Rigidity Of Matrices Research Articles

Directional Cell Migration Guided by a Strain Gradient.

Strain gradients widely exist in development and physiological activities. The directional movement of cells is essential for proper cell localization, and directional cell migration in responses to gradients of chemicals, rigidity, density, and topography of extracellular matrices have been well-established. However; it is unclear whether strain gradients imposed on cells are sufficient to drive directional cell migration. In this work, a programmable uniaxial cell stretch device is developed that creates controllable strain gradients without changing substrate stiffness or ligand distributions. It is demonstrated that over 60% of the single rat embryonic fibroblasts migrate toward the lower strain side in static and the 0.1Hz cyclic stretch conditions at ≈4% per mm strain gradients. It is confirmed that such responses are distinct from durotaxis or haptotaxis. Focal adhesion analysis confirms higher rates of contact area and protrusion formation on the lower strain side of the cell. A 2D extended motor-clutch model is developed to demonstrate that the strain-introduced traction force determines integrin fibronectin pairs' catch-release dynamics, which drives such directional migration. Together, these results establish strain gradient as a novel cue to regulate directional cell migration and may provide new insights in development and tissue repairs.

Matrix and tensor rigidity and Lp-approximation

We apply methods originated in Complexity theory to some problems of Approximation. We notice that the construction of Alman and Williams that disproves the rigidity of Walsh-Hadamard matrices, provides good ℓp-approximation for p<2. Hence the first n functions of Walsh system can be approximated by a low-dimensional linear space:dn1−δ({w1,…,wn},Lp[0,1])⩽n−δ,1≤p<2, where δ=δ(p)>0. We do not know if this is possible for the trigonometric system. We show that the algebraic method of Alon–Frankl–Rödl for bounding the number of low-signum-rank matrices, works for tensors: almost all signum-tensors have large signum-rank and can't be ℓ1-approximated by low-rank tensors. This implies lower bounds for Θm — the error of m-term approximation of multivariate functions by sums of tensor products u1(x1)⋯ud(xd). For the set of trigonometric polynomials with spectrum in ∏j=1d[−nj,nj] and of norm ‖t‖∞⩽1, and m⩽c(d)∏nj/max⁡{nj}, we haveΘm(T(n1,…,nd)∞,L1[−π,π]d)⩾c1(d)>0. Sharp bounds follow for classes of dominated mixed smoothness in the case 2⩽p⩽∞,1⩽q⩽2:Θm(Wp(r,r,…,r),Lq[0,1]d)≍m−rdd−1.

Spectral rigidity of non-Hermitian symmetric random matrices near the Anderson transition

We study the spectral rigidity of the non-Hermitian analog of the Anderson model suggested by Tzortzakakis, Makris and Economou (TME). This is a $L\times L \times L$ tightly bound cubic lattice, where both real and imaginary parts of on-site energies are independent random variables uniformly distributed between $-W/2$ and $W/2$. The TME model may be used to describe a random laser. In a recent paper we proved that this model has the Anderson transition at $W= W_c \simeq 6$ in three dimension. Here we numerically diagonalize TME $L \times L \times L$ cubic lattice matrices and calculate the number variance of eigenvalues in a disk of their complex plane. We show that on the metallic side $W < 6$ of the Anderson transition, complex eigenvalues repel each other as strongly as in the complex Ginibre ensemble only in a disk containing $N_c(L,W)$ eigenvalues. We find that $N_c(L,W)$ is proportional to $L$ and grows with decreasing $W$ similarly to the number of energy levels $N_c$ in the Thouless energy band of the Anderson model.

Stereolithographic 3D Printing of Bioceramic Scaffolds of a Given Shape and Architecture for Bone Tissue Regeneration

Abstract—A standard series of ceramic scaffolds for bone tissue regeneration with Kelvin architecture was developed, providing matrix water permeability not less than 900 Darcy and relative rigidity of matrices of not more than 0.2. The possibility of producing bioceramic scaffolds is shown using stereolithographic 3D printing of light-cured slurries containing a mixed calcium sodium phosphate Ca2.5Na(PO4)2 composition. A production technology of bioceramic scaffolds is developed and tested, including the development of photocurable slurries; 3D printing modes are also worked out, as well as conditions of heat treatment of printed models and sintering. The proposed methods of stereolithography formation followed by heat treatment of printed models make it possible to produce ceramic scaffolds with lateral resolution no worse than 50 μm and 50 μm layering. The dimensions of the bioceramic scaffold differ from the reference dimensions of the initial model by no more than 10%; the degree of macroporosity is not less than 70% and the pore size is 500 μm. It is shown that the obtained bioceramic scaffolds are compatible with the human fibroblast cell culture, are not cytotoxic, do not contain components inhibiting fibroblast adhesion, spreading, and proliferation, and can be used in tissue engineering.

