The theory of infinite-dimensional lie groups and its applications

The purpose of this paper is to survey the theory of regular Fréchet-Lie groups developed in [1–10]. Such groups appear and are useful in symplectic geometry and the theory of primitive infinite groups of Lie and Cartan [11]. From the group theoretical standpoint, general relativistic mechanics is a more closed system than Newtonian mechanics. Quantized objects of these classical groups are closely related to the group of Fourier integral operators [12]. These can also be managed as regular Fréchet-Lie groups. However, there are many Fréchet-Lie algebras which are not the Lie algebras of regular Fréchet-Lie groups [13]. Thus, the enlargeability of the Poisson algebra is discussed in detail in this paper. Enlargeability is relevant to the global hypoellipticity [14, 15] of second-order differential operators.     

Majorizing functions and two-point Newton-type methods

The semi-local convergence of a Newton-type method used to solve nonlinear equations in a Banach space is studied. We also give, as two important applications, convergence analyses of two classes of two-point Newton-type methods including a method mentioned in [5] and the midpoint method studied in [1], [2] and [12]. Recently, interest has been shown in such methods [3] and [4].     

有关行列式和迹的极值以及典则变量的最优性的补注

二次型极值已被一些文献诸如[1]和[3]讨论,一些相近的题目可在文献[2]中见到.文献[1]得到与二次型有关的行列式极值和典则变量的最优性质,本文对[1]给出补注.     

Ultrafilters and Regular Extensions of Re-proximities

There is a one-to-one correspondence between the proximities on a set X and some symmetric relations on ultrafilters, the so-called "nasses" of [9] and [10]. It is proved in [4] that one gets a new characterization of RI-proximities from this correspondence. The purpose of this paper is to show that the same method is also of interest for RE-proximities which were studied by Császár in [6]. By the introduction of the equivalence kernel and domain of a nasse, we give a new characterization and new regular extensions for these proximities.     

Hydraulic Theory and Particle-Driven Gravity Currents

Hydraulic theory, as it has been applied to compositionally driven gravity flows, involves the single simplifying assumption that the pressure in the fluid is hydrostatic [1]. This assumption provides, as a consequence, a depth independent horizontal velocity field. This approach has led to a greatly increased understanding of many of the phenomena associated with these complex flows, including issues surrounding internal hydraulic jumps and energy loss [2]. Recently, investigations into flow and deposition of particles from particle-driven gravity currents have been carried out using an approach that employs the hydraulic theory that had proved so successful in the case of homogeneous flows [3]–[7]. Unfortunately, as we show here, there is a fundamental contradiction in adopting this simplifying assumption when particles drive the flow. This contradiction is essentially that one cannot have a hydrostatic pressure that arises from the presence of particles while at the same time maintaining a depth-independent horizontal velocity field, as was assumed in references [3]–[7].     

Inclusion of the roots of a polynomial based on Gerschgorin's theorem

Summary In this note a new companion matrix is presented which can be interpreted as a product of Werner's companion matrices [13]. Gerschgorin's theorem yields an inclusion of the roots of a polynomial which is best in the sense of [4] and generalizes a result of L. Elsner [5]. This inclusion is better than the one due to W. Börsch-Supan in [1].Dedicated to Professor E. Stein on the occasion of his 60th birthday.     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

Minimal geodesics on groups of volume‐preserving maps and generalized solutions of the Euler equations

The three‐dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations but can also be seen, following Arnold [1], as a geodesic on a group of volume‐preserving maps. Local existence and uniqueness of minimal geodesics have been established by Ebin and Marsden [16]. In the large, for a large class of data, the existence of minimal geodesics may fail, as shown by Shnirelman [26]. For such data, we show that the limits of approximate solutions are solutions of a suitable extension of the Euler equations or, equivalently, are sharp measure‐valued solutions to the Euler equations in the sense of DiPerna and Majda [14]. © 1999 John Wiley & Sons, Inc.     

An efficient implementation for SSOR and incomplete factorization preconditionings

We investigate methods for efficiently implementing a class of incomplete factorization preconditioners which includes Symmetric Gauss Seidel [9]. SSOR [9], generalized SSOR [1]. Dupont Kendall Rachford [4]. ICCG(0) [7], and MICCG(0) [6]. Our techniques can be extended to similar methods for nonsymmetric matrices.     

On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1–3, 11, 12]. The issue of their computational complexity has received at-tention in the past, and was raised recently by E. Rassart in [11]. We prove that the problem of computing either quantity is #P-complete. Thus, unless P = NP, which is widely disbelieved, there do not exist efficient algorithms that compute these numbers.     

