Using combinatorial methods we study the structural coefficients of the formal homogeneous universal enveloping algebra (U) over cap (h)(sl(2)) of the special linear algebra sl(2) over a characteristic zero field. We ...
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Using combinatorial methods we study the structural coefficients of the formal homogeneous universal enveloping algebra (U) over cap (h)(sl(2)) of the special linear algebra sl(2) over a characteristic zero field. We provide explicit formulae for the product of generic elements in (U) over cap (h)(sl(2)), and construct combinatorial objects giving flesh to these formulae;in particular, we provide explicit formulae and combinatorial interpretations for the structural coefficients of divided power Poincare-Birkhoff-Witt basis.
We discuss a simple toy model which allows, in a natural way, for deriving central facts from thermodynamics such as its fundamental laws, including Carnot's version of the second principle. Our viewpoint represen...
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ISBN:
(数字)9781665483414
ISBN:
(纸本)9781665483414
We discuss a simple toy model which allows, in a natural way, for deriving central facts from thermodynamics such as its fundamental laws, including Carnot's version of the second principle. Our viewpoint represents thermodynamic systems as binary strings, and it links their temperature to their Hamming weight. From this, we can reproduce the possibility of negative temperatures, the notion of equilibrium as the coincidence of two notions of temperature - statistical versus structural -, as well as the zeroth law of thermodynamics (transitivity of the thermal-equilibrium relation), which we find to be redundant, as other authors, yet at the same time not to be universally valid.
Motivated by the combinatorial properties of the protein-design problem and the specific and non-specific interactions in biomolecular systems, we build exactly-solved models for the statistical physics of the symmetr...
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Motivated by the combinatorial properties of the protein-design problem and the specific and non-specific interactions in biomolecular systems, we build exactly-solved models for the statistical physics of the symmetric group, permutation glasses, and the self-assembly of dimer systems. The first two models are studied for their statistical physics properties apart from the motivating system, and the third model is used to better understand the constraints of correct dimerization in biomolecular systems. These models are exactly-solved in the sense that the sum-over-states defining their partition functions can be reduced to analytically more tractable expressions, and unlike most exactly-solved models in statistical physics whose motivations lie in condensed matter scenarios, these models are found by abstractly considering the combinatorial properties of biomolecular systems. This work suggests that there is a class of interesting but unexplored models in the statistical physics of biomolecules. We conclude by suggesting extensions to our presented models and starting points for new ones.
The work is intended to combinatorial properties of filters on natural numbers as an introduction and motivation to the problematics between definability of the filters and its combinatorics. Basic filter types: P-fil...
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The work is intended to combinatorial properties of filters on natural numbers as an introduction and motivation to the problematics between definability of the filters and its combinatorics. Basic filter types: P-filter, Q-filter, Rapid filter; orders: Rudin-Kiesler, Rudin-Blass, Katětov and Tukey; filter kon- structions; basic definitions related to combinatorics on ω; introduction to basic descriptive set theory and topology and some specific results. 1
In recent decades, tropical mathematics gradually evolved as a field of study in mathematics and it has more and more interactions with other fields and applications. In this thesis we solve combinatorial and computat...
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In recent decades, tropical mathematics gradually evolved as a field of study in mathematics and it has more and more interactions with other fields and applications. In this thesis we solve combinatorial and computational problems concerning implicitization, linear systems and phylogenetic trees using tools and results from tropical mathematics. In Chapter 1 we briefly introduce basic aspects of tropical mathematics, including tropical arithmetic, tropical polynomials and varieties, and the relationship between tropical geometry and phylogenetic trees. In Chapter 2 we solve the implicitization problem for almost-toric hypersurfaces using results from tropical geometry. With results about the tropicalization of varieties, we present and prove a formula for the Newton polytopes of almost-toric hypersurfaces. This leads to a linear algebra approach for solving the implicitization problem. We implement our algorithm and show that it has better solving ability than some existing methods. In Chapter 3 we compute linear systems on metric graphs. In algebraic geometry, linear systems on curves have been well-studied in the literature. In comparison, linear systems on metric graphs are their counterparts in tropical mathematics. We introduce the anchor divisors and anchor cells in order to compute the polyhedral cell complex structure of the linear system |D| and the tropical semi-module R(D). We develop algorithms, discuss their efficiency and present computational results for canonical linear systems on metric graphs. In Chapter 4 we extend the previous chapter by introducing the tropical Hodge bundle, which is the counterpart of the Hodge bundle in tropical geometry. We study properties of tropical Hodge bundles and present computational results for some examples. In Chapter 5 we study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The CAT(0)-metric of Billera-H
We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the n-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use o...
