Invertible Square Research Articles

Structure‐preserving invariant interpolation schemes for invertible second‐order tensors

AbstractTensor interpolation is an essential step for tensor data analysis in various fields of application and scientific disciplines. In the present work, novel interpolation schemes for general, that is, symmetric or non‐symmetric, invertible square tensors are proposed. Critically, the proposed schemes rely on a combined polar and spectral decomposition of the tensor data , followed by an individual interpolation of the two rotation tensors and and the positive definite diagonal eigenvalue tensor resulting from this decomposition. Two different schemes are considered for a consistent rotation interpolation within the special orthogonal group , either based on relative rotation vectors or quaternions. For eigenvalue interpolation, three different schemes, either based on the logarithmic weighted average, moving least squares or logarithmic moving least squares, are considered. It is demonstrated that the proposed interpolation procedure preserves the structure of a tensor, that is, and remain orthogonal tensors and remains a positive definite diagonal tensor during interpolation, as well as scaling and rotational invariance (objectivity). Based on selected numerical examples considering the interpolation of either symmetric or non‐symmetric tensors, the proposed schemes are compared to existing approaches such as Euclidean, Log‐Euclidean, Cholesky and Log‐Cholesky interpolation. In contrast to these existing methods, the proposed interpolation schemes result in smooth and monotonic evolutions of tensor invariants such as determinant, trace, fractional anisotropy (FA), and Hilbert's anisotropy (HA). Moreover, a consistent spatial convergence behavior is confirmed for first‐ and second‐order realizations of the proposed schemes. The present work is mainly motivated by the frequently occurring necessity for remeshing or mesh adaptivity when applying the finite element method to complex problems of nonlinear continuum mechanics with inelastic constitutive behavior, which requires the consistent interpolation of tensor‐valued history data for the transfer between different meshes. However, the proposed schemes are very general in nature and suitable for the interpolation of general invertible second‐order square tensors independent of the specific application.

Triquintic interpolation in three dimensions

This paper develops a local triquintic interpolation method for a function defined over a unit cube mesh element. The resulting polynomial interpolant is isotropic and has global C2 continuity. The triquintic polynomial coefficients are related to the function and its derivatives at the eight corners of a mesh element by an invertible square matrix of rank 216. The proposed method is an extension of a previously developed tricubic interpolation scheme with global C1 continuity.

From Fractal Behavior of Iteration Methods to an Efficient Solver for the Sign of a Matrix

Investigating the fractal behavior of iteration methods on special polynomials can help to find iterative methods with global convergence for finding special matrix functions. By employing such a methodology, we propose a new solver for the sign of an invertible square matrix. The presented method achieves the fourth rate of convergence by using as few matrix products as possible. Its attraction basin shows larger convergence radii, in contrast to its Padé-type methods of the same order. Computational tests are performed to check the efficacy of the proposed solver.

Projective free algebras of bounded holomorphic functions on infinitely connected domains

The algebra H ∞ ( D ) H^\infty (D) of bounded holomorphic functions on D ⊂ C D\subset \mathbb {C} is projective free for a wide class of infinitely connected domains. In particular, for such D D every rectangular left-invertible matrix with entries in H ∞ ( D ) H^\infty (D) can be extended in this class of matrices to an invertible square matrix. This follows from a new result on the structure of the maximal ideal space of H ∞ ( D ) H^\infty (D) asserting that its covering dimension is 2 2 and the second Čech cohomology group is trivial.

Jordan structures of irreducible totally nonnegative matrices with a prescribed sequence of the first p-indices

Let A in {mathbb {R}}^{n times n} be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, A is irreducible with all minors nonnegative, r is the size of the largest invertible square submatrix of A and p is the size of its largest invertible principal submatrix. We consider the sequence {1,i_2,ldots ,i_p} of the first p-indices of A as the first initial row and column indices of a p times p invertible principal submatrix of A. A triple (n, r, p) is called (1,i_2,ldots ,i_p)-realizable if there exists an irreducible totally nonnegative matrix A in {mathbb {R}}^{n times n} with rank r, principal rank p, and {1,i_2,ldots ,i_p} is the sequence of its first p-indices. In this work we study the Jordan structures corresponding to the zero eigenvalue of irreducible totally nonnegative matrices associated with a triple (n, r, p) (1,i_2,ldots ,i_p)-realizable.

