In this paper, Hybrid Automata, which is a formal model for hybrid systems, has been introduced. A summary of its theory is presented and some of its special and important classes are listed and some properties that can be studied and checked for it are mentioned. Finally, the purposes of use, the most widely used areas, and the tools that provide H.A. Support are addressed.

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter and singular in a complex time variable at the origin. These equations combine differential operators of Fuchsian type in time and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the complex parameter known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic type functions in time and are bounded holomorphic in space. A set of logarithmic type solutions are shaped by means of Laplace transforms relatively to time and parameter and Fourier integrals in space. Furthermore, a formal logarithmic type solution is modeled which represents the common asymptotic expansion of Gevrey type of the genuine solutions with respect to the complex parameter on bounded sectors at the origin.

The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalized entropies crucially depend on the number of allowed number degrees of freedom $N$. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume $W$ on $N$. We review the ensuing entropies, discuss the composability axiom, relate to the Gibbs' paradox discussion and explain why group entropies may be particularly relevant from an information theoretic perspective.

Wireless Sensor Networks (WSNs) are being increasingly adopted in critical applications, where verifying the correct operation of sensor nodes is a major concern. Undesired events may undermine the mission of the WSNs. Hence their effects need to be properly assessed before deployment to obtain a good level of expected performance and during the operation in order to avoid dangerous unexpected results. In this paper we propose a methodology to support design and deployment of dependable WSNs by means of an event-based formal verification technique. The methodology includes a process to guide designers towards the realization of a dependable WSN and a tool ("ADVISES") to simplify its adoption. The tool allows to generate automatically formal specifications used to check correctness properties and evaluate dependability metrics at design time and at runtime. During the runtime we can check the behavior of theWSN accordingly to the results obtained at design time and we can detect sudden and unexpected failures, in order to trigger recovery procedures. The effectiveness of the methodology is shown in the context of two case studies, aiming to illustrate how the tool is helpful to drive design choices and to check the correctness properties of theWSN at runtime

We investigate how formal institutional distance (FID) moderates the relationship between cul-tural distance (CD) and the financial performance of foreign subsidiaries firms. Following recent research, we estimate the asymmetric effects of CD by considering its size and direction towards host countries on the opposite poles of each cultural dimension` scale. We propose that a limited understanding of the formal institutions in the host country, as measured by the magnitude and direction of the FID, can have a positive moderating effect, increasing the impact of CD on finan-cial performance. This is mainly because foreign subsidiary firms may be more reliant on their ca-pacity to navigate the less formal (and more implicit) aspects of the host country's institutional environment, such as their ability to cope with the CD. We use foreign subsidiary data from the Orbis database including 22 developed and 22 developing home countries and over 1400 foreign subsidiaries during a period of 3 consecutive years operating in 10 of the largest economies (host countries) in Latin America including: Argentina, Brazil, Colombia, Chile, Ecuador, Mexico, Panama, Peru, Uruguay, and Venezuela. Findings confirm the asymmetric effects of CD, howev-er, by considering the direction of FID, our findings reveal that the higher the FID towards less developed host countries, the more significant the effects of CD on the financial performance. These findings contribute to the knowledge of how formal and informal institutional distances in-teract by showing that the greater the FID towards less developed host countries, to higher the impact of CD on the financial performance of foreign subsidiary firms.

Misuse of measurement units and orientations leads to errors in scientific applications, Cyber Physical Systems (CPS), and IoT C/C++ programs. Standard type system are inadequate in preventing such errors. Although dimensional and orientational analysis in physics can manually detect these errors in equations, analyzing complex code with intricate physical computations is impractical. To overcome this challenge, we propose an advanced type system that incorporates units and orientations as integral components within a specialized type library. Our enhanced type system automatically detects potential errors during compile time by representing physical quantities as types and utilizing dimensional analysis, orientational analysis, and metaprogramming techniques. Our improved type system enables formal verification of C++ software, successfully verifying programs with extensive codebases. We also employ it for runtime verification of dynamic linking and pointer operations in C++ programs. The integration of compile-time verification, dimensional analysis, orientational analysis, and advanced type system enhances the robustness and accuracy of scientific applications, CPS, and IoT C/C++ programs. By leveraging these approaches, we ensure precise calculations and prevent errors related to measurement units and orientations, resulting in substantial improvements in reliability and accuracy.

Alzheimer’s disease is a debilitating neurodegenerative condition which is known to be the most common cause of dementia. Despite its rapidly growing prevalence, medicine still lacks a comprehensive definition of the disease. As a result, Alzheimer’s disease remains neither preventable nor curable. In recent years, broad interdisciplinary collaborations in Alzheimer’s disease research are becoming more common. Furthermore, such collaborations have already demonstrated their superiority in addressing the complexity of the disease in innovative ways. However, establishing effective communication and optimal knowledge distribution between researchers and specialists with different expertise and background is not a straightforward task. To address this challenge, we propose the Alzheimer’s disease Ontology for Diagnosis and Preclinical Classification (AD-DPC) as a tool for effective knowledge sharing in interdisciplinary/multidisciplinary teams working on Alzheimer’s disease. It covers six major conceptual groups, namely Alzheimer's disease pathology, Alzheimer's disease spectrum, Diagnostic process, Symptoms, Assessments, and Relevant clinical findings. All concepts were annotated with definitions or elucidations and in some cases enriched with synonyms and additional resources. The potential of AD-DPC to support non-medical experts is demonstrated through an evaluation of its usability, applicability and correctness. The results show that the participants in the evaluation process who lack prior medical knowledge can successfully answer Alzheimer’s disease-related questions by interacting with AD-DPC. Furthermore, their perceived level of knowledge in the field increased leading to effective communication with medical experts.

In recent years, the internet of things has been widely utilized in various fields, such as in smart factories or connected cars. As its domain of application has expanded, it has begun to be employed using multi-server architectures for a more efficient use of resources. However, because users wishing to receive IoT services connect to multi-servers over wireless networks, this can expose systems to various attacks and result in serious security risks. To protect systems (and users) from potential security vulnerabilities, a secure authentication technology is necessary. In this paper, we propose a smart card-based authentication protocol, which performs the authentication for each entity by allowing users to go through the authentication process using a smart card transmitted from an authentication server, and to login to a server connected to the IoT. Furthermore, the security of our proposed authentication protocol is verified by simulating a formal verification scenario using AVISPA, a security protocol-verification tool.

With the rise of Internet of Things (IoT) devices and Cyber-Physical Systems, the demand for more functional and high-quality software has increased tremendously. To meet this need, we must reengineer and reuse existing software, as well as develop new software formal verification methods. One such method is based on physical quantities defined by the System International, which have physical dimensions. The homogeneity of physical dimensions in software code enables us to check the software code in the space of base units, making it the first basis of the new software verification method. However, this method cannot check expressions with angles, angle speed, and other similar features. To address this, a transformation for physical value orientation introduced by Siano allows us to check software code for orientational, stabilization, and other related branches. This makes the orientational homogeneity the second basis of the new software verification method. To assess the effectiveness of the proposed method, we developed special software defect models based on the statistical characteristics of software code. We used a special statistical analysis tool to define the statistical characteristics of modern software and analyzed over 2 GB of C++ GITHUB code for drones. Based on the actual distribution of software characteristics, the proposed method can detect over 60% of latent software defects. Implementing this method can significantly reduce testing time, improve reliability, and enhance overall software quality.

Integrated Model of Distributed Systems is used for modeling and verification. In formalism, the distributed system is modeled as a collection of server states and agent messages. The evolution of the system takes the form of actions that transform the global system configuration (states and messages) into a new configuration. Formalism is used in the Dedan verification environment for finding different kinds of deadlocks: communication deadlocks in the server view and resource deadlocks in the agent view. For other purposes, a conversion has been developed to equivalent models: to Petri nets for structural analysis and do Distributed Autonomous and Asynchronous Automata (DA3) for easy graphical modeling in terms of system components. In addition, it is possible to simulate a verified system on distributed components in DA3. The automata have two forms: Server-DA3 (S-DA3) for the server view and Agent-DA3 (A-DA3) for the agent view. DA3 formalism is compared to other concepts of distributed automata known from the literature.

We examine a family of nonlinear q-difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation q-difference operators, recently investigated by the author, and quasi-linear Kowalevski type problems that involve contraction q-difference operators. We build up local holomorphic solutions to these problems. Two aspects of these solutions are explored. One facet deals with asymptotic expansions in the complex time variable for which a mixed type Gevrey and q-Gevrey structure is exhibited. The other feature concerns the problem of confluence of these solutions as q tends to 1.

