Variety Of Rank Research Articles

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

DEGENERATE SCHUBERT VARIETIES IN TYPE A

We introduce rectangular elements in the symmetric group. In the framework of PBW degenerations, we show that in type A the degenerate Schubert variety associated with a rectangular element is indeed a Schubert variety in a partial ag variety of the same type with larger rank. Moreover, the degenerate Demazure module associated with a rectangular element is isomorphic to the Demazure module for this particular Schubert variety of larger rank. This generalises previous results by Cerulli Irelli, Lanini and Littelmann for the PBW degenerate ag variety in [CLL].

Real identifiability vs. complex identifiability

A real tensor T of rank r is identifiable if it has a unique decomposition with rank 1 tensors. Sometimes identifiability fails over , for general tensors of fixed rank. This behaviour is peculiar when r is sub-generic. In several cases, the failure is due to an elliptic normal curve passing through general points of the corresponding variety of rank 1 tensors (Segre, Veronese, Grassmann). When this happens, we prove the existence of nonempty euclidean open subsets of some varieties of rank r tensors, whose elements have several complex decompositions, but only one of them is real.

A fuzzy approach to service quality diagnosis

Purpose – The purpose of this paper is to identify service quality dimensions that could enhance the level of services provided by a variety of carriers and rank them, accordingly. Design/methodology/approach – This research aims to identify service quality dimensions that could enhance the level of services provided by a variety of carriers and rank them, accordingly. Findings – The results show that “safety,” “timeliness,” and “variety and type of airplane” are the most important service quality criteria from the passengers’ viewpoint. It also indicates that the most and least important dimensions of service quality are “tangibles” and “responsiveness,” respectively. The results ranked the airlines from one to three as Emirates Airline, Mahan Air, and Iran Air. Originality/value – Most of the service quality criteria are descriptive and are stated using lingual expressions (cardinal or ordinal scales). Therefore, evaluation of perceptions and expectations of customers regarding service quality using non-fuzzy methods ignores the ambiguity involved in individual judgments.

Homogeneous projective varieties with semi-continuous rank function

Let $${\mathbb{X}\subset\mathbb{P}(V)}$$ be a projective variety, which is not contained in a hyperplane. Then every vector v in V may be written as a sum of vectors from the affine cone over $${\mathbb{X}}$$ . The minimal number of summands in such a sum is called the rank of v. In this paper, we classify all equivariantly embedded homogeneous projective varieties $${\mathbb{X}\subset\mathbb{P}(V)}$$ whose rank function is lower semi-continuous. Classical examples are: the variety of rank one matrices (Segre variety with two factors) and the variety of rank one quadratic forms (quadratic Veronese variety). In the general setting, $${\mathbb{X}}$$ is the orbit in $${\mathbb{P}(V)}$$ of a highest weight line in an irreducible representation V of a reductive algebraic group G. Thus, our result is a list of all irreducible representations of reductive groups, for which the corresponding rank function is lower semi-continuous.

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA * = B

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equations AX = B and AXA * = B. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations AX = B and AXA * = B, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.

The rank-one limit of the Fourier-Mukai transform

We give a formula for the specialization of the Fourier-Mukai transform on a semi-abelian variety of torus rank 1.

