The main goal of this article is to understand the trace properties
of nonlocal minimal graphs in , i.e. nonlocal minimal surfaces with a graphical structure.
We establish that at any boundary points at which
the trace from inside happens to coincide with
the exterior datum, also the tangent planes
of the traces necessarily coincide with those of the exterior datum.
This very rigid geometric constraint is in sharp contrast with the case
of the solutions of the linear equations driven by the fractional Laplacian,
since we also show that, in this case, the fractional normal
derivative can be prescribed arbitrarily, up to a small error.
We remark that, at a formal level,
the linearization of the trace of a nonlocal minimal graph
is given by the fractional normal derivative of a fractional Laplace problem,
therefore the two problems are formally related. Nevertheless, the nonlinear equations of fractional mean curvature type
present very specific properties which are strikingly different from
those of other problems of fractional type
which are apparently similar, but
diverse in structure, and the nonlinear
case given by the nonlocal minimal graphs
turns out to be significantly more rigid than its linear counterpart.
Rickard complexes in the context of categorified quantum groups can be used to construct braid group actions. We define and study certain natural deformations of these complexes which we call curved Rickard complexes. One application is to obtain deformations of link homologies which generalize those of Batson–Seed  [J. Batson and C. Seed, A link-splitting spectral sequence in Khovanov homology,
Duke Math. J. 164 2015, 5, 801–841] and Gorsky–Hogancamp [E. Gorsky and M. Hogancamp, Hilbert schemes and y-ification of Khovanov–Rozansky homology,
preprint 2017] to arbitrary representations/partitions. Another is to relate the deformed homology defined algebro-geometrically in [S. Cautis and J. Kamnitzer,
Knot homology via derived categories of coherent sheaves IV, colored links,
Quantum Topol. 8 2017, 2, 381–411] to categorified quantum groups (this was the original motivation for this paper).
We study deformations of rational curves and their singularities
in positive characteristic.
We use this to prove that if a smooth and proper surface
in positive characteristic p is dominated by a family of rational curves
such that one member has all δ-invariants (resp. Jacobian numbers)
strictly less than (resp. p),
then the surface has negative Kodaira dimension.
We also prove similar, but weaker results hold for higher-dimensional varieties.
Moreover, we show by example that our result is in some sense optimal.
On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.
We give a restriction formula on jumping numbers
which is a reformulation of Demailly–Ein–Lazarsfeld’s important restriction formula for multiplier ideal sheaves
and a generalization of Demailly–Kollár’s important restriction formula on complex singularity exponents,
and then we establish necessary conditions for the extremal case in the reformulated formula;
we pose the subadditivity property on the complex singularity exponents of plurisubharmonic functions
which is a generalization of Demailly–Kollár’s fundamental subadditivity property,
and then we establish necessary conditions for the extremal case in the generalization.
We also obtain two sharp relations on jumping numbers, introduce a new invariant of plurisubharmonic singularities and get its decreasing property for consecutive differences.
In this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.
An iterative method (AIM) is one of the numerical method, which is easy to apply and very time convenient for solving nonlinear differential equations. However, if we want to work in a large interval, sometimes it may be difficult to apply AIM. Therefore, a multistage AIM named Multistage Modified Iterative Method (MMIM) is introduced in this article to work in a large computational interval. The applicability of MMIM for increasing the solution domain of the given problems is construed in this article. Some problems are solved numerically using MMIM, which provides a better result in the extended interval as compared to AIM. Comparison tables and some graphs are included to demonstrate the results.