP_d(N,q),\quad Z_E=\prod_{k=1}^N\frac{1}{1-q^k},
\label{zdnq}
\end{equation}
where $P_d(N,q)$ is a polynomial in $q$ which satisfies \emph{Svrtan's
recursion}
\begin{equation}
NP_d(N,q)=\sum_{k=1}^{N}(\pm 1)^{k+1}\left[C^{N}_{k}(q)\right]^dP_d(N-k,q),
\label{recurPd}
\end{equation}
with the upper sign for bosons, and the lower for fermions. Here
\be
C^N_k(q)=\frac{(1-q^N)\cdots (1-q^{N-k+1})}{(1-q^k)}
\ee
is a polynomial, and $P_d(0,q)=P_d(1,q)=1$.

The Hilbert series~\eqref{zdnq} already reveals a lot about the structure of
$N$-body Hilbert space. In particular, the polynomial $P_d(N,q)$ counts the
$d$-dimensional states which \emph{cannot} be induced from the one-dimensional
ones by simple combinatorial tricks, such as
\be
\Psi_a(x_1,\ldots,x_N)
\to\Psi_a(x_1,\ldots,x_N)\Psi_b(y_1,\ldots,y_N)\Psi_c(z_1,\ldots,z_N).
\label{triv}
\ee
Namely, these trivial extensions of 1D states are counted by the other factor,
$Z_E^d$. Furthermore, comparing Eqs.~\eqref{e123} and~\eqref{zdnq} shows that
$P_d(N,q)$ is the numerator in these expressions, hence it counts the
secondary invariants. Thus the form of Eq.~\eqref{zdnq} identifies $N$-body
Hilbert space as a finitely generated graded algebra. Because the number of
generators of each grade [coefficient of a power of $q$ in $P_d(N,q)$] must be
a nonnegative integer, inserting $q=1$ into the recursion~\eqref{recurPd}
gives the total dimension of the algebra, $P_d(N,1)=N!^{d-1}$. For example,
in the case of three-fermion wave functions in two dimensions,
$P_2(3,q)=1+4q+q^2$ and $P_2(3,1)=6=3!^1$.

\subsection{Wave functions}

The generators of the Hilbert-space algebra are called \emph{shapes}. Any
state in Hilbert space has a wave function of the form
\be
\Psi=\sum_{i=1}^{N!^{d-1}} \Phi_i\Psi_i,
\label{genspan}
\ee
where $\Psi_i$ are the shapes, while the $\Phi_i$ are polynomials in the
primary invariants. These polynomials are trivially extended from the 1D case
along the lines of Eq.~\eqref{triv}. Explicitly, let $t_m$ denote one-node
single-particle wave functions along the $x$-axis, e.g. $t_m=e^{2i\pi x_m/L}$
with $0<x<L$ for $N$ particles in a box, $m=1,\ldots,N$. Then all the primary
invariants in the $x$-direction are generated by Euler's
expression~\eqref{eulergen}, in other words they are all unrestricted
polynomials in the elementary symmetric functions $e_k$~\eqref{ek}. Replacing
$t$ by $u$ for the $y$-direction, and so on, the independent primary
invariants are just all the monomials generated by multiplying together $d$
copies of Eq.~\eqref{eulergen}, one for each direction in space. The functions
$\Phi_i$ above are arbitrary polynomials in these monomials, with complex
coefficients. [It is most convenient, although not necessary, to interpret
powers of the $e_k$ in these monomials without cross terms, e.g.
$e_1^2(t_1,t_2)\to t_1^2+t_2^2$.]

In this context the $e_k$ are called \emph{Euler bosons}. They are the same
for bosons and for fermions. Thus the shapes $\Psi_i$ alone encode both the
space dimension and the distinction between bosons and fermions. Euler bosons
have no zero-point energy, because it is contained in the shapes, which play
the role of vacua to Euler-boson excitations.

The insights above, based on the Hilbert series~\eqref{zdnq}, are restrictive
enough to obtain the wave functions of the shapes explicitly.\cite{Sunko16-1}
The construction is recursive and exhaustive. For each grade, one first
constructs all states based on shapes of lower grade, using the
formula~\eqref{genspan} with all possible $\Phi_i\neq 1$. The shapes are then
obtained as the orthogonal complement in the Hilbert space of that grade,
which is independently generated by Slater determinants\footnote{Slater
determinants are replaced by permanents (lose the alternating sign) for
identical bosons.} in the standard way. This algorithm makes it immediately
obvious that the dimension of the Hilbert-space algebra is indeed $N!^{d-1}$,
i.e.\ the number of shapes cannot be further reduced. Namely, no shape at a
given grade can be expressed in the form~\eqref{genspan} with $\Phi_i\neq 1$,
because all the states which can be so expressed are constructed explicitly
\emph{before} taking their orthogonal complement. Of course, the
orthogonal-complement procedure defines the shapes of a given grade only up to
linear basis transformations among themselves. That freedom is trivial
theoretically, but may be important in applications.

Unfortunately the above general algorithm is quite inefficient in practice,
because it requires the computation of the complete Hilbert spaces of a given
grade, which are much larger than the number of shapes. There is a much more
efficient algorithm, which is however only applicable to fermions in odd
dimensions.\cite{Sunko16-3}

\section{Discussion}

The developments reviewed above are quite recent, so that the present
discussion is necessarily more prospective than retrospective. There seems to
be more work than any one person or group can do to explore all possible
ramifications of the shape paradigm. Here only three which come to mind
immediately are briefly mentioned.

\subsection{Slater determinants in finite systems}

While Slater determinants still appear as an auxiliary tool in the algorithm
which constructs shapes, one feels that they have become redundant in physical
descriptions, at least of finite systems. There is a mathematical way to
express this feeling. Specialize to one dimension, where the only shape is the
ground-state Slater determinant $\Psi_0$. One may ask, what is $\Phi$ in the
expression $\Psi=\Phi\Psi_0$, if $\Psi$ is an excited-state Slater
determinant? We have seen above that the most convenient way to write $\Phi$
is in terms of Euler bosons, but then $\Psi$ is \emph{not} a Slater
determinant, in general. If it is, then the ratio $\Psi/\Psi_0$ is called a
Schur function, which is the generating function of non-standard Young
tableaux.\cite{Stanley99} The physical meaning of such an object is quite
obscure, and so is, by extension, the physical meaning of a generic Slater
determinant. By contrast, the elementary symmetric functions (Euler bosons)
have simple physical connotations in general. Thus $e_1$ in Eq.~\eqref{ek} is
a sum of one-body wave functions, which is physically a liquid, while $e_N$ is
their product, physically a gas.

While the lowest-graded shapes are always just the Slater determinants of the
(possibly degenerate) ground state, for $d>1$ shapes are generally
superpositions of Slater determinants with roughly equal amplitudes. With
complexity increasing by grade, they are the natural choices for strongly
correlated trial wave functions. Conversely, if the DFT approach gives
realistic one-body wave functions, the shape paradigm reduces to the
Slater-determinant one, at least for the ground state. Even in that case,
shapes may increase the scope of calculations of excited states and related
radiation transitions, which are an independent indicator of how realistic the
wave function is.

\subsection{Bands in spectra}

A spectrum is organized into bands when states of the same band are easily
connected by excitations, while states of different bands are not, i.e.\ they
behave as if they were of qualitatively different kinds. The lowest state in
each band is called a bandhead. There is a natural physical picture behind
this scheme: bandheads correspond to different configuratons of particles in
laboratory space. Some of these may be easy to excite by vibration, others by
rotation, while it is difficult to change a configuration from one prone to
vibration to another, prone to rotation.

The abstract structure of ideals of many-body Hilbert space, described in
Eq.~\eqref{ideal}, matches the above physical pattern. The band may be
understood mathematically as the ideal generated by the bandhead. The ideal is
closed under wave-function superposition, and excitations within it can be
described by multiplying the bandhead with Euler bosons. Physically, this
multiplication corresponds to bosonic excitations of the bandhead. In this way
one can distinguish theoretically between excitations within a band, which are
bosons, and excitations from one band to another, which involve
reconfigurations of the bandhead. The latter cannot be obtained
multiplicatively, because shapes are like prime numbers, $\Psi\neq\Phi\Psi'$
for any two distinct shapes $\Psi$ and $\Psi'$: one is not divisible by the
other.

