Maximal Rank Research Articles

Generalized Rank Weights of Reducible Codes, Optimal Cases, and Related Properties

Reducible codes for the rank metric were introduced for cryptographic purposes. They have fast encoding and decoding algorithms, include maximum rank distance (MRD) codes, and can correct many rank errors beyond half of their minimum rank distance, which makes them suitable for error correction in network coding. In this paper, we study their security behavior against information leakage on networks when applied as coset coding schemes, giving the following main results: 1) we give lower and upper bounds on their generalized rank weights (GRWs), which measure worst case information leakage to the wire tapper; 2) we find new parameters for which these codes are MRD (meaning that their first GRW is optimal) and use the previous bounds to estimate their higher GRWs; 3) we show that all linear (over the extension field) codes, whose GRWs are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence; and 4) we show that the information leaked to a wire tapper when using reducible codes is often much less than the worst case given by their (optimal in some cases) GRWs. We conclude with some secondary related properties: conditions to be rank equivalent to Cartesian products of linear codes and conditions to be rank degenerate, duality properties, and MRD ranks.

General single-index survival regression models for incident and prevalent covariate data and prevalent data without follow-up.

This article mainly focuses on analyzing covariate data from incident and prevalent cohort studies and a prevalent sample with only baseline covariates of interest and truncation times. Our major task in both research streams is to identify the effects of covariates on a failure time through very general single-index survival regression models without observing survival outcomes. With a strict increase of the survival function in the linear predictor, the ratio of incident and prevalent covariate densities is shown to be a non-degenerate and monotonic function of the linear predictor under covariate-independent truncation. Without such a structural assumption, the conditional density of a truncation time in a prevalent cohort is ensured to be a non-degenerate function of the linear predictor. In light of these features, some innovative approaches, which are based on the maximum rank correlation estimation or the pseudo least integrated squares estimation, are developed to estimate the coefficients of covariates up to a scale factor. Existing theoretical results are further used to establish the -consistency and asymptotic normality of the proposed estimators. Moreover, extensive simulations are conducted to assess and compare the finite-sample performance of various estimators. To illustrate the methodological ideas, we also analyze data from the Worcester Heart Attack Study and the National Comorbidity Survey Replication.

Some new non-additive maximum rank distance codes

In this paper, a construction of maximum rank distance (MRD) codes as a generalization of generalized Gabidulin codes is given. The family of the resulting codes is not covered properly by additive generalized twisted Gabidulin codes, and does not cover all twisted Gabidulin codes. When the basis field has more than two elements, this family includes also non-affine MRD codes, and such codes exist for all parameters. Therefore, these codes are the first non-additive MRD codes for most of the parameters.

Local Demailly-Bouche's holomorphic morse inequalities

Let $(X,\omega)$ be a Hermitian manifold and let $(E,h^E)$, $(F,h^F)$ be two Hermitian holomorphic line bundle over $X$. Suppose that the maximal rank of the Chern curvature $c(E)$ of $E$ is $r$, and the kernel of $c(E)$ is foliated, i.e. there is a foliation $Y$ of $X$, of complex codimension $r$, such that the tangent space of the leaf at each point $x\in X$ is contained in the kernel of $c(E)$. In this paper, local versions of Demailly-Bouche's holomorphic Morse inequalities (which give asymptotic bounds for cohomology groups $H^{q}(X,E^k\otimes F^l)$ as $k,l,k/l\rightarrow \infty$) are presented. The local version holds on any Hermitian manifold regardless of compactness and completeness. The proof is a variation of Berman's method to derive holomorphic Morse inequalities on compact complex manifolds with boundary.

Low-Rank Model for Wideband Electromagnetic Induction Sensors

Wideband electromagnetic induction (WEMI) sensors are used to detect, classify, and locate obscured targets. WEMI sensors can be designed to measure the magnetic fields either in the time domain or frequency domain. They also capture multiple location measurements from a target by either a physical sensor array, a synthetic sensor array created by scanning the sensor, or a combination of the two. This letter uses a physical model for the sensor to develop a low-rank model for WEMI data. The data are organized into a matrix where the rows contain the time/frequency measurements and the columns contain the multiple locations so all of the data can be exploited jointly. It is shown that this data matrix can be processed directly with singular value decomposition (SVD) to extract three independent terms—one for target signature, one related to location, and one for tensors that define the orientation. The low-rank model predicts a maximum rank for WEMI measurements and is exploited to provide a relationship between the rank of the measurements and the number of linearly independent tensors for the target. Results are presented from the laboratory data to validate the low-rank model and demonstrate the connection between the data’s rank and the physical target. The low-rank model leads to a new “filterless” processing paradigm for exploiting WEMI data by using the SVD to perform sensor calibration, hardware debugging, as well as target detection.