The border rank of the multiplication of $2\times 2$ matrices is seven

One of the leading problems of algebraic complexity theory is matrix multiplication. The naive multiplication of two n× n matrices uses n multiplications. In 1969, Strassen [20] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [12] and Winograd [22] proved independently that there is no algorithm for multiplying 2×2 matrices using only six multiplications. The precise number of multiplications needed to execute matrix multiplication (or any given bilinear map) is called the rank of the bilinear map. A related problem is to determine the border rank of matrix multiplication (or any given bilinear map), first introduced in [6, 5]. Roughly speaking, some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small “good” perturbations of the map. These perturbed maps can give rise to fast exact algorithms for matrix multiplication; see [7]. The border rank made appearances in the literature in the 1980s and early 1990s (see, e.g., [6, 5, 19, 8, 15, 2, 10, 11, 4, 3, 17, 16, 18, 1, 9, 21]), but to our knowledge there has not been much progress on the question since then. More precisely, for any complex projective variety X ⊂ CP = PV and point p ∈ PV , define the X-rank of p to be the smallest number r such that p is in the linear span of r points of X. Define σr(X), the r-th secant variety of X, to be the Zariski closure of the set of points of X-rank r, and define the X-border rank of p to be the smallest r such that p ∈ σr(X). The terminology is motivated by the case X = Seg(Pa−1 × Pb−1) ⊂ P(C ⊗ C), the Segre variety of rank one matrices. Then the X-rank of a matrix is just its usual rank. Let A∗, B∗, C be vector spaces and let f : A∗ ×B∗ → C be a bilinear map, i.e., an element of A⊗B⊗C. Let X = Seg(PA×PB×PC) ⊂ P(A⊗B⊗C) denote the Segre variety of decomposable tensors in A⊗B⊗C. The border rank of a bilinear map is its X-border rank. While for the Segre product of two projective spaces, border rank coincides with rank, here they can be quite different. In this paper we prove the theorem stated in the title. Let MMult ∈ C ⊗C ⊗ C denote the matrix multiplication operator for 2 × 2 matrices. Strassen [18] showed that MMult / ∈ σ5(P × P × P). Our method of proof is to decompose

Singular Configurations of Parallel Manipulators and Grassmann Geometry

Parallel manipulators have a specific mechanical architecture where all the links are connected both at the base and at the gripper of the robot. By changing the lengths of these links we are able to control the position1 of the gripper. In general, for a given set of links lengths there is only one position for the gripper. However, it may be suspected that in some cases more than one solution may be found for the position of the gripper: the robot is in a singular configuration. To determine these singular configurations the classical method is to find the roots of the determinant of the jacobian matrix. In our case the jacobian matrix is complex, and it seems not to be possible to find these roots. We propose here a new method based on Grassmann line geometry. The set of lines, P3, constitutes a linear variety of rank 6. We show that a singular configuration is obtained when the variety spanned by the lines associated to the robot links has a rank less than 6. An important feature of the varieties of this geometry is that they can be described by simple geometric rules. Thus to find the singular configurations of parallel manipulators we have to find the configuration in which the robot matches these rules. Such an analysis is performed on a special parallel manip ulator. We show that we find all the well-known singular configurations, and also some new ones.

Communicating with patients about antipsychotic drugs

T HE QUALITY of communication between doctor and patient on the issue of medication has become the focus of increasing attention in recent years. A number of factors have given impetus to this, including efforts to restore a greater degree of humanity to the practice of medicine, and, from the other side of the relationship, a rising influence of the consumer movement in our society. In the field of clinical drug research, pharmacologists are increasingly recognizing the need to study the so-called “nondrug factors” in pharmacotherapy.’ In the treatment of schizophrenia with antipsychotic medications, communication appears to be of special importance. Despite the fact that clinicians in a wide variety of private-practice, institutional, and academic settings rank the administration of antipsychotic drugs as the most useful treatment available for schizophrenic illness.2 this enthusiasm is not reflected in studies of patient compliance. Studies of inpatients have produced up to a 20% noncompliance figure,” and studies of outpatients show that between 40% and 50% of schizophrenic patients do not take prescribed medications.4*5 In a review of medication compliance problems (which are not limited to schizophrenic patients), Blackwell has cited several high-risk factors, including chronicity of illness, symptom suppressive as opposed to curative action of the medication, delayed relapse with cessation of medications, ambivalent feelings about dependency issues, multiple medications, and social isolation.” Many of these factors are particularly applicable to schizophrenic patients receiving antipsychotic medications. Their relevance is confirmed by the marked dis: crepancy between intention and performance noted in one study of compliance in schizophrenic outpatients.’ These serious compliance problems may represent effects of the schizophrenia itself, the reaction to inevitable aversive drug effects, or reactions to problems within the doctor-patient relationship. The general thrust of much consumer-oriented literature has been that doctors should communicate fully with their patients concerning any proposed treatment, especially being sure that patients at-e aware of the benefits to be derived from the medication, and the side effects and risks involved, so that they can provide a meaningful informed consent and interact with the physician in a participatory model.