Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

A recurring theme in attempts to break the curse of dimensionality in the numerical approximation of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class \(\Sigma _n\) of functions, which can be written as a sum of rank-one tensors using a total of at most \(n\) parameters, and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side \(f\) of the elliptic PDE can be approximated with a certain rate \(\mathcal {O}(n^{-r})\) in the norm of \({\mathrm H}^{-1}\) by elements of \(\Sigma _n\), then the solution \(u\) can be approximated in \({\mathrm H}^1\) from \(\Sigma _n\) to accuracy \(\mathcal {O}(n^{-r'})\) for any \(r'\in (0,r)\). Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second “basis-free” model of tensor-sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent to which such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.     

Two-sided a posteriori error bounds for incompressible quasi-Newtonian flows

Endre Süli ^{We develop a posteriori upper and lower error bounds for mixedfinite-element approximations of a general family of steady,viscous, incompressible quasi-Newtonian flows in a bounded Lipschitzdomain ; thefamily includes degenerate models such as the power law model,as well as non-degenerate ones such as the Carreau model. Theunified theoretical framework developed herein yields residual-baseda posteriori bounds which measure the error in the approximationof the velocity in the W1, r() norm and that of the pressurein the Lr'() norm, 1/r + 1/r' = 1, r (1, ).}     

Neighborhood hypergraphs of bipartite graphs

Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity and relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular, we recall results from the literature showing that matrix symmetrization is in fact NP‐hard; furthermore, it is equivalent with the problem of recognizing whether a hypergraph can be realized as the neighborhood hypergraph of a graph. There are several variants of the latter problem corresponding to the concepts of open, closed, or mixed neighborhoods. While all these variants are NP‐hard in general, one of them restricted to the bipartite graphs is known to be equivalent with graph isomorphism. Extending this result, we consider several other variants of the bipartite neighborhood recognition problem and show that they all are either polynomial‐time solvable, or equivalent with graph isomorphism. Also, we study uniqueness of neighborhood realizations of hypergraphs and show that, in general, for all variants of the problem, a realization may be not unique. However, we prove uniqueness in two special cases: for the open and closed neighborhood hypergraphs of the bipartite graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 69–95, 2008     

Introductory remarks

Separate terms to distinguish the matrix between the cells that grow in vitro and the matrix that separates and surrounds cells in the living body are being proposed. The currently used designations: pericellular, extracellular, or intercellular matrix should be used only for the substance produced by cells in vitro (cultures). The well organized, highly specific, and stable structure with significant topographic variations between the cells of living tissues and organs should be called biological matrix or briefly as biomatrix. My experiments published in the early 1940s on plant cell walls illustrate some similarities between biomatrix of animal tissues and plant cell wall. When cells of plant epithelial membranes undergo plasmolysis, the space between the contracting cell and cellulose cell wall is filled with filaments called Hechtian filaments or strands. With silver impregnation, these filaments become a fibrous network, filling the space between the cell wall and the contracting plasma (cell) membrane. A similar extracellular fibrous network has been described earlier after silver impregnation in some animal tissues. I interpreted my finding that a pectin-like polysaccharide in the cell wall biomatrix and hyaluronan, the only polysaccharide present in the biomatrix of animal tissues, are responsible for this fibrous network artifact.     

Cafiero approach to the Dieudonné type theorems

An algebra of subsets of a normal topological space containing the open sets is considered and in this context the uniform exhaustivity and uniform regularity for a family of additive functions are studied. Based on these results the Cafiero convergence theorem with the Dieudonné type conditions is proved and in this way also the Nikodým-Dieudonné convergence theorem is obtained.     

