Maximal Rank Research Articles

Chow–Witt rings and topology of flag varieties

AbstractThe paper computes the Witt‐sheaf cohomology rings of partial flag varieties in type A in terms of the Pontryagin classes of the subquotient bundles. The proof is based on a Leray–Hirsch‐type theorem for Witt‐sheaf cohomology for the maximal rank cases, and a detailed study of cohomology ring presentations and annihilators of characteristic classes for the general case. The computations have consequences for the topology of real flag manifolds: we show that all torsion in the integral cohomology is 2‐torsion, which was not known in full generality previously. This allows for example to compute the Poincaré polynomials of complete flag varieties for cohomology with twisted integer coefficients. The computations also allow to describe the Chow–Witt rings of flag varieties, and we sketch an enumerative application to counting flags satisfying multiple incidence conditions to given hypersurfaces.

Integrable geodesic flow in 3D and webs of maximal rank

Integrable geodesic flow in 3D and webs of maximal rank

Random survival forest algorithm for risk stratification and survival prediction in gastric neuroendocrine neoplasms.

This study aimed to construct and assess a machine-learning algorithm designed to forecast survival rates and risk stratification for patients with gastric neuroendocrine neoplasms (gNENs) after diagnosis. Data on patients with gNENs were extracted and randomly divided into training and validation sets using the Surveillance, Epidemiology, and End Results database. We developed a prediction model using 10 machine learning algorithms across 101 combinations to forecast cancer-related mortality in patients with gNENs, selecting the best model using the highest mean over a sequence of time-dependent area under the receiver operating characteristic (ROC) curve (AUC). The performance of the final model was assessed through time-dependent ROC curves for discrimination and calibration curves for calibration. The maximum selection rank method was used to determine the best prognostic risk score threshold for classifying patients into high- and low-risk groups. Afterward, Kaplan-Meier analysis and log-rank test were used to compare survival rates among these groups. Our study examined 775 patients with gNENs, dividing them into training and validation sets. A training set comprised 543 patients, with a median follow-up of 42 months and cumulative mortality rates of 40.0% at 1 year, 48.6% at 3 years, and 54.0% at 5 years. A validation set comprised 232 patients, with cumulative mortality rates of 29.1% at 1 year, 43.5% at 3 years, and 53.2% at 5 years. The optimal random survival forest (RSF) model (mtry = 4, node size = 5) achieved an AUC of 0.839 for survival prediction in the training set. Comprising 11 variables such as demographics, treatment details, tumor characteristics, T staging, N staging, and M staging, the RSF model revealed high predictive accuracy with AUCs of 0.92, 0.96, and 0.96 for 1-, 3-, and 5-year survival, respectively, which was consistently reflected in the validation set with AUCs of 0.88, 0.92, and 0.89, respectively. Moreover, patients were risk-stratified. Although our RSF model effectively stratified patients into different prognostic groups, it needs external validation to confirm its utility for noninvasive prognostic prediction and risk stratification in gNENs. Further research is required to verify its broader clinical applicability.

Counting maximal isotropic subbundles of orthogonal bundles over a curve

Let C be a smooth projective curve and V an orthogonal bundle over C. Let IQe(V) be the isotropic Quot scheme parameterizing degree e isotropic subsheaves of maximal rank in V. We give a closed formula for intersection numbers on components of IQe(V) whose generic element is saturated. As a special case, for g≥2, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in [20], and of the symplectic case studied in [8].

Biderivations, commuting linear maps, post-Lie algebra structure on solvable Lie algebras of maximal rank

In this paper, all biderivations of solvable Lie algebras of maximal rank g = Q ⊕ N are characterized. Namely, we consider a solvable Lie algebra of the form g = Q ⊕ N , where Q is the maximal torus subalgebra of g , N is the nilradical of g and dim Q = dim N / N 2 .In the case dim Q = N we characterize the form of skew-symmetric and symmetric biderivations of g . As applications, the forms of linear commuting (skew-commuting) maps and the commutative post-Lie algebra structures on g are given.

The strong maximal rank conjecture and moduli spaces of curves

The strong maximal rank conjecture and moduli spaces of curves

Higher Tate traces of Chow motives

Abstract We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

Trade-Offs Between Energy and Depth of Neural Networks.

