The images of non-commutative polynomials evaluated on 2 × 2 matrices over an arbitrary field

2014 ◽  
Vol 13 (06) ◽  
pp. 1450004 ◽  
Author(s):  
Sergey Malev
Keyword(s):  
Partial Solution ◽  
Arbitrary Field ◽  
Multilinear Polynomial ◽  
Homogeneous Polynomials ◽  
Quadratic Extensions

Let p be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field K. Kaplansky conjectured that for any n, the image of p evaluated on the set Mn(K) of n × n matrices is either zero, or the set of scalar matrices, or the set sl n(K) of matrices of trace 0, or all of Mn(K). This conjecture was proved for n = 2 when K is closed under quadratic extensions. In this paper, the conjecture is verified for K = ℝ and n = 2, also for semi-homogeneous polynomials p, with a partial solution for an arbitrary field K.

scholarly journals The images of noncommutative polynomials evaluated on the quaternion algebra

2020 ◽  
pp. 2150074
Author(s):  
Sergey Malev
Keyword(s):  
Vector Space ◽  
Quaternion Algebra ◽  
Arbitrary Field ◽  
Multilinear Polynomial ◽  
Noncommutative Polynomials

Let [Formula: see text] be a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field [Formula: see text]. Kaplansky conjectured that for any [Formula: see text], the image of [Formula: see text] evaluated on the set [Formula: see text] of [Formula: see text] by [Formula: see text] matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

Big Questions in Ecology and Evolution

2009 ◽  
Author(s):  
Thomas N. Sherratt ◽  
David M. Wilkinson
Keyword(s):  
Partial Solution ◽  
Fundamental Importance ◽  
Evolutionary Perspective ◽  
Text Book ◽  
Broad Audience ◽  
The Future ◽  
The Tropics ◽  
Made In ◽  
Do So

Why do we age? Why cooperate? Why do so many species engage in sex? Why do the tropics have so many species? When did humans start to affect world climate? This book provides an introduction to a range of fundamental questions that have taxed evolutionary biologists and ecologists for decades. Some of the phenomena discussed are, on first reflection, simply puzzling to understand from an evolutionary perspective, whilst others have direct implications for the future of the planet. All of the questions posed have at least a partial solution, all have seen exciting breakthroughs in recent years, yet many of the explanations continue to be hotly debated. Big Questions in Ecology and Evolution is a curiosity-driven book, written in an accessible way so as to appeal to a broad audience. It is very deliberately not a formal text book, but something designed to transmit the excitement and breadth of the field by discussing a number of major questions in ecology and evolution and how they have been answered. This is a book aimed at informing and inspiring anybody with an interest in ecology and evolution. It reveals to the reader the immense scope of the field, its fundamental importance, and the exciting breakthroughs that have been made in recent years.

scholarly journals Graph Learning for Combinatorial Optimization: A Survey of State-of-the-Art

2021 ◽  
Author(s):  
Yun Peng ◽  
Byron Choi ◽  
Jianliang Xu
Keyword(s):  
Machine Learning ◽  
Combinatorial Optimization ◽  
Graph Embedding ◽  
Partial Solution ◽  
Complex Data ◽  
Learning Methods ◽  
Graph Learning ◽  
Second Stage ◽  
End To End ◽  
Embedding Methods

AbstractGraphs have been widely used to represent complex data in many applications, such as e-commerce, social networks, and bioinformatics. Efficient and effective analysis of graph data is important for graph-based applications. However, most graph analysis tasks are combinatorial optimization (CO) problems, which are NP-hard. Recent studies have focused a lot on the potential of using machine learning (ML) to solve graph-based CO problems. Most recent methods follow the two-stage framework. The first stage is graph representation learning, which embeds the graphs into low-dimension vectors. The second stage uses machine learning to solve the CO problems using the embeddings of the graphs learned in the first stage. The works for the first stage can be classified into two categories, graph embedding methods and end-to-end learning methods. For graph embedding methods, the learning of the the embeddings of the graphs has its own objective, which may not rely on the CO problems to be solved. The CO problems are solved by independent downstream tasks. For end-to-end learning methods, the learning of the embeddings of the graphs does not have its own objective and is an intermediate step of the learning procedure of solving the CO problems. The works for the second stage can also be classified into two categories, non-autoregressive methods and autoregressive methods. Non-autoregressive methods predict a solution for a CO problem in one shot. A non-autoregressive method predicts a matrix that denotes the probability of each node/edge being a part of a solution of the CO problem. The solution can be computed from the matrix using search heuristics such as beam search. Autoregressive methods iteratively extend a partial solution step by step. At each step, an autoregressive method predicts a node/edge conditioned to current partial solution, which is used to its extension. In this survey, we provide a thorough overview of recent studies of the graph learning-based CO methods. The survey ends with several remarks on future research directions.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Irreducible modules with highest weight vectors over modular Witt and special Lie superalgebras

2019 ◽  
Vol 17 (1) ◽  
pp. 1381-1391
Author(s):  
Keli Zheng ◽  
Yongzheng Zhang
Keyword(s):  
Lie Superalgebras ◽  
Arbitrary Field ◽  
Special Linear ◽  
Highest Weight ◽  
Irreducible Modules

Abstract Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

scholarly journals Designing a Bonus-Malus system reflecting the claim size under the dependent frequency–severity model

2021 ◽  
pp. 1-25
Author(s):  
Rosy Oh ◽  
Joseph H.T. Kim ◽  
Jae Youn Ahn
Keyword(s):  
Recent Literature ◽  
Insurance Industry ◽  
Random Effect ◽  
Random Effect Model ◽  
Partial Solution ◽  
Risk Classification ◽  
Transition Rule ◽  
Auto Insurance ◽  
Dependent Frequency ◽  
Effect Model

In the auto insurance industry, a Bonus-Malus System (BMS) is commonly used as a posteriori risk classification mechanism to set the premium for the next contract period based on a policyholder's claim history. Even though the recent literature reports evidence of a significant dependence between frequency and severity, the current BMS practice is to use a frequency-based transition rule while ignoring severity information. Although Oh et al. [(2020). Bonus-Malus premiums under the dependent frequency-severity modeling. Scandinavian Actuarial Journal 2020(3): 172–195] claimed that the frequency-driven BMS transition rule can accommodate the dependence between frequency and severity, their proposal is only a partial solution, as the transition rule still completely ignores the claim severity and is unable to penalize large claims. In this study, we propose to use the BMS with a transition rule based on both frequency and size of claim, based on the bivariate random effect model, which conveniently allows dependence between frequency and severity. We analytically derive the optimal relativities under the proposed BMS framework and show that the proposed BMS outperforms the existing frequency-driven BMS. Later, numerical experiments are also provided using both hypothetical and actual datasets in order to assess the effect of various dependencies on the BMS risk classification and confirm our theoretical findings.

An Inequality for Homogeneous Polynomials on ℝn

2005 ◽  
Vol 112 (3) ◽  
pp. 264-266
Author(s):  
Luo Xuebo ◽  
Zhu-Jun Zheng
Keyword(s):  
Homogeneous Polynomials
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