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Abstract: Remark on Polickýʼs paper on circular units of a compositum of quadratic number fields is given. [Copyright &y& Elsevier]

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. [ABSTRACT FROM AUTHOR]

Even in the highest-level mathematics courses at undergraduate institutions, most students do not read mathematical research from professional journals. Such depth of content is often reserved for “independent studies” or summer REU's. In this article I describe a course that required students to read research papers, all of which centered on a single theme. The course covered advanced group theory topics while conveying the notion that there is a mathematical community that works together to advance mathematical knowledge. Using a single story as an integrative thread is an approach that can be adapted to any topic in mathematics. [ABSTRACT FROM AUTHOR]

In 1975, Symons [[5]] introduced a subsemigroup of HT ht , defined as HT ht , for a nonempty subset I Y i of I X i , determining all its automorphisms. The first paper [[21]] introduces a construction of a new graph associated with a semihypergroup, using the fundamental relation HT ht . In particular, it is shown that the Cayley graphs generated by transposition trees on the set HT ht are HT ht -extendable and their extendability number is HT ht for any integer HT ht . Connections between algebraic structure theory and graph theory have been established in order to solve open problems in one theory with the help of the tools existing in the other, emphasizing the remarkable properties of one theory with techniques involving the second. [Extracted from the article]

Leibniz algebras are non-antisymmetric generalizations of Lie algebras. In this paper, we investigate the properties of complete Leibniz algebras under certain conditions on their extensions. Additionally, we explore the properties of derivations and direct sums of Leibniz algebras, proving several results analogous to those in Lie algebras. [ABSTRACT FROM AUTHOR]

The set of polynomial mappings of n-dimensional complex space into itself the Jacobian matrix of which has a constant nonzero determinant is known to be very vast in any dimension that exceeds one. The well-known Jacobian conjecture states that any such mapping is polynomially invertible. Even though the computation of the determinant of the Jacobian matrix is very well supported in modern computer algebra systems, the algorithmic inversion of a polynomial mapping is still a problem of considerable computational complexity. In this paper, we present a Mathematica package JC that can be used for construction and inversion of polynomial mappings and more general analytic mappings with the unit determinant of the Jacobian matrix. The package includes functions that allow one to algorithmically construct these mappings for a given dimension of the space of variables and a given degree of mapping components. The package, together with a library of datasets for testing it and results of computational experiments, is available for free public use at https://www.researchgate.net/publication/358409332_JC_Package_and_Datasets. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0. We study some cohomological properties of Lie subalgebras of the Witt algebra W = Der (k [ t , t − 1 ]) and the one-sided Witt algebra W ≥ − 1 = Der (k [ t ]). In the first part of the paper, we consider finite codimension subalgebras of W ≥ − 1. We compute derivations and one-dimensional extensions of such subalgebras. These correspond to Ext U (L) 1 (M , L) , where L is a subalgebra of W ≥ − 1 and M is a one-dimensional representation of L. We find that these subalgebras exhibit a kind of rigidity: their derivations and extensions are controlled by the full one-sided Witt algebra. As an application of these computations, we prove that any isomorphism between finite codimension subalgebras of W ≥ − 1 extends to an automorphism of W ≥ − 1. The second part of the paper is devoted to explaining the observed rigidity. We define a notion of "completely non-split extension" and prove that W ≥ − 1 is the universal completely non-split extension of any of its subalgebras of finite codimension. In some sense, this means that even when studying subalgebras of W ≥ − 1 as abstract Lie algebras, they remember that they are contained in W ≥ − 1. We also consider subalgebras of infinite codimension, explaining the similarities and differences between the finite and infinite codimension situations. Almost all of the results above are also true for subalgebras of the Witt algebra. We summarise results for W at the end of the paper. [ABSTRACT FROM AUTHOR]

Fostering students' relational understanding of the equals sign is a challenge for math educators that begins in the primary levels and persists into tertiary education. In this paper, I develop an entry point, especially for students who only have an operational understanding of the equals sign, to the core idea of equivalence in linear algebra. My approach is informed by the history of mathematics: In the 17th and 18th centuries, mathematics research underwent an algebraicization, with mathematicians replacing their classical geometric questions with novel algebraic investigations. In this paper, I will offer geometric interpretations of two operations developed at the precipice of this monumental shift: algebraic substitution and Gaussian elimination. I will then utilize Lakoff & Johnson's theory of conceptual metaphor to compare and contrast this historically-grounded geometric re-interpretation of modern linear algebra to the direct algebraic interpretation taken in most modern textbooks. [ABSTRACT FROM AUTHOR]

