In the last 30 days

• Open/close CRITICAL POINTS OF SMOOTH FUNCTIONS AND THEIR NORMAL FORMS

  Vladimir I Arnol'd 1975 Russ. Math. Surv. 30 1 Tag this article

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  This paper contains a survey of research on critical points of smooth functions and their bifurcations. We indicate applications to the theory of Lagrangian singularities (caustics), Legendre singularities (wave fronts) and the asymptotic behaviour of oscillatory integrals (the stationary phase method). We describe the connections with the theories of groups generated by reflections, automorphic forms, and degenerations of elliptic curves. We give proofs of the theorems on the classification of critical points with at most one modulus, and also a list of all singularities with at most two moduli. The proofs of the classification theorems are based on a geometric technique associated with Newton polygons, on the study of the roots of certain Lie algebras resembling the Enriques-Demazure technique of fans, and on spectral sequences that are constructed with respect to quasihomogeneous filtrations of the Koszul complex defined by the partial derivatives of a function.

• Open/close SINGULARITIES OF SMOOTH MAPPINGS

  Vladimir I Arnol'd 1968 Russ. Math. Surv. 23 1 Tag this article

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  CONTENTS
  Chapter 1. The structure of singularities
  § 1. Examples
  § 2. The classes Σ ^I
  § 3. The quadratic differential
  § 4. The local ring of a singularity and the Weierstrass preparation theorem
  Appendix. A proof of the Weierstrass preparation theorem
  Chapter 2. Deformations of singularities
  § 5. "Infinite-dimensional Lie groups" acting on "infinite-dimensional manifolds"
  § 6. The stability theorem
  § 7. Proof of convergence
  § 8. In the neighbourhood of an isolated critical point every analytic function is equivalent to a polynomial
  References

• Open/close NORMAL FORMS OF FUNCTIONS IN NEIGHBOURHOODS OF DEGENERATE CRITICAL POINTS

  Vladimir I Arnol'd 1974 Russ. Math. Surv. 29 10 Tag this article

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  An analysis of the normal forms to which functions can be reduced in neighbourhoods of degenerate critical points shows that many of them are quasihomogeneous or semiquasihomogeneous.
  A semiquasihomogeneous function is a sum of a quasihomogeneous (or weighted homogeneous) polynomial with an isolated critical point and summands of a higher degree of quasihomogeneity. The normal form to which a semiquasihomogeneous function can be reduced is described in terms of the local ring of the gradient mapping given by the quasihomogeneous part of the function. The number of parameters in this normal form is called the inner modality of the quasihomogeneous part.
  A classification is given of all quasihomogeneous critical points of inner modality 1: up to stable equivalence they are exhausted by three one-parameter families of parabolic singularities and 14 exceptional polynomials, 8 of which are functions of two variables, and 6 functions of three variables.

• Open/close INDICES OF SINGULAR POINTS OF 1-FORMS ON A MANIFOLD WITH BOUNDARY, CONVOLUTION OF INVARIANTS OF REFLECTION GROUPS, AND SINGULAR PROJECTIONS OF SMOOTH SURFACES

  Vladimir I Arnol'd 1979 Russ. Math. Surv. 34 1 Tag this article

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  CONTENTS
  Introduction
  § 1. Singularities and their indices
  § 2. A theorem on the sum of indices
  § 3. Indices of singular points of adapted vector fields
  § 4. Connection between the indices of fields and forms
  § 5. Covering indices
  § 6. Algebraic definitions
  § 7. Indices of critical points of functions on the boundary of a manifold
  § 8. Indices of simple boundary singularities
  § 9. Duality
  § 10. Some applications
  References

• Open/close CRITICAL POINTS OF FUNCTIONS ON A MANIFOLD WITH BOUNDARY, THE SIMPLE LIE GROUPS B_k, C_k, AND F₄ AND SINGULARITIES OF EVOLUTES