In order for such a mathematical interpretation to have physical meaning, the
interaction matrix elements between the shapes should be small. This
proposition has been checked\cite{Sunko16-1} in small numerical examples in
$d=2$ and $d=3$ with the Coulomb interaction $V_C$, for a harmonic oscillator
potential (spin-polarized electrons in a quantum dot). It has been found that
$\left<\Psi_i\right|V_C\left|\Psi_j\right>\ll
\left<S_i\right|V_C\left|S_j\right>$, typically by about two orders of
magnitude, between any two shapes $\Psi_{i,j}$ and any two Slater determinants
$S_{i,j}$ of the same grade. This result is of course just a sanity check, as
the opposite outcome would have doomed the interpretation immediately.

The ideal structure explains why the original idea of Bloch\cite{Bloch34} to
represent excitations of fermionic systems as density waves was generically
successful only in one dimension.\cite{Tomonaga50} Bosonisation corresponds to
multiplying the ground state by a symmetric function. It can describe all
excitations in one dimension, where there is only one shape, the ground-state
Slater determinant, or all within one band for $d>1$. In general, however,
excitations may also involve a reconfiguration of the ground state, which is
an \emph{additive} correction as explained above, e.g.
$\Psi\to\Psi+\Phi\Psi'$, where $\Psi$ and $\Psi'$ are ``relatively prime.''
Thus the difficulty in representing excitations of fermionic systems by bosons
appears as just another case of the incompatibility between addition and
multiplication, ubiquitous throughout mathematics, of which, say, translation
and rotation are a particularly well-known example.

\subsection{The fermion sign problem}

The fermion sign problem\cite{Loh90} appears in simulations of fermionic
many-body systems in real space. It is most easily understood in a
sum-over-histories formulation. When a transition matrix element is formed as
a sum over trajectories in fermion many-body space, one has to distinguish
contributions which interfere destructively for physical reasons, from those
which should interfere constructively but have opposite signs because a phase
convention for the many-body wave function has not been implemented
consistently across many incremental updates of different
trajectories.\cite{Iglovikov15}

The sign problem appears only because there is more than one shape. In one
dimension, where there is only one shape $\Psi_0$, the factorization
$\Psi=\Phi\Psi_0$ immediately translates the fermionic problem into a bosonic
one involving the symmetric functions $\Phi$. Physically, in one dimension it
is possible to fix a phase convention simply by forbidding particles to skip
over each other in a simulation. If they never exchange places, the difference
between bosons and fermions cannot manifest itself either, so the orbital
solution is actually a symmetric function for both fermions and bosons, the
difference between the two cases being in the fixed factor $\Psi_0$.

The representation~\eqref{genspan} provides a generalization of this insight
to $d>1$. Instead of writing the wave function as in~\eqref{genspan}, one can
list the coefficients $\Phi_i$ as a vector-like object,
\be
\left(\Phi_1,\Phi_2,\ldots,\Phi_{N!^{d-1}}\right).
\label{fmod}
\ee
When the $\Phi_i$ are symmetric polynomials, such a structure is called a
\emph{free module} (as distinct from a vector space, where the $\Phi_i$ would
be just numbers). It carries the idea that the shapes are a basis to its
natural conclusion, proposing to calculate in the space~\eqref{fmod} of
``coordinates'' $\Phi_i$ instead of directly in the space~\eqref{genspan} of
wave functions $\Psi$. Calculations in the free module have no sign problem
\emph{a priori}, because the $\Phi_i$ are symmetric functions. It remains to
be seen whether they will be practical. At least, there is a chance to manage
complexity, because one can imagine\cite{Sunko16-1} truncating the
list~\eqref{fmod} to limit the calculation to one ideal, or band of
excitations. Such an approach is similar in spirit to a fixed-node
approximation\cite{Bressanini12,Filinov14} in usual simulations.

The proposal to use the free module in order to avoid the sign problem can be
placed in the more general setting of \emph{constraint
elimination}.\cite{Derksen02} In some design problems constraints can be
encoded as requirements that certain polynomials $g_i$ simultaneously evaluate
to zero, leading to the expression of allowed solutions in the
form~\eqref{palg}. Eliminating constraints means mapping the original space of
the problem to some reduced space in which they are satisfied automatically.
The whole geometric locus of zeros in the original space (\emph{kernel} of
this mapping) maps to a single point, zero, in the reduced space. This is the
case when wave functions are mapped to the free module: $\Psi=0$ in
Eq.~\eqref{genspan} corresponds to $(0,0,\ldots,0)$ in Eq.~\eqref{fmod}.

Hilbert's \emph{Nullstellensatz}\cite{Derksen02} (``zeroplacesclaim'') is an
either/or proposition about constraint elimination: either the constraints
$g_i=0$, $i=1,\ldots,D$, are incompatible, or every solution $f$ of the
problem can be written as an algebraic root of the expression~\eqref{palg},
$f^m=p$ with an integer $m\ge 1$. The many-body problem is an interesting
variation of this general framework. Normally the simultaneous solutions of a
set of polynomial equations are a discrete set of points, called an
\emph{algebraic variety}. Each shape vanishes according to the Pauli principle
when $\vec{r}_i=\vec{r}_j$ with $i\neq j$ for any two particles,
$i,j=1,\ldots,N$. These points of coincidence are the simultaneous zeros of
the shapes, corresponding to the algebraic variety of the problem. However, as
noticed long ago in the many-body context,\cite{Ceperley91} when $d>1$ the
algebraic variety is a subset of a larger object, the \emph{nodal surface} of
the total wave function $\Psi$. The nodal surface is a (possibly
multiply-connected) differentiable manifold defined by $\Psi=0$, on which each
individual shape need not vanish. It appears only because the Pauli principle
is triggered by an equality of \emph{vector} variables, a weaker condition for
zeros than explicitly envisaged by Hilbert. The nodal surface is the reason
for the practical difficulty in defining the wave-function phase consistently
in simulations, because it is not known in advance. Thus it is of considerable
interest to understand its relation to the algebraic variety. Of course, a
polynomial like~\eqref{palg} will in general have more zeros than there are
common zeros of the $g_i$. However, for $d=1$ the extra zeros are discrete and
separate from the algebraic variety. The specific feature of the problem for
$d>1$ is that the algebraic variety naturally generates a whole
differential-geometric ``zero object,'' the nodal surface, which contains it.

The free module~\eqref{fmod} is a structured implementation of Heisenberg's
matrix mechanics, which refines Dirac's vector-space interpretation.
Heisenberg's algebraic approach\cite{Heisenberg26} eschewed wave-function
\emph{arguments} --- real-space coordinates --- in favor of wave-function
\emph{values}, as encoded by the matrix elements. The textbook harmonic
oscillator is a one-body realization of this program, where replacing $x\pm
d/dx$ by the creation and destruction operators faithfully maps laboratory
space onto wave-function space, so that one never needs to revert to the
variable $x$ for concrete calculation. In close analogy, the $\Phi_i$ are
polynomials in the elementary symmetric functions~\eqref{ek}, which are
themselves expressed directly in terms of the \emph{values} of the one-body
wave functions. Changing powers of the $e_i$ in this formalism corresponds to
harmonic-oscillator ladder operations. The representation~\eqref{fmod} of the
free module thus opens the possibility of practically implementing
Heisenberg's approach in a general many-body setting. This program has a much
larger scope than the motivating fermion sign problem.

\section{Conclusions}

The general algebraic structure of many-body Hilbert space described in this
review provides a firm mathematical foundation to study the physics of finite
systems in a pleasantly intuitive language, with suggestive correspondences
between mathematical structures and observations. Physical vacua appear
mathematically as generators of ideals in the algebra of many-body invariants,
while these ideals physically correspond to bands in the excitation spectra.
Although the required mathematics far exceeds contemporary training of physics
students in abstract algebra, such is remarkably not the case with students of
engineering. Industrial design optimization uses engineering constraints as
generators of allowed solutions in much the same way as shapes are used
here.\cite{Lv12} Such a large body of practical knowledge gives rise to
optimism for the proposed physical applications.