Advanced mathematics is seen as an integral component of secondary teacher preparation, and thus most secondary teacher preparation programs require their students to complete an array of advanced mathematics courses. In recent years, though, researchers have questioned the utility of proposed connections between advanced and secondary mathematics. It is simply not clear in many cases—to researchers, teacher educators, and teachers themselves—exactly how advanced mathematics content is related to secondary content. In this paper, we propose using a conceptual analysis—a form of theory in which one explicitly describes ways of reasoning about a particular mathematical idea—to address this issue. Specifically, we use conceptual analyses for the foundational notions of equivalence and inverse to illustrate how the ways of reasoning needed to support productive engagement with tasks in advanced mathematics can mirror and reinforce those that are similarly productive in school mathematics. To do so, we propose conceptual analyses for the key concepts of equivalence and inverse and show how researchers can use these conceptual analyses to identify connections to school mathematics in advanced mathematical tasks that might otherwise be obscured and overlooked. We conclude by suggesting ways in which conceptual analyses might be productively used by both teacher educators and future teachers. [ABSTRACT FROM AUTHOR]

Francis Sowerby Macaulay began his career working on Brill and Noether's theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay's work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected. [ABSTRACT FROM AUTHOR]

Copyright of Ñawi: Arte, Diseño y Comunicación is the property of Escuela Superior Politecnica del Litoral and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We show functoriality of γα in our joint paper with Á. Császár “Further remarks on δ- and θ-modifications”. [ABSTRACT FROM AUTHOR]

Cell average decomposition (CAD) plays a critical role in constructing boundpreserving (BP) high-order discontinuous Galerkin and finite volume methods for hyperbolic conservation laws. Seeking optimal CAD (OCAD) that attains the mildest BP Courant--Friedrichs--Lewy (CFL) condition is a fundamentally important yet difficult problem. The classic CAD, proposed in 2010 by Zhang and Shu using the Gauss--Lobatto quadrature, has been widely used over the past decade. Zhang and Shu only checked for the 1D P2 and P3 spaces that their classic CAD is optimal. However, we recently discovered that the classic CAD is generally not optimal for the multidimensional P2 and P3 spaces. Yet, it remained unknown for a decade what CAD is optimal for higher-degree polynomial spaces, especially in multiple dimensions. This paper presents the first systematical analysis and establishes the general theory on the OCAD problem, which lays a foundation for designing more efficient BP schemes. The analysis is very nontrivial and involves novel techniques from several branches of mathematics, including Carath\'eodory's theorem from convex geometry, and the invariant theory of symmetric group in abstract algebra. Most notably, we discover that the OCAD problem is closely related to polynomial optimization of a positive linear functional on the positive polynomial cone, thereby establishing four useful criteria for examining the optimality of a feasible CAD. Using the established theory, we rigorously prove that the classic CAD is optimal for general 1D Pk spaces and general 2D Qk spaces of an arbitrary k\geq 1. For the widely used 2D Pk spaces, the classic CAD is, however, not optimal, and we develop a generic approach to find out the genuine OCAD and propose a more practical quasi-optimal CAD, both of which provide much milder BP CFL conditions than the classic CAD yet require much fewer nodes. These findings notably improve the efficiency of general high-order BP methods for a large class of hyperbolic equations while requiring only a minor adjustment of the implementation code. The notable advantages in efficiency are further confirmed by numerical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the q-analog of Kostant’s Partition Function of Lie algebras sl4(C) and sp6(C) and present a closed formula for the values of these functions. [ABSTRACT FROM AUTHOR]

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included. [ABSTRACT FROM AUTHOR]

In this paper, we study smooth Lie algebra bundles of finite Type. We discuss tangent bundle and Lie algebra bundle induced by Lie group bundle of finite type. Finite type property of top space bundles and RMS bundle for are also examined. [ABSTRACT FROM AUTHOR]

