Hendrik Schön
email: hendrik.schoen@tu-dresden.de

Jelena Zdravkovic
email: jelenaz@dsv.su.se

Janis Stirna

Susanne Strahringer
email: susanne.strahringer@tu-dresden.de

A Role-based Capability Modeling Approach for Adaptive Information Systems

Keywords: Capability Modeling, Roles, Context, Business Requirements

Most modeling approaches lack in their ability to cover a full-fledged view of a software system's business requirements, goals, and capabilities and to specify aspects of flexibility and variability. The modeling language Capability Driven Development (CDD) allows modeling capabilities and their relation to the execution context. However, its context-dependency lacks the possibility to define dynamic structural information that may be part of the context: persons, their roles, and the impact of objects that are involved in a particular execution occurrence. To solve this issue, we extended the CDD method with the BROS modeling approach, a role-based structural modeling language that allows the definition of context-dependent and dynamic structure of an information system. In this paper, we propose the integrated combination of the two modeling approaches by extending the CDD meta-model with necessary concepts from BROS. This combination allows for technical development of the information system (BROS) by starting with capability modeling using CDD. We demonstrate the combined meta-model in an example based on a real-world use case. With it, we show the benefits of modeling detailed business requirements regarding context comprising environment-and object-related information.

Introduction

Organizations need a rapid response to changes in the business environment in terms of new legislation, changes in customer and supplier behavior, new and often adverse events. Such change cannot always be foreseen at the time of information system (IS) development and hence the current approach that is based on implementing change by redesigning and redeploying applications is no longer sufficient. A strand of approaches aims at continuous development and tightening the gap between development and operations [START_REF] Hüttermann | DevOps for Developers[END_REF]. This is, however, not suitable for developing and customizing enterprise applications that need to respond to change both on the business and IS level. That is, a congruent approach that supports responsiveness to changes in the application context and facilitates the responses to transcend from the business to the information system is needed. Sandkuhl and Stirna [START_REF] Sandkuhl | Capability Management in Digital Enterprises[END_REF] contributed to making IS more flexible with respect to the adaptation to context. The concept of capability was used for this purpose because it unifies the business aspects traditionally used in areas such as enterprise modeling like goals and processes with execution context [START_REF] Zdravkovic | A Comparative Analysis of Using the Capability Notion for Congruent Business and Information Systems Engineering[END_REF]. Furthermore, it connects context with the specification of algorithms for adjusting the IS once the context changes. The stance of CDD is that any information that influences the IS is to be modeled as context. BROS (Business Role-Object Specification) [START_REF] Schön | Business Role-Object Specification: A Language for Behavior-aware Structural Modeling of Business Objects[END_REF] is a structural modeling language for design time specification of business objects concerning a domain model as well as specific business logic. Role-fulfilling objects cover the static specification part regarding the separation of concerns, whereas the dynamic specification part of the business logic is expressed via events. The final BROS model serves as a blueprint for development and can be implemented in role-based modeling languages. BROS supports the specification of system-internal variability that is induced by, e.g., the change of role fulfillment. In case of human or organizational roles, changes of this kind often require adaptations in terms of business process variants because the same roles can be fulfilled by several actors each of which having a different skill profile. This aspect has not been elaborated in the CDD approach. Therefore, the objective of this paper is to explore the integration of the CDD and BROS for the purpose of supporting role-based capability modeling and IS design. Among the motivators for the CDD [START_REF] Bērziša | Deliverable 1.4: Requirements specification for CDD, CaaS-capability as a service for digital enterprises[END_REF] are the following goals, to which the proposed integration of the two approaches is set to contribute:  To allocate resources to process execution tasks and to provision human resources to process execution. The integrated proposal addresses this goal by explicitly modeling skill profiles of actors and skill requirements of roles, which allows specifying the actor-role fulfillment by using the concept of scene in BROS.  To customize services according to context. The integrated proposal allows designing and monitoring changes in the context caused by actor-role fulfillment.  To monitor process execution. The CDD approach supports model-driven generation of a monitoring application, Capability Navigation Application (CNA) for overseeing context elements, KPIs, and triggering capability adjustments. The proposal allows integration of actor-role fulfillment and skill monitoring in the CNA.

The remainder of the paper is structured as follows. Section 2 covers the related background of our research. Section 3 provides a conceptual overview of the suggested role-based capability approach. Further, in section 4, the abstract and theoretical part of our research is demonstrated via the introduction of the extended meta-model, a core part of the paper. Section 5 demonstrates our new approach by applying it to a real-world use case, a lecture management scenario in higher education, followed by the conclusion in Section 6 with summary and outlook.

Background

The enterprise modeling discipline endeavors to support businesses by means of IS, which imposes supporting some low-volatile business processes and concepts, but lately even more is required -coping with dynamically changing business environments requiring adaptations of IS at execution time. In this regard, adaptability is seen as an architectural property, enabling a system to efficiently adjust to different or evolving operational or usage circumstances [START_REF] Morin | Models@Run.time to Support Dynamic Adaptation[END_REF][START_REF] Engel | Designing Products for Adaptability: Insights from Four Industrial Cases[END_REF].

To achieve adaptability, organizations should be able to, by the support of modeling, master different variations of their businesses, such as user preferences, environmental variations, changes on partners' sides, legislations, and other [START_REF] Koutsopoulos | Dynamic Adaptation of Capabilities: Exploring Meta-model Diversity[END_REF]. This study also investigates the area of dynamic adaptions of IS and finds that there is a plethora of capability modeling approaches that depict adaptability elements in different ways. Many of the existing approaches address capability delivery by means of, for example, services, business processes, or actions. Nevertheless, the current state in capability design does not offer a transition to tasks associated with IS development. The CDD approach (section 2.1) relies on enterprise models for designing IS based business capabilities with inbuilt support for adaptation to changing contexts at the execution time [START_REF] Sandkuhl | Capability Management in Digital Enterprises[END_REF]. Amongst Enterprise Architecture frameworks and languages, including TOGAF, Archimate, DODAF, NAF and MODAF, the NAF framework [START_REF]NATO Architecture Framework v4[END_REF] is the closest to CDD in its ability to define local conditions in design, but it does not have a method for capability adjustments at run-time. Also, the work of Rodriguez et al. [START_REF] Rodrigues | Reliability Support for the Model Driven Architecture[END_REF] is related to CDD and includes context-dependency as well. However, this approach focuses more on reliability modeling and transformation with replicas at design time. The specifications of the other frameworks provide methods neither for the use of capability at runtime nor for adjustments [START_REF] Zdravkovic | A Comparative Analysis of Using the Capability Notion for Congruent Business and Information Systems Engineering[END_REF]. However, its methodology and the underlying architecture for designing variability for the purpose of adaptation lack the support for dynamic roles of the entities being involved in the implementation of the capabilities, such as subjects (persons, organizations) and objects. The BROS language (section 2.2) uses business scenarios as a fitting complement to support the specification of system variability induced by the change of role fulfillment.

Capability-Driven Development (CDD)

The foundation for CDD is provided by the conceptual Capability Meta-Model (CMM). CMM was developed on the basis of industrial requirements and related research on capabilities. In brief, it consists of the three main parts of the meta-model: The meta-model in Fig. 1 is a simplified version of CMM showing the key components of CDD; the full version with complete element definitions is available in [START_REF] Grabis | Deliverable 5.3: The Final Version of Capability Driven Development Methodology[END_REF]. The CDD methodology combines three interconnected cycles of working -design, delivery, and feedback. Design starts with configuring existing or creating new enterprise goals and processes combined with captured business contexts and eliciting required capabilities. This is followed by delivery of the capability requiring composition and integration of existing technologies and applications, such as ERP systems. During the execution of the application, the changes of context are monitored, and runtime adjustment algorithms are used to calculate if the context's changes require another capability pattern. Feedback is achieved by monitoring defined KPIs, which enable capability refinement and pattern updating.



Business Role-Object Specification (BROS)

Roles and the related concepts were investigated in various research areas during the last decades (e.g., theories [START_REF] Steimann | On the representation of roles in object-oriented and conceptual modelling[END_REF], modeling languages [START_REF] Kühn | A Metamodel Family for Role-Based Modeling and Programming Languages[END_REF], programming languages [START_REF] Herrmann | Programming with roles in ObjectTeams/Java[END_REF][START_REF] Leuthäuser | A Pure Embedding of Roles[END_REF], runtime environments [START_REF] Taing | A Dynamic Instance Binding Mechanism Supporting Run-time Variability of Role-based Software Systems[END_REF], or enterprise modeling [START_REF] Almeida | Applying and extending a semantic foundation for role-related concepts in enterprise modelling[END_REF][START_REF] Frank | Delegation: An important concept for the appropriate design of object models[END_REF]). Roles extend the established object-oriented paradigm by the ability to represent an object in different contexts and by changing its behavior and characteristics accordingly. Roles qre described in terms of (a) behavioral, (b) relational, and (c) context-dependent properties [START_REF] Steimann | On the representation of roles in object-oriented and conceptual modelling[END_REF][START_REF] Kühn | A Metamodel Family for Role-Based Modeling and Programming Languages[END_REF]. This serves a more accurate description of the domain's entities with their context-dependent structure and behavior. BROS uses this advantage of roles to model software based on required business needs. The BROS modeling language [START_REF] Schön | Business Role-Object Specification: A Language for Behavior-aware Structural Modeling of Business Objects[END_REF] was originally developed for an easy adaptation of (structural) reference models [START_REF] Schön | Role-based Adaptation of Domain Reference Models: Suggestion of a Novel Approach[END_REF], it can also be used for creating role-based software in general. It utilizes the role-paradigm to specify mainly structural models. Via roles, however, BROS (in contrast to traditional modeling languages such as UML) is able to include the behavior-aware specifications in structural models non-invasively.

BROS does not focus on process modeling itself; it explicitly includes events induced in the respective background processes, nevertheless. Via events, temporality, and role-based context-dependent behavior, BROS allows for behavioral modeling constructs within a mainly structural modeling language. Thus, BROS benefits from CDD due to its ability to define the complex business constraints (i.e., when to choose a scene) as a background source of these events.

The main concepts of BROS are objects, roles, scenes, and events (Fig. 2). Objects are selected from an underlying structural domain model and are the target of any use case or enterprise-specific adaptation done by using the remaining concepts. BROS utilizes roles as specific representations of objects in certain scenes (the role's context). The enterprise-specific processes are the main drivers of the adaptation and serve two kinds of information: (a) the scene as an encapsulation context of a use case or task, and (b) the events as certain points in time affecting the roles. The details of the language, as well as an example, are described in [START_REF] Schön | Business Role-Object Specification: A Language for Behavior-aware Structural Modeling of Business Objects[END_REF], based on the research of CROM [START_REF] Kühn | A Metamodel Family for Role-Based Modeling and Programming Languages[END_REF]. For the purpose of this research, the BROS concepts were introduced in the CDD meta-model with the knowledge implied by the BROS meta-model.

Conceptual Approach

According to the motivation for this research, we strive for a framework that extends the CDD approach with the role-based paradigm provided by the BROS approach.

Although both approaches are settled on different levels and phases of the software development stack (see Fig. 3), the role concepts introduced by BROS are suitable to be used for fine-grained capability design. CDD and BROS have been developed independently of each other. Nevertheless, they share a common concept of "dedicated context": the process variant in CDD and the scene in BROS. Both are representatives of a special, single task or execution, dependent on the chosen environment. This task may change during runtime since one or more requirements from the given environment is dropped. In CDD, a process variant is derived from a general process description (e.g., giving a lecture). The concrete process variant is then chosen by variation points based on the environment requirements. Thus, CDD focuses on the conceptual view of the requirements, capabilities, and goals of the respective IS. In contrast, BROS utilizes scenes to describe the behavior of roles for certain tasks or executable procedures. The scene defines the context-dependent boundary of a role's validity (e.g., the role "Teacher" is only valid in the context of the scene "Giving Lecture") in combination with a start and an end as specific points in time. BROS is intended to be an extension to CDD, hence, we integrated its concepts (see Fig. 4) into the already existing CDD methodology. We state that, with the BROS concepts, it is possible, to describe the capabilities of an enterprise with regard to performers that are able to play certain roles (or not). CDD, as presented in [START_REF] Sandkuhl | Capability Management in Digital Enterprises[END_REF], is able to model the capability dependence on static environment information (e.g., resource utilization, calendar time or the weather condition), while including BROS enables the modeling of capabilities that depend on the participating performers (that is, actors and objects with abilities), illustrated in Fig. 5.

Technically, our proposal is realized as a meta-model extension to the CDD metamodel. Extending the meta-model also allows maintaining the adaptation and decision mechanisms of CDD. Thus, we strived for a non-invasive adaptation to implement the BROS features for two reasons: (a) to use CDD as a new source of business knowledge usable in BROS, and (b) to include the structural modeling concepts (scenes, roles, and objects) into CDD to provide a more powerful modeling approach.

Meta-model Extension

This section introduces the full (extended) meta-model, describes the respective model elements as well as their relationships and purposes.

To achieve the envisioned integration of CDD and BROS and keep the existing CDD method components and method extensions intact, the meta-model has been extended "non-invasively" (c.f. [START_REF] Sandkuhl | Capability Management in Digital Enterprises[END_REF] for more information about CDD methods components). The CDD-BROS integration is intended as a method extension. For this purpose, we extended the complete CDD meta-model with a set of new meta-model elements that override or extend already defined elements. As a result, the CDD meta-model ensures that the new extension is compatible with the CDD environment and other CDD extensions. The meta-model depicting CDD with the BROS extension is shown in Fig. 6. The set of BROS elements contains the newly developed elements. While ProcessVariant and ProcessVariantVariationPoint are overridden (i.e., marked as abstract) and not usable together with the BROS extension, the ContextElement and MeasurableProperty are extended and can be used simultaneously with the related BROS elements. Apart from the inherited elements, several new elements are used to model the BROS part of the combined approach. The newly introduced BROS elements are responsible for dedicated context decisions based on provided skills by entities. For that reason, we use the semantics of roles, objects, and scenes established in BROS.

A Performer is an entity (type) of the real world that is allowed to take over a Role in a Scene via Fulfillment to provide certain Abilities. For example, the performer "professor" is able to fulfill the role "teacher" in the scene "giving lecture" to achieve the ability (and responsibility) of "teaching". This is also possible for non-human objects like, e.g., the entity "room" that may fulfill the role "lecture hall." However, the performer and its roles depend on the use case that has to be modeled. With the meta-model extension, it is stated that the scene (a BROS concept) replaces the Pro-cessVariant. Thus, the process defined by CDD now uses scenes to describe different flows dependent on the chosen context. Scenes contain roles that are determined for a specific temporal execution (e.g., the "giving lecture" scene requires the role "teacher"). The variation mechanisms in CDD then do not point to process variants but specific scenes with roles. The roles are fulfilled by performers (real-world entities). The fulfillment between a performer and a role is annotated with the fulfillment value that quantifies the ability of the performer to fulfill a specific role. E.g., various instances of a performer "professor" fulfill the "lecturer" role in the scene "giving lecture" with their own specific FulfillmentValue in term of a skill profile. This value is inherited from MeasurableProperty, i.e., the fulfillment value is given by, e.g., a database in the ERP system that lists the employees with their skill profiles and may be timedependent. There must be at least one role (provided by a scene) that is responsible for providing the necessary abilities. However, at runtime, the concrete AbilityValue for the abilities is derived via ContextCalculation from the fulfillment value, i.e., dependent on the entity that takes over the specific role. Since the ability is inherited from ContextElement, the ContextElementRange (from CDD) is assigned to the abilities to limit the possible value range for the context. At runtime, those are concrete value boundaries. If the range is violated (due to not fitting ability values) the adaptation part of CDD uses the SceneVariationPoint to define another scene that is able to be used for the new context. However, this paper does not focus on the adaptation part, which is defined in the full CDD meta-model. In the proposed meta-model, we use Events since the BROS scene definition includes, inter alia, a start and end via an event. Thus, we use an event as an interface from the scene variation point towards the scene. This allows the start of multiple scenes with triggering a single start event.

The new meta-model elements are listed in Table 2. The M1 level is used for capability design, e.g., specifying that a lecturing capability is based on performers such as professors and roles such as teachers, students, and course assistants. The M0 level of a capability model materializes once the lecturing capability is executed and runtimespecific professors, e.g., "John" and "Alice", perform specific roles for specific scenes. An M1 instantiation example for the new elements in this meta-model is given in the next section. Due to the non-invasive changes to the original meta-model, all mechanisms of CDD, like the capability adjustment algorithms and calculations of KPIs and context, are still operational.

Use Case -Provisioning of Subjects in Higher Education

This section demonstrates the proposed CDD and BROS integration with an example case of the teaching environment at a large university in Sweden.

Use Case Description

The provisioning of the subjects in Higher Education requires substantial planning and effort. That includes organizing the lecturers' team, scheduling, admission of students, and publishing course materials. Once a course starts, the major activities are teaching sessions, exercises, supervision, and examinations. Requirements Engineering is a standard subject offered at both the undergraduate and graduate levels to about 250 students in total. The course is given in Swedish at the undergraduate level, and in English at the graduate level. The team of teachers includes several roles: lectures and Q&A seminars are given by professors; exercises are supervised by teaching assistants, PhD students, and professors; tool tutoring and supervision is done in the computer labs and led by teaching assistants, PhD students, and research assistants. The course material includes lectures, tutorials, reading material, and media. It is published on the Moodle online education portal. The platform is managed by the whole teaching team according to the assigned roles and responsibilities. During the course execution, the portal is also used for managing communication among the students and the teachers, management of quizzes, grading of exercises, as well as other activities. Since we investigate a Swedish university, there is the possibility of a sudden and severe snowfall in colder seasons. Hence, the local traffic information system and the weather forecast are analyzed for possible general delays. If severe delays throughout the city are to be expected or are occurring, the course events might be cancelled, rescheduled, repeated, and/or switched to online delivery.

Concerning course scheduling, each classroom has a limit for the maximum number of persons. Because the classrooms are a resource constraint, they need to be booked well in advance. If the number of students exceeds the size of the classroom, it is possible to stream a lecture to another classroom in real-time. This, however, poses additional tasks related to the management of the teaching process. It is not easy to re-schedule the rooms in cases when more students than estimated register for the course (the deadline is the day when the course starts), as well as when additional tutoring (and thereby rooms) becomes needed. The final exam is classroom-based and as such requires a sufficient number of places and invigilators for each of the examination rooms, which requires engaging both the teachers as well as additional staff.

Meta-model Instance Design

According the defined schema of model layers by Object Management Group [START_REF]Object Management Group: Meta Object Facility (MOF) Core Specification v2[END_REF][START_REF] Atkinson | Model-driven development: a metamodeling foundation[END_REF], the meta-model is on the M2 layer. An instance of this layer is the M1 layer, which is a model that uses the M2 defined concepts to specify the targeted "real model." An instance of M1 is on the layer M0, which represents "real items" like "Alice" as a person or "INF003" as a room. However, since the CDD and BROS extensions are on M2, we need to define the capability design on the M1 layer before considering runtime items. However, not all M2 concepts need to be instantiated on the M1 level because concepts that denote runtime concepts, like values of context elements or ability values, belong to M0 (i.e., fulfillment, fulfillment value, ability value, and event). For these values, we model the M1 pendants as a type that needs to be expressed at runtime. Thus, as elaborating the M0 is not our primary goal, we do not go into detail of their runtime assignments. Fig. 7. An example instance of the meta-model for the use case Fig. 7 shows the M1 instance of the M2 meta-model, including the CDD part (white background) and the new BROS concepts (gray background). We derived this instance example from the use case as only one of many different possibilities. The respective M0 types from the meta-model are annotated to the M1 model elements.

The use case requirements are encoded in the context set named "Req. Eng. Lecture", which serves the overall CDD capability of "Teach Req. Eng." The ranges (specified on M0 with concrete range boundaries later on) set the parameters of deciding between different scenes. For this use case, there are several of such ranges as parameters: a specific schedule delay, a required teaching language, a device that is able to display the slides, and so forth. If something happens, e.g., there is too little capacity available in the room for the lecture, the adaptation part of CDD (not shown in the meta-model) adapts towards this new situation with triggering changes. By using BROS, the capacity of a lecture room is modeled in Fig. 7 as an ability of a role that is fulfilled by a performer, i.e., any room (e.g., "INF003") that plays the role of the lecture hall at runtime so that its capacity is used for the lecture's capacity.

If at runtime this capacity is outside the range set in the capability design, the adaptation part has two options:

1. It uses a different performer (that is, a new room with higher capacity) that fulfills the role in the same scene so that the scene does not change; or 2. If there is no other performer available, the scene is switched to another scene (e.g., a scene that streams the lecture to various locations) that meets the range requirements with possibly other roles (e.g., a stream receiving device).

If neither of the two possibilities can be applied, then an error occurs since there is no available solution to the new context. The real-world entities, the performers, have to fulfill the roles in a scene, i.e., the CDD environment is able to perform calculations deriving the ability value out of their profile since the real room "INF003" at runtime does not know which abilities one wants to derive (e.g., its capacity or its ability to be ready for exams). The CDD environment delivers the actual runtime value for the ability (e.g., "15" for capacity) that gets checked against the range boundaries, which can be Boolean, lists, formal expressions or simple number ranges (e.g., "1 to 20"). The context set may also contain ranges that are not dependent on entities but on environmental states. In Fig. 7, we modeled the traffic situation and the weather forecast as measurable properties, so that the calculation results in the value of expected delay. This is checked against the range in the context set to decide whether it is possible to hold a lecture or whether one should start streaming (or skip the lecture). This expected delay is an environment-based state and independent from any concrete performers and roles (for demonstration purposes on how to model BROS-independent context elements). Thus, when designing capabilities, one has to decide between environment-or entity-based context elements and their ranges. Regarding the modeling complexity, we only designed a simple CDD-BROS model for one capability with limited scene-based variability. There are plenty of options to extend this design, e.g., multiple abilities or performers per role, performers that fulfill a set of roles in certain scenes, involving different IT supporting tools for teaching and other variants.

Use Case Discussion

With the modeled use case stated in Fig. 7, we argue that the modeling of entity-based context elements receives more attention in the capability design. Previously, every context element was handled as external. Thus, the modeling of capabilities is enhanced due to additional modeling constructs:

 The construct of scene allows the definition of concrete variations of an executable, providing a set of necessary roles;  Roles (encapsulated in scenes) enable modeling the necessary entity-based context elements (i.e., abilities of performers);  Performers are the main constructs to define the concrete entities that are responsible for fulfilling a context element range (indirectly via roles).

This trinity of the role-based BROS paradigm (scene, roles, and performers) is the tool for switching between contexts and related situations at runtime. When encountering an unmet range condition, the new possibility, to switch between fulfillments instead of switching to a whole new scene, is an important improvement. As such, the same context scene may be continued with only changing the performers, who are fulfilling the needed roles and their abilities for the scene. The combination of CDD and BROS allows modeling on multiple levels, i.e., subsequent stages along the model-driven development lifecycle (as shown in Fig. 5). With the usage of the BROS concepts in CDD, the transition towards implementation is simplified. BROS, as intended to be on the technical level, is tightly connected to the underlying software development. Assuming the fact that the CDD modeling would define several scenes, roles, and performers for its contexts, these can also be used for a related BROS model. Fig. 8 shows a possible BROS model for software construction that is derived from the CDD model in Fig. 7 with the same scene, its roles, and the related performers. This interrelation of the different abstraction levels while developing IS via CDD and BROS benefits model-driven development.

Conclusion

We have shown two existing approaches, CDD and BROS, which were developed independently and for different purposes, that have be combined to complement each other. This was done by integrating the needed BROS elements into the CDD metamodel as an extension. The combined approach was demonstrated with a use case and allowed for a more realistic and fine-grained modeling of an enterprise's capabilities. The introduction of entity-based abilities, related to roles, grasp the nature of capability responsibility and liability. Thus the CDD-BROS integration contributes to an enterprise modeling approach that can be used for early business-focused modeling as well as for later specification of technical details in a seamless manner. The different facets of adaptation and variation are covered through the combined approach encompassing adaptations at runtime due to resource allocation via performers. This, in general, supports role-based capability modeling and IS design as stated in Section 1.

Comparing this overall contribution to Enterprise Architecture approaches, we conclude that the suggested combination supports modeling on the level of detail that is needed for seamless IS development even encompassing runtime aspects like accounting for performers. However, in comparison to MDA-like approaches, which are (by nature) aligned to seamless integration along the development lifecycle, our suggestion is stronger when it comes to early capability driven modeling and contextdependent adaptations. One limitation of our current work is that we cannot ensure that other methodological enhancements in the enterprise architecture, enterprise modeling, or MDA domain may have achieved comparable goals. Also, the CDD-BROS-integration still needs to be fully implemented and supported with tools. This will require that the BROS extension to the CDD meta-model is implemented as a CDD method component. In the realm of a full-fledged integration, examples at the level of complexity of our use case could then be used to demonstrate the full potential of using CDD with BROS -not only at the modeling level but also within the accordingly developed IS.

Fig. 1 .

 1 Fig. 1. A conceptual meta-model supporting Capability Driven Development

  Set) should match the context in which the Pattern is applicable. Process Variant Process variant is a part of the Process using the same input and delivers the same outcome as the Process in a different way.

Fig. 2 .

 2 Fig.2. The basic BROS meta-model[START_REF] Schön | Business Role-Object Specification: A Language for Behavior-aware Structural Modeling of Business Objects[END_REF]

Fig. 3 .

 3 Fig. 3. The connection between both areas

Fig. 4 .

 4 Fig. 4. The set of common concepts

Fig. 5 .

 5 Fig. 5. Adding BROS to CDD

Fig. 6 .

 6 Fig. 6. The small CDD meta-model with the BROS extension

Fig. 8 .

 8 Fig.8. An example BROS model with objects, roles, events, and a scene

Table 1 .

 1 Concepts of the core CDD meta-model Value is a value of a specific Context Element at a given the runtime situation. It can be calculated from several Measurable Properties.GoalGoal is a desired state of affairs that needs to be attained. Goals can be refined into sub-goals. Goals should typically be expressed in measurable terms such as KPIs.ProcessProcess is series of actions that are performed in order to achieve particular result. A Process supports Goals and has input and produces output in terms of information and/or material. A process is perceived to consume resources.PatternPatterns are reusable solutions for reaching business Goals under specific situational contexts. The context defined for the

	Measurable	Context	KPI
		Property	Indicator
	Context Type	Context Element	Context Element Range	Indicator
	Concept Capability KPI Context Set Context ment Range Context ment Measurable Property Context Context Ele-Ele-Ele-Element Value Context Description Capability is the ability and capacity that enable an enterprise to achieve a business Goal in a certain context (represented by Context Set). Key Performance Indicators (KPIs) are measurable properties that can be seen as targets for achievement of Goals. Context Set describes the set of Context Elements that are rele-vant for design and delivery of a specific Capability. Context Element Range sets boundaries of permitted values for a specific Context Element and for a specific Context Set. A Context Element is representing any information that can be used to characterize the situation of an entity. Measurable Property is any information about the organiza-tion's environment that can be measured. Process Process Variant Capability Delivery Pattern Resource Situation Context Element Capability Goal Context Set
	ment Value

Table 2 .

 2 New meta-model elements within the BROS method component

	Concept	Description	M1 Example	M0 Example
	Performer	A real-world entity on	Person, Room,	Alice, INF003
		type-level that is able to	Computer
		do something
	Role	A context-specific be-	Attendant, Teacher,	Alice's Teacher
		havior that may be	Lecture Hall	role, INF003's
		adopted by a performer		Lecture Hall role
	Scene	A contextual boundary	Giving Lecture,	Lecture ID 5
		that denotes a temporal	Checking Exams
		execution
	Fulfillment The process of a per-	Employee-Teacher-	Alice playing the
		former playing a role	Fulfillment	Teacher
	Fulfillment	A runtime value that is	--	Profile of room
	Value	related to the profile of a		INF003 when
		role.		fulfilling Lecture
				Hall
	Event	A type of point in time	Start, End, Interrup-	9am at 24. Dec
		when something may	tion	2019, Incoming
		happen		Call ID 42

Acknowledgements

This work is partially funded by the German Research Foundation (DFG) within the Research Training Group "Role-based Software Infrastructures for continuouscontext-sensitive Systems" (GRK 1907).


Matthieu Josuat-Verg Ès
email: matthieu.josuat-verges@u-pem.fr

COMBINATORIAL PROOF OF AN IDENTITY ON GENOCCHI NUMBERS BE ÁTA B ÉNYI AND

Keywords: Mathematics Subject Classification. 05A19, 11B68 Genocchi numbers, poly-Bernoulli numbers, combinatorial identity, bijections, Callan sequences

In this note we present a combinatorial proof of an identity involving poly-Bernoulli numbers and Genocchi numbers. We introduce the combinatorial objects, m-barred Callan sequences and show that the identity holds in a more general manner.

Introduction

In this paper, we present a combinatorial proof of an identity that establishes an interesting relation between the C-poly-Bernoulli numbers and Genocchi numbers. The Genocchi numbers G n can be defined for instance by the generating function [START_REF] Stanley | Enumerative Combinatorics[END_REF]Exercise 5.8]:

∞ n=0 G n t n n! = 2t e t + 1
.

The first values are 0, 1, -1, 0, 1, 0, -3, 0, 17, 0, -155, . . .

(see A036968 in [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF]).

The Poly-Bernoulli numbers B (k)

n are defined for k ∈ Z by Li k (1 -e -t ) 1 -e -t = ∞ n=0 B (k) n t n n! , (1)
where Li k (z) is the polylogarithm function given by Li

k (z) = ∞ m=1 z m m k (|z| < 1).
For negative k indices, the poly-Bernoulli numbers are integers (A099594 in [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF]). Several combinatorial objects are enumerated by these numbers, for instance lonesum matrices [START_REF] Brewbaker | Lonesum (0, 1) matrices and the poly-Bernoulli numbers of negative index[END_REF], matrices uniquely reconstructible from their row and column sum vectors. For further objects see [START_REF] Bényi | Combinatorics of poly-Bernoulli numbers[END_REF][START_REF] Bényi | Combinatorial properties of poly-Bernoulli relatives[END_REF]. It is proven in [START_REF] Arakawa | On poly-Bernoulli numbers[END_REF] analytically and in [START_REF] Bényi | Combinatorics of poly-Bernoulli numbers[END_REF] combinatorially that n j=0 (-1) j B (-j) n-j = 0.

(2) Arakawa -Kaneko introduced a C-version of poly-Bernoulli numbers in [START_REF] Arakawa | Multiple Zeta values, poly-Bernoulli numbers, and related zeta functions[END_REF] as

Li k (1 -e -t ) e t -1 = ∞ n=0 C (k) n t n n! . (3)
In particular, C

n is the ordinary Bernoulli numbers B n defined by t e t -1 = ∞ n=0 B n t n n! .

For negative k, these numbers

C (k)
n are again positive integers. They are the array A136126 in [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF], see also Table 1 below. In [START_REF] Bényi | Combinatorial properties of poly-Bernoulli relatives[END_REF], this integer array is introduced as the number of lonesum matrices without any all-zero columns.

Kaneko -Sakurai -Tsumura [START_REF] Kaneko | On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application[END_REF] proved the C-version of the identity (2) using analytical methods.

n j=0 (-1) j C (-j-1) n-j = -G n+2 (n ≥ 0). (4)
In this note we present a combinatorial proof of the identity (4). Recently, Matsusaka [START_REF] Matsusaka | Symmetrized poly-Bernoulli numbers and combinatorics[END_REF] showed that both (2) and ( 4) are special cases of a more general polynomial identity between symmetrized poly-Bernoulli numbers and Gandhipolynomials.

Definitions

In this section we give the necessary definitions and notation. To prove the identity, we establish a connection between barred Callan sequences and Dumont permutations of the first kind defined in [START_REF] Dumont | Interprétations combinatoires des nombres de Genocchi[END_REF].

In the entry A099594 of the OEIS [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF], Callan defines a permutation class that are enumerated by the poly-Bernoulli numbers. These are essentially the same as the Callan sequences that were defined in [START_REF] Bényi | Bijective enumeration of Γ-free 0-1 matrices[END_REF]. Based on this notion, the authors introduced barred Callan sequences in [START_REF] Bényi | On the combinatorics of symmetrized poly-Bernoulli numbers[END_REF]. Here we recall these definitions.

Let k, n ≥ 0 and consider the sets K = {1, . . . , k} ∪ { * } (referred to as blue elements), and N = {1, . . . , n} ∪ { * } (referred to as red elements).

Definition 1. A Callan sequence of size k × n is a sequence (B 1 , R 1 )(B 2 , R 2 ) • • • (B m , R m )(B * , R * ),
for some m with 0 ≤ m ≤ n, such that:

• {B 1 , . . . , B m , B * } form a set partition of K into m + 1 non-empty blocks,

• {R 1 , . . . , R m , R * } form a set partition of N into m + 1 non-empty blocks,

• * ∈ B * , and * ∈ R * .

Note that a Callan sequence can be encoded in an obvious way by the following data:

• an integer m with 0 ≤ m ≤ min(k, n),

• an ordered partition (B 1 , . . . , B m ) of some subset K ⊂ {1, . . . , k},

• an ordered partition (R 1 , . . . , R m ) of some subset N ⊂ {1, . . . , n}. This is done by letting K = K \ B * and N = N \ R * .

To deal with these objects, it will be convenient to use the following terminology: . In particular, we have Remark 5. We note here that Callan sequences are equivalent to the special class of partial permutations where elements greater than n are excedances and the elements smaller than or equal to n are deficiencies. The number C k n also counts alternative tableaux of rectangular shape k × n, and other tableaux of rectangular shapes that are in bijection with alternative tableaux such as permutation tableaux or tree-like tableaux. See [START_REF] Bényi | Combinatorics of poly-Bernoulli numbers[END_REF][START_REF] Bényi | Combinatorial properties of poly-Bernoulli relatives[END_REF] The other number sequence that is involved in the identity we want to prove is the Genocchi numbers. Genocchi numbers can be defined in different ways, here we focus on the combinatorial point of view. There are several combinatorial objects that are enumerated by the Genocchi numbers. The first known interpretation was Dumont permutations of the first kind.

C k n = |C k n |.

Definition 6 ([8]

). A Dumont permutation of the first kind is a permutation π ∈ S 2n , such that each even entry begins a descent and each odd entry begins an ascent or ends the string, i.e., for every i = 1, 2, . . . , 2n:

π(i) even =⇒ i < 2n and π(i) > π(i + 1), π(i) odd =⇒ i < 2n and π(i) < π(i + 1), or i = 2n.
Let D 2n denote the set of Dumont permutations on 2n elements. In general, we have the following:

Theorem 8 (Dumont [START_REF] Dumont | Interprétations combinatoires des nombres de Genocchi[END_REF]). The cardinality of D 2n is (-1) n+1 G 2n+2 .

Main result

Using our notation, Identity (4) can be reformulated as follows:

Theorem 9 ([10]). For n ≥ 0, it holds that:

n j=0 (-1) j C j n-j = -G n+2 . (5)
We introduce some combinatorial objects, that will be at the core of our combinatorial proof of a generalization of the identity above. They can be seen as barred Callan sequences with extra structure. We could say, that they interpolate in a certain sense between barred Callan sequences and Dumont permutations. • A blue bar with label i is followed by a bar with label strictly smaller than i.

• A red bar with label i is followed either by a Callan pair, or by a bar with label strictly greater than i. The elements of C k n (m) are called m-barred Callan sequences, and the cardinality of

C k n (m) is denoted C k n (m). Remark 11.
The m-barred Callan sequences are defined otherwise in [START_REF] Bényi | On the combinatorics of symmetrized poly-Bernoulli numbers[END_REF], we call our special objects m-barred Callan sequence for the sake of convenience.

Remark 12. The numbers C k n (1) are the so-called poly-Bernoulli D-relatives, also introduced in [START_REF] Bényi | Combinatorial properties of poly-Bernoulli relatives[END_REF] as the number of lonesum matrices that do not contain any all-zero column and any all-zero row.

Note that a bar is always followed by another element in the sequence, so the last element of the sequence is a Callan pair (more precisely, it is necessarily the extra pair of the underlying Callan sequence). For example, an element of C 6 4 (3) is:

| 3 | 2 | 1 | 2 (5, 45)| 3 (79, 7)(4, 6)| 1 | 0 (68 * , * ).
Note 13. Sometimes it is convenient to think that each Callan pair is associated to the (possibly empty) sequence of consecutive bars preceding it. In the previous example, we have the four "groups":

| 3 | 2 | 1 | 2 (5, 45) | 3 (79, 7) (4, 6) | 1 | 0 (68 * , * ).
Moreover, the last bar of each group is a red bar.

Let us first record some properties that are simple consequences of the definition.

Proposition 14. We have the following identities:

C k n (m) = C n k (m), (6)
C k n (0) = C k n , (7)
C 0 0 (m) = (-1) m+1 G 2m+2 . (8)
Proof. There is a simple bijection between C k n (m) and C n k (m): it consists in exchanging blue blocks with red blocks in each Callan pair. As a consequence, we get the first identity.

The elements in C n k (0) contain a unique bar, | 0 . We thus have a simple bijection between C n k (0) and C n k by changing | 0 into |. Hence, we get the second identity. The elements in C 0 0 (m) contain a unique Callan pair, namely ( * , * ), in last position. This Callan pair is preceded by | m , as there is no label strictly greater than m. Now, there is a simple bijection between C 0 0 (m) and D 2m : remove | m ( * , * ), transform | i into 2i and | i into 2i + 1.

Our main result is that the numbers C k n (m) also have an alternating sum equal to Genocchi numbers: Theorem 15. For n, m ≥ 0, we have:

n j=0 (-1) j C j n-j (m) = (-1) m+1 G n+2m+2 . (9)
Note that when n is odd, both sides of (9) are 0 (the sum has an even number of terms, and they can be paired using the symmetry

C k n (m) = C n k (m))
. So only the case where n is even is of interest. Remark that the case m = 0 in the previous theorem is precisely Theorem 9.

We will use the following identity:

Proposition 16. For n ≥ 2 and m ≥ 0, we have:

n j=0 (-1) j C j n-j (m) = - n-2 j=0 (-1) j C j n-2-j (m + 1) (10)
The combinatorial proof of Proposition 16 will be done in the next two sections. To finish the present section, let us prove that Theorem 15, and therefore Theorem 9, are consequences of Proposition 16.

Proof of Theorem 15. As noted above, we can assume n is even. Denote n = 2n . We can iterate the identity in Proposition 16 successively with (n, m), then (n -2, m + 1), etc. Each iteration gives a sum with one less term, and we eventually get:

n j=0 (-1) j C j n-j (m) = (-1) n C 0 0 (m + n ).
From the third equation in Proposition 14, this is also (-1) n +n +m+1 G 2m+2n +2 which simplifies to (-1) m+1 G n+2m+2 .

The bijection φ

Here we use the map introduced in [START_REF] Bényi | Combinatorics of poly-Bernoulli numbers[END_REF] in order to prove [START_REF] Arakawa | On poly-Bernoulli numbers[END_REF]. We slightly modify this map, to reformulate it on the set of m-barred Callan sequences. This map is a bijection between two particular subsets of m-barred Callan sequences. Suppose that n ≥ 1 and k ≥ 0 are fixed throughout this section. We define the map B1) m + n is in an ordinary block as a singleton, i.e., there is an ordinary pair (B i , R i ) with R i = {m + n}. B2) m + n is in an ordinary block with other red elements, i.e., there is an ordinary pair replace the extra red block R * with { * }, replace (B i , R i ) with two pairs (B i , R * \{ * }) and ({m }, R ), move the new pair (B i , R * \{ * }) in front of the sequence, together with the group of consecutive bars preceding it (as in Remark 13).

φ : C k n (m, R * ) → C k+1 n-1 (m,
(B i , R i ) with m + n ∈ R i and R i \ {m + n} = R = ∅. Accordingly, φ(α) is
Example 18. Suppose that m, n, k are such that we remove 9 and add 8. The four cases can be illustrated as follows: Note that when k = 0 and n > 0, we have C 0 n (m, * ) = ∅. Indeed, the only 0 × n Callan sequence is ( * , 123 . . . n * ), which has a nonempty extra red block. Accordingly, we also take the convention C 0 n (m, * , |m ) = ∅. Note also that C k 0 (m, * , |m ) = ∅. Indeed, the only k × 0-Callan pair is (123 . . . k * , * ), so it is not possible to have m as a singleton in an ordinary pair.

Lemma 20. The map φ defined as above is a bijection between the sets C k n (m, R * ) and

C k+1 n-1 (m, * ) \ C k+1 n-1 (m, * , |m ). Proof. Let us first check that φ(α) ∈ C k+1
n-1 (m, * ). Note that each group of consecutive bars is unchanged (though these groups are possibly permuted in case B1 and B2), so we only need to check that the pairs form a valid Callan sequence. This is straightforward.

Then, let us check that φ(α) / ∈ C k+1 n-1 (m, * , |m ). In cases A1 and B1, the blue element m is added to a blue block (which is nonempty by definition) so m is not a single element in its block. In cases A2 and B2, m is in a singleton block, so we need to check that there is no bar preceding the pair containing m . In case A2, this is because this pair is at the beginning of the sequence. In case B2, the new pair ({m }, R ) is created with another pair to its left, and since the pair to its left is moved together with the associated group of bars, there is again another pair to the left of ({m }, R ) in φ(α).

In the previous paragraph, we observed the following properties in φ(α):

• m is in the first pair iff the sequence was obtained from the cases A1 or A2.

• m is alone in its block iff the sequence was obtained from the cases A2 or B2. So, the four cases can be distinguished based on which of the above properties they satisfy. In each case, knowing the position of m it is straightforward to describe the inverse procedure, so that φ is injective.

The bijection φ can be used to obtain pairwise cancellations in the alternating sums, and we get the following: Lemma 21. We have: by the C-poly-Bernoulli number, we applied the involution φ that reduced the number of elements to obtain a new set, on which we again applied the involution, until we reached the set that is known to be counted by the Genocchi number. This tricky solution shows somewhat the difficulty of the problem. It would be also interesting to find a direct bijection on any of the corresponding combinatorial objects that proves Theorem 9 without any iteration steps needed. Moreover, it is still an open problem to find a combinatorial proof for the more general identity related with symmetrized poly-Bernoulli numbers and Gandhi polynomials (see [START_REF] Matsusaka | Symmetrized poly-Bernoulli numbers and combinatorics[END_REF]).

Example 4 .

 4 The 7 barred Callan sequences for k = 2 and n = 1 are |(12 * , 1 * ); |(12, 1)( * , * ); |(2, 1)(1 * , * ); |(1, 1)(2 * , * ); (1, 1)|(2 * , * ); (2, 1)|(1 * , * ); (12, 1)|( * , * ). It is shown in [5] that the number of barred Callan sequences of size k × n is the C-poly-Bernoulli number C (-k-1) n . To avoid negative indices in what follows, we denote C k n = C (-k-1) n

Example 7 .

 7 For 2n = 4 we have D 4 = {2143, 3421, 4213}. For 2n = 6 we have D 6 = {642135, 634215, 621435421365, 342165, 214365, 564213, 563421, 562143, 216435, 435621, 215643, 436215, 364215, 421563, 356421, 421635.}

Definition 10 .

 10 Let C k n (m) denote the set of sequences defined by the following conditions. First, the elements in the sequence are:• m blue bars labelled with 1, . . . , m, and m + 1 red bars labelled with 0, . . . , m,• the Callan pairs of a k × n Callan sequence, shifted up by m (i.e., with base sets K = {m + 1, ..., m + k} ∪ { * } and N = {m + 1, ..., m + n} ∪ { * }). Second, these elements (labelled bars and Callan pairs) are ordered in a way such that:

Definition 17 .

 17 Let C k n (m, * ) denote the subset of C k n (m) containing elements α such that the extra red block is empty (by convention, this means it contains only * ). Let C k n (m, R * ) = C k n (m)\C k n (m, * ) denote the complementary subset. Following the previous convention, let C k n (m, * ) (respectively, C k n (m, R * )) denote the cardinality of C k n (m, * ) (respectively, C k n (m, R * )).

  * ) as follows. Let α be an m-barred Callan sequence with a non-empty extra red block, α ∈ C k n (m, R * ). The general idea is to remove the maximal red element m + n and add a new blue element m = m + k + 1. Distinguish four cases according to the location of the maximal red element m + n: A1) m + n is in the extra block as a singleton, R * = {m + n, * }. A2) m + n is in the extra block with other red elements, R * \ {m + n} = R = { * }.

  defined as the result of the following procedure: A1) Delete m + n and add the new element m to the first Callan pair. A2) Replace the extra red block R * with { * }, and place ({m }, R ) in front of the Callan sequence as the first element. B1) The successive steps are as follows:replace the extra red block R * with { * }, replace R i = {m + n} with R * \{ * }, move the new pair (B i , R * \{ * }) in front of the Callan sequence, together with the group of bars to its left (as in Note 13), add m to the (i + 1)st blue block (this means either B i+1 , or the extra blue block B * if B i was in the last ordinary pair). B2) The several steps are as follows:

A1) ( 5 ,

 5 35)| 1 (47, 247)| 1 | 0 (23, 68)(6 * , 9 * ) → (58, 35)| 1 (47, 247)| 1 | 0 (23, 68)(6 * , * ) A2) (5, 35)| 1 (47, 2)| 1 | 0 (23, 68)(6 * , 479 * ) → (8, 47)(5, 3, 5)| 1 (47, 2)| 1 | 0 (23, 68)(6 * , * ) B1) (5, 358)| 1 (47, 24)| 1 | 0 (23, 9)(6 * , 67 * ) → | 1 | 0 (23, 67)(5, 358)| 1 (47, 24)(68 * , * ) B2) (5, 38)| 1 (47, 4)| 1 | 0 (23, 259)(6 * , 67 * ) → | 1 | 0 (23, 67)(5, 38)| 1 (47, 4)(8, 25)(6 * , * ) Definition 19. For n ≥ 0 and k ≥ 1, let C k n (m, * , |m ) denote the subset of C k n (m, * ), containing elements α such that the maximal blue element m = m + k is in an ordinary Callan pair as a singleton, and there is at least one bar preceding this pair. Following the previous convention, let C k n (m, * , |m ) denote the cardinality of C k n (m, * , |m ).

(- 1 )(- 1 )

 11 j C j n-j (m, * , |m ).Proof. The set C j n-j (m) can be partitioned asC j n-j (m) = C j n-j (m, R * ) C j n-j (m, * ) \ C j n-j (m, * , |m ) C j n-j (m, * , |m ), j C j n-j (m, * ) \ C j n-j (m, * , |m ) + n j=0(-1) j C j n-j (m, * , |m ).

  • B * and R * are called extra blocks, while B i and R i are called ordinary blocks, • each pair (B i , R i ) or (B * , R * ) is a Callan pair, moreover, the former are called ordinary pairs and the latter is called the extra pair. A Callan sequence is thus a linear arrangement of Callan pairs, with the extra pair at the end.Definition 3. A barred Callan sequence of size k × n is a sequence obtained by inserting a bar (denoted |) into a Callan sequence of size k × n, with the restriction that the bar cannot be at the end of the sequence. We denote by C k n the set of barred Callan sequences with k blue elements and n red elements.

	Example 2. The Callan sequences with k = 2 and n = 2 are:
	(12 * , 12 * ),	(12, 12)( * , * ),	(1, 12)(2 * , * ),	(2, 12)(1 * , * ),	(12, 1)( * , 2 * ),
	(12, 2)( * , 1 * ),	(1, 1)(2 * , 2 * ),	(2, 1)(1 * , 2 * ),	(1, 2)(2 * , 1 * ),	(2, 2)(1 * , 1 * ),
	(1, 1)(2, 2)( * , * ), (1, 2)(2, 1)( * , * ), (2, 1)(1, 2)( * , * ), (2, 2)(1, 1)( * , * ).

Table 1

 1 gives the first few values of the numbers C k

n . Note that the symmetry C k n = C n k is clear from the combinatorial interpretation in terms of barred Callan sequences, as one can exchange red and blue blocks to get a n × k Callan sequence from a k × n Callan sequence.

Table 1 .

 1 C-poly-Bernoulli numbers C k n .

	for

Acknowledgements

The authors would like to thank the anonymous referees for helpful comments and suggestions.

The bijection φ readily shows that the first and second sum cancel each other out, upon checking the boundary terms.

The first boundary term is j = n in the first sum. The set C n 0 (m, R * ) is empty (as there are no red element besides * , the extra red block cannot be nonempty), so the corresponding term is 0. The second boundary term is j = 0 in the second sum. As noted after Definition 19, C 0 n (m, * ) is empty, C 0 n (m, * ) \ C 0 n (m, * , |m ) is empty as well, and the corresponding term is 0.

Using the previous lemma to prove Proposition 16 it remains only to show:

This will be done in the next section.

The bijection ψ

To prove [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF], it is convenient to use another set.

containing elements α such that the minimal blue element m + 1 is in an ordinary Callan pair as a singleton, and there is at least one bar preceding this pair.

There is a bijection from C k n (m, * , |m ) to C k n (m, * , |m + 1), via a relabelling of blue elements that exchanges m + k and m + 1. Using this, [START_REF] Sloane | The on-line encyclopedia of integer sequences[END_REF] is clearly equivalent to:

In order to show (13), we give a bijection ψ between the sets C k n (m, * , |m + 1) and Definition 23. For α ∈ C k n (m, * , |m + 1), note that α contains an ordinary pair (m + 1, R). Let w 1 (respectively, w 2 ) denote the maximal subsequence of consecutive bars that are directly to the left (respectively, to the right) of (m + 1, R). Note that w 1 is nonempty by definition, but w 2 is possibly empty. We define ψ b (α) as follows.

• Replace w 1 (m + 1, R) w 2 with w 2 | m+1 w 1 in the sequence.

• Replace the extra red block { * } with R ∪ { * }.

Example 24.

Secondly, we check that the labelled bars in ψ(α) sastisfy the conditions so that ψ(α) ∈ C k-1 n-1 (m + 1). We begin with labelled bars in ψ b (α). There are two locations to check:

• The element after | m+1 should be a labelled bar with label strictly smaller than m + 1. This element is the first one of w 1 , as w 1 is nonempty. By definition, it satisfies the required properties. • If w 2 is empty, either | m+1 begins the sequence or it is preceded by a Callan pair.

Otherwise, the last element of w 2 is a bar | i with i ≤ m, as it is followed by a Callan pair in α. In both cases, this is a valid configuration. Then, ψ(α) is obtained by adding | m+1 to the left of a Callan pair. Thus, either it is a new group of bars in itself, or it is added at the end of an existing group of bars. In the second case, as the last element of this group is a red bar with label i at most m, the configuration | i | m+1 is valid.

We thus have ψ(α) ∈ C k-1 n-1 (m + 1). Now it remains to describe the inverse bijection ψ -1 and check that it is indeed the left and right inverse of ψ. In the definition of ψ r , the cases can be distinguished: the first one (respectively, second one) results in | m+1 being to the left of the extra pair (respectively, of an ordinary pair). Using that, the inverse bijection is straightforward to describe explicitly.

Remark 28. Our bijections provided a combinatorial proof for the fact that the alternating diagonal sum of C-poly-Bernoulli numbers is given by the Genocchi number (Theorem 9). We used a special technique: instead of defining a direct involution on the set that is enumerated by the C-poly-Bernoulli numbers, we introduced a kind of interpolating sets, the m-barred Callan sequences. Starting with the set of Callan sequences, enumerated


Sylvain Iseni

Peter J Bruggeman

Klaus-Dieter Weltmann

Stephan Reuter

Nitrogen metastable (N 2 (A 3 Σ + u )) in a cold argon atmospheric pressure plasma jet: shielding and gas composition

Keywords: nitrogen metastable, laser-induced fluorescence, argon, atmopsheric pressure plasma jet, plasma medicine

N 2 (A 3 Σ + u ) metastable species are detected and measured in a non-equilibrium atmospheric pressure plasma jet by laser induced fluorescence (LIF). A shielding device is used to change the ambient conditions additionally to the feeding gas composition. Varying the amount of N 2 and air admixed to the feeding gas as well as changing the shielding gas from N 2 to air reveals that the highest N 2 (A 3 Σ + u ) is achieved in the case of air admixtures in spite of the enhanced collisional quenching due the presence of O 2 . The reasons for these observations are discussed in detail.

The interest in cold atmospheric pressure plasma jets (APP jets) has been growing continuously over the last decade, particularly in view of therapeutic applications 1 . APP jets are often obtained by noble gases -mostly helium (He), argon (Ar)-flowing through a capillary tube. This yields a low temperature reactive plasma plume expanding into the surrounding air 2 . Recent publications reporting the interaction of APP jets with biological tissues 3 evidence the real potential of such plasma sources to trigger beneficial biological responses. The impact of plasma on cells has been largely attributed to the presence of reactive oxygen and nitrogen species 4 (RONS) resulting from the non-equilibrium plasma chemistry reacting with ambient humid air 5 . Among these RONS, nitric-oxide (NO) is of significant importance due to its high reactivity and key role in biological pathways. NO plays a key role in cell signaling and vascularization 4 . The production of such RONS by cold Ar APP jets results from a complex non-equilibrium chemistry involving several reaction pathways 6 . Van Gaens et al. 7,8 have identified that NO production is not governed by the Zel'dovich reactions -as it is the case at elevated gas temperatures for combustion conditions-but rather via electronically excited N 2 (A 3 Σ + u ) -metastable state-reacting with atomic oxygen (O). As N 2 (A 3 Σ + u ) is a precursor species for the production of NO 7 , the obtained results in this manuscript are valuable to gain a better understanding of the NO production in cold plasma jets.

Several groups have already studied atomic He 9-11 and Ar 9 metastables in APP jets. With a strong focus on Hea) Corresponding author: sylvain.iseni@univ-orleans.fr O 2 and Ar-O 2 mixtures, these metastables are important precursors for RONS formation. However metastable molecules including the N 2 (A 3 Σ + u ) have not been investigated in cold APP jets. Investigations of N 2 (A 3 Σ + u ) in atmospheric pressure discharges have been performed in corona 12 , streamer 13 and dielectric barrier discharge (DBD) 14 . The direct quantitative measurement of N 2 (A 3 Σ + u ) remains challenging. Several approaches are reported in literature involving different diagnostic techniques such as optical emission spectroscopy 15 (OES), cavity ring-down spectroscopy 16 (CRDS) and laser-induced fluorescence 13,17 (LIF). Common challenges for those diagnostics are the absolute calibration and the often weak fluorescence or absorption signals. Optical-optical double resonance LIF (OODR-LIF) was introduced by Dilecce et al. 14 to enhance the weak fluorescence signal but is a complex technique to implement.

In this letter, we report on the detection and measurement of the relative density of N 2 (A 3 Σ + u ) in the plasma plume of a radio-frequency driven Ar APP jet with a coaxial geometry: the kinpen 18 . The plasma dissipates a power ranging between 0.8 W to 2.2 W and the gas temperature is less than 500 K 19 . The source is operated at the flow rate of 1.0 standard liter per minute (slm). The feeding gas is Ar with small admixtures (up to 1 %) of N 2 and air. The 1.0 slm flow rate is chosen to ensure a laminar gas flow 20 and to prevent excessive mixing of the effluent gas containing RONS with the surrounding gas 19 . Additionally, a custom-made nozzle covers the device head and generates -via an external gas flow-a gas shielding enveloping the plasma plume 21,22 . This controls the gas species surrounding and diffusing into the plasma effluent. In this work, LIF is used for its good spatial and temporal resolution. A scheme of the experimental setup is shown in figure 1. A dye laser (Sirah Lasertechnik, Cobra-Stretch) with pyridine 1 is pumped by a frequency doubled Nd:YAG laser (Spectra-Physics, Quanta-Ray Pro-230). The laser has a pulse a duration of ∼11.9 ns (FWHM) and operates at 10 Hz. The dye laser is tuned to 687.44 nm (with a linewidth of ∆λ=2.2 pm FWHM) to pump the lower rotational levels (J ≤ 4) of the P 11 and Q 12 branch from the transition N 14 . The laser beam is cropped with an iris to produce a 2.0 mm diameter spot to probe a plasma plume volume 2.0 mm away from the capillary exit if not stated otherwise. The laser irradiance (I L ) is (60.0 ± 1.6) mJ/cm 2 . We have checked that the LIF intensity scales linearly with the laser energy at this irradiance. The LIF signal (S LIF ) of the transition N

2 (B 3 Π g , v = 3) ← N 2 (A 3 Σ + u , v = 0)
2 (B 3 Π g , v = 3) → N 2 (A 3 Σ + u , v = 1
) is imaged with a lens on the slit of a monochromator (Andor Technology ltd., Shamrock 750, 1200 grooves ruled grating Blazed at 500 nm) fine-tuned on 762 nm. The later is also equipped with a bandpass filter (cw:760 nm, FWHM:10 nm) to block the stray light. The detection range of this system is 9.1 nm allowing to record the entire vibrational band. The signal is recorded by a multichannel plate photo multiplier (MCP-PMT, Hamamatsu, R5916U-51) read out by an oscilloscope (Tektronix, DPO 70000, 8 GHz bandwidth). The LIF technique is applied in this work under the following assumptions:

• The laser energy is low enough to excite a small fraction of N 2 (A 3 Σ + u , v = 0) so that its population is considered constant and the population of the excited state is much smaller than the ground state. Thus, the effect of stimulated emission can be neglected.

• The emission coefficients and quenching rates are assumed to be identical for all rotational states of N 2 (B 3 Π g , v = 3) and the rotational energy transfer is fast compared with the fluorescence time scale.

• Both rotational levels of the ground state and the metastable state N 2 (A 3 Σ + u ) are in equilibrium with the gas temperature and can be described with a Boltzmann distribution. This formalism was also used by van Gessel et. al. for the diagnostic of NO under similar conditions 23 . The density of N 2 (A 3 Σ + u ), n N2(A) , is in this case proportional to the ratio of the fluorescence intensity and is the effective fluorescence lifetime,

n N2(A) ∝ S LIF τ • I L , ( 1
)
where τ = (A B + Q) -1 the fluorescence lifetime. A B (in 1/s) is the spontaneous emission rate of the v = 3 level (2.30 × 10 6 /s) 24 and Q = i q i • n i (1/s) the quenching rate with q i (cm 3 /s) the quenching rate coefficient and n i (1/cm 3 ) the density of the quencher i.

Figure 2 presents a typical time evolution of the fluorescence signal recorded by the MCP-PMT after an averaged acquisition of 10 4 shots. Compared with the laser excitation pulse I L (t) , the signal has a clear asymmetric evolution and results from the convolution of I L (t) with an exponential decay. Since the jitter of the laser is less than 0.5 ns, the time instrument response function 2 are not significantly altered by the detection system. A numerical convolution between I L (t) and an exponential function, A LIF • exp -t τ , is computed and optimized with a Levenberg-Marquardt algorithm to obtain the best fit of the recorded signal. The amplitude, A LIF and the decay time τ are the fitting parameters. An example is shown in figure 2. S LIF = A LIF • τ is the integration of the fluorescence signal. This provides the necessary input for equation 1 to calculate n N2(A) .

We measured N 2 (A 3 Σ + u ) relative densities using both N 2 and air as admixture and shielding gas. Table I introduces the measured values of the fluorescence decay time and the deduced quenching rate (coefficient) for different gas admixture/gas shielding parameter sets. The case with N 2 admixtures and N 2 shielding has a longer radiative lifetime of the N 2 (B 3 Π g ) state, while an admixture of air in the plasma plume, or air gas shielding reduces the fluorescence lifetime by a factor 3. This is explained by the O 2 fraction in the buffer gas. O 2 quenches the N 2 (B 3 Π g ) state much more effectively compared with N 2 and Ar. Table II reports quenching rate coefficients of the N 2 (B 3 Π g ) state for the main gases used in this work. In the cases where air and N 2 are simultaneously used as admixture and/or shielding gas, quenching rates are similar. The influence of the gas temperature on the quenching rate is estimated to be about 15 % between positions 1.5 mm and 5.0 mm due to a temperature difference of ∼50 K 19 .

Relative densities of N 2 (A 3 Σ + u ) as a function of air or N 2 admixture are presented in figure 3 shielding. n N2(A) can vary by a factor 7 depending on the type of admixture ranging from 0 % to 1 % and the air or N 2 shielding gas diffusing in the plasma plume. As a result, an increase of molecular gas fraction up to 1.0 % induces a decrease in metastable density exceeding a factor 2.5 for all conditions. It is not excluded that the molecular admixtures affect the deposited power in the plasma. This may result in a density drop such as observed between 0.6 % and 0.8 %. This was already observed by Verreycken et. al 31 in the case of OH production. Densities obtained by chemical kinetics model of this discharge 8 follow the same trend and are in good agreement with the case of N 2 admixtures and air shielding presented in figure 3. Variations of n N2(A) without admixture of air or N 2 result from the fact that experimentally, pure Ar conditions are challenging to achieve due to inherent impurities in the feeding gas. Interestingly, a significant increase of n N2(A) is observed when the plasma plume is surrounded by air while no molecular admixture is supplied. This means that the diffusion of air into the plasma plume influences the plasma chemistry enhancing the generation of N 2 (A 3 Σ + u ). In addition, the presence of O 2 while admixing air was expected to negatively impact n N2(A) since O 2 is an efficient quencher of N 2 (A 3 Σ + u ) (cf. table II). Recently, van Gessel et. al 32 published an experimental study of electron properties of an Ar APP jet similar to the plasma device used in this work. The authors report similar electron temperatures (T e ) with admixtures of N 2 and air in the Ar jet while the electron density (n e ) is higher for the air admixture than N 2 admixture case. As in this discharge the formation mechanism of N 2 (A 3 Σ + u ) is mostly driven by direct electron impact 8 , a strong correlation between n N2(A) and n e is expected. A similar dependence was notified by the same group 33

Figure 1 .

 1 Figure 1. Schematic of the experimental setup used for the measurement of N2(A 3 Σ + u ) in a cold Ar APP jet with shielding gas.

Figure 2 .

 2 Figure 2. Time resolved LIF signal and Rayleigh signal in air as recorded by the MCP-PMT. The fluorescence curve of N2(B 3 Πg) is calculated from the fit of the LIF signal by a convolution of the laser pulse with an exponential decay (τ =11.4 ns). The data is recorded for Ar + 1.0 % air and N2 as a shielding gas.

Figure 3 .

 3 Figure 3. N2(A 3 Σ + u ) relative densities versus feeding gas admixture fractions for different types of gas shielding. The measurements are space averaged over the section of the plasma plume (2 mm) and time averaged over 10 4 laser shots.

Table I .

 I Measured values of τ and Q for different combinations of feeding gas admixtures and shielding gas.

	Admix.	Shield.	τ / 10 -9 s	Q / 10 7 /s
	N2	N2	19.7 ± 1.5	5.1 ± 0.4
	N2	air	8.6 ± 0.6	11.6 ± 0.8
	air	N2	10.8 ± 1.9	9.4 ± 1.6
	air	air	6.1 ± 0.9	16.4 ± 2.4
	a N2	air	8.5 ± 0.6	11.7 ± 0.8

a Value obtained at 5.0 mm from the capillary.

Table II .

 II Quenching rate coefficients (qi) of N2(A 3 Σ + u ) and N2 (B 3 Πg) for different gases.

	Quencher	Rate / 10 -11 cm 3 /s
		N2(A 3 Σ + u )	ref.	N2 (B 3 Πg)	ref.
	Ar+ 0.3 % N2	-		>1	25
	N2	10 -9	26	3 ± 0.2	27,28
	O2	0.50 ± 0.03	29	20 ± 0.5	28
	N2(A 3 Σ + u )	19 ± 0.5	27	-
	NO	8.0 ± 1.5	29	-
	O	3.1 ± 0.5

30

of our MCP-PMT is 85 ps and the time constant of the measuring circuit with 50 Ω load is 1.3 ns, one can safely assume that both signals presented in figure

with the observation of NO(A) emission resulting from an excitation of NO(X ) by N 2 (A 3 Σ + u ). The NO(A) emission intensity gives a relative estimation of n N2(A) and correlates with the measured n e during one modulation cycle 32 . Moreover, higher n e in the presence of air may also compensate the quenching of N 2 (A 3 Σ + u ) by O 2 . According to figure 3, the nature of the gas shielding has a larger influence on n N2(A) than the reduction of O 2 in the feed gas while admixing air. Surprisingly, n N2(A) is higher when the plasma plume is surrounded by air with air admixture ( ) than in conditions with air substituted by N 2 ( ). One infers from this result that a reduction of O 2 in the shielding gas is not conducive to the production of N 2 (A 3 Σ + u ). Schmidt-Bleker et. al. published the results of a combined theoretical and experimental work and suggest an explanation for the enhancement of the discharge intensity in the presence of O 2 in the shielding gas 34 . The presence of electronegative molecules, such as O 2 , in the vicinity of the plasma plume is prone to form anions. Their lower mobility compared with electrons induces an additional electrostatic focusing mechanism 34 which confines the electric field parallel to the gas flow and leads to the generation of a more intense discharge. The case of air used as a feeding gas admixture and as shielding gas ( ) may benefit from the combination of a higher n e with the contribution of an electrostatic focusing mechanism. This results in the highest production of N 2 (A 3 Σ + u ) for this condition. Figure 4 shows the axial evolution of n N2(A) . In view of the lifetime of N 2 (A 3 Σ + u ) which equals approximately 5 µs (for Ar + 1 % air), the N 2 (A 3 Σ + u ) is confined to the active plasma zone. Figure 4 even shows that the N 2 (A 3 Σ + u ) density is reduced by 2 orders of magnitude within the active plasma effluent. This is in agreement with prior numerical chemistry kinetic analysis 8 on N 2 (A 3 Σ + u ) ranging from 10 14 /cm 3 to 10 12 /cm 3 . Van Gessel et. al. found similarly a strong correlation between the NO(A) emission and the active plasma effluent.

In conclusion, N 2 (A 3 Σ + u ) relative densities have been measured by LIF in a low temperature Ar APP jet (<500 K). The N 2 (A 3 Σ + u ) density strongly correlates with the discharge intensity and is confined to the active plasma zone. Interestingly, a reduction of O 2 in the feeding gas and surrounding gas is not prone to the production of N 2 (A 3 Σ + u ). The maximum production is achieved in the case of air that enhances the discharge intensity. This is explained by a higher n e and electrostatic focusing mechanism for this condition. N 2 (A 3 Σ + u ) is an important intermediate specie responsible for the production of NO in many biomedical applications.


Mohammad Abobala

On the Representation of Winning Strategies of Finite Games by Groups and Neutrosophic Groups

Keywords: Group, Neutrosophic group, Winning strategy 1-Introduction :

In this paper, we show that for a finite game with two players A , B:

Each winning strategy of the first player A can be represented by a neutrosophic subgroup of the neutrosophic group ( , and each winning strategy of the second player B can be represented by an elementary abelian group .

Also, we introduce the concept of algebraically relative games and present some examples on it.

Neutrosophy as a branch of philosophy introduced by F.Smarandache has many applications in the real world and the mathematical concepts. The concept of neutrosophic group had been defined in [START_REF] Kandasamy | F , some neutrosophic algebraic structures and neutrosophic N-algebraic structures[END_REF] as a generalization of classical groups, subgroups and normal subgroups also were defined and studied.

The most useful understanding of neutrosophic group has been written in [START_REF] Chalapathi | neutrosophic graphs of finite groups[END_REF], we consider N(G) as a union of G and GI i.e N(G)= We will use a neutrosophic subgroup to represent every winning strategy of player A, and a classical group to represent every winning strategy of player B. This research maybe very useful in the progression of game theory by algebraic views.

2-Preliminaries :

Definition 2.1 : [START_REF] Kandasamy | F , some neutrosophic algebraic structures and neutrosophic N-algebraic structures[END_REF] Let (G,*) be a group . Then the neutrosophic group is generated by G and I under * denoted by N(G)= .

I is called the indeterminate element (neutrosophic element) with the property .

The most useful understanding of this definition has been written in [START_REF] Chalapathi | neutrosophic graphs of finite groups[END_REF], we consider N(G) as a union of G and GI i.e N(G)= .

Definition 2.2 :[2]

Let N(G) be a neutrosophic group, then a neutrosophic subgroup is a subset of N(G) contains a proper subgroup of G.

Remark2.3 :[2]

Neutrosophic subgroup is not a group but contains a group.

Definition 2.4 :[5]

An abelian group G is called elementary abelian if it is isomorphic to for such a positive integer n.

For concepts like game, analyzing game, and combinatorial game see [START_REF] Dabash | Create HIM , the adjusted NIM game , and analyses its winning strategy[END_REF].

3-Main results and discussion :

Suppose that G is a game with finite steps. Two players A , B play this game, they make their steps alternately, i.e (their choices) from a finite set of objects S={ .

If we reach to a position which A cannot chose any object then B is the winner, and conversely A is the winner.

Without affecting the generality we can suppose that the alternating choices of two players can be realized as :

B A . … ….
We say that a step i is complete if both players were able to chose objects without being losers.

For each complete step, we can represent it by a bijective map f wich permutes the chosen objectives in this step an fixes the rest of unchosen objectives, i.e if the player A chooses the element and B chooses , then we represent this complete step by the map : f with f( )= and f( )= and f( )= for each , we can use algebraic symbol as: f = ( .

Theorem 3.1 :

Let be the representation of the complete step i, then =I (of order 2) , where I is the identity map on S.

Proof :

It is easy to see that .

We represent the beginning position of the game by I (identity map).

Theorem 3.2 :

Each winning strategy of second player B can be represented by a group with type Proof :

If B has a winning strategy, then we will reach to a position that B can choose and player A cannot, as follows :

B A . ……………………………. ………………………….. We assume that the number of steps is k, we remark that all steps are complete and each step's representation is a bijective with order 2, so the group generated by all representations is .

We call the previous group by a strategy representation.

Definition 3.3 :

If we reach to a position which A can chose and B cannot, we represent it by the indeterminate map J, which it means that A can pick an object and B cannot.

Remark: The indeterminate map J has the property , we mean by this property that if we reach to a winning position of player A , then the next position is the same.

Theorem 3.4 :

Each winning strategy of first player A can be represented by a neutrosophic subgroup with type .

Proof :

If A has a winning strategy then we will reach to a position that A can choose and then B cannot , as the following : B A

We assume that the number of steps is k+1 , we remark that all steps are complete unless the last step. The group generated by all steps unless the last one is .

For the last step we can represent it by the indeterminate J, thus the strategy representation is the neutrosophic subgroup of N( which is set :

If A is the winner then the strategy representation is a neutrosophic group , and if B is the winner then the strategy representation is a classical group Definition 3.6 :

(a) If the player B has a winning strategy, then the winning strategy with minimum representation group order is called the perfect strategy of B.

(b) If the player A has a winning strategy , then the winning strategy with minimum representation group order is called the perfect strategy of A.

Definition 3.7 :

If H , K are two finite games , we say that H is algebraically relative or (H-ar-K), if there is a perfect strategy of the Player A in both games with the same representation neutrosophic group, or a perfect strategy of the player B in both games with same representation group .

Remark : The essential meaning of algebraically relative games is that they have winning strategies with the same number of steps.

Example 3.8 :

Suppose that we have two players A , B which they are playing Wythoff game with (3,2) as a beginning position , A at least needs two steps to win , we can clarify it by the following example:

2) (after the choice of A ) A chooses (1,1) and wins

The representation neutrosophic subgroup is Let the same players play the HIM-Game defined in [START_REF] Dabash | Create HIM , the adjusted NIM game , and analyses its winning strategy[END_REF]. The beginning position is [START_REF] Kandasamy | F , some neutrosophic algebraic structures and neutrosophic N-algebraic structures[END_REF][START_REF] Dabash | Create HIM , the adjusted NIM game , and analyses its winning strategy[END_REF][START_REF] Haushi | Algebraic structures[END_REF][START_REF] Abobala | n-Refined Neutrosophic Groups I[END_REF] , A has a perfect strategy as B A

(2,4) after B choice (2,4,2,2) after A choice A chooses [START_REF] Kandasamy | F , some neutrosophic algebraic structures and neutrosophic N-algebraic structures[END_REF][START_REF] Dabash | Create HIM , the adjusted NIM game , and analyses its winning strategy[END_REF]

and wins

The representation neutrosophic subgroup is , thus the previous two games are algebraically relative.

4-Conclusion

In this research, we have introduced a representation of winning strategies of finite alternating games by groups and neutrosophic groups. Also, we have introduced the notion of algebraically relative games and gave many examples.