Properties Of Quotients Research Articles

Pseudo-Schur complements and their properties

The notion of Schur complement of a partitioned matrix with a square nonsingular block is well known and it has many applications in various branches of mathematics. When the block is rectangular or singular, pseudo-Schur complements can be defined and studied. In particular, they satisfy an extension of the quotient property for Schur complements. A new proof of this property is given in this paper and various related topics are discussed.

The geometric minimal models of analytic spaces

This paper shows that an analytic space X has a unique maximal model through which every proper surjective morphism from a non-singular analytic space to X factors. This is called the geometric minimal model of X and characterized by the contraction property of rational curves. Some other properties such as functoriality, the direct product property and the quotient property of the geometric minimal model are also studied here. The relation of the geometric minimal model with Mori's minimal model is discussed.

Hückel spectra of Möbius π systems

Any Huckel π system with the Mobius-strip topology has a quotient property whereby its Huckel spectrum is the ungerade, and its adjacency spectrum the gerade half of the spectrum of a centrosymmetric cylindrical graph with twice as many vertices. Consequences include the spectral complementarity of Huckel and anti-Huckel monocycles, spectral embedding of twisted in untwisted cyclic polyacenes, and spectral inversion between in- and out-of-plane π sub-systems of odd-numbered rings in cluster molecules. Cartesian coordinates for Mobius graphs can be derived from those of the cylindrical parent.

Goldie dimensions of quotient modules

AbstractFor an infinite cardinal ℵ an associative ring R is quotient ℵ<-dimensional if the generalized Goldie dimension of all right quotient modules of RR are strictly less than ℵ. This latter quotient property of RR is characterized in terms of certain essential submodules of cyclic modules being generated by less than ℵ elements, and also in terms of weak injectivity and tightness properties of certain subdirect products of injective modules. The above is the higher cardinal analogue of the known theory in the finite ℵ = ℵ0 case.

On some properties of the quasisymmetry quotients of functions

We solve functional equations of the form¶¶\( k(x,h) = {f(x+h)- f(x)\over f(x) - f(x-h)}, \,\, x\,\in\,{\Bbb R},\,h > 0 \),¶in the class of injective functions \( f:{\Bbb R} \to {\Bbb R} \), provided \( k:{\Bbb R} \times (0,\infty) \to {\Bbb R} \) is a function of only one variable.

On some properties of the quasisymmetry quotients of functions

On some properties of the quasisymmetry quotients of functions

Schur complements obey Lambek's categorial grammar: Another view of Gaussian elimination and LU decomposition

For three decades Schur complements have seen increasing applications in linear algebra, often as abstractions of Gaussian elimination. It is known that they obey certain nontrivial identities, such as Crabtree and Haynsworth's quotient property. We began this work asking if there were a theory for deciding their properties in general. Lambek's Categorial Grammar is a deductive system formalized in 1958 by Lambek as a mathematical foundation for a syntactic calculus of language. We show that Categorial Grammar gives a deductive system for deriving identities obeyed by LU-and UL-decompositions, Gaussian elimination, and Schur complements. At first impression this seems to be a strange result, connecting two unrelated topics. In retrospect, though, it is a consequence of the way both use quotients. It may have applications in developing grammatical formalisms and numerical algorithms.

Quotients of polytopes and C-groups

Abstract polytopes are combinatorial and geometrical structures with a distinctive topological flavor, which resemble the convex polytopes. C-groups are generalizations of Coxeter groups and are the automorphism groups of abstract polytopes which are regular. We investigate general properties of quotients of abstract polytopes and C-groups.

Symmetric quotients and domain constructions

We introduce the symmetric quotient of two relations as a new construct in abstract relational algebra generalizing the notion of a “noyau” of Riguet. After exhibiting the main properties of symmetric quotients we study applications in domain theory. In particular, we give a monomorphic characterization of powersets and function domains.

A new proof of the cross-rule for the ϵ-algorithm based on Schur-complements

Matrices of the form H — GE −1 F are called Schur-complements. They are closely related to elimination procedures and to determinantal identities. In this paper, the quotient property for Schur-complements is used to establish the ‘Schur-complement-method’, a useful tool for proving certain formulas. As an example, a new proof of the cross-rule for the ϵ-algorithm is presented.

Partitioned matrices satisfying certain null space properties

Partitioned matrices satisfying certain null space properties for all leading principal submatrices are shown to be equivalent to a sequence of generalized Schur complements satisfying the same null space properties with respect to their (1, 1) entry. It is shown that a matrix with these null space properties has nonsingular principal submatrices of all orders less than or equal to its rank. Also, a theorem of Carlson, Haynsworth, and Markham concerning the quotient property for Schur complements is extended.