At the same time, specifics of the physical many-body problem should not be
overlooked. In particular, the arrangement of $Nd$ variables in $N$ columns
(vectors) of $d$ rows, however natural when describing particles, does not
seem to have been pursued in the mathematical literature on invariants.
Monographs\cite{Derksen02} and textbooks\cite{Stanley99} typically treat
all variables on an equal footing, corresponding to the many-body problem in
one dimension. Due to this additional structure in the variables when $d>1$,
the shape invariants described here appear to be new to mathematics as well,
although the author cannot claim so with authority.

Perhaps the most interesting insight based on the shape paradigm at present is
the reinterpretation of matrix mechanics in terms of a free module rather than
a vector space. It is tantalizing to speculate how far the young Heisenberg
would have pursued his algebraic approach if Schr\"odinger's analytic work had
never appeared, or if Hilbert or Noether had been the one to search for their
common ground. Hopefully the present brief review will encourage readers to
explore shapes from the point of view of their own expertise.

\section*{Acknowledgements}

I thank D.~Svrtan for his help and interest. This work was supported by the
Croatian Ministry of Science grant 119-1191458-0512 and by the University of
Zagreb grant 202759.

redpajama_set_name: RedPajamaArXiv


text:
Archbp. Chaput: "We are Catholics before we are Americans"
Posted on 25 October 2012 by Fr. John Zuhlsdorf
More from Archbp. Chaput:
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This entry was posted in "How To..." - Practical Notes, Cri de Coeur, Our Catholic Identity, Religious Liberty, The future and our choices, Year of Faith and tagged abortion, Archbp. Chaput, Catholic and Faithful American and Free, Catholic Democrats, Democrats. Bookmark the permalink.
Card. Burke's contribution at the Synod of Bishops: "antinomianism …infected post-Council ecclesial life"
Of processions, flower girls, quzzing altar boys, and the use of the biretta outside
dnicoll says:
And in England during the Reformation that is why we were considered Traitors.
I agree 100%. I am Catholic… then I am American. Or should I say, I am an American Catholic, not a Catholic American. The American part can (though I hope it never will) change. The Catholic will NEVER change.
We say "God and Country", we do not say "Country and God."
Texana says:
"(Abortion) is a very serious issue that requires absolute adherence on the part of Catholics, and if we don't stand united on this issue we're bound to failure…". Because it is an intrinsic evil, abortion must not be supported by Catholics as legal in the instances of rape, incest, and the physical/psychological health of the mother.
SKAY says:
Obama is now talking about his ability to nominate judges to the Supreme Court who will uphold Roe v Wade so that abortion on demand will continue—if he is re-elected. Apparently he is trying to recapture the lead he had with the womens vote.
For any Catholic it is clear that a vote for Obama is a vote for abortion and the HHS mandate that will eliminate our first ammendment freedom of religion.
Forty years ago Brent Bozell was being excoriated for saying the same thing (albeit a bit more strongly) in the pages of Triumph. Are American Catholics ready for it now?
fib09002 says:
Muslims say this kind of thing all the time, that their faith comes before their national identity. Which is why we frequently hear of all these "Americans" joining terrorist groups in Somalia and elsewhere. Obviously it is considered "Islamophobia" to point this out, however. When a Catholic says the same thing, though, we can expect him to endure some criticism.
frjim4321 says:
Wow, he really tipped his hand and revealed himself as an in-the-tank-for-the-RNC zealot when he referred wrongly to the "democrat" party rather than the proper term, "Democratic" party. He either does not know the proper nomenclature (doubtful) or his has embraced the insulting practice of purposely not calling his partisan competitors by their proper name.
I don't know how he would expect any Catholic member of the Democratic party to take him at all seriously. I certainly don't
Further he makes the same error as do some of his cohort by failing to properly understand the nature of political discourse and putting more stock in what people say about abortion versus the actual incidence of abortion.
I am so glad he is not my ordinary.
Okay, that might have been overly harsh and I retract it.
If he were my ordinary, I don't think we would get along very well.
David Collins says:
Democrat party or democratic party: only an in-the-tank-DNC-zealot would think there's an important difference between the two.
How can Americans have a national identity? A person's family can be from any where and still be American. You can have any religion or no religion at all and still be American. You can grow up speaking Urdu in the home and still be an American. There is no historical memory to speak of. So being American is nothing more than what government do you pay taxes to.
David, (1) I'm not getting up there using twisted logic to get people to vote for my candidate, (2) it's pretty clear what the name of the party is, just look at the website. It's the Democratic National Committee, and the challengers have been playing this game for a long time purposely not using the right name.
I don't know if we will have a clear winner on the morning after but (1) I certainly hope SCOTUS doesn't have a coronation this year and (2) if my candidate is the winner it will be hard for me to be sportsmanlike after all the dirty tricks and disrespect from the other side.
Southern Catholic says:
You continue to use this twisted logic to justify your pro-death position, even though you have been called out on it multiple times by Father Z and other posters on here. My question to you, if you truly are a priest, how can you morally and consentingly support a position that is taught to be instinctively evil by the Church you claim to serve?
b/c I don't love everything about the President, in fact I know that both candidates are far from perfect. Nevertheless I see the President as being the lesser of two evils; I give the challenger no points for having the upper hand with regard to morality because (1) he has changed his position regarding choice in the past, (2) he has changed his position on a raft of other issues recently and (3) the RNC has bent over backwards for the past two years trying make sure the poor, blacks and the young would not be able to vote. The flagrant efforts of the RNC to corrupt the democratic process disqualifies them from claiming the ethical high ground in this election.
Suburbanbanshee says:
26 October 2012 at 12:53 AM
I'm distressed that someone would nitpick the local dialectal speech of Archbishop Chaput. I'm also surprised that anybody would not know that certain regions in this country have the dialectal habit of referring to it as the Democrat Party, including lifelong Democrats of yellow-dog loyalty. The -ic part is more easily swallowed and elided than pronounced, in these dialectal areas. But alas, such is the lot of the undiscovered America between the two coasts.
Frankly, if a dialect has to choose between "Democrat Party" and "Democrack Party," I think the more flattering elision is quite clear.
Phil_NL says:
26 October 2012 at 3:06 AM
@Suburbanbanshee
Yes. The more accurate one is very much debatable though.
OK, Fr. Jim, OK. It just seems nitpicking to insist on Democratic party.
It is certainly insane to think Romney is really means it when he says he is pro-life. How convenient it was for him to have a road-to-Damascus experience prior to running in Republican primaries.
If Romney wins, it will probably be, given the changing demographics of the US, the last hurrah for the Grand Old Party.
ckdexterhaven says:
I am shocked that a Catholic priest is supporting a political party that spent their whole convention as a tribute to abortion. A political party that purposely left God out of their platform. This party didn't even want the word God in the platform.
The kill lists that have killed hundreds of innocent Pakistanis, talk about disenfranchised minorities…. Seems to bother not at all.
FrJim, maybe Archbishop Chaput was talking to Democratics as well as Republicans?
JonPatrick says:
I guess I have a hard time understanding how anyone can support a politician who is so overtly pro-abortion. To me this is like the Christians in the 18th and 19th century that justified slavery. It is not just one of many issues, it stands alone as it is about whether or not a human being is allowed to live. The ultimate disenfranchisement.
One thing that I find frustrating about people in the secular world (and unfortunately some in the Catholic world also such as Fr. Jim) is the tendency to see everything as a political issue, left vs. right. In fact Catholic doctrine aligns with some "left wing" stands such as caring for the poor, as well as "right wing" stands such as pro-life, pro-traditional marriage. The Church should be above aligning with any particular party, because neither party perfectly aligns with Catholic doctrine, plus the fact that the realities of political life and the compromises that are perceived to be necessary in a democratic society in order to get elected and to govern, causes most politicians to vary from even the compromised (from a Catholic standpoint) positions that they take.
In the past the bishops for various reasons ended up aligning themselves more with the Democratic party due to the perceived good by working with government programs. They now see that tying themselves to that party meant also tying themselves to social policies that violate key Catholic doctrine, and are attempting to return to a more politically neutral stand, which is interpreted by those who see things only through a political lens as siding with the conservatives.
Imrahil says:
Well, dear @JonPatrick, whatever to be said about seeing things as a political issue (my own concept is the 1. a thing becomes political the instant it is made an issue by campaigners and politicians, 2. that even about political things there may be absolute truth), caring for the poor is not left wing; and being pro-life is only a very accidental position of present right-wing parties, who justly never have lost the wonder about this being an issue at all.
Caring for the poor is, if anything, right-wing, because the words as they stand would speak about "caring", a personal thing, and "the poor", a quite politically incorrect expression. Providing for the lower classes would be the left-wing thing, of course (by which I do not mean necessarily an evil thing); and especially (though I do not buy into the doctrine that all left-wingism is eo ipso Socialism, and this would be) the background thought that all citizens should in effect be civil-servants.
It is the regrettable fact (but not a necessity, and even less something to be accepted) that the parties do not all of them differ somewhat from Catholic doctrine. But that means not that the Catholic Church is above parties (except, of course, in rank of her mission, holiness, etc.), especially as that breeds the other thought that if a thing is contested politically, both positions would be of equal worth to the Church. It means that the Catholic Church is all with this party on this point and all with that on the other – of course only as far as the accordance goes.
And even aligning with a party is sometimes a natural event. I see the Catholic Church in America aligned to the Republican Party for all foreseeable time (which, technically, equals zero, as "prognoses are hard especially if they concern the future", it's said; but you see what I mean).
robtbrown says:
kdexterhaven says:
I am shocked that a Catholic priest is supporting a political party that spent their whole convention as a tribute to abortion.
Ever hear of the Stockholm Syndrome?
Marion Ancilla Mariae says:
"As for me and my house, we will serve the Lord." Joshua 24:15
When many of the people of England (16th c.) chose to follow their monarch into rebellion against Christ's representative on Earth, and therefore against Christ Himself, it was only to be expected that they would attack the Catholics who remained faithful to the Pope and to the Lord.
"Traitors," "not loyal Englishmen," "Papists," etc. Many English and Irish men and women, while loyal to their homelands, but to God first, willingly shed their blood rather than to reverse these two allegiances.
Are American Catholics of today willing to uphold the same standard that the Catholic martyrs of the Reformation died for? No compromise when it comes to the Law of God.
"God and country." God first. Country second.
wmeyer says:
Nevertheless I see the President as being the lesser of two evils…
Which alone calls into question your reasoning, and your commitment to Church teaching. Obama is the most determined pro-abortion president in history, and is bent on the destruction of our economy, as well.
frjim
"RNC has bent over backwards for the past two years trying make sure the poor, blacks and the young would not be able to vote." How?
Is it your belief that non citizens should be able to vote in our elections in this country?State ID's are free in my state.
"(1) he has changed his position regarding choice in the past"
So did I. Since it happened to me–I know that it CAN and DOES happen and for a v. How sad that – as a priest-that is your view but It does sound like the Democrat/Democratic(whatever) talking points.
"The flagrant efforts of the RNC to corrupt the democratic process disqualifies them from claiming the ethical high ground in this election."
