数学・物理の英語の例文集

英数

Recall that $E_N(Z, \vectorR)$ denotes the ground state energy in the fixed nuclear case, and $E_{N, M}(Z)$ is the ground state energy in the case of dynamic nuclei.

Let us focus on the case $\gamma = 1$.

Let $k$ be a field of characteristic zero and let $\rbk{x_i}_{i \in I}$ be a family of independent variables ed by a finite set $I$.

Derivation of the LT inequalities.

$M$ admits a good filtration if and only if $M$ is finitely generated.

Let $M$ be a left $W_n(k)$-module endowed with a good filtration $\Gamma$.

Normalization conventions other than $N!/(N-k)!$ are often used in the literature.

Since $\abs{p}$ and $1 / \abs{x}$ both scale like an inverse length, there is a critical coupling constant above which even stability of the first kind fails.

We conclude that if $\psi$ is a miniminzer for $\calE$ so is $\psi_{s} / \norm{\psi_s}$.

Let $R$ be a ring (not necessarily commutative).

For our purposes, $\Trace^{(N-k)}$ can be thought of simply as a mnemonic device for (3.1.27), with the relation between kernel and operator given by (3.1.22).

Again new inequalities were needed when it was realized that magnetic fields could also cause instabilities, even for just one atom, if $Z \alpha$ is too large.

The reader is referred to Chapter 13.

Let us briefly sketch how (4.4) is proved in the notationally simple situation.

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What effect does Einstein's relativistic kinematics have?

For given nonnegative $\psi$, we write $\psi = \psi_{s} + \psi_{r}$.

It is a basic postulate of quantum mechanics that the allowed wave functions for a given particle species must belong to some definite permutation symmetry type which is characteristic of that species.

Cryptographically secure pseudo random number generator.

Even today hardly any physics textbook discusses, or even raises this question, even though the basic conclusion of stability is subtle and not easily derived using the elementary means available to the usual physics student.

Without loss of generality.

In other words.

As shall be explained below.

Using a different method from the one presented in the proof below she has shown the following inequality in the relativistic case.

In what follows. In the sequel.

In the sequel we will merely write $\bf{Ch}$ for $\bf{Ch} \rbk{\calA}$ when $\calA$ is understood.

The currently best known constant for $d=3$ is $C$.

The dynamic nuclei are coupled to a magnetic field.

Much of the power and usefulness of ergodic theory is due to the following probabilistic interpretation of the abstract set up discussed above.

They can refer to a huge number of elementary quantum mechanics texts, some of which present the subject in a way that is congenial to mathematicians.

Quite a few mathematicians have devoted a lot of time to explore generalizations of the inequalities and bounds on the optimal constants.

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One can easily see that the kernel forms a vector bundle.

One can ask if. One can ask whether.

A ring $R$ is simple if $R$ is not $0$ and has no two sided ideals except $0$ and $R$.

Let $R$ be commutative. Then $R$ is simple if and only if $R$ is a field.

With $F(x, \dot{x}, t)$ given, the above expression is a system of second order differential equations which together with the initial conditions $x(t_0)$ and $v(t_0)$ determine $x(t)$ and thus $v(t)$ for all times.

Assume, for simplicity, that the system under consideration contains only one species of particles, either fermions or bosons.

Simply speaking. For simplicity.

We shall slightly abuse notaion.

In the interest of keeping the notation simple, we shall regist the temptation to write down the most general formula in the following treatment.

We fix notation.

The teacher may pick and choose topics according to interest and time constraints.

It is easy to see that strong mixing is an invariant of measure theoretic isomorphism.

Another fact about the Coulomb potential, which will be of great importance, is Newton's theorem. It states that outside a charge distribution that is rotationally symmetric at the point $x_0$, the associated potential looks as if the charge is concenttated at the point $x_0$.

See A for details.

A group $G$ acts on a space $S$.

The basic question that has to be resolved in order to understand the existence of atoms and stability of our world is: Why don't the point-like electrons fall into the (nearly) point-like nuclei?

We thus have the following expression, a fact that will be useful later.

This latter quantity equals the number of eigenvalues.

From this point on, until the end of this book, we will restrict our attention to the physical case of $\bbRthree$.

