Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings

We investigate how the integrability of the derivatives of Orlicz-Sobolev mappings defined on open subsets of Rn affect the sizes of the images of sets of Hausdorff dimension less than n. We measure the sizes of the image sets in terms of generalized Hausdorff measures.     

Integrability of derivatives of inverses of maps of exponentially integrable distortion in the plane

The K-quasiconformal maps form a category which is invariant under inversion, i.e. f and f⁻¹ are simultaneously K-quasiconformal. Maps of exponentially integrable distortion are a useful class for extending the Beltrami equation to a degenerate setting. This class is not invariant under inversion. In this note we show that the inverses of homeomorphisms of exponentially p-integrable distortion have β-integrable distortion for all β<p, but not necessarily for β=p.     

Slow mappings of finite distortion

We examine mappings of finite distortion from Euclidean spaces into Riemannian manifolds. We use integral type isoperimetric inequalities to obtain Liouville type growth results under mild assumptions on the distortion of the mappings and the geometry of the manifolds.     

Generalized nonisospectral super integrable hierarchies

For the Lie superalgebra B(0,1), we choose a set of basis matrices. Then we consider a linear combination of the basis matrices, which is exactly the spectral matrix of the spatial part for the super Ablowitz‐Kaup‐Newell‐Segur (AKNS) hierarchy. The compatible condition of the spatial and temporal spectral problems leads to the well‐known zero curvature equation. Here, when the spectral parameter is independent (dependent) of temporal parameter, we obtain isospectral (nonisospectral) super AKNS hierarchy. Furthermore, we derive the generalized nonisospectral super AKNS hierarchy (GNI‐SAKNS). As another example, similar method is successfully applied to the super Dirac hierarchy, and we obtain the generalized nonisospectral super Dirac hierarchy (GNI‐SD).     

Graded Lie algebras, representation theory, integrable mappings, and integrable systems

A new class of integrable mappings and chains is introduced. The corresponding 1+2 integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple A_n algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 251–271, February, 2000.     

Generalized stability of multi-additive mappings

In this paper we unify the system of Cauchy functional equations defining multi-additive mapping to obtain a single equation and prove the generalized Hyers–Ulam stability both of this system and this equation using the so-called direct method.     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

Quasiconformal mappings and singularity of boundary distortion

We extend a well-known theorem of Jones and Makarov [8] on the singularity of boundary distortion of planar conformal mappings. Using a different technique, we recover the previous result and generalize the result to quasiconformal mappings of the unit ball ⊂_∝ ⁿ , n ≥ 2. We also establish an estimate on the Hausdorff (gauge) dimension of the boundary of the image domain outside an exceptional set of given size on the sphere ∂ and show that this estimate is essentially sharp.     

Hull mappings and dimension effect algebras

An effect algebra (EA) is a partial algebraic structure, originally formulated as an algebraic base for unsharp quantum measurements. The class of EAs includes, as special cases, several partially ordered algebraic structures, including orthomodular lattices (OMLs) and orthomodular posets (OMPs), hitherto used as mathematical models for experimentally verifiable propositions pertaining to physical systems. Moreover, MV-algebras, which are mathematical models for many-valued logics, are special cases of EAs. The present paper studies generalizations to EAs of the hull mapping featured in L. Loomis’s dimension theory for complete OMLs and develops a theory of direct decomposition for EAs with a hull mapping. A. Sherstnev and V. Kalinin have extended Loomis’s dimension theory to orthocomplete OMPs, and here it is further extended to orthocomplete EAs; moreover, a corresponding direct decomposition into types I, II, and III is obtained using the hull mapping induced by the dimension equivalence relation.     

On the convergence of mappings with k-finite distortion

Mathematical Notes -     

Boundary property of n-dimensional mappings with bounded distortion

The following assertion is proved: letf:BRⁿ be an arbitrary (in general, not single-sheeted) mapping with bounded distortion of an n-dimensional sphere B, satisfying the conditions: A) the setf(B) is bounded; B) the partial derivatives are summable with respect to B with degree (1< <- n). Then the mappingf has angular boundary values everywhere on the boundary of the sphere with the possible exception of a set of-capacity zero.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 159–164, February, 1972.