Search

This paper is dedicated to the classification of uniform vector bundles of rank d + 1 over the Grassmannian G(d,n) (d ≤ n − d) over an algebraically closed field in characteristic 0. Specifically, we show that all uniform vector bundles with rank d + 1 over G(d,n) are homogeneous. [ABSTRACT FROM AUTHOR]

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras. [ABSTRACT FROM AUTHOR]

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

As shown by Melnikov, the orbits of a Borel subgroup acting by conjugation on upper-triangular matrices with square zero are indexed by involutions in the symmetric group. The inclusion relation among the orbit closures defines a partial order on involutions. We observe that the same order on involutive permutations also arises while describing the inclusion order on B-orbit closures in the direct product of two Grassmannians. We establish a geometric relation between these two settings. [ABSTRACT FROM AUTHOR]

We introduce the concept of shape operator and rotating blade (also known in the theory of embedded Riemannian manifolds as the second fundamental form and the Gauss map) in the realm of Yang–Mills theories. Hence we arrive at new gauge-invariant variables, which can serve as an alternative to the usual gauge potentials. [ABSTRACT FROM AUTHOR]

We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GL n case, we classify the type (1 , ... , 1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4 n -spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators. [ABSTRACT FROM AUTHOR]

We extend the periodicity of birational rowmotion for rectangular posets to the case when the base field is replaced by a noncommutative ring (under appropriate conditions). This resolves a conjecture from 2014. The proof uses a novel approach and is fully self-contained. Consider labelings of a finite poset \(P\) by \(\left\vert P\right\vert + 2\) elements of a ring \(\mathbb{K}\): one label associated with each poset element and two constant labels for the added top and bottom elements in \(\widehat{P}\). Birational rowmotion is a partial map on such labelings. It was originally defined by Einstein and Propp for \(\mathbb{K}=\mathbb{R}\) as a lifting (via detropicalization) of piecewise-linear rowmotion, a map on the order polytope \(\mathcal{O}(P) := \{\text{order-preserving } f: P \to[0,1]\}\). The latter, in turn, extends the well-studied rowmotion map on the set of order ideals (or more properly, the set of order filters) of \(P\), which correspond to the vertices of \(\mathcal{O}(P)\). Dynamical properties of these combinatorial maps sometimes (but not always) extend to the birational level, while results proven at the birational level always imply their combinatorial counterparts. Allowing \(\mathbb{K}\) to be noncommutative, we generalize the birational level even further, and some properties are in fact lost at this step.In 2014, the authors gave the first proof of periodicity for birational rowmotion on rectangular posets (when \(P\) is a product of two chains) for \(\mathbb{K}\) a field, and conjectured that it survives (in an appropriately twisted form) in the noncommutative case. In this paper, we prove this noncommutative periodicity and a concomitant antipodal reciprocity formula. We end with some conjectures about periodicity for other posets, and the question of whether our results can be extended to (noncommutative) semirings.It has been observed by Glick and Grinberg that, in the commutative case, periodicity of birational rowmotion can be used to derive Zamolodchikov periodicity in the type \(AA\) case, and vice-versa. However, for noncommutative \(\mathbb{K}\), Zamolodchikov periodicity fails even in small examples (no matter what order the factors are multiplied), while noncommutative birational rowmotion continues to exhibit periodicity. Thus, our result can be viewed as a lateral generalization of Zamolodchikov periodicity to the noncommutative setting.Mathematics Subject Classifications: 06A07, 05E99Keywords: Rowmotion, posets, noncommutative rings, semirings, Zamolodchikov periodicity, root systems, promotion, trees, graded posets, Grassmannian, tropicalization

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins are defined by taking a closure. We study the set obtained before making the closure (often called the open part of the join) and the set added after making the closure (called the boundary of the join). For a point q of the open part, we give conditions for the uniqueness of the set proving that q is in the open part. [ABSTRACT FROM AUTHOR]

We construct the parity–time symmetric solitons of the complex KP equation using the totally nonnegative Grassmannian. We obtain that every element in the totally nonnegative orthogonal Grassmannian corresponds to a parity–time symmetric soliton solution. [ABSTRACT FROM AUTHOR]

I revisit the condition number of computing left and right singular subspaces from J.-G. Sun (1996) [19]. For real and complex matrices, I present an alternative computation of this condition number in the Euclidean distance on the input space of matrices and the chordal, Grassmann, and Procrustes distances on the output Grassmannian manifold of linear subspaces. Up to a small factor, this condition number equals the inverse minimum singular value gap between the singular values corresponding to the selected singular subspace and those not selected. [ABSTRACT FROM AUTHOR]

We describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K -theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K -theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K -theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters. [ABSTRACT FROM AUTHOR]

This note is devoted to a special Fano fourfold defined by a four-dimensional space of skew-symmetric forms in five variables. This fourfold appears to be closely related with the classical Segre cubic and its Cremona–Richmond configuration of planes. Among other exceptional properties, it is infinitesimally rigid and has Picard number six. We show how to construct it by blow-up and contraction, starting from a configuration of five planes in a four-dimensional quadric, compatibly with the symmetry group S5. From this construction, we are able to describe the Chow ring explicitly [ABSTRACT FROM AUTHOR]

For the symplectic Grassmannian SpG (2 , 2 n) of 2-dimensional isotropic subspaces in a 2 n -dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an n -dimensional torus T on it, we characterize its irreducible T -invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank 2 which are representable over C. [ABSTRACT FROM AUTHOR]

Let O (3) denote the group of orthogonal 3 × 3 real matrices, and M the 5-dimensional real vector space of all 3 × 3 real symmetric matrices with trace zero. Let λ 1 (A) ≤ λ 2 (A) ≤ λ 3 (A) be the eigenvalues of A ∈ M and Ξ - = { A ∈ M ∣ λ 1 (A) = λ 2 (A) } . M is an inner product space with the inner product ⟨ A , B ⟩ = trace (A B) . Let G 3 (M) be the set of all 3-dimensional subspaces of M , a 6-dimensional Grassman manifold. O (3) acts on M on the left by conjugation via inner product preserving linear isomorphisms, which map any 3-dimensional subspace into another 3-dimensional subspace; thus G 3 (M) also has a left action of O (3) . G 3 (M) becomes a category, an action groupoid, with morphisms (V , M , W) ∈ G 3 (M) × O (3) × G 3 (M) , where W = M V M T . Composition of morphisms is (V 1 , N , V 2) ∘ (V 0 , M , V 1) = (V 0 , N M , V 2) . Let C be a category whose objects (V, S) consist of a real inner product space V and S ⊂ V , and whose arrows (V , S) → (W , T) consist of f : V → W , an inner product preserving real linear mapping such that f (S) ⊂ T . We have the functor V ↓ (V , M , W) W G 3 (M) ⟶ F - (V , Ξ - ∩ V) ↓ F - (M) (W , Ξ - ∩ W) C where F - (M) : V → W : A ↦ M A M T . Suppose further that S 3 denotes the group of permutations of { 1 , 2 , 3 } , and ρ : S 3 → O (3) denotes a group homomorphism which is isomorphic as a group representation to the natural representation of S 3 on R 3 (which permutes the coordinates). Let Obj (L S) denote the set of all V ∈ G 3 (M) whose isotropy subgroup contains S = ρ (S 3) as a subgroup. This paper completely describes the full subcategory L S of G 3 (M) with object set Obj (L S) , as well as the details of the above functor restricted to L S . Thus all the members V ∈ Obj (L S) are determined, as well as the smooth manifold structure on Obj (L S) ; it is embedded as a one-dimensional submanifold of G 3 (M) . The isotropy subgroups of all V ∈ Obj (L S) are computed and all pairs V , W ∈ Obj (L S) which are isomorphic via some M ∈ O (3) are determined. The sets Ξ - ∩ V are all determined, and the functorial mappings on morphism sets are computed. However, L S is not a Lie groupoid. The image of Obj (L S) under the functor π 1 F - is the collection of fibres of the smooth manifold ∐ V ∈ Obj (L S) V , which is the total space of the canonical vector bundle over the base manifold Obj (L S) . The bifurcation points of the family of subsets Ξ - ∩ V as V ranges over Obj (L S) (within this total space) are seen to be the points of Obj (L S) with infinite isotropy subgroups. We also show how this mathematical problem arises naturally from a problem in mathematical chemistry. Hence certain features of numerical calculations of energy eigenvalue intersection patterns of the simple chemical system H3 are rationalized through linearization about the triple intersection point. [ABSTRACT FROM AUTHOR]

In a paper by Kodiyalam and Raghavan, they provide an explicit combinatorial description of the Hilbert function of the tangent cone at any point on a Schubert variety in the Grassmannian, by giving a certain "degree-preserving" bijection between a set of monomials defined by an initial ideal and a "standard monomial basis". We prove here that this bijection is in fact a bounded RSK correspondence. As an application, we prove that the bijection given in a paper of Ghorpade and Raghavan (for the symplectic Grassmannian) is also a bounded RSK correspondence. [ABSTRACT FROM AUTHOR]

Let $G_{n,2n}$ be the Grassmannian parameterizing the $n$-dimensional subspaces of $\mathbb{C}^{2n}$. The Picard group of $G_{n,2n}$ is generated by a unique ample line bundle $\mathcal{O}(1)$. Let $T$ be a maximal torus of $\mathrm{SL}(2n,\mathbb{C})$ which acts on $G_{n,2n}$ and $\mathcal{O}(1)$. By [10, Theorem 3.10, p. 764], $2$ is the minimal integer $k$ such that $\mathcal{O}(k)$ descends to the GIT quotient. In this article, we prove that the GIT quotient of $G_{n,2n}$ ($n\ge 3$) by $T$ with respect to $\mathcal{O}(2)=\mathcal{O}(1)^{\otimes 2}$ is not projectively normal when polarized with the descent of $\mathcal{O}(2)$.

This paper is devoted to the differential geometry of ρ-dimensional complexes Cρ of m-dimensional planes in the projective space Pn containing a finite number of torsos. We find a necessary condition under which the complex Cρ contains a finite number of torsos. We clarify the structure of ρ-dimensional complexes Cρ for which all torsos belonging to the complex Cρ have one common characteristic (m+1)-dimensional plane that touches the torso along an m-dimensional generator. Such complexes are denoted by Cρ(1). Also, we determine the image of complexes Cρ(1) in the (m+1)(n−m)- dimensional algebraic variety Ω(m, n) of the space PN, where N = m + 1 n + 1 - 1 , which is the image of the variety G(m, n) of m-dimensional planes of the projective space Pn under the Grassmann mappping. [ABSTRACT FROM AUTHOR]

We prove a strong theorem on the partial non-defectivity of secant varieties of embedded homogeneous varieties developing a general set-up for families of subvarieties of Grassmannians. We study this type of problem in the more general set-up of joins of embedded varieties. Joins are defined by taking a closure. We study the set obtained before making the closure (often called the open part of the join) and the set added after making the closure (called the boundary of the join). For a point q of the open part, we give conditions for the uniqueness of the set proving that q is in the open part.

In order to incorporate group information for different groups, we extend the partial envelopes (Su and Cook, Statistica Sinica 23:213–30, 2011) to the groupwise partial envelopes which can improve the efficiency of parameter estimation and enlarge the scope of partial envelope model. It maintains the potential of the original partial envelope methods to increase efficiency and allows for both different regression coefficients and different error structures for diverse groups. Further, we demonstrate the maximum likelihood estimation under the groupwise partial envelope model. Meanwhile, we give asymptotic distribution and theoretical properties. At last, simulation studies are carried out to compare our proposed groupwise partial envelope model with the other three methods, including the standard model, the partial envelope model and the separate partial envelope model. From the simulation results and real data analysis, we can see that the performance of the groupwise partial envelope estimators is much better than that of the standard model estimators, the partial envelope estimators and the separate partial envelope estimators. [ABSTRACT FROM AUTHOR]

The second part of our study is devoted to a detailed convergence analysis of two extensions of the well-known DCA method for solving DC (Difference of Convex functions) optimization problems to the case of general cone constrained DC optimization problems. We study the global convergence of the DCA for cone constrained problems and present a comprehensive analysis of a version of the DCA utilizing exact penalty functions. In particular, we study the exactness property of the penalized convex subproblems and provide two types of sufficient conditions for the convergence of the exact penalty method to a feasible and critical point of a cone constrained DC optimization problem from an infeasible starting point. In the numerical section of this work, the exact penalty DCA is applied to the problem of computing compressed modes for variational problems and the sphere packing problem on Grassmannian. [ABSTRACT FROM AUTHOR]

