Questions of existence and uniqueness

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Existence And Uniqueness ◽  
Symplectic Topology ◽  
Symplectic Structures ◽  
Open Problems ◽  
Precise Formulation ◽  
The Third

This chapter discusses the fundamental existence and uniqueness questions in symplectic topology: which manifolds admit symplectic structures, and to what extent they are unique. There are partial answers for some classes of manifolds and many related open problems. The chapter begins with a precise formulation of some relevant questions and continues by describing some related examples. Much of what is known is in dimension four, largely because of the existence of the powerful theory developed by Taubes and Seiberg–Witten in the 1990s. The third section outlines this theory, while the last section applies it in various explicit cases.

scholarly journals Introduction to Symplectic Topology

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Existence And Uniqueness ◽  
Holomorphic Curves ◽  
Symplectic Topology ◽  
Open Problems ◽  
Global Theory ◽  
New Methods ◽  
Witten Theory ◽  
New Material ◽  
The Subject ◽  
Developing Area

Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.

scholarly journals Reduction of Neighborhood-Based Generalized Rough Sets

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Zhaohao Wang ◽  
Lan Shu ◽  
Xiuyong Ding
Keyword(s):  
Information Systems ◽  
Set Theory ◽  
Rough Set ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Open Problems ◽  
The Third ◽  
Generalized Rough Sets

Rough set theory is a powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. This paper discusses five types of existing neighborhood-based generalized rough sets. The concepts of minimal neighborhood description and maximal neighborhood description of an element are defined, and by means of the two concepts, the properties and structures of the third and the fourth types of neighborhood-based rough sets are deeply explored. Furthermore, we systematically study the covering reduction of the third and the fourth types of neighborhood-based rough sets in terms of the two concepts. Finally, two open problems proposed by Yun et al. (2011) are solved.

scholarly journals Determination of an Unknown Coefficient in the Third Order Pseudoparabolic Equation with Non-Self-Adjoint Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Yashar T. Mehraliyev ◽  
Gulshan Kh. Shafiyeva
Keyword(s):  
Boundary Conditions ◽  
Classical Solution ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Unknown Coefficient ◽  
Pseudoparabolic Equation ◽  
Third Order ◽  
System Of Integral Equations ◽  
Inverse Boundary ◽  
The Third

The solvability of the inverse boundary problem with an unknown coefficient dependent on time for the third order pseudoparabolic equation with non-self-adjoint boundary conditions is investigated in the present paper. Here we have introduced the definition of the classical solution of the considered inverse boundary value problem, which is reduced to the system of integral equations by the Fourier method. At first, the existence and uniqueness of the solution of the obtaining system of integral equations is proved by the method of contraction mappings; then the existence and uniqueness of the classical solution of the stated problem is proved.

On the definition of post-newtonian approximations

1992 ◽  
Vol 438 (1903) ◽  
pp. 341-360 ◽  
Keyword(s):  
General Relativity ◽  
Newtonian Limit ◽  
Minimal Set ◽  
Asymptotically Flat ◽  
Precise Formulation ◽  
The Third ◽  
Newtonian Approximation ◽  
Definition Of

A definition of post-newtonian approximations is presented where the whole formalism is derived from a minimal set of axioms. This establishes a link between the existing precise formulation of the newtonian limit of general relativity and the post-newtonian equations which are used in practical calculations. The breakdown of higher post-newtonian approximations is examined within this framework. It is shown that the choice of harmonic gauge leads to equations which do not admit asymptotically flat solutions at the second post-newtonian level if one starts with a generic newtonian solution. The most simple choice of gauge gives equations which are solvable at the 2PN level but which in general have no solutions in the case of the third post-newtonian approximation.