We present an investigation on threshold circuits and other discretized neural networks in terms of the following four computational resources-size (the number of gates), depth (the number of layers), weight (weight resolution), and energy-where the energy is a complexity measure inspired by sparse coding and is defined as the maximum number of gates outputting nonzero values, taken over all the input assignments. As our main result, we prove that if a threshold circuit C of size s, depth d, energy e, and weight w computes a Boolean function f (i.e., a classification task) of n variables, it holds that log( rk (f))≤ed(logs+logw+logn) regardless of the algorithm employed by C to compute f, where rk (f) is a parameter solely determined by a scale of f and defined as the maximum rank of a communication matrix with regard to f taken over all the possible partitions of the n input variables. For example, given a Boolean function CD n(ξ) =⋁i=1n/2ξi∧ξn/2+i, we can prove that n/2≤ed( log s+logw+logn) holds for any circuit C computing CD n. While its left-hand side is linear in n, its right-hand side is bounded by the product of the logarithmic factors of s,w,n and the linear factors of d,e. If we view the logarithmic terms as having a negligible impact on the bound, our result implies a trade-off between depth and energy: n/2 needs to be smaller than the product of e and d. For other neural network models, such as discretized ReLU circuits and discretized sigmoid circuits, we also prove that a similar trade-off holds. Thus, our results indicate that increasing depth linearly enhances the capability of neural networks to acquire sparse representations when there are hardware constraints on the number of neurons and weight resolution.

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial f in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.

A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a $$\mathbb {C}$$-convex ring

A maximum rank theorem for solutions to the homogenous complex Monge–Ampère equation in a $$\mathbb {C}$$-convex ring

Computational methods for periodic systems: Recent developments

In the past few decades, there has been a significant focus on computational methods for periodic systems. In some of these works, various considerations were outlined to establish and categorize algorithms for designing periodic discrete‐time state‐space systems. In a recent work, a direct algorithm was proposed to obtain low‐dimensional realizations from a proper lifted system in state‐space form. This algorithm, which lies in a category often called “fast” algorithms, takes advantage on the structure of lifted representations being highly efficient. In a 1993 work by Ching‐An Lin and Chwan‐Wen King, another direct algorithm was proposed that appears to be quite similar to the algorithm outlined recently. These two algorithms generate state‐space periodic realizations that are both uniform, meaning that the dimension of the state remains constant at each time instant. The algorithms proposed in these two works require both a rank test applied to two different types of matrices, and , respectively (in chronologic order). Although the former algorithm establishes an upper bound for the state dimension à priori, by determining the maximum rank of the matrices , the latter obtains an upper bound à posteriori, based on the rank of matrices . In this work, we proved that these matrices and have the same rank for all , and we use this information to revise the latter algorithm and assess whether it represents an improvement. The algorithm presented as well its modified versions that were coded using the Python programming language. Surprisingly, the revised algorithm did not demonstrate any significant improvement in terms of computational efficiency.

Semiparametric estimation of average treatment effects in observational studies

AbstractWe propose a semiparametric method to estimate average treatment effects in observational studies based on the assumption of unconfoundedness. Assume that the propensity score model and outcome model are a general single index model, which are estimated by the kernel method and the unknown index parameter is estimated via linearized maximum rank correlation method. The proposed estimator is computationally tractable, allows for large dimension covariates and not involves the approximation of link functions. We showed that the proposed estimator is consistent and asymptotically normally distributed. In general, the proposed estimator is superior to existing methods when the model is incorrectly specified. We also provide an empirical analysis on the average treatment effect and average treatment effect on the treated of 401(k) eligibility on net financial assets.

On the finite generation of valuation semigroups on toric surfaces

We provide a combinatorial criterion for the finite generation of a valuation semigroup associated with an ample divisor on a smooth toric surface and a non-toric valuation of maximal rank. As an application, we construct a lattice polytope such that none of the valuation semigroups of the associated polarized toric variety coming from one-parameter subgroups and centered at a non-toric point are finitely generated.

The description of solvable Lie superalgebras of maximal rank

In the paper we give some basic properties of the superderivations of Lie superalgebras. Under certain condition, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to the dimensions of complementary subspaces to the nilradicals. Moreover, under the condition that an analogue of Lie's theorem is true, the description of solvable Lie superalgebras of maximal rank is obtained. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank under the mentioned condition is isomorphic to the maximal solvable extension of nilradical of maximal rank. Finally, along with effective method of construction of solvable Lie superalgebras of maximal rank we present the description of special type of maximal solvable extension of nilpotent Lie superalgebras.

