scholarly journals The Arithmetic Tutte polynomial of two matrices associated to Trees

2018 ◽  
Vol 6 (1) ◽  
pp. 310-322
Author(s):  
R. B. Bapat ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Analogous Result ◽  
Polynomial Matrix ◽  
Tutte Polynomial ◽  
Distance Matrix ◽  
Minimum Volume ◽  
Maximum Volume ◽  
Recent Interest ◽  
Minimum Volume Ellipsoid ◽  
Tutte Polynomials ◽  
Exponential Distance

Abstract Arithmetic matroids arising from a list A of integral vectors in Zn are of recent interest and the arithmetic Tutte polynomial MA(x, y) of A is a fundamental invariant with deep connections to several areas. In this work, we consider two lists of vectors coming from the rows of matrices associated to a tree T. Let T = (V, E) be a tree with |V| = n and let LT be the q-analogue of its Laplacian L in the variable q. Assign q = r for r ∈ ℤ with r/= 0, ±1 and treat the n rows of LT after this assignment as a list containing elements of ℤn. We give a formula for the arithmetic Tutte polynomial MLT (x, y) of this list and show that it depends only on n, r and is independent of the structure of T. An analogous result holds for another polynomial matrix associated to T: EDT, the n × n exponential distance matrix of T. More generally, we give formulae for the multivariate arithmetic Tutte polynomials associated to the list of row vectors of these two matriceswhich shows that even the multivariate arithmetic Tutte polynomial is independent of the tree T. As a corollary, we get the Ehrhart polynomials of the following zonotopes: - ZEDT obtained from the rows of EDT and - ZLT obtained from the rows of LT. Further, we explicitly find the maximum volume ellipsoid contained in the zonotopes ZEDT, ZLT and show that the volume of these ellipsoids are again tree independent for fixed n, q. A similar result holds for the minimum volume ellipsoid containing these zonotopes.

scholarly journals Allochthonous sediment in till near a lithological boundary in central Ontario

2002 ◽  
Vol 51 (1) ◽  
pp. 17-27 ◽  
Author(s):  
J. Graham Cogley ◽  
M. Aikman ◽  
D. J. A. Stokes
Keyword(s):  
Distance Scale ◽  
Sand Fraction ◽  
Precambrian Rocks ◽  
Gradual Loss ◽  
Intimate Mixture ◽  
Spatially Homogeneous ◽  
Exponential Distance

Abstract Clast counts, and measurements of carbonate abundance in the sand fraction, show that little of the till covering a portion of central Ontario was carried across the boundary between Precambrian rocks (up-ice) and Palaeozoic limestone (downice). Seven eighths of the pebble fraction is local, from within ~2-5 km of the site of deposition. The distantly-derived component becomes gradually less abundant down-ice from the lithological boundary, with an exponential distance scale of about 30 km. We ascribe this gradual loss to a combination of comminution and depletion by deposition.However it is not possible to map variations in the abundance of erratics; the pattern is spatially homogeneous and random. The same is true of the abundance of insoluble sand which, moreover, is highly variable.The role of transport of allochthonous sand from the Shield cannot be separated from those of comminution, erosion of local bedrock, postglacial alteration and inheritance of pre-glacial material. The till is best understood as an intimate mixture of "frictional gouge", still more or less in situ , and englacially-transported sediment. Thus it is neither lodgement till nor meltout till; it may be a deformation till, but if so the episode of deformation can have lasted only a few hundred years

PYTHAGOREAN FUZZY MULTI SET AND ITS APPLICATIONS IN FISH FEED FOR INDIAN MAJOR CARP

2020 ◽  
Vol 9 (11) ◽  
pp. 9803-9811
Author(s):  
R. Sophia Porchelvi ◽  
V. Jayapriya
Keyword(s):  
Decision Making ◽  
Fuzzy Set ◽  
Fish Feed ◽  
Uncertain Environments ◽  
Pythagorean Fuzzy Set ◽  
Indian Major Carp ◽  
Set Operations ◽  
Basic Set ◽  
Decision Making Problem ◽  
Exponential Distance

Pythagorean fuzzy set is an extension of Intutionistic fuzzy set, which is more capable of expressing and handling the uncertainty under uncertain environments, so that it was broadly applied in various fields. In this paper, we explored the concept of Pythagorean fuzzy multi set (PFMS). We describe some basic set operations of Pythagorean fuzzy multi set and also, we proposed sine exponential distance function. Finally, through an illustrative example it is shown how the proposed distance works in decision-making problem.

RNA-Mediated Electron Transfer: Double Exponential Distance Dependence

2009 ◽  
Vol 131 (37) ◽  
pp. 13188-13189 ◽  
Author(s):  
Kenji Maie ◽  
Kazuyuki Miyagi ◽  
Tadao Takada ◽  
Mitsunobu Nakamura ◽  
Kazushige Yamana
Keyword(s):  
Electron Transfer ◽  
Double Exponential ◽  
Mediated Electron Transfer ◽  
Distance Dependence ◽  
Exponential Distance

scholarly journals Clustering Longitudinal Life-Course Sequences using Mixtures of Exponential-Distance Models

2019 ◽  
Author(s):  
Keefe Murphy ◽  
Brendan Murphy ◽  
Raffaella Piccarreta ◽  
Isobel Claire Gormley
Keyword(s):  
Life Course ◽  
Hamming Distance ◽  
Sequence Data ◽  
Clustering Algorithms ◽  
Model Fitting ◽  
Northern Irish ◽  
Sampling Weights ◽  
Data Set ◽  
Model Based Clustering ◽  
Exponential Distance

Sequence analysis is an increasingly popular approach for the analysis of life courses represented by an ordered collection of activities experienced by subjects over a given time period. Several criteria exist for measuring pairwise dissimilarities among sequences. Typically, dissimilarity matrices are employed as input to heuristic clustering algorithms, with the aim of identifying the most relevant patterns in the data.Here, we propose a model-based clustering approach for categorical sequence data. The technique is applied to a survey data set containing information on the career trajectories of a cohort of Northern Irish youths tracked between the ages of 16 and 22.Specifically, we develop a family of methods for clustering sequences directly, based on mixtures of exponential-distance models, which we call MEDseq. The use of the Hamming distance or weighted variants thereof as the distance metrics permits closed-form expressions for the normalising constant, thereby facilitating the development of an ECM algorithm for model fitting. Additionally, MEDseq models allow the probability of component membership to depend on fixed covariates. Sampling weights, which are often associated with life-course data arising from surveys, are also accommodated. Simultaneously including weights and covariates in the clustering process yields new insights on the Northern Irish data.

On the exponential distance effect

1976 ◽  
Vol 10 (6) ◽  
pp. 357-358 ◽  
Author(s):  
Martin J. Beckmann
Keyword(s):  
Distance Effect ◽  
Exponential Distance
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