The h-vector of coned graphs

AbstractThe coned graph Gˆ on a finite graph G is obtained by joining each vertex of G to a new vertex p with a simple edge. In this work we show a combinatorial interpretation of each term in the h-vector of Gˆ in terms of partially edge-rooted forests in the base graph G. In particular, our interpretation does not require edge ordering. For an application, we will derive an exponential generating function for the sequence of h-polynomials for the complete graphs. We will also give a new proof for the number of spanning trees of the wheels

Weighted Tree-Numbers of Matroid Complexes

International audienceWe give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le $\beta$;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes

A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube and the Crapo $\beta$-invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified and condensed. Final version to appear in J. Combin. Theory Ser.

Weighted Tree-Numbers of Matroid Complexes

We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes

Simplicial effective resistance and enumeration of spanning trees

A graph can be regarded as an electrical network in which each edge is a resistor. This point of view relates combinatorial quantities, such as the number of spanning trees, to electrical ones such as effective resistance. The second and third authors have extended the combinatorics/electricity analogy to higher dimension and expressed the simplicial analogue of effective resistance as a ratio of weighted tree enumerators. In this paper, we first use that ratio to prove a new enumeration formula for color-shifted complexes, confirming a conjecture by Aalipour and the first author, and generalizing a result of Ehrenborg and van Willigenburg on Ferrers graphs. We then use the same technique to recover an enumeration formula for shifted complexes, first proved by Klivans and the first and fourth authors. In each case, we add facets one at a time, and give explicit expressions for simplicial effective resistances of added facets by constructing high-dimensional analogues of currents and voltages (respectively homological cycles and cohomological cocycles).Comment: 27 pages, minor revisions from v

A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube

HOW CAN WE TEACH STUDENT TO ESTIMATE VERTICAL JUMP HEIGHTS USING GROUND REACTION FORCE DATA

The purpose of this study was to estimate vertical jump heights using ground reaction force (GRF) data and to suggest one practical example of biomechanical theory application to a real human motion. Vertical jump heights of impulse and flight time method were statistically smaller than three-dimensional video method. The causes of height differences seemed mainly from the fact that impulse was used to move jumper into the horizontal direction as well as into the vertical direction. Other important factors for accurate height calculation are jumper's mass and threshold value of GRF data collection. Vertical jump height calculation with GRF data showed an example of practical application of biomechanical theory to human motion and demonstrated a way of GRF equipment use for effective biomechanical theory education

Co-occurrence matrix analysis-based semi-supervised training for object detection

One of the most important factors in training object recognition networks using convolutional neural networks (CNNs) is the provision of annotated data accompanying human judgment. Particularly, in object detection or semantic segmentation, the annotation process requires considerable human effort. In this paper, we propose a semi-supervised learning (SSL)-based training methodology for object detection, which makes use of automatic labeling of un-annotated data by applying a network previously trained from an annotated dataset. Because an inferred label by the trained network is dependent on the learned parameters, it is often meaningless for re-training the network. To transfer a valuable inferred label to the unlabeled data, we propose a re-alignment method based on co-occurrence matrix analysis that takes into account one-hot-vector encoding of the estimated label and the correlation between the objects in the image. We used an MS-COCO detection dataset to verify the performance of the proposed SSL method and deformable neural networks (D-ConvNets) as an object detector for basic training. The performance of the existing state-of-the-art detectors (DConvNets, YOLO v2, and single shot multi-box detector (SSD)) can be improved by the proposed SSL method without using the additional model parameter or modifying the network architecture.Comment: Submitted to International Conference on Image Processing (ICIP) 201