Enhance capability of separable ball criterion for the bipartite quantum states

The largest separable ball is a meaningful geometric sufficient condition for separability of bipartite quantum states which tells that all the states in the unit ball in Frobenius norm centered at the identity matrix are separable. In this paper, we propose an algorithm that can improve the capability of this criterion to detect the separability outside the unit ball by solving an optimization problem based on the invertible local operators, that is, to optimize with respect to the invertible square matrices A and B. The numerical examples have shown that our algorithm is more powerful than the original separable ball criterion.

Sunklodas’ Approach to Normal Approximation for Time-Dependent Dynamical Systems

We consider time-dependent dynamical systems arising as sequential compositions of self-maps of a probability space. We establish conditions under which the Birkhoff sums for multivariate observations, given a centering and a general normalizing sequence $b(N)$ of invertible square matrices, are approximated by a normal distribution with respect to a metric of regular test functions. Depending on the metric and the normalizing sequence $b(N)$, the conditions imply that the error in the approximation decays either at the rate $O(N^{-1/2})$ or the rate $O(N^{-1/2} \log N)$, under the additional assumption that $\Vert b(N)^{-1} \Vert \lesssim N^{-1/2}$. The error comes with a multiplicative constant whose exact value can be computed directly from the conditions. The proof is based on an observation due to Sunklodas regarding Stein's method of normal approximation. We give applications to one-dimensional random piecewise expanding maps and to sequential, random, and quasistatic intermittent systems.

Irreducible totally nonnegative matrices with a prescribed Jordan structure

Let A∈Rn×n be an irreducible totally nonnegative matrix with rank r and principal rank p, that is, all minors of A are nonnegative, r is the size of the largest invertible square submatrix of A and p is the size of its largest invertible principal submatrix. A triple (n,r,p) is called realizable if there exists an n×n irreducible totally nonnegative matrix A with rank r and principal rank p.In this work we present a method to construct irreducible totally nonnegative matrices associated with a realizable triple (n,r,p) and with a prescribed Jordan canonical form corresponding to its zero eigenvalue.

A comparative study on existing and new methods to design internal model controllers for non-square systems

This paper focuses on a challenging problem in the internal model control (IMC) strategy: the model inversion to design the IMC controller for non-square systems. Several existing approaches for the synthesis of a specific inversion of the identified model will be presented in this paper to deal with the differences between the system’s inputs and outputs numbers. The non-square effective relative gain is firstly presented. It consists of the measurement of interactions between the loops of the system in order to square the system and make it invertible. The equivalent transfer function method is presented as well. It is based on tuning the pseudo-inverse of the process to design the internal model controller. These methods are then compared with a novel proposed model inversion approach based on virtual outputs method. Virtual adding is considered in order to obtain an invertible square transfer matrix to design the internal model controller. This simple yet effective method ensures robust control performance. Its efficiency and availability, as compared with other presented methods, is illustrated through simulations on an overactuated system with three inputs/two outputs.

