Remarks on the Liechti-Strenner's examples having small dilatations

Platonic maps of low genus

Algorithm for obtaining characteristic polynomial (CP) coefficients of an alternant edge-weighted cycle is used to arrive at the algorithm for that of the cycloparaphenylene (CPP) graphs in matrix product form. The algorithm gives a recursive relation in expressing the sum of the CP coefficients of a CPP in terms of that of its two immediately preceding analogues which ultimately ends up with the use of transfer matrix in an analytical form. The sum of CP coefficients, being combinatorial in nature, is found to be used as a topological index showing much similarity with Hosoya index (sum all matching polynomial coefficients), cardinality and number of Kekulé valence structures of CPP graphs compared to the Wiener index which is the distance sum of all pairs of vertices in the graph. The sum of CP coefficients has been found to model the physical properties like strain energy and diameter of CPPs that are verified by the respective excellent correlations.

Analysis of distributed lag Statistics is a branch of science that discuss the case of time series data. The method can be used in a distributed lag analysis there are two Koyck Method and Almon Method. At Koyck Method of regression coefficients are assumed to have the same sign and decreases geometrically. In this method there is the dependent variable at a time ago as the independent variable so that the equation is autoregressive. By using this method to analyze the effect of the exchange rate against IHSG. Data used were 35 data. The results showed that the regression coefficient does not decrease geometrically, so that Koyck Method can not be used to resolve this case. To resolve this case, the used Almon Method. In the Almon Method assumed regression coefficient can be approximated by a polynomial has degree. Before applying this method to be specified maximum length of lag and the degree polynomial. To determine performed several experiments using lag length and degree polynomial different. Through these experiments the best results are obtained with a lag of three and a maximum length of second degree polynomials. The results indicate that the effect of the exchange rate against IHSG inversely. Keywords : Distributed Lag, Koyck Method, Almon Method, Autoregressive, Lag, Polynomial

Stocks are securities that can be bought or sold by individuals or institutions as a sign of participating or possessing a company in the amount of its proportions. From the lens of market capitalization values, stocks are divided into 3 groups: large capitalization (Big-Cap), medium capitalization (Mid-Cap) and small capitalization (Small-Cap). Longitudinal data is observation which is conducted as n subjects that are independent to each subject observed repeatedly in different periods dependently. Smoothing technique used to estimate the nonparametric regression model in longitudinal data is local polynomial estimator. Local polynomial estimator can be obtained by WLS (Weighted Least Square) methods. Local polynomial estimator is very dependent on optimal bandwidth. Determination of the optimal bandwidth can be obtained by using GCV (Generalized Cross Validation) method. Among the Gaussian kernel, Triangle kernel, Epanechnikov kernel and Biweight kernel, it is obtained the best model using Gaussian kernel. Based on the application of the model simultaneously, it is obtained coefficient of determination of 97,80174% and MSE values of 0,03053464. Using Gaussian kernel, MAPE out sample of data is obtained as 11,74493%. Keywords: Longitudinal Data, Local Polynomial, Stocks

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard'' on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs.

This thesis studies exponential time algorithms that give optimum solutions to optimization problems which are highly unlikely to be solvable to optimality in polynomial time. Such algorithms are designed and upper bounds on their running times are analyzed, for pro-blems from graph theory, constraint satisfaction, and computational biology. It is studied how restrictions on one computationally hard parameter affect the running time of algorithms for computing another parameter. Algorithms are obtained to compute minimum dominating sets, minimum bandwidth and topological bandwidth layouts, and other hard parameters, in time exponential only in the maximum number of leaves in a spanning tree of a graph. For sparse graphs, algorithms find minimum connected dominating sets in time exponential only in the optimum value. These algorithms are based on kernels of small size, meaning that arbitrary instances are polynomial-time compressible to equivalent instances whose size depends on the parameter alone. Lower and upper bounds on the number and size of certain combinatorial objects are proved. For minimal feedback vertex sets in tournaments, improved bounds on their number and a polynomial-space polynomial-delay algorithm for their enumeration are given. For subcubic planar graphs, fast algorithms finding large induced matchings are given. Fixed-parameter algorithms are presented for maximization problems whose optimum solution value grows as an unbounded function with the instance size. Polynomial kernels are obtained for all permutation constraint satisfaction problems with ternary constraints, and independent set problems in restricted graph classes, when these problems are parameterized above tight lower bounds on their solution value. Polynomial space algorithms with subexponential time requirement are obtained for the minimum triplet inconsistency problem in phylogenetics, and basically all bidimensional problems in sparse graphs such as graphs excluding a fixed minor.

The motivation to write this paper arose out of the following problem which was posed in a recent mathematical olympiad: Given a polynomial P(X) with integer coefficients, show that there exist non-zero polynomials Q(X), R(X) with integer coefficients such that P(X)Q(X) is a polynomial in X 2 and P(X)R(X) is a polynomial in X 3 . For instance, if P(X) = 2 − 5X + 3X 2 + 12X 3 , then we notice that Q (X) = 2 + 5X + 3X 2 − 12X 3 serves the purpose for the first part, viz. A moment's thought makes it fairly evident that this trick easily solves the first part of the problem for a general polynomial P(X) .

