Primitive Matrix Research Articles

Short‐circuit current distribution analysis using a network model based on a 5×5 primitive matrix

Short‐circuit current distribution analysis using a network model based on a 5×5 primitive matrix

Reconciling petrological and isotopic mixing mechanisms in the Pitcairn mantle plume using stable Fe isotopes

Ocean-island basalts (OIB) are the products of intra-plate mantle melting above mantle plumes. These hot, buoyant upwellings rise from the deep mantle, possibly the core-mantle boundary, to the Earth's surface. Radiogenic isotope signatures of trace elements in OIB are consistent with a variety of crustally-derived recycled components in their source that are suggested to reside in the plume as eclogite or their pyroxenitic melt-reaction products. The global OIB compositional spectrum exhibits a continuum of radiogenic isotope characteristics (Sr-Nd-Pb-Hf) that indicate mixing of these plume components. However, the petrological mechanisms underpinning this mixing remain elusive. Here, we report stable Fe isotope data for plume-derived lavas of Pitcairn Island from the southwest Pacific. Iron isotope compositions, corrected for olivine fractionation, show a strong co-variation with radiogenic Nd-Sr-Pb isotopes. At Pitcairn, two isotopically distinct components have been identified, one being an enriched-mantle 1 (EM-1) component and the other being a more primitive component, which plots in multiple radiogenic isotope space in the so-called focal zone (FOZO), tentatively identified here as the plume matrix. We suggest a two-stage scenario in which Fe isotopes are first fractionated towards heavier values during formation of a reaction-zone pyroxenite. Partial silicate melt derived from a recycled crustal eclogite, assumed here to be of a subducted sediment origin, reacts with ambient peridotite to form an isotopically heavier Fe isotope pyroxenite with memory of its precursor in radiogenic isotopes. Subsequently, both, isolated and enriched reaction zone pyroxenites and ambient plume matrix reach their solidus and mixing of their partial melts to variable proportions create the spread in radiogenic and stable isotope systematics. This mixing of components is coherent with the widely observed disparity in trajectories between plume matrix and enriched components in long-lived (Sr-Nd-Pb-Hf) and extinct (W) radiogenic and stable (Fe-Mg) isotope systematics and marks a fundamental process for chemical diversity in OIB. It supports a model of isolated plume components in a primitive mantle matrix. Enriched components in the plume that contribute to this mixing appear to be solely pyroxenitic with no or very little direct contribution from the original eclogite. Timing and loci of pyroxenite formation remain to be elucidated.

An Eigenvector Centrality for Multiplex Networks with Data

Networks are useful to describe the structure of many complex systems. Often, understanding these systems implies the analysis of multiple interconnected networks simultaneously, since the system may be modelled by more than one type of interaction. Multiplex networks are structures capable of describing networks in which the same nodes have different links. Characterizing the centrality of nodes in multiplex networks is a fundamental task in network theory. In this paper, we design and discuss a centrality measure for multiplex networks with data, extending the concept of eigenvector centrality. The essential feature that distinguishes this measure is that it calculates the centrality in multiplex networks where the layers show different relationships between nodes and where each layer has a dataset associated with the nodes. The proposed model is based on an eigenvector centrality for networks with data, which is adapted according to the idea behind the two-layer approach PageRank. The core of the centrality proposed is the construction of an irreducible, non-negative and primitive matrix, whose dominant eigenpair provides a node classification. Several examples show the characteristics and possibilities of the new centrality illustrating some applications.

Companion matrices and Golden-Fibonacci sequences

We assign a companion sequence to a given companion matrix. Given a companion matrix \(\mathbf{C}\), we make use of the associated companion sequence to provide a simple closed-form expression for \(\mathbf{C}^n\), the nth power of \(\mathbf{C}\). We determine conditions under which a given companion matrix \(\mathbf{C}\) is a primitive matrix. A systematic method for obtaining the limit values of a companion sequence is provided. A new class of primitive companion matrices is introduced for which the limit values of the related companion sequences are connected with the Golden ratio \(\uptau =\frac{1+\sqrt{5}}{2}\). In fact, a new generalization of the well-known \(\mathbf{Q}\)-matrix and the ordinary Fibonacci numbers are presented in this paper. This generalized form of the Fibonacci numbers will be referred to as the Golden-Fibonacci sequence. We show that the limit values of a Golden-Fibonacci sequence are powers of the Golden ratio. We apply primitive companion matrices as encoder matrices and introduce a type of error-correcting codes to be called companion coding. We show that the error-correcting relations of companion coding are connected with the limit values of the related companion sequence.

Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

A NOTE ON RÉNYI'S ENTROPY RATE FOR TIME-INHOMOGENEOUS MARKOV CHAINS

AbstractIn this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

A note on matrices mapping a positive vector onto its element-wise inverse

For any primitive matrix M∈Rn×n with positive diagonal entries, we prove the existence and uniqueness of a positive vector x=(x1,…,xn)t such that Mx=(1x1,…,1xn)t. The contribution of this note is to provide an alternative proof of a result of Brualdi et al. (1966) [1] on the diagonal equivalence of a nonnegative matrix to a stochastic matrix.

Breaking Megrelishvili protocol using matrix diagonalization

In this article we conduct a theoretical security analysis of Megrelishvili protocol—a linear algebra-based key agreement between two participants. We study the computational complexity of Megrelishvili vector-matrix problem (MVMP) as a mathematical problem that strongly relates to the security of Megrelishvili protocol. In particular, we investigate the asymptotic upper bounds for the running time and memory requirement of the MVMP that involves diagonalizable public matrix. Specifically, we devise a diagonalization method for solving the MVMP that is asymptotically faster than all of the previously existing algorithms. We also found an important counterintuitive result: the utilization of primitive matrix in Megrelishvili protocol makes the protocol more vulnerable to attacks.

Amyloids Are Novel Cell-Adhesive Matrices.

Amyloids are highly ordered peptide/protein aggregates traditionally associated with multiple human diseases including neurodegenerative disorders. However, recent studies suggest that amyloids can also perform several biological functions in organisms varying from bacteria to mammals. In many lower organisms, amyloid fibrils function as adhesives due to their unique surface topography. Recently, amyloid fibrils have been shown to support attachment and spreading of mammalian cells by interacting with the cell membrane and by cell adhesion machinery activation. Moreover, similar to cellular responses on natural extracellular matrices (ECMs), mammalian cells on amyloid surfaces also use integrin machinery for spreading, migration, and differentiation. This has led to the development of biocompatible and implantable amyloid-based hydrogels that could induce lineage-specific differentiation of stem cells. In this chapter, based on adhesion of both lower organisms and mammalian cells on amyloid nanofibrils, we posit that amyloids could have functioned as a primitive extracellular matrix in primordial earth.

On rank-critical matrix spaces

A matrix space of size m×n is a linear subspace of the linear space of m×n matrices over a field F. The rank of a matrix space is defined as the maximal rank over matrices in this space. A matrix space A is called rank-critical, if any matrix space which properly contains it has rank strictly greater than that of A.In this note, we first exhibit a necessary and sufficient condition for a matrix space A to be rank-critical, when F is large enough. This immediately implies the sufficient condition for a matrix space to be rank-critical by Draisma (2006) [5], albeit requiring the field to be slightly larger.We then study rank-critical spaces in the context of compression and primitive matrix spaces. We first show that every rank-critical matrix space can be decomposed into a rank-critical compression matrix space and a rank-critical primitive matrix space. We then prove, using our necessary and sufficient condition, that the block-diagonal direct sum of two rank-critical matrix spaces is rank-critical if and only if both matrix spaces are primitive, when the field is large enough.

Hypoxia promotes primitive glycosaminoglycan-rich extracellular matrix composition in developing heart valves.

During postnatal heart valve development, glycosaminoglycan (GAG)-rich valve primordia transform into stratified valve leaflets composed of GAGs, fibrillar collagen, and elastin layers accompanied by decreased cell proliferation as well as thinning and elongation. The neonatal period is characterized by the transition from a uterine environment to atmospheric O2, but the role of changing O2 levels in valve extracellular matrix (ECM) composition or morphogenesis is not well characterized. Here, we show that tissue hypoxia decreases in mouse aortic valves in the days after birth, concomitant with ECM remodeling and cell cycle arrest of valve interstitial cells. The effects of hypoxia on late embryonic valve ECM composition, Sox9 expression, and cell proliferation were examined in chicken embryo aortic valve organ cultures. Maintenance of late embryonic chicken aortic valve organ cultures in a hypoxic environment promotes GAG expression, Sox9 nuclear localization, and indicators of hyaluronan remodeling but does not affect fibrillar collagen content or cell proliferation. Chronic hypoxia also promotes GAG accumulation in murine adult heart valves in vivo. Together, these results support a role for hypoxia in maintaining a primitive GAG-rich matrix in developing heart valves before birth and also in the induction of hyaluronan remodeling in adults.NEW & NOTEWORTHY Tissue hypoxia decreases in mouse aortic valves after birth, and exposure to hypoxia promotes glycosaminoglycan accumulation in cultured chicken embryo valves and adult murine heart valves. Thus, hypoxia maintains a primitive extracellular matrix during heart valve development and promotes extracellular matrix remodeling in adult mice, as occurs in myxomatous disease.

