arXiv subjects

Charles Johnson

Publications and source records attributed to Charles Johnson.

16 recordsLinked to original sources

arXiv2026

Comments on "Condensation of the digraph associated with a reciprocal matrix and a vector", arXiv:2607.10279

Since manuscript \cite{R}, recently posted on arXiv, is closely related to our previous work, we would like to clarify some facts about it. Namely, we would like to point out that several results in \cite{R} are already known (Theorem 1 and Corollaries 2, 3 and 4) and some observations regarding our work \cite{FJ} are not correct. We also notice that R\'{e}dei's Theorem, a classical result in graph theory, gives a simple tool to show Theorems 5 and 9.

math.CO↗

arXiv2026

Existence of reciprocal matrices with specified orders for the right and inverse left Perron eigenvectors

Here we give a procedure to construct a reciprocal matrix for which the right and entrywise inverse left Perron eigenvectors have any pair of given orders. An explicit example when the matrix is of size 4 is presented. In particular, it gives an afirmative answer to the question posed in a recent manuscript by Boz\'oki and Csat\'o (2026) about the existence of a reciprocal matrix of size 4 such that the right and entrywise inverse left Perron eigenvectors have reverse orders.

math.CO↗

arXiv2025

Completions of pairwise comparison data that minimize the triad measure of inconsistency

We consider incomplete pairwise comparison matrices and determine exactly when they have a consistent completion and, if not, when they have a nearly consistent completion. We use the maximum 3-cycle product as a measure of inconsistency and show that, when the graph of the specified entries is chordal, a completion in which this measure is not increased is always possible. Methodology to produce such completions is developed. Such methodology may also be used to reduce inconsistency with few changes of comparisons.

math.CO↗

arXiv2025

Exact bounds for efficient consistent matrices obtained from a reciprocal matrix

For a given reciprocal matrix A, we give a union of matrix intervals in which any consistent matrix obtained from an efficient vector for A lies, and, conversely, any consistent matrix in this union comes from an efficient vector for A. The maximal sets of entries in the lower and upper bound matrices of each interval that are attainable by some consistent matrix in the interval are described. This allows us to understand which subsets of the alternatives lie above which other subsets in all efficient orders for each interval. As a result, the partial order on the alternatives dictated by the efficient vectors follows. Then, we use the tools developed to also show that, when the n-by-n reciprocal matrices A,B are simple perturbed consistent matrices, or n=4, the sets of efficient vectors for A and B coincide only if A=B.

math.CO↗

arXiv2025

Through Silicon Via Aware Design Planning for Thermally Efficient 3-D Integrated Circuits

3-D integrated circuits (3-D ICs) offer performance advantages due to their increased bandwidth and reduced wire-length enabled by through-silicon-via structures (TSVs). Traditionally TSVs have been considered to improve the thermal conductivity in the vertical direction. However, the lateral thermal blockage effect becomes increasingly important for TSV via farms (a cluster of TSV vias used for signal bus connections between layers) because the TSV size and pitch continue to scale in {\mu}m range and the metal to insulator ratio becomes smaller. Consequently, dense TSV farms can create lateral thermal blockages in thinned silicon substrate and exacerbate the local hotspots. In this paper, we propose a thermal-aware via farm placement technique for 3-D ICs to minimize lateral heat blockages caused by dense signal bus TSV structures.

cs.AR↗

arXiv2024

Efficiency of the convex hull of the columns of certain triple perturbed consistent matrices

In decision making a weight vector is often obtained from a reciprocal matrix A that gives pairwise comparisons among n alternatives. The weight vector should be chosen from among efficient vectors for A. Since the reciprocal matrix is usually not consistent, there is no unique way of obtaining such a vector. It is known that all weighted geometric means of the columns of A are efficient for A. In particular, any column and the standard geometric mean of the columns are efficient, the latter being an often used weight vector. Here we focus on the study of the efficiency of the vectors in the (algebraic) convex hull of the columns of A. This set contains the (right) Perron eigenvector of A, a classical proposal for the weight vector, and the Perron eigenvector of AA^{T} (the right singular vector of A), recently proposed as an alternative. We consider reciprocal matrices A obtained from a consistent matrix C by modifying at most three pairs of reciprocal entries contained in a 4-by-4 principal submatrix of C. For such matrices, we give necessary and sufficient conditions for all vectors in the convex hull of the columns to be efficient. In particular, this generalizes the known sufficient conditions for the efficiency of the Perron vector. Numerical examples comparing the performance of efficient convex combinations of the columns and weighted geometric means of the columns are provided.

math.CO↗

arXiv2024

Efficiency analysis for the Perron vector of a reciprocal matrix

In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is necessary to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. One of the most used ranking methods employs the (right) Perron eigenvector of the reciprocal matrix as the vector of weights. It is known that the Perron vector may not be efficient. Here, we focus on extending arbitrary reciprocal matrices and show, constructively, that two different extensions of any fixed size always exist for which the Perron vector is inefficient and for which it is efficient, with the following exception. If B is consistent, any reciprocal matrix obtained from B by adding one row and one column has efficient Perron vector. As a consequence of our results, we obtain families of reciprocal matrices for which the Perron vector is inefficient. These include known classes of such matrices and many more. We also characterize the 4-by-4 reciprocal matrices with inefficient Perron vector. Some prior results are generalized or completed.

