Degenerate Matrices Research Articles

Degenerate cases in discrete Lotka – Volterra dynamical systems

The purpose of the work is to study the asymptotic behavior of trajectories of interior points of discrete Lotka – Volterra dynamical systems with degenerate skew-symmetric matrices operating in two-dimensional and three-dimensional simplexes. It turned out that in a number of applied problems, the Lotka – Volterra mappings of this type arise and the simplex points in this case are considered as the state of the system under study. In this case, the mapping preserving the simplex determines the discrete law of evolution of this system. For an arbitrary starting point, we can construct a sequence – an orbit that determines its evolution. And if in this case the mapping in question is an automorphism, we can define both a positive and a negative orbit for the point in question. At the same time, the limiting sets of positive and negative orbits are of particular interest. Methods. It is known that for Lotka – Volterra mappings it is possible to define limit sets, which in the case of non-degenerate mappings consist of a single point. In this paper, we define these sets for degenerate LotkaVolterra mappings by constructing the Lyapunov function and applying Jacobian spectrum analysis. It should be noted that these sets allow us to describe the dynamics of the systems under consideration. Results. Taking into account that the considered mappings are automorphisms, using the Lyapunov functions and applying the analysis of the Jacobian spectrum, sets of limit points of both positive and negative trajectories are constructed and it is proved that in the degenerate case they are infinite. It is also shown that partially oriented graphs can be constructed for degenerate mappings. Conclusion. Degenerate cases of Lotka – Volterra mappings have not been considered by other authors before us. These mappings are interesting because they can be considered as discrete models of epidemiological situations, in particular, for studying the course of airborne viral infections. The results obtained in this work provide a detailed description of the dynamics of the trajectories of Lotka – Volterra mappings with degenerate matrices. In addition, partially oriented graphs were constructed for the systems under consideration in order to visually represent the dynamics of epidemiological situations.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Ideals of spaces of degenerate matrices

The variety Singn,m consists of all tuples X=(X1,…,Xm) of n×n matrices such that every linear combination of X1,…,Xm is singular. Equivalently, X∈Singn,m if and only if det(λ1X1+…+λmXm)=0 for all λ1,…,λm∈Q. Makam and Wigderson [12] asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Singn,m lies in the ideal generated by the polynomials det(λ1X1+…+λmXm). We answer this question in the negative by determining the vanishing ideal of Sing2,m for all m∈N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m≥n2−n+1, we prove there are equations vanishing on Singn,m that are not in the ideal generated by polynomials of type det(λ1X1+…+λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.

Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions

We provide a natural generalization of a Riemannian structure, i.e., a metric, recently introduced by Sj\"{o}qvist for the space of non degenerate density matrices, to the degenerate case, i.e., the case in which the eigenspaces have dimension greater than or equal to 1. We present a physical interpretation of the metric in terms of an interferometric measurement. We apply this metric, physically interpreted as an interferometric susceptibility, to the study of topological phase transitions at finite temperatures for band insulators. We compare the behaviors of this susceptibility and the one coming from the well-known Bures metric, showing them to be dramatically different. While both infer zero temperature phase transitions, only the former predicts finite temperature phase transitions as well. The difference in behaviors can be traced back to a symmetry breaking mechanism, akin to Landau-Ginzburg theory, by which the Uhlmann gauge group is broken down to a subgroup determined by the type of the system's density matrix (i.e., the ranks of its spectral projectors).

Oscillator realisations associated to the D-type Yangian: Towards the operatorial Q-system of orthogonal spin chains

We present a family of novel Lax operators corresponding to representations of the RTT-realisation of the Yangian associated with D-type Lie algebras. These Lax operators are of oscillator type, i.e. one space of the operators is infinite-dimensional while the other is in the first fundamental representation of so(2r). We use the isomorphism between the first fundamental representation of D3 and the 6 of A3, for which the degenerate oscillator type Lax matrices are known, to derive the Lax operators for r=3. The results are used to generalise the Lax matrices to arbitrary rank for representations corresponding to the extremal nodes of the simply laced Dynkin diagram of Dr. The multiplicity of independent solutions at each extremal node is given by the dimension of the fundamental representation. We further derive certain factorisation formulas among these solutions and build transfer matrices with oscillators in the auxiliary space from the introduced degenerate Lax matrices. Finally, we provide some evidence that the constructed transfer matrices are Baxter Q-operators for so(2r) spin chains by verifying certain QQ-relations for D4 at low lengths.