Shells, probably like no other product of nature, have played an important role in the history of mankind. The pre-Hispanic civilizations of Ibero-America also used certain type of shells profusely in their religious ceremonies, in particular, in Ecuador there were two species of main importance, the Spondylus princeps and the Spondylus calcifer broadly employed to manufacture ornaments that possess a strong symbolic, religious and social meaning and that were almost exclusively used by ruling classes. Among these ornaments, the faces carved on the Spondylus shells are little known. In the present study, we chose a total of fifteen (15) pieces from the Pastor Restrepo Lince´s archaeomalacology collection to understand the possible uses of these objects, through the interpretation of the gestures represented on the faces, their dimensions, and their geographical distribution in pre-Hispanic Ecuador. To achieve the proposed objective, we approach the present investigation from the perspective of the formal analysis of concepts, which is a mathematical theory of representation of knowledge, finding that these faces carved in Spondylus, were used daily or in special ceremonial occasions and that its use was common in all the regional cultures of ancient Ecuador, from the oldest such as Valdivia, and for more than 2000 years, indicating a long tradition of the use of Spondylus as an object of great symbolic and economic value until the arrival of the Spanish

A linear Cauchy problem with polynomial coefficients wich combines q-difference operators for q>1 and differential operators of irregular type is examined. A finite set of sectorial holomorphic solutions w.r.t the complex time is constructed by means of classical Laplace transforms. These functions share a common asymptotic expansion in the time variable which turns out to carry a double layers structure which couples q-Gevrey and Gevrey bounds. In the last part of the work, the problem of confluence of these solutions as q tends to 1 is investigated.

In the achievement of the Sustainable Development Goals (SDGs), education plays a fundamental role. However, traditional methodologies do not favor the enrichment and personal development essential to promote global awareness. The use of active methodologies based on experiences improve the quality of learning. This work describes the design, implementation, and evaluation of the acquired knowledge of a didactic proposal for non-formal education as a support for regulated education based on botany content. Firstly, a workshop was held, where young people participated directly in developing field work with a real scientific methodology. Subsequently, a group of students was chosen to be interviewed to obtain a global vision of the learning they obtained. The motivation of the students was quite positive, which allowed us to obtain voluntary participation in the field work and gave the students a participative attitude throughout the development of the workshops. Four months later, this positive attitude remained during their direct involvement in various activities, and the students still remembered the fundamental content discussed. Relating the didactic proposal to its immediate environment was shown to increase interest in learning and value in its own context. The results of this educational experience have been very positive, as knowledge was acquired, and interest in the preservation of the environment and the profession of a researcher was promoted.

This case study assessed marketing arrangements used by small scale farmers in the Lake Basin and Lower Eastern bean corridors of Kenya to determine which markets work for rural producers and what changes are needed to produce and supply sufficient quantities for trade. Using exploratory research, data was collected through focus group discussions with six farmer groups representing a total of 1255 bean farmers and key informant interviews with extension staff. The results indicated that 94% of the farmers produced beans before identifying buyers with only 6% participating in group marketing. Though spot-market transactions with brokers and traders provided ready cash for the farmers, formal buyers were perceived to be more reliable but difficult to find and, operated stringent requirements which were a barrier to entry. A theory of change to integrate smallholders into formal markets to sustainably produce and supply sufficient volumes for trade should entail a transformation agenda at four levels of the value chain: intensification of production through pure stand models with greater use of certified high yielding varieties; stable price guarantees; a market-driven research and extension service and; an enabling political, policy and business environment in the bean value chain. Further research is needed to pilot these changes in a case control study.

This research contributes to the overall debate on education for sustainable development (ESD) by shed- ding lights on the contributing role of formal education to the contemporaneous dynamics of literacy, labor market participation and poverty reduction in Africa, with a focus on Burkina Faso. The study uses a semi-parametric recursive trivariate probit modeling approach, and data from the 2014 National Survey on Household Living Conditions in Burkina Faso. The results show that the embraced systemic approach in this analysis is statistically signicant as shown by the 95% condence intervals on the three correlation coeffcients in the model. Furthermore, education does improve literacy skills, however improved literacy skills in itself does not guaranty active labor market participation in Burkina Faso. Active labor market participation seem to be affected by labor market rates of return, and individual reservation wage (or income). When labor market rate of return is short of high literacy skilled individuals' reservation wage, then the natural response is a choice of inactivity in the labor market, by the later group. Simultaneously however, it is found that active labor market participation leads to poverty reduction; therefore, in addition to new industrial policies for structural transformation of the economy, policy makers in Burkina Faso should consider education and minimum wage reforms to give highly literate household members the incentive to be active in the labor market.

