Coordinate Subspaces Research Articles

On a cube and subspace projections

We consider the arrangement of vertices of a unit multidimensional cube, an affine subspace, and its orthogonal projections onto coordinate subspaces. Upper and lower bounds on the subspace dimension are given under which some orthogonal projection always preserves the incidence relation between the subspace and cube vertices. Some oblique projections are also considered. Moreover, a brief review of the history of the development of multidimensional descriptive geometry is given. Analytic and synthetic methods in geometry diverged since the 17th century. Although both synthesis and analysis are tangled, from this time forth many geometers as well as engineers keep up a nice distinction. One can find references to the idea of higher-dimensional spaces in the 18th-century works, but proper development has been since the middle of the 19th century. Soon such works have appeared in Russian. Next, mathematicians generalized their theories to many dimensions. Our new results are obtained by both analytic and synthetic methods. They illustrate the complexity of pseudo-Boolean programming problems because reducing the problem dimension by orthogonal projection meets obstacles in the worst case.

Equations of the multi-Rees algebra of fattened coordinate subspaces

In this paper, we describe the equations defining the multi-Rees algebra [Formula: see text] ([Formula: see text]), where the ideals are generated by subsets of [Formula: see text]. We also show that a family of binomials whose leading terms are squarefree, form a Gröbner basis for the defining equations with lexicographic order. We show that if we remove binomials that include [Formula: see text]’s, then remaining binomials form a Gröbner basis for the toric ideal associated to the multi-fiber ring. However binomials, including [Formula: see text]’s, in Gröbner basis of defining equations of the multi-Rees algebra are not necessarily defining equations of corresponding symmetric algebra. Despite this fact, we show that this family of ideals is of multi-fiber type.

A reduced variational approach for searching cycles in high-dimensional systems

Searching recurrent patterns in complex systems with high-dimensional phase spaces is an important task in diverse fields. In the current work, an improved scheme is proposed to accelerate the recently designed variational approach for finding periodic orbits in systems with chaotic dynamics based on the existence of inertial manifold widely observed in various spatially extended systems, especially those with high dimensions. On the premise of keeping exponential convergence of the variational method, an effective loop evolution equation is derived to greatly reduce the storage and computation time. With repeated modification of local coordinates and evolution of the guess loop being carried out alternately, the rapid convergence and the stability of the reduction scheme are effectively achieved. The dimension of local coordinate subspaces is generally larger than the number of nonnegative Lyapunov exponents to ensure the exponential convergence. The proposed scheme is successfully demonstrated on several well-known examples and expected to supply a powerful tool in the exploration of high-dimensional nonlinear systems.

Invariant Coordinate Subspaces of Normal Form of a System of Ordinary Differential Equations

For systems of ordinary differential equations (ODEs) with a nondegenerate linear part in the general and Hamiltonian cases, the problem of finding invariant coordinate subspaces in the coordinates of the normal form calculated in the vicinity of the equilibrium is stated. Conditions for the existence of such invariant subspaces in terms of the resonant relations between the eigenvalues of the linear part of the system are obtained. An algorithm for finding the resonant relations between the eigenvalues without their explicit calculation is described; this algorithm substantially uses computer algebra methods and the q-analog of the polynomial subresultants. The implementation of this algorithm in three popular computer algebra systems—Mathematica, Maple, and SymPy—is discussed. Interesting model examples are provided.

A vanishing theorem for the canonical blow-ups of Grassmann manifolds

AbstractLet 𝒯s,p,nbe the canonical blow-up of the Grassmann manifoldG(p,n) constructed by blowing up the Plücker coordinate subspaces associated with the parameters. We prove that the higher cohomology groups of the tangent bundle of 𝒯s,p,nvanish. As an application, 𝒯s,p,nis locally rigid in the sense of Kodaira-Spencer.

Two-weight norm inequalities for product fractional integral operators

In the open range 1<p<q<∞, under a certain geometrical condition on weights, two-weight norm inequality for the product fractional integral operator, whose kernel has singularity appeared on each of the coordinate subspaces, is shown to follow from the Fefferman–Phong-type characteristic for the product cubes. This geometrical condition turns out to be the testing condition of Carleson-type embedding for product dyadic cubes.

Robertson – Walker Metric in (2 + 1) – Dimensions: The 2 – D Coordinate Subspaces and Their Curvature

In this paper, we will first construct a Robertson – Walker like metric in (2 + 1) – dimensional space. The easiest way of doing this is to consider a 2-dimensional coordinate space as a space embedded in a 3-dimensional hypersurface. The curvature of each surface is determined using the spatial part of the Robertson – Walker like metric constructed. Our main goal is to find out if the Robertson – Walker like metric in (2 + 1) – dimensional space can be used as a prototype model to study Robertson – Walker in (3 + 1) dimensions since calculations involved in higher dimensions are tedious.

Инвариантные координатные подпространства нормальной формы системы обыкновенных дифференциальных уравнений

For systems of ordinary differential equations (ODEs) with a nondegenerate linear part in the general and Hamiltonian cases, the problem of finding invariant coordinate subspaces in the coordinates of the normal form calculated in the vicinity of the equilibrium is stated. Conditions for the existence of such invariant subspaces in terms of the resonant relations between the eigenvalues of the linear part of the system are obtained. An algorithm for finding the resonant relations between the eigenvalues without their explicit calculation is described; this algorithm substantially uses computer algebra methods and the q-analog of the polynomial subresultants. The implementation of this algorithm in three popular computer algebra systems—Mathematica, Maple, and SymPy—is discussed. Interesting model examples are provided.

Rational approximation of affine coordinate subspaces of Euclidean space

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.

The homotopy type of spaces of coprime polynomials revisited

The purpose of this paper is to study the topology of certain toric varieties XI, arising as quotients of the action of C⁎ on complements of arrangements of coordinate subspaces in Cn, and to improve the homotopy stability dimension for the inclusion map id:Hold⁎(S2,XI)→Mapd⁎(S2,XI) given in [11] by making use of the Vassiliev spectral sequence.We also improve the homotopy stability dimension of this inclusion given by G. Segal [18] for XI=CPn−1 and n≥3.

Toric graph associahedra and compactifications of $$M_{0,n}$$

To any graph G, one can associate a toric variety $$X(\mathcal {P}G)$$X(PG), obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of G. The polytopes of these toric varieties are the graph associahedra, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $$X(\mathcal {P}{G})$$X(PG) is isomorphic to a Hassett compactification of $$M_{0,n}$$M0,n precisely when G is an iterated cone over a discrete set. This may be viewed as a generalization of the well-known fact that the Losev---Manin moduli space is isomorphic to the toric variety associated with the permutohedron.

A Comprehensive Pythagorean Theorem for All Dimensions

We use a 200-year-old theorem on determinants to prove a very general version of the Pythagorean theorem. It relates the square of the n-dimensional volume of an n-parallelotope in ℝm to the sum of the squares of the n-dimensional volumes of the orthogonal projections of the parallelotope onto the k-dimensional coordinate subspaces of ℝm.

Monotone functions and maps

In [S. Basu, A. Gabrielov, N. Vorobjov, Semi-monotone sets. arXiv:1004.5047v2 (2011)] we defined semi-monotone sets, as open bounded sets, definable in an o-minimal structure over the reals, and having connected intersections with all translated coordinate cones in R^n. In this paper we develop this theory further by defining monotone functions and maps, and studying their fundamental geometric properties. We prove several equivalent conditions for a bounded continuous definable function or map to be monotone. We show that the class of graphs of monotone maps is closed under intersections with affine coordinate subspaces and projections to coordinate subspaces. We prove that the graph of a monotone map is a topologically regular cell. These results generalize and expand the corresponding results obtained in Basu et al. for semi-monotone sets.

Restriction of hypergeometric 𝒟-modules with respect to coordinate subspaces

We compute the restriction of an A A -hypergeometric D \mathscr {D} -module with respect to a coordinate subspace under certain genericity conditions on the parameter.

Construction of Optimal Cubature Formulas Related to Computer Tomography

We study the problem of the optimization of approximate integration on the class of functions defined on the parallelepiped Πd=[0,a1]×⋅⋅⋅×[0,ad], a1,…,ad>0, having a given majorant for the modulus of continuity (relative to the l1-metric in ℝd). An optimal cubature formula, which uses as information integrals of f along intersections of Πd with n arbitrary (d−1)-dimensional hyperplanes in ℝd (d>1) is obtained. We also find an asymptotically optimal sequence of cubature formulas, whose information functionals are integrals of f along intersections of Πd with shifts of (d−2)-dimensional coordinate subspaces of ℝd (d>2).

Irregular hypergeometric [formula omitted]-modules

We study the irregularity of hypergeometric D -modules M A ( β ) via the explicit construction of Gevrey series solutions along coordinate subspaces in X = C n . As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of M A ( β ) given by M. Schulze and U. Walther (2008) in [23] still holds for any full rank integer matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters. Holomorphic solutions of M A ( β ) at nonsingular points can be understood as Gevrey solutions of order one along X at generic points and so they are included as a particular case.

Irregularity of hypergeometric systems via slopes along coordinate subspaces

We study the irregularity sheaves attached to the $A$-hypergeometric $D$-module $M_A(\beta)$ introduced by Gel'fand et al., where $A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and $\beta\in\mathbb{C}^d$. More precisely, we investigate the slopes of this module along coordinate subspaces. In the process we describe the associated graded ring to a positive semigroup ring for a filtration defined by an arbitrary weight vector $L$ on torus equivariant generators. To this end we introduce the $(A,L)$-umbrella, a simplicial complex determined by $A$ and $L$, and identify its facets with the components of the associated graded ring. We then establish a correspondence between the full $(A,L)$-umbrella and the components of the $L$-characteristic variety of $M_A(\beta)$. We compute in combinatorial terms the multiplicities of these components in the $L$-characteristic cycle of the associated Euler-Koszul complex, identifying them with certain intersection multiplicities. We deduce from this that slopes of $M_A(\beta)$ are combinatorial, independent of $\beta$, and in one-to-one correspondence with jumps of the $(A,L)$-umbrella. This confirms a conjecture of Sturmfels and gives a converse of a theorem of Hotta: $M_A(\beta)$ is regular if and only if $A$ defines a projective variety.

The geometric structure of Chebyshev sets in ℓ∞(n)

A subset M of a normed linear space X is called a Chebyshev set if each x ∈ X has a unique nearest point in M. We characterize Chebyshev sets in l∞(n) in geometric terms and study the approximative properties of sections of Chebyshev sets, suns, and strict suns in l∞(n) by coordinate subspaces.

Linear preservers and diagonal hypergraphs

In this paper we elucidate a connection between linear preservers on coordinate subspaces that permute the elements of a matrix and diagonal hypergraph isomorphisms. Under certain circumstances, the only permutation linear preservers are the special ones which permute rows and columns, and possibly interchange rows with columns as well.

Subspaces where an immanant is convertible into its conjugate*

Let n≥5 and let be an irreducible nonlinear character of Sn such that whenever σ is a transposition or a cycle of length three; furthermore let Tn be the (0, 1)-matrix of order n that has ones exactly on and below the upper neighbours of the main diagonal and denote by Eij the matrix of order n with 1 in position (i, j) and 0 elsewhere. Given i,jϵ{1,…,n}, with i+1<j, we prove that if j−i≠3, then in the subspace Mn (Tn +Eij there exist matrices for which the immanant is not convertible into the immanant by sign-affixing. Abusing language, we say that the space is -inconvertible, and show that spaces Mn (Tn +E25 ) and Mn (Tn +En−3,n ). We also state some sufficient fonditions for the subspace Mn (Tn ) to be external convertible. With some exceptions our theorems say that the coordinate subspaces found for the conversion of the permanent into the determinant by Gibson around 1970 are also best possible for other immanants.