SYNOPTIC ABSTRACTStructural connections between m × n(−1,1)-matrices A such that A has the maximum number of (2 × 2)-submatrices with nonzero determinant and m × n (−1,1)-matrices B that maximize det (Bt B) are explored. The two types of matrices can be defined as solutions to two instances of the same optimization problem. The focus is on matrices satisfying the former property, specifically the most difficult case when A is n × n with n = 4k + 1, k a natural number. Constructive approaches and a tabu search are developed and are found to provide near-optimal solutions for 13 < n ≤ 41. The best known solutions to date are found for n = 4k + 1 ≤ 41.

Determinant Optimization on Binary Matrices

Consider an m×n zero-one matrix A. An s×t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m=n and s=t=2. The maximum number M(n) of even 2×2 submatrices of A is clearly ** and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turan numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that ** for some constant B. In this paper we show that if the matrix A=An is considered to be induced by an infinite zero one matrix obtained at random, then ** where En denotes the number of even 2×2 submatrices of An. Results such as these provide us with specific information about the tightness of the concentration of En around its expected value of **.

Even 2 × 2 Submatrices of a Random Zero-One Matrix

For the list object, introduced in Chapter 5, it was shown that each data element contains at most one predecessor element and one successor element. Therefore, for any given data element or node in the list structure, we can talk in terms of a next element and a previous element.

Graphs and Digraphs

Let A be an m×n, (0, 1)-matrix. A submatrix of A is odd if the sum of its entries is an odd integer and even otherwise. The maximum number of 2×2 odd submatrices in a (0, 1)-matrix is related to the existence of Hadamard matrices and bounds on Turan numbers. Pinelis [On the minimal number of even submatrices of 0-1 matrices, Designs, Codes and Cryptography, 9:85–93, 1994] exhibits an asymptotic formula for the minimum possible number of p × q even submatrices of an m × n (0, 1)-matrix. Assuming the Hadamard conjecture, specific techniques are provided on how to assign the 0’s and 1’s, in order to yield the maximum number of 2 × 2 odd submatrices in an m × n (0, 1)-matrix. Moreover, formulas are determined that yield the exact maximum counts with one exception, in which case upper and lower bounds are given. These results extend and refine those of Pinelis.

/pdf/the-maximum-number-of-2-by-2-odd-submatrices-in-0-1-matrices-f2mleuf70q.pdf

The maximum number of 2 by 2 odd submatrices in (0,1)-matrices

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Michael Marks

Author Tools

Chat about Author

Papers

The maximum number of 2 by 2 odd submatrices in (0,1)-matrices