This paper directly addresses a long-standing issue that affects the development of many complex distributed software systems: how to establish quickly, cheaply, and reliably whether they can deliver their intended performance before expending significant time, effort and money on detailed design and implementation. We describe ∆QSD, a novel metrics-based and quality-centric paradigm that uses formalised outcome diagrams to explore the performance consequences of design decisions, as a performance blueprint of the system. The distinctive feature of outcome diagrams is that they capture the essential observational properties of the system, independent of the details of system structure and behaviour. The ∆QSD paradigm derives bounds on performance expressed as probability distributions encompassing all possible executions of the system. The ∆QSD paradigm is both effective and generic: it allows values from various sources to be combined in a rigorous way, so that approximate results can be obtained quickly and subsequently refined. ∆QSD has been successfully used by Predictable Network Solutions for consultancy on large-scale applications in a number of industries, including telecommunications, avionics, and space and defence, resulting in cumulative savings worth billions of US dollars. The paper outlines the ∆QSD paradigm, describes its formal underpinnings, and illustrates its use via a topical real-world example taken from the blockchain/cryptocurrency domain. ∆QSD has enabled challenging throughput targets to be met for a globally distributed blockchain operating on the public internet.

Sometimes there are clear and natural limits to the scope of action of a science, and in other cases they are simply convenient ones. Geographic Information Science (GISc) is a transversal science, with contacts with all geosciences but also with various formal sciences such as Mathematics, Logic and Computer Science. A first approach to specifying the limits of a science is through its definition. Definitions of GISc are often so expansive that they have been rightly criticized for practicing gerrymandering, in particular with the rest of the geosciences. To avoid this, an operational definition is proposed that places GISc among the sciences that handle Data and not Information. This solves the gerrymandering problem without really implying a significant cut of what is usually considered within GISc. As an unforeseen consequence, this delimitation will allow it to be characterized as Formal Science, leaving it as the only geoscience with this characteristic.

We examine a nonlinear initial value problem both singularly perturbed in a complex parameter and singular in complex time at the origin. The study undertaken in this paper is the continuation of a joined work with A. Lastra published in 2015. A change of balance between the leading and a critical subdominant term of the problem considered in our previous work is performed. It leads to a phenomenon of coalescing singularities to the origin in the Borel plane w.r.t time for a finite set of holomorphic solutions constructed as Fourier series in space on horizontal complex strips. In comparison to our former study, an enlargement of the Gevrey order of the asymptotic expansion for these solutions relatively to the complex parameter is induced.

We examine a family of linear partial differential equations both singularly perturbed in a complex parameter and singular in complex time at the origin. These equations entail forcing terms which combine polynomial and logarithmic type functions in time and that are bounded holomorphic on horizontal strips in one complex space variable. A set of sectorial holomorphic solutions are built up by means of complete and truncated Laplace transforms w.r.t time and parameter and Fourier inverse integral in space. Asymptotic expansions of these solutions relatively to time and parameter are investigated and two distinguished Gevrey type expansions in monomial and logarithmic scales are exhibited.

The notion of hierarchy is a key characteristic of any complex system. This paper explores which notions of hierarchy are being used in the field of management and organization studies. Four distinct types of hierarchy are identified: a ladder of formal decision-making authority, a ladder of achieved status, a self-organized ladder of responsibility, and an ideology-based ladder. A social mechanism-based perspective serves to define and distinguish these four types. Subsequently, the typology is further developed by comparing the four hierarchy types in terms of their tacit/explicitness, (in)transitivity, and behavior- versus cognition-centeredness. This review article contributes to the literature by dissecting the general metaphor of hierarchy into four different constructs and their social mechanisms, which serves to create a typology of the various ways in which complex systems can be characterized as being hierarchical. This typology can inform future research drawing on any type of hierarchy.

