Index matrices are an extension of the ordinary matrices with explicitly assigned index sets on both their rows and columns, forming an advanced mathematical structure used for specialized data modeling and problem-solving. In the present paper, a new operator, called “Inverse” operator, is defined over hierarchical index matrices with a description of its software implementation. This operator has no analogue in the classical theory of matrices. Its application allows various restructuring of datasets with multiple criteria, in order to outline data stratifications in the best possible way serving the particular decision maker’s needs. As an illustrative example for the new operator, a dataset of the recorded blood donors in Bulgaria is provided, with discussions of the different stratifications and perspectives in which the dataset can be rearranged using the “Inverse” operator.

The concept of multi-dimensional modal topological structures was introduced in the middle of 2025. One of the special forms of these structures are the intuitionistic fuzzy multi-dimensional modal topological structures. Here, we give examples of intuitionistic fuzzy multi-dimensional modal topological structures, related to extended intuitionistic fuzzy index matrices and to multi-person multi-criteria decision making procedures with intuitionistic fuzzy evaluations of the experts, who have intuitionistic fuzzy scores and use criteria having intuitionistic fuzzy evaluations.

Two arithmetic operations are introduced and some of their properties are studied. It is proved that they can be operations of semi-groups, but not of monoids. It is shown that their reverse operations are not one-valued. Some connections between the new operations and the well-known arithmetic functions phi, psi, sigma are shown.

The present paper proposes a generalized net model of the male urinary system that reflects the principal anatomical and functional relations among the kidneys, ureters, urinary bladder, urethra, nervous system, muscle system, and prostate gland, both in norm and pathology. Particular attention is given to the role of the prostate gland and to changes in urinary flow and bladder emptying associated with prostate-related outflow obstruction. By representing material and informational flows and their interactions across several physiological subsystems, the model is conceptually aligned with the perspective of network physiology. The proposed framework provides a formal basis for the qualitative comparison of normal and pathological states and for future extension, parameterization, scenario-based simulation, and clinical validation.

The lower urinary tract symptoms (LUTS) examined in the study are a common problem in elderly patients and are associated with disorders in the functioning of the bladder, urethra and prostate. The article presents a Generalized Net model for assessing lower urinary tract symptoms based on seven main criteria, including PSA, residual urine volume and prostate volume. The patient’s age plays a key role in the development and progression of these symptoms. The model, developed with data from the Urology Clinic of UMBAL Burgas, supports the diagnosis and decision-making in the treatment of LUTS and the risk of prostate cancer.

A Generalized net model of QoS/QoE estimation and management of an overall telecommunication system is proposed. It is based on a GN model of an overall telecommunication system including a queuing system at the Switching stage. The GN model of the overall system is represented as a transition of the GN model of the management of the system. The model can be used for real-time monitoring and management of the system and for implementation of dynamic pricing policies.

A new extension of the class of the ordinary Generalized Nets (GNs) is defined - GNs with Inhibitor Arcs (GNIA). In a GNIA, some of the arcs can inhibit the tokens that are transferred through them. Algorithm for transfer of tokens in GNIA and algorithm for GNIA functioning are described. It is proved that the class of all GNIAs is a conservative extension of the class of the ordinary GNs. The new extension can be used in the modelling of information service networks, such as overall telecommunication systems.

In the current investigation, a modifying operator over temporal intuitionistic fuzzy index matrices (TIFIM) is presented. Thereafter, an example of autonomous priority-based self-managing database system using intuitionistic fuzzy logic is developed. The modifying operator is implemented for priorities updating. The priorities are changed according to three cases. In the first case, the modifying operator increases the degree of membership and decreases the degree of non-membership if the record is frequently searched. In the second case, the modifying operator decreases the degree of membership and increases the degree of non-membership if the record is rarely searched. In the third case, the modifying operator set the record priority to ⟨0,1⟩ and deleted it the record is not searched more than n days.

In the theory of the Intuitionistic Fuzzy Sets (IFS) there exist operators of modal type, of level type, and of topological type. The modal operators themselves are from three different types. Initially, an intuitionistic fuzzy analogues of the classical modal operators "necessity" and "possibility" were introduced. Following that, they were generalized to new modal-type operators specific to IFSs. In the present paper is considered the most general (at least until this moment) of these operators. It is shown that with appropriate choice of its parameters, this operator can transform the intuitionistic fuzzy values of an arbitrary IFS to the intuitionistic fuzzy values of another arbitrarily chosen IFS.

A new, general approach to the feeble modal topological structure is introduced and these structures are illustrated with five groups of examples using intuitionistic fuzzy objects.

In the current research, we introduce two operations of a "symmetric difference" type over three-dimensional extended index matrices, and investigate some of their basic properties. An example of the implementation of symmetric difference-type operations is presented in the field of relational databases, aiming to demonstrate the operations' efficiency by comparing sets with dissimilar attributes.

All modern process simulators rely on thermodynamic methods to estimate physical properties and calculate phase equilibria. The critical properties of individual components or pseudo-components, which represent undefined mixtures, play a crucial role in these calculations. However, the chemical compositions and characteristics of whole crude oils, petroleum fractions, and fuels, which are very complex mixtures of individual hydrocarbons, can vary significantly depending on the specific crude oil and the processing involved. For instance, straight-run petroleum fractions differ from those obtained through cracking processes due to differences in unsaturated hydrocarbon content. Consequently, effective methods for predicting critical temperature and pressure must account for a wide range of compositional scenarios. To address this challenge, we utilized a database of 176 individual hydrocarbons to evaluate the existing correlations for critical temperature and pressure calculations. Intercriteria analysis was performed to evaluate the relations between the different variables to be used for critical temperature and pressure predictions. Additionally, we proposed new correlations and ANN models for these properties and assessed their performance. Our study aims to provide robust predictive models that can accurately estimate critical properties across diverse petroleum fractions and compositions.

Ebullated bed vacuum residue hydrocracking is a well-established technology providing a high conversion level of low-value residue fractions in high-value light fuels. The main challenge in this technology when processing vacuum residues derived from different crude oils is the sediment formation rate that leads to equipment fouling and cycle length shortening. With the severity enhancement, the asphaltenes become more aromatic and less soluble which leads to sediment formation when the difference between solubility parameters of asphaltenes and maltenes goes beyond a threshold value. Although theoretical models have been developed to predict asphaltene precipitation, the great diversity of oils makes it impossible to embrace the full complexity of oil chemistry by any theoretical model making it impractical for using it in all applications. The evaluation of process data of a commercial ebullated bed vacuum residue hydrocracker, properties of different feeds, and product streams by intercriteria and regression analyses enabled us to decipher the reason for hydrocracked oil sediment content rising from 0.06 to 1.15 wt.%. The ICrA identified the presence of statistically meaningful relations between the single variables, while the regression analysis revealed the combination of variables having a statistically meaningful effect on sediment formation rate. In this study, vacuum residues derived from 16 crude oils have been hydrocracked as blends, which also contain fluid catalytic cracking heavy cycle oil and slurry oil (SLO), in a commercial H-Oil plant. It was found that the hydrocracked oil sediment content decreased exponentially with fluid catalytic cracking slurry oil augmentation. It was also established that it increased with the magnification of resin and asphaltene and the reduction in sulfur contents in the H-Oil feed.

In the middle of 2022, the concept of a Modal Topological Structure (MTS) was introduced by the author and in the next three years, it has been developed in various research directions. In the present paper, a new direction of development of the MTSs is introduced: multi-dimensional MTSs. It is illustrated with examples from the area of the intuitionistic fuzziness.

The formula ¬A = (A → ((A → A) ∧ ¬(A → A))) is a tautology in the classical propositional logic. In this paper, we determine all intuitionistic fuzzy implications that satisfy this formula together with the classical intuitionistic fuzzy negation or with the negation generated by this implication.

The production of very-low-sulfur residual fuel oil is a great challenge for modern petroleum refining because of the instability issues caused by blending incompatible relatively high-sulfur residual oils and ultra-low-sulfur light distillates. Another obstacle in the production of very-low-sulfur residual fuel oil using hydroprocessing technology is the contradiction of hydrodesulfurization with hydrodemetallization, as well as the hydrodeasphaltization functions of the catalytic system used. Therefore, the production of very-low-sulfur residual fuel oil by employing hydroprocessing could be achieved by finding an appropriate residual oil to be hydroprocessed and optimal operating conditions and by controlling catalyst system condition management. In the current study, data on the characteristics of 120 samples of heavy fuel oils produced regularly over a period of 10 years from a high-complexity refinery utilizing H–oil vacuum residue hydrocrackers in its processing scheme, the crude oils refined during their production, the recipes of the heavy fuel oils, and the level of H–oil vacuum residue conversion have been analyzed by using intercriteria and regression analyses. Artificial neural network models were developed to predict the characteristics of hydrocracked vacuum residues, the main component for the production of heavy fuel oil. It was found that stable very-low-sulfur residual fuel oil can be manufactured from crude oils whose sulfur content is no higher than 0.9 wt.% by using ebullated bed hydrocracking technology. The diluents used to reduce residue viscosity were highly aromatic FCC gas oils, and the hydrodemetallization rate was higher than 93%.

This paper introduces a variant of the classic Towers of Hanoi (TH) puzzle in which parallel moves—simultaneous transfers of multiple discs—are permitted. The problem is formalized with Generalized Nets (GN), an extension of Petri nets (PN) whose tokens and transitions encode the ordering and movement of the n discs among three rods under the usual constraints. The resulting GN model, implemented on the OnlineGN platform, provides a clear, precise, and systematic representation that highlights the role of parallelism and supports interactive experimentation. This framework enables the exploration of strategies that reduce the number of parallel steps (PS) and, more broadly, illustrates how GN-captured parallelism can shorten the sequential depth for selected problems with exponential-time solutions.

Refining light and heavy oils in different proportions seems attractive, especially in cases of geopolitical, economic, environmental, and logistical constraints. The economical attractiveness could be undermined in cases where incompatibility occurs. The current study explores a highly complex refinery performance during processing a blend consisting of 17 crude oils of which one was extra light, five were light, nine were medium, and two were heavy. A n-heptane dilution test, using centrifugation, was employed to assess the colloidal stability of crude oils. In addition, a previously established correlation to relate crude oil vacuum residue fraction Conradson carbon content to asphaltene peptizability Sa according to ASTM D 7157 was also availed for the purpose of evaluating colloidal stability. It was found that the crude desalter amperage increases with the SBNIBN ratio and Sa reduction, reaching its maximum allowable value of 180 A at the SBNIBN ratio of 1.35, and Sa of 0.64. The SBNIBN ratio was found more reliable in predicting oil compatibility than the other SBNINmax ratio used to assess colloidal stability in various research. Along with the increase in crude desalter amperage, fouling of the heat exchangers of a crude oil distillation plant was also recorded. An intercriteria analysis of process data together with crude composition data, and compatibility indices revealed that the amperage enhancement is statistically meaningfully related to an increase in the heaviest crude oil content in the process blend and the compatibility indices SBNIBN ratio and Sa, while the fouling was related only to the content of one of the light crude oils in the processed blend.

Efficient and fair allocation of donated blood depends on multiple factors, like medical urgency, donor/recipient compatibility, blood availability, geographic location, limited shelf life, etc. Due to the limited supply of blood and its critical role in healthcare, fair distribution protocols are essential. This study builds upon previous authors’ research that proposed a general mathematical model for fair blood allocation, taking as inputs the universal blood compatibility chart and the assumption of allocating equal shares of the donated blood from each blood type to recipients with respectively compatible blood types. The sum normalization technique was performed (twice, first per recipients and then per donors) for the purpose of balancing between donation needs and options. The result was an indicative blood allocation reference framework in support of the decision making in transfusion haematology. In the present paper, we tailor that general model by introducing as model variables the actual blood group frequencies of a given population. Additional customization is proposed by adding weight coefficients to the values along the framework’s main diagonal that represent ABO-identical transfusions, preferred to non-identical transfusions for minimizing the risks of hemolytic reactions. The model is further elaborated via intervalization of the estimations in the resultant blood allocation framework, thus making the model more flexible and usable. While demonstrated with Bulgarian blood group distributions from 2023, the model can be adapted to other populations and contexts.

A new intuitionistic fuzzy extended modal operator from a first type is introduced and some of its properties are discussed. It is shown that it represents all hitherto existing intuitionistic fuzzy extended modal operators from a first type. Some open problems are formulated