A Classification Theorem for SKT-Modules

Projective equivalence of homogenous binary polynomials

Local invariants for a class of mixed states

We study the equivalence of bipartite mixed states under local unitary transformations. A complete set of invariants under local unitary transformations is presented for a class of mixed states in CM ⊗ CN composite systems. It is shown that two density matrices in this class are locally equivalent if and only if all these invariants have equal values for these density matrices.

Feedback invariants for linear dynamical systems over a principal ideal domain

We provide in this paper a complete set of invariants for differentiable conjugacies of Morse-Smale diffeomorphisms. In the setting of topological conjugacies, these diffeomorphisms play an important role similar to that of gradients whose singularities are non-degenerate and have their stable and unstable manifolds in general position: they exhibit a simple and yet stable dynamics, and they exist on every manifold [PS]. Our main global invariant corresponds to a generalization of Mather's invariant for diffeomorphisms of the interval.

The global identifiability problem of linear systems from a tuple of invariants of the system is separated into two different problems. Once we fix a nominal point of interest /spl theta//spl circ/, we can check the local identifiability at that point. In addition we provide two algorithms for checking the existence of other remote points /spl theta/ in the parameter domain, indistinguishable from /spl theta//spl circ/. The approach is based on the assumptions that there exists a finite complete set of invariants and that both the invariants and feasible parameter domain are given by polynomials.

This chapter is a brief introduction into the structure of algebras, mostly finite dimensional, over any field k. The main contents are the Wedderburn theorems for a finite dimensional algebras A over an algebraically closed field k. If A has no nilpotent ideals ≠ 0, then A is a finite product of total matrix algebras over k. In this case, the set d (A) of degrees of the total matrix algebras is a complete set of invariants of A. Thus, 13.7 two finite dimensional semiprime algebras A and B over k are isomorphic if and only if d (A) = d (B).

We construct an algebraic-combinatorial model of the SOS compartment of the EGFR biochemical network. A Petri net is used to construct an initial representation of the biochemical decision making network, which in turn defines a hyperdigraph. We observe that the linear algebraic structure of each hyperdigraph admits a canonical set of algebraic-combinatorial invariants that correspond to the information flow conservation laws governing a molecular kinetic reaction network. The linear algebraic structure of the hyperdigraph and its sets of invariants can be generalized to define a discrete algebraic-geometric structure, which is referred to as an oriented matroid. Oriented matroids define a polyhedral optimization geometry that is used to determine optimal subpaths that span the nullspace of a set of kinetic chemical reaction equations. Sets of constrained submodular path optimizations on the hyperdigraph are objectively obtained as a spanning tree of minimum cycle paths. This complete set of subcircuits is used to identify the network pinch points and invariant flow subpaths. We demonstrate that this family of minimal circuits also characteristically identifies additional significant biochemical reaction pattern features. We use the SOS Compartment A of the EGFR biochemical pathway to develop and demonstrate the application of our algebraic-combinatorial mathematical modeling methodology.

§1. Introduction.The motivation of this work comes from two different directions: infinite abelian groups, and ordered algebraic structures. A challenging problem in both cases is that of classification. In the first case, it is known for example (cf. [KA]) that the classification of abelian torsion groups amounts to that of reducedp-groups by numerical invariants called theUlm invariants(given by Ulm in [U]). Ulm's theorem was later generalized by P. Hill to the class of totally projective groups. As to the second case, let us consider for instance the class of divisible ordered abelian groups. These may be viewed as ordered ℚ-vector spaces. Their theory being unstable, we cannot hope to classify them by numerical invariants. On the other hand, being o-minimal, the theory enjoys several good model theoretic properties (cf. [P-S]), so the search for some reasonable invariants is well motivated. The common denominator of the two cases, as well as of many others, is valuation theory. Indeed given an ordered vector space, one can consider it as a valued vector space, endowed with the natural valuation. Also, the socleG[p] of a reduced abelianp-groupG, endowed with the height functionhG, is a valued vector space over(the prime field of characteristicp) with values in the ordinals (cf. [F]).

The representation of a ring as the ring of all the automorphisms of a primary abelian operator group expresses significant inner properties of this ring, since there exists essentially at most one such representation of a given ring. Thus it is the object of this investigation to expose invariant qualities of these automorphism rings. They are found to be pecularities of their ideal theory (Chapter III); and from these we select a complete set of postulates for the class of all the automorphism rings of primary abelian operator groups (Sections 12 and 13); thus proving incidentally that the structure of these rings is completely determined by their ideal theory (Section 14). The properties of the ideals in these rings reflect so completely the structure of the underlying operator group that we are able to prove-provided the rank of this group is at least three-the essential identity of the group of a-utomorphisms of the ring and of the group of projectivities (= biunivoque and monotone increasing transformations) of the system of admissible subgroups of the underlying operator group (Chapter II). Two [extreme] special cases may serve as an-i illustration for these theorems. Every projective geometry of finite dimension not less than three may be represented as the set of admissible subgroups of a suitable primary abelian operator group; and our characterization of the automorphisms of the automophism ring specializes in this case to the (well known) theorem that each automorphism of the automorphism ring (= ring of square matrices with coefficients from a suitable (not necessarily commutative) field) is induced by a so-called seini-linear transformation. If secondly G is an ordinary primary abelian group with the property that the least common multiple of the orders of the elements in G is a prime power p' and that G contains at least three independent elements of maximum order, then our theorem states that every automorphism of the automorphism ring is an inner auomorphism.

