Spaces Of Rank Research Articles

Nontrivial t-designs in polar spaces exist for all t

AbstractA finite classical polar space of rank n consists of the totally isotropic subspaces of a finite vector space over $$\mathbb {F}_q$$ F q equipped with a nondegenerate form such that n is the maximal dimension of such a subspace. A t-$$(n,k,\lambda )$$ ( n , k , λ ) design in a finite classical polar space of rank n is a collection Y of totally isotropic k-spaces such that each totally isotropic t-space is contained in exactly $$\lambda $$ λ members of Y. Nontrivial examples are currently only known for $$t\le 2$$ t ≤ 2 . We show that t-$$(n,k,\lambda )$$ ( n , k , λ ) designs in polar spaces exist for all t and q provided that $$k>\frac{21}{2}t$$ k > 21 2 t and n is sufficiently large enough. The proof is based on a probabilistic method by Kuperberg, Lovett, and Peled, and it is thus nonconstructive.

Wave equation on general noncompact symmetric spaces

abstract: We establish sharp pointwise kernel estimates and dispersive properties for the wave equation on noncompact symmetric spaces of general rank. This is achieved by combining the stationary phase method and the Hadamard parametrix, and in particular, by introducing a subtle spectral decomposition, which allows us to overcome a well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general. Consequently, we deduce the Strichartz inequality for a large family of admissible pairs and prove global well-posedness results for the corresponding semi-linear equation with low regularity data as on hyperbolic spaces.

RBSS: A fast subset selection strategy for multi-objective optimization

Multi-objective optimization problems (MOPs) aim to obtain a set of Pareto-optimal solutions, and as the number of objectives increases, the quantity of these optimal solutions grows exponentially. However, a plethora of optimal solutions can impose significant decision stress on decision-makers. Subset selection, as the extension of a model, can extract a representative set of solutions, thereby alleviating the decision-makers’ choice pressure. In addition, extending a model undoubtedly incurs additional time costs. To cope with the foregoing issues, a fast subset selection method named ranking-based subset selection (RBSS) is proposed in this paper. It can efficiently select a small number of optimal solutions within an unbounded external archive and can be directly applied to any multi-objective evolutionary algorithm. This allows it to maintain good distribution and diversity with very little time investment. We employed a ranking-based approach to map the objective space to a ranking space (an integer space) defined by us and then selected the corresponding subset in the ranking space. The well-behaved mathematical properties of the ranking space and the advantages of using integer calculations accelerated the subset selection process. Experimental results indicate that compared to several state-of-the-art subset selection methods, RBSS is capable of selecting a set of representative and diverse solutions across different types of MOPs, while consuming significantly less time. Specifically, for problems where the Pareto front is a two-dimensional manifold and a one-dimensional manifold, the time consumption of RBSS is approximately only 0.028% to 27.5% and 4.6e−4% to 0.15% of that required by other algorithms, respectively.

Linking integrals on rank-1-symmetric spaces

In symmetric spaces of rank one we obtain a right inverse of the Cartan differential on exact differential forms as an integral operator. As a corollary we extend Gauss’ linking integral beyond space forms.

Irreducibility of moduli of vector bundles over a very general sextic surface

Let S be a very general smooth hypersurface of degree 6 in P3. In this paper we will prove that the moduli space of μ-stable rank 2 torsion free sheaves with respect to hyperplane section having c1=OS(1), with fixed, c2≥27 is irreducible.

Limiting Distribution of Dense Orbits in a Moduli Space of Rank m Discrete Subgroups in (m+1)-Space

Abstract We study the limiting distribution of dense orbits of a lattice subgroup $\Gamma \leq \textrm{SL}(m+1,\mathbb{R})$ acting on $H\backslash \textrm{SL}(m+1,\mathbb{R})$, with respect to a filtration of growing norm balls. The novelty of our work is that the groups $H$ we consider have infinitely many non-trivial connected components. For a specific such $H$, the homogeneous space $H\backslash G$ identifies with $X_{m,m+1}$, a moduli space of rank $m$-discrete subgroups in $\mathbb{R}^{m+1}$. This study is motivated by the work of Shapira-Sargent who studied random walks on $X_{2,3}$.

New and improved bounds on the contextuality degree of multi-qubit configurations

AbstractWe present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al. [(2022). Journal of Physics A: Mathematical and Theoretical55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

On rank 3 instanton bundles on P3$\mathbb {P}^3$

AbstractWe investigate rank 3 instanton vector bundles on of charge and its correspondence with rational curves of degree . For , we present a correspondence between stable rank 3 instanton bundles and stable rank 2 reflexive linear sheaves of Chern classes and we use this correspondence to compute the dimension of the family of stable rank 3 instanton bundles of charge 2. Finally, we use the results above to prove that the moduli space of rank 3 instanton bundles on of charge 2 coincides with the moduli space of rank 3 stable locally free sheaves on of Chern classes . This moduli space is irreducible, has dimension 16 and its generic point corresponds to a generalized't Hooft instanton bundle.

PRINCIPLES OF ORGANIZATION OF THE MATERIAL WORLD

An attempt has been made to present and describe the PRINCIPLES OF ORGANIZATION OF THE MATERIAL WORLD using the “Scheme of the structure of our world”, which is the foundation of the philosophy of Hermeticism. In it, the three parts of the philosophy of the Universe are designated as the three Kingdoms that make up our world - Abstractions, Mass and Gravity, as well as Radiation. A diagram is shown for converting time into mass and back to implement phase transitions of matter into various states in the series: radiation – free space – dark matter – baryonic matter – black holes. The Scheme points to the location of dark energy. The geometric body of the rank space and the fractal principle of its construction are demonstrated. The previously proposed concepts were repeated: a) material zero as an elementary/electromagnetic quantum of space EQS; b) space-time complex of STC inside the quantum of space; c) the axis of the speed of time as an indicator for phase transitions of space quanta.