Since you were unaware of the attempt by the Black Panthers to intimidate voters during the last
election you are probably not keeping up with the Democrat/Democratic(whatever) activities to corrupt the democratic process.
Full disclosure–I was a Democra(voted for some Democrats)t until I realized how undemocratic(corrupt) they had become. Yes–another change. I don't agree with all Republican all the time–but it is clear where the Democrat/Democratic(whatever) party and Obama are going and it has nothing to do with freedom.
Sorry –I did not finish the sentence.
–I know that it CAN and DOES happen and for a variety of reasons.
Wow, for the record, I did not hear Archbishop Chaput say anything about a Catholic having to be in the tank for either party. He did say that partisanship can be a danger to the Catholic faith regardless of which party one associates with and that our loyalty is with Christ and the Church. While a Democrat Governor in New York pushed for homosexual so-called marriage in New York, it did not pass without the votes of Republicans and Catholic Republican representatives at the state level.
http://www.christianpost.com/news/new-york-senate-approves-gay-marriage-bill-51559/
For the record, I am not a partisan voter, I am a discriminate voter. I often wish there were no R or D after a purported representatives name. I remember this being proposed in a state, at the state level, but being decried because people would not know how to vote. Wow!
Giuseppe says:
In response to the silly-sounding "Democrat" Party, a neighbor has starting calling the other party the Republic Party. Mind you, she supports the Republic candidate Mitt Romney. But she has gotten people to say the correct name of the Democratic Party. Despite how much one may loathe the opposition, why not call them by their proper name?
This election is going to be close, Congress may still be split, and I fear that neither candidate can temper the loathing and lack of cooperation that exists between die-hard members of each party.
Late for heaven says:
It occurs to me that despite any and all political structures that may be voted on or imposed by fiat, from the point of view of saving my soul it is still MY personal obligation to care for the needy. Nor is there ANY political structure that could prevent me from fulfilling my obligation to those in need. I am all in favor of the efficiencies that federal programs can often provide, but such programs do not substitute for my own acts of mercy. Nor can they address the more important imperative to save souls. It is undeniable that there is far less poverty in this country now than 50 years ago for example, but also evident that we have lost many many souls during the same time.
The first thing anyone needs is life itself, our heavenly gift of being. The primary act of mercy is to protect life.
As for the actual numbers of abortions I believe that both sides agree on that data. And if you are referring to the effects of abortion on the mother, there is now much damning evidence about that as well.
Daniel Hamiche says:
Thank you Archbishop Chaput — with such a French flavored name… As said St Joan of Arch « Our Lord is first served » (by the way Joan was borned in 1412, that's six centuries ago… Please keep praying for France as we are facing dangerous challenges).
happyCatholic says:
First, frjim's comment is ironic, because on another message board a poster excoriated a journalist who used the term "Democratic Party" as being completely ignorant, as the proper term was "Democrat Party." A quick google search on my part found support for using either term.
If any of the erudite :-) posters here are so inclined, could you possibly add some comments to this article on the WaPo website? The vitriol and bigotry over there spun me up so much yesterday that I signed up both on that board (pretty painless) as well as signing up here (thanks, Father Z!) so as to defend (over there) and recruit (over here) some who will defend Holy Mother Church. I was so spun up I "replied" to so many posts on the WaPo website that I think it would strengthen the replies if it were not just coming from me! There was a smattering of a few other posts defending Archbishop Chaput, too. But seriously, folks, these people want to strip the Archbishop (and us) of our citizenship. And their failure in historical knowledge and logical thinking is at times breathtaking — and I don't think mine is all that stellar. However, my thinking is while, many of those posters over there may be a lost cause, readers who might be a bit more open-minded may be persuadable. I have seen that happen on other boards where people of good will were actually open to being persuaded by patient but firm explications of the truth.
And with all due respect, maybe I am not asking for frjim to answer this appeal for "posting help." ;-) I think his views may already be well represented there.
frjim4321 said: " the RNC has bent over backwards for the past two years trying make sure the poor, blacks and the young would not be able to vote"
Father Jim, I've seen you make this claim repeatedly, and yet, you never cited a single piece of evidence as proof. Isn't it uncharitable to make false accusations? Exactly what has the Republican National Committee done to "make sure" anyone can't vote?
catholicmidwest says:
David Collins,
Maybe you don't have an American identity but I do. I'm descended from colonists that fought the Revolutionary War on both sides of my family. This may be trivial to you, but not to many of the oldest American families. That said, it's not the same thing now to say you're American that it was just a few short years ago. The condition of the country is disgraceful. The country lies face down in the gutter. It's very painful to watch.
It's also very painful to have to make a choice between being Catholic and being American. It's not like we will run off to some other country though. Patriots stand their ground and do the right thing. A person can be both Catholic and American and that must always be the case. Otherwise America no longer exists, and this is just real estate.
frjim4321,
It's too bad Archbishop Chaput isn't your ordinary.
Father Jim, I've seen you make this claim repeatedly, and yet, you never cited a single piece of evidence as proof.
On the other hand, Sissy, it's the same level of proof he offered on the evils of the RNC. I'm no big fan of either party at this point, but with the Republicans, we have some chance to undo the damage done; with the Dems, the damage will increase.
wmeyer, we have also asked for and never receive any evidence that supports Fr. Jim's repeated claims that there will be fewer abortions under the Obama regime. He claims he is basing his support for Obama on this slender reed -that fewer children will be murdered during an Obama administration – a suggestion that is preposterous. I can't comprehend that a priest could continue to support this man.
Father Jim, I've seen you make this claim repeatedly, and yet, you never cited a single piece of evidence as proof. Sissy
You must be kidding . . . it's on the news every day. The Speaker of the House of PA is even on video bragging that voter suppression there will assure a Romney win!
For more: http://www.aclu.org/maps/battle-protect-ballot-voter-suppression-measures-passed-2011
we have also asked for and never receive any evidence that supports Fr. Jim's repeated claims that there will be fewer abortions under the Obama regime.
Perhaps he's simply observed that in the last 20 years, there has been a gradual decline in abortions in the USA: http://www.guttmacher.org/pubs/fb_induced_abortion.html
But that decline seems to have ended: http://www.mccl.org/us-abortion-stats.html
Clinton R. says:
Amen! God will not judge us on our national origin, but He will judge His people on how they followed His teachings and that of His Holy Church.
Father Jim, Illinois is a Democrat-run state. Chicago has long been governed by the Democrat machine. Yet, a poll worker in Chicago asked President Obama to show photo ID in order to vote yesterday. Was that an attempt to suppress a minority from voting? Why aren't you exercised about Democrats doing that? What is your evidence that asking anyone for identification is voter suppression? Just saying it doesn't make it so. Are you similarly concerned about US citizens being asked for photo ID to buy a handgun? Is that racist?
Voting is almost the only legal action a citizen can take without photo ID. These same people for whom it "would be a hardship" to obtain a free photo ID almost certainly have done other things for which they were required to produce such ID.
The rationale against requiring photo ID for voters is so strained as to be utterly preposterous, except to the mind of a liberal.
It's too bad Archbishop Chaput isn't your ordinary. – Cmw
Tho I would not be a good fit in that see. A bit right of center for me in the 70's.
And still!
My posts are prob looking bad – learning a new Device. Learning curve
bookworm says:
"a poll worker in Chicago asked President Obama to show photo ID in order to vote yesterday."
In Illinois you have to show photo ID to vote EARLY, but it is not required for voting on Election Day itself. I've never been asked for photo ID when I vote on Election Day; the election judges just ask my name and check to see whether it is on the rolls for my precinct. If I vote early (which I did 2 years ago, and plan to do again this year) I must go to the local courthouse and there I am asked to show ID.
(1) he has changed his position regarding choice in the past, (2) he has changed his position on a raft of other issues recently and (3) the RNC has bent over backwards for the past two years trying make sure the poor, blacks and the young would not be able to vote.
(1) So? Obama has been the greatest abortion supporter of all presidents, and has increased the funding to those in the business of providing abortions. The only way Romney could be worse is if he rounded up and killed the first born of all Americans. (2) Diversion from the issue at hand and (3) another diversion. If you seriously believe that "voter suppression" is on the level or greater than abortion, you clearly should evaluate your conscious and what the church teaches about forming said conscious.
Dear catholicmidwest, it is not trivial to me that your family goes all the way back to the Revolutionary War. I mourn the loss of American identity.
Otherwise America no longer exists, and this is just real estate. Yes, indeed. God bless you sir in your effort to remain a Catholic American.
bookworm said: "In Illinois you have to show photo ID to vote EARLY, but it is not required for voting on Election Day itself. I've never been asked for photo ID when I vote on Election Day;"
Well, thanks for that clarification, bookworm. So, only those minorities voting early experience voter suppression. How can Democrats permit this?????
"Patriots stand their ground and do the right thing. A person can be both Catholic and American and that must always be the case."
That is true. However, does this mean that the many Religious and lay persons who emigrated to America from their home countries to escape persecution, oppression, or just lack of economic opportunity were NOT doing the right thing? If it was OK for our ancestors to flee when things got bad, why would it not be OK for us to do so — though I'm not sure where we would go.
frjim-In reference to your link–it would be nice if the ACLU cared as much about Democrat voter fraud .
'Bridgeport Mayor Bill Finch is at it again. The Democrat who presided over a 2010 vote casting and counting scandal in the tight race for governor was caught on video Friday guaranteeing Democratic U.S. Senate candidate Christopher Murphy a big result in the state's largest city–even if it takes days to count.
Bridgeport provided a crucial and dubious margin of victory to Democratic and Working Families Governor Dannel P. Malloy in 2010 amidst unprecedented chaos at the city's polling places. The 2010 Bridgeport assault on democracy included photocopied ballots, altered hours at polling places, a mysterious bag of votes and Finch's abuse of the city's emergency notification system to increase turnout on Election Day.
Murphy, locked in a dead heat with Republican Linda McMahon, needs the grimy Bridgeport Guarantee. He got it from Finch on Friday while visiting small businesses in the poverty-wracked city. A laughing Finch boasts even if it takes a couples of days to get the results,"You can be guaranteed you're going to get the vote." Two years after the 2010 fiasco, it remains no laughing matter to the rest of Connecticut."
Gee, I hope the ACLU is on the case in Ohio. This Republican voter fraud must be investigated.
http://www.humanevents.com/2012/10/26/is-voter-fraud-being-committed-in-ohio/
One wonders if they are even citizens, if they don't speak English. But I hope we don't disenfranchise them by asking for an I.D. Consarn it! These dastardly Republicans must be stopped.
COLUMBUS, Ohio — Two volunteer poll workers at an Ohio voting station told Human Events that they observed van loads of Ohio residents born in Somalia — the state is home to the second-largest Somali population in the United States — being driven to the voting station and guided by Democratic interpreters on the voting process. No Republican interpreters were present, according to these volunteers.A source, who wishes to remain anonymous, is a volunteer outside the Morse Road polling center. She has witnessed Somalis who cannot speak English come to the polling center. They are brought in groups, by van or bus. The Democrats hand them a slate card and say, "vote Brown all the way down." Given that Sherrod Brown is the incumbent Democrat Senator in Ohio, one can assume that this is the reference.

