Sections of a differential spectrum and factorization-free computations

We construct sections of a differential spectrum using only localization and projective limits. For this purpose we introduce a special form of multiplicative systems generated by one differential polynomial and call it D-localization. Owing to this technique one can construct sections of a differential spectrum of a differential ring without computation of diffspec . We compare our construction with Kovacic’s structure sheaf and with the results obtained by Keigher [J. Pure Appl. Algebra, 27, 163–172 (1983)]. We show how to compute sections of factor-rings of rings of differential polynomials. All computations in this paper are factorization-free. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 133–144, 2003.     

Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map , i.e., of a subsetF ⊂ Z/A such thatf can be globally extended to its complement. It is shown that ifZ is a fixed metric space with dimZ ≤ n andp, q ≥−1 are fixed numbers, then obstructions to extending all partial maps can be concentrated in preselected fairly thin subsets ofZ. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 803–812, December 1997 Translated by O. V. Sipacheva     

Cauchy problem for an essentially infinite-dimensional parabolic equation with variable coefficients

The Cauchy problem for the equation with positive essentially infinite-dimensional functionalsj(x) is studied in a properly chosen Banach space of functions on an infinite-dimensional separable real Hilbert space. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 6, pp. 663–670, June, 1994.     

Holomorphic synthesis of monogenic functions of several quaternionic variables

The system of differential equations for polymonogenic functions of several quaternionic variables is an analogue of the -equation in complex analysis. We give a representation of polymonogenic functions by means of integration of a family of σ-holomorphic functions as σ runs over the variety Σ of all complex structures ℍ ≅ ℂ² which are consistent with the metric and an orientation in ℍ. The variety Σ is isomorphic to the manifold of all proper right ideals in the complexified quaternionic algebra and has a natural complex analytic structure. We construct a -complex on Σ that provides a resolvennt for the sheaf of polymonogenic functions.     

What we know about the second adjunction mapping

Let be an algebraic submanifold of complex projective space ℙ ^N . Assume thatn:=dim and let denote the restriction of the hyperplane section bundle to . The meromorphic map associated to fork≥1 ties together the pluricanonical maps of the surface sections of . Known results show that the behavior of is far better than one would expect from experience with the pluricanonical mappings of algebraic surfaces. In this article we discuss the known results on the structure of the mappings and describe the open problems. Conferenza tenuta il 5 giugno 1997     

On the structure of the spectrum of a linear time periodic wave equation

In this paper we study the structure of the spectrum of the evolution operator generated by the time periodic wave equation with 2π periodicity inx as boundary condition. We show that such structure depends heavily on the rationality or irrationality of the rotation numbers of the characteristics deiffential equations as well as on the phase space.     

Self-dual Sheaves on Reductive Borel—Serre Compactifications of Hilbert Modular Surfaces

The main theorem of this article asserts that the category of self-dual sheaves compatible with the intersection chain sheaves (for upper/lower middle perversity) on the reductive Borel—Serre compactification of a Hilbert modular surface is nonempty. Also we prove that the direct image of such a sheaf under the canonical map to the Baily—Borel compactification is isomorphic (in the derived category) to the intersection chain sheaf for upper and lower middle perversity. As a consequence of the main theorem, there exist characteristic L-classes of these sheaves in the rational homology of . In fact, these classes do not depend on the choice of a self-dual sheaf and hence are invariants of the compactification .     

Relationship between manifolds of solutions to generic and model generalized Cauchy-Riemann systems with a singular point

The unique solvability of a specific two-dimensional integral equation is proved. It is used to establish a one-to-one correspondence between the sets of continuous solutions to two systems of the form and defined in the same domainG (z=0∈G). Translated fromMatermaticheskie Zametki, Vol. 66, No. 2, pp. 302–307, August, 1999.     