CAT(0) Spaces of Higher Rank I

A CAT(0) space has rank at least n if every geodesic lies in an n-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is rigid – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least n≥2 is rigid if it contains a periodic n-flat and its Tits boundary has dimension (n−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called Morse flats. We show that the Tits boundary ∂TF of a periodic Morse n-flat F contains a regular point – a point with a Tits-neighborhood entirely contained in ∂TF. More precisely, we show that the set of singular points in ∂TF can be covered by finitely many round spheres of positive codimension.

Stochastic simulation of storm surge extremes along the contiguous United States coastlines using the max-stable process

Extreme sea levels impact coastal society, property, and the environment. Various mitigation measures are engineered to reduce these impacts, which require extreme event probabilities typically estimated site-by-site. The site-by-site estimates usually have high uncertainty, are conditionally independent, and do not provide estimates for ungauged locations. In contrast, the max-stable process explicitly incorporates the spatial dependence structure and produces more realistic event probabilities and spatial surfaces. We leverage the max-stable process to compute extreme event probabilities at gridded locations (gauged and ungauged) and derive their spatial surfaces along the contiguous United States coastlines by pooling annual maximum (AM) surges from selected long-record tide gauges. We also generate synthetic AM surges at the grid locations using the predicted distribution parameters and reordering them in the rank space to integrate the spatiotemporal variability. The results will support coastal planners, engineers, and stakeholders to make the most precise and confident decisions for coastal flood risk reduction.

An upper bound on the generic degree of the generalized Verschiebung for rank two stable bundles

In the present paper, we give an upper bound for the generic degree of the generalized Verschiebung between the moduli spaces of rank two stable bundles with trivial determinant.

Isoparametric hypersurfaces in symmetric spaces of non-compact type and higher rank

We construct inhomogeneous isoparametric families of hypersurfaces with non-austere focal set on each symmetric space of non-compact type and rank ${\geq }3$. If the rank is ${\geq }4$, there are infinitely many such examples. Our construction yields the first examples of isoparametric families on any Riemannian manifold known to have a non-austere focal set. They can be obtained from a new general extension method of submanifolds from Euclidean spaces to symmetric spaces of non-compact type. This method preserves the mean curvature and isoparametricity, among other geometric properties.

Two Series of Components of the Moduli Space of Semistable Reflexive Rank 2 Sheaves on the Projective Space

Two Series of Components of the Moduli Space of Semistable Reflexive Rank 2 Sheaves on the Projective Space

The Coble quadric

Abstract Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in ${\mathbb {P}}^8$ as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$ , the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2,8)$ . In fact, each point $p\in C$ defines a natural embedding of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$ . We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$ .

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

Ulrich bundles on some threefold scrolls over [formula omitted

We investigate the existence of Ulrich vector bundles on suitable 3-fold scrolls Xe over Hirzebruch surfaces Fe, for any e⩾0, which arise as tautological embeddings of projectivization of very-ample vector bundles on Fe that are uniform in the sense of Brosius and Aprodu–Brinzanescu, cf. [10] and [3] respectively.We explicitly describe components of moduli spaces of rank r⩾1 vector bundles which are Ulrich with respect to the tautological polarization on Xe and whose general point is a slope-stable, indecomposable vector bundle. We moreover determine the dimension of such components, proving also that they are generically smooth. As a direct consequence of these facts, we also compute the Ulrich complexity of any such Xe and give an effective proof of the fact that these Xe's turn out to be geometrically Ulrich wild.At last, the machinery developed for 3–fold scrolls Xe allows us to deduce Ulrichness results on rank r⩾1 vector bundles on Fe, for any e⩾0, with respect to a naturally associated (very ample) polarization.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We expect that every Cartan–Münzner polynomial of degree four can be described as a squared-norm of a moment map for a Hamiltonian action. Our expectation is known to be true for Hermitian cases, that is, those obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. In this paper, we prove that our expectation is true for the Cartan–Münzner polynomials obtained from the isotropy representations of Grassmannian manifolds of rank two over R, C or H. The quaternion cases are the first non-Hermitian examples that our expectation is verified.

The Euler characteristic of the moduli space of graphs

The moduli space of rank n graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex have the same rational cohomology. We show that the associated Euler characteristic grows like −e−1/4(n/e)n/(nlog⁡n)2 as n goes to infinity, and thereby prove that the total dimension of this cohomology grows rapidly with n.

Α ranking model based on user generated content and fuzzy logic

User Generated Comments and ranking based on it, has been studied extensively due to its impact on consumers and businesses. Literature highlights that platforms’ rankings might be deceiving due to fake reviews or rates that do not correspond to reality, but existing ranking models are a ‘black box’ thus, they cannot be assessed. This study fills in this gap by proposing a step-by-step ranking model which is based on comments rather than rates or the theoretically established criteria (e.g., food, service, atmosphere), which are not capable to portray the complexity of the dining experience. The proposed ranking model extracts topics from qualitative data (documents) and converts them into quantitative values through the lens of fuzzy logic. The topics are used as the ranking space while, pairwise comparisons between items (restaurants) are performed to extract a ranking ladder. We applied our model to TripAdvisor’s documents from restaurants in Athens, Greece.