text:
Q: Как создать динамический трехмерный массив и заполнить его случайными значениями? Я больше по Java, но сейчас надо сделать пару задачек по плюсам. Одна из них звучит как-то так:

Введя значения m,n,r, определить
трехмерный массив P = размерами i,j,k,
где i=1,n;j=1,m;k=1,r. Значения
элементов которого определяются
случайно.

Упустим тот момент, что преподаватель не знает, что в плюсах массив считается с нуля...
С malloc/free не работал в плюсах, а тем более с трехмерными массивами, так что понятия не имею, как это делается. Надо использовать malloc/free.
A: Трехмерный массив можно представить как правильный многогранник или же двумерные массивы, уложенные по слоям.

Соответственно легче ориентироваться как в декартовой системе координат, проставив оси X,Y,Z на соответствующих направлениях. Сначала создаются слои, от ближнего к дальнему:
int ***arr = new int**[zindex];

Ну а каждый из слоев, это просто обычный уже двумерный массив. Получается такая ссылка на массив двумерных массивов. Остается выделить память под строки и столбцы для каждого слоя.
for(int i=0; i<zindex; i++){
    arr[i] = new int*[yindex]; //выделяем память на столбцы

    for(int j=0; j<yindex;j++){
        arr[i][j] = new int[xindex]; //выделяем память на строки.
    }
}

Вот в общем-то и все, остается только заполнить рандомными числами. Полный пример(если я нигде не намудрил с памятью):
int main(){
    setlocale(LC_ALL,"Russian");

    int zindex = 10, yindex = 5, xindex = 7;

    int ***arr = new int**[zindex];

    for(int i=0; i<zindex; i++){
        arr[i] = new int*[yindex];

        for(int j=0; j<yindex;j++){
            arr[i][j] = new int[xindex];
        }
    }

    for(int z=0;z<zindex;z++)
        for(int y=0;y<yindex;y++)
            for(int x=0;x<xindex;x++)
                arr[z][y][x] = rand();

    for(int z=0;z<zindex;z++)
        for(int y=0;y<yindex;y++)
            for(int x=0;x<xindex;x++)
                cout << arr[z][y][x] << endl;

    system("pause");

    return 0;
}

Учитывая, что malloc/free практически ничем не отличаются в плане выделения памяти(вообще это сишный вариант выделения памяти), стоит только заменить new на его эквивалент. Вот тут не уверен, но возможно это будет выглядеть так:
arr = (int ***) malloc(zindex*sizeof(int**));
for (i = 0; i < zindex; i++) {
    arr[i] = (int **) malloc(yindex*sizeof(int*));
    for(j = 0; j < yindex; j++) {
        arr[i][j] = (int *) malloc(xindex*sizeof(int));
    }
}

A: Не знаю, как другие компиляторы, а gcc (но не g++) позволяет передавать в функцию размерности измерений матрицы в ее аргументах.
Это позволяет не моделировать 2-х и 3-х мерные массивы, как показал в своем ответе @Alex Krass массивами указателей, а выделить одну последовательную область памяти нужного размера и разместить многомерный массив в ней. Т.е. практически так же, как делает компилятор, когда размер каждого измерения известен при компиляции (константа).
Вот пример:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>

void
rfill (int n, int m, int r, int a[n][m][r], int d)
{
  srand(time(0)); // выбросьте, если хотите повторяемости результата в разных запусках

  for (int i = 0; i < n; i++)
    for (int j = 0; j < m; j++)
      for (int k = 0; k < r; k++)
        a[i][j][k] = rand() % d;
}

void
print (int n, int m, int r, int a[][m][r]) // а старшую размерность можно опустить !!!
{
  for (int i = 0; i < n; i++) {
    printf("slice %d\n", i);
    for (int j = 0; j < m; j++) {
      for (int k = 0; k < r; k++)
        printf ("%5d ", a[i][j][k]);
      puts("");
    }
  }
}

int
main (int ac, char *av[])
{
  int n, m, r, nn;
  scanf("%d %d %d", &n, &m, &r);
  nn = n * m * r;
  printf("size %d (%dx%dx%d)\n", nn, n, m, r);
  int *a = malloc(sizeof(*a) * nn);

  rfill(n, m, r, (int (*)[m][r])a, 1000);
  print(n, m, r, (int (*)[m][r])a);
}

Обратите внимание на приведение  типа при передаче указателя в функции.
Транслируем (ключ -std=c99 нужен исключительно для описаний индексов в стиле C++) и запускаем
avp@avp-xub11:hashcode$ gcc -std=c99 c.c
avp@avp-xub11:hashcode$ ./a.out
2 3 4
size 24 (2x3x4)
slice 0
  819   953   252   669
  287   460     8   582
  154   112   346   220
slice 1
  385   608   815   953
  333   808   582   233
  304   746   445   773
avp@avp-xub11:hashcode$

Возможно, это не то, что хотел преподаватель, но как и многое в GNU достаточно практично.
Конечно, можно и просто моделировать обращения к элементам такой матрицы, самостоятельно вычисляя адрес нужного элемента.
Добавим в конец main() вот такой код:
  printf ("Enter i, j, k: "); fflush(stdout);
  int i, j, k;
  scanf("%d %d %d", &i, &j, &k);

  printf ("%d\n", a[i * m * r + j * r + k]);

и в результате увидим:
....
slice 1
  215   877   552   933
  458   501   568   727
  485   975   235    39
Enter i, j, k: 1 1 3
727
avp@avp-xub11:hashcode$

Надеюсь, что все это достаточно интересно.


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As usual, a big crowd is expected for Saturday's annual crosstown rivalry game between UCLA and USC at the Los Angles Memorial Coliseum.
If you don't feel like driving and getting stuck in traffic and paying a small ransom for parking, the Expo Line and Silver Line are both a short stroll to the stadium. All the details are here and some basic tips are below.
Too bad the best deal in downtown LA program got less good with the increase in parking fee to $8 at the Union Station parking garage.
6 minute headways on Expo on USC game days? Extra service on the Silver Line? Who pays for the additional service? You? Me? or USC? Seriously, I want to know! Are we giving them extra service for free, at taxpayer (and transit user) expense? Remember, this is the same USC that is cutting back on transit subsidies for their staff so that they can justify building more parking structures.
I was on the 110 Freeway South on Saturday and was very angry when I saw that the fastrack lanes were open to Manchester after the end of the USC/UCLA football game at the Coliseum. I often travel the 110 on weekends and traffic in the general purpose lanes are worse than during weekday rush hour while no one is in the fastrack lanes. If the lanes can be open to everyone for a "football" game, then why aren't they open every Saturday and Sunday for the benefit of everyone? Why is the benefit granted to USC?
More extra service for USC fans at the expense of the general transit public. Does USC pay anything for this extra service?
I disagree that it's at the expense of the general public. None of the local sports teams pay for service. But one reason to have a mass transit rail system is to be able to provide service to big events and to help give some of the people attending an option other than traffic. That benefits everyone, I think. I also am aware that if we didn't provide service to the games, we'd have a lot of people complaining about that.


NEW DELHI: Claiming that Prime Minister Narendra Modi would have been "number one accused" in the Rafale issue had Lokpal been implemented, the Congress on Monday said in Lok Sabha that his broad shoulders cannot stand the "bullet of corruption".
If anybody has to be held guilty in Rafale deal, "it is the PM and PM alone", Congress leader Veerappa Moily said launching a scathing attack against PM Modi and the Union government during a discussion on the interim budget.
He alleged that the Rafale issue has become a "household scandal" and that the Prime Minister may "escape" his responsibility today but not in the days to come.
While the defence budget saw only a paltry increase in absolute terms and the allocation to the sector as percentage of the GDP has gone down, he alleged the ruling BJP is using money to "buy" MLAs, build "palatial" offices and filling its coffers.
On the Rafale deal, in which the Congress has accused the government of corruption, Moily said, "Now it is crystal clear why Lokpal (law) is not being implemented, if it had been in place, the PM would be number one accused."
Due to this "guilty conscience", the Union government has been using investigative agencies against its rivals, he alleged.
In an apparent reference to Modi's comments about his "56 -inch chest", Moily said, "When bullet of corruption is hit, his broad shoulders cannot bear it."
Modi had earlier hit out at the Congress over the Rafale issue, saying the party did not want the country's armed forces to be strengthened, and had asked if it was bidding for any private player.
During his reply to the discussion on motion of thanks on the President's address, the Prime Minister had alleged that no defence deal under the Congress was without kickbacks.
Moily on Monday also targeted the government over unemployment issue and accused it of "fudging" statistics.
If the GDP is growing at 7.5 per cent, "can it be without creating jobs," he said, adding according to latest NSSO estimate, the country's unemployment stood at a 45-year high.
There is a "job crisis", which they are not prepared to accept, Moily said.
"Now Rafael will fail and Rahul will win," he said, alleging the ruling government has no respect for Parliament and Constitution.
Supporting the demand of the TDP, he said the NDA government denied the "legitimate support" to Andhra Pradesh and did not grant it special status which was promised on the floor of Parliament.
"This shows they have scant regard for Parliament, scant regard for Constitution of India. I want to tell you (TDP), the Congress party is fully with you and today or tomorrow we will implement it," said Moily.
"They (the government) believe only in Reliance, their integrity lies in relying on Reliance," he said.
He said that Congress leader Sonia Gandhi has promised that "we will implement that".
"After demonetisation, Rs 2 lakh crore worth of GDP was lost, industrial output and MSMEs suffered and now the government will pay for the sin committed against poor people," he said.
They need to prove that they are a clean government and that is why they spend on advertisements, he said adding "If the youth are so enamoured by Mudra Yojana, then why are youths seeking jobs?"
"40 per cent of Mudra funds are lying idle, Mudra loans are leading to rising NPAs. In five years of this government, NPAs have risen from Rs 2 lakh crore to Rs 12 lakh crore.
"If you give them another five years, then the public sector banks will be wiped out or liquidated," he claimed.
Moily said that the government is accumulating "sin after sin".
"They do not care about women....This is the impression... if women started cursing you, you will never come again," he said.
He also said that funds for 'Beti Bachao and Beti Padhao' have also decreased and it has been reduced to a slogan only as the amount was mainly spent on advertisements.
"It is only for PM bachao," he said, adding huge amounts were spent on media advertisement and on chosen channels.
Taking a swipe at the government, he said it is "good at manufacturing news, lies and web of lies. Nobel prize should be given to them for the web of lies".


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text:
Let's Travel Light! The Slow Travel Movement of Backpackers in Taitung
About Chihshang Township
Chihshang Township of Taitung County is a flood plain formed by Wanchao River and Sinwulyu River which separately flows down from the north and south of Central Range. Rural settlements where farming is feasible have been built thanks to the sources of water, and agriculture has naturally become the major industry. More than a hundred years ago, immigrants migrated to this land from the west in succession. Since most of them gathered up around the area of Dapo Pond, the place was called "the new countryside". The region was renamed "Chihshang" during Japanese-Occupied Period later on.

