Rank Inequalities Research Articles

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

Biases among classes of rank-crank partitions (mod 11)

In this paper, we prove inequalities for ranks, cranks, and partitions among different classes modulo 11. These were conjectured by Borozenets.

A Spectral Sequence from Khovanov Homology to Knot Floer Homology

A well-known conjecture of Rasmussen states that for any knot K K in S 3 S^{3} , the rank of the reduced Khovanov homology of K K is greater than or equal to the rank of the reduced knot Floer homology of K K . This rank inequality is supposed to arise as the result of a spectral sequence from Khovanov homology to knot Floer homology. Using an oriented cube of resolutions construction for a homology theory related to knot Floer homology, we prove this conjecture.

Deciding Together as Faculty: Narratives of Unanticipated Consequences in Gendered and Racialized Departmental Service, Promotion, and Voting

Abstract Workplace inequalities scholarship often assumes making people aware of problems will lead to change, although gendered and racialized organizations theories show systemic problems beyond individual awareness. Still, not enough research analyzes the narratives of savvy organizational actors – like university faculty aware of inequalities – to understand the mechanisms operating against leveraging that knowledge for change. Data consist of 10 group interviews with 45 faculty across departments in one US public university, supplemented by content analysis of 56 departments’ written bylaws. Findings focus on three common shared decisions: committee service, hiring/promotion, and voting practices. We find awareness of inequality may actually reinforce the status quo when narratives about gendered and racialized processes feature decoupling from formal bylaws, and when narratives about outcomes relate to multiple layers of unanticipated consequences favoring whiteness and men. Specifically, inequality is reproduced when narratives about gendered and racialized unanticipated consequences: 1) highlight the imperviousness of change, as in the difficulty of allocating service work equitably, 2) lack reflexivity and shift responsibility to ‘other’ groups – ‘faculty’ or ‘administrators’ – as in unequal hiring and promotion decisions, and 3) focus on standard old boy stories which obscure other inequalities, as in faculty voting where non-tenure track rank inequality obscures race/gender inequalities. When unanticipated consequences narratives have dimensions of fatalism, finger pointing, and blindness to intersectionality, white men may continue to benefit. This study shows how formal policies and awareness of inequalities may still fail to produce change.

Access structures for finding characteristic-dependent linear rank inequalities

Access structures for finding characteristic-dependent linear rank inequalities

White psychodrama

White psychodrama

Lower bounds on the rank and symmetric rank of real tensors

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.

A generalization of Kruskal’s theorem on tensor decomposition

Abstract Kruskal’s theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. We prove a ‘splitting theorem’ for sets of product tensors, in which the k-rank condition of Kruskal’s theorem is weakened to the standard notion of rank, and the conclusion of uniqueness is relaxed to the statement that the set of product tensors splits (i.e., is disconnected as a matroid). Our splitting theorem implies a generalization of Kruskal’s theorem. While several extensions of Kruskal’s theorem are already present in the literature, all of these use Kruskal’s original permutation lemma and hence still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization uses a completely new proof technique, contains many of these extensions and can certify uniqueness below this threshold. We obtain several other useful results on tensor decompositions as consequences of our splitting theorem. We prove sharp lower bounds on tensor rank and Waring rank, which extend Sylvester’s matrix rank inequality to tensors. We also prove novel uniqueness results for nonrank tensor decompositions.

On the comparison of inequality measures: evidence from the world values survey

ABSTRACT This paper overviews the well-known inequality measures and applies information-theoretic tools to compare inequality between income levels among nations. Specifically, we use the World Values Survey data on households’ income levels in 57 countries to rank inequality. We then examine the effects of macroeconomic outcomes on inequality. Our findings suggest that the Gini coefficient with -divergence provides further insight into inequality rankings. In addition, the results indicate that higher inflation and GDP growth lead to lower inequality. Finally, we find a positive relationship between income inequality and money growth.

Abelian sharing, common informations, and linear rank inequalities

Abstract Dougherty et al. introduced the common information (CI) method as a method to produce non-Shannon inequalities satisfied by linear random variables, which are called linear rank inequalities. This method is based on the fact that linear random variables have CI. Dougerthy et al. asked whether this method is complete, in the sense that it can be used to produce all linear rank inequalities. We study this question, and we attack it using the theory of secret sharing schemes. To this end, we introduce the notions of Abelian secret sharing scheme and Abelian capacity. We prove that: If there exists an access structure whose Abelian capacity is smaller than its linear capacity, then the CI method is not complete. We investigate the existence of such an access structure.

Un teorema sobre desigualdades rango lineales que dependen de la característica del cuerpo finito

A linear rank inequality is a linear inequality that holds by dimensions of vector spaces over any finite field. A characteristic-dependent linear rank inequality is also a linear inequality that involves dimensions of vector spaces but this holds over finite fields of determined characteristics, and does not in general hold over other characteristics. In this paper, using as guide binary matrices whose ranks depend on the finite field where they are defined, we show a theorem which explicitly produces characteristic-dependent linear rank inequalities; this theorem generalizes results previously obtained in the literature.

Indefinite linear quadratic optimal control for continuous-time rectangular descriptor Markov jump systems: infinite-time case

The indefinite linear quadratic (ILQ) optimal control problem has many important applications in financial, economic systems, etc., which has been widely researched in stochastic systems and descriptor systems, etc. However, for ILQ problem of rectangular descriptor Markov jump systems (DMJSs), because there are impulses simultaneously in descriptor subsystems and at the switching time, it is theoretically difficult to research and there is no result yet. This paper discusses the ILQ optimal control problem for continuous-time linear rectangular DMJSs. Firstly, under some rank conditions and inequality conditions, the ILQ problem for rectangular DMJSs can be equivalently transformed into standard LQ problem for Markov jump systems (MJSs) by using elementary linear algebra method. Then based on the LQ theory of MJSs, the solvable sufficient condition of the ILQ problem for rectangular DMJSs and the non-negative optimal cost value are obtained. The optimal control can be synthesised as state feedback, and the resulting optimal closed-loop system has the stochastically stable solution. In addition, with some rank inequality assumptions, the differential subsystem of the resulting optimal closed-loop system can be ensured to have a unique solution. Finally, two numerical examples are provided to illustrate the effectiveness of the methods proposed in this paper.