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We illustrate a rigorous approach to express the totally symmetric isotropic tensors of arbitrary rank in the n-dimensional Euclidean space as a linear combination of products of Kronecker deltas. By making full use of the symmetries, one can greatly reduce the efforts to compute cumbersome angular integrals into straightforward combinatoric counts. This method is generalised into the cases in which such symmetries are present in subspaces. We further demonstrate the mechanism of the tensor-integral reduction that is widely used in various physics problems such as perturbative calculations of the gauge-field theory in which divergent integrals are regularised in d = 4 - 2 epsilon space-time dimensions. The main derivation is given in the ndimensional Euclidean space. The generalisation of the result to the Minkowski space is also discussed in order to provide graduate students and researchers with techniques of tensor-integral reduction for particle physics problems.
This thesis is concerned with the connection between Lie algebras with multiple brackets and the topology of partially ordered sets. From a partially ordered set (poset) one obtains a simplicial complex, called the or...
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This thesis is concerned with the connection between Lie algebras with multiple brackets and the topology of partially ordered sets. From a partially ordered set (poset) one obtains a simplicial complex, called the order complex, whose faces are the chains of the poset. There is a long tradition of using topological properties of the order complex to study various geometric and algebraic structures. (Abstract shortened by UMI.)
In this paper, we explore some of the methods that are often used to solve combinatorial problems by proving Cayley’s theorem on trees in multiple ways. The intended audience of this paper is undergraduate and gradua...
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In this paper, we explore some of the methods that are often used to solve combinatorial problems by proving Cayley’s theorem on trees in multiple ways. The intended audience of this paper is undergraduate and graduate mathematics students with little to no experience in combinatorics. This paper could also be used as a supplementary text for an undergraduate combinatorics course.
Polymer physics models suggest that chromatin spontaneously folds into loop networks with transcription units (TUs), such as enhancers and promoters, as anchors. Here we use combinatoric arguments to enumerate the eme...
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Polymer physics models suggest that chromatin spontaneously folds into loop networks with transcription units (TUs), such as enhancers and promoters, as anchors. Here we use combinatoric arguments to enumerate the emergent chromatin loop networks, both in the case where TUs are labeled and where they are unlabeled. We then combine these mathematical results with those of computer simulations aimed at finding the inter-TU energy required to form a target loop network. We show that different topologies are vastly different in terms of both their combinatorial weight and energy of formation. We explain the latter result qualitatively by computing the topological weight of a given network—i.e., its partition function in statistical mechanics language—in the approximation where excluded volume interactions are neglected. Our results show that networks featuring local loops are statistically more likely with respect to networks including more nonlocal contacts. We suggest our classification of loop networks, together with our estimate of the combinatorial and topological weight of each network, will be relevant to catalog three-dimensional structures of chromatin fibers around eukaryotic genes, and to estimate their relative frequency in both simulations and experiments.
Marsdenia tenacissima is a traditional Chinese medicinal plant used for treating cancer, and its main medicinal part is the stem. Considering the resource shortage of M. tenacissima, it is of great significance to imp...
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Marsdenia tenacissima is a traditional Chinese medicinal plant used for treating cancer, and its main medicinal part is the stem. Considering the resource shortage of M. tenacissima, it is of great significance to improve its utilization efficiency. Steroids and caffeoylquinic acids, the two main components of M. tenacissima, are composed of several basic structures. Based on this rule, a novel strategy of combinatorics-based chemical characterization was proposed to analyze the constituents of roots, stems and leaves of M. tenacissima. combinatorics was used to generate a compound library for structure alignment, which has the advantages of wide coverage and high specificity. Steroids are composed of four basic parts: core skeleton (C), substituent at position 11 (A), substituent at position 12 (B) and sugar moiety (S). Based on combinatorics, a compound library consisting of 1080 steroids was generated. Diagnostic neutral loss has been used to effectively predict the sub-stituents at position 11 and 12 of steroids, including acetyl, 2-methylpropionyl, tigloyl, 2-methylbutyryl and benzoyl. As a result, 131, 131 and 99 components were detected from the roots, stems and leaves of M. tenacissima, respectively. Principal component analysis (PCA) was used to analyze the differences of roots, stems and leaves, and orthogonal partial least squares-discriminant analysis (OPLS-DA) was further applied to find differential components. Tenacissoside H, a critical indicator component for quality evaluation of the stem, has been proved to be a differential component between roots and stems. Notably, the relative content of tenacissoside H in the roots was significantly higher than that in the stems. The bioactivity comparison showed that roots, stems and leaves of M. tenacissima had similar scavenging activity on 1,1-diphenyl-2-picrylhydrazyl (DPPH) radical. However, their a-glucosidase inhibitory activity was ranked as leaves > stems > roots. There-fore, besides stems, the othe
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