redpajama_set_name: RedPajamaCommonCrawl


text:
Ixigo appoints 6 Independent Directors on its board
The new independent members of the board are Arun Seth, Mahendra Pratap Mall, Shubha Rao Mayya, Frederic Lalonde, Rahul Pandit, and Rajesh Sawhney, according to a statement.
In a Twitter post, Aloke Bajpai, Group CEO and Co-Founder, ixigo, said, "We are thrilled to welcome 6 new independent board members bringing rich, diverse and relevant experience to our Board. Together, we will take our iconic travel-tech offerings to the next level !"
Travel app ixigo (Le Travenues Technology) on Monday announced the appointment of six independent members to its board of directors.
Arun Seth was associated with Alcatel-Lucent India as a non-executive chairman from May 2011 until May 2014. Previously, he has worked for the BT Group in India in a variety of positions for over 17 years, retiring in July 2012. He also serves as an Independent Director on the Boards of Jubilant Pharmova, Jubliant Ingrevia, Usha Breco and Sify Technologies. He is also a trustee of the Nasscom Foundation.
Mahendra Pratap Mall was an officer of the Indian Railway Accounts Service (1984 batch). Until recently, he was serving as the Chairman and Managing Director of the Indian Railway Catering and Tourism Corporation (IRCTC). Before that, he was working with IRCTC as a Director of Finance. Before joining IRCTC, he was working as an executive director, Accounting Information Management System, Ministry of Railways.
Shubha Rao Mayya has 30 years of experience in the banking and insurance sector. Previously, she served as Vice President at ICICI, Senior Vice President and Head – CSO branch operations at ICICI Prudential Life Insurance Company and as General Manager at Tata Consultancy Services (formerly known as TCS eServe). She also serves as an Independent Director on the Boards of Happiest Minds Technologies, Ace Manufacturing System, and Stovekraft.
Rahul Pandit is MD and CEO at Hamstede Living. He was a founding team member, President and Executive Director of Lemon Tree Hotels. He subsequently led the performance turnaround at Tata Group's Ginger Hotels as its MD and CEO. Before that, he has worked and trained across India, the USA and Singapore with Choices Hotels, Taj Hotels and Intercontinental Hotels. He also serves as a Member of the National Tourism Committee – Confederation of India Industry.
Rajesh Sawhney is the Co-Founder of InnerChef and founder of GSF. He serves on multiple boards, including that of Indiamart and advises Trident Capital.
Frederic Lalonde has been associated with Le Travenues Technology since 2011. He is the founder and CEO of Hopper and previously co-founded Newtrade, which was acquired by Expedia in 2002. Before that, he has also served as a vice president at Expedia, where he was responsible for developing the Direct Connect product strategy and Expedia M&A activities.
Launched in 2007, ixigo is an AI-based travel app based in Gurugram, India. Founded by Aloke Bajpai and Rajnish Kumar, ixigo is backed by affiliates of Sequoia Capital and Elevation Capital among other investors.
Travel and Hospitality News
ixigo news
Ixigo Independent Directors
Toyota to replace Akio Toyoda as president and CEO
Provident Financial CEO to step down
How HRs can welcome laid-off employees, what strategies to follow, and advantages & disadvantages of getting them aboard
World Entertainment Forum
Scalability and data breach are the biggest HR tech concerns, says DXC Technology HR Head
Is the HR department still playing the administrative role?
Batting Order: Tapping the Right Talent at the Right Time
We Talk Talent Season 3 Episode 28 | Thales India
ETHRWorld in conversation with Sekhar Sahay, Head of HR, Thales India #ETHRWTT
We Talk Talent Season 3 Episode 27 | Deloitte India
We Talk Talent Season 3 Episode 26 | Nagarro
We Talk Talent Season 3 Episode 25 | Khatabook


