Grid-block difference families and related combinatorial structures

<p>As a powerful technique for holding relations in things, combinatorics has experienced rapidly development in the past century, particularly,enumeration of configurations, combinatorial design and graph theory. How-<br /> ever, the main objective for mathematics is to bring about a quantitative analysis for other sciences, which implies a natural question on combinatorics. Thus, how combinatorics can contributes to other mathematical sciences, not just in discrete mathematics, but metric mathematics and physics? After a long time speculation, I brought the CC conjecture for advancing mathematics by combinatorics, i.e., any mathematical science can be reconstructed from or made by combinatorialization in my postdoctoral report for Chinese Academy of Sciences in 2005, and then reported it at a few conferences of China. Clearly,<br /> CC conjecture is in fact a combinatorial notion and holds by a philosophical law, i.e., all things are inherently related, not isolated. The main purpose of this report is to survey the roles of CC conjecture in developing mathematical sciences with notions, such as those of its contribution to algebra, topology,Euclidean geometry and differential geometry, non-solvable differential equations or classical mathematical systems with contradictions to mathematics,quantum fields after it appeared 10 years ago. All of these show the importance of combinatorics to mathematical sciences in the past and future.</p>

This paper investigates the connections between combinatorial design theory and the creation of new forms of poetry through a specific combinatorial structure called Steiner triple systems. We introduce five original poems constructed using variations of Steiner triple systems on seven and nine words, illustrating how mathematical structures can inform and inspire new poetic forms. The work includes a reflective discussion from dual creative perspectives; one emphasizing structural design and the other, literary expression, highlighting how formal constraints can foster occasional frustrations, but also unexpected artistic freedoms. This study demonstrates the potential of mathematical frameworks as generative tools in literary creativity.

Combinatorial designs with two singular values II. Partial geometric designs

Experimental design focuses on describing or explaining the multifactorial interactions that are hypothesized to reflect the variation. The design introduces conditions that may directly affect the variation, where particular conditions are purposely selected for observation. Combinatorial design theory deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. In this work, borrowing the concept of “balance” in combinatorial design theory, a novel method for multifactorial bio-chemical experiments design is proposed, where balanced templates in combinational design are used to select the conditions for observation. Balanced experimental data that covers all the influencing factors of experiments can be obtianed for further processing, such as training set for machine learning models. Finally, a software based on the proposed method is developed for designing experiments with covering influencing factors a certain number of times.

In a nesting of a balanced incomplete block design (or BIBD), we wish to add a point (the nested point) to every block of a ‐BIBD in such a way that we end up with a partial ‐BIBD. In the case where the partial ‐BIBD is in fact a ‐BIBD, we have a perfect nesting. We show that a nesting is perfect if and only if . Perfect nestings were previously known to exist in the case of Steiner triple systems (i.e., ‐BIBDs) when , as well as for some symmetric BIBDs. Here we study nestings of ‐BIBDs, which are not perfect nestings. We prove that there is a nested ‐BIBD if and only if , . This is accomplished by a variety of direct and recursive constructions.

A graph $G$ with vertex set $V$ is said to be $n$-existentially closed if, for every $S \subset V$ with $|S|=n$ and every $T \subseteq S$, there exists a vertex $x \in V-S$ such that $x$ is adjacent to each vertex of $T$ but is adjacent to no vertex of $S-T$. Given a combinatorial design ${\cal D}$ with block set ${\cal B}$, its block-intersection graph $G_{{\cal D}}$ is the graph having vertex set ${\cal B}$ such that two vertices $b_1$ and $b_2$ are adjacent if and only if $b_1$ and $b_2$ have non-empty intersection. In this paper we study BIBDs (balanced incomplete block designs) and when their block-intersection graphs are $n$-existentially closed. We characterise the BIBDs with block size $k \geq 3$ and index $\lambda=1$ that have 2-e.c. block-intersection graphs and establish bounds on the parameters of BIBDs with index $\lambda=1$ that are $n$-e.c. where $n \geq 3$. For $\lambda \geq 2$ and $n \geq 2$, we prove that only simple $\lambda$-fold designs can have $n$-e.c. block-intersection graphs. In the case of $\lambda$-fold triple systems we show that $n \geq 3$ is impossible, and we determine which 2-fold triple systems (i.e., BIBDs with $k=3$ and $\lambda=2$) have 2-e.c. block-intersection graphs.