Agriculture is the backbone of Kenya’s economy, supporting up to 80% of the rural livelihoods. Kenya’s export horticulture is currently the leading Agriculture subsector in Kenya has evolved from small-holder farming to agro-industrial large-scale export farming dominated by multinational companies. It is regarded as an agro-industrial food system based on the economies of scale producing for mass markets outside of the production area. Much of the food consumed from this food system has undergone multiple transformations and been subject to a host of formal and informal insitutions (rules, regulations, standards, norms and values). An Anthropological study of export horticulture in Northwest Mount Kenya was carried out utilizing qualitative data collection methods in Northwest Mount Kenya region. Data was coded and analysed thematically based on grounded theory approach. The study described the institutional settings of export horticulture from an emic perspective as changing and defining the operations of the food system access and management of common pool resources, namely water and land. With the agro-industrial food system competing for these scarce resources in a semi-arid zone, there is potential for conflict and also reduced production and overall benefits to the different actors in the study area.

Formalized Calculation, using auxiliary form to calculate various nested sums, based on binary or Gaussian coefficients and providing three different expressions. In addition to computation, it introduces a large number of new concepts and methods for analysis. With just a little skill, it can easily obtain dozens of theorems. It has undergone three years of development. This article summarizes it and provides a large number of latest achievements, showcasing many analytical methods. This article includes various relationships between two types of Stirling numbers, associated Stirling numbers, generalized Euler numbers and polynomials, and provides a theorem of symmetry and extended Wolstenholme theorems.

Mass plays a strange multiple role in classical physics. This results in a difference between gravity and all other forces - a difference that stood at the beginning of Einstein's development of his general theory of relativity. The equivalence principle (EP) deals with homogenious gravitational fields, which, in fact, don‘t exist in nature. There is only an approximation for this field in Einstein-Lift possible for local situations. Nevertheless this EP can be true, because in classical formal binary logic there can be formulated a true conclusion from a wrong premise or assumption.

Discrete, logic-based models are increasingly used to describe biological mechanisms. Initially introduced to study gene regulation, these models evolved to cover various molecular mechanisms, such as signalling, transcription factor cooperativity, and even metabolic processes. The abstract nature and amenability of discrete models to robust mathematical analyses make them appropriate for addressing a wide range of complex biological problems. Recent technological breakthroughs have generated a wealth of high throughput data. Novel, literature-based representations of biological processes and emerging algorithms offer new opportunities for model construction. Here, we review up-to-date efforts to address challenging biological questions by incorporating omic data into logic-based models, and discuss critical difficulties in constructing and analysing integrative, large-scale, logic-based models of biological mechanisms.

A family of linear singularly perturbed Cauchy problems is studied. The equations defining the problem combine both partial differential operators together with the action of linear fractional transforms. The exotic geometry of the problem in the Borel plane, involving both sectorial regions and strip-like sets, gives rise to asymptotic results relating the analytic solution and the formal one through Gevrey asymptotic expansions. The main results lean on the appearance of domains in the complex plane which remain intimately related to Lambert W function, which turns out to be crucial in the construction of the analytic solutions. On the way, an accurate description of the deformation of the integration paths defining the analytic solutions and the knowledge of Lambert W function are needed in order to provide the asymptotic behavior of the solution near the origin, regarding the perturbation parameter. Such deformation varies depending on the analytic solution considered, which lies in two families with different geometric features.

The work is devoted to the study of a family of linear initial value problems of partial differential equations in the complex domain, dealing with two complex time variables. The use of a truncated Laplace-like transformation in the construction of the analytic solution allows to overcome a small divisor phenomenon arising from the geometry of the problem and represents an alternative approach to the one proposed in a recent work by the first two authors. The result leans on the application of a fixed point argument and the classical Ramis-Sibuya theorem.

A nonlinear singularly perturbed Cauchy problem with confluent fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has been investigated by the author in a recent work, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