math.CO↗

arXiv2023

Cycle products and efficient vectors in reciprocal matrices

We focus upon the relationship between Hamiltonian cycle products and efficient vectors for a reciprocal matrix $A$, to more deeply understand the latter. This facilitates a new description of the set of efficient vectors (as a union of convex subsets), greater understanding of convexity within this set and of order reversals in efficient vectors. A straightforward description of all efficient vectors for an $n$-by-$n$, column perturbed consistent matrix is given; it is the union of at most $(n-1)$ choose $2$ convex sets.

math.CO↗

arXiv2023

Efficient vectors for block perturbed consistent matrices

In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is important to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. Recently a method to generate inductively all efficient vectors for any reciprocal matrix has been discovered. Here we focus upon the study of efficient vectors for a reciprocal matrix that is a block perturbation of a consistent matrix in the sense that it is obtained from a consistent matrix by modifying entries only in a proper principal submatrix. We determine an explicit class of efficient vectors for such matrices. Based upon this, we give a description of all the efficient vectors in the 3-by-3 block perturbed case. In addition, we give sufficient conditions for the right Perron eigenvector of such matrices to be efficient and provide examples in which efficiency does not occur. Also, we consider a certain type of constant block perturbed consistent matrices, for which we may construct a class of efficient vectors, and demonstrate the efficiency of the Perron eigenvector. Appropriate examples are provided throughout.

math.CO↗

arXiv2023

The complete set of efficient vectors for a reciprocal matrix

Efficient vectors are the natural set from which to choose a cardinal ranking vector for a pairwise comparison matrix. Such vectors are the key to certain business project selection models. Many ways to construct specific efficient vectors have been proposed. Yet, no previous method to produce all efficient vectors was known. Here, using some graph theoretic ideas, as well as a numerical extension technique, we show how to generate inductively all efficient vectors for any given pairwise comparison matrix. We apply this method to give a matricial proof of the fact that the set of efficient vectors and other related sets are piecewise linearly connected. In addition, we determine explicitly all efficient vectors for a 4-by-4 pairwise comparison matrix. Several examples are provided.

math.CO↗

arXiv2023

Efficient vectors in priority setting methodology

The Analytic Hierarchy Process (AHP) is a much discussed method in ranking business alternatives based on empirical and judgemental information. We focus here upon the key component of deducing efficient vectors for a reciprocal matrix of pair-wise comparisons. It is not yet known how to produce all efficient vectors. It has been shown that the entry-wise geometric mean of all columns is efficient for any reciprocal matrix. Here, by combining some new basic observations with some known theory, we 1) give a method for inductively generating large collections of efficient vectors, and 2) show that the entry-wise geometric mean of any collection of distinct columns of a reciprocal matrix is efficient. We study numerically, using different measures, the performance of these geometric means in approximating the reciprocal matrix by a consistent matrix.

math.OC↗

arXiv2022

An Atomic Viewpoint of the TP Completion Problem

We present two complementary techniques called catalysis and inhibition which allow one to determine if a given pattern is TP completable or TP non-completable, respectively. Empirically, these techniques require considering only one unspecified entry at a time in a vast majority of cases, which makes these techniques ripe for automation and a powerful framework for future work in the TP completion problem. With small modifications, these techniques are also applicable to the TN completion problem. We provide two major applications. First, we characterize all 4-by-4 patterns by completability. There are a total of 78 new obstructions of this size, six times as many as the 3-by-$n$ case for all $n$ combined. Second, we provide a characterization of the so-called 1-variable obstructions in the TN case, which includes as a corollary a characterization of patterns with a single unspecified entry. This also provides a novel partial result towards proving the conjecture that all TN-completable patterns are TP-completable.

math.CO↗

arXiv2021

The Complexity of Checking Partial Total Positivity

We prove that checking if a partial matrix is partial totally positive is co-NP-complete. This contrasts with checking a conventional matrix for total positivity, for which we provide a cubic time algorithm. Checking partial sign regularity with any signature, including partial total nonnegativity, is also co-NP-complete. Finally, we prove that checking partial total positivity in a partial matrix with logarithmically many unspecified entries may be done in polynomial time.

cs.CC↗

arXiv2019

Characteristics of Eigenvalues Realized by Path-Connected Sets of Matrices

We consider path-connected sets of matrices and the induced paths between eigenvalues. We discuss the equivalence relation generated by these paths, and how it relates to the presence of higher multiplicity eigenvalues realized by the set. Particular interest is applied to the convex hull of matrices, where additional characterizations are provided of this phenomena, and computational methods are given for further study.

math-ph↗

arXiv2019

Wannier90 as a community code: new features and applications

Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronic-structure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials.

cond-mat.mtrl-sci↗

arXiv2014

The critical exponent for generalized doubly nonnegative matrices

It is known that the critical exponent (CE) for conventional, continuous powers of $n$-by-$n$ doubly nonnegative (DN) matrices is $n-2$. Here, we consider the larger class of diagonalizable, entry-wise nonnegative $n$-by-$n$ matrices with nonnegative eigenvalues (GDN). We show that, again, a CE exists and are able to bound it with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for $n=3$. There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being $n-2$ for DN matrices.

math.RA↗