ABSTRACTOrthogonal arrays (OAs) are a key research focus in combinatorial design theory and experimental design theory. They play a very important role in an increasing number of interdisciplinary fields, such as computer science, cryptography, multipartite quantum states and quantum information. And the open problems in many fields are closely related to the construction of orthogonal arrays with high strength and large size. At present, some open problems have been solved by using orthogonal arrays. There is an urgent need for developing such orthogonal arrays. This article introduces a general method for obtaining asymmetric orthogonal arrays with high strength. The proposed method is straightforward and constructive. The constructed OAs can be utilized to generate more OAs. Some selective OAs are tabulated for practical applications.

Partially Balanced Incomplete Block (PBIB) designs are the primary subject of this investigation of independent dominance in block designs. In the combinatorial structure of PBIB-designs, where treatments and blocks are depicted as vertices and edges, respectively, the graph-theoretic parameter known as independent dominance is applied. Within the framework of PBIB-designs defined by different association schemes, the study delves into the theoretical foundations, computation, and consequences of independent domination. Finding autonomous dominating sets that effectively represent all treatments while minimizing redundancy is the goal of the study, which offers a paradigm for doing so by examining the links between blocks and treatments. Experiment design optimization using independent dominating parameters is demonstrated in case studies of 2-associate and 3-associate PBIB-designs. This study sheds light on the structural features of PBIB-designs by demonstrating the interaction between graph theory and combinatorial design theory. In situations where resources are limited, it highlights the value of independent dominance as a method for assessing and improving block designs’ efficiency. Our current understanding of block designs and how to optimize them for use in experimental planning, data analysis, and resource allocation is enhanced by these findings.

If we agree to use one of v possible messages to communicate one of k possible source states, then an opponent can successfully impersonate a transmitter with probability at least k / v, and can successfully substitute a message with a fraudulent one with probability at least (k - 1) / (v - 1). We wish to limit an opponent to these bounds. In addition, we desire that the observation of any two messages in the communication channel will give an opponent no clue as to the two source states. We describe a construction for a code which achieves these goals, and which does so with the minimum possible number of encoding rules (namely, v(v -1) / 2). The construction uses a structure from combinatorial design theory known as a perpendicular array.

An application of splittable 4-frames to coloring of Kn, n

This chapter outlines some of the main computational methods that have been successful in solving existence and enumeration problems in combinatorial design theory. Issues related to the validation of search results are also discussed.

Applying computer algebra systems with SAT solvers to the Williamson conjecture

In this paper, we present some well known results in combinatorial design theory. Next, we will use these results to construct a balanced three-parallel session schedule for a conference. A conference normally contains a set of sessions (each one of them involving three fields) that has to be scheduled within a time period of k days. The problem consists of finding a balanced schedule for each field.

On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects

Compressed sensing is a novel sampling technique that provides a fundamentally new approach to data acquisition. Comparing with the traditional method, compressed sensing asserts that a sparse signal can be reconstructed from very few measurements. A central problem in compressed sensing is the construction of sensing matrices. While random sensing matrices have been studied intensively, only a few deterministic constructions are known. As a long-standing subject in combinatorial design theory, the packing design gives rise to deterministic sparse matrices with low coherence. With this framework in mind, we investigate a series of packing designs originated from finite geometry. The connection between the sensing matrix and finite geometry is revealed, using the packing design as a bridge. More specifically, we construct a series of m ×n binary sensing matrices with sparsity order k=Θ(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/3</sup> ) or k=Θ(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> ). Moreover, we use an embedding operation to merge our binary matrices with matrices having low coherence. Comparing with the original binary matrices, this embedding operation generates modified matrices with better recovery performance. Numerical simulations show that our binary and modified matrices outperform several typical sensing matrices. The sparse property of our matrices helps to accelerate the recovery process, which is very preferable in practice.