COMPLEX CLASSIFIERS OF FORMAL CLASSIFICATION THEORY OF TECHNICAL SYSTEMS: HIERARCHO-MATRIX, ZONE, CYCLIC

The article highlights the different aspects of complex technical systems that can be ordered and classified in accordance with the totality, structure and values of the attributes characterizing these systems by a unified approach to various types of classifications. The most complex classifiers studied in this work are hierarchical-matrix, cyclic and zonal classifiers. Zones are the areas identified in the space of classification attributes that characterized by a similar value of an additional target classification attribute. The dimension of the zonal classification is equal to the number of descriptive classification attributes, the zonal dimension is equal to the number of zones. Adding the zones is carried out according to the target classification criterion, multiplying the zones consists in introducing the new descriptive classification attributes. Cyclicity is repetition of the similar elements that occurs in the space of physical quantities or other parameters. The concept of cycle stages is defined for all cycles, which is a specific (target) classification attribute. The internal dimension of the cycle is equal to the number of stages, the external dimension is equal to the number of acts of the cycle, the descriptive dimension is equal to the number of descriptive classification attributes. Addition of cycles can be carried out both by stages and by descriptive features and consists in increasing the number of values of classification attribute. Multiplication of cycles consists in the introduction of new descriptive classification attributes. Zonal and cyclic classifiers are widely used in the practice of describing and planning technical energy systems. A wide range of classifiers ordered from the standpoint of a unified formal theory of classification will take into account the features of specific technical systems, the conditions for the objects functioning, the context of the interpretation area. As a re-sult, the degree of adequacy of classifiers to the diversity of the interpretation area objects and the representativeness of models based on classifiers will increase

Approximating inverse FEM matrices on non-uniform meshes with $${\mathcal{H}}$$-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse $${\mathcal{H}}$$ H -matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the $${\mathcal{H}}$$ H -matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Delineating the heterogeneity of matrix-directed differentiation towards soft and stiff tissue lineages via single-cell profiling

SummaryMesenchymal stromal/stem cells (MSCs) are a heterogeneous population of multipotent progenitors that contribute to tissue regeneration and homeostasis. MSCs assess extracellular elasticity by probing resistance to applied forces via adhesion, cytoskeletal, and nuclear mechanotransducers, that direct differentiation toward soft or stiff tissue lineages. Even under controlled conditions, MSC differentiation exhibits substantial cell-to-cell variation that remains poorly characterized. By single-cell transcriptional profiling of naïve, matrix-conditioned, and early differentiation state cells, we identified distinct MSC subpopulations with distinct mechanosensitivities, differentiation capacities, and cell cycling. We showed that soft matrices support adipogenesis of multipotent cells and endochondral ossification of non-adipogenic cells, whereas intramembranous ossification and pre-osteoblast proliferation are enhanced by stiff matrices. Using diffusion pseudotime mapping, we delineated hierarchical matrix-directed differentiation and identified mechanoresponsive genes. We found that tropomyosin-1 (TPM1) is highly sensitive to stiffness cues both at RNA and protein levels and that changes in expression of TPM1 determine adipogenic or osteogenic fates. Thus, cell-to-cell variation in tropomyosin-mediated matrix-sensing contributes to impaired differentiation with implications to the biomedical potential of MSCs.

Block-Adaptive Cross Approximation of Discrete Integral Operators

In this article, we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems, it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the blockwise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.