text:
Q: Why EL function not working in this case? I am working with Java 1.7, Eclipse Luna and JBoss 4.2.3.
I created a simple Dynamic Web Project named "Test001_WAR". In it:
I created a StringUtil class where I define a public static method:
package com.srh.base.util;

public final class StringUtil {

    public static String notNullTrim(String string) {
        System.out.println("StringUtil.java: notNullTrim(String): string=" + string);
        if (string == null) {
            return "";
        } else {
            return string.trim();
        }
    }

}

I created a myfunction.tld file where I mapped the above Java method to an EL function:
<?xml version="1.0" encoding="UTF-8"?>
<taglib version="2.1"
        xmlns="http://java.sun.com/xml/ns/javaee"
        xmlns:xml="http://www.w3.org/XML/1998/namespace"
        xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
        xsi:schemaLocation="http://java.sun.com/xml/ns/javaee http://java.sun.com/xml/ns/javaee/web-jsptaglibrary_2_1.xsd ">
    <display-name>Custom Functions</display-name>
    <tlib-version>1.0</tlib-version>
    <short-name>NMTOKEN</short-name>
    <uri>http://test001.srh.com/functions</uri>

    <function>
        <name>notNullTrim</name>
        <function-class>com.srh.base.util.StringUtil</function-class>
        <function-signature>java.lang.String notNullTrim(java.lang.String)</function-signature>
    </function>
</taglib>

I created a index.jsp where I use the above EL function for testing. In it I define a string with spaces " abc ", store it in a hidden variable, call the EL function on it and store the return value in another hidden variable, and then after load alert values of both hidden variables:
<%@ page language="java" contentType="text/html; charset=UTF-8" pageEncoding="UTF-8"%>
<%@ taglib uri="http://test001.srh.com/functions" prefix="fr" %>

<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>

<head>

<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Insert title here</title>
<script type="text/javascript">
function afterLoad() {
    alert("h_mystring_org=" + document.getElementById("h_mystring_org").value);
    alert("h_mystring_new=" + document.getElementById("h_mystring_new").value);
}
</script>

</head>

<body onload="afterLoad();">

<%
String mystring = " abd ";
System.out.println("index.jsp: mystring=" + mystring);
%>

<form action="">
    <input type="hidden" id="h_mystring_org" name="h_mystring_org" value="<%=mystring%>">
    <input type="hidden" id="h_mystring_new" name="h_mystring_new" value="${fr:notNullTrim(mystring)}">
</form>

</body>

</html>

I deploy the WAR to JBoss. Then go to URL
http://localhost:8080/Test001_WAR/index.jsp

It did not work. The function in StringUtil class was called but the parameter value it received was not the same as being passed from the JSP.
In the server.log file I see:

2016-01-10 10:58:37,295 INFO  [STDOUT] index.jsp: mystring= abd
2016-01-10 10:58:37,298 INFO  [STDOUT] StringUtil.java:
notNullTrim(String): string=

The alert values were

h_mystring_org= abd
h_mystring_new=

What I am doing wrong here?

A: Why don't you use the standard JSTL fn:trim() function that does exactly that?
Anyway, the JSP EL uses page/request/session/application attributes. Not scriptlet local variables.
Your code would (probably) work if the scriptlet did
page.setAttribute("mystring", " abd ");

or, if you did the right thing and forgot completely about scriptlet:
<c:set var="mystring" value=" abd "/>

redpajama_set_name: RedPajamaStackExchange


The deterioration of teeth is often an unavoidable consequence for many. Even those with the best hygiene habits suffer from an eventual buildup of plaque, tartar, and wear and tear of the teeth from everyday use that can result in the need for a filling. At Severns Dentistry and Orthodontics, we offer tooth fillings in McMurray to help you restore your teeth to their full functionality and shape for a confident smile.
A tooth filling is commonly used to treat a cavity, but they can also be used to repair teeth that have worn down over time due to everyday use or to repair teeth that are broken or cracked.
If the tooth decay or crack has damaged most of the tooth, a cap or crown may be recommended instead. Only a dentist can determine if you have a cavity that needs to be filled. During your checkup with a dentist at our McMurray office, we will closely examine the surface of each tooth for cavities or other damage that may require a filling.
Typically, we conduct an initial examination, including x-rays, to determine which fillings are the most effective form of treatment. No one else but you should know that you have a filling. Our natural, tooth-colored fillings are completely discrete and blend seamlessly with the surrounding tooth structure.
This allows you to feel completely confident showing off your smile in all situations. During the procedure, the decayed or damaged portion of the tooth is removed. The hole is then filled with durable and discrete material, restoring the structure and function of the tooth while also preventing bacteria from entering the tooth. By removing the damaged portion, your tooth is protected from further damage.
Our expert team will work to ensure you are comfortable before your appointment and throughout the entire procedure and will be happy to answer any questions you may have. Contact Severns Dentistry and Orthodontics today to schedule your appointment!


text:
<TITLE>STIX Font Table for STIXNonUnicode-Bold (E000-F8FF)</TITLE>

STIXNonUnicode-Bold Character Tables

Private Use Area: E000 - F8FF

0	1	2	3	4	5	6	7

8	9	A	B	C	D	E	F

E00






E01



E02








E03



E04



E05









E06













E07





E08







E09

E0A



E0B








E0C

E0D









E0E




E0F

E10






E11

E12

E13

E14

E15





E16

E17

E18

E19

E1A

E1B

E1C

E1D

E1E

E1F

E20

E21

E22

E23

E24

E25

E26









E27









E28









E29


E2A

E2B

E2C

E2D

E2E

E2F





E30

















E31

















E32













E33

E34

E35

E36







E37





E38








E39


















E3A
















E3B


















E3C





Latin-1 Supplement
/\

NonUnicode
<

O

>
NonUnicode-Bold-Italic

\/
All

redpajama_set_name: RedPajamaGithub


text:
Q: Only display codenarc priorities 1 and 2 in Jenkins violations report Is there a way I can exclude priority 3 violations from the Jenkins graph? The other codenarc report types like 'html' or 'sortable' have a maxPriority option which does exactly what I want, but that option doesn't exist for the type=xml report.
Help?