Chihshang Township covers an area of only 82.6 square kilometers, which is about 30% of the area of Taipei city, and it has a population of 8,300, which accounts for merely 0.3% of the one of Taipei City. Just like most of the towns in Huadong Valley, the residents here comprise 30% of Taiwanese aborigines, 30% of Hakka, and 30% of Pingpu tribes. The three cultural elements integrate mutually here in Chihshang Township that culture fusion is consequently perceivable from time to time. In the year of 2004, Council of Agriculture established "Chihshang Township Jinyuan Wanan Agri-tourism Area", which was renamed "Chihshang Rice Leisure Farming Industrial Zone" in 2007, accentuating the farming features of Chihshang area.
Be Wise and Make the Pilgrimage to Mr. Brown Avenue at Dawn
One is able to recognize the famous Mr. Brown Avenue and Takeshi Kaneshiro Tree by looking ahead from Chihshang Bridge on Provincial Highway No. 9. There are many tandem bicycles traversing the fields, and what comes into view is the visual impact brought by large paddy fields and rice ears. The picture frame set up at the entrance of Mr. Brown Avenue is the pilgrimage spot for tourists to take selfies as mementos. Wanan Village located close to the foot of Coastal Mountain Range is one of the earliest-developed villages in Chihshang Township. Many old houses have been renovated into bicycle rentals as tourists and tour buses pour into the area one after the other since a few years ago.
The burning sun of Taitung is fierce and powerful. Would you like to ride as dashingly as Kaneshiro Takeshi on Mr. Brown Avenue? The best time to visit the scenic site is at dawn. You will be able to enjoy proper temperature, golden sunlight, and the undisturbed background without people and cars.

Appreciate the trees, the fields, and local hospitality; these are exactly the most splendid highlights of Mr. Brown Avenue. Many farming chores are practiced starting from early in the morning, and the farmers will always welcome you warmly as long as the progress of the work is not interrupted. Local peasants tend to stop their trucks at times to offer advice on the secret points and ideal shot angles. You will be stunned by their ability of talking joyfully without reserve when speaking of farm work.

Join a local guiding tour is a recommendable way to dig deeper into the land of Chihshang. Set off early in the morning, ride on a bicycle, and learn more about the trifles of Chihshang and other secret sites which are not put down in travel books through the eyes of a local resident.
Farming Experience
Chihshang is a town where glamorous and picturesque sceneries vary along with the change of seasons. After rice ears' turning from green into yellow, the charm of rape flowers of winter takes over the shift. Dissimilar scenic sights on both sides of the windy "Heaven's Path" can be seen even in the month of November and December.

The reputation of Chihshang's breathtaking landscape has already spread throughout the countries in Southeast Asia, especially tourists from Hong Kong and Singapore where large fields are rarely seen due to small but densely populated living environment. In addition to marveling at the views and relishing gourmet dishes, foreign visitors are inclined to enjoy the experience of working in the field. The schedule of farming follows the routine and law of nature that different agricultural jobs are carried out at different times. The travelers must put aside the notion of "designated item" and walk into the fields in the wake of the young farmers. The farming experience program arranged by "Good Farming Day" takes place after three o'clock in the afternoon. As clouds often converge and moderate the sultriness that comes along with the scorching sun in summer afternoon, it is suitable for going down into the fields. After sweating like a pig for about an hour, it is time to relish the desserts and dunch made from rice crops. The food seems to be more delicious than ever.

Tian Ma Ma Restaurant in Rice Village is Genuinely Authentic
Rice is the most famous delicacy in Chihshang Township, together with the enticing by-products made from rice crops. If you wish to have a taste of the genuinely authentic rice meals, "Tian Ma Ma Restaurant" would definitely be the first choice. Single-serving set menu suitable for individual and small group dining is served apart from table d'hote.
Tian Ma Ma Restaurant has always claimed that the ingredients they apply are local and of low-carbohydrate miles. Rice produced by Chihshang Farmer's Association is grainy, aromatic, and lip-smacking in big portion. The fried pork ribs served with toothsome side dishes is crispy with plenty of flavor. A variety of culinary methods are applied to prepare local vegetables for realizing the enjoyment of multiple senses of taste aside from daintiness.
The Interior of the restaurant has been redecorated during the recent years, and the bright and cozy space is now brimming with a sense of modernity. Hiding inside a room with air conditioning and have an appetizing meal which replenishes our physical strength at noon is undoubtedly a brilliant idea. Some say that people mostly have additional appetite for desserts. After gratifying the stomach, buy a cone of ice cream made from Chihshang rice at the neighboring Chihshang Farmer's Association for dessert. The rice ice cream of authentic flavor holds both the fragrance and texture of rice grains. The snack which represents the regional good taste of Chihshang is rarely-seen in other places.
Located on Zhongshan Road, the so-called "main street" known by the locals, Tianweijia always commences its business in the afternoon with constant visitors from all parts of the country. The female shopkeeper has inherited excellent cooking skills of Hakka from her mother; all Hakka snacks sold in the stores are bestsellers. There are many customers who specifically come for eating ice in summer afternoon. The highly asked item of "Niu Wenshui" is a kind of Hakka snack made with mochi, on top of which spread peanut powder, and sweet soup. Though being extremely simple, it carries the savor and feelings of traditional Hakka dim sum. It is exactly because the appearance of mochi floating in sugar water resembles the back of a cattle rising to the water surface that the dessert has got its name. Why not try imagining the tableau with attentive care before relishing? In addition, the Almond black tea particularly developed by the firm is of adequately strong drupe flavor. Its rich texture and down-to-earth portion makes it more than worth sampling.
Many stores tend to make use of rice crop to develop new delicacies owing to the cultivation of origin. "Good Farming Day" is one of the pioneers that have developed the diversity of rice products: blending rice into breakfast steamed buns, for instance. You can discover the taste of fillings and the al dente texture of rice grains at the same time. Unlike the regular products of high sweetness, the simmered rice and peanut milk is of pure and thick flavor. The rice cake is a dessert which can only be encountered by the lucky birds. In addition to silky and delicate texture, a pleasant smell of rice is distributed within the mouth.
The Black Riders
Writer Jiang Xun once mentioned "the black riders" formed by a group of youngsters from Chihshang in "The Impression of Chihshang". While some of them are the youths who have returned home from other places, some are the ones who have always been stuck to the soil of Chihshang Township. Wei Wen-xuan at "Good Farming Day", Zhang Li-you at "Tianweijia", Luo Zheng-jie at "Chihshang Promenade", and Hao Chao-yang at "Bike de Koffie" are several of the members who run small businesses in local area.
Each of them possesses his own outdated black bicycle, the so-called "martial ride" which they usually place inside the store. These riders are often sighted carrying the snacks or sweets of their store on the back of the bikes when special occasions or events are held. The original intention of the black riders is to hopefully arouse those amazing memories everybody has for the good old days upon seeing these well-worn bicycles. The union they have established is like the vital spark that hides indistinctly behind the main street. You can feel the buried energy of Chihshang upon noticing this small, insignificant group. If you happen to visit the shop where an iron plate that reads "the black rider" is flagged, inquire whether or not there are relevant activities going on at that moment so that you can get the chance of partaking in one and experience the dynamic vigor of Chihshang Township.
During the evening of solo travel, visit Chihshang Promenade and Jiece Used Bookstore to halt, have a little chit chat with strangers in the store, and hear about what stories of Chihshang other people can share. Renovated and rebuilt by Luo Zheng-jie, a designer who has stayed in Chihshang, Chihshang Promenade on the first floor is a coffee shop which seemingly serves customers with coffee drinks but indeed gives out more quality of Chishang Township. For example, the postcards which made from the photos shot by himself, the pictures on top of which are not necessarily the common spectacles that are often seen but the revelation of private travels accompanied by the details of Chishang life.
"Chihshang Rice Leisure Farming Industrial Zone" is suitable for going on a solo trip. Remember to slow down the pace and immerse yourself in the local living culture. Talk to more local residents and enable yourself to find diversified beauties of Chishang Township.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}
In non-centrosymmetric magnetic systems,
antisymmetric \textit{Dzyaloshinskii-Moriya} (DM) exchange
causes particular magnetic couplings
and inhomogeneous states \cite{Dz64}.
These \textit{chiral} interactions  stabilize
two- and three- dimensional localized structures
\cite{JETP89,Nature06,JPC10}.
Phenomenologically the inhomogeneous
Dzyaloshinskii energy contributions \cite{Dz64,PRB02}
are described by Lifshitz invariants,
antisymmetric differential forms
linear in first spatial derivatives
of the magnetization  $\mathbf{M}$
\begin{eqnarray}
\Lambda_{ij}^{(k)} = M_i \frac{\partial M_j}{\partial x_k}
-M_j\frac{\partial M_i}{\partial x_k}
\label{Lifshitz}
\end{eqnarray}
where $x_k$ are Cartesian components of the
spatial variable $\mathbf{r}$.
Solutions for chiral magnetic {\em Skyrmions}
as static states localized in two dimensions
have been derived first in 1989 \cite{JETP89}.
It was shown that these topological
solitonic field configurations exist in
magnetic systems for all non-centrosymmetric
crystallographic classes that
allow Lifshitz invariants in their magnetic
free energy \cite{JETP89}.
The unique character of these textures stems from the
general instability of multidimensional solitonic states
in field theories \cite{Nature06,Derrick64}.
Nonlinear continuum models for condensed
matter systems
do not contain solutions for
static and smooth multidimensional localized states.
Such states appear only as dynamic excitations,
while static configurations are generally unstable
and collapse spontaneously into topological singularities.