On the Relation between the Properties of a Degenerate Linear-Quadratic Control Problem and the Euler–Poisson Equation

A quadratic functional with linear constraints in the form of differential equations with identically degenerate matrices multiplying the derivative of the state vector function is considered. The structure of the general solutions of such systems and some of their properties are discussed. On this basis, the conditions for the non-negativity of the objective functional and small deviation of the control from the stationary point for small values of the functional are obtained.

Ideology and problems of systems and economic optimization of NPPs

Iterative technique for optimization of components of nuclear fuel and energy complex (NFEC) is described. Nuclear power (NP) is one of such NFEC components. Optimization of nuclear power plant (NPP) is iteratively interconnected with optimization of the structure NP as the component within the NFEC system. Correlation functions between the components describing material flows and their costs as time functions are obtained by sequential optimization of components of the NFEC system. External flows and costs for each iteration step during optimization of specific component are taken from other components of the NFEC system optimized during the same iteration step. Flows and costs obtained during optimization of the component of the NFEC system examined during the consecutive iteration step are transmitted for optimization of other system components during the same iteration step. Specific NPPs are optimized following the same principle. Convergence of plans (solutions) of development and arrangement of all components is achieved by sequential optimization of all components within the NFEC system. Thus, convergence of all time functions including material flows and their costs is achieved allowing finding one or several locally optimal plans of development and arrangement of the fuel and energy complex (FEC) of Russia, as well as power plants (PPs) fueled with coal, gas, NPPs and other PPs. If the number of locally optimal plans is countable, then it is possible to select the best among them by comparing their functionals (values of the objective function). Upon obtaining new information with regard to any of the factors of mathematical models of components of the NFEC system optimization of components of the system must be repeated. In this case optimization must be performed taking into consideration the incurred costs under construction and already commissioned PPs.Problems of systems optimization of components of the NFEC system as the functional of NPP optimization are identified. The task of optimization of complex systems refers to the most complex class of degenerate optimization problems [1]. No theory of degenerate optimization problems exists as of the present moment, although specifically these problems are real. Degenerate optimization problems should not be associated with degenerate matrices. Degenerate optimization problems deal with non-singular matrices having inverse matrices and, consequently, having solutions. However, the nature of these problems is such, that these solutions are degenerate (one or more components of the basis vector of the solution are equal to zero). The problem of convergence in the optimization of arbitrary component of the NFEC system to locally optimal plan coordinated with locally optimal plans for other components of the NFEC system is one of the main problems of degenerate optimization problems. Appearance of iterating loops during internal iterations in the process of optimization of plans for separate component of the NFEC system is not less important for problems of this class. This feature requires including incorporation of additional algorithms for exiting from the loops in the optimization algorithms.

How the Structural Integrity of the Matrix Can Influence the Microstructural Response of Articular Cartilage to Compression

This study investigated how the structural integrity of healthy, surface-removed (healthy), and degenerate matrices can modify the response of cartilage to compression. Six groups of specimens were loaded up to the onset of consolidation or at full consolidation (N = 30, 5 per group, respectively) and then subsequently chemically fixed to capture the deformed state of the tissues. Creep compression was applied through an 8 mm flat-ended indenter containing a 450 μm diameter central pore, providing a region of high stress that also allowed the tissue samples to deform freely around the indenter pore during compression. Differential interference contrast microscopy was used in order to explore the microstructural responses of the tissues. The results demonstrated that superficial layer removal or tissue degeneration can reduce the observed deformation within the tissue region corresponding to the central pore of the loading indenter. Fibril crimping within the central pore matrix and matrix shear at the indenter edge regions are also reduced by both superficial layer removal and by tissue degeneration. These findings suggest that surface removal or tissue degeneration renders the matrix more susceptible to deformation and can also reduce the tissue’s ability to transfer forces over a greater surface area and induce stress within the matrix.

Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix

Nous étudions l’homogénéisation d’opérateurs paraboliques du second ordre sous forme divergence à coefficients localement stationnaires. Ces coefficients présentent deux échelles d’évolution: une évolution microscopique presque constante et une évolution macroscopique régulière. La théorie de l’homogénéisation consiste à donner une approximation macroscopique de l’opérateur initial qui tient compte des hétérogénéités microscopiques. Cet article fait suite à [Probab. Theory Related Fields (2009)] et généralise ce dernier en considérant des matrices de diffusion pouvant dégénérer.

Mechanical systems, equivalent in Lyapunov's sense to systems not containing non-conservative positional forces

Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535–539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774–780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698–704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 (6): 906–913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.