Algebraic hyperstructures are the classical generalizations of algebraic structures which has several applications in uncertainity theory [6], rough set theory [7], lattice based probability theories, analysis etc. Davvaz et.al.[5] extensively studied the chemical and biological applications of hyperstructures by exploring several inheritance examples of algebraic hyperstructures. This paper focusses on the occurences of twin zeros and triple zeros in Hyperlattices with respect to hyperideals. A lattice is a partially ordered set in which every pair of elements has a least upper bound (supremum or join) and a greatest lower bound (infimum or meet). Multilattice is a generalization of a lattice introduced by Benado [3]. They extended the concept of supremum and infimum to"multi" versions, allowing for the consideration of suprema and infima over multiple elements instead of just pairs. This provides a more flexible framework for dealing with larger collections of elements. A lattice can also be viewed as an algebraic structure with two binary operations: join (supremum) and meet (infimum). These operations are used to define the least upper bound and greatest lower bound of elements in the lattice, respectively. Konstantinidou [12], further generalized lattices by replacing the binary operations of join and meet with hyperoperations. However, with these generalizations some properties are not retained. Later, Konstantinidou [11] discussed the concept of distributivity of hyperlattices, particularly of P-hyperlattices. Rasouli and Davvaz [17] considered special relations on hyperlattices, called regular relations and showed that the quotient structure with respect to regular relations form a lattice. Rasouli and Davvaz [16] defined a topology on the spectrum of join hyperlattices and showed that it forms a T0-space. Ameri [2] and others have explored the distributivity and dual distributivity of elements in a hyperlattice. [ABSTRACT FROM AUTHOR]

Algebraic topology studies topological spaces with the help of tools from abstract algebra. The main focus of this paper is to show that many concepts from algebraic topology can be conveniently expressed in terms of (normal) factor graphs. As an application, we give an alternative proof of a classical duality result of Kramers and Wannier, which expresses the partition function of the 2-D Ising model at a low temperature in terms of the partition function of the 2-D Ising model at a high temperature. Moreover, we discuss analogous results for the 3-D Ising model and the Potts model. [ABSTRACT FROM AUTHOR]

Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A. The characterization of normal block Toeplitz matrices with entries from a commutative algebra A of normal matrices is also obtained. [ABSTRACT FROM AUTHOR]

The idea of integrating the history of mathematics content in mathematics teaching and learning is not new. Researchers stress many benefits of using history of mathematics in mathematics education. They suggest that it is very important for mathematics teachers to be familiar with the genesis of mathematical concepts and statements, since these might help them to better understand the difficulties students face. As a matter of fact, history of mathematics shows us that students very often form mathematical concepts in the similar way these concepts have been formed through the history of mankind. The main purpose of this study is to introduce some pre-algebra concepts, such as systems of linear equations, to primary-grade students by using an ancient Chinese method. In the first part of the paper we give an overview of the Fangcheng method. The Fangcheng method is presented in chapter 8 of the book The Nine Chapters on the Art and Calculation which is one of the most important and most influential mathematical works in the long history of China. Originally, the method was used for solving some real-life problems, such as calculating the yields of rice, prices of different products and numbers of animals. It deals with the solution of simultaneous linear equations with two to five unknowns by placing them in a table, and operating with columns in a way identical to the row transformations of the modern matrix algebra. In Serbia, students do not learn how to solve systems of linear equations until the 8th grade of primary school. The aim of the study was to investigate the possibility for fourth grade students to use an adapted Fangcheng method as a tool for solving word problems. The second part of the paper consists of the research methodology, results and discussion. We used the quasi-experimental one-group design with post-test only. There was no justified reason to include a control group since the comparison would not be possible considering that the contents presented to the experimental group were not usually taught in first four primary school grades. Furthermore, the pre-test could not be monitored since no student had had previous experience in using the presented method for solving systems of linear equations. The first research task was to determine if fourth grade students were able to learn, understand, and use the Fangcheng method when solving systems of linear equations with two unknowns. The second research task was to examine if students were able to learn, understand, and use the same method in solving systems of linear equations with three unknowns. The sample included 48 fourth grade students. The research had two phases, and at the end of each phase post-tests were conducted. In the first phase, all students participated in the intervention program, while the second phase included only those students who performed well on the first post-test (14 students). The study results indicate that students who show greater interest in mathematics successfully adopt the procedures necessary for the performance of the Fangcheng method. Furthermore, the majority of students use the Fangcheng method without any difficulties when solving text-based systems of linear equations with two unknowns within the given formed initial table. Difficulties arise when students need to mark the values and form a table on their own. A large number of students manages to understand the technique used, but due to frequent computational errors, they are unable to accurately determine the values of the unknowns. The findings of the study cannot be generalized considering the fact that there are certain limitations, such as a small sample size and quasi-experimental design. Therefore, some further research should be performed with a larger sample of students. However, since there are not many empirical researches which explore the effects of applying the history of mathematics in math teaching, we believe that our study contributes to the field. In this regard, it would be important in future studies to examine the views of teachers as to whether they apply some segments of the history of mathematics in their teaching work, for what purpose, in what parts of the class, whether their application sufficiently arouses students' interest. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Qadisiyah Computer Science & Mathematics is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In the present paper we give a matrix form of local derivations of the complex finite dimensional simple (non-Lie) Malcev algebra M, and a direct proof of the statement that every 2-local derivation of M is a derivation. We have some description of local and 2-local derivations of complex finite-dimensional semisimple binary Lie algebras. [ABSTRACT FROM AUTHOR]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

