Decompositions of Nakano norms by ODE techniques

We study decompositions of Nakano type varying exponent Lebesgue norms and spaces. These function spaces are represented here in a natural way as tractable varying $\ell^p$ sums of projection bands. The main results involve embedding the varying Lebesgue spaces to such sums, as well as the corresponding isomorphism constants. The main tool applied here is an equivalent variable Lebesgue norm which is defined by a suitable ordinary differential equation introduced recently by the author. We also analyze the effect of transformations changing the ordering of the unit interval on the values of the ODE-determined norm

Using qualitative reasoning in modelling consensus in group decision-making

Ordinal scales are commonly used in rating and evaluation processes. These processes usually involve group decision making by means of an experts’ committee. In this paper a mathematical framework based on the qualitative model of the absolute orders of magnitude is considered. The entropy of a qualitatively described system is defined in this framework. On the one hand, this enables us to measure the amount of information provided by each evaluator and, on the other hand, the coherence of the evaluation committee. The new approach is capable of managing situations where the assessment given by experts involves different levels of precision. The use of the proposed measures within an automatic system for group decision making will contribute towards avoiding the potential subjectivity caused by conflicts of interests of the evaluators in the group.Postprint (published version

Potts Model Partition Functions for Self-Dual Families of Strip Graphs

We consider the $q$-state Potts model on families of self-dual strip graphs $G_D$ of the square lattice of width $L_y$ and arbitrarily great length $L_x$, with periodic longitudinal boundary conditions. The general partition function $Z$ and the T=0 antiferromagnetic special case $P$ (chromatic polynomial) have the respective forms $\sum_{j=1}^{N_{F,L_y,\lambda}} c_{F,L_y,j} (\lambda_{F,L_y,j})^{L_x}$, with $F=Z,P$. For arbitrary $L_y$, we determine (i) the general coefficient $c_{F,L_y,j}$ in terms of Chebyshev polynomials, (ii) the number $n_F(L_y,d)$ of terms with each type of coefficient, and (iii) the total number of terms $N_{F,L_y,\lambda}$. We point out interesting connections between the $n_Z(L_y,d)$ and Temperley-Lieb algebras, and between the $N_{F,L_y,\lambda}$ and enumerations of directed lattice animals. Exact calculations of $P$ are presented for $2 \le L_y \le 4$. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), $W(q)$. Generalizing $q$ from ${\mathbb Z}_+$ to ${\mathbb C}$, we determine the continuous locus ${\cal B}$ in the complex $q$ plane where $W(q)$ is singular. We find the interesting result that for all $L_y$ values considered, the maximal point at which ${\cal B}$ crosses the real $q$ axis, denoted $q_c$ is the same, and is equal to the value for the infinite square lattice, $q_c=3$. This is the first family of strip graphs of which we are aware that exhibits this type of universality of $q_c$.Comment: 36 pages, latex, three postscript figure

A framework for semiqualitative reasoning in engineering applications

In most cases the models for experimentation, analysis, or design in engineering applications take into account only quantitative knowledge. Sometimes there is a qualitative knowledge that is convenient to consider in order to obtain better conclusions. These qualitative concepts can be labels such as ``high,’ ’ ``very negative,’ ’ ``little acid,’ ’ ``monotonically increasing’ ’ or symbols such as ¾; º, etc. . . Engineers have already used this type of knowledge implicitly in many activities. The framework that we present here lets us express explicitly this knowledge. This work makes the following contributions. First, we identify the most important classes of qualitative concepts in engineering activities. Second, we present a novel methodology to integrate both qualitative and quantitative knowledge. Third, we obtain signi® cant conclusions automatically. It is named semiqualitative reasoning. Qualitative concepts are represented by means of closed real intervals. This approximation is accepted in the area of Arti® cial Intelligence. A modeling language is speci® ed to represent qualitative and quantitative knowledge of the model. A numeric constraint satisfaction problem is obtained by means of corresponding rules of transformation of the semantics of this language. In order to obtain conclusions, we have developed algorithms that treat the problem in a symbolic and numeric way. The interval conclusions obtained are transformed into qualitative labels through a linguistic interpretation. Finally, the capabilities of this methodology are illustrated on different problems