The universal group of homogeneous quotients

This paper constructs from the homogeneous quotients of an arbitrary semigroupS a universal group (G(S), γ) onS. If S is left reversible and cancellative, thenG(S) coincides with the embedding group of quotients of S due to Ore. If S is an inverse semigroup, G(S) coincides with the maximum group homomorphic image of S due to Munn. In these cases, γ coincides with the embedding and canonical homomorphism respectively ofS intoG(S). In general (G(S), γ) is equivalent to the universal group on S due to N. Bouleau. A universal group constructed from the set of Lambek ratios had earlier been exhibited by A.H. Clifford and G.B. Preston for cancellative semigroups satisfying the condition Z of Malcev. No previous construction has, however, emerged as a direct generalisation of both the work of Ore and Munn as does the present. Elementary properties of homogeneous quotients are employed to illuminate Bouleau's counter-example on why certain Malcev conditions are insufficient to guarantee the embeddability of a semigroup in a group.

A Study of Dynamic Bilinear Stochastic Models with Applications to Economic Processes

A new result on the multiplicative and quotient properties of scalar bilinear Ito equations is presented. Employing this result and the well-known closed form solutions for the scalar cases, some Lie theory is used to obtain closed form solutions for a large class of multivariable bilinear Ito differential equations. The closed form solutions are employed to obtain discrete time representations for those systems that can be directly employed in computations. Some computational results for economic system examples are presented.

On manifestations of the Schur complement

In this paper the author is concerned with some of the ways in which the Schur complement can be used in numerical linear algebra. Variable elimination and block pivoting are first outlined: the quotient property is next used in order to test whether the leading principal minors of a matrix are nonzero or of a particular sign. Another useful application is in computing the inertia of a real symmetric matrix: the inertia is instrumental in checking such matrices for positive (semi-) definiteness. This can be exploited in mathematical programming problems in order to check a non convex quadratic function for quasi-convexity (pseudo-convexity) on the nonnegative orthant.

On strictly cyclic algebras, $\mathcal{P}$-algebras and reflexive operators

An operator algebra $\mathfrak {A} \subset \mathcal {L}(\mathcal {X})$ (the algebra of all operators in a Banach space $\mathcal {X}$ over the complex field C) is called a “strictly cyclic algebra” (s.c.a.) if there exists a vector ${x_0} \in \mathcal {X}$ such that $\mathfrak {A}({x_0}) = \{ A{x_0}:A \in \mathfrak {A}\} = \mathcal {X};{x_0}$ is called a “strictly cyclic vector” for $\mathfrak {A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak {A}$ (i.e., if $A \in \mathfrak {A}$ and $A{x_0} = 0$, then $A = 0$), then $\mathfrak {A}$ is called a “separated s.c.a." $\mathfrak {A}$ is a $\mathcal {P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal {X}$, there exists ${x_0} \in \mathcal {X}$ such that $\left \| {A{x_j}} \right \| \leq \left \| {A{x_0}} \right \|$, for all $A \in \mathfrak {A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak {A}’$ of the algebra $\mathfrak {A}$ is an s.c.a., then $\mathfrak {A}$ is a $\mathcal {P}$-algebra and the strong and the uniform operator topology coincide on $\mathfrak {A}$; these results are specialized for the case when $\mathfrak {A}$ and $\mathfrak {A}’$ are separated s.c.a.’s. (Here, and in what follows, algebra means strongly closed subalgebra on $\mathcal {L}(\mathcal {X})$ containing the identity I on $\mathcal {X}$.) In the second part of the paper, it is shown that a large class of bilateral weighted shifts (which includes all the invertible ones) in a Hilbert space are reflexive. The result is used to show that “reflexivity” is neither a “restriction property” nor a “quotient property." Recall that an algebra $\mathfrak {A}$ is called reflexive if, whenever $T \in \mathcal {L}(\mathcal {X})$ and the lattice of invariant subspaces of T contains the corresponding lattice of $\mathfrak {A}$, then $T \in \mathfrak {A}$.

Liapunov Stability Criteria for Continuous Systems Under Parametric Excitation

A Liapunov direct method is developed here to derive stability criteria for continuous systems under parametric excitation. The method makes use of time-dependent Liapunov functionals and the extremal properties of Rayleigh quotients of self-adjoint operators. The application of the method entails solving an eigenvalue problem with the variable t appearing in the problem strictly as a parameter. As examples of application the method is applied to (a) a clamped column under the action of periodic axial load, and (b) the panel flutter problem with the panel also subjected to periodic in-plane load. The calculated results for the first example show improvement over those obtained previously.

Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem

Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem

Proof of Horner’s Method of Approximation to a Numerical Root of an Equation by the Properties of Algebraical Quotients and Remainders

This short paper is supplementary to that read to the Association by Mr. Langley (Jan. 1892); that also being supplementary to one read by Mr. Hayward (Jan. 1889).