Lefschetz property and powers of linear forms in 𝕂[x,y,z

Abstract In [9], Migliore, Miró-Roig and Nagel proved that if R = 𝕂 ⁢ [ x , y , z ] {R=\mathbb{K}[x,y,z]} , where 𝕂 {\mathbb{K}} is a field of characteristic zero, and I = ( L 1 a 1 , … , L 4 a 4 ) {I=(L_{1}^{a_{1}},\dots,L_{4}^{a_{4}})} is an ideal generated by powers of four general linear forms, then the multiplication by the square L 2 {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of R / I {R/I} . More recently, Migliore and Miró-Roig proved in [7] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjectured that the same holds for arbitrary powers. In this paper, we will prove that this conjecture is true, that is, we will show that if I = ( L 1 a 1 , … , L r a r ) {I=(L_{1}^{a_{1}},\dots,L_{r}^{a_{r}})} is an ideal of R generated by arbitrary powers of any set of general linear forms, then the multiplication by the square L 2 {L^{2}} of a general linear form L induces a homomorphism of maximal rank in any graded component of R / I {R/I} .

Lazy orbits: An optimization problem on the sphere

Non-transitive subgroups of the orthogonal group play an important role in the non-Euclidean geometry. If $G$ is a closed subgroup in the orthogonal group such that the orbit of a single Euclidean unit vector does not cover the (Euclidean) unit sphere centered at the origin then there always exists a non-Euclidean Minkowski functional such that the elements of $G$ preserve the Minkowskian length of vectors. In other words the Minkowski geometry is an alternative of the Euclidean geometry for the subgroup $G$. It is rich of isometries if $G$ is "close enough" to the orthogonal group or at least to one of its transitive subgroups. The measure of non-transitivity is related to the Hausdorff distances of the orbits under the elements of $G$ to the Euclidean sphere. Its maximum/minimum belongs to the so-called lazy/busy orbits, i.e. they are the solutions of an optimization problem on the Euclidean sphere. The extremal distances allow us to characterize the reducible/irreducible subgroups. We also formulate an upper and a lower bound for the ratio of the extremal distances. As another application of the analytic tools we introduce the rank of a closed non-transitive group $G$. We shall see that if $G$ is of maximal rank then it is finite or reducible. Since the reducible and the finite subgroups form two natural prototypes of non-transitive subgroups, the rank seems to be a fundamental notion in their characterization. Closed, non-transitive groups of rank $n-1$ will be also characterized. Using the general results we classify all their possible types in lower dimensional cases $n=2, 3$ and $4$. Finally we present some applications of the results to the holonomy group of a metric linear connection on a connected Riemannian manifold.

Homogeneity for a class of Riemannian quotient manifolds

We study Riemannian coverings φ:M˜→Γ\\M˜ where M˜ is a normal homogeneous space G/K1 fibered over another normal homogeneous space M=G/K and K is locally isomorphic to a nontrivial product K1×K2. The most familiar such fibrations π:M˜→M are the natural fibrations of Stiefel manifolds SO(n1+n2)/SO(n1) over Grassmann manifolds SO(n1+n2)/[SO(n1)×SO(n2)] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ:M˜→Γ\\M˜ are the universal Riemannian coverings of spherical space forms. When M=G/K is reasonably well understood, in particular when G/K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M˜. In other words we show that Γ\\M˜ is homogeneous if and only if every γ∈Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M˜ we work out the full isometry group of M˜, extending Élie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.

Testing the maximal rank of the volatility process for continuous diffusions observed with noise

In this paper, we present a test for the maximal rank of the volatility process in continuous diffusion models observed with noise. Such models are typically applied in mathematical finance, where latent price processes are corrupted by microstructure noise at ultra high frequencies. Using high frequency observations we construct a test statistic for the maximal rank of the time varying stochastic volatility process. Our methodology is based upon a combination of a matrix perturbation approach and pre-averaging. We will show the asymptotic mixed normality of the test statistic and obtain a consistent testing procedure.

Hadamard full propelinear codes of type Q; rank and kernel

Hadamard full propelinear codes (HFP-codes) are introduced and their equivalence with Hadamard groups is proven (on the other hand, it is already known the equivalence of Hadamard groups with relative $(4n,2,4n,2n)$-difference sets in a group and also with cocyclic Hadamard matrices). We compute the available values for the rank and dimension of the kernel of HFP-codes of type Q and we show that the dimension of the kernel is always 1 or $2$. We also show that when the dimension of the kernel is 2 then the dimension of the kernel of the transposed code is 1 (so, both codes are not equivalent). Finally, we give a construction method such that from an HFP-code of length $4n$, dimension of the kernel $k=2$, and maximum rank $r=2n$, we obtain an HFP-code of double length $8n$, dimension of the kernel $k=2$, and maximum rank $r=4n$.

The role of phases in detecting three-qubit entanglement

We propose separability criteria for three-qubit states in terms of diagonal and anti-diagonal entries to detect entanglement with positive partial transposes. We report that the phases, that is, the angular parts of anti-diagonal entries, play a crucial role in determining whether a given three-qubit state is separable or entangled, and they must obey even an identity for separability in some cases. These criteria are strong enough to detect PPT (positive partial transpose) entanglement with nonzero volume. In several cases when all the entries are zero except for diagonal and anti-diagonal entries, we characterize separability using phases. These include the cases when anti-diagonal entries of such states share a common magnitude, and when ranks are less than or equal to six. We also compute the lengths of rank six cases and find three-qubit separable states with lengths 8 whose maximum ranks of partial transposes are 7.

High-rank ternary forms of even degree

We exhibit, for each positive even degree, a ternary form of rank strictly greater than the maximum rank of monomials. Together with an earlier result in the odd case, this gives a lower bound of $$\begin{aligned} \left\lfloor \frac{d^2+2d+5}{4}\right\rfloor \;, \end{aligned}$$ for $$d\ge 2$$ , on the maximum rank of degree d ternary forms with coefficients in an algebraically closed field of characteristic zero.

Ranking the Scientific-Technology Literacy Indexes of Students from the Viwepoint of Faculty Members

Background: Increasing scientific-technology literacy is a major educational purpose around the world. This concept is an umbrella that covers everything related to science and knowledge. The aim of this research is its scientific ranking - technological literacy indicators of students from the viewpoint of faculty members of Islamic Azad University, Tabriz Unit. Methods: The research method is descriptive and survey. The population consisted of all faculty members of the Islamic Azad University of Tabriz, which included 350 people in 2015. The sample size was estimated according to the Cochran formula. A total of, 186 individuals were selected by using the stratified random sampling. The measuring tool includes a Researcher questionnaire of scientific - IT literacy Miller (2006) (10 components and 40 questions); its validity was confirmed by experts and its reliability was assessed by using Cronbach’s alpha coefficient to 0.81 values. Data were analyzed with SPSS 18 software. Results: Data analysis showed that the viewpoints of faculty members and scientific literacy indicators have a significant difference between the students (χ2 = 127.5; P = 0.00). The maximum rank was related to basic knowledge as well as foresight indexes and the lowest was related to the creativity index, however, no significant difference was found between the students for technology indexes (χ2 = 2.667; P = 0.264). Conclusions: The students are in high rank in terms of basic and theoretical knowledge, which is a requisite for scientific literacy, however, in terms of creativity and curiosity, thinking manners (tendency to act) is ranked lower. Attention to critical thinking development and knowledge application in performance and real-life was recommended

Rank/inertia approaches to weighted least-squares solutions of linear matrix equations

The well-known linear matrix equation AX=B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q1−XP1X′ and Q2−X′P2X subject to the restriction trace[(AX−B)′W(AX−B)]=min, where both Pi and Qi are real symmetric matrices, i=1,2,W is a positive semi-definite matrix, and X′ is the transpose of X. We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of AX=B, including necessary and sufficient conditions for weighted least-squares solutions of AX=B to satisfy the quadratic symmetric matrix equalities XP1X′=Q1 an X′P2X=Q2, respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP1X′≻Q1 (≽Q1, ≺Q1, ≼Q1) and X′P2X≻Q2 (≽Q2, ≺Q2, ≼Q2) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q1−XP1X′ and Q2−X′P2X subject to weighted least-squares solutions of AX=B.