Understanding the pavement project failures in Pakistan: identifying causes and solutions

Pavement deterioration is a major issue in Pakistan's road and highway infrastructure. The current study attempts to clarify the most important factors of road degradation in Pakistan using a questionnaire prepared and distributed to engineers, contractors, architects, surveyors, and project managers involved in road development and upkeep. The study had 147 replies. This data was collected from employees from various road construction sectors in Pakistan. Convenience sampling, a sort of non-probability sampling, is used for data gathering due to time and resource constraints. The data were analysed through SPSS and Microsoft Excel. The responses were gathered using a 10-point Likert scale. The standard (inadequate density in surface or sub-base) has the maximum rank of 9.1, while the lowest factor (use of naturally smooth uncrushed aggregate) is 2.0. Furthermore, the causes of deterioration were reorganised into eight consistent groups of pertinent causes. The first group (effect of traffic load and volume) has a severity level of 6.5, while the last group (effect of bond between layers) ranks 3.22 in group comparison. The study is helpful in identifying the causes of road deterioration in Pakistan and avoiding or mitigating their effects during design, construction, and maintenance through operation.

On Low-Rank Directed Acyclic Graphs and Causal Structure Learning.

Despite several advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high-dimensional settings when the graphs to be learned are not sparse. In this article, we propose to exploit a low-rank assumption regarding the (weighted) adjacency matrix of a DAG causal model to help address this problem. We utilize existing low-rank techniques to adapt causal structure learning methods to take advantage of this assumption and establish several useful results relating interpretable graphical conditions to the low-rank assumption. Specifically, we show that the maximum rank is highly related to hubs, suggesting that scale-free (SF) networks, which are frequently encountered in practice, tend to be low rank. Our experiments demonstrate the utility of the low-rank adaptations for a variety of data models, especially with relatively large and dense graphs. Moreover, with a validation procedure, the adaptations maintain a superior or comparable performance even when graphs are not restricted to be low rank.

Computational complexity of normalizing constants for the product of determinantal point processes

We consider the product of determinantal point processes (DPPs), a point process whose probability mass is proportional to the product of principal minors of multiple matrices, as a natural, promising generalization of DPPs. We study the computational complexity of computing its normalizing constant, which is among the most essential probabilistic inference tasks. Our complexity-theoretic results (almost) rule out the existence of efficient algorithms for this task unless the input matrices are forced to have favorable structures. In particular, we prove the following:•Computing ∑Sdet⁡(AS,S)p exactly for every (fixed) positive even integer p is UP-hard and Mod3P-hard, which gives a negative answer to an open question posed by Kulesza and Taskar [51].•∑Sdet⁡(AS,S)det⁡(BS,S)det⁡(CS,S) is NP-hard to approximate within a factor of 2O(|I|1−ε) or 2O(n1/ε) for any ε>0, where |I| is the input size and n is the order of the input matrix. This result is stronger than the #P-hardness for the case of two matrices derived by Gillenwater [36].•There exists a kO(k)nO(1)-time algorithm for computing ∑Sdet⁡(AS,S)det⁡(BS,S), where k is the maximum rank of A and B or the treewidth of the graph formed by nonzero entries of A and B. Such parameterized algorithms are said to be fixed-parameter tractable. These results can be extended to the fixed-size case. Further, we present two applications of fixed-parameter tractable algorithms given a matrix A of treewidth w:•We can compute a 2n2p−1-approximation to ∑Sdet⁡(AS,S)p for any fractional number p>1 in wO(wp)nO(1) time.•We can find a 2n-approximation to unconstrained maximum a posteriori inference in wO(wn)nO(1) time.

On anti-quasi-Sasakian manifolds of maximal rank

We discuss the existence of invariant anti-quasi-Sasakian (aqS) structures of maximal rank on compact homogeneous Riemannian manifolds and on nilpotent Lie groups. In the former case we obtain a non-existence result, while in the latter case we provide a complete classification. We also show that every compact aqS manifold has nonvanishing second Betti number.

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

Positive Intermediate Ricci Curvature with Maximal Symmetry Rank

RANKS OF SUBMATRICES IN THE REFLEXIVE SOLUTIONS OF SOME MATRIX EQUATIONS

Maximal and minimal ranks of the two submatrices X₁ and X₂ in the (skew-) Hermitian reflexive solution X=U[X₁ 00 X₂]U^{∗} of the matrix equation AXA^{∗}=C, in the reflexive solution of the matrix equation AXB=C are derived. Then necessary and sufficient conditions for these reflexive solutions to have special forms, and the general expressions of these reflexive solutions are achieved.