Linear codes with complementary duals, or LCD codes, have recently been applied to side-channel and fault injection attack-resistant cryptographic countermeasures. We explain that over characteristic two fields, they exist whenever the permanent of any generator matrix is non-zero. Alternatively, in the binary case, the matroid represented by the columns of the matrix has an odd number of bases. We explain how Grassmannian varieties as well as linear and quadratic complexes are connected with LCD codes. Accessing the classification of polarities, we relate the binary LCD codes of dimension k to the two kinds of symmetric non-singular binary matrices, to certain truncated Reed–Muller codes, and to the geometric codes of planes in finite projective space via the self-orthogonal codes of dimension k. [ABSTRACT FROM AUTHOR]

Let G k , n be the Grassmannian of oriented subspaces of dimension k of R n with its canonical Riemannian metric. We study the energy of maps assigning to each P ∈ G k , n a unit vector normal to P. They are sections of a sphere bundle E k , n 1 over G k , n . The octonionic double and triple cross products induce in a natural way such sections for k = 2 , n = 7 and k = 3 , n = 8 , respectively. We prove that they are harmonic maps into E k , n 1 endowed with the Sasaki metric. This, together with the well-known result that Hopf vector fields on odd dimensional spheres are harmonic maps into their unit tangent bundles, allows us to conclude that all unit normal sections of the Grassmannians associated with cross products are harmonic. In a second instance we analyze the energy of maps assigning an orthogonal complex structure J P on P ⊥ to each P ∈ G 2 , 8 . We prove that the one induced by the octonionic triple product is a harmonic map into a suitable sphere bundle over G 2 , 8 . This generalizes the harmonicity of the canonical almost complex structure of S 6 . [ABSTRACT FROM AUTHOR]

A family of probability density functions (pdfs) is defined on the unit hypersphere S n . The parameter space for the pdfs is G (d , n + 1) × R ≥ 0 , for 1 ≤ d ≤ n , where G (d , n + 1) is the Grassmannian of d-dimensional linear subspaces in R n + 1 and R ≥ 0 is the range of values for a concentration parameter. This family of pdfs generalises the Watson distribution on the sphere S 2 . It is shown that the pdfs are tractable, in that (i) a given pdf can be sampled efficiently, (ii) the parameters of a pdf can be estimated using maximum likelihood, and (iii) the Kullback–Leibler divergence and the Fisher–Rao metric on G (d , n + 1) × R ≥ 0 have simple forms. A wide range of shapes of the pdfs can be obtained by varying d and the concentration parameter. The pdfs are used to model clusters of feature vectors on the hypersphere. The clusters are compared using the Kullback–Leibler divergences of the associated pdfs. Experiments with the mnist, Human Activity Recognition and Gas Sensor Array Drift datasets show that good results can be obtained from clustering algorithms based on the Kullback–Leibler divergence, even if the dimension n of the hypersphere is high. [ABSTRACT FROM AUTHOR]

Inference based on the partial envelope model is variational or nonequivariant under rescaling of the responses, and tends to restrict its use to responses measured in identical or analogous units. The efficiency acquisitions promised by partial envelopes frequently cannot be accomplished when the responses are measured in diverse scales. Here, we extend the partial envelope model to a scaled partial envelope model that overcomes the aforementioned disadvantage and enlarges the scope of partial envelopes. The proposed model maintains the potential of the partial envelope model in terms of efficiency and is invariable to scale changes. Further, we demonstrate the maximum likelihood estimators and their properties. Lastly, simulation studies and a real-data example demonstrate the advantages of the scaled partial envelope estimators, including a comparison with the standard model estimators, partial envelope estimators, and scaled envelope estimators. [ABSTRACT FROM AUTHOR]

