Gas emission into the excavated space during the mining of coal seams

Using the method of successive approximations we construct a numerical/analytic solution, of the nonlinear filtration equation that describes the process of gas emission from a coal seam that is being mined. It is assumed that in the opening and excavation of a deposit a finite filtration region forms near the exposed coal face, and that the gas emission process is accompanied by the appearance of an unknown moving boundary. It is established that this boundary moves slowly toward the end of the filtration region and the determination of its location must be taken into account in computing the load on the drift from the gas factor. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 112–115.     

On the embedding of BV space into Besov-Orlicz space

We give a sufficient (and, in the case of a compact domain, a necessary) condition for the embedding of Sobolev space of functions with integrable gradient into Besov-Orlicz spaces to be bounded. The condition has a form of a simple integral inequality involving Young and weight functions. We provide an example with Matuszewska-Orlicz indices of involved Orlicz norm equal to one. The main tool is the molecular decomposition of functions from a BV space.     

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\) , provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

On the coincidence of the canonical embeddings of a metric space into a Banach space

Recall the two classical canonical isometric embeddings of a finite metric space X into a Banach space. That is, the Hausdorff–Kuratowsky embedding x → ρ(x, ⋅) into the space of continuous functions on X with the max-norm, and the Kantorovich–Rubinshtein embedding x → δ _x (where δ _x , is the δ-measure concentrated at x) with the transportation norm. We prove that these embeddings are not equivalent if |X| > 4. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 153–161.     

Complete self-shrinkers confined into some regions of the space

We study geometric properties of complete non-compact bounded self-shrinkers and obtain natural restrictions that force these hypersurfaces to be compact. Furthermore, we observe that, to a certain extent, complete self-shrinkers intersect transversally a hyperplane through the origin. When such an intersection is compact, we deduce spectral information on the natural drifted Laplacian associated to the self-shrinker. These results go in the direction of verifying the validity of a conjecture by H.D. Cao concerning the polynomial volume growth of complete self-shrinkers. A finite strong maximum principle in case the self-shrinker is confined into a cylindrical product is also presented.     

Compression of uniform embeddings into Hilbert space

If one tries to embed a metric space uniformly in Hilbert space, how close to quasi-isometric could the embedding be? We answer this question for finite dimensional CAT(0) cube complexes and for hyperbolic groups. In particular, we show that the Hilbert space compression of any hyperbolic group is 1.     

On the isometric minimal immersion into a euclidean space

We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space. A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given. This work is partially supported by the National Science Foundation of China     

Direct and inverse problems of gas emission and the sorptive deformation of coal beds

Using the equations of state for fractured-porous media that describe the sorptioninduced deformation of coal, we develop a geomechanical model for radial gas influx to a borehole drilled in a coal bed with the concurrent evolution of stress field in the borehole environment. A numerical-and-analytical method is proposed for solving the corresponding system of equations for poroelastic media. A relation is found between the volume of slack withdrawn in the borehole (when opening up the gas-bearing seams), the sorption-and-storage capacities of coal, the permeability k, and the horizontal component σ _h of the natural stress field. We demonstrate the solvability of the inverse boundary-coefficient problem of determining k and σ _h on the basis of pressure in the closed borehole. We substantiate an express-method for estimating the permeability by the measurements of pressure in the borehole operating in the “pressure drop” mode.     

Thermoelastic contact during pressing of a heated elliptical cross-section die into a transversely isotropic piezoceramic half space

The three-dimensional thermoelastic contact as a nonplanar elliptical cross-section die is pressed into a transversely isotropic half space is examined under conditions such that the die has a previously specified temperature while the boundary of the half space outside the die is kept at zero temperature. Contact between the die and the half space is assumed to be ideal. The stressed state in the elastic medium is characterized by potential functions which are constructed explicity. Expressions are given for determining the translational displacement of points on the base of the stamp as well as the major half axis and eccentricity of the contact area. Some possible forms of boundary conditions for the electrical variables at the contact area are examined. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 37–53, 1999.     

Special embeddings of graphs into a three-dimensional space

Embeddings of graphs into ?³ such that the minimal possible number of points are placed on each line are considered. A theorem stating that for any embedding into ?³ of a pair of circumferences with a nonzero linking number there exists a line crossing this set at not less than 4 points is proved. Using this theorem, it is proved that Petersen graphs are minimal 3-nonembeddable.     

A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft