Related papers: Resolution and alteration with ample exceptional d…
Regular resolution is a refinement of the resolution proof system requiring that no variable be resolved on more than once along any path in the proof. It is known that there exist sequences of formulas that require exponential-size proofs…
Let $X$ be an integral affine or projective scheme of finite presentation over a perfect field. We prove that $X$ admits a resolution, that is, there exists a smooth scheme $\widetilde X$ and a projective birational morphism from…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
The dual complex can be associated to any resolution of singularities whose exceptional set is a divisor with simple normal crossings. It generalizes to higher dimensions the notion of the dual graph of a resolution of surface singularity.…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
Let $X$ be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of $X$ by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove…
In this paper, we study the existence and uniqueness of periodic solutions of the differential equation of the form . Here, we obtain some sufficient conditions which guarantee the existence of periodic solutions. This equation is a quite…
Let $(Z,o)$ be a three-dimensional terminal singularity of type $cA/r$. We prove that all exceptional divisors over $o$ with discrepancies $\le 1$ are rational.
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…
We describe the resolution of singularities of a threefold which has minimal Picard number. We describe the relation between this minimal resolution and an arbitrary resolution of singularities.
Consider a projective variety $X \subset \mathbb{P}^n$ (over an algebraically closed field of characteristic zero), together with a (reduced) simple normal crossings divisor $E \subset \mathbb{P}^n$, where the degrees of both $X$ and $E$…
An ordinary differential equation is said to have a superposition formula if its general solution can be expressed as a function of a finite number of particular solution. Nonlinear ODE's with superposition formulas include matrix Riccati…
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…
In this note we study the connection between the existence of a projective reconstruction and the existence of a fundamental matrix satisfying the epipolar constraints.
We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.
A rationality condition is derived for the existence of odd perfect numbers involving the square root of a product, which consists of a sequence of repunits, multiplied by twice the base of one of the repunits. This constraint also provides…
Given a singular curve on a smooth surface, we determine which exceptional divisors on the minimal resolution of that curve contribute toward its jumping numbers.
Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the…
This article shall serve as a quick reference for somebody who needs precise information on concepts and results related to resolution of singularities. As such, it is more a technical manual than a bedtime story. Topics which are covered:…
Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent -2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.