The genealogical decomposition of a matrix population model with applications to the aggregation of stages

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

A note on the multiple-recursive matrix method for generating pseudorandom vectors

The multiple-recursive matrix method for generating pseudorandom vectors was introduced by Niederreiter (1993). We propose an algorithm for finding an efficient primitive multiple-recursive matrix method. Moreover, for improving the linear complexity, we introduce a tweak on the contents of the primitive multiple-recursive matrix method.

A Markov chain representation of the normalized Perron–Frobenius eigenvector

We consider the problem of finding the Perron–Frobenius eigenvector of a primitive matrix. Dividing each of the rows of the matrix by the sum of the elements in the row, the resulting new matrix is stochastic. We give a formula for the normalized Perron–Frobenius eigenvector of the original matrix, in terms of a realization of the Markov chain defined by the associated stochastic matrix. This formula is a generalization of the classical formula for the invariant probability measure of a Markov chain.

On recurrence and transience of multivariate near-critical stochastic processes

We obtain complementary recurrence/transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ \xi _{n+1}$. Here $M$ denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and $g$ denotes some function. The conditional expectation and variance of the noise $(\xi _{n+1})_{n \ge 0}$ are such that $X$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].

Strong shift equivalence and the Generalized Spectral Conjecture for nonnegative matrices

Given matrices A and B shift equivalent over a dense subring R of R, with A primitive, we show that B is strong shift equivalent over R to a primitive matrix. This result shows that the weak form of the Generalized Spectral Conjecture for primitive matrices implies the strong form. The foundation of this work is the recent result that for any ring R, the group NK1(R) of algebraic K-theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over R.

Strengthening Crypto-1 Cipher Against Algebraic Attacks

In the last few years, several studies addressed the problem of data security in Mifare Classic. One of its weaknesses is the low random number quality. This causes SAT solver attacks to have lower complexity. In order to strengthen Crypto-1 against SAT solver attacks, a modification of the feedback function with better cryptographic properties is proposed. It applies a primitive polynomial companion matrix. SAT solvers cannot directly attack the feedback shift register that uses the modified Boolean feedback function, the register has to be split into smaller groups. Experimental testing showed that the amount of memory and CPU time needed were highest when attacking the modified Crypto-1 using the modified feedback function and the original filter function. In addition, another modified Crypto-1, using the modified feedback function and a modified filter function, had the lowest percentage of revealed variables. It can be concluded that the security strength and performance of the modified Crypto-1 using the modified feedback function and the modified filter function are better than those of the original Crypto-1.

Predicting online ratings based on the opinion spreading process

Predicting users’ online ratings is always a challenge issue and has drawn lots of attention. In this paper, we present a rating prediction method by combining the user opinion spreading process with the collaborative filtering algorithm, where user similarity is defined by measuring the amount of opinion a user transfers to another based on the primitive user-item rating matrix. The proposed method could produce a more precise rating prediction for each unrated user-item pair. In addition, we introduce a tunable parameter λ to regulate the preferential diffusion relevant to the degree of both opinion sender and receiver. The numerical results for Movielens and Netflix data sets show that this algorithm has a better accuracy than the standard user-based collaborative filtering algorithm using Cosine and Pearson correlation without increasing computational complexity. By tuning λ, our method could further boost the prediction accuracy when using Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) as measurements. In the optimal cases, on Movielens and Netflix data sets, the corresponding algorithmic accuracy (MAE and RMSE) are improved 11.26% and 8.84%, 13.49% and 10.52% compared to the item average method, respectively.

Commutative Standard Table Algebras with at Most One Nontrivial Multiplicity

We classify commutative standard table algebras (STA) with at most one nontrivial multiplicity. The main result shows that there exists exactly one nontrivial multiplicity if and only if the table basis is the wreath product of a two-dimensional subalgebra and an abelian group. The theorem applies to adjacency algebras of commutative association schemes with exactly one primitive idempotent matrix of rank greater than one. A theorem of Seitz that characterizes finite groups with exactly one irreducible representation of degree greater than one is another corollary of the main theorem.

Strong shift equivalence and positive doubly stochastic matrices

We give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over R+ to a positive doubly stochastic matrix through matrices of the same size. We also prove that every positive stochastic matrix is strong shift equivalent over R+ to a positive doubly stochastic matrix. Consequently, the set of nonzero spectra of primitive stochastic matrices over R with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over R are identical. We exhibit a class of 2×2 matrices, pairwise strong shift equivalent over R+ through 2×2 matrices, for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size. In contrast, we show for any n×n primitive matrix of positive trace that the set of positive n×n matrices similar to it contains only finitely many SSE–R+ classes.