In this book the authors introduce the new notions of subset polynomial semirings and subset matrix semirings. Solving subset polynomial equations is an interesting exercise. Open problems about the solution set of subset polynomials are proposed.

Generalized Polynomial Chaos (gPC) is known to fail for problems involving strong nonlinear dependencies on stochastic inputs, which especially arise in the context of long term integration or stochastic discontinuities. There are various attempts in the literature which address these difficulties, such as the time-dependent generalized Polynomial Chaos (TD-gPC) and the multi-element generalized Polynomial Chaos (ME-gPC) both leading to higher accuracies but higher numerical costs in comparison to the classical gPC approach. A combination of these methods is introduced, which leads to a powerful solution method since high accuracies can be maintained and computational cost can be distributed by utilizing parallel computation. However, to be able to apply the hybrid method to all types of ordinary differential equations subject to random inputs, new modifications with respect to TD-gPC are carried out by creating an orthogonal tensor basis consisting of the random input variable as well as the solution itself. Such modifications allow TD-gPC to capture the dynamics of the solution by increasing the approximation quality of its time derivatives.

The novel contributions of this paper are twofold. First, we demonstrate how to characterize unweighted graphs in a permutation-invariant manner using the polynomial coefficients from the Ihara zeta function, i.e., the Ihara coefficients. Second, we generalize the definition of the Ihara coefficients to edge-weighted graphs. For an unweighted graph, the Ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph. Since the Ihara zeta function has poles that give rise to infinities, the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial. Moreover, the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph. To generalize the representation to edge-weighted graphs, we make use of the reduced Bartholdi zeta function. We prove that the computation of the Ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights. We also present a spectral analysis of the Ihara coefficients and indicate their advantages over other graph spectral methods. We apply the proposed graph characterization method to capturing graph-class structure and clustering graphs. Experimental results reveal that the Ihara coefficients are more effective than methods based on Laplacian spectra.

Some results on the signless Laplacians of graphs

This paper considers three separate matrices associated to graphs and (each dimension of) cell complexes. It relates all the coefficients of their respective characteristic polynomials to the geometric and combinatorial enumeration of three kinds of sub-objects. The matrices are: the mesh matrix for integral d-cycles, the mesh matrix for integral d-boundaries, and the Kirchhoff matrix, i.e., the combinatorial Laplacian, for integral $$(d-1)$$ -chains. Trent’s theorem states that the determinant of the mesh matrix on 1-cycles of a connected graph is equal to the number of spanning trees (Trent in Proc Nat Acad Sci USA 40:1004–1007, 1954; J Acoust Soc Am 27(3):500–527, 1955). Here this theorem is extended to the mesh matrix on d-cycles in an arbitrary cell complex and, new even for graphs, to enumerative combinatorial interpretation of all of the coefficients of its characteristic polynomial. This last is well defined once a basis for the integral d-cycles is chosen. Additionally, a parallel result for the mesh matrix for integral d-boundaries is proved. Kirchhoff’s classical theorem on graphs (Kirchhoff in Ann Phys Chem 72:497–508, 1847) states that the product of the non-zero eigenvalues of the Kirchhoff matrix, i.e., combinatorial Laplacian, for connected graphs equals n times the number of spanning trees, with n the number of vertices. Kalai (Isr J Math 45(4): 337–351, 1983) investigated counting spanning trees in some higher dimensional settings including simplices here weighting by order of torsion homology groups entered. Lyons has generalized Kirchhoff’s result on the product of the non-zero eigenvalues of the Kirchhoff or combinatorial Laplacian on $$(d-1)$$ -chains to cell complexes for $$d >1$$ (Lyons in J Topol Anal 1(2), 153–175, 2009; Lyons and Peres in Probability on trees and networks. Cambridge series in statistical and probabilistic mathematics, vol 42, Cambridge University Press, New York, 2016). The present analysis extends this to all coefficients of the complete characteristic polynomial. An evaluation of the Reidemeister–Franz torsion of the cell complex with respect to its integral basis gives relations between these combinatorial invariants.

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve. This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids. When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Many interesting properties of polynomials are closely related to the geometry of their Newton polytopes. In this dissertation thesis, we analyze the growth properties of real multivariate polynomials in terms of their so-called Newton polytopes at infinity. We show some applications of our results, relate them to the existing literature and illustrate them with several examples.

The family of q-Konhauser matrix polynomials have been extended to Konhauser matrix polynomials. The purpose of the present work is to show that an extension of the explicit forms, generating matrix functions, matrix recurrence relations and Rodrigues-type formula for these matrix polynomials are given, our desired results have been established and their applications are presented.