Narrow Results

In this paper, we classify the gradings up to equivalence on the filiform Lie superalgebras $L_{m,n}$ over an algebraically closed field of characteristic 0. Moreover, by the relation between a grading and a standard grading, we describe all pairwise inequivalent gradings and their universal groups of $L_{m,n}$.

A computer-aided diagnosis system was developed for assisting brain astrocytomas malignancy grading. Microscopy images from 140 astrocytic biopsies were digitized and cell nuclei were automatically segmented using a Probabilistic Neural Network pixel-based clustering algorithm. A decision tree classification scheme was constructed to discriminate low, intermediate and high-grade tumours by analyzing nuclear features extracted from segmented nuclei with a Support Vector Machine classifier. Nuclei were segmented with an average accuracy of 86.5%. Low, intermediate, and high-grade tumours were identified with 95%, 88.3%, and 91% accuracies respectively. The proposed algorithm could be used as a second opinion tool for the histopathologists.

We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.

We study commutative associative polynomial operations ${𝔸}^n\times {𝔸}^n\to {𝔸}^n$ with unit on the affine space ${𝔸}^n$ over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

Zonotopal algebra studies pairs of dual algebraic structures that are associated with a linear matroid X and are connected to corresponding dual geometries. Both the geometry and the algebra encode in their statistics combinatorial properties of the matroid. We provide in this paper a general, unified, framework for the chapter in zonotopal algebra that is known as external. The approach is critically based on employing simultaneously the two dual algebraic constructs and invokes the underlying matroidal and geometric structures in an essential way.

An algebra $A$ with a generalized $H$-action is a generalization of an $H$-module algebra where $H$ is just an associative algebra with $1$ and a relaxed compatibility condition between the multiplication in $A$ and the $H$-action on $A$ holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if $A$ is a finite dimensional (not necessarily associative) algebra over a field of characteristic $0$ and $A$ is simple with respect to a generalized $H$-action, then there exists ${lim}_{n\to \infty }\sqrt[n]{c_n^H(A)}\in {ℝ}_{+}$ where ${(c_n^H(A))}_{n=1}^{\infty }$ is the sequence of codimensions of polynomial $H$-identities of $A$. In particular, if $A$ is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists ${lim}_{n\to \infty }\sqrt[n]{c_n^{\mathrm{\text{gr}}}(A)}\in {ℝ}_{+}$ where ${(c_n^{\mathrm{\text{gr}}}(A))}_{n=1}^{\infty }$ is the sequence of codimensions of graded polynomial identities of $A$. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized $H$-actions.

The aim of the paper is to introduce a new class of rings — the inner generalized Weyl algebras (IGWA) — and to give simplicity criteria for them. For each IGWA $A$ a derivative series of IGWAs, $A\to A^{\prime }\to A^{″}\to \cdots \to A^{(\alpha )}\to \cdots$, is attached where $\alpha$ is an arbitrary ordinal. In general, all rings $A^{(\alpha )}$ are distinct. A new construction of rings, the inner $(\sigma ,\tau ,a)$-extension of a ring, is introduced (where $\sigma$ and $\tau$ are endomorphisms of a ring $D$ and $a\in D$).

Let UT_n be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UT_n by a finite abelian group G commuting with an involution (involution gradings).

We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either charF = 0 or charF > dimA, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.

In the study of the structure of graded algebras (such as graded ideals, graded subspaces, and radicals) or graded polynomial identities, the grading group can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G; however, it turns out that there is one distinguished group among them, called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support, and show that the universal grading group functor has neither left nor right adjoint.

Due to the pivotal role of environmental objectives, targets and indicators in Strategic Environmental Assessment (SEA) to assess environmental impact and monitor environmental condition resulting from a proposed plan it is crucial to ensure high quality in their development. Here we present a quality assurance checklist for practitioners to ensure compliance with (i) the minimum requirements of the SEA Directive and (ii) guidelines established by the Irish government for SEA of land use development plans. Furthermore, we propose additional recommended tasks based on expertise and experience gained in carrying out SEA to ensure high quality. Some of the questions posed in the checklist include, for example, "Are the environmental indicators capable of demonstrating the likely significant environmental impact(s) of the implementation of the proposed plan or programme?", "Have limits and thresholds been established for each environmental target?" and "Is responsibility for implementing the monitoring programme clearly defined?". We applied the completed checklist to 10 environmental reports from Ireland, Wales, England, Malta and Germany and proposed a grading system for comparative purposes. The results showed varying quality among the ERs from Ireland, UK and Malta. In addition, we found that the checklist did not directly apply to the German ERs because of the different approach taken to SEA whereby environmental targets and indicators were used in a more implicit manner than elsewhere. However, as environmental objectives and the monitoring programmes were similar for all ERs we concluded that different approaches still resulted in a manageable and focused monitoring system.

Early diagnosis of cancer continues to be a major challenge in the field of cancer prevention and management as it can decrease the severity and also improve the life of patients. Rapid progress in the past few decades in molecular- and immunological-based methods has helped to alleviate the problem to a certain extent. However, a rapid, reagent free method to identify cancers in situ or ex vivo, to make screening objective and swift still remains unachieved. At this point, the utilization of methods based on IR spectroscopy is expected to provide the breakthrough and make diagnosis and grading of cancers/malignancies a simple and inexpensive affair. In the present chapter, we deal with the utilization of FTIR-microspectroscopy for the diagnosis of colon cancer, which is a preliminary step towards its implications for future oncology. We highlight a few technological aspects and also the enormous potential of IR spectroscopy in colon cancer specifically, and other cancers in general.

An experimental machine vision system was designed and implemented for automatic external quality inspection and grading of apples with diameters from 50 mm to 100 mm in a speed of 6 apples per second. A novel roller conveyor was designed to convey and rotate apples regardless of size with an approximately equal rotational speed. During the apples were conveyed literally through the camera's field of view by the roller conveyor, each apple was captured 6 images, and rolled approximately 60 degrees between the exposures of each frame, and got one complete revolution in the camera's field of view.