Subtle Regulation of Scaffold Stiffness for the Optimized Control of Cell Behavior.

The rigidity of extracellular matrices can impact cell fate, guide tissue development, and initiate tumor formation. Scaffolds such as hydrogels with tunable levels of stiffness have been developed to control cell adhesion, migration, and differentiation, providing suitable microenvironments for different tissue outcomes. However, studies of cell-material interactions are largely confined to biomaterials with stiffness values that are coarsely regulated, so refinements in sensitive cellular responses and optimal stiffness values that determine cell fate remain elusive. Here, a freezing temperature, as a tunable regulating factor, was introduced to freeze-drying processes to form silk fibroin (SF) scaffolds with refined control of stiffness values. Due to this control of intermediate structural conformations of SF, the scaffolds exhibited differences in stiffness values to permit refined assessments of impact on cell behavior on cell-friendly surfaces. Both in vitro and in vivo results with these scaffolds exhibited gradually changeable cell migration and differentiation outcomes, as well as differences in tissue ingrowth, demonstrating the sensitivity of cellular responses to such refined mechanical cues. The optimal vascularization capacity of these SF scaffolds was in the 3-7.4 kPa range, suggesting a key range to develop bioactive biomaterials. Systematic fine regulation of scaffold rigidity based on the present strategy provides a platform for an improved understanding of cell-material interactions and also for generating optimized microenvironments for tissue regeneration.

New applications of the polynomial method: The cap set conjecture and beyond

The cap set problem asks how large can a subset of ( Z / 3 Z ) n (\mathbb {Z}/3\mathbb {Z})^n be and contain no lines or, more generally, how can large a subset of ( Z / p Z ) n (\mathbb {Z}/p\mathbb {Z})^n be and contain no arithmetic progressions. This problem was motivated by deep questions about structure in the prime numbers, the geometry of lattice points, and the design of statistical experiments. In 2016, Croot, Lev, and Pach solved the analogous problem in ( Z / 4 Z ) n (\mathbb {Z}/4\mathbb {Z})^n , showing that the largest set without arithmetic progressions had size at most c n c^n for some c &gt; 4 c &gt; 4 . Their proof was as elegant as it was unexpected, being a departure from the tried and true methods of Fourier analysis that had dominated the field for half a century. Shortly thereafter, Ellenberg and Gijswijt leveraged their method to resolve the original cap set problem. This expository article covers the history and motivation for the cap set problem and some of the many applications of the technique: from removing triangles from graphs, to rigidity of matrices, and to algorithms for matrix multiplication. The latter application turns out to give back to the original problem, sharpening our understanding of the techniques involved and of what is needed for showing that the current bounds are tight. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, and the integers modulo N N .

Rigidity and Shrinkability of Diagonalizable Matrices

Rigidity and Shrinkability of Diagonalizable Matrices

Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

A square matrix V is called rigid if every matrix $${V^\prime}$$V? obtained by altering a small number of entries of V has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper, we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field $${\mathbb{F}_2.}$$F2. Stated informally, we conjecture that under a suitable embedding of $${\mathbb{F}_2^n}$$F2n into $${\mathbb{R}^n,}$$Rn, vectors arising from a low-dimensional $${\mathbb{F}_2}$$F2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low-dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above.

Plasticization of Poly(vinylpyrrolidone) Thin Films under Ambient Humidity: Insight from Single-Molecule Tracer Diffusion Dynamics