The Structure of Fuzzifying Measure Space and Fuzzifying Measure

2010 ◽  
Vol 108-111 ◽  
pp. 844-849
Author(s):  
Shi Hua Luo
Keyword(s):  
Measure Space ◽  
Structural Features ◽  
Fuzzy Measure ◽  
Open Problems ◽  
The Third

A new so-called fuzzifying measurable theory that generalizes the classical measurable theory is established, the essence of which is a fuzzy measure on a multiple-valued algebra. First, the semantics method of continuous-valued logic is used to describe the new measure succinctly. Then, the structures of the new theory are discussed in detail and some of the key structural features of the classic measure can be successfully extended to the new theory. Lastly, the product of the two fuzzifying measures is studied and a problem, which is similar to the third of the open problems in fuzzy measure by Z. Wang, is considered.

Open problems

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Symplectic Topology ◽  
Open Problems ◽  
Open Questions

The final chapter of this book discusses some open questions and conjectures that either have served as guiding lights or have emerged in the study of symplectic topology over the last quarter of a century. The wide variety of problems, though inevitably incomplete, provides a snapshot of where the field is at the time of writing.

Metamathematical problems

1973 ◽  
Vol 38 (3) ◽  
pp. 500-516 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
External World ◽  
Open Problems ◽  
The Third ◽  
The World

When a logician approaches the world of mathematics, he may have in mind one or more of several purposes. He may try to find in mathematics a framework for formalizing commonly accepted laws of thought or perhaps laws of thought that are not commonly accepted. He may want to assist the mathematician by providing him with firm foundations for his theories. But it may also be the case that the logician wishes to use his own characteristic tools—formalized languages, explicit relations between symbols and objects, rigidly expressed and controlled rules of deduction—in order to gain a better understanding of the various and variegated kinds of structures, methods, theories and theorems that are to be found in mathematics. We may then expect him to adopt the attitude of the physicist or psychologist who (whatever his professed philosophy) feels that he deals with phenomena of the external world, whose rules cannot be imposed by him arbitrarily. He, or those that come after him, may indeed use the understanding thus gained in order to modify these phenomena, but as a scientist he would not regard this possibility as his only justification.For many years now, I have concentrated on the third of the lines of approach sketched above, and it seemed natural that I should discuss it again on the present occasion. However, today I do not wish to emphasize past developments but, using some of them as a background, I propose to enumerate a number of open problems. These problems seemed to me of some interest not only for their own sake but also because their solution might well require weapons whose introduction would close definite gaps in our armory.

scholarly journals Existence and uniqueness of solutions to weakly singular integral-algebraic and integro-differential equations

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Mikhail Bulatov ◽  
Pedro Lima ◽  
Ewa Weinmüller
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Solution ◽  
Main Stream ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
The Third ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Dimensional Integral

AbstractWe consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

scholarly journals A Precise Formulation of the Third Law of Thermodynamics

2009 ◽  
Vol 134 (4) ◽  
pp. 781-792 ◽  
Author(s):  
Walter F. Wreszinski ◽  
Elcio Abdalla
Keyword(s):  
Precise Formulation ◽  
The Third ◽  
Third Law Of Thermodynamics ◽  
Third Law

European Coal and Steel Community

1955 ◽  
Vol 9 (3) ◽  
pp. 450-452
Keyword(s):  
Working Party ◽  
Common Market ◽  
Precise Formulation ◽  
The Third ◽  
Universal Suffrage ◽  
The Common

Common AssemblyThe Common Assembly of the European Coal and Steel Community (ECSC) held its second extraordinary session in Stras-bourg from May 6 to 9, 1955, and its third ordinary session in the same place, May 10 ot 14; the third ordinary session was scheduled to reconvene on June 21. At the extraordinary session, the Assembly heard a report by its president, Mr. Pella (Italy), on his talks with the governments of ECSC members on various political matters. The Assembly approved a provisional budget of 78 million Belgian francs and also decided to establish a working party with 26 members to report to the Assembly on the procedure which might be proposed to ensure a more precise formulation of the Assembly's powers of control over the Community and the common market and to study the problems which would arise in the event the members of the Assembly were elected by universal suffrage.

Sign in / Sign up

Export Citation Format

Share Document