For a long time investigations of chiral Skyrmions
have been restricted to theoretical studies
\cite{Nature06,JMMM94,PRB10,confinement10,PRL01}.
The solutions for two-dimensional localized
and bound states (isolated Skyrmions and Skyrmion lattices)
have been studied in non-centrosymmetric
ferromagnets \cite{JMMM94,PRB10} and antiferromagnets \cite{PRB02},
in cubic helimagnets \cite{Nature06,PRB10,confinement10},
and in confined centrosymmetric
magnetic systems with surface/interface-induced chiral interactions
(e.g. nanolayers of magnetic metals) \cite{PRL01}.
In \cite{Nature06}, we formulated
the idea of Skyrmionic matter in
non-centrosymmetric crystals
predicting the existence of mesophases,
composed of Skyrmions as 'molecular units',
similar to vortex matter
in type-II superconductors \cite{Blatter94}.
Various effects observed in MnSi and other
cubic helimagnets with B20 structure \cite{Lebech89}
indicate multidimensionally modulated magnetic states
conforming with our theoretical predictions of Skyrmions
and their properties \cite{Nature06,JMMM94}.
The theoretical ideas of Skyrmions in chiral magnets  have
triggered various experimental efforts to find evidence
for these twisted textures \cite{Lamago06,Muhlbauer09}.
These experiments collected an impressive range
of data that suggest complex magnetic order phenomena.
But, mainly using diffraction or indirect evidence
by transport measurements, these experimental results
remained essentially inconclusive and have been contested,
see, e.g., \cite{Maleyev10}.
Moreover, the interpretation of the experimental
data has been based on approximate solutions \cite{Muhlbauer09,Binz06}
to the Dzyaloshinskii model by using variational approaches
in terms of a {\em mode instability}.
The corresponding results do not describe the properties of Skyrmions
and the phase transition behavior of chiral magnets,
which is governed by the {\em nucleation} of
a localized mesoscale entity \cite{Dz64,JMMM94,DeGennes75}.
Our theoretical developments \cite{PRB10,confinement10}
show that multiply modulated chiral states
in non-centrosymmetric magnets
are composed of localized solitonic states
with particle-like behavior.
The thermodynamic stability of condensed Skyrmionic phases
has been shown \cite{Nature06,PRB10,confinement10}
for the standard and modified Dzyaloshinskii models devised
for cubic helimagnets \cite{Dz64,Bak80}.
Meanwhile, experimental efforts culminated in
the direct microscopic observations of chiral Skyrmions
in thin layers of (Co,Fe)Si \cite{Yu10}.
This break-through is the first clear experimental
evidence for the existence of Skyrmions
as \textit{axisymmetric chiral localized states} that
are stabilized by a complex interplay of
nonlinear and chiral effects, as predicted earlier
\cite{JETP89,Nature06}.

Here, we address the problem of multidimensional
solitonic states in nonlinear systems lacking inversion symmetry.
This emerging field of nonlinear physics
is based on solutions of nonlinear
partial differential equations \cite{Nature06,JMMM94,confinement10},
and intricate mathematical methods of micromagnetics \cite{Hubert98},
and physics of solitons \cite{Novikov84}.
Concentrating on the physical side of the problem
rather than on mathematical details we give
an elementary introduction into the properties
of chiral \textit{Skyrmions} in magnetism.

\section{Dzyaloshinskii theory for cubic helimagnets}

\subsection{Phenomenological energy and equations}

Within the phenomenological theory
introduced by Dzyaloshinskii \cite{Dz64}
the magnetic energy density of
a  cubic non-centrosymmetric ferromagnet
with the magnetization $\mathbf{M}$
can be written as \cite{Dz64,Bak80}

\begin{equation}
w=\underbrace{A  \left(\mathrm{grad}\mathbf{M} \right)^2
-D\,\mathbf{M}\cdot \mathrm{rot}\mathbf{M}
-\mathbf{M}\cdot\mathbf{H}}_{w_0}
-\sum_{i=1}^{3}\left[B  \left(\partial M_i / \partial x_i \right)^2
+K_c M_i^4 \right] - f(M)
\label{density}
\end{equation}
including the exchange
stiffness with constant $A$.
The inhomogeneous DM exchange with constant $D$
is a combination of Lifshitz
invariants (\ref{Lifshitz}),
$w_D = D\,(\Lambda_{yx}^{(z)}+\Lambda_{xz}^{(y)}+\Lambda_{zy}^{(x)})$
=$D\,\mathbf{M}\cdot \mathrm{rot}\mathbf{M}$.
These isotropic interactions
together with the Zeeman energy
are the \textit{essential}
couplings to stabilize
chiral  modulations ($w_0$).
The sum in Eq.~(\ref{density})
includes exchange ($K$) and
cubic ($K_c$) magnetic anisotropy
contributions; $f(M)$ comprises
magnetic interactions imposed
by variation of the magnetization
modulus $M \equiv |\mathbf{M}|$.
In a broad temperature range
the magnetization vector practically does not
change its length, and nonuniform magnetic states
display only rotation of $\mathbf{M}$.
The low temperature properties, thus,
can be described by
the model (\ref{density})
with a fixed magnetization
modulus $M$ = const.

\subsection{One-dimensional chiral modulations: cones and helicoids}

In non-centrosymmetric magnets,
chiral couplings of type (\ref{Lifshitz}) favour
rotation of the magnetization vector.
They destabilize the homogeneous magnetic structure
and induce long-range modulations of the magnetization \cite{Dz64}.
At zero field helical modulations
\begin{equation}
\mathbf{M}= M \left[ \mathbf{n}_1 \cos
\left(\mathbf{k} \cdot \mathbf{r} \right)
+ \mathbf{n}_2 \sin  \left(\mathbf{k}
\cdot \mathbf{r} \right) \right],
\quad  |\mathbf{k}| = 2 \pi/L_D,
\quad  L_D = 4 \pi A/|D|
\label{helix0}
\end{equation}
with period $L_D$ and wave vector
$\mathbf{k}$  correspond to the absolute
minimum of the  isotropic functional $w_0$
(Fig. \ref{helices} a, b).
In Eq. (\ref{helix0}) $\mathbf{n}_1$, $\mathbf{n}_2$
are orthogonal unit vectors
in the plane of the magnetization rotation (Fig. 1).
The modulations (\ref{helix0})
have a fixed rotation sense determined by the sign
of constant $D$ and are continuously degenerate
with respect to propagation directions in the space.

\begin{figure*}
\includegraphics[width=16cm]{F1.pdf}
\caption{
\label{helices}
One-dimensional chiral modulations in cubic helimagnets.
In a helical "array" (a) the magnetization
rotates in the plane spanned by the orthogonal
unity vectors
$\mathbf{n}_1$ and $\mathbf{n}_2$
and the rotation sense is determined
by the sign of Dzyaloshinskii constant $D$.
Chiral \textit{helices} (b) are composed
of arrays (a) with the same propagation
direction.
Under the influence of the applied field
the helix (b) transforms
into longitudinal distorted \textit{cones} (c)
or into transversally distorted
\textit{helicoids}(d).
}
\end{figure*}
An applied magnetic field
lifts the degeneracy of the helices (\ref{helix0})
and stabilizes a {\em cone} solution with the propagation
direction along the magnetic field
\begin{equation}
\cos \theta = |\mathbf{H}|/H_D,  \quad \psi = 2 \pi z/L_D,
\quad H_D = D^2 M/(2A),
\label{cone1}
\end{equation}
where $\mathbf{M}=M(\sin\theta\cos\psi;
\sin\theta\sin\psi;\cos\theta)$ is written
in spherical coordinates.
In such a helix the magnetization component along
the applied field has a fixed value
$M_{\bot} = M \cos \theta = M(H/H_D)$, and
the magnetization vector $\mathbf{M}$ rotates
within a cone surface.
The critical value $H_D$ marks the saturation
field of the cone phase.
The helix periods ($L_D$) and critical fields ($H_D$)
for some non-centrosymmetric
cubic ferromagnets (\textit{helimagnets}) are
presented in Table~\ref{table1}.

\begin{table}[h]
\caption{
\label{table1}
N{\'e}el temperatures ($T_N$), helix periods ($L_D$), and saturation fields ($H_D$) \\
for some cubic helimagnets, data from Refs. \cite{Lebech89}.}
\begin{center}
\begin{tabular}{llllllllll}
\br
Compound \quad \quad & MnSi \quad \quad & FeGe & Fe$_{0.3}$Co$_{0.7}$Si \quad \quad
 & Fe$_{0.5}$Co$_{0.5}$Si \quad \quad & Fe$_{0.8}$Co$_{0.2}$Si \\
 \mr
$T_N$ [K] & 29.5& 278.7& 8.8 & 43.5& 32.2\\
$L_D$ [nm] & 18.0 & 68.3- 70.0 & 230&  90.0 & 29.5 \\
$H_D$ [T] & 0.62 & 0.2 & (6.0 $\pm$ 1.5)$\cdot 10^{-3}$ &  (4.0 $\pm$ 0.5)$\cdot 10^{-2}$  & 0.18 \\
\br
\end{tabular}
\end{center}
\end{table}
In real non-centrosymmetric magnets anisotropic
forces  fix the propagation of chiral modulations
along certain \textit{easy axes} directions
imposed by the crystal symmetry.
A magnetic field perpendicular
to the propagation direction violates
the uniform rotation of the magnetization
($\psi \propto z$, Eq.~(\ref{helix0})).
In a sufficiently high magnetic field
$H = H_H$  distorted \textit{helicoids}
infinitely expand their period
into a system of isolated 2$\pi$- domain
walls separating domains with the magnetization
along the applied field
(Figs. \ref{lattice} (d), \ref{mcurves} (a))
\cite{Dz64,JMMM94}.