Analysis of the nucleotide context of Arabidopsis thaliana mitochondrial mRNA editing sites

C → T deamination is among the most widespread types of mitochondrial mRNA editing in higher plants. In the majority of cases, the first and second codon positions are edited, which leads to “correction” of the codon, substitution in amino acid sequence, and synthesis of a fully functional protein. Numerous experimental works have demonstrated that the 5′-and 3′-flanks of the editing site are important for precise and efficient editing. However, no significant motifs have been detected in the flanking regions, and the editing mechanism is still unknown. We analyzed the editing sites of Arabidopsis thaliana mitochondrial mRNA using region-specific degenerate oligonucleotide motifs and trinucleotide weight matrices. In addition, we compared the mRNA folding energies in the regions of editing sites. Significant oligonucleotide patterns were discovered in the region [−50; 50] relative to the editing site. Jackknife assessment suggested that such signals could play an important role in the functioning of editing sites. However, the detected characteristics alone were insufficient for efficient recognition of editing sites in A. thaliana mitochondrial mRNA.

Comparative analysis by means of finite differences and DM methods for linearized problem of gyrotrons

Abstract The problem of Schrodinger equation with complex boundary conditions for modelling a motion of electrons in gyrotrons is considered. Numerical results obtained by using Fourier, Finite Differences (FD) and Degenerate Matrices (DM) methods are compared in the simplest case. For DM methods they are analysed also in more general cases, when FD can not be applied because of fast oscillations of the solution.

Degenerate matrices methods by splines for boundary values problems of ordinary differential equations

Abstract A method for numerical solving of boundary values problems of ordinary differential equations based on the use of splines and differentiation matrices with nodes as zeroes of classical orthogonal polynomials is considered. Possibilities of the method are shown by means of different examples. The method essentially uses the results obtained by Degenerate Matrices methods and it is applied for solving initial values problems of ordinary differential equations.

Dynamics of small bubble interface perturbations in vertical hele‐shaw cell with magnetic liquid under the action of normal magnetic field

Abstract A linearized problem of dynamics for small perturbations of the gas bubble rising in the Hele‐Shaw cell filled with magnetic liquid is considered. It is reduced to searching of eigenvalues and eigenfunctions for a linear operator with periodic boundary conditions. The obtained operator is presented as a sum of two linear operators: the second order differential operator with varying coefficients and the integro ‐ differential operator with the singularity of the Cauchy type. The spectral problem is solved by the Degenerate Matrices (DM) method using Chebyshev polynomials of the first and second kind.

Coincidence of ESAD and ESS in dominant–recessive hereditary systems

The paper deals with the following question: when do the phenotypic evolutionarily stable state (ESS) and the evolutionarily stable allele distribution (ESAD) coincide? It is supposed that for a sexual population, in dominant–recessive inheritance system, n allele at one autosomal locus determine n possible pure individual phenotypes and each pure phenotype is obtained as the phenotype of a homozygote. Under these conditions, earlier results of the authors imply that, if a phenotype distribution is an ESS then the allele distribution generating it is an ESAD. In this paper, apart from a certain degenerate pay-off matrices, the inverse statement is also proved: if a distribution is an ESAD then the corresponding phenotypic distribution is an ESS.

Unstable hyperplanes for Steiner bundles and mulitdimensional matrices

We study some properties of the natural action of SLOV0U SLOVpU on non- degenerate multidimensional complex matrices A A POV0 n n VpU of boundary format (in the sense of Gelfand, Kapranov and Zelevinsky); in particular we characterize the non-stable ones as the matrices which are in the orbit of a ''triangular'' matrix, and the matrices with a stabilizer containing C as those which are in the orbit of a ''diagonal'' matrix. For pa 2i t turns out that a non-degenerate matrix A A POV0 n V1 n V2U detects a Steiner bundle SA (in the sense of Dolgachev and Kapranov) on the projective space P n , na dimOV2Uˇ1. As a consequence we prove that the symmetry group of a Steiner bundle is contained in SLO2U and that the SLO2U-invariant Steiner bundles are exactly the bundles introduced by Schwarzen- berger (Schw), which correspond to ''identity'' matrices. We can characterize the points of the moduli space of Steiner bundles which are stable for the action of AutOP n U, answering in the first nontrivial case a question posed by Simpson. In the opposite direction we obtain some results about Steiner bundles which imply properties of matrices. For example the number of unstable hyperplanes of SA (counting multiplicities) produces an interesting discrete invariant of A, which can take the values 0; 1; 2; .. .; dim V0a 1o ry; the y case occurs if and only if SA is Schwarzenberger (and A is an identity). Finally, the Gale transform for Steiner bun- dles introduced by Dolgachev and Kapranov under the classical name of association can be understood in this setting as the transposition operator on multidimensional matrices.