Determination of polyphenolic compounds by liquid chromatography–mass spectrometry in Thymus species

Polyphenolic compounds represent a wide group of phytochemicals, including well-known subgroups of phenolic acids, flavonoids, natural dyes, lignans etc., which are produced by plants. These natural bioactive compounds possess a variety of beneficial effects including antioxidant and anticarcinogenic activities, protection against coronary diseases as well as antimicrobial properties. Thymus species have already been reported as sources of different phenolic acids and flavonoids. Moreover, the composition and content of flavonoids in Thymus species play important role as taxonomic markers providing distinction of species. High-performance liquid chromatography (HPLC) coupled with diode array detector (DAD) and on-line mass spectrometry (ESI-MS) method was used for analysis. The method was evaluated for a number of validation characteristics (repeatability and intermediate precision, LOD, LOQ, calibration range, and recovery). The polyphenolic pattern of five native Hungarian Thymus species (T. glabrescens Willd., T. pannonicus All., T. praecox Opiz, T. pulegioides L., and T. serpyllum L.) was characterized. The dominant compound was rosmarinic acid, which ranged between 83.49 μg g⁻¹ and 1.436 mg g⁻¹. Other phenolic acids (ferulic acid, caffeic acid and its other derivatives, chlorogenic acid and p-coumaric acids) were present in every examined Thymus species, as well as flavanones: naringenin, eriodictyol and dihydroquercetin; flavones: apigenin and apigenin-7-glucoside, flavonols: quercetin and rutin. The polyphenolic pattern was found to be a useful additional chemotaxonomic tool for classification purposes and determination of the locality of origin.     

Evaluation of the H2)dedpa scaffold and its cRGDyK conjugates for labeling with 64Cu

Studies of the acyclic ligand scaffold H(2)dedpa and its derivatives with the peptide cRGDyK for application in copper radiopharmaceuticals are described. Previously shown to be a superb ligand for (67/68)Ga, the chelate is now shown to coordinate (64)Cu in its derivatized and nonderivatized forms rapidly under mild reaction conditions (10 min, RT, pH 5.5 10 mM sodium acetate buffered solution). The hexadentate, distorted octahedral coordination of H(2)dedpa is confirmed in the corresponding solid state X-ray crystal structure of [Cu(dedpa)]. Cyclic voltammetry determined the reduction potential of [Cu(dedpa)] to be below values found for common bioreductants. Reduction and reoxidation were irreversible but reproducible, indicating a potential change of coordination mode upon reduction of Cu(II) to Cu(I). The thermodynamic stability constant log K(CuL) was determined to be 19.16(5), comparable to other frequently used (64)Cu chelates. Serum stability of the (64)Cu labeled chelate revealed only 3% transchelation/association to serum proteins after 2 h, while the conjugates reveal 10% ([Cu(RGD1)]) and 6% ([Cu(RGD2)]) transchelation at the same time point.     

Therapeutic use of hyaluronan

Hyaluronan used today as a therapeutic agent in human and animal medicine must be in a highly purified form, free of immunologically active protein molecules, from endotoxin and from inflammatory molecules originating from the tissues or bacterial cultures—the source of hyaluronan. All hyaluronans, whether in liquid or in covalently crosslinked gel form, must have certain elastoviscous properties to be usable for current therapeutic applications. Elastoviscous properties are especially important in the ophthalmic viscosurgical use. The metabolism of hyaluronan is very fast in most tissues, except in the vitreus. The ability of injected hyaluronan with a short half-life time in the joint to accomplish long-lasting analgesia represents a challenge for the design of the various products. The history of the therapeutic use of hyaluronan and its present status is described, with emphasis on its use in ophthalmic surgery and for a long-lasting analgesia in arthritic joints in humans and animals. The use of hyaluronan gels is described in viscoaugmentation as injected into the dermal layer of the skin. Hyaluronan gels were also used as viscosupplements injected into sphincter muscles to improve their function in aging or disease. The use of hyaluronan and its gel for drug delivery was suggested several decades ago, and is also mentioned.     

The hereditary monocoreflective subcategories of Abelian groups and R-modules

The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ?? Ob Ab | G is a torsion group, and for all g ?? G the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, ??,??}. (o(·) means the order of an element, and n ?? ?? means n < ??.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ?? Ob CompAb | G has a neighbourhood subbase {G _?? } at 0, consisting of open subgroups, such that G/G _?? is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.