(0,1)-matrices and Discrepancy

Let $m$ and $n$ be positive integers, and let $R =(r_1, \ldots, r_m)$ and $S =(s_1,\ldots, s_n)$ be nonnegative integral vectors. Let $A(R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$. Let $R$ and $S$ be nonincreasing, and let $F(R)$ be the $m \times n$ $(0,1)$-matrix where for each $i$, the $i^{th}$ row of $F(R,S)$ consists of $r_i$ 1's followed by $n-r_i$ 0's. Let $A\in A(R,S)$. The discrepancy of A, $disc(A)$, is the number of positions in which $F(R)$ has a 1 and $A$ has a 0. In this paper, we investigate the possible discrepancy of $A^t$ versus the discrepancy of $A$. We show that if the discrepancy of $A$ is $\ell$, then the discrepancy of the transpose of $A$ is at least $\frac{\ell}{2}$ and at most $2\ell$. These bounds are tight.

ANALYSIS OF THE PRE-INJECTION CONFIGURATION IN A MARINE ENGINE THROUGH SEVERAL MCDM TECHNIQUES

The present manuscript describes a computational model employed to characterize the performance and emissions of a commercial marine diesel engine. This model analyzes several pre-injection parameters, such as starting instant, quantity, and duration. The goal is to reduce nitrogen oxides (NOx), as well as its effect on emissions and consumption. Since some of the parameters considered have opposite effects on the results, the present work proposes a MCDM (Multiple-Criteria Decision Making) methodology to determine the most adequate pre-injection configuration. An important issue in MCDM models is the data normalization process. This operation is necessary to convert the available data into a non-dimensional common scale, thus allowing ranking and rating alternatives. It is important to select a suitable normalization technique, and several methods exist in the literature. This work considers five well-known normalization procedures: linear max, linear max-min, linear sum, vector, and logarithmic normalization. As to the solution technique, the study considers three MCDM models: WSM (Weighted Sum Method), WPM (Weighted Product Method) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). The linear max, linear sum, vector, and logarithmic normalization procedures brought the same result: -22º CA ATDC pre-injection starting instant, 25% pre-injection quantity and 1-2º CA pre-injection duration. Nevertheless, the linear max min normalization procedure provided a result, which is different from the others and not recommended.