A graph approach for the construction of high order divergence-free Raviart–Thomas finite elements

We propose and analyze an efficient algorithm for the computation of a basis of the space of divergence-free Raviart–Thomas finite elements. The algorithm is based on graph techniques. The key point is to realize that, with very natural degrees of freedom for fields in the space of Raviart–Thomas finite elements of degree $$r+1$$ and for elements of the space of discontinuous piecewise polynomial functions of degree $$r \ge 0$$ , the matrix associated with the divergence operator is the incidence matrix of a particular graph. By choosing a spanning tree of this graph, it is possible to identify an invertible square submatrix of the divergence matrix and to compute easily the moments of a field in the space of Raviart–Thomas finite elements with assigned divergence. This approach extends to finite elements of high degree the method introduced by Alotto and Perugia (Calcolo 36:233–248, 1999) for finite elements of degree one. The analyzed approach is used to construct a basis of the space of divergence-free Raviart–Thomas finite elements. The numerical tests show that the performance of the algorithm depends neither on the topology of the domain nor or the polynomial degree r.

An infinite C*-algebra with a dense, stably finite *-subalgebra

We construct a unital pre-C*-algebra $ A_0$ which is stably finite, in the sense that every left invertible square matrix over $ A_0$ is right invertible, while the C*-completion of $ A_0$ contains a nonunitary isometry, and so it is infinite.

Fabrication of disposable flexible SERS substrates by nanoimprint

In this work, we report on the fabrication of disposable flexible substrates dedicated to the Surface-Enhanced Raman Scattering (SERS) technique. The transfer of pyramidal features elaborated on silicon wafers towards flexible plastic foil is performed via a two-steps nanoimprint process. Gold coated normal and inverted square pyramids patterned on the master silicon molds are reproduced with fidelity into a double-layer flexible plastic foil. SERS spectroscopy characterization was performed on the master SERS devices as well as on the final polymeric duplicated structures. Both, masters and replica, were shown to exhibit a good Raman signal enhancement. Moreover, this fabrication procedure was shown to be compatible with the use of large size substrates and well-mastered collective fabrication techniques. As a consequence, the developed SERS-active polymeric devices combine cost effectiveness and a high application potential in various detection and process control domains.

Disposable plasmonic plastic SERS sensor

The 'Klarite™' SERS sensor platform consisting of an array of gold coated inverted square pyramids patterned onto a silicon substrate has become the industry standard over the last decade, providing highly reproducible SERS signals. In this paper, we report successful transfer from silicon to plastic base platform of an optimized SERS substrate design which provides 8 times improvement in sensitivity for a Benzenethiol test molecule compared to standard production Klarite. Transfer is achieved using roll-to-roll and sheet-level nanoimprint fabrication techniques. The new generation plastic SERS sensors provide the added benefit of cheap low cost mass-manufacture, and easy disposal. The plastic replicated SERS sensors are shown to provide ~10(7) enhancement factor with good reproducibility (5%).

The inverse of banded matrices

The inverses of r-banded matrices, for r=1,2,3 have been thoroughly investigated as one can see from the references we provide. Let Br,n (1≤r≤n) be an n×n matrix of entries {aji}, −r≤i≤r, 1≤j≤r, with the remaining un-indexed entries all zeros. In this paper, generalizing a method of Mallik (1999) [5], we give the LU factorization and the inverse of the matrix Br,n (if it exists). Our results are valid for an arbitrary square matrix (taking r=n), and so, we will give a new approach for computing the inverse of an invertible square matrix. Our method is based on Hessenberg submatrices associated to Br,n.

Rings over which the transpose of every invertible matrix is invertible

We prove that the transpose of every invertible square matrix over a ring R is invertible if and only if R / rad ( R ) is commutative. Many other characterizations are obtained for such rings R in terms of U ( R ) (the group of units of R), including, for instance, c + b a ∈ U ( R ) ⇒ c + a b ∈ U ( R ) , and 1 + a b c − c b a ∈ U ( R ) (for all a , b , c ∈ R ). We also consider a natural weakening of these conditions, namely, 1 + a b c ∈ U ( R ) ⇒ 1 + c b a ∈ U ( R ) , and show that, for von Neumann regular rings, this is a (necessary and) sufficient condition for the commutativity of R.