Regularity of solutions of certain parabolic system with nonstandard growth condition

By the use of Moser iteration and Campanato space estimate, theL ^∞ andC ^α regularity estimate of the gradient of solutions of nonlinear parabolic system with nonstandard growth conditions are obtained under the natural structure constraints. This work was supported by NNSF of China under Grant 39570223 and Grant 19675005     

Projective line over the finite quotient ring GF(2)[x]/〈x3 ? x〉 and quantum entanglement: Theoretical background

We consider the projective line over the finite quotient ring R_⋄ ≡ GF(2)[x]/〈x ³ − x〉. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because R_⋄ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical of the ring. The quotient ring is isomorphic to GF(2) ⊗ GF(2). The projective line over features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 44–53, April, 2007.     

The mathieu-hill operator equation with dissipation and estimates of its instability index

The behavior ast→∞ of solutions of the equation

The Neumann boundary value problem for a semilinear elliptic equation in a thin cylinder. The least energy solutions

It is proved that the least energy solution of the BVP

On a problem of integer convex interpolation

The study of a generalized projective scheme closely related to the binary projective space P ⁽²⁾ = Proj \mathbbF₀[t]{\mathbb{F}}_0[t], where t is a free binary variable, is started, In particular, the global sections of the structure sheaf on elements of a certain affine covering are computed, and the scheme points of the unary version are described. Bibliography: 4 titles.     

Transformation of measures generated by jump Lévy processes

Let ξ(t), t ∈ [0, 1], be a jump Lévy process. We denote by the law of ξ in the Skorokhod space [0, 1]. Under some nondegeneracy condition on the Lévy measure Λ of the process, we construct a group of -preserving transformations of the space [0, 1]. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 175–188.     

On the correct solvability of the Dirichlet problem for operator differential equations in a Banach space

We investigate the structure of solutions of an equation y″(t) = By(t), where B is a weakly positive operator in a Banach space , on the interval (0, ∞) and establish the existence of their limit values as t → 0 in a broader locally convex space containing as a dense set. The analyticity of these solutions on (0, ∞) is proved and their behavior at infinity is studied. We give conditions for the correct solvability of the Dirichlet problem for this equation and substantiate the applicability of power series to the determination of its approximate solutions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1462–1476, November, 2006.     

The Geometry of Polygons in ℝ5 and Quaternions

We consider the moduli space _r of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between _r and a weighted quotient of n-fold products of the quaternionic projective line ¹ by the diagonal PSL(2, )-action. We explore the relation between _r and the fixed point set of an anti-symplectic involution on a GIT quotient Gr_ℂ(2, 4) ⁿ /SL(4, ℂ). We generalize the Gel'fand—MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space _r as a quotient of a subspace in the quaternionic Grassmannian Gr_ℍ(2, n) by the action of the group Sp(1) ⁿ . We also give analogues of the Gel'fand—Tsetlin coordinates on the space of quaternionic Hermitean marices and briefly describe generalized action—angle coordinates on _r .     

Behavior of Solutions of Quasilinear Elliptic Inequalities

The paper considers solutions of the coercive inequalities

On the\bar \partial equation in weightedL 2 norms in ℂ1equation in weightedL 2 norms in ℂ1

For a large class of subharmonicφ, the equation is studied in . Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal projection onto the space of entire functions inH. Existence theorems inL ^p norms are derived as a corollary. A class of counterexamples, related to the failure of to be analytic-hypoelliptic on certain CR manifolds, is discussed. Communicated by Steven Krantz     

Estimates of the rate of decay of solutions to the impedance mixed problem for the wave equation in a region with noncompact boundary

The paper is devoted to the study of the behavior of the following mixed problem for large values of time:

Auslander–Reiten sequences on schemes

Auslander–Reiten sequences are the central item of Auslander–Reiten theory, which is one of the most important techniques for the investigation of the structure of abelian categories. This note considers X, a smooth projective scheme of dimension at least 1 over the field k, and , an indecomposable coherent sheaf on X. It is proved that in the category of quasi-coherent sheaves on X, there is an Auslander–Reiten sequence ending in .