nils-herrmann/clean_scirepeval huggingface dataset:
Topological classification of trigonometric polynomials and combinatorics of graphs with an equal number of vertices and edges

title: Spaces of functions with moderate singularities

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Wave front evolution and equivariant Morse lemma

title: Singularities of smooth mappings with additional structures : collected papers
From Hilbert's Superposition Problem to Dynamical Systems
Today I will try to explain the diversity of subjects I have worked on. In fact, I have followed one line from the very beginning, and there has been essentially one problem I have worked on all my life. This fact seems strange even to me, but I will try to explain it. When you are collecting mushrooms, you see only the upper part of the mushroom. But if you are a mycologist, you know that the real mushroom is in the earth. There's an enormous thing down there, and you just see the fruit, the part that you eat.

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Wave front evolution and equivariant Morse lemma

title: Spaces of functions with moderate singularities

title: Singularities of smooth mappings with additional structures : collected papers
Remarks on the perturbation theory for problems of Mathieu type
for perturbations of a simple eigenvalue ??2. Perturbations of a multiple eigenvalue ??3. Widths of forbidden zones of even Hill's equations of Mathieu type ??4. Formulae for the denominators of terms in the perturbation theory series ??5. Conclusion of the investigation of forbidden zones ??6. Widths of resonance zones for mappings of the circleReferences

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Wave front evolution and equivariant Morse lemma

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Spaces of functions with moderate singularities

title: Singularities of smooth mappings with additional structures : collected papers
Characteristic class entering in quantization conditions

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Wave front evolution and equivariant Morse lemma

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Spaces of functions with moderate singularities

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Singularities of smooth mappings with additional structures : collected papers
Cohomology classes of algebraic functions invariant under Tschirnhausen transformations

title: Spaces of functions with moderate singularities

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Wave front evolution and equivariant Morse lemma

title: Singularities of smooth mappings with additional structures : collected papers
Local normal forms of functions

title: Spaces of functions with moderate singularities

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Wave front evolution and equivariant Morse lemma

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Singularities of smooth mappings with additional structures : collected papers
Remarks on the Extatic Points of Plane Curves
There exists an interesting relation between the points where a plane curve has an unusually high tangency with an algebraic curve of degree n and the flattening points of space curves (points where the highest torsion vanishes). For the cases n = 1,2 and 3 this relation leads directly to the minorations of the numbers of the points of higher order tangency, but for n ≥ 4 the problem of finding such a minoration is wide open (even for the curves close to a circle).

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Spaces of functions with moderate singularities

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Wave front evolution and equivariant Morse lemma

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Singularities of smooth mappings with additional structures : collected papers
The Sturm theorems and symplectic geometry
A variable inductor includes an input external electrode, an output external electrode, and a coil formed by electrically connecting at least two spiral coil pattern portions in series between the input external electrode and the output external electrode. At least one trimming electrode is further provided in each of the spiral coil pattern portions. A first end and a second end of each trimming electrode are connected to the spiral coil pattern portion, and a lead out electrode, respectively, so that the trimming electrode bridges between the lead out electrode and the coil. The trimming electrodes are sequentially trimmed one-by-one, by, for example, irradiating a laser beam, starting from a trimming electrode closer to an edge, whereby the inductance of the coil is increased accordingly.

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Wave front evolution and equivariant Morse lemma

title: Spaces of functions with moderate singularities

title: Singularities of smooth mappings with additional structures : collected papers
Singularities in variational calculus
A survey is given of theories of singularities of systems of rays and wave fronts, that is, singularities of systems of extremals of variational problems and solutions of the Hamilton-Jacobi equations near caustics. The problem of passing about an obstacle bounded by a smooth surface of general position is studied in detail. Theorems are proved on the normal forms of Lagrangian manifolds with singularities formed by rays of the system of extremals of a variational problem in the symplectic space of all oriented lines which tear off from the surface of the obstacle as well as theorems on Legendre manifolds with singularities formed by contact elements of a wave front and 1-jets of a solution of the Hamilton-Jacobi equation.

title: Spaces of functions with moderate singularities

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Wave front evolution and equivariant Morse lemma

title: Singularities of smooth mappings with additional structures : collected papers
Topological problems of the theory of wave propagation
Contents §1. An apologia for Applied Mathematics §2. Theorems on four cusps §3. The Hurwitz-Kellogg-Tabachnikov theorem of Sturm type §4. Trigonometric approximations §5. Lagrangian intersections in symplectic topology §6. Legendrian links in contact topology §7. Lagrangian collapse and the cusps of caustics §8. Legendrian collapse and the cusps of fronts §9. Space curves and their points of flattening §10. Vertices of convex space curves §11. Applications to the theory of extatic points of plane curves §12. Multidimensional generalizations of Sturm theory Bibliography

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Spaces of functions with moderate singularities

title: Wave front evolution and equivariant Morse lemma

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Singularities of smooth mappings with additional structures : collected papers
Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves

title: The Theory of Singularities and its Applications
abstract: Part I. The Zoo of Singularities: 1. Morse theory of functions 2. Whitney theory of mappings 3. The Whitney-Cayley umbrella 4. The swallowtail 5. The discriminants of the reflection groups 6. The icosahedron and the obstacle by-passing problem 7. The unfurled swallowtail 8. The folded and open umbrellas 9. The singularities of projections and of the apparent contours Part II. Singularities of Bifurcation Diagrams: 10. Bifurcation diagrams of families of functions 11. Stability boundary 12. Ellipticity boundary and minima functions 13. Hyperbolicity boundary 14. Disconjugate equations, Tchebyshev system boundaries and Schubert singularities in flag manifolds 15. Fundamental system boundaries, projective curve flattenings and Schubert singularities in Grassmann manifolds.

title: Poisson structures on the plane and other powers of volume forms
abstract: The hierarchy of formsf(dx1∧...∧dxn)α (α fixed) starts with the series A, D, and E. Their singular hypersurfaces admit quasihomogéneous polynomials g as representatives. The dimension of the space of moduli of forms with fixed g is calculated; it is equal to the number of monomials h, for which g(hx)α has weight zero. For Poisson structures on the plane (α=−1, n=2) this dimension is one less than the number of irreducible components of the curve g=0, i.e., is equal to the number h1,0 of mixed Hodge structures.

title: Spaces of functions with moderate singularities

title: Weak asymptotics for the numbers of solutions of diophantine problems

title: Wave front evolution and equivariant Morse lemma

title: Singularities of smooth mappings with additional structures : collected papers