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Vol 213, No 1 (2022)
- Year: 2022
- Articles: 5
- URL: https://journal-vniispk.ru/0368-8666/issue/view/7485
Global and semilocal theorems on implicit and inverse functions in Banach spaces
Abstract
We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates for solutions are given. As an application of these results, we obtain, in particular, a theorem on extending an implicit function from a given closed set to the whole parameter space and a theorem on coincidence points of mappings. Bibliography: 32 titles.
Matematicheskii Sbornik. 2022;213(1):3-45
3-45
On the local and boundary behaviour of inverse maps on Riemannian manifolds
Abstract
Results on the local behaviour of maps between Riemannian manifolds such that their inverses satisfy upper bounds on the distortion of the moduli of families of curves are obtained. For families of such maps theorems on their equicontinuity at interior points and boundary points of the domain are established. Bibliography: 30 titles.
Matematicheskii Sbornik. 2022;213(1):46-68
46-68
Nonlocal balance equations with parameters in the space of signed measures
Abstract
A parametric family of nonlocal balance equations in the space of signed measures is studied. Under assumptions that cover a number of known conceptual models we establish the existence of the solution, its uniqueness and continuous dependence on the parameter and the initial distribution. Several corollaries of this theorem, which are useful for control theory, are discussed. In particular, this theorem yields the limit in the mean field of a system of ordinary differential equations, the existence of the optimal control for an assembly of trajectories, Trotter's formula for the product of semigroups of the corresponding operators, and the existence of a solution to a differential inclusion in the space of signed measures. Bibliography: 33 titles.
Matematicheskii Sbornik. 2022;213(1):69-94
69-94
On singular log Calabi-Yau compactifications of Landau-Ginzburg models
Abstract
We consider the procedure that constructs log Calabi-Yau compactifications of weak Landau-Ginzburg models of Fano varieties. We apply it to del Pezzo surfaces and coverings of projective spaces of index $1$. For coverings of degree greater than $2$ the log Calabi-Yau compactification is singular; moreover, no smooth projective log Calabi-Yau compactification exists. We also prove, in the cases under consideration, the conjecture that the number of components of the fibre over infinity is equal to the dimension of an anticanonical system of the Fano variety. Bibliography: 46 titles.
Matematicheskii Sbornik. 2022;213(1):95-118
95-118
More about sparse halves in triangle-free graphs
Abstract
One of Erdős's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish the new bound $27n^2/1024$ on the number of edges in the general case. We completely prove the conjecture for graphs of girth $\geq 5$, for graphs with independence number $\geq 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph. Bibliography: 21 titles.
Matematicheskii Sbornik. 2022;213(1):119-140
119-140