IMAGES OF MENTAL VISUALIZATION IN BIOETHICS

В статье даётся краткий очерк антиномической природы биоэтического дискурса и возможностей его геометрической визуализации. Рассматриваются два варианта визуализации. Первый связан с представлением той или иной ситуации как системы полярностей, которая в свою очередь моделируется в рамках векторной модели. В простейшем случае тезис и антитезис рассматриваются как два перпендикулярных вектора, а синтез – как их векторная сумма. В этом случае можно ввести и более количественную оценку «меры многомерности» полярной системы – как величины проекции её векторного представления на суммарный вектор. С использованием этих конструкций разбирается один пример из биоэтики, связанный со столкновением принципов милосердия и правдивости (проблема «лжи во спасение»). Деяние (действие или бездействие) интерпретируется как своеобразный оператор на событиях, который переводит одни события в другие. Предполагается, что субъект в своих деяниях рассматривает различные возможности и выбирает те из них, которые максимизируют ту или иную ценностную меру субъекта, в данном случае – меру векторной проекции полярного вектора ситуации на суммарный вектор – вектор синтеза базисных полярностей. Второй вариант визуализации связан с понятием антиномий в биоэтике – таких противоречий, которые не являются формально-логическими ошибками. В отличие от последних, в антиномиях как тезис, так и антитезис имеют свой момент оправдания в рамках тех или иных условий. Используется также понятие «антинома» – логического субъекта антиномии, который предицируется тезисом и антитезисом антиномии. Редукции антиномии соответствуют двум крайним аспектам антинома, которые называются его «редуктами» – по аналогии с редукцией волновой функции в квантовой механике. Приводятся различные примеры антиномов: биоэты, глоболоки, холомеры. В биоэтах один редукт выражает в большей мере биологические (биоредукт), второй – этические (эторедукт) определения антинома. В глоболоках выделяются глобальный (глоборедукт) и локальный (локоредукт) виды редуктов: первый выражает более глобальные (универсальные) этические определения, второй – более локальные, связанные с ценностями и нормами того или иного сообщества. Наконец, холомеры – вид антиномов, где антиномически соединяются определения целого (холоредукт) и части (мероредукт). Даётся их интерпретация как многомерных ментальных объектов в некотором обобщённом пространстве, так что крайние их аспекты (редукции антиномии) можно представить как проекции более многомерного состояния. В заключении делается предположение о связи биоэтических проблем с идеей ментальной многомерности, что составляет основу возможной визуализации как интерпретации ментальной многомерности на векторном её представлении. The article provides a brief outline of the antinomic nature of bioethical discourse and the possibilities of its geometric visualization. Two visualization options are considered. The first is associated with the representation of a particular situation as a system of polarities, which in turn is modeled in the framework of a vector model. In the simplest case, the thesis and the antithesis are considered as two orthogonal vectors P1 and P2, and the synthesis is considered as their vector sum S = P1+P2. In this case, we can also introduce a more quantitative estimate of the “measure of multidimensionality” M(P) of the polar system – as the magnitude of the projection of its vector representation P on the sum vector S, i.e. M(P) = (P,es), where es = S/|S| is the unit vector of the vector S, and (P,es) is the scalar product of the vectors P and es. Using these constructs, the author analyzes one example from bioethics related to the clash of the principles of mercy and truthfulness (the problem of “lying for salvation”). An act (action or omission) is interpreted as a kind of an operator on events that transforms some events into others. It is assumed that the subject considers various possibilities in their actions and chooses those that maximize a particular value measure of the subject, in our case, the measure M(P) of the vector projection of the polar vector P of the situation on the sum vector S – the vector of synthesis of basic polarities. The second version of visualization is related to the concept of antinomies – such contradictions that are not formal logical errors – in bioethics. In contrast to errors, in antinomies, both the thesis and the antithesis have their moment of justification within the framework of certain conditions. The concept “antinome” is also used; it is the logical subject of antinomy, which is predicated by the thesis and the antithesis of antinomy. Antinomy reductions correspond to two extreme aspects of the antinome, which are called its “reducts” – by analogy with the reduction of the wave function in quantum mechanics. Various examples of antinomes are given: bioets, globolocs, and holomers. In bioets, one reduct expresses the biological (bioreduct) definition of the antinome, another the ethical (ethoreduct) one. In globolocs, global (globoreduct) and local (locoreduct) types of reducts are distinguished: the former expresses more global (universal) ethical definitions, the latter more local ones, related to the values and norms of a particular local community. Finally, holomers are a kind of antinomes in which the definitions of the whole (holoreduct) and the part (meroreduct) are antinomically connected. They are interpreted as multidimensional mental objects in some generalized space, so that their extreme aspects (antinomy reductions) can be represented as generalized projections of a more multidimensional state within certain constricted conditions (reduction intervals). In this case, it is possible to geometrically visualize such states as, for example, three-dimensional objects in space, through which antinomes can be modeled, and their reducts as two-dimensional projections of a three-dimensional body on certain projection planes (intervals of reducts). In this case, one of the central tasks of bioethics is to determine the boundaries of the demarcation of some intervals from others. For example, in solving the problem of abortion and the status of the human embryo, such a demarcation is expressed in the search for a time point that would separate the phase of a more biological definition (bioreduct) of the embryo from its more ethical state (ethoreduct). In conclusion, the author suggests that bioethical problems are connected with the idea of mental multidimensionality, which forms the basis of a possible visualization as an interpretation of mental multidimensionality in its vector representation.

Extreme Points of Certain Transportation Polytopes with Fixed Total Sums

Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.

On the little secondary Bruhat order

Let $R$ and $S$ be two sequences of positive integers in nonincreasing order having the same sum. We denote by ${\cal A}(R,S)$ the class of all $(0,1)$-matrices having row sum vector $R$ and column sum vector $S$. Brualdi and Deaett (More on the Bruhat order for $(0,1)$-matrices, Linear Algebra Appl., 421:219--232, 2007) suggested the study of the secondary Bruhat order on ${\cal A}(R,S)$ but with some constraints. In this paper, we study the cover relation and the minimal elements for this partial order relation, which we call the little secondary Bruhat order, on certain classes ${\cal A}(R,S)$. Moreover, we show that this order is different from the Bruhat order and the secondary Bruhat order. We also study a variant of this order on certain classes of symmetric matrices of ${\cal A}(R,S)$.

Discrepancy of Matrices of Zeros and Ones

Let $m$ and $n$ be positive integers, and let $R=(r_1,\ldots, r_m)$ and $ S=(s_1,\ldots, s_n)$ be non-negative integral vectors. Let ${\cal A} (R,S)$ be the set of all $m \times n$ $(0,1)$-matrices with row sum vector $R$ and column vector $S$, and let $\bar A$ be the $m \times n$ $(0,1)$-matrix where for each $i$, $1\le i \le m$, row $i$ consists of $r_i$ $1$'s followed by $n-r_i$ $0$'s. If $S$ is monotone, the discrepancy $d(A)$ of $A$ is the number of positions in which $\bar A$ has a $1$ and $A$ has a $0$. It equals the number of $1$'s in $\bar A$ which have to be shifted in rows to obtain $A$. In this paper, we study the minimum and maximum $d(A)$ among all matrices $A \in {\cal A} (R,S)$. We completely solve the minimum discrepancy problem by giving an explicit formula in terms of $R$ and $S$ for it. On the other hand, the problem of finding an explicit formula for the maximum discrepancy turns out to be very difficult. Instead, we find an algorithm to compute the maximum discrepancy.