  Vladimir I Arnol'd 1978 Russ. Math. Surv. 33 99 Tag this article

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  Contents
  Introduction
  § 1. Classification of simple critical points of functions on the boundary of a manifold
  § 2. Versal deformations and bifurcation diagrams
  § 3. Vanishing cycles and half-cycles
  § 4. The form of intersections
  § 5. The monodromy group
  § 6. Elliptic singularities
  § 7. Classification of singularities
  § 8. The determinant of singularities
  § 9. Some corollaries
  § 10. Singularities of a family of evolutes
  References

• Open/close The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups

  Vladimir I Arnol'd 1992 Russ. Math. Surv. 47 1 Tag this article

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  CONTENTS
  Introduction
  Chapter I. The enumerative combinatorics of snakes
  § 1. Combinatorics of Springer cones
  § 2. Numbers and triangles of snakes
  § 3. Combinatorial identities for triangles of snakes
  § 4. Combinatorial identities for numbers of A-snakes
  § 5. Combinatorial identities for numbers of D-snakes and B-snakes
  § 6. Exponential generating functions of triangles of snakes
  Chapter II. Proofs of combinatorial theorems
  § 7. Proof of recurrence relations
  § 8. Proofs of the relations between triangles b and β
  § 9. Numbers of assignments
  § 10. Proof of the relations between triangles b and d
  § 11. Proofs of the theorems on D-snakes
  § 12. Comparison of the numbers of D-snakes and B-snakes with a given beginning
  § 13. Properties of numbers of snakes modulo p ^r
  Chapter III. Springer cones and bifurcation diagrams of singularities
  § 14. Morsification spaces and Maxwell strata
  § 15. Modifications of singularities of B-series
  § 16. Topological classification of odd functions
  References
  

• Open/close ON MATRICES DEPENDING ON PARAMETERS

  V I Arnold 1971 Russ. Math. Surv. 26 29 Tag this article

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  Given a family of matrices smoothly depending on parameters of endomorphisms of a complex linear space, it is shown that there is a normal form to which the family can be reduced by the choice of a base smoothly depending on the parameters. The formulae obtained are applied to the investigation of bifurcation diagrams of families of matrices.

• Open/close A property of the set of prime numbers

  Anatolii A Karatsuba 2011 Russ. Math. Surv. 66 209 Tag this article

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  A phenomenon of interdependency between the structure of positive integers and the form of their prime factors is discovered. This paper was prepared for publication by E.A. Karatsuba and M.E. Changa, based on A.A. Karatsuba's drafts and notes from 2007-2008. Details of calculations are due to Changa.

  Bibliography: 10 titles.

• Open/close REMARKS ON THE STATIONARY PHASE METHOD AND COXETER NUMBERS

  Vladimir I Arnol'd 1973 Russ. Math. Surv. 28 19 Tag this article

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  We study integrals of rapidly oscillating functions. Such integrals tend to zero when the length of the oscillation wave tends to zero through wave fronts of constant form. The asymptotic decrease of the integral is determined by the character of the critical points of the function describing the front. If all of these critical points are non-degenerate (Morse), then the integral tends to zero like the wave length raised to the power of half the dimension of the space, and indeed this is the asymptotic behaviour of the integral for functions in general position.
  However, if the integral depends on additional parameters, then for certain "caustic" parameter values there arise non-Morse critical points and the integral decreases slowly. The investigation of the asymptotic behaviour of the integral of an oscillating function in caustic cases can be regarded as a generalization of the theory of Airy functions; it is closely connected with Artin's braid theory, and the answer in the case of few parameters is expressed in terms of the Coxeter numbers of the Weyl groups of the series A, D, E, and in the case of many parameters it is expressed in terms of generalizations of them.

• Open/close Metric regularity and subdifferential calculus

  A D Ioffe 2000 Russ. Math. Surv. 55 501 Tag this article

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  The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem. Developments in non-smooth analysis in the 1980s and 1990s paved the way for a number of far-reaching extensions of these results. It was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure. At the same time it became clear that some basic hypotheses of the subdifferential calculus are closely connected with the metric regularity of certain set-valued maps. The survey is devoted to the metric theory of metric regularity and its connection with subdifferential calculus in Banach spaces.