Here $x_i \in \bbR^3$ is the (spatial) coodinate of the $i$-th particle.

Here we use the same notation as in Chapter 3, Eq. (3.1.5), with $\hat{dx_i}$ meaning that integration is over all variables but $x_i$.

The material presented here consists of a more or less self-contained advanced course in complex algebraic geometry.

So far we have considered the non-relativistic kinetic energy, $T_{\psi} = (1/2) \bkt{\psi}{p^2 \psi}$.

This problem of classical mechanics wa nicely summarized by Jeans in 1915.

The understading of this strange, and totally unforeseen, fact requires the knowledge that the appropriate Schrödinger equation has \coloredtextbf{zero-modes}.

In the remainder of the present chapter we limit the discussion to a single particle.

The chapter also contains several results that will be used repeatedly in the chapters to follow.

The results of this section can be summarized in the following statement.

The main point of this section will be that chain complexes form an abelian category.

In this section, we will describe in detail some of the many-body Hamiltonians whose ground state energy will be studied in this book.

The rest of this subsection is devoted to their deduction and illustration.

In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation.

In this introduction we shall explain the various inequalities.

Then a map $f$ is the chain map given by the formula.

These notes are based on courses given in the fall of 1992 at the University of Leiden.

This filtration is compatible with the multiplication in $W_n(k)$ in the sense that $\scrF_d \cdot \scrF_e \subset \scrF_{d+e}$.

The striking feature of this inequality is that $(N+M)^2$ terms on the left side are bounded by only $N+M$ terms on the right.

Throughout this book.

The topics covered in this book include quantum electrodynamics, stability of large Coulomb systems, gravitational statbility of stars, basics of equilibrium statistical mechanics, and the existence of the thermodynamic limit.

The main purpose of this book is to study the ground state energies of quantum mechanical systems of particles interacting via electric and magnetic forces.

This book aims to present those ideas and methods that can now be effectively used by mathematicians working in a variety of other fields of mathematical research.

The book is directed towards researchers on various aspects of quantum mechanics.

The aim of the theory is to describe the behavior of $T^n(x)$ as $n \to \infty$.

This led to the invention of interesiting new inequalities to simplify and improve his result.

This book is aimed at a second- or third-year graduate student.

This is the minimal setup needed to prove the theorem.

The reader is invited to prove this.

This leads to the central notion of holonomic $W_n(k)$-modules.

The following simple quesion illustrates these points.

We emphasize that the LT inequalities with these constants hold for arbitrary magnetic vector potentials $A$.

These inequalities play a crucial role in our understanding of stability of matter.

To see this, let $\calA$ be the smallest abelian subcategory of $\calC$.

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In combination with the argument above this actually implies the following proposition.

The operator $A$ is not normalized to have trace equal to one.

An operator valued function.

Moreover, for the same reason the following expression holds if $H$ is invariant under permutations.

Inequality (5.2.6) is of no use in this case, however, because of the infinite self-energy of $\mu$.

It is easily verified that $m$ does not depend on $f_D$ and the chosen point $p \in M$.

we denote by $\mathrm{gr}_{\Gamma} (M)$ (or simply $\mathrm{gr} (M)$) if no confusion can arise) the graded vector space.

It turns out that the desired inequalities can be achieved if and only if $\gamma$ and $d$ satisfy certain conditions (stated in the theorem below).

Two numbers, generalizing the genus of a projective curve, play an important rose in higher dimensions.

As already said before, the collection of holomorphic line bundles on a complex manifold $M$ modulo isomorphism form a group under the tensor product.

It is convenient to work throughout with the stalk of the sheaf of holomorphic functions. This way we avoid shrinking open neighbourhoods explicitly again and again. Let us introduce the necessary notations.

The proof of the Lieb-Thirring inequalities presented in the previous section carries over to the magnetic case essentially without change.

As remarked in the previous section.

It is straightforward to describe relativistic mechanics in the Hamiltonian formalism.

There is no difficulty in describing many-body systems in the Hamitonian formalism -- with either relativistic or non-relativistic kinematics.

In order to avoid measure theoretic pathologies, we will always assume that $\rbk{X, \calB, \mu}$ is the completion of a standard measure space.