Details:
I use codenarc as a lint-like source code analysis tool for my Groovy / Grails project.
The Jenkins Violations Plugin supports codenarc (among others) and allows the Jenkins job to not only display an HTML codenarc report, but also Fail the build if, say, you have 1 or more high priority violations.
This is all great but I don't want any priority 3 (low level) violations to appear in the nifty graph which the violations plugin creates in Jenkins.
It's more an issue of visual appeal than anything else. I don't want low level violations to appear at all.
Am I left with only the option of removing all priority 3 references by post-processing the xml file that is initially generated? Regex to the rescue. In that case I guess I add a Build Step that is a Groovy Script which runs before the Violations are evaluated?


text:
Q: Unable to reference class in main The first two parameters seem to work but when I add the string I get an error (Line 17 cannot convert string to double). What am I missing here? From everything I've read in my book it seems like this should work so I'm guessing it's a stupid error but I've been looking up and down the code for the last 3 hours and haven't found anything. Thank you for reading this far!
    using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ConsoleApplication14
{
class Program
{
    static void Main(string[] args)
    {

        SimpleCalc Calc = new SimpleCalc("{0.0}", "{0.0}", "{0}");
        Console.WriteLine(Calc);

    }
  }
}

This is the class
    using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;

namespace ConsoleApplication14
{
class SimpleCalc
{

        public SimpleCalc(double num1, double num2, string oper)
        {

            Console.Write("Enter first integer: ");
            num1 = Convert.ToDouble(Console.ReadLine());

            Console.Write("Enter operator (+,-,*, / or %)");
            oper = Convert.ToString(Console.ReadLine());

            Console.Write("Enter second integer: ");
            num2 = Convert.ToDouble(Console.ReadLine());

            if (oper == "+")
                Console.Write("Answer is: {0}", num1 + num2);

            if (oper == "-")
                Console.Write("Answer is: {0}", num1 - num2);

            if (oper == "*")
                Console.Write("Answer is: {0}", num1 * num2);

            if (oper == "/")
                Console.Write("Answer is: {0}", num1 / num2);

            if (oper == "%")
                Console.Write("Answer is: {0}", num1 % num2);

            Console.ReadKey();
        }
    }
}

A: The problem is that your constructor takes two doubles and a string:
public SimpleCalc(double num1, double num2, string oper)

But you're calling it with three strings:
SimpleCalc Calc = new SimpleCalc("{0.0}", "{0.0}", "{0}");

Change that to:
SimpleCalc calc = new SimpleCalc(0.0, 0.0, "{0}");

and it should be fine, in terms of compiling. (It's not good that you're interacting with the user in a constructor, and ignoring the values of the parameters that were passed in, but that's a different matter.)
Also note that I've changed the name of the local variable from Calc to calc to follow normal C# conventions.

A: Your SimpleCalc constructor expects doubles and a single string at the end.. you're passing it strings only.
So, instead of this:
SimpleCalc Calc = new SimpleCalc("{0.0}", "{0.0}", "{0}");

You should be doing something like this:
SimpleCalc Calc = new SimpleCalc(0.0, 0.0, "0");

A: Well firstly you are some converion errors string to double and overload with invalid args.
Secondly you are passing parameters to constructor but not using them?
Maybe you want
    static void Main(string[] args)
    {
        SimpleCalc Calc = new SimpleCalc();
        Console.WriteLine(Calc);
    }

and consructor as
    public SimpleCalc()
    {
        double num1;
        double num2;
        string oper;

        Console.Write("Enter first integer: ");
        num1 = Convert.ToDouble(Console.ReadLine());

        Console.Write("Enter operator (+,-,*, / or %)");
        oper = Convert.ToString(Console.ReadLine());

        Console.Write("Enter second integer: ");
        num2 = Convert.ToDouble(Console.ReadLine());

        if (oper == "+")
            Console.Write("Answer is: {0}", num1 + num2);

        if (oper == "-")
            Console.Write("Answer is: {0}", num1 - num2);

        if (oper == "*")
            Console.Write("Answer is: {0}", num1 * num2);

        if (oper == "/")
            Console.Write("Answer is: {0}", num1 / num2);

        if (oper == "%")
            Console.Write("Answer is: {0}", num1 % num2);

        Console.ReadKey();
    }

A: your code had multiple bugs. If you are sending parameters from Main function from where you get this parameters. And If you send parameters from Main function why are you reading it from console in SimpleCalc function.
First remove reading part from SimpleCalc function to Main.
static void Main(string[] args)
{
    double num1;
    double num2;
    string oper;

    Console.Write("Enter first integer: ");
    num1 = Convert.ToDouble(Console.ReadLine());

    Console.Write("Enter operator (+,-,*, / or %)");
    oper = Convert.ToString(Console.ReadLine());

    Console.Write("Enter second integer: ");
    num2 = Convert.ToDouble(Console.ReadLine());

    SimpleCalc Calc = new SimpleCalc(num1,num2,oper);
    Console.WriteLine(Calc);
}

and remove reading part from function and make it simple and clear
    public SimpleCalc(double num1, double num2, string oper)
    {

        if (oper == "+")
            Console.Write("Answer is: {0}", num1 + num2);

        if (oper == "-")
            Console.Write("Answer is: {0}", num1 - num2);

        if (oper == "*")
            Console.Write("Answer is: {0}", num1 * num2);

        if (oper == "/")
            Console.Write("Answer is: {0}", num1 / num2);

        if (oper == "%")
            Console.Write("Answer is: {0}", num1 % num2);

        Console.ReadKey();
    }

A: You need to pass correct type of argument what you have initiated in the method. Refer below code.
SimpleCalc Calc = new SimpleCalc(0.0, 0.0, "0");
or
You can pass like below.
SimpleCalc Calc = new SimpleCalc("{0.0}", "{0.0}", "{0}");
But you need create the method as required like below
class SimpleCalc
{
    public SimpleCalc(string num1, string num2, string oper)
    {
        //............
    }
}

A: As everyone already pointed you to the problem with the parameters type passed to the constructor of your class I wish to point a logical problem in your code
The parameters passed to the SimpleCalc constructor becomes local variable inside the constructor and are not used at all because you ask you user to type the value for them. So why passing them in the first instance? I also find not a good example the fact that all the works of your class is executed in the constructor (taking also the input without any validation).
Perhaps you should use a different approach
static void Main(string[] args)
{
    // To keep short, I have not added any validation, but the user input
    // should always validated See double.TryParse

    Console.Write("Enter first integer: ");
    double num1 = Convert.ToDouble(Console.ReadLine());

    Console.Write("Enter operator (+,-,*, / or %)");
    string oper = Convert.ToString(Console.ReadLine());

    Console.Write("Enter second integer: ");
    double num2 = Convert.ToDouble(Console.ReadLine());
    SimpleCalc calc = new SimpleCalc(num1, num2, oper);
    double result = calc.Execute();
    Console.WriteLine("Answer is: {0}", result);
}

class SimpleCalc
{
    double _num1;
    double _num2;
    string _oper;
    public SimpleCalc(double num1, double num2, string oper)
    {
        _num1 = num1;
        _oper = oper;
        _num2 = num2;
    }
    public double Execute()
    {
        if (oper == "+")
            return num1 + num2;

        else if (oper == "-")
            return num1 - num2;

        else if (oper == "*")
            return num1 * num2;

        else if (oper == "/")
            return num1 / num2;

        else if (oper == "%")
            return num1 % num2;

        else
             throw new ArgumentException("Invalid operator: " + oper);
    }
}

A: You Should use it as below
    SimpleCalc Calc = new SimpleCalc(0.0, 0.0, "{0}");
As your Class expects first two params as double and 3rd one as a string