Rank inequalities on knot Floer homology of periodic knots

Let $\widetilde{K}$ be a 2-periodic knot in $S^3$ with quotient $K$. We prove a rank inequality between the knot Floer homology of $\widetilde{K}$ and the knot Floer homology of $K$ using a spectral sequence of Hendricks, Lipshitz and Sarkar. We also conjecture a filtered refinement of this inequality, for which we give computational evidence, and produce applications to the Alexander polynomials of $\widetilde{K}$ and $K$.

Rank inequalities for the Heegaard Floer homology of branched covers

Given a double cover between 3-manifolds branched along a nullhomologous link, we establish an inequality between the dimensions of their Heegaard Floer homologies. We discuss the relationship with the L-space conjecture and give some other topological applications, as well as an analogous result for sutured Floer homology.

Optimizing over the Closure of Rank Inequalities with a Small Right-Hand Side for the Maximum Stable Set Problem via Bilevel Programming

In the context of the maximum stable set problem, rank inequalities impose that the cardinality of any set of vertices contained in a stable set be, at most, as large as the stability number of the subgraph induced by such a set. Rank inequalities are very general, as they subsume many classical inequalities such as clique, hole, antihole, web, and antiweb inequalities. In spite of their generality, the exact separation of rank inequalities has never been addressed without the introduction of topological restrictions on the induced subgraph and the tightness of their closure has never been investigated systematically. In this work, we propose a methodology for optimizing over the closure of all rank inequalities with a right-hand side no larger than a small constant without imposing any restrictions on the topology of the induced subgraph. Our method relies on the exact separation of a relaxation of rank inequalities, which we call relaxed k-rank inequalities, whose closure is as tight. We investigate the corresponding separation problem, a bilevel programming problem asking for a subgraph of maximum weight with a bound on its stability number, whose study could be of independent interest. We first prove that the problem is [Formula: see text]-hard and provide some insights on its polyhedral structure. We then propose two exact methods for its solution: a branch-and-cut algorithm (which relies on a family of faced-defining inequalities which we introduce in this paper) and a purely combinatorial branch-and-bound algorithm. Our computational results show that the closure of rank inequalities with a right-hand side no larger than a small constant can yield a bound that is stronger, in some cases, than Lovász’s Theta function, and substantially stronger than bounds obtained with standard inequalities that are valid for the stable set problem, including odd-cycle inequalities and wheel inequalities. Summary of Contribution: This paper proposes two original methods for solving a challenging cut-separation problem (of bilevel type) for a large class of inequalities valid for one of the key operations research problems, namely, the max stable set problem. An extensive set of experimental results validates the proposed methods. All the source code and data sets are available online on GitHub.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

Inequalities for odd ranks of odd Durfee symbols

Andrews introduced odd Durfee symbols to give an interesting combinatorial interpretation of $$\omega (q)$$ invoked by MacMahon’s modular partitions, where $$\omega (q)$$ is one of the mock theta functions defined by Watson. In analogy with Dyson’s rank, Andrews defined the odd rank of an odd Durfee symbol as the number of entries in the top row minus the number of entries in the bottom row. Let $$N^0(m,n)$$ be the number of odd Durfee symbols of n with odd rank m. In this paper, we employ Wright’s circle method to give an asymptotic formula for $$N^0(m,n)$$ which implies that the inequalities $$\begin{aligned} N^0(m,n)\ge N^0(m+2,n) \end{aligned}$$ and $$\begin{aligned} N^0(m,n)\le N^0(m,n+2) \end{aligned}$$ hold for sufficient large n. Motivated by the work of Chan and Mao, we proved that the above inequalities hold for all nonnegative integers m and n.

Common information, matroid representation, and secret sharing for matroid ports

Linear information and rank inequalities as, for instance, Ingleton inequality, are useful tools in information theory and matroid theory. Even though many such inequalities have been found, it seems that most of them remain undiscovered. Improved results have been obtained in recent works by using the properties from which they are derived instead of the inequalities themselves. We apply here this strategy to the classification of matroids according to their representations and to the search for bounds on secret sharing for matroid ports.

Characteristic-dependent linear rank inequalities via complementary vector spaces

A characteristic-dependent linear rank inequality is a linear inequality that holds by ranks of subspaces of a vector space over a finite field of determined characteristic, and does not in general hold over other characteristics. In this paper, we produce new characteristic- dependent linear rank inequalities by an alternative technique to the usual Dougherty’s inverse function method [9]. We take up some ideas of Blasiak [4], applied to certain complementary vector spaces, in order to produce them. Also, we present some applications to network coding. In particular, for each finite or co-finite set of primes P, we show that there exists a sequence of networks in which each member is linearly solvable over a field if and only if the characteristic of the field is in P, and the linear capacity, over fields whose characteristic is not in P, → 0 as k → ∞.

Earrings, Sutures, and Pointed Links

Abstract We prove a rank inequality on the instanton knot homology and the Khovanov homology of a link in $S^3$. The key step of the proof is to construct a spectral sequence relating Baldwin–Levine–Sarkar’s pointed Khovanov homology to a singular instanton invariant for pointed links.