Weakly injective and weakly projective modules, which are extensions of the classes of quasi-injective modules and quasi-projective modules respectively, are studied. Descriptions are obtained for weakly injective and weakly projective Abelian groups, as well as for some classes of Abelian groups closely related to them.

Several years ago, my colleagues and I developed a project-based assignment where students create a fictional animal and predict its ability to thrive in a hypothetical environment by examining the interactions between different combinations of designs for selected body systems. This assignment, which was implemented at different types of institutions and for different levels of students, can be readily adapted to either lecture-based or laboratory courses. Here I describe how students design a fictional animal as a semester-long collaborative group project in an interactive lecture course on comparative animal physiology of vertebrates. The fictional animal project aligns with several course learning goals including 1) Acquire a fundamental knowledge of “how animals work” and 2) Exercise responsibility and teamwork. Students learn integration of body systems as they choose a random fictional animal, consider if this animal can survive in a specific environment, and create the animal. Creation of the animal, which cannot closely resemble an extant animal, is scaffolded throughout the semester as students complete question sets where they describe the structure and function of the systems and consider potential trade-offs and physiological constraints. At the end of the semester, we hold a fictional animal showcase where students share highlights about their animal and its environment. Student learning is assessed through the questions sets, which are submitted as homework, and the class presentation; this group project contributes to 25% of the overall course grade and replaces more traditional assessments such as exams. Examples of fictional animals demonstrate how students must consider interactions and functions between different systems in their design; in addition, they must think about the metabolic requirements of their animal as they decide its lifestyle. Reflection data from students strongly supports that designing a fictional animal helps them understand how physiological systems function together; students gain a new or deeper awareness and appreciation of trade-offs in function as well as constraints on physiological processes. As this project continues to evolve, students will have the freedom to choose a creative format for the fictional animal showcase, such as a recorded or live oral presentation, a podcast, a tri-fold brochure, a poster, an educational game, or an illustrated story. All protocols were approved by Rice University IRB (Protocol FY2017-294). Rice University Office of Undergraduate Research and Inquiry (OURI). This is the full abstract presented at the American Physiology Summit 2023 meeting and is only available in HTML format. There are no additional versions or additional content available for this abstract. Physiology was not involved in the peer review process.

A new method is presented for solving equivalence problems for the extended jet of finite order of the scale-space corresponding to a 2-dimensional input image and the groups of spatially homogeneous affine and orthogonal transformations of local cartesian coordinate frames. By means of this method complete and irreducible sets of algebraic invariants are found that may describe any local and (or) multilocal affine or orthogonal invariant of scale-space. Consequently these sets may form bases for topological descriptions of scale-space.

This paper elaborates the theoretical foundations of a semi-differential framework for invariance. Semi-differential invariants combine coordinates and their derivatives with respect to some contour parameter at several points of the image contour, thus allowing for an optimal trade-off between identification of points and the calculation of derivatives. A systematic way of generating complete and independent sets of such invariants is presented. It is also shown that invariance under reparametrisation can be cast in the same framework. The theory is illustrated by a complete analysis of 2D affine transformations. In a companion paper (Pauwels et al. 1995) these affine semi-differential invariants are implemented in the computer program FORM (Flat Object Recognition Method) for the recognition of planar contours under pseudo-perspective projection.

The systems considered in the paper are m-input m-output, linear, time invariant decouplable systems S = (A, B, C) which can be described by the set of differential equations $ where A∊R n×n , B∊R n×m and C∊R m×n . In § 1 the decoupling problem is stated and some results relevant to our discussion are reviewed. In § 2 we analyse the effect of an integrator decoupling feedback law (IDF) upon the controllability and the observability of the system under consideration, showing that observability is preserved under IDF if and only if $. Furthermore it is shown that if a decouplable system satisfies this condition, then it is controllable and observable. In § 3 we give our main result : for the class Ω of decouplable systems which preserve observability under IDF, the set of integers {di } ; i = 1, 2,…, m is a complete set of independent invariants under feedback and similarity transformations. Finally a canonical form (unique) is given for the elements of Ω, completely determined by the set {di } ; i = 1,…, m.