Mathematics and early state formation, or the Janus face of early Mesopotamian mathematics: bureaucratic tool and expression of scribal professional autonomy: revised contribution to the symposium Mathematics and the State. I think linear algebra is a representative test case, being one of the core courses of undergraduate mathematics and a mathematical discipline with a millennia-long historical tradition. All that debate is about secondary school mathematics teaching, but the tide of changes is already reaching university mathematics as well. He emphasized the role of elementary matrices, introduced the I LU i -factorization of matrices, and suggested solving systems of simultaneous linear equations HT ht for a nondegenerate square matrix I A i not by Gaussian elimination, but by inversion of I A i : HT ht . [Extracted from the article]

This paper is concerned with basic concepts and some results on regular Γ-nearness semigroup and ideals of a Γ-nearness semigroup. Also, it is given some properties about ideals of a regular Γ-nearness semigroup and an example about the subject. Furthermore, we study relations among ideals of a Γ-nearness semigroup. [ABSTRACT FROM AUTHOR]

Modulo lattice additive noise (MLAN) channels appear in the analysis of structured binning codes for Costa's dirty-paper channel and of nested lattice codes for the additive white Gaussian noise (AWGN) channel. In this paper, we derive a new lower bound on the error exponents of the MLAN channel. With a proper choice of the shaping lattice and the scaling parameter, the new lower bound coincides with the random-coding lower bound on the error exponents of the AWGN channel at the same signal-to-noise ratio (SNR) in the sphere-packing and straight-line regions. This result implies that, at least for rates close to channel capacity, 1) writing on dirty paper is as reliable as writing on clean paper; and 2) lattice encoding and decoding suffer no loss of error exponents relative to the optimal codes (with maximum-likelihood decoding) for the AWGN channel. [ABSTRACT FROM AUTHOR]

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings. [ABSTRACT FROM AUTHOR]

We discuss some of the work of Laci Kovács on representation theory and related topics. [ABSTRACT FROM PUBLISHER]

Nowadays, mathematicians are very interested in discovering new and advanced methods for determining the solution of Diophantine equations. Diophantine equations are those equations that have more unknowns than equations. Diophantine equations appear in astronomy, cryptography, abstract algebra, coordinate geometry and trigonometry. Congruence theory plays an important role in finding the solution of some special type Diophantine equations. The absence of any generalized method, which can handle each Diophantine equation, is challenging for researchers. In the present paper, the authors have discussed the existence of the solution of exponential Diophantine equation ( ) ( ) where are whole numbers. Results of the present paper show that the exponential Diophantine equation ( ) ( ) where are whole numbers, has no solution in the whole number. [ABSTRACT FROM AUTHOR]

Copyright of Iraqi Journal of Science is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Sequence knitting comes from a straightforward directive; determine a short basic sequence of stitches and create a piece by continuous repetition of that sequence. This paper determines the number of sequence knitting patterns. We restrict our attention to sequences containing only knit and purl stitches and to flat knitting. We further assume that a sequence is restarted at the beginning of each row of knitting, but we do not assume that the number of stitches cast on is a multiple of the sequence length. We consider the case that a pattern and its left/right mirror image are the same, and the case that a pattern and its left/right mirror image are not necessarily the same. [ABSTRACT FROM AUTHOR]

In tertiary mathematics courses, students often have difficulties acquiring an understanding of the mathematical concepts covered. One approach to address this problem is to implement so-called Concept-Tests. These are multiple-choice questions whose distractors represent common problems and misconceptions related to the concepts. While there exist lots of such questions for calculus, Concept-Test questions focusing on basic concepts of abstract algebra are still rare, although previous research has shown that students have many problems with these. We therefore developed such questions for important concepts of basic group and ring theory in the years 2020–2022. In this paper, we first want to present the questions and the developmental process. Furthermore, we want to present an empirical study investigating to what extent the questions helped students in a proof-oriented abstract algebra course to acquire an understanding of the concepts covered. This study especially indicates that the developed Concept-Test questions provided good starting points for conceptual changes. [ABSTRACT FROM AUTHOR]