A Generalized Green Operator for a Boundary Value Problem with Impulse Action

with the impulse action `iz(·) = ai, i = 1, . . . , p. (2) Here A(t) is a continuous n×n matrix function on [a, b]; f(t) is an n-vector function from the class of functions that are continuous with respect to t ∈ [a, b], t 6= τi, and have jump discontinuities at t = τi : f(·) ∈ C {[a, b]\ {τi}I}; ai and c are constant column vectors from R, i = 1, . . . , p; `iz(·) are linear vector functionals `iz(·) : C {[a, τi+1[ \ {τ1, . . . , τi}I} → R, `iz(·) = ∑i j=0 ` (j) i z(·), where ` i z(·) : C [a, τ1[→ R, . . . , ` (i) i z(·) : C [τi, τi+1[→ R, i = 1, . . . , p− 1, . . . , ` p z(·) : C [a, τ1[→ R, . . . , ` p z(·) : C [τp, b]→ R are linear bounded functionals. For the functional `iz(·) = z (τi + 0)− (In + Si) z (τi − 0) in which all In +Si are nondegenerate matrices, we obtain the problem investigated in [1, 2]; in particular, if Si = 0, then we have the problem considered in [3]. If In +Si are degenerate matrices for some i, then we have a degenerate impulse action. If `iz(·) = Miz (τi + 0) +Niz (τi − 0), then we obtain the problem analyzed in [4]; in particular, for the case in which ai ∈ R (Ni), R (Ni) = R (Mi), and N (Ni) = ∅, the problem was investigated in [5], for the case k = n in [6, 7], and for the case of nondegenerate matrices Mi and Ni in [8]. Here R (Ni) and N(Ni) are the ranges and null spaces, respectively, of the matrices Ni.

Null Model Analysis of Species Co-Occurrence Patterns

The analysis of presence–absence matrices with “null model” randomization tests has been a major source of controversy in community ecology for over two decades. In this paper, I systematically compare the performance of nine null model algorithms and four co-occurrence indices with respect to Type I and Type II errors. The nine algorithms differ in whether rows and columns are treated as fixed sums, equiprobable, or proportional. The three models that maintain fixed row sums are invulnerable to Type I errors (false positives). One of these three is a modified version of the original algorithm of E. F. Connor and D. Simberloff. Of the four co-occurrence indices, the number of checkerboard combinations and the number of species combinations may be prone to Type II errors (false negatives), and may not reveal significant patterns in noisy data sets. L. Stone and A. Robert's checkerboard score has good power for detecting species pairs that do not co-occur together frequently, whereas D. Schluter's V ratio reveals nonrandom patterns in the row and column totals of the matrix. Degenerate matrices (matrices with empty rows or columns) do not greatly alter the outcome of null model analyses. The choice of an appropriate null model and index may depend on whether the data represent classic “island lists” of species in an archipelago or standardized “sample lists” of species collected with equal sampling effort. Systematic examination of a set of related null models can pinpoint how violation of the assumptions of the model contributes to nonrandom patterns.

NULL MODEL ANALYSIS OF SPECIES CO-OCCURRENCE PATTERNS

The analysis of presence-absence matrices with ''null model'' randomization tests has been a major source of controversy in community ecology for over two decades. In this paper, I systematically compare the performance of nine null model algorithms and four co-occurrence indices with respect to Type I and Type II errors. The nine algorithms differ in whether rows and columns are treated as fixed sums, equiprobable, or proportional. The three models that maintain fixed row sums are invulnerable to Type I errors (false positives). One of these three is a modified version of the original algorithm of E. F. Connor and D. Simberloff. Of the four co-occurrence indices, the number of checkerboard com- binations and the number of species combinations may be prone to Type II errors (false negatives), and may not reveal significant patterns in noisy data sets. L. Stone and A. Robert's checkerboard score has good power for detecting species pairs that do not co- occur together frequently, whereas D. Schluter's V ratio reveals nonrandom patterns in the row and column totals of the matrix. Degenerate matrices (matrices with empty rows or columns) do not greatly alter the outcome of null model analyses. The choice of an ap- propriate null model and index may depend on whether the data represent classic ''island lists'' of species in an archipelago or standardized ''sample lists'' of species collected with equal sampling effort. Systematic examination of a set of related null models can pinpoint how violation of the assumptions of the model contributes to nonrandom patterns.

The canonical form of reducible discrete systems with degenerate coefficient matrices

The canonical form of reducible discrete systems with degenerate coefficient matrices