Fundamental lemma

About: Fundamental lemma is a research topic. Over the lifetime, 229 publications have been published within this topic receiving 4989 citations.

Small denominators and problems of stability of motion in classical and celestial mechanics

Abstract: CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincare variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences

Numerical inverting of matrices of high order

Abstract: PREFACE 188 CHAPTER VIII. Probabilistic estimates for bounds of matrices 8.1 A result of Bargmann, Montgomery and von Neumann 188 8.2 An estimate for the length of a vector 191 8.3 The fundamental lemma 192 8.4 Some discrete distributions 194 8.5 Continuation 196 8.6 Two applications of (8.16) 198 CHAPTER IX. The error estimates 9.1 Reconsideration of the estimates (6.42)-(6.44) and their consequences.. 199 9.2 The general Ai 200 9.3 Concluding evaluation 200

On siegel modular forms of genus two (ii).1

Abstract: Introduction. The main subject we shall discuss in this second paper is the Siegel modular forms of genus two with levels. The method we used in the first paper [5] did not give sufficient information even for level two. Therefore the problem (raised by Grothendieck) whether modular varieties become non-singular or not for higher levels was beyond our reach. With some other applications in mind, we therefore investigated "theta-constants" as modular forms and proved, among others, a fundamental lemma in our recent paper [6]. Using the results in that paper, we shall show that modularvarieties of high levels do not have non-singular coverings even locally around their singular points. Also we shall determine how r2(1)/1'2(2) = Sp(2, Z/2Z) acts on the ring of modular forms A (r2 (2)) and obtain the characters of its action on the homogeneous parts A (r2 (2) ) k for k = 0, 1, 2, . In this way, we shall determine A (IF2 (1)) reproducing our earlier structure theorem on A (r2 (1 )) (2). Furthermore, the polynomial expressions of the four basic Eisenstein series of level one by theta-constants and a known identity of this kind (between a certain Eisenstein series of level two and the eighth power of Riemann's theta-constant [1]) will be obtained. We note that this identity was previously obtained using the Siegel main theorem on quadratic forms.

On Yao's XOR-lemma

Abstract: A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao's Lemma and present a third alternative proof. The third proof proceeds by first proving that a function constructed by concatenating the values of the original function on several independent instances is much more unpredictable, with respect to specified complexity bounds, than the original function. This statement turns out to be easier to prove than the XOR-Lemma. Using a result of Goldreich and Levin (1989) and some elementary observation, we derive the XOR-Lemma.

Le lemme fondamental pour les algebres de Lie

Abstract: We propose a proof for conjectures of Langlands, Shelstad and Waldspurger known as the fundamental lemma for Lie algebras and the non-standard fundamental lemma. The proof is based on a study of the decomposition of the l-adic cohomology of the Hitchin fibration into direct sum of simple perverse sheaves.