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Let p_[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p_[1,r;t](n) is the generalisation of the two-colour partition function p_[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

This chapter proposes a multiobjective model to design a Closed Loop Supply Chain (CLSC) network. The first objective is to minimize the total cost of the