\section{Chiral flux-lines: the building blocks for Skyrmionic textures}

\subsection{Chiral Skyrmions in cubic helimagnets}

The equations minimizing energy (\ref{density})
include axisymmetric localized
solutions $\psi = \varphi + \pi/2$,
$\theta = \theta(\rho)$, $M$ = const
(Fig. \ref{Skyrmion}, (a), (b))
where the spatial variable
$\textit{\textbf{r}}=
(\rho\cos\varphi;\rho\sin\varphi;z)$
is written in cylindrical coordinates and
the magnetization in spherical coordinates
(\ref{cone1}).
The Euler equation for $\theta(\rho)$
to determine the equilibrium structure of isolated Skyrmions
 \cite{JETP89,JMMM94} is
\begin{eqnarray}
\label{eq}
A \left( \frac{d^2 \theta}{d \rho ^2}
+ \frac{1}{\rho}\frac{d \theta}{d \rho} \right.
\left. - \frac{1}{\rho ^2}\sin \theta \cos \theta  \right)
-\frac{H}{2M}\sin \theta
-\frac{D}{\rho} \sin^2 \theta=0
\end{eqnarray}
with boundary conditions
$\theta(0)=\pi,\, \theta(\infty)=0$.
\begin{figure*}
\includegraphics[width=16cm]{F2v9.pdf}
\caption{Isolated skyrmions in  a nanolayer (a)
and a bulk sample (b) of a cubic helimagnet.
The magnetization profiles $ \theta (\rho)$ (c)
and corresponding  energy densities
$\varepsilon(\rho)$ (d) for
typical solutions of Eq. \ref{eq}.
A cross-section through an isolated skyrmion
shows axisymmetric distribution of the
magnetization (shaded area indicates the core
with the diameter $D_0$ (Inset (c)).
Phase trajectories of skyrmions correspond
to separatrix curves in the phase space
(curve $b-o$)(e).
Typical eigen functions for
radial excitations (f).
}
\label{Skyrmion}
\end{figure*}
\begin{figure*}
\includegraphics[width=16cm]{F2bv3.pdf}
\caption{Magnetization arrangement in Skyrmions
of uniaxial ferromagnets with $n$mm (C$_{nv}$) (a,b),
$\bar{4}$2m (D$_{2d}$)(c), and $n$ (C$_n$) (d) symmetries.
}
\label{uniaxial2}
\end{figure*}
In \textit{non-centrosymmetric uniaxial magnets}  functionals $w_D$
have anisotropic forms depending on
crystallographic class and may include
several terms \cite{JETP89}.
For example, in ferromagnets belonging to
$C_n$ crystallographic classes ($n = 3,4,6$)
the chiral energy $w_D$ includes three Dzyaloshinskii constants
$D_i$ \cite{JETP89},
\(w_D=D_1\,(\Lambda_{xz}^{(x)}+\Lambda_{yz}^{(y)}) +
D_2\,(\Lambda_{xz}^{(y)}-\Lambda_{xz}^{(y)}) +
D_3\,\Lambda_{xy}^{(z)}\).
For all non-centrosymmetric ferromagnets
the equation for isolated Skyrmions $\theta (\rho)$
have the same functional form as Eq. (\ref{eq}),
however, $\psi(\varphi)$ have different solutions
(Fig. \ref{uniaxial2})
\begin{eqnarray}
\label{uniskyrm1}
\psi = \varphi,  \   (nmm), \quad
\psi = -\varphi+ \pi/2, \ (\bar{4}2m),
\quad
\psi = \varphi + \pi/2,  \ (n22),
 \quad
\psi = \varphi + \gamma,  \ (n)
\end{eqnarray}
where $\gamma = \arctan (D_1/D_2)$.
For a complete list of $w_D$ functionals and solutions $\psi (\varphi)$,
see Ref.~\cite{JETP89}).
\subsection{Solutions for Skyrmions}
Typical solutions of Eq.~(\ref{eq}), $\theta (\rho)$,
consist of arrow-like cores
($ \pi - \theta \propto \rho$ for  $\rho \leq L_D$)
and  exponential "tails",
$ \theta \propto  \exp{(-\rho)}$ for
$\rho \gg L_D$ \cite{JETP89,JMMM94}
(Fig. \ref{Skyrmion}).
In  \textit{phase space}
$(\theta, d \theta / d \rho)$
these localized solutions correspond
to \textit{separatrix} trajectories
(e.g. to the curve with initial derivative
$(d \theta / d \rho)_0 = b$ in Fig. \ref{Skyrmion}(e)).
A Skyrmion core diameter $D_0$ can be defined
as two times the value of $R_0$, which is the coordinate
of the point where the tangent at the inflection
point ($\rho_0, \theta_0$) intersects the $\rho$-axis
(Fig.~\ref{Skyrmion}(d)),
in analogy to definitions for domain wall width \cite{Hubert98}.
Skyrmion energy per unit length and diameter are
\begin{eqnarray}
\label{energySkyrmion}
E = 2\pi \int_{0}^{\infty} \varepsilon (\theta, \rho) d \rho,
\quad \quad
D_0= 2L_0[\rho_0 - \theta_0 (d \theta / d \rho)^{-1}_{\rho = \rho_0}]
\end{eqnarray}
where $\varepsilon (\theta, \rho)$ is the "linear"  energy density
\cite{JMMM94}.
The energy density distributions $\varepsilon ( \rho)$
(Fig.~\ref{Skyrmion}(f)) reveal two distinct
regions: positive energy "bags" are concentrated
in the Skyrmion center and are surrounded by
extended areas with negative energy density,
where the DM exchange dominates.
The radial stability of Skyrmion solutions has been
proved by solving the corresponding spectral
problem \cite{JMMM94}, which also yields
excitation modes of Skyrmion cores, as
in Fig. \ref{Skyrmion}(f),

\subsection{Analytical results for the linear ansatz}
For localized solutions $\theta (\rho)$ a linear trial
function has proved to be a suitable approximation \cite{JETP89}.
With ansatz \(\theta = \pi (1 - \rho/R) \ \  ( 0 < \rho < R), \quad
\theta = 0  \  \  (\rho > R)\)
the Skyrmion energy (\ref{energySkyrmion}) is
reduced to a quadratic potential
\begin{eqnarray}
\label{ansatz2}
E (R)  = E_0 + \frac{\alpha H}{M} R^2  - \frac{\pi}{2} |D| R,
\quad  \quad
 R_{\mathrm{min}} = 0.42 L_D  H_D/H,
\quad
E_{\mathrm{min}} = E_0 - 4.15 H_D/H
\end{eqnarray}
where
$E_0 = [\pi^2 + \mathrm{Cin}\,2 \pi]A/2 = 6.154\,A$,
$\alpha = (1-  4/\pi^2)/2 =0.297$,
and the parabola vertex
$(R_{\mathrm{min}}, E_{\mathrm{min}})$
determines the minimum of energy (\ref{ansatz2}).
This simplified model offers
an important insight into the mechanisms underlying
the formation of chiral Skyrmions.
The exchange energy $E_0$ does not depend
on the Skyrmion size and contributes a
positive energy associated with its distorted core.
The equilibium Skyrmion size
arises as a result of the competition between
chiral and Zeeman energies:
$R_{\mathrm{min}} \propto |D|/H$,
and becomes zero in centrosymmetric systems ($D =0$).
Isolated solutions of Eq.~(\ref{eq}) with
positive energy exist at high applied magnetic
fields. With decreasing fields
equilibrium energy $E_{\mathrm{min}})$ for Eq.~(\ref{ansatz2})
decreases and becomes \textit{negative}
at a critical value $H_S$.
Below $H_S$ the Skyrmions condense into a lattice
at the exact value $H_S$=0.80132 (Table \ref{table2})) \cite{JMMM94}.

\section{Skyrmion lattices: where
does "double-twist" become beneficial?}
\subsection{Solutions for bound Skyrmions}
The equilibrium parameters of a
Skyrmion lattice are derived from a system
of differential equations for
$\theta (x,y)$, $\psi (x,y)$ minimizing
the system energy.
For Skyrmions at low temperatures
the \textit{circular cell} approximation
provides a very elegant and reliable
approximation \cite{JMMM94}.
In this method the lattice cell
is replaced by a circle of equal area.
The equilibrium parameters of Skyrmion lattices
are derived by integration of Eq.~(\ref{eq})
with boundary conditions $\theta(0) = \pi$,
$\theta(R) = 0$ and a subsequent minimization
of the lattice energy density
$W_{S} =(2/R^2)\int^{R}_{0} w(\rho)\rho d\rho$
with respect to the cell radius $R$ \cite{JMMM94}.
The solutions for magnetization profiles $\theta( \rho)$ (c) and
the lattice periods (d) are plotted in Fig. \ref{lattice}.
The reduced perpendicular magnetization of
a Skyrmion lattice averaged over the Skyrmion
cell $m_{\small{S}} = (2/R^2)\int_0^{R} \cos(\theta) \rho d \rho$
is plotted together with the magnetization of the helicoid,
$m_{\small{H}} $  (Fig.~\ref{mcurves}).
Contrary to the helicoid the Skyrmion lattice has a finite
magnetization at zero field, $m_{\small{S}}  (0)$ = 0.124.
\begin{figure*}
\includegraphics[width=16cm]{F3v5.pdf}
\caption{Skyrmion lattices:
(a) unit cells have axisymmetric
magnetization structure near the center;
(b) a thin layer with a hexagonal lattice.
The magnetization
profiles  $\theta(\rho/L_D)$ of a
Skyrmion cell (solid blue lines)
and of an isolated Skyrmion
(dashed red lines) for different values
of applied magnetic field (c).
Equilibrium sizes of the cell core ($D_0$)
and lattice period compared to helicoid periods (d).
Inset shows the transformation
of the hexagonal lattice into a
set of isolated Skyrmions in
increasing magnetic field.
Snapshots correspond to the
field values indicated
in the diagram for period $L/L_D$ vs. applied field $(H/H_D)$
by hollow circles (d).
}
\label{lattice}
\end{figure*}

\begin{table}[h]
\caption{
\label{table2}
Critical fields and characteristic parameters of the hexagonal Skyrmion lattice:
$H_1$ transition field between the helicoid and Skyrmion lattice;
$H_S$ saturation field of the Skyrmion lattice; last column for
isolated Skyrmions as excitations of the saturated state.
}
\begin{center}
\begin{tabular}{llllll}
\br
&& $H_1$& $H_{S}$& \\
 Reduced magnetic field, $H/H_D \quad \quad $&$0$&$0.216 \quad \quad$&$0.801 \quad \quad$&1.4\\
 \mr
Lattice cell period, $L/L_D$&$1.376 \quad \quad$&1.270& $\infty$ & - \\
Core diameter, $D_0/L_D$&1.362&1.226& 0.920 & 0.461 \\
Averaged magnetization, $m_{\small{S}}$ &0.124& 0.278&1&1  \\
\br
\end{tabular}
\end{center}
\end{table}
\subsection{Skyrmions compete with helicoids. Transition field $H_1$}
In the Skyrmion lattices rotation of the magnetization in two directions
leads to a larger reduction in the Dzyaloshinskii-Moriya energy than
single-direction rotation in helical phases.
On the other hand, such
\textit{double-twist} modulations increase the exchange energy.
The equilibrium energy of the Skyrmion cell at zero field
$\widetilde{w}_{S}(\zeta )=(2/\zeta^2)\int^{\zeta}_{0} w(\rho)\rho d\rho$
plotted as a function of the distance from the center $\zeta$
(Fig. \ref{uniaxial0} (a)) shows that an energy excess near
the border outweighs the energy gain
at the Skyrmion center (for details see \cite{Nature06}).
At higher magnetic fields, however, the Skyrmion lattice has
lower energy than the helicoid. The first order transition
between these two modulated states occurs at $H_1 = 0.2168 H_D$  \cite{JMMM94}.
\begin{figure}[h]
\includegraphics[width=16 cm]{F5v6.pdf}
\caption{\label{uniaxial0}
Local energies $\widetilde{w}(\zeta)$ of
the Skyrmion lattice and helicoid at zero field (a).
Below the critical field $H_S$ the negative
internal energies of Skyrmions overcome the repulsive
Skyrmion-Skyrmion interaction and a dense packed
(hexagonal) lattice is created (b).
Under the influence of induced uniaxial anisotropies ($E_u$)
the Skyrmion lattice becomes globally stable in
a broad range of the applied fields
($K_0$ is the critical value for uniaxial
distortions suppressing the cones in zero field)
(adopted from \cite{PRB10}) (c).
}
\end{figure}

\subsection{"Receding" Skyrmions. Critical field $H_S$}
Properties of the Skyrmion lattice solutions
are collected in Figs.~\ref{lattice}, \ref{mcurves}
and in Table \ref{table2}.
With increasing magnetic field,
a gradual localization of the Skyrmion core $D_0$
is accompanied  by the expansion of the lattice period.
The lattice transforms
into the homogeneous state by infinite
expansion of the period at the critical field $H_S = 0.80132 H_D$.
Remarkably,
the Skyrmion core retains a finite size,
$D_0 (H_S) = 0.920 L_D$ and
the lattice releases a set
of \textit{repulsive} isolated Skyrmions
at the transition field $H_S$,
owing to their topological stability.
These free Skyrmions can exist
far above $H_S$.
On decreasing the field again below $H_S$,
they can re-condense into a Skyrmion lattice
(Fig. \ref{uniaxial0} (b)).
A similar type of sublimation and resublimation
of particle-like textures occurs in
helicoids at the critical field
$ H =H_H /H_D= \pi^2/8 = 0.6168$:
the period infinitely expands and the helicoid
splits into a set of isolated 2$\pi$ domain
walls or kinks \cite{Dz64,JMMM94}.

