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Search: All articles in the CJM digital archive with keyword singularity

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1. CJM Online first

Hakl, Robert; Zamora, Manuel
Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere
We study a second-order ordinary differential equation coming from the Kepler problem on $\mathbb{S}^2$. The forcing term under consideration is a piecewise constant with singular nonlinearity which changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

Keywords:singular differential equation, indefinite singularity, periodic solution, Kepler problem on $\mathbb{S}^1$, degree theory
Categories:34B16, 34C25, 70F05, 70F15

2. CJM 2010 (vol 62 pp. 787)

Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.
An Explicit Treatment of Cubic Function Fields with Applications
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Keywords:cubic function field, discriminant, non-singularity, integral basis, genus, signature of a place, class number
Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29

3. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point
This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.

Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity
Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

4. CJM 2008 (vol 60 pp. 379)

rgensen, Peter J\o
Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes
A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth. This paper proves a non-commutative analogue. Namely, if $A$ is a (non-commutative) graded Artin--Schelter \CM\ algebra which is fully bounded Noetherian and has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.

Keywords:Artin--Schelter Cohen--Macaulay algebra, Artin--Schelter Gorenstein algebra, Auslander's theorem on finite Cohen--Macaulay type, Cohen--Macaulay ring, fully bounded Noetherian algebra, isolated singularity, maximal Cohen--Macaulay module, non-commutative
Categories:14A22, 16E65, 16W50

5. CJM 2001 (vol 53 pp. 834)

Veys, Willem
Zeta Functions and `Kontsevich Invariants' on Singular Varieties
Let $X$ be a nonsingular algebraic variety in characteristic zero. To an effective divisor on $X$ Kontsevich has associated a certain motivic integral, living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational (smooth, projective) Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the invariant {\it motivic (Igusa) zeta function}, associated to a regular function on $X$, which specializes to both the classical $p$-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization to singular varieties. Batyrev already considered such a `Kontsevich invariant' for log terminal varieties (on the level of Hodge polynomials of varieties instead of in the Grothendieck ring), and previously we introduced a motivic zeta function on normal surface germs. Here on any $\bbQ$-Gorenstein variety $X$ we associate a motivic zeta function and a `Kontsevich invariant' to effective $\bbQ$-Cartier divisors on $X$ whose support contains the singular locus of~$X$.

Keywords:singularity invariant, topological zeta function, motivic zeta function
Categories:14B05, 14E15, 32S50, 32S45

6. CJM 2000 (vol 52 pp. 1149)

Ban, Chunsheng; McEwan, Lee J.
Canonical Resolution of a Quasi-ordinary Surface Singularity
We describe the embedded resolution of an irreducible quasi-ordinary surface singularity $(V,p)$ which results from applying the canonical resolution of Bierstone-Milman to $(V,p)$. We show that this process depends solely on the characteristic pairs of $(V,p)$, as predicted by Lipman. We describe the process explicitly enough that a resolution graph for $f$ could in principle be obtained by computer using only the characteristic pairs.

Keywords:canonical resolution, quasi-ordinary singularity
Categories:14B05, 14J17, 32S05, 32S25

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