A hybrid ring coupler quasi-optical antenna-mixer

This paper proposes a hybrid ring coupler quasi-optical antenna-mixer for mitigating local oscillator retransmission. By demonstrating at K-band, the antenna element consists of back-to-back aperture coupled inverted square patch antenna to couple the RF signal at 18.8 GHz to the sigma port of a hybrid ring mixer while the LO signal at 17.5 GHz is coupled to the delta port. The HSCH-9101 Schottky diodes are used to transform the RF signal to the intermediate frequency signal at 1.3 GHz. The results show that the RF/LO isolation is better than 29 dB at 18.19 GHz, and the isotropic conversion loss of the down converted signal is better than 16 dB at 19.25 GHz. The application of the interest is an inverse measurement technique for dielectric property determination.

A numerically reliable solution for the squaring-down problem in system design

In this paper, matrix pencil theory is used to study the squaring-down problem where a linear time-invariant system with an unequal number of inputs and outputs is turned into an invertible square system with an equal number of inputs and outputs. Both static and dynamic compensators are considered for squaring down. In the case of static compensation, the infinite-zero structure of the original system is preserved after squaring down. In the case of dynamic compensation, key system properties including stabilizability, detectability and the infinite-zero structure of the original system are also preserved after squaring down. Furthermore, one can additionally assign the invariant zeros induced by squaring down to the open left half plane, provided that the original system is stabilizable and detectable. This means that squaring down by dynamic compensation preserves minimum phaseness as well. The preservation of these system properties is highly desirable for subsequent feedback design of the squared-down system. Unlike existing squaring-down methods which do not address the issue of numerical properties, our solution is based on a condensed form derived using only orthogonal transformations which are numerically stable. Explicit formulas which can be implemented in a numerically reliable manner are given for determining the squaring-down compensators. Examples are presented to illustrate the numerical superiority of the proposed method.

Quantum yields of hydroxyl radicals in illuminated TiO 2 nanocrystallite layers

The yield of hydroxyl radicals has been determined by illumination of TiO 2 layers immersed in air saturated aqueous methanol solutions. This yield is equal to half the measured formaldehyde yield in the pH range 7–13. A detailed mechanism is proposed, accounting for the lack of accumulation of hydrogen peroxide. The effect of changing methanol concentration, pH and light intensity (the latter by three orders of magnitude) is in agreement with a very simple mechanism. In contrast to hydroxyl radicals, which react via hydrogen abstraction, leading to formation of HCHO, there is no sign for reaction of methanol with mobile holes. Thus, the limiting quantum yield observed at high methanol concentration is related to the maximum yield of OH ads under air saturated conditions at the given pH and light intensity. The effect of light intensity shows the expected inverted square root dependency. The yield of OH ads is nearly constant in the range 7<pH<12. This system may be useful for comparative tests of different TiO 2 preparations.

K1of separative exchange rings and C*-algebras with real rank zero

For any (unital) exchange ring R whose finitely generated projective modules satisfy the separative cancellation property (AAABBB =) AB), it is shown that all invertible square matrices over R can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism GL1(R) ! K1(R) is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra A with real rank zero, the topological K1(A) is naturally isomorphic to the unitary group U(A) modulo the connected component of the identity. This verifies, in the separative case, a conjecture of Shuang Zhang.

Partial Contact Force Controllability in Active Wheeled Vehicles

This paper presents a geometric approach for solving the force distribution problem in active wheeled vehicles (AWVs) moving on uneven surfaces. Here an active vehicle is defined as a system that includes independent actuators for all its internal joints. In general, AWVs do not possess omni-directional mobility, and they possess fewer actuators than the number of wheel-ground contact force components. This article presents an approach for separating the contact force vectors into active and passive components such that there exists an invertible square matrix that maps the active contact forces to the actuator efforts. An appropriate force allocation algorithm can then be developed for these systems. The concepts introduced in this article are demonstrated via an example of AWVs on uneven terrain. An example of force distribution in active legged vehicles (ALVs) that possess the same number of actuators as contact forces is also provided for comparison.