It was William of Wykeham, Bishop of Winchester, who took as his motto "Manners makyth Man" as long ago as the fourteenth century. His chosen maxim was fairly revolutionary at the time as it implied that good manners, including adherence to norms of politeness and good conduct are what distinguish someone and make them a rounded, attractive and successful individual. The words meant that an individual is defined, not by birth, money, or property, but by how he or she behaves towards other people. The maxim seems to have worked for William, because although he was born into a peasant family, after a long and varied career, he ended up as one of the richest men in England.
I wondered what William would have made of the results of a survey conducted two or three years ago, which found that forty per cent of those polled thought that 'thank you' sounds too formal and would rather express their gratitude with terms like 'cheers', 'cool', or 'nice one'. In New Zealand of course, we have the ubiquitous 'ta', which in my youth in Manchester was considered to be a very unrefined way of acknowledging a favour, even worse if someone chose to expand it to 'ta muchly'. Other preferred alternatives mentioned in the survey are 'awesome' and 'wicked' and rather surprisingly 'much appreciated'. Surely a monosyllabic 'thanks' would be simpler than this mouthful of six syllables. Even more out of the ordinary, some people, it seems, prefer to resort to other languages to communicate that they are grateful and say 'merci', 'danke', or 'gracias'.
William might have pondered on why people have such difficulty nowadays with those simple words 'thank you' or 'thanks' which articulate so precisely what we want to say and why they prefer to employ such apparently casual alternatives. It's fair to say that at least they are using an alternative and not letting kindness or helpfulness or service go unnoticed. I've been thinking about this a good deal in the last few days while trying to encourage my young granddaughters aged 5 and 3 to say 'please' and 'thank you' occasionally. Their natural way, in common with most young children I have met, is to say 'I want x, y or z' and to accept what they are given without acknowledgment. It was hard going, even with the promise of a visit to the two dollar shop to inspire them. But I have begun to notice the odd 'please' and 'thank you' creeping in from time to time, so I'm sure that with the persistence of their parents, they'll get there sooner rather than later.
It's true that in other countries, manners take different forms. I was surprised when I first visited Spain to notice how little the Spanish use 'por favor' and 'gracias' in everyday situations. They're just not considered necessary and I remember how our Spanish friends used to tease us for being too polite by half for using them all the time. In other countries, other rules apply. For example at the table, things we judge to be rude are considered good behaviour: if you're eating noodles in Japan, it's fine to slurp them and some say it makes them taste better; in China it's good to burp after a meal because it means you enjoyed the food; in India, if you clean your plate, your host will think the meal didn't satisfy you so you should leave a little food to show that you're well-fed.
But we have to live in the framework of the manners of the country we live in. Emily Post, the well-known American writer on manners and etiquette, gives an interesting definition of what constitutes good manners: 'Manners are a sensitive awareness of the feelings of others. If you have that awareness, you have good manners, no matter what fork you use.' Her point is that good manners are more about attitude to others than empty etiquette. Courtesy, kindness and respect for others are the key values. I would only take issue with Emily on the matter of the fork, and suggest that respect for others should include finding out about customs in other countries before we visit them and risk causing offence by applying our own standards rather than those of the country we're visiting.
Where do good manners come from? As we all know, they have to be taught as well as absorbed from the example of all those around us. It is parents who mainly have that huge responsibility, but it's up to all of us to stick to a code of good manners so that it remains the norm, because children copy what they hear and see around them. It's not a quick fix either, it's a daily commitment over many years, gently persisting, encouraging and reminding. But once inculcated, good manners lead to a genuine awareness of the other person and a lessening of the natural selfish impulse that motivates us all on some level. They contribute to a true sense of gratitude and the courtesy which goes with it. They make life more pleasant: a warm greeting instead of a blank stare, a goodnight with real feeling, appreciation for the meal provided, a little note to thank for a favour done; all these things oil the wheels of life and make others feel appreciated. I wonder if it's even possible that those with good manners are liked more by others; wouldn't you rather have polite children round to your place than those who have no manners? And in the workplace, isn't it more pleasurable doing business with courteous colleagues rather than those whose cut throat attitude makes your nerve ends jangle?
It's been said by some that good manners are a sign of weakness and that being polite implies a kind of passivity that is a flaw, not a strength. Rudeness is perceived as a kind of assertiveness, as a way of getting on in the world and promoting one's own concerns, particularly in the workplace. This view seems to me to boil down to selfishness which will always detract from the aspiration to be a rounded person who respects the feelings of others while at the same time pursuing their own concerns. Good manners do not rule out ambition and aspiration. Empathy and awareness of others are likely to lead to better relationships, successful careers and increased self-confidence.
That all brings me back to the alternatives to 'thank you' identified in the survey mentioned above and to conclude that they all bring with them the same genuine sentiment as the conventional two words. People, especially young people, have always invented new ways of saying things and these new forms enrich the language rather than detract from it. So next time someone says to me 'cool', 'cheers' or 'awesome', I'll be glad, because in their own way they're showing gratitude, respect and courtesy. And I think William of Wykeham would agree.


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Toronto woman's 'Win a Restaurant' raffle hits snag
A Toronto woman is hoping to relight the fire under the contest to raffle off her restaurant.
Ruthie Cummings, owner of Das Gasthaus German eatery, hit a roadblock when the website Eventbrite pulled the plug on her Win a Restaurant draw in September, after 500 tickets were sold.
U.S.-based Eventbrite was prohibited from operating a contest on this side of the border, according to Canadian Restaurant News. Cummings had to refund about half the tickets while she searched for another host, but last month she teamed up with Moneris to process online payments and still plans to sell tickets up to the original deadline, Dec. 24, or until she sells 4,000 tickets — whichever comes first.
Tickets are $150 apiece. A winner will be randomly drawn on Dec. 30. Cummings renovated the 1,200-square-foot restaurant when she took over in 2012. It comes fully equipped, furnished and staffed, but the new owner will be free to change the name, the menu, the decor and anything else he or she wishes, according to the detailed rules and regulations posted on her website.
It's a bittersweet endeavour.
"I'm selling the restaurant not because I want to but because I have to," she says in a video on her site. She says she is putting her dream "on the backburner" to care for her elderly parents.
They live about three hours from Toronto, and aren't well.
"I would love to see someone with the same kind of passion I had being able to walk in here and take it over and create what they want to do, make it their dream," says Cummings, who saved for 10 years to open her business, which she thought would be her "forever restaurant."
Cummings says the assets of Das Gasthaus — which is German for "the guest house" — have been valued at $144,550 by an "independent professional." There are no liens, taxes or debts associated with the restaurant. The winner can "start making money the moment you open the doors," she says.
Source Sherry Noik, Daily Brew


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Ilari Kaila
Lecturer, Composer-in-Residence
The Hong Kong University of Science and Technology
Dr. Ilari Kaila works as Composer-in-Residence/lecturer on the faculty of The Hong Kong University of Science and Technology. His recent works have been presented in a MATA composer portrait in New York with the Aizuri Quartet; at the Chelsea Music Festival in New York City and Taipei as the 2015 festival's Composer-in-Residence; at the Metropolis Festival in Australia by the Melbourne Symphony Orchestra; on the Avanti Chamber Orchestra's tour of Japan; in Finland by the Joensuu Symphony Orchestra; as a Composer Fellow at the Intimacy of Creativity 2014 festival in Hong Kong; at the Banff Centre Summer Arts Festival in Canada; and at the 2014 MATA Festival in New York City. He has worked with artists and ensembles including Olli Mustonen, the Escher String Quartet, the Tanglewood New Fromm Players, the Albany Symphony Orchestra, the Kuopio Symphony Orchestra, and the Helsinki Philharmonic Orchestra's chamber ensembles, among others, and has been awarded in the Mellon Foundation/ASO "Composer to Center Stage" competition, in the IC2014 World Premiere Concert Audience Vote in Hong Kong, and in the Composer Competition of the 9th International Piano Festival in Espoo. Kaila received his PhD in Music Composition from Stony Brook University in 2011, having previously studied at the Sibelius Academy in Finland. Before moving to Hong Kong, he taught at Columbia University in New York, and worked as a teaching artist with the New York Philharmonic.
Intimacy of Creativity: Entering the Minds of Composers


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Airlines Disad Negative
HSR trades off with air routes and outcompetes due to access to city centers—China and Europe prove
Page 11/47
1 ... 7 8 9 10 11 12 13 14 ... 47
This month, passengers could buy tickets
Chinas airlines will be put further on the back foot by new train routes.
The National 10 [Daniel Bardsley, Foreign Correspondent, March 20th http://www.thenational.ae/business/full-throttle-on-high-speed-rail]
While high-speed trains may be popular with passengers, they can cause turbulence to the airline industry. The rolling stock may not be as fast as an aircraft, but as the trains run directly into city centres they can be more attractive than flying, even for business travellers. No wonder then that airlines have cut prices to stay competitive. China Southern Airlines used to charge a reported 700 yuan to fly between Guangzhou and Changsha, which lies on the line to Wuhan. This month, passengers could buy tickets online from the carrier for as little as 170 yuan. In Europe, airlines have dropped some routes between major cities altogether as a result of competition from high-speed railways. Mr Sangiambut believes China's airlines will be put further on the back foot by new train routes. Flights of less than two hours, he says, would be "very much impacted" if high-speed trains start operating the same route. "They will come under pressure when these high-speed networks become more fully operational," he says. "I don't think they will be closed entirely, but frequency could be reduced." The price of a Beijing-Shanghai high-speed train ticket has not been announced yet, but Mr Sangiambut says the ministry of railways will ensure it is "rather competitive" with flying. As a result, he thinks the Beijing-to-Shanghai air route will suffer when the high-speed rail line opens and cuts the rail trip from 10 hours to four hours. "There will be some impact for sure," he says.
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