Studies on diffusion dynamics of single molecules (SMs) have been useful in revealing inhomogeneity of polymer thin films near and above the glass-transition temperature (T(g)). However, despite several applications of polymer thin films where exposure to solvent (or vapor) is common, the effect of absorbed solvent molecules on local morphology and rigidity of polymer matrices is yet to be explored in detail. High-T(g) hydrophilic polymers such as poly(vinylpyrrolidone) (PVP) are used as pharmaceutical coatings for drug release in aqueous medium, as they readily absorb moisture, which results in effective lowering of the T(g) and thereby leads to plasticization. The effect of moisture absorption on swelling and softening of PVP thin films was investigated by visualizing the diffusion dynamics of rhodamine 6G (Rh6G) tracer molecules at various ambient relative humidities (RH). Wide-field epifluorescence microscopy, in conjunction with high-resolution SM tracking, was used to monitor the spatiotemporal evolution of individual tracers under varied moisture contents of the matrix. In the absence of atmospheric moisture, Rh6G molecules in dry PVP films are translationally inactive, suggestive of rigid local environments. Under low moisture contents (RH 30-50%), translational mobility remains arrested but rotational motion is augmented, indicating slight swelling of the polymer network which marks the onset of plasticization. The translational mobility of Rh6G was found to be triggered only at a threshold ambient RH, beyond which a large proportion of tracers exhibit extensive diffusion dynamics. Interestingly, SM tracking data at higher moisture contents of the film (RH ≥ 60%) reveal that the distributions of dynamic parameters (such as diffusivity) are remarkably broad, spanning several orders of magnitude. Furthermore, Rh6G molecules display a wide variety of translational motion even at a fixed ambient RH, clearly pointing out the extremely inhomogeneous environment of plasticized PVP network. Intriguingly, it is observed that a majority of tracers undergo anomalous subdiffusion even under high moisture contents of the matrix. Analyses of SM trajectories using velocity autocorrelation function reveal that subdiffusive behaviors of Rh6G are likely to originate from fractional Brownian motion, a signature of tracer dynamics in viscoelastic medium.

Using Elimination Theory to Construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.

Transport of nitrogen gas in glassy maltodextrins

Glassy maltodextrins are amorphous carbohydrates characterized by very high barrier properties for low molecular weight gases. Although they are commonly used for the encapsulation of active ingredients in food and pharmaceutics, the microscopic description of the processes which limit the diffusion of molecules inside the polymeric glassy matrix is still not fully understood. In this paper, we study sorption and diffusion of nitrogen molecules in glassy maltodextrins at negligible water content. The solution–diffusion model for the transport of small molecules is combined with recent positron annihilation lifetime spectroscopy data on the microscopic structure of the matrices. We find that both sorption and diffusion are strongly suppressed by the reduced size of the hole volumes and by the enhanced rigidity of the glassy carbohydrates matrices. Values of the diffusion coefficient of nitrogen gas in glassy maltodextrin calculated using the transition state theory were found to correspond well with experimentally obtained values. Also, our calculations can explain the recently observed remarkably high stability of pressurized nitrogen encapsulated in maltodextrin.

Rigidity in Finite-Element Matrices: Sufficient Conditions for the Rigidity of Structures and Substructures

We present an algebraic theory of rigidity for finite-element matrices. The theory provides a formal algebraic definition of finite-element matrices; notions of rigidity of finite-element matrices and of mutual rigidity between two such matrices; and sufficient conditions for rigidity and mutual rigidity. We also present a novel sparsification technique, called fretsaw extension, for finite-element matrices. We show that this sparsification technique generates matrices that are mutually rigid with the original matrix. We also show that one particular construction algorithm for fretsaw extensions generates matrices that can be factored with essentially no fill. This algorithm can be used to construct preconditioners for finite-element matrices. Both our theory and our algorithms are applicable to a wide range of finite-element matrices, including matrices arising from finite-element discretizations of both scalar and vector partial differential equations (e.g., electrostatics and linear elasticity). Both the theory and the algorithms are purely algebraic-combinatorial. They manipulate only the element matrices and are oblivious to the geometry, the material properties, and the discretization details of the underlying continuous problem.

Circuit lower bounds and linear codes

In 1977, Valiant proposed a graph-theoretical method for proving lower bounds on algebraic circuits with gates computing linear functions. He used this method to reduce the problem of proving lower bounds on circuits with linear gates to proving lower bounds on the rigidity of a matrix, a notion that he introduced in that paper. The largest lower bound for an explicitly given matrix is due to J. Friedman, who proved a lower bound on the rigidity of the generator matrices of error-correcting codes over finite fields. He showed that the proof can be interpreted as a bound on a certain parameter defined for all linear spaces of finite dimension. In this note, we define another parameter that can be used to prove lower bounds on circuits with linear gates. Our parameter may be larger than Friedman’s, and it seems incomparable with rigidity, hence it may be easier to prove a lower bound using this notion. Bibliography: 14 titles.