In this paper we examine four different models for the realization space of a polytope: the classical model, the Grassmannian model, the Gale transform model, and the slack variety. Respectively, they identify realizations of the polytopes with the matrix whose columns are the coordinates of their vertices, the column space of said matrix, their Gale transforms, and their slack matrices. Each model has been used to study realizations of polytopes. In this paper we establish very explicitly the maps that allow us to move between models, study their precise relationships, and combine the strengths of different viewpoints. As an illustration, we combine the compact nature of the Grassmannian model with the slack variety to obtain a reduced slack model that allows us to perform slack ideal calculations that were previously out of computational reach. These calculations allow us to answer the question of Criado and Santos (Exp. Math. (2019). https://doi.org/10.1080/10586458.2019.1641766), about the realizability of a family of prismatoids, in general in the negative by proving the non-realizability of one of them. [ABSTRACT FROM AUTHOR]

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and 1432-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015). [ABSTRACT FROM AUTHOR]

A remarkable connection between the cohomology ring H*(Gr(d,d + r),Z) of the Grasssmannian Gr(d, d + r) and the lattice points of the dilation rΔd of the standard d-simplex is investigated. The natural grading on the cohomology induces different gradings of the lattice points of rΔd. This leads to different refinements of the Ehrhart polynomial LΔd (r) of the standard d-simplex. We study two of these refinements which are defined by the weights (1,1, ∙∙∙, 1) and (1, 2, ∙∙∙,d). One of the refinements interprets the Poincare polynomial P(Gr(d, d + r), z) as the counting of the lattice points which lie on the slicing hyperplanes of the dilation rΔd. Therefore, on the combinatorial level the Poincare polynomial of the Grassmannian Gr(d, d + r) is a refinement of the Ehrhart polynomial LΔd(r) of the standard d-simplex Δd. [ABSTRACT FROM AUTHOR]

We study vector bundles on flag varieties over an algebraically closed field k. In the first part, we suppose G=Gk(d,n)$G=G_k(d,n)$(2≤d≤n−d)$(2\le d\le n-d)$ to be the Grassmannian parameterizing linear subspaces of dimension d in kn$k^n$, where k is an algebraically closed field of characteristic p>0$p>0$. Let E be a uniform vector bundle over G of rank r≤d$r\le d$. We show that E is either a direct sum of line bundles or a twist of the pullback of the universal subbundle Hd$H_d$ or its dual Hd∨$H_d^{\vee }$ by a series of absolute Frobenius maps. In the second part, splitting properties of vector bundles on general flag varieties F(d1,...,ds)$F(d_1,\ldots ,d_s)$ in characteristic zero are considered. We prove a structure theorem for bundles over flag varieties which are uniform with respect to the ith component of the manifold of lines in F(d1,...,ds)$F(d_1,\ldots ,d_s)$. Furthermore, we generalize the Grauert–Mü$\ddot{\text{u}}$lich–Barth theorem to flag varieties. As a corollary, we show that any strongly uniform i‐semistable (1≤i≤n−1)$(1\le i\le n-1)$ bundle over the complete flag variety splits as a direct sum of special line bundles. [ABSTRACT FROM AUTHOR]

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that π0 of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable C-valued Z-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators. [ABSTRACT FROM AUTHOR]

The present article studies holomorphic isometric embeddings of the complex two-plane Grassmannian into quadrics. We discuss the moduli space of these embeddings up to gauge and image equivalence using a generalisation of do Carmo–Wallach theory. [ABSTRACT FROM AUTHOR]

The Peterson comparison formula proved by Woodward relates the three-pointed Gromov-Witten invariants for the quantum cohomology of partial flag varieties to those for the complete flag. Another such comparison can be obtained by composing a combinatorial version of the Peterson isomorphism with a result of Lapointe and Morse relating quantum Littlewood-Richardson coefficients for the Grassmannian to k-Schur analogs in the homology of the affine Grassmannian obtained by adding rim hooks. We show that these comparisons on quantum cohomology are equivalent, up to Postnikov's strange duality isomorphism. [ABSTRACT FROM AUTHOR]

Based on the study of quantum calculus, we construct a multicomponent -KP hierarchy and its additional symmetries. The additional symmetries form a multifold algebra and the generating operator of the additional symmetries can be shown to have a concise form in terms of wave functions. Furthermore, the string equation and the action of additional symmetries of the multicomponent -KP hierarchy on the Grassmannian are considered. After quantization, we derive the corresponding quantum torus symmetry, whose flows constitute an interesting multifold quantum torus type Lie algebra. [ABSTRACT FROM AUTHOR]

The solutions sets to linear PDEs with constant coefficients are naturally parameterised by polynomial rows. In this note, introducing appropriate topologies, this parametrization is made a quotient map (continuous, in particular). [ABSTRACT FROM AUTHOR]

The Gauss map g of a surface Σ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2 , 4) . We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the area of the Gauss map | g (Σ) | of a minimal surface is smaller than 2 π then the surface is stable by deformations which fix the boundary of the surface. This answers the question of Barbosa and Do Carmo (Math Z 173:13–28, 1980) in R 4 . [ABSTRACT FROM AUTHOR]

In this note, we study the Radon transform and its dual on the discrete matrices by defining hyperplanes as being infinite sets of solutions of linear Diophantine equations. We then give an inversion formula and a support theorem.

Let H be a hyperplane of the Grassmannian of the k-dimensional subspaces of the projective space PG (n , F) , 0 ≤ k ≤ n - 1 , and let H ~ denote the subgeometry of the Grassmannian induced on the point set H. We prove that the embedding and generating ranks of H ~ are always equal to n + 1 k + 1 - 1 . [ABSTRACT FROM AUTHOR]

It is a well-known fact that there exists a standard minimal embedding map for the Grassmannians of n-planes G (n , m ; F) (F = R , C , H) and Cayley plane O P 2 into Euclidean spheres, then a natural question is that if the cones over these embedded Grassmannians are area-minimizing? In this paper, detailed descriptions for this embedding map are given from the point of view of Hermitian orthogonal projectors which can be seen as a direct generalization of Gary R. Lawlor’s([23]) original considerations for the case of real projective spaces, then we re-prove the area-minimization of those cones which was gradually obtained in [18-19], and [30] from the perspectives of isolated orbits of adjoint actions or canonical embedding of symmetric R-spaces, all based on the method of Gary R. Lawlor’s Curvature Criterion. Additionally, area-minimizing cones over almost all common Grassmannians have been given by Takahiro Kanno, except those cones over oriented real Grassmannians G ~ (n , m ; R) which are not Grassmannians of oriented 2-planes. The second part of this paper is devoted to complement this result, a natural and key observation is that the oriented real Grassmannians can be considered as unit simple vectors in the exterior vector spaces, we prove that all their cones are area-minimizing except G ~ (2 , 4 ; R) . [ABSTRACT FROM AUTHOR]

Basic algebraic and combinatorial properties of finite vector spaces in which individual vectors are allowed to have multiplicities larger than 1 are derived. An application in coding theory is illustrated by showing that multispace codes that are introduced here may be used in random linear network coding scenarios, and that they generalize standard subspace codes (defined in the set of all subspaces of F q n) and extend them to an infinitely larger set of parameters. In particular, in contrast to subspace codes, multispace codes of arbitrarily large cardinality and minimum distance exist for any fixed n and q. [ABSTRACT FROM AUTHOR]

The impressive success of style-based GANs (StyleGANs) in high-fidelity image synthesis has motivated research to understand the semantic properties of their latent spaces. In this paper, we approach this problem through a geometric analysis of latent spaces as a manifold. In particular, we propose a local dimension estimation algorithm for arbitrary intermediate layers in a pre-trained GAN model. The estimated local dimension is interpreted as the number of possible semantic variations from this latent variable. Moreover, this intrinsic dimension estimation enables unsupervised evaluation of disentanglement for a latent space. Our proposed metric, called Distortion , measures an inconsistency of intrinsic tangent space on the learned latent space. Distortion is purely geometric and does not require any additional attribute information. Nevertheless, Distortion shows a high correlation with the global-basis-compatibility and supervised disentanglement score. Our work is the first step towards selecting the most disentangled latent space among various latent spaces in a GAN without attribute labels. • We propose a scheme to estimate local intrinsic dimensions in GAN latent spaces. • Our local dimension estimate provides an upper bound on local semantic perturbations. • We propose a layer-wise unsupervised disentanglement score, called Distortion. • We analyze the layers of the mapping network in StyleGANs through Distortion metric. [ABSTRACT FROM AUTHOR]