Examples of non-Kähler Hamiltonian torus actions

An important question with a rich history is the extent to which the symplectic category is larger than the K?hler category. Many interesting examples of non-K?hler symplectic manifolds have been constructed [T] [M] [G]. However, sufficiently large symmetries can force a symplectic manifolds to be K?hler [D] [Kn]. In this paper, we solve several outstanding problems by constructing the first symplectic manifold with large non-trivial symmetries which does not admit an invariant K?hler structure. The proof that it is not K?hler is based on the Atiyah-Guillemin-Sternberg convexity theorem [At] [GS]. Using the ideas of this paper, C. Woodward shows that even the symplectic analogue of spherical varieties need not be K?hler [W]. Oblatum IX-1995 & 3-III-1997     

Approximations and Mittag-Leffler conditions the tools

Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [20], [14], [19]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [16], and it does not provide for approximations when R has cardinality ≤ ?₀, [8]. We remove the cardinality restriction on R in the latter result. We also prove an extension of the Countable Telescope Conjecture [23]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits.In order to prove these results, we develop new general tools combining relative Mittag-Leffler conditions with set-theoretic homological algebra. They make it possible to trace the above facts to their ultimate, countable, origins in the properties of Bass modules. These tools have already found a number of applications: e.g., they yield a positive answer to Enochs’ problem on module approximations for classes of modules associated with tilting [4], and enable investigation of new classes of flat modules occurring in algebraic geometry [26]. Finally, the ideas from Section 3 have led to the solution of a long-standing problem due to Auslander on the existence of right almost split maps [22].     

Mathematical models of the Bandpass problem and OrderMatic computer game

A new combinatorial optimization problem, the Bandpass problem, was defined in Bell and Babayev (2004) [4]. Recently, this problem was investigated in detail in Babayev et al. (2009) [5]. In this paper, we first present some new mathematical models of the Bandpass problem. Then related to this problem, we introduce a software called OrderMatic which is very useful for teaching permutations.     

Concerning the semilocal convergence of Newton’s method and convex majorants

We approximate a locally unique solution of an equation on a Banach space setting using Newton’smethod.Motivated by the work by Ferreira and Svaiter [5] but using more precise majorization sequences, and under the same computational cost we provide: a larger convergence region; finer error bounds on the distances involved, and an at least as precise information on the location of the solution than in [5]. The results can also compare favorably to the corresponding ones given byWang in [10]. Finally we complete the study with two concrete applications.      

Divisors,measures and critical functions

In [4] we have introduced a new distance between Galois orbits over ℚ. Using generalized divisors, we have extended the notion of trace of an algebraic number to other transcendental quantities. In this article we continue the work started in [4]. We define the critical function for a class of transcendental numbers, that generalizes the notion of minimal polynomial of an algebraic number. Our results extend the results obtained by Popescu et al [5].     

Inconsistency of GPK + AFA

M. Forti and F. Honsell showed in [4] that the hyperuniverses defined in [2] satisfy the anti-foundation axiom X₁ introduced in [3]. So it is interesting to study the axiom AFA, which is equivalent to X₁ in ZF, introduced by P. Aczel in [1]. We show in this paper that AFA is inconsistent with the theory GPK. This theory, which is first order, is defined by E. Weydert in [6] and later by M. Forti and R. Hinnion in [2]. It includes all general hyperuniverses as defined in [5]. In order to achieve our aim, we need to define ordinals in GPK and to study some of their properties. Mathematics Subject Classification: 03E70, 03E10.     

Generalized uniform distribution of hasse invariants

Kaplansky asked if in a Noetherían domain the intersection of two height 2 primes must contain a non-zero prime. This paper presents a counterexample. Some positive results are given in [2]. The construction in the example proper considerably simpli-fies the argument of [3-Theorem 2.5]. We assume familiarity with [1, Section 1-5].     

Frustration and stability in random boolean networks

Discrete iterations of boolean mappings are known to yield to limit cycles [3, 8]. These limit cycles share a common stable part: the stable core which never oscillate along the different limit cycles.We show that non-frustrated circuits (defined as an extension of [7, 10]) are part of this core. We then characterize non-frustration — thus stability — in terms of the discrete derivative as introduced in [6, 11, 12].     

Schensted Algorithms for Dual Graded Graphs

This paper is a sequel to [3]. We keep the notation and terminology and extend the numbering of sections, propositions, and formulae of [3].The main result of this paper is a generalization of the Robinson-Schensted correspondence to the class of dual graded graphs introduced in [3], This class extends the class of Y-graphs, or differential posets [22], for which a generalized Schensted correspondence was constructed earlier in [2].The main construction leads to unified bijective proofs of various identities related to path counting, including those obtained in [3]. It is also applied to permutation enumeration, including rook placements on Ferrers boards and enumeration of involutions.As particular cases of the general construction, we re-derive the classical algorithm of Robinson, Schensted, and Knuth [19, 12], the Sagan-Stanley [18], Sagan-Worley [16, 29] and Haiman's [11] algorithms and the author's algorithm for the Young-Fibonacci graph [2]. Some new applications are suggested.The rim hook correspondence of Stanton and White [23] and Viennot's bijection [28] are also special cases of the general construction of this paper.In [5], the results of this paper and the previous paper [3] were presented in a form of extended abstract.