On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes

This paper discusses theoretical limitations of classification systems that are based on feature maps and use a separating hyper-plane in the feature space. In particular, we study the embeddability of a given concept class into a class of Euclidean half spaces of low dimension, or of arbitrarily large dimension but realizing a large margin. New bounds on the smallest possible dimension or on the largest possible margin are presented. In addition, we present new results on the rigidity of matrices and briefly mention applications in complexity and learning theory.

Cycles of Nonzero Elements in Low Rank Matrices

We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices.

Spectral Methods for Matrix Rigidity with Applications to Size–Depth Trade-offs and Communication Complexity

The rigidity of a matrix measures the number of entries that must be changed in order to reduce its rank below a certain value. The known lower bounds on the rigidity of explicit matrices are very weak. It is known that stronger lower bounds would have important consequences in complexity theory. We consider some restricted variants of the rigidity problem over the complex numbers. Using spectral methods, we derive lower bounds on these variants. Two applications of such restricted variants are given. First, we show that our lower bound on a variant of rigidity implies lower bounds on size–depth trade-offs for arithmetic circuits with bounded coefficients computing linear transformations. These bounds generalize a result of Nisan and Wigderson. The second application is conditional; we show that it would suffice to prove lower bounds on certain restricted forms of rigidity to conclude several separation results such as separating the analogs of PH and PSPACE in the model of two-party communication complexity. Our results complement and strengthen a result of Razborov. We introduce a combinatorial complexity measure, called AC0-dimension, of sets of Boolean functions. While high rigidity implies large AC0-dimension, large AC0-dimension for explicit sets would already give explicit languages outside the analog of PH in two-party communication complexity. Moreover, the concept of AC0-dimension allows us to formulate interesting combinatorial problems which may be easier than rigidity and which would still have consequences to separation questions in communication complexity.

A note on the use of determinant for proving lower bounds on the size of linear circuits

We consider computations of linear forms over R by circuits with linear gates where the absolute value coefficients are bounded by a constant. Also we consider a related concept of restricted rigidity of a matrix. We prove some lower bounds on the size of such circuits and the restricted rigidity of matrices in terms of the absolute value of the determinant of the matrix.

On the rigidity of Vandermonde matrices

The rigidity function R A(r) of a matrix A is the minimum number of entries of A that must be changed to reduce the rank of A to less than or equal to r. While almost all matrices have rigidity close to (n−r) 2 , proving strong lower bounds on the rigidity of explicit matrices is a fundamental open question with several consequences in complexity theory. A natural class of matrices expected to have high rigidity is that of Vandermonde matrices V=(x i j−1) 1⩽i, j⩽n . However, even when the x i are algebraically independent, it was not known if R V(r)= Ω(n 2) for nonconstant r. We prove that for any constant c<1, there exists a constant ε>0 such that if r⩽ε n , then R V(r)⩾cn 2 , when the x i are algebraically independent. Although not explicit, this provides a natural n-dimensional manifold in the space of n×n matrices with Ω(n 2) rigidity for nonconstant r. Our proof is based on a technique due to Shoup and Smolensky (Comput. Complexity 6(4) (1997) 301–311). For explicit Vandermonde matrices, the best-known lower bound is R V(r)= Ω(n 2/r log(n/r)) , when log 2 n⩽r⩽n/2 , which follows from a result of Shokrollahi et al. (Inform. Process. Lett. 64(6) (1997) 283–285).

Some structural properties of low-rank matrices related to computational complexity

We consider the problem of the presence of short cycles in the graphs of nonzero elements of matrices which have sublinear rank and nonzero entries on the main diagonal, and analyze the connection between these properties and the rigidity of matrices. In particular, we exhibit a family of matrices which shows that sublinear rank does not imply the existence of triangles. This family can also be used to give a constructive bound of the order of k 3/2 on the Ramsey number R(3,k), which matches the best-known bound. On the other hand, we show that sublinear rank implies the existence of 4-cycles. Finally, we prove some partial results towards establishing lower bounds on matrix rigidity and consequently on the size of logarithmic depth arithmetic circuits for computing certain explicit linear transformations.

Matrix rigidity

The rigidity of a matrix M is the function R M(r) , which, for a given r, gives the minimum number of entries of M which one has to change in order to reduce its rank to at most r. This notion has been introduced by Valiant in 1977 in connection with the complexity of computing linear forms. Despite more than 20 years of research, very little is known about the rigidity of matrices. Nonlinear lower bounds on matrix rigidity would lead to new lower bound techniques for the computation of linear forms, e.g., for the computation of the DFT, as well as to more general advances in complexity theory. We put forward a number of linear algebra research issues arising in the above outlined context.