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.

On the strong Lefschetz problem for uniform powers of general linear forms in 𝑘[𝑥,𝑦,𝑧

Schenck and Seceleanu proved that if R = k [ x , y , z ] R = k[x,y,z] , where k k is an infinite field and I I is an ideal generated by any collection of powers of linear forms, then multiplication by a general linear form L L induces a homomorphism of maximal rank from any component of R / I R/I to the next. That is, R / I R/I has the weak Lefschetz property. Considering the more general strong Lefschetz problem of when × L j \times L^j has maximal rank for j ≥ 2 j \geq 2 , we give the first systematic study of this problem. We assume that the linear forms are general and that the powers are all the same, i.e. that I I is generated by uniform powers of general linear forms. We prove that for any number of such generators, × L 2 \times L^2 always has maximal rank. We then specialize to almost complete intersections, i.e. to four generators, and we show that for j = 3 , 4 , 5 j = 3,4,5 the behavior depends on the uniform exponent and on j j in a way that we make precise. In particular, there is always at most one degree where × L j \times L^j fails maximal rank. Finally, we note that experimentally all higher powers of L L fail maximal rank in at least two degrees.

The Maximum Rank of the Transfer Matrix in 1-D Mirrored Interferometric Aperture Synthesis

Because the principle of mirrored interferometric aperture synthesis (MIAS) is different from that of the conventional interferometric aperture synthesis, the antenna array for 1-D MIAS should be redesigned. And in order to get a precise estimation of the cosine visibilities, the maximum rank of the transfer matrix should be set as the key constraint condition of the array optimization model, but this has not been clearly presented before. In this letter, the maximum rank of the transfer matrix is discussed and proved by the mathematical induction method. When the position of each antenna in the array is an even multiple of 0.5, the value for the maximum rank is proven to be $M-2$ , and when the position of each antenna is an odd multiple of 0.5, the value for the maximum rank is proven to be $M-1$ , where M is the number of the spatial frequencies provided by the array. This conclusion is significant for the array design of 1-D MIAS.

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

Multi-homogeneous polynomial systems arise in many applications. We provide bit complexity estimates for solving them which, up to a few extra other factors, are quadratic in the number of solutions and linear in the height of the input system, under some genericity assumptions. The assumptions essentially imply that the Jacobian matrix of the system under study has maximal rank at the solution set and that this solution set is finite. The algorithm is probabilistic and a probability analysis is provided.Next, we apply these results to the problem of optimizing a linear map on the real trace of an algebraic set. Under some genericity assumptions, we provide bit complexity estimates for solving this polynomial minimization problem.

On the locus of points of high rank

Given a closed subvariety X in a projective space, the rank with respect to X of a point p in this projective space is the least integer r such that p lies in the linear span of some r points of X. Let W_k be the closure of the set of points of rank with respect to X equal to k. For small values of k such loci are called secant varieties. This article studies the loci W_k for values of k larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to X in special cases, including when X is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any mathbf {GL}_n orbit of a homogeneous form.

Budget transparency in Croatian counties, cities and municipalities (November 2016 – March 2017)

The overall average level of budget transparency in Croatian local government units, measured by the number of budget documents published on their respective official websites, has improved year by year. In the current research cycle, it averages 3.1 (the maximum rank is 5), with counties already being very transparent (4.6 on average), cities fairly good (3.7 on average), but municipalities still inadequately transparent (2.8 on average). Despite the relatively solid averages, there are too many extremely non-transparent local government units. Four cities (Gospic, Imotski, Valpovo and Vrgorac) and as many as 39 municipalities have not published a single budget document that we were searching for, and another seven cities and 50 municipalities have published only one such document each. As in previous years, the most transparent local government units include very sparsely populated ones, such as Ribnik and Dekanovec, and local government units with low per capita income (e.g. Jarmina and Đelekovec). On the other hand, among the least transparent local government units there are units with very high total and per capita budget revenues (e.g. Sutivan, Solta, Dugopolje and Bol). Detailed results for all counties, cities and municipalities are given later in the text and are also available on the interactive map and in the Excel table.