Validation and verification are the critical requirements in the knowledge acquisition method for the clinical decision support system (CDSS). After acquiring the medical knowledge from diverse sources, the rigorous validation and formal verification process are required before creating the final knowledge model. Previously, we have proposed a hybrid knowledge acquisition method for acquiring medical knowledge from clinical practice guidelines (CPGs) and patient data in the Smart CDSS for treatment of oral cavity cancer. The final knowledge model was created by combining knowledge models obtained from CPGs and patient data after passing through a rigorous validation process. However, detailed analysis shows that due to lack of formal verification process, it involves various inconsistencies in knowledge relevant to the formalism of knowledge, conformance to CPGs, quality of knowledge, and complexities of knowledge acquisition artifacts. Therefore, it is required to enhance a hybrid knowledge acquisition method that thwarts the inconsistencies using formal verification. This paper presents the verification process using the Z formal method and its outcome as an enhanced acquisition method – known as the refined knowledge acquisition (ReKA) method. The ReKA method adopted verification method and explored the mechanism of theorem proving using the Z notation. It enables to identify inconsistencies in the validation process used for hybrid knowledge acquisition. Additionally, it refines the hybrid knowledge acquisition method by discovering the missing steps in the current validation process at the acquisition stage. Consequently, ReKA adds a set of nine additional criteria to be used to have a final valid refined clinical knowledge model. The criteria ensure the validity of final knowledge model concerning formalism of knowledge, conformance to GPGs, quality of the knowledge, usage of stringent conditions and treatment plans, and inconsistencies possibly resulting from the complexities. Evaluation, using four medical knowledge acquisition scenarios, shows that newly added knowledge in CDSS due to the addition of criteria by ReKA method always produces a valid knowledge model. The final knowledge model was also evaluated with 1229 oral cavity patient cases, which outperformed with an accuracy of 72.57\% compared to a similar approach with an accuracy of 69.7\%. Furthermore, ReKA method identified a set of decision paths (about 47.8%) in the existing approach, which results in a final knowledge model with low quality, non-conformed from standard CPGs. In conclusion, ReKA is formally proved method which always yields valid knowledge model having high quality, supporting local practices, and influenced from standard guidelines.

This paper is a slightly modified, abridged version of a previous work ``Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference ``Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.

This paper is a slightly modified, abridged version of a previous work "Parametric Gevrey asymptotics in two complex time variables through truncated Laplace transforms'' motivated by our contribution in the conference "Formal and Analytic Solutions of Diff. (differential, partial differential, difference, q-difference, q-difference-differential) Equations on the Internet'' (FASnet20). It aims to clarify and give further detail at some crucial points concerning the asymptotic behavior of the solutions of the problems studied in that work.

Electrical microgrids are deemed to be the future of modern power systems. Microgrids are sophisticated, decentralized, and self-sufficient small-scale power systems consisting of various resources ranging from wind turbines, solar photovoltaics, electric vehicles, smart energy storage, and complex communication infrastructure. However, renewable energy sources such as solar photovoltaics and wind-based generators are highly intermittent, and if not appropriately planned, they can compromise the stability of the grid. Formal methods can define and analyze the functionality and behavior of any system and show if the system design is correct before the actual system is implemented. Although formal methods have been around for many years, it is surprising that little to none are utilized in the design of safety-critical electrical power systems. Currently, in modeling microgrids, few to no attempts of formalization are being used to improve the design reliability and reduce system operating costs and time. This work demonstrates how complex systems such as microgrids can be modeled elegantly using a formal specification method. In this work, the Z state-based formal specification language (Z-Method) is used to model and verify microgrid designs. In this work, 3-interconnected microgrid systems with a high penetration level of solar and wind-based renewable energy sources with plugin hybrid electric vehicles (PHEV) as battery energy storage systems (BESS) are modeled using the Z-method. To the best of authors, the knowledge presented formal method is one of the first reported attempts in modeling microgrid communities using Software Engineering formalism.

We consider a family of nonlinear singularly perturbed PDEs whose coefficients involve a logarithmic dependence in time with confluent Fuchsian singularities that unfold an irregular singularity at the origin and rely on a single perturbation parameter. We exhibit two distinguished finite sets of holomorphic solutions, so-called outer and inner solutions, by means of a Laplace transform with special kernel and Fourier integral. We analyze the asymptotic expansions of these solutions relatively to the perturbation parameter and show that they are (at most) of Gevrey order 1 for the first set of solutions and of some Gevrey order that hinges on the unfolding of the irregular singularity for the second.

The II law of thermodynamics is most often given in three supposedly equivalent formulations: two Clausius (I and II) and one Kelvin. The most general and indisputable entropy formulation belong to Clausius (II). The earlier Clausius I principle determines the natural direction heat flow between bodies at different temperatures. On the other hand, the Kelvin principle states that it is impossible to completely convert heat into work. The author argues that the Kelvin principle is a weaker statement (or more strictly non-equivalent) than Clausius I principle, and the latter is a weaker statement than Carnot principle, which is equivalent to Clausius II principle. As a result, the Kelvin principle and the Clausius I principle are not exhaustive formulations of the II law of thermodynamics. At the same time, it turns out that the Carnot principle becomes such a formulation. Apart from providing a complete set of proof and disproof, the author, indicates where the methodological errors were made in the alleged proof of the equivalence of the Kelvin principle and both Clausius principles.