A Note on the Erdős-Szekeres Theorem in Two Dimensions

Burkill and Mirsky, and Kalmanson prove independently that, for every $r\ge 2, n\ge 1$, there is a sequence of $r^{2^n}$ vectors in $\mathbb R^n$, which does not contain a subsequence of $r+1$ vectors $v^1, v^2,\dots,v^{r+1}$ such that, for every $i$ between 1 and $n$, $(v^{j}_i)_{1\le j\le r+1}$ forms a monotone sequence. Moreover, $r^{2^n}$ is the largest integer with this property. In this short note, for two vectors $u = (u_1, u_2,\dots, u_n)$ and $v = (v_1, v_2, \dots, v_n)$ in $\mathbb{R}^n$, we say that $u\le v$ if, for every $i$ between 1 and $n$, $u_i\le v_i$. Just like Burkill and Mirsky, and Kalmanson, for every $k, \ell\ge 1, d\ge 2$ we find the maximal $N_1, N_2$ (which turn out to be equal) such that there are numerical two-dimensional arrays of size $(k+\ell-1)\times N_1$ and $(k+\ell)\times N_2$, which neither contain a subarray of size $k\times d$, whose columns form an increasing sequence of $d$ vectors in $\mathbb{R}^k$, nor contain a subarray of size $\ell\times d$, whose columns form a decreasing sequence of $d$ vectors in $\mathbb{R}^{\ell}$. In a consequent discussion, we consider a generalisation of this setting and make a connection with a famous problem in coding theory.

Extended conditional G-expectations and related stopping times

<p style='text-indent:20px;'>In this paper, we extend the definition of conditional <inline-formula> <tex-math id="M2">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> to a larger space on which the dynamical consistency still holds. We can consistently define, by taking the limit, the conditional <inline-formula> <tex-math id="M3">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula> for each random variable <inline-formula> <tex-math id="M4">\begin{document}$ X $\end{document}</tex-math> </inline-formula>, which is the downward limit (respectively, upward limit) of a monotone sequence <inline-formula> <tex-math id="M5">\begin{document}$ \{X_{i}\} $\end{document}</tex-math> </inline-formula> in <inline-formula> <tex-math id="M6">\begin{document}$ L_{G}^{1}(\Omega) $\end{document}</tex-math> </inline-formula>. To accomplish this procedure, some careful analysis is needed. Moreover, we present a suitable definition of stopping times and obtain the optional stopping theorem. We also provide some basic and interesting properties for the extended conditional <inline-formula> <tex-math id="M7">\begin{document}$ G{\text{-}}{\rm{expectation}} $\end{document}</tex-math> </inline-formula>. </p>

A constructive method for convex solutions of a class of nonlinear Black-Scholes equations

In this work, we are concerned with the theoretical study of a nonlinear Black-Scholes equation resulting from market frictions. We will focus our attention on Barles and Soner’s model where the volatility is enlarged due to the presence of transaction costs. The aim of this paper is to give a constructive mathematical approach for proving the existence of convex solutions to a non degenerate fully nonlinear deterministic problem with nonlinear dependence upon the highest derivative. The existence of a strong solution to the original equation is shown by considering a monotone sequence satisfying an abstract Barenblatt equation and converging toward the solution of a limit problem.

Real-Option Valuation in Multiple Dimensions Using Poisson Optional Stopping Times

We provide a new framework for valuing multidimensional real options where opportunities to exercise the option are generated by an exogenous Poisson process, which can be viewed as a liquidity constraint on decision times. This approach, which we call the Poisson optional stopping times (POST) method, finds the value function as a monotone sequence of lower bounds. In a case study, we demonstrate that the frequently used quasi-analytic method yields a suboptimal policy and an inaccurate value function. The proposed method is demonstrably correct, straightforward to implement, reliable in computation, and broadly applicable in analyzing multidimensional option-valuation problems.

More Precise Decision-Making Methodology in the Temporalized Body of Evidence. Application in the Information Technology Management

In this paper, we perform the analysis of temporalized structure of a body of evidence and possibilistic Extremal Fuzzy Dynamic System (EFDS) for the construction of more precise decisions based on the expert knowledge stream. The process of decision precision consists of two stages. In the first stage the relation of information precision is defined on a monotone sequence of bodies of evidence. The principle of negative imprecision is developed, as the maximum principle of knowledge ignorance measure of a body of evidence. Corresponding mathematical programming problem is constructed. On the output of the first stage we receive the expert knowledge precision stream of the criteria with respect to any decision. In the second stage the constructed stream is an input trajectory for the finite possibilistic model of EFDS. A genetic algorithm approach is developed for identifying of the EFDS finite model. The modelling process gives us the more precise decisions as a prediction of a temporalization procedure. The constructed technology is applied in the non-probabilistic utility theory for the information technology management problem.

Temporalized Structure of Bodies of Evidence in the Multi-Criteria Decision-Making Model

In this paper, we perform the analysis of temporalized structure of bodies of evidence to construct more precise decisions based on the mathematical model of experts’ evaluations. The relation of information precision is defined on a monotone sequence of the bodies of evidence. For determining of a body of evidence the maximum principles of nonspecificity measure, the Shannon and Shapley entropies are applied. Corresponding mathematical programming problems are constructed. A new approach for the numerical solution of these problems is developed. The temporalized structure of bodies of evidence is used for precising the decision in the well-known Kaufmann’s theory of expertons. A measure of increase of decision precision is introduced, which takes into account all steps of temporalization. The temporalized method of expertons is applied to the problem of decision risk management, where the investment fund expert commission provides evaluation of competition results. In our specially created decision-making model, the goal of the expert technology is to aggregate and refine subjective evaluations provided by the expert commission members. The model performs as an adviser that assists the expert commission in selecting of decision with a minimum risks. The results of developed method are then compared with other well-known methods and aggregation operators such as: mean, median, ordered weighed averaging (OWA) and method of expertons.

Index Summability of an Infinite Series Using δ-Quasi Monotone Sequence

A result concerning absolute indexed Summability factor of an infinite series using δ-Quasi monotone sequence has been established.

A Note on a Ramsey-Type Problem for Sequences

Two sequences $\{x_i\}_{i=1}^{t}$ and $\{y_i\}_{i=1}^t$ of distinct integers are similar if their entries are order-isomorphic. Let $f(r,X)$ be the length of the shortest sequence $Y$ such that any $r$-coloring of the entries of $Y$ yields a monochromatic subsequence that is also similar to $X$. In this note we show that for any fixed non-monotone sequence $X$, $f(r,X)=\Theta(r^2)$, otherwise, for a monotone $X$, $f(r,X)=\Theta(r)$.