This peculiar transformation of chiral modulations
into homogeneous states constitutes
a \textit{nucleation} type of continuous phase transition,
according to a classification introduced by
De Gennes \cite{DeGennes75}.
In contrast to the common \textit{instability}
type of continuous (2nd order) transitions
that is described by the amplitude of a sole
fundamental mode acting as the order parameter,
the nucleation transitions retain a full spectrum
of lattice modes at the transition.
Magnetic-field-driven transitions of multidomain
states into the homogeneous phase also belong
to the nucleation type of phase transformations \cite{Hubert98}.

\subsection{Stabilization effects of magnetic anisotropy}
For isotropic model $w_0$ (\ref{density}) the cone
phase (\ref{cone1}) is the global minimum in the whole range
of the applied fields where the modulated
states exist ($ 0 < H < H_D$).
The helicoids and Skyrmion lattices can exist
only as metastable states.
Under the influence of anisotropies (\ref{density}),
however, the Skyrmion
lattices may gain thermodynamic stability
in a certain interval of applied
fields, for details see \cite{PRB10,confinement10}.
E.g., uniaxial anisotropies can be induced
in bulk helimagnets by uniaxial stresses or they
arise due to surface effects in thin layers.
These anisotropies are described by
$ E_u = K (\mathbf{M}\cdot \mathbf{n})^2$,
where $K$ is a constant of uniaxial anisotropy,
$\mathbf{n}$ is a unity vector along the distortion axis).
Such uniaxial distortions
suppress the cone phase in cubic helimagnets
and establish global stability of the Skyrmion lattice in broad
ranges of the thermodynamic parameters
(Fig. \ref{uniaxial0} (c)) \cite{PRB10}.

\section{On the observation of  Skyrmionic and helical
textures in Fe$_{0.5}$Co$_{0.5}$Si nanolayers}

\vspace{2mm}
Real-space images of Skyrmion states
in a thin layer of cubic helimagnet Fe$_{0.5}$Co$_{0.5}$Si
have recently been obtained by using Lorentz transmission
electron microscopy \cite{Yu10}.
This is the first clear experimental
manifestation of \textit{chiral Skyrmion states}.
The first-order transition
of a helicoid into a Skyrmion lattice and  its subsequent transformation
into a system of isolated Skyrmions  observed in bias magnetic
fields (Figs. 1, d-f, 2, 3 (a-d) in \cite{Yu10})
are in excellent agreement with the theoretical
predictions on the behavior of Skyrmions and the field-driven
transitions into densely packed Skyrmion lattices
according to the magnetic phase diagrams calculated
earlier \cite{JMMM94,Nature06} (Fig.~\ref{mcurves}).

In the experiments, the thickness 20~nm of
the magnetic layer Fe$_{0.5}$Co$_{0.5}$Si
is much smaller than the helix period $L_D$ = 90 nm \cite{Yu10}.
But, even for such a small thickness, the conical state
propagating only for a fraction of a period perpendicularly
through the layer has lower energy than a Skyrmion lattice,
absent additional effects that stabilize it in applied fields.
Usually in magnetic nanolayers strong perpendicular
uniaxial anisotropy arises, either as a result of surface
effects \cite{Johnson96} or of lattice strains.
Thus, induced anisotropies give a possible explanation
for the experimental observation of the Skyrmions in
these Fe$_{0.5}$ Co$_{0.5}$Si layers, in accordance with
the phase diagram for cubic helimagnets with uniaxial
distortions (Fig.~\ref{uniaxial0} (c)) \cite{PRB10}.
\begin{figure*}
\includegraphics[width=16cm]{F4v9.pdf}
\caption{The ideal magnetization curves
for a bulk sample (based on results of
\cite{JMMM94}) (a) and for a thin layer (b)
of a cubic helimagnet with suppressed cone phases.
Solid lines indicate the thermodynamically
stable states; dashed lines in Fig. (a), metastable
configurations.
The 1st order transition line at $H_1$
in the thin layer
is expanded into a region of multidomain states.
Fragments of experimentally observed images
\cite{Yu10} are in complete agreement
with theoretically calculated magnetization
curves. The patterns marked by ``(!)''
display \textit{isolated chiral Skyrmions}.
}
\label{mcurves}
\end{figure*}
Fig.~\ref{mcurves} presents the magnetization curve for a bulk isotropic
helimagnet (a) (based on results of \cite{JMMM94}, Fig. 12)
and the corresponding magnetization curve for a thin layer
involving demagnetization effects \cite{Hubert98} (b).
Compared to theoretically calculated
values in a bulk material ($H_S$, $H_H$) the
corresponding critical fields in a thin layer
are shifted, and their values can be estimated
as $\widetilde{H}_{S(H)} = H_{S(H)} + 4\pi M$.
Due to demagnetization effects multidomain states
can be stabilized in the vicinity of the transition
field $H_1$ \cite{JMMM94}. The boundaries of these
regions with coexisting phases can be estimated as
$\widetilde{H}_{1H}= H_1 + 4\pi M m_{\small{H}}(H_1) $,
$\widetilde{H}_{1S}= H_1 + 4\pi M m_{\small{H}}(H_1) $.
The magnetizations of the competing phases at the transition field
equal  $m_{\small{H}}  (H_1)$ = 0.111 and $m_{\small{S}}  (H_1)$ = 0.278.
The jump of the magnetization during the transition equals
$\Delta M = [ m_{\small{S}} (H_1) - m_{\small{H}} (H_1)] M = 0.167 M$,
i.e., it reaches about 17 \% of the saturation value.

The magnetization curves in Fig.~\ref{mcurves} are constructed
for ideally soft magnetic material under the condition that only
the equilibrium states are realized in the magnetic sample.
In real materials the formation of the equilibrium
states  is often hindered (especially during the phase transitions),
and evolution of metastable states and hysteresis effects play
an important role in the magnetization processes.
Particularly, the formation of the Skyrmion lattice below $H_S$
can be suppressed. Then isolated Skyrmions exist below this
critical field. At a critical field H$_{El}$
the Skyrmions become unstable with respect to elliptical
deformations and "strip-out" into isolated 2$\pi$ domain walls.
In a bulk material $H_{El} = 0.534 \, H_D$
(indicated in Fig.~\ref{mcurves} with a red arrow).
In a thin layer, one estimates
$\widetilde{H}_{El} = H_{El} + 4 \pi M$.
As discussed earlier \cite{JETP89,Nature06,JMMM94}
the evolution of chiral Skyrmions in magnetic fields
has many features in common with that of bubble domains
in perpendicular magnetized films,\cite{Hubert98} and
with Abrikosov vortices in superconductors \cite{Blatter94}.

The fragments of images from Ref. \cite{Yu10} (Fig.\ref{mcurves} (b))
reflect in details theoretically
predicted evolution of the chiral modulations
in the applied magnetic field:
the helicoid phase is realized at low fields (region (I));
at higher field this transforms into the Skyrmion lattice
(region (II)) via an intermediate state
($\widetilde{H}_{1H}< H < \widetilde{H}_{1S}$);
finally the Skyrmion lattice by extension of the
period transforms into the homogeneous phase
where isolated Skyrmions still exist as topologically
stable 2D solitons.

Two patterns indicated in Fig.~\ref{mcurves} (b)
with exclamation mark manifest the main result
of Ref.~\cite{Yu10}:  the first images of
\textit{static two-dimensional localized states}
aka \textit{chiral Skyrmions}!
In Ref.  \cite{Yu10} this result has been overlooked
and misinterpreted as a coexisting ferromagnetic
and Skyrmion lattice phases.
As it was expounded in the previous section, the
transition of the Skyrmion lattice into the
homogeneous state is a \textit{continuous} transition,
but of the particular \textit{nucleation} type.
Such transitions exclude the formation of coexisting states.

The condensed Skyrmion phases in the micrograph of  Ref.~\cite{Yu10}
also appear as heavily distorted densely packed two-dimensional
lattice configurations. This is expected for Skyrmionic matter.
As these mesophases are composed from elastically
coupled radial strings, dense Skyrmion configurations
generally do not form ideal crystalline lattices
but various kinds of partially ordered states,
e.g. hexatic ordering implying only orientational order
of bonds without positional long-range order, or other
glassy  arrangement following standard arguments
put forth for the similar vortex matter in type-II superconductors
\cite{Blatter94}.
The observation derives from the particle-like (or string-like)
nature of Skyrmions and suggests that Skyrmionic mesophases may
display rich phase diagrams.

\section{Chiral modulations near the ordering temperature}

\subsection{Solutions for high-temperature Skyrmions.
Confinement temperature $T_p$}

Near the ordering temperature the magnetization
amplitude varies under the influence of the applied
field and temperature. Basic properties of chiral
modulations in this region can be derived by
minimization of isotropic model
$w = w_0 - f(M)$ (\ref{density}) with respect
to \textit{all} components of the magnetization vector
($M, \theta, \psi$).
Within this model  axisymmetric isolated structures
are described by equations
$\psi (\varphi)$ (\ref{uniskyrm1})
and the solutions of the Euler equations
for $\theta(\rho)$, $M(\rho)$ \cite{Nature06,confinement10}
\begin{eqnarray}
A \left( \frac{d^2 \theta}{d \rho^2} +  \frac{1}{\rho} \frac{ d \theta}{d \rho}
-\frac{\sin \theta \cos \theta}{\rho^2} \right)
 - D\frac{2}{\rho}\sin ^2 \theta - H\sin \theta
 + A \frac{2}{M} \frac{d M}{d\rho} \left( \frac{ d \theta}{d \rho} -1 \right) = 0,
\nonumber \\
A \left[\frac{1}{M} \left(\frac{d^2 M}{d \rho^2} +  \frac{1}{\rho} \frac{ d M}{d \rho} \right)
+\left(\frac{ d \theta}{d \rho}\right)^2 + \frac{\sin^2 \theta}{\rho^2} \right]
+D \left[ \frac{ d \theta}{d \rho} +\frac{\sin \theta \cos \theta}{\rho} \right]
-\frac{H}{M} \cos \theta + \frac{1}{M} \frac{\partial f }{\partial M }= 0
\label{Euler1}
\end{eqnarray}
With $ f(M) = a_1 M^2 + a_2 M^4$
($a_1 = J (T_c-T)$, $a_2 > 0$) which is commonly
used to describe magnetization processes in cubic helimagnets at
high temperatures the solutions of Eqs. (\ref{Euler1}) have been
investigated in \cite{Nature06,confinement10}.
With boundary conditions $\theta(0) = \pi$, $\theta(\infty) = 0$,
$M(\infty) = M_0 $, Eqs.~(\ref{Euler1}) describe the structure of isolated Skyrmions.
The magnetization of the homogeneous
phase $M_0$ is derived from equation  $2a_1 M+ 4 a_2 M^3 - H =0$.
It is convenient to introduce new parameters,
an effective temperature $a = (T_c - T)/T_D$ and
$H_0 =  \left(D^2/2 A\right)^{3/2}/\sqrt{a_2}$ where
"chiral shift", $T_D = D^2/(8 A J)$ is a characteristic
temperature equal to the difference between the ordering temperature
of the helimagnet ($T_N = T_c + T_D$) and the ferromagnet
with $D =0$ ($T_c$).

