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Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, , can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous — and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πħ where is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.

If a mathematical model contains many different scales the computational cost for its numerical solution is very large. The smallest scale must be resolved over the distance of the largest scale. A huge number of unknowns are required and until recently many such problems could not be treated computationally. We will discuss a new set of numerical techniques that couples models for different scales in the same simulation in order to handle many realistic multi-scale problems. In most of this presentation we shall survey existing methods but we shall also give some new observations.

In this chapter, you’ve looked at the JSP EL. This EL is largely intended to replace scriptlets and to be used in combination with custom tags.

You’ve examined the following topics in this chapter:

In the next chapter, you’ll learn about the JSTL and the tags contained within it.

If a mathematical model contains many different scales the computational cost for its numerical solution is very large. The smallest scale must be resolved over the distance of the largest scale. A huge number of unknowns are required and until recently many such problems could not be treated computationally. We will discuss a new set of numerical techniques that couples models for different scales in the same simulation in order to handle many realistic multi-scale problems. In most of this presentation we shall survey existing methods but we shall also give some new observations.

If a mathematical model contains many different scales the computational cost for its numerical solution is very large. The smallest scale must be resolved over the distance of the largest scale. A huge number of unknowns are required and until recently many such problems could not be treated computationally. We will discuss a new set of numerical techniques that couples models for different scales in the same simulation in order to handle many realistic multi-scale problems. In most of this presentation we shall survey existing methods but we shall also give some new observations.

Among seven problems, proposed for the XXI century by the Clay Mathematical Institute [], there are two stemming from physics. One of them is called “Yang-Mills Existence and Mass Gap”. The detailed statement of the problem, written by A. Jaffe and E. Witten [], gives both motivation and exposition of related mathematical results, known until now. Having some experience in the matter, I decided to complement their text by my own personal comments aimed mostly to mathematical audience.

In this chapter, you’ve looked at the JSP EL. This EL is largely intended to replace scriptlets and to be used in combination with custom tags.

You’ve examined the following topics in this chapter:

In the next chapter, you’ll learn about the JSTL and the tags contained within it.

In this chapter, you’ve looked at the JSP EL. This EL is largely intended to replace scriptlets and to be used in combination with custom tags.

You’ve examined the following topics in this chapter:

In the next chapter, you’ll learn about the JSTL and the tags contained within it.

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, , can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4–5 decades ago, and to explore new regimes not treated before. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous — and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πħ where is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, several other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the ‘traps’ actually used in the experiments. For this system it is also possible to prove complete Bose condensation and superfluidity as we have shown. On the frontier of experimental developments is the possibility that a dilute gas in an elongated trap will behave like a one-dimensional system; we have proved this mathematically. Another topic is a proof that Foldy’s 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy; using this we can also prove the formula for the ground state energy of the two-component charged Bose gas proposed by Dyson in 1967. All of this is quite recent work and it is hoped that the mathematical methodology might be useful, ultimately, to solve more complex problems connected with these interesting systems.

In this chapter, you’ve looked at the JSP EL. This EL is largely intended to replace scriptlets and to be used in combination with custom tags.

You’ve examined the following topics in this chapter:

In the next chapter, you’ll learn about the JSTL and the tags contained within it.