This paper is on the research of Wilhelm von Waldenfels in the mathematical field of quantum (or non-commutative) probability theory. Wilhelm von Waldenfels certainly was one of the pioneers of this field. His idea was to work with moments and to replace polynomials in commuting variables by free algebras which play the role of algebras of polynomials in non-commuting quantities. Before he contributed to quantum probability he already worked with free algebras and free Lie algebras. One can imagine that this helped to create his own special algebraic method which proved to be so very fruitful. He came from physics. His PhD thesis, supervised by Heinz König, was in probability theory, in the more modern and more algebraic branch of probability theory on groups. Maybe the three, physics, abstract algebra and probability, must have been the best prerequisites to become a pioneer, even one of the founders, of quantum probability. We concentrate on a small part of the scientific work of Wilhelm von Waldenfels. The aspects of physics are practically not mentioned at all. There is nothing on his results in classical probability on groups (Waldenfels operators). This is an attempt to show how the concepts of non-commutative notions of independence and of Lévy processes on structures like Hopf algebras developed from the ideas of Wilhelm von Waldenfels. [ABSTRACT FROM AUTHOR]

Reid had interviewed Taussky-Todd for her I Hilbert, i though she could have gotten this version of the story from Courant or possibly from Olga's coeditor, Wilhelm Magnus, who then taught at the Courant Institute. Although "occasional frictions" occurred when their views clashed, "he was really a good friend and ... a welcome visitor in our home ...." Olga Taussky-Todd probably first met Constance Reid through Reid's sister Julia, a well-known mathematician. Its content shows that Olga Taussky-Todd had been, in fact, the primary source for Reid's account of the errors in Hilbert's number-theoretic papers, for this letter ends with a short remark about Reid's interview with her for I Hilbert. i However, certain details in Reid's rendering of that story led Olga to suspect that Courant had tampered with her version of the tale, in particular by suggesting that Hilbert had ridiculed her in a light-hearted way. The original account Constance Reid gave on pages 200-201 of I Hilbert i in (Reid [9]) actually differs only slightly from the revised version found in (Reid [12]), but those small changes are telling when seen from the vantage point of this correspondence during the mid-1970s. [Extracted from the article]

Vector linear solvability of non-multicast networks depends upon both the characteristic of the finite field and the dimension of the vector linear network code. In the literature, the dependency on the characteristic of the finite field and the dependency on the dimension have been studied separately. In this paper, we show the interdependency between the characteristic of the finite field and the dimension of the vector linear network code that achieves a vector linear network coding (VLNC) solution in non-multicast networks. For any given network $\mathcal {N}$ , we define $P(\mathcal {N},d)$ as the set of all characteristics of finite fields over which the network $\mathcal {N}$ has a $d$ -dimensional VLNC solution. To the best of our knowledge, for any network $\mathcal {N}$ shown in the literature, if $P(\mathcal {N},1)$ is non-empty, then $P(\mathcal {N},1) = P(\mathcal {N},d)$ for any positive integer $d$. We show that, for any two non-empty sets of primes $P_{1}$ and $P_{2}$ , there exists a network $\mathcal {N}$ such that $P(\mathcal {N},1) = P_{1}$ , but $P(\mathcal {N},2) = \{P_{1},P_{2} \}$. We also show that there are networks exhibiting a similar advantage (the existence of a VLNC solution over a larger set of characteristics) if the dimension is increased from 2 to 3. However, such behaviour is not universal, as there exist networks which admit a VLNC solution over a smaller set of characteristics of finite fields when the dimension is increased. Using the networks constructed in this paper, we further demonstrate that: (i) a network having an $m_{1}$ -dimensional VLNC solution over a finite field of some characteristic and an $m_{2}$ -dimensional VLNC solution over a finite field of some other characteristic may not have an $(m_{1} + m_{2})$ -dimensional VLNC solution over any finite field; (ii) there exist a class of networks for which scalar linear network coding (SLNC) over non-commutative rings has some advantage over SLNC over finite fields: the least sized non-commutative ring over which each network in the class has an SLNC solution is significantly lesser in size than the least sized finite field over which it has an SLNC solution. [ABSTRACT FROM AUTHOR]