Typical solutions  $\theta (\rho)$, $M (\rho)$ of Eqs. (\ref{Euler1})
demonstrate an inhomogeneity of $M$ associated with the
Skyrmion cores, Fig.~\ref{conf1}(a).
Detailed analysis of the solutions for
isolated skyrmions has revealed a number
of remarkable phenomena imposed by interplay
between  the angular ($\theta$)
and modulus ($M$) order parameters
\cite{confinement10}.

\begin{figure*}
\includegraphics[width=16cm]{F6v4.pdf}
\caption{Typical solutions
for magnetization profiles $\theta (\rho)$,
$M(\rho)$ (Eqs. \ref{Euler1}) (a).
Magnetic phase diagram of  an \textit{isotropic}
helimagnet  near the ordering temperature
includes areas with \textit{repulsive} (I),
\textit{attractive} (II) skyrmions, and
\textit{confined} skyrmion states (III) (b).
\textit{Confinement temperature} $T_L$ separates
the main part of phase diagram with regular
chiral modulations ($T < T_L$) from the region
of "precursor state" ($T_L < T < T_N$).
The confined skyrmion textures include square
(\textit{half}-skyrmion) (c) and hexagonal
(d) lattices.
}
\label{conf1}
\end{figure*}

(i) \textit{Collapse of Skyrmions at high fields.}
The solutions of Eqs.~(\ref{Euler1}) exist only
below a critical \textit{collapse} line (Fig. \ref{conf1}).
As the applied field approaches this line the magnetization
in the Skyrmion center $M(0)$ gradually shrinks, and
as $M(0)$ becomes zero the Skyrmion collapses.
This is in contrast with low-temperature
Skyrmions, which exist without collapse even at very large
magnetic fields \cite{JMMM94} because the stiffness
of the magnetization modulus maintains topological stability of Skyrmions.
At high temperatures the softening of the magnetization
allows the order parameter to pass through zero
and to unwind the Skyrmion core.

(ii) \textit{Crossover of Skyrmion-Skyrmion interactions.}
The coupling of Skyrmions is repulsive in a broad
temperature range, but it starts to oscillate
at high temperatures,  Fig. \ref{conf1} (b).
The equation to describe this crossover in
the inter-Skyrmion coupling is
\begin{eqnarray}
\label{crossover}
\eta H^* = \sqrt{2 \pm \sqrt{3+4 a}} \;( a + 1 \pm \sqrt{3+4 a}/2 ),
\quad
T_L =  T_N - 4 T_D,
\quad
T_D = D^2/(8 A J).
\end{eqnarray}
For $a < -0.5$ this yields a crossover line in the
phase diagram, Fig.~\ref{conf1}(b).
This is ended in turning point (-0.5, 0).
Another turning point (-0.75, $\sqrt{2} \eta/4$)
indicates the lowest temperature where
Skyrmions can attract each other
at certain distances,
owing to the oscillatory character of
their interactions, the \textit{confinement temperature},
$T_L$ (\ref{crossover}).
This results in energetic confinement of Skyrmions,
as they can form clusters to lower their energy.

(iii) \textit{Confinement.}
For $-0.5 < a < 0.25$
line (\ref{crossover}) delimits
a small pocket (III) in the
vicinity of the ordering temperature.
Within this region Skyrmions
can exist only as bound states.
In the \textit{confinement} region
Skyrmionic states drastically differ
from those in the main part of the phase diagram.
Confined Skyrmion textures
(hexagonal and square half-Skyrmion lattices,
Fig. \ref{conf1} (c))
arise from the disordered
state through a rare case of an
\textit{instability-type nucleation transition},
and their field-driven transformation
evolves by distortions of the modulus
profiles $M(\rho)$ while the equilibrium
periods of the lattices do not
change strongly with increasing
applied field (for details
see \cite{confinement10}).
\subsection{Confinement phenomenon and precursor states}
The magnetic phase diagram of isotropic
helimagnet (Fig. \ref{conf1} (b))
includes two distinct temperature intervals:
(i) in the main part ($0 < T < T_L$)
the rotation of the local magnetization vector
determines the chiral modulation, while
the magnetization amplitude remains constant.
(ii) At high temperatures ($T_L < T < T_N$)
spatial variation of the magnetization modulus
becomes a sizeable effect, and strong interplay
between $M$ and angular variables is the main
factor in the formation and peculiar behaviour
of chiral modulations in this region.
The \textit{confinement temperature} $T_L$
\cite{confinement10} provides the scale
that delineates the border between these
two regimes in the phase diagram.
The characteristic temperature $T_L$
is of fundamental importance for chiral magnets.
It is of the same order of magnitude as
the temperature interval $(T_N- T_c) \propto D^2/A$,
where chiral couplings cause inhomogeneous
precursor states around the magnetic order temperature.
Due to the relativistic origin
and corresponding weakness of the DM exchange
the shift $\Delta T =T_N - T_L$ is small
(for MnSi this is estimated to be about 2 K \cite{confinement10}).

Since discovery of chiral modulations
in MnSi family of B20 compounds
numerous magnetic anomalies have been observed
near the ordering temperature
of these helimagnets \cite{Lebech89}.
However, the nature of the chiral
spin textures in this region of the phase diagram
is largely unresolved in experiment.
During last years these \textit{precursor} effects
have become a subject of intensive investigations
\cite{Lamago06,Muhlbauer09}
motivated by the expectation to identify Skyrmionic states \cite{Nature06}.

The rigorous solutions for helical and Skyrmionic
modulations recently derived within
the basic model $w = w_0 +f(M)$ (\ref{density})
and the theoretical description
of novel phenomena attributed
to this region \cite{confinement10}
give a first explanation of these anomalies.
The stabilization of Skyrmionic textures and
other chiral modulations near magnetic ordering
involve the confinement of localized state, but also
a strong influence of minor energy contributions.
E.g., the phase diagram of cubic
helimagnets contains defected half-Skyrmion
lattices at small applied fields and densely
packed full-Skyrmion lattices.
However, the conformation of these mesophases
and their relative stability will depend on various
small additional effects, such as dipolar couplings,
thermal fluctuations, quenched defects etc.
It would be naive to expect these phase diagrams to
be simple and to be determined by one dominating mechanism
able to stabilize Skyrmions over helicoids.
Owing to the hierarchy of interactions, the properties and
stability of the Skyrmion cores will always be provided
by exchange and Dzyaloshinskii-Moriya couplings via
the double-twist mechanism.
But the mesophase formation will be ruled by much
weaker couplings, owing to the localized and frustrated
nonlinear character of these solitonic entities.

\section{Topogical solitons, vortices, Skyrmions...}

\subsection{How the chiral Skyrmion got its name}

Localized solutions of Eq.~(\ref{eq}) have been initially introduced
under name \textit{magnetic vortices} \cite{JETP89} because, to a certain
extent, they are similar to 2D topogical defects investigated
in magnetism and known as "two-dimensional topological solitons" or "vortices"
(e.g.  well-known Belavin-Polyakov solutions
for magnetic vortices \cite{Belavin75}).
On the other hand, the term \textit{skyrmion}
has been conceived in a field rather distant from condensed-matter physics
and  initially was related to the localized solutions derived by
Skyrme within his model for low-energy dynamics of mesons and baryons \cite{Skyrme61}.
In fact, the Skyrme model \cite{Skyrme61}
comprises three spatial dimensions,
and the name ``baby skyrmion'' was used
by some field theorists to distinguish
two-dimensional localized states from ``mature''
three-dimensional solutions in the original Skyrme model \cite{Skyrme61},
both types of them being topological static solitons.
During the last decades the "skyrmion" has
progressively won currency in general physics
to designate \textit{any} non-singular localized
and topologically stable field configuration.
Complying with this trend, in 2002 we have renamed
"chiral magnetic vortices" into "chiral skyrmions" \cite{PRB02}.
Ironically, the fate of the localized states
near ordering transition, as they can decay
by longitudinal magnetization processes,
betrays that these ``chiral Skyrmions''
are not topologically stable, at high enough temperature.

\subsection{'What exactly is a chiral skyrmion'?}

Thus, the term \textit{skyrmion}
is an umbrella title for smooth localized
structures to distinguish them from
singular localized states,
e.g., disclinations in liquid crystal textures \cite{DeGennes}.
This convention provides only
a formal \textit{label} for a large
variety of very different solitonic states
from many fields of physics \cite{skyrmion10}.
With respect to the subject of this paper
a "skyrmion"  designates well-defined solutions
of Eq.~(\ref{eq}) which are
\textbf{ (i) localized}, \textbf{(ii) axisymmetric},
and have \textbf{ (iii)  fixed rotation sense}.
Examples of chiral skyrmions have been presented
in Figs. \ref{Skyrmion}, \ref{uniaxial0}.
The axisymmetric structure of
the skyrmion core and its localized character are retained
in bound states as skyrmion lattices \cite{Nature06,JPC10}.
This reflects the particle-like character
of chiral skyrmions and the most general
features of their energetics (Fig. \ref{uniaxial0} (a),
and Refs. \cite{Nature06,JPC10}).
Skyrmions as countable entities can be arranged in
various ways to create dense magnetic textures.
This is the essence of Skyrmionic matter and entails
the possibility to form a variety of mesophases
with crystalline, but also with liquid-like or
glassy large-scale structure in chiral magnetism.

Alternative approaches construct skyrmionic
textures from crossing plane waves (helices) as fundamental
modes of so-called spin-spiral crystals \cite{Muhlbauer09,Binz06}.
In \cite{Muhlbauer09}  a skyrmion lattice is composed
of three helices superimposed under an angle of 120 degree.
According to \cite{Muhlbauer09} such a "triple-Q antiskyrmion lattice"
ansatz even reaches the global free energy minimum in the A phase of MnSi.
This construction is inconsistent with the properties of Skyrmions.
Chiral skyrmionic textures incorporate isolated or embedded axisymmetric lines.
The notions of a "spin-spiral crystal" \cite{Binz06} or
"multi-Q skyrmions"  \cite{Muhlbauer09}
are misconceptions because they blend
mutually exclusive ideas of particle-like localized
skyrmions from one side and delocalized plane waves from the other.
The radial structure of the chiral Skyrmion cannot
be reduced to a superposition of harmonic helical waves.
Calculations based on such multi-Q ansaetze
predict incorrect phase diagrams.
E.g., the hexagonal Skyrmion lattice is stable in remanent state
at $H=0$ \cite{JMMM94}, but in \cite{Muhlbauer09} an instability at a finite
field is presented for the variational hexagonal spin-spiral solution.
The idea of a spin-spiral crystal
ruled by one harmonic mode, where contributions of
higher-order modes are small in some sense \cite{Muhlbauer09,Binz06},
also is at variance with the nucleation type of transition
actually observed in the chiral Skyrmion textures.
This idea wrongly places the Dzyaloshinskii theory
in a class of models for modulated states
with instability type of transitions.

\section{Conclusions: What makes  chiral skyrmions interesting?}

In non-centrosymmetric magnets
chiral magnetic skyrmions arise as a result of
the specific stabilization mechanism imposed by the
handedness of the underlying crystal structure
\cite{Dz64,JETP89,Nature06}.
In centrosymmetric
magnets such solutions are radially unstable and
collapse spontaneously under the influence
of the applied magnetic field or intrinsic
short-range interactions.
In nonlinear field models skyrmion states
can be stabilized by higher order spatial derivatives
(often this is refered to \textit{Skyrme mechanism}).
In condensed-matter systems
there are no physical interactions providing such
energy contributions. \textit{Chiral interactions}
present a unique mechanism to stabilize skyrmion
states in ordered condensed-matter systems.
This singles out chiral systems (including non-centrosymmetric
magnets, multiferroics, liquid crystals, and metallic nanostructures
with induced chiral interactions) into a particular class of materials
with skyrmionic states.

\textit{Acknowledgment.}
We thank
S. Bl\"{u}gel, S.V. Grigoriev, G. G\"{u}ntherodt, R. Sch\"{a}fer, W. Selke,
and  H. Wilhelm, for discussions.
Support by DFG project RO 2238/9-1 is gratefully acknowledged.

\section*{References}

redpajama_set_name: RedPajamaArXiv


I'm honored to have been asked to speak to you in this lecture series dedicated to Rick Blumberg, a highly respected colleague I worked with for many years in FDA's Office of the Chief Counsel (OCC). I'd like to focus today on a topic I discussed many times with new attorneys joining OCC—the role of the federal agency attorney.
Of course, federal agency attorneys have more than one "role" or function, and over the decades there have been interesting articles reflecting different perspectives. I spoke with OCC attorneys on the topic against a backdrop of articles written in the second half of the 20th century—in the 1950's by Alanson Willcox,1 General Counsel of the Federal Security Agency (a predecessor agency to HHS); in the early 1980's by Fred Degnan,2 an OCC lawyer who later has spent many years in the private sector; and in the late 1990's by Thomas McGarity,3 law professor at the University of Texas. We also considered the recommendations for Offices of General Counsel related to the National Performance Review4 conducted by the Vice President's office in 1993. Although the authors wrote with different focuses and perspectives, their views reflect ongoing interest in the particular roles of federal agency attorneys for more than half a century. I'm certain the debate began well before the 1950's and continues today. Looking back at issues raised during the second half of the 20th century helps us consider similar issues facing federal agency attorneys now.
So, what about the "what," "how," and "why" of agency lawyers' roles? Elements of these aspects are inevitably intertwined and not always distinguishable. At the highest level of generality "what" agency lawyers do is to provide legal services. The tasks performed are familiar to you all. Here at an enforcement conference you might think especially about agency attorneys' work related to enforcement actions: analyzing statutes, regulations, and judicial opinions; reviewing evidence to determine whether an action is supportable; determining the likelihood of success in court; presenting the matter in a referral to the Department of Justice (DOJ); working with the DOJ attorneys to prepare witnesses, to conduct discovery, to draft complaints or indictments, briefs, arguments to the court or jury, appeal papers; and much more. Throughout, OCC's litigators apply facts to legal standards, draft, edit, construct compelling narratives so that the judicial decisionmakers can understand the public health and legal import of what has happened, and otherwise advocate and negotiate resolutions. OCC litigators similarly play key roles in defensive litigation, again working closely with DOJ attorneys and applying those same outstanding skill sets throughout the course of a case. OCC attorneys also have major roles in administrative litigation, such as civil money penalty proceedings, which have become much more numerous in the tobacco product regulatory context.
At this conference you might also think about the ongoing advice OCC attorneys give to FDA's various compliance programs, including by reviewing the agency's manuals on inspections and on regulatory programs. Especially when there is a new program because of a change in the law or a public health tragedy, or both, agency attorneys help construct strategic plans designed to ensure compliance by regulated parties. Examples that come readily to mind include FDA's plans for inspection and regulation of tobacco-related facilities after the Tobacco Control Act was enacted and of compounding facilities after the Compounding Quality Act became law. Of course, agency attorneys play a major role in advising FDA officials about ongoing inspectional and other individual compliance activities. In addition to having central roles in decisions related to prosecutorial discretion involving particular cases, OCC attorneys provide significant advice on broader enforcement discretion approaches embodied in enforcement policy documents.
As Deputy Chief Counsel for Program Review, I also had the privilege of seeing OCC attorneys apply their considerable knowledge and skills to FDA's non-litigation related matters. OCC's counselors also analyze statutes, regulations, and court opinions as they advise FDA officials on programs throughout the agency. They review and revise many different agency documents: regulations, guidance documents, citizen petition responses, warning letters, other correspondence, press releases, congressional testimony and responses to congressional inquiries, proposed legislation, and more. OCC counselors also advise FDA officials in the context of certain individual product reviews, where again the facts presented are analyzed in relation to the legal standard and the adequacy of the administrative record is attended to.
In various contexts, OCC attorneys prepare legal memos in response to agency officials' inquiries or proposed courses of action. OCC attorneys also help agency officials with strategic planning for important agency initiatives. For example, Commissioner Gottlieb has identified battling the current opioid epidemic and strengthening the generic drug review program as key agency initiatives. OCC attorneys have worked with agency officials to help develop strategic plans to advance these extremely important initiatives. OCC attorneys also have a vitally important role in helping FDA respond to public health emergencies. Whether the situation involves foodborne illness outbreaks, natural disasters, intentionally harmful actions such as terrorism, or serious risks of infectious disease such as pandemic influenza, OCC lawyers work in close cooperation with FDA and other federal agency officials to protect the public health. In all of these tasks, OCC attorneys apply their well-honed skills, such as critical thinking, writing, editing, determining multiple legal options, and applying facts within the legal framework.
In an agency like FDA, there are generally perspectives focused on three main elements: science, policy, and law. Agency scientists tend to describe FDA as a "science-based" agency. Policy makers usually call it a "public health" agency. And lawyers most often call it a "regulatory" agency. Of course, everyone would agree that FDA is all of these, and most agency decisions inevitably involve science, policy, and law. Depending on the situation, one element or another may dominate, but in general the goal is to have a positive public health outcome that is scientifically sound and legally supportable. Who gets to decide in any given case that the decision satisfies the relevant criteria can vary.
The predominant thinking has been that the agency's lawyers should not be policy makers. As the HHS Office of the General Counsel (OGC) philosophy statement puts it: "Our job is to provide legal advice and not to make policy."5 But this is not to say that agency lawyers have no role in policy decision-making. At one end of the spectrum, Willcox, writing more than half a century ago, argued that even in what he called "nonregulatory programs," those providing benefits such as Social Security benefits, what "may well be the most important [function of lawyers] … lies in the making of a substantive contribution to administrative policy itself. . . .[L]awyers should play a part in shaping policy."6 Willcox took it for granted that in a regulatory agency the lawyers would make a substantive contribution to policy, likely "automatically or unconsciously, as a result of the agency's constant concern with the courts."7 Maybe the essence of this matter lies not in asking whether lawyers should have any role in policy matters, but in asking what that role should be and how it should be implemented.
This question brings us to consideration of the organizational structure of most agencies. That is, in most agencies, including HHS and FDA, those providing legal services are centrally organized within the office of the agency head. HHS OGC is part of the Office the Secretary and FDA's Office of the Chief Counsel is simultaneously the Food and Drug Division of OGC and part of FDA's Office of the Commissioner. This structure reinforces the importance of consistency in legal advice and defines the hierarchy of decision-making about legal issues. But it also sets the OGC attorneys structurally apart from those assigned particular policymaking functions.
The HHS OGC philosophy statement acknowledges attorneys' role in policy by declaring "[t]o the extent that we provide advice about policy, we make it clear that we understand that it is not the responsibility of attorneys at [OGC] to make policy decisions." During my years at FDA's OCC, I was on occasion asked my views about particular policy choices. If I thought the options under consideration were legally supportable, I—like my OCC colleagues—tried hard to be clear that what I was being asked about and expressing a view on was not a legal issue, but a policy one. Agency decisionmakers need to be confident that agency lawyers are not cloaking personal policy views in the guise of legal advice.
The lines among legal, scientific, and policy issues are not always bright, and these issues are often necessarily intertwined in ways that demand careful, arduous analyses. Building on some work by Rich Cooper, Degnan wrote also about the relationship between law and science and its effects on the relationship between lawyers and scientists.11 Starting from the premise that law is essentially a decision-making device and that science is a process of consensus-building, Degnan argued that, especially in the face of scientific uncertainty, there is an inescapable tension between the lawyer and the scientist.12 Decisionmakers at FDA and other regulatory agencies have to make regulatory decisions based on the facts, scientific understanding, and legal framework existing at the time of the decision. Lawyers often significantly contribute to resolution by asking probing questions about all aspects of the matter, including the nature of the evidence and the scientific support for any position. The lawyers, scientists, and policymakers need to work closely together to understand how the facts of a particular situation fit together within the regulatory framework and how a potential decision makes coherent, defensible sense.
Which brings us to consider the "how" of the agency lawyer's role. Willcox emphasized the importance of "close and informal day-to-day contact"13 between lawyers and administrators. He argued that such a close and collaborative working relationship, in which the lawyer knows the program thoroughly, is essential to providing sound legal advice.14 In the federal government, it can also be critically important for attorneys from various agencies to work collaboratively as the agencies work together to solve major problems. The effective collaborations of agency attorneys from FDA, CDC, USDA, and other agencies were essential to the government's successful protection of Americans from the risks of monkeypox (after imported African rodents had infected prairie dogs in the U.S.) and from Creutzfeldt-Jakob disease (after BSE, or "mad cow's disease," became widespread in other countries).
The much more controversial recommendations of the National Performance Review report concerned the definition of "clients" or "customers" and the use of "franchising and other market mechanisms" to encourage OGC staffs to respond appropriately to their clients.18 The report noted that a federal agency attorney could view the client as, among others, the President, the Secretary of the department, the head of an agency within the department, the mid-level official posing a question to OGC, or the public at large. The report concluded that OGCs have many customers and that confusion among OGC staff about who the client is leads to dissatisfaction among program managers who believe they are the client. The resulting recommendation was to subject the OGC legal advice function to "market forces" by allowing line managers to select who they want to receive their legal advice from.
As you can imagine, the general counsels assembled by ACUS took strong exception to this recommendation. The general counsels argued that defining the federal attorney's client using the private bar's concept of "client" is not the best way to increase the effectiveness of federal OGCs.19 They stated that "the 'client' concept is less suitable to the public sector, where the relationship among agency officials, managers, and lawyers is inherently complex, and the identity of the clients may vary according to the nature or stage of the matter, even in a single case."20 Many of you are familiar, for example, with FDA's separation of functions regulations21 and OCC's practice of assigning different attorneys to represent the initiating FDA Center and the Commissioner in an administrative action involving a hearing. The general counsels also did not believe that "franchising"—permitting program managers to shop for legal services—would "ensure consistent, reliable, high quality legal services."22 These goals take us back to the main reasons for having centralized legal services with OGCs as separate entities within the office of the Secretary and the general counsels in charge of legal opinions.
Which also brings us to the "why" of agency attorneys' functions. When you examine particular OGC statements of purpose you generally see various aspects of the role of agency attorneys combined. For example, the HHS OGC philosophy statement begins: "Our ultimate philosophy is to provide the highest quality legal services in order to assist the Department in establishing and maintaining programs that advance health and human services."23 Similarly, FDA's OCC vision statement begins "The Office of the Chief Counsel promotes FDA's public health mission by providing the highest quality legal services from a staff that is dedicated to integrity, excellence, and respect."24 FDA's statutory mission25 to promote and protect the public health guides not only agency officials' policy views, but also agency lawyers' interpretations of the law.
Degnan pointed to various definitions of the role of lawyers from the courts and from the bar associations' professional responsibility codes.30 He also referred to Supreme Court Justice Brennan's view that the "ultimate role and responsibility of any lawyer are to uphold the law."31 Rick Blumberg believed in vigorously and responsibly upholding the law and inspired generations of OCC lawyers to do so.
Of course, most people inside or outside of the U.S. government are not trying to overthrow the government. But most people do want to do what advances goals they care about, and that desire can make some want to downplay or even disregard legal constraints on how they want to proceed. Here's the bottom line: The role of the federal agency attorney matters because in a democracy the law matters.
Ann H. Wion was in the Office of the General Counsel, Department of Health and Human Services (DHHS), Food and Drug Division, from 1979 to 2017. She received a PhD from Cornell University and a JD from Stanford Law School. After joining the DHHS Office of General Counsel in 1979, she specialized primarily in drug and biologics law. From 1993 to 2013 she was Deputy Chief Counsel for Program Review and oversaw the attorneys providing legal counsel in DHHS on FDA-related matters. From 2014 to 2017 she was Senior Advisor to the Chief Counsel.
Alanson W. Willcox, "The Lawyer in the Administration of Nonregulatory Programs," 13 Public Administration Review 12 (1953) (Willcox).
Frederick H. Degnan, "The Role of the Lawyer in Agency Decision-Making," 36 Food Drug Cosmetic Law Journal 510 (1981) (Degnan).
Thomas O. McGarity, "The Role of Government Attorneys in Regulatory Agency Rulemaking," 61 Law and Contemporary Problems 19 (1998) (McGarity).
"SMC Recommendation 04: Increase the Effectiveness of Offices of General Counsel," Office of the Vice President, Accompanying Report of the National Performance Review, Streamlining Management Control (SMC) (September 1993) (National Performance Review Report); "Recommendations To Increase the Effectiveness of Offices of General Counsel," Office of the Chairman, Administrative Conference of the United States (October 1995) (ACUS Implementation Group Recommendations).
"Philosophy of OGC," HHS OGC New Employee Orientation Manual at 17 (July 2014) (HHS OGC Manual).
Degnan at 512-514 (referring to Richard M. Cooper, "Scientists and Lawyers in the Legal Process" (unpublished paper)).
National Performance Review Report at 27.
ACUS Implementation Group Recommendations at 6.
ACUS Implementation Group Recommendations at 6.National Performance Review Report at 28-29.
ACUS Implementation Group Recommendations at 3-4.
ACUS Implementation Group Recommendations at 3.
ACUS Implementation Group Recommendations at 4.
HHS OGC Manual at 17.
FDA Office of the Chief Counsel, Vision Statement (1999).
See 21 U.S.C. 393 (section 1003 of the Federal Food, Drug, and Cosmetic Act).
Degnan at 515-516 (referring to S.J. Friedman, An Affair with Freedom: William J. Brennan, Jr., p. 328, Atheneum, New York (1967)).


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HomeResourcesSoftware DevelopmentJavaScript Tutorial: Learn JavaScript from ScratchThe Best Guide on How to Implement JavaScript Closures
Lesson 8 of 26By Simplilearn
Programmers frequently use JavaScript Closures. Many might not be aware of exactly what a closure is, although its results are present. Nevertheless, it's always better to have practical knowledge of such concepts.
What Is Variable Scope?
The location of the variable declaration generally defines the scope of a variable. It defines the part of the program where a particular variable can be accessed. JavaScript supports two types of variable scopes -
Global Scope/Variables
Local Scope/Variables
A variable declared outside of a function is attributed to the global scope and is hence accessible from anywhere in your code.
Local Variables or Function Variables are variables declared within a function. These variables are only accessible from that function and its nested functions.
Consider this code to understand the working of global and local variables:
var a = 10
function function_Name(){
var b = 20;
document.write("The sum is " + (a+b) + '<br>')
function_Name()
The output of the above code is:
Variable Scope
In the above example, the variable "a" is a global variable and can be accessed from anywhere in the code. In contrast, variable "b" is a local variable and can be accessed within the function "function_Name" and perhaps its nested functions.
The scope chain is a stack consisting of all the references to the variables for the function being executed.
Now that you know what variable scope let's understand what exactly is JavaScript Closure
What Is a Closure?
Closures in JavaScript is a feature where an inner function has access to the outer function's variables. Closures are important as they define what is and what isn't in the scope of functions. They also control the variables that are being shared with the nested functions.
A closure –
Has access to the variables of its scope.
Has access to the variables of the outer functions.
Has access to the global variables.
function First_func()
function Second_func()
var c = 20+a+b
return Second_func();
var sum = First_func()
The variable "a" is a global variable and can be accessed from anywhere within the program. The variable "b" is declared within the function "First_func" and the variable "c" is declared within the function "Second_func" which is a nested function.
Since a closure is a function that has access to the variable from another function's scope, function "Second_func" can access all the variables a,b and c.
The output of the code above is shown below:
JavaScript Closures
This was a simple introduction to Closures in JavaScript. We recommend you play around with variables and understand their behavior within different scopes.
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Just a wee little bit of Thai politics. Wee little bit!
JAKARTA, Indonesia, Sept. 24 — When soldiers and tanks rolled onto the streets of Bangkok last week and the king appeared on television with the generals, it was not the first time Thailand's wildly popular monarch had given his blessing to a military takeover.
A new and comprehensive history of the Thai modern monarchy, written by an American journalist, Paul M. Handley, and banned in Thailand, argues that in his 60-year reign King Bhumibol Adulyadej has generally exercised a preference for order over democracy.
In doing so, Mr. Handley said, the king has put the preservation of the institution of the monarchy ahead of a democratic Thailand.
The book, "The King Never Smiles," presents a direct counterpoint to years of methodical royal image-making that projects a king beyond politics, a man of peace, good works and Buddhist humility. It also runs counter to how most Thais see their king, as a man of mystique and charisma but also as a bastion of Thailand's moves to modernity.
The book's publisher, Yale University Press, said it came under heavy pressure from the Thai government not to publish it.
The director of Yale University Press, John Donatich, said the pressure included a visit to New Haven by a delegation of Thai officials, including the cabinet secretary general, Bowornsak Uwanno, and the Thai ambassador to the United States, Virasakdi Futrakul.
Mr. Donatich said he ruled out canceling publication of the book, and copies are now on sale in Asian capitals and the United States. But he did agree, he said, to their request that publication be delayed until July, a month after the June 9 celebrations in Bangkok of King Bhumibol's 60th anniversary on the throne and his 80th birthday.
"We didn't want to be accused of exploiting the event," Mr. Donatich said.
The televised coverage of the gala provided an unusual look at the court's unyielding protocol that emphasizes a godly king above ordinary mortals. In one live segment, white-liveried attachés could be seen running ahead of the king to open an elevator door, and then lying prostrate on the floor as the king and his wife passed by.
But his book does note that the king sided with a brutal army takeover in 1976, and in 1992 waited three days before stopping a four-star general from ordering troops to fire on demonstrators.
Much of what Mr. Handley writes is not new, and most of the facts are not in dispute, reviewers and Thai historians say. It is the book's interpretation of the facts that can be disputed, said Thitinan Pongsudhirak, the director of the Institute of Security and International Studies at Chulalongkorn University in Bangkok.
He disagreed, he said, with the argument that the trouble with Thailand's democracy lay with the king.
That Constitution worked well, he said, until Prime Minister Thaksin Shinawatra, ousted last Tuesday, was acquitted of asset concealment by the constitutional court in 2001, a decision the king had nothing to do with.
The Yale press agreed to consider some factual errors that the Thais said were of concern. In the end, Mr. Donatich said, the Thais submitted only three or four minor corrections, like the correct title of a royal daughter's thesis. "He did his homework," Mr. Donatich said of the author.
A portion of a document from the Thai cabinet that appeared on a Thai Web site and appearing, by all accounts, to be authentic, listed the ways the Thais tried to prevent publication, and if it went ahead, how to block the book's distribution in Thailand.
According to the document, the government contacted the American law firm LeBoeuf, Lamb, Greene and MacRae, which told it that publication would be impossible to stop on constitutional grounds. Such efforts would generate unwanted publicity, the lawyers advised.
The document said the authorities had banned the book in Thailand on the grounds that it was a threat to stability. It said Thai officials had contacted the Yale University president, Richard Levin, and had sought the help of former President George H. W. Bush, an alumnus of Yale.
King Bhumibol was born in the United States, grew up in Switzerland and married the lithe, pretty Princess Sirikit, a favorite of the 1960's jet set, who by the 1980's had weathered into a much more fulsome version of a queen with her own court favorites, expensive tastes and pet charities.
The book describes their only son, Vajiralongkorn, as a willful man prone to violence, fast cars and dubious business deals. It may well be, Mr. Handley suggests, that the king's favorite daughter, Princess Sirindhorn, who is a dutiful royal with wide-ranging interests in rural development and architecture, will be his successor.
"Bhumibol's most fundamental failing is the Achilles' heel of every monarchy: he has been unable to guarantee an orderly succession to a wise, selfless, and munificent king like himself," the book concludes.
Oakley, I would really like to hear your comments about the stuff you've read in this book. I read some reviews of the book myself, and refuse to touch it. It would be like consuming poison and my immune system can't stomach that just yet.
Have you read "The Revolutionary King", written by William Stevenson? It's difficult to find, out of print but can probably find used ones.