Cartan matrices and quasi-Cartan matrices play an important role in such areas as Lie theory, representation theory, and algebraic graph theory. It is known that each (connected) positive definite ...quasi-Cartan matrix A\in \mathbb{M}_n(\mathbb{Z}) is \mathbb{Z}-equivalent with the Cartan matrix of a Dynkin diagram, called the Dynkin type of A. We present a symbolic, graph-theoretic algorithm to compute the Dynkin type of A, of the pessimistic arithmetic (word) complexity \mathcal {O}(n^2), significantly improving the existing algorithms. As an application we note that our algorithm can be used as a positive definiteness test for an arbitrary quasi-Cartan matrix, more efficient than standard tests. Moreover, we apply the algorithm to study a class of (symmetric and non-symmetric) quasi-Cartan matrices related to Nakayama algebras.
World Class Manufacturing system consists of ten technical and ten managerial pillars. These, impacting directly and indirectly on each other, generate the flow of internal processes. Two of the ...mentioned pillars, Early Product Management (EPM) and Cost Deployment (CD) play a special role in the system, because they create a future strategic management of a company influencing design engineering, manufacturing and economy 1, 2. Referring to the author’s previous publications on Early Product Management methodology 3, 4, the role of Cost Deployment pillar in the new product launch remains an important issue. Additionally, there is a noticeable lack of publications in this specific field of the WCM system. Therefore, a proper understanding of the relationship between these two technical pillars is the basis for effective project management for the implementation of new products. In this article, the correlation between EPM and CD will be highlighted whereas some critical remarks will be indicated. The main part of the article will describe: the current approach to project management according to the standards set by the WCM system and recommended improvements originated from EPM and CD pillars. The quality scientific methods used in this article are based on a case study of internal processes in an international plant specializing in agriculture machinery production and include elements of direct observation and theoretical analysis and synthesis. This paper refers to the presented issues in practical terms on the example of the methodology of managing of new launch product projects in terms of cost management. The purpose of this paper is to draw attention to the problem of the cost factor generated during the design phase and early implementation of the new product into production, which will enable effective cost management of new implementation projects.
The modern companies, which are competing on product market, need to use innovative solutions, in order to become potential leaders. One of the modernization methods is rearrangement of ...organizational structure and redistribution of competence. The article describes the Advanced Manufacturing Engineering Department in production plant, which is an innovative initiative in worldwide organizational management. Some aspects including AME application in plant processes are highlighted. Some advanced techniques are presented. In the article summary, perspectives for the development of AME are included.
Approximately 90% of colorectal cancer (CRC) deaths are caused by tumors ability to migrate into the adjacent tissues and metastase into distant organs. More than 40 genes have been causally linked ...to the development of CRC but no mutations have been associated with metastasis yet. To identify molecular basis of CRC metastasis we performed whole-exome and genome-scale transcriptome sequencing of 7 liver metastases along with their matched primary tumours and normal tissue. Multiple, spatially separated fragments of primary tumours were analyzed in each case. Uniformly malignant tissue specimen were selected with macrodissection, for three samples followed with laser microdissection.
> 100 sequencing coverage allowed for detection of genetic alterations in subpopulation of tumour cells. Mutations in KRAS, APC, POLE, and PTPRT, previously associated with CRC development, were detected in most patients. Several new associations were identified, including PLXND1, CELSR3, BAHD1 and PNPLA6.
We confirm the essential role of inflammation in CRC progression but question the mechanism of matrix metalloproteinases activation described in other work. Comprehensive sequencing data made it possible to associate genome-scale mutation distribution with gene expression patterns. To our knowledge, this is the first work to report such link in CRC metastasis context.
We study the complexity of Bongartzs algorithm for determining a maximal common direct summand of a pair of modules M, N over k-algebra ; in particular, we estimate its pessimistic computational ...complexity (rm6n2(n + m log n)), where m = dimkM n = dimkN and r is a number of common indecomposable direct summands of M and N. We improve the algorithm to another one of complexity (rm4n2(n+m log m)) and we show that it applies to the isomorphism problem (having at least an exponential complexity in a direct approach). Moreover, we discuss a performance of both algorithms in practice and show that the average complexity is much lower, especially for the improved one (which becomes a part of QPA package for GAP computer algebra system).
Coxeter energy of graphs Mróz, Andrzej
Linear algebra and its applications,
10/2016, Letnik:
506
Journal Article
Recenzirano
Odprti dostop
We study the concept of the Coxeter energy of graphs and digraphs (quivers) as an analogue of Gutman's adjacency energy, which has applications in theoretical chemistry and is a recently widely ...investigated graph invariant. Coxeter energy CE(G) of a (di)graph G is defined to be the sum of the absolute values of all complex eigenvalues of the Coxeter matrix associated with G. Our main inspiration for the study comes from the Coxeter formalism appearing in group theory, Lie theory, representation theory of algebras, mathematical physics and other contexts. We focus on the Coxeter energy of trees and we prove that the path (resp. the maximal star) has the smallest (resp. the greatest) Coxeter energy among all trees (resp. two large subclasses of trees) with fixed number of vertices. We provide several other related results, as the characterization of trees with second smallest and second greatest Coxeter energy, bounds for Coxeter energy, and general facts on Coxeter spectra of graphs extending known results e.g. for Salem trees and for certain special real Coxeter eigenvalues of trees. Additionally, we discuss few other energy-like quantities for the Coxeter spectra of (di)graphs, including Coxeter energy “normalized” by the trace of the Coxeter matrix and the quantities derived from variants of Coulson integral formula.
Cartan matrices, quasi-Cartan matrices and associated upper triangular Gram matrices control important combinatorial aspects of Lie theory and representation theory of associative algebras. We ...provide a graph theoretic proof of the fact that the absolute values of the coefficients of a non-negative quasi-Cartan matrix A as well as of its (minimal) symmetrizer D are bounded by 4, and that the analogous bound in case of the associated Gram matrix GˇA is 8. Moreover, we show that D (and GˇA) has at least one diagonal coefficient equal to 1. We describe some other restrictions and interrelations between the coefficients of A, D and GˇA, and the corank and other properties of A relevant in Lie theory. We apply our results to construct an algorithm by which we classify all non-negative quasi-Cartan matrices of small sizes.
Cartan matrices, quasi-Cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like Lie theory, representation theory and algebraic graph theory. ...We study quasi-Cartan matrices by means of the inflation algorithm, an idea used in Ovsienko's classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results. We prove that it is enough to perform linear number of inflations to reduce positive or non-negative principal quasi-Cartan matrix to its canonical form, i.e., to the Cartan matrix of a Dynkin or Euclidean diagram, respectively. Moreover, we show that in the positive case the length of every sequence of inflations has quadratic bound, and that the only other “natural” non-negative class having such universal bound is the class of so-called pos-sincerely principal quasi-Cartan matrices (and in this case the bound is linear). We obtain such low bounds by applying, among others, some new observations on the properties of the reduced root systems (in the sense of Bourbaki) associated with positive quasi-Cartan matrices.
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