In the meantime, homological algebra continued to evolve.

Elementary particles have an internal degree of freedom called spin which is characterized by a specific number that can take one of the values $S = 0, 1/2, 1, 3/2, \dots$.

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Symmetry has always played a fundamental role in quantum theory.

The generalization of the following concepts to multiple species is obvious and left to the reader.

It remains to show the following statement.

Because $\abs{p} \geq \sqrt{p^2 + m^2} - m \geq \abs{p} - m$, it suffices to consider the ultra-relativistic energy $T_{\psi} = \bkt{\psi}{\abs{p} \psi}$.

This expression is unchanged under the transformation $A \mapsto A^*, \lambda \mapsto \overline{\lambda}$.

We can endow $\mathrm{gr}_{\Gamma} (M)$ with a natural structure of a graded $\mathrm{gr} W_n(k)$-module as follows.

This theorem 3.2.3 has various interesting consequences for affine maps of states and for one-parameter groups of automorphisms.

The fact that $E_0 > - \infty$ under the assumption on $V$ stated in Theorem 4.1 already follows from the discussion in Section 2.2.1, Eq. (2.2.14).

Charge distributions may be continuous or discrete and hence it is convenient to consider them as Borel measures.

The electric force is attractive between oppositely charged particles and repulsive between like-charged particle

A Riemann surface with marked points.

In a similar vain.

The reader will find introductions to several other subjects.

In particular. Especially.

Unless otherwise stated.

The modern treatment of the question \coloredtextbf{how come a deterministic system can behave randomly} is based on this idea.

The historical connections with topology, regular local rings, and semisimple Lie algebras are also described.

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We first prove that (3.1.35) suffices to insure the existence of a $\Gamma$.

First we need to recall what an abelian category is.

We shall first discuss the LT inequalities in the non-relativistic case.

Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has recently obtained new impetus from physicists' interest in questions related to mirror symmetry.

I recall the basic objects and maps one works with in complex algebraic geometry.

Generalities on complex and projective manifolds.

We begin by reviewing basic notions and conventions to set the stage.

The basic constituents of ordinary matter are electrons and atomic nuclei.

The fundamental theory that underlies the physicist's description of the material world is quantum mechanics.

Physicists firmly believe that quantum mechanics is a \coloredtextbf{theory of everything} at the level of atoms and molecules.

It is common in the physics literature to refer to the choice of symmetry type as the *statistics of the particles or of their wave function.

Before presenting the inequalities let us discuss their \coloredtextbf{semiclassical} interpretation, which will make them more transparent and natural.

The variables $z_{k+1}, \dots, z_N$ are integrated out.

There are many useful function spaces and we select a proper space as the situation demands.

Homological algebra is a tool used in several branches of mathematics: algebraic topology, group theory, commutative ring theory, and algebraic geometry.

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As explained in the previous section, the coordinate of a particle is a point in $\bbRthree \times \cbk{1, \dots, q}$, i.e., a point $x \in \bbRthree$ and a point $\sigma \in \cbk{1, \dots, q}$.

We first consider the case $e > 0$.

Our goal is to use (4.2.1) to bound $T$ instead of the $E_j$.

We note that if $H$ is permutation invariant, the $\phi$ that yields the lowest value of $\rbkt{\phi}{H \phi}$ among all possible (normalized) functions can be taken to have a definite symmetry type without loss of generality, just as an energy minimizing state for a rotation invariant $H$ can be taken to have a definite angular momentum.

At its simplest form, a dynamical system is a function $T$ defined on a set $X$.

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A statistics (of a quantum particle).

The connection between quantum mechanics and classical mechanics becomes apparent in the semiclassical limit.

According to the semiclassical approach, which goes back to the earliest days of quantum mechanics, each volume $\rbk{2 \pi}^d$ in $2d$-dimensional phase space (consisting of pairs of points $(p, x)$ with $p \in \bbRd$ and $x \in \bbRd$) can support one quantum state.

As an example, consider the problem of $N$ electrons and $M$ nuclei interacting with each other via the Coulomb force.

We will not necessarily follow the historical route.

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Recall that the Weyl algebra $W_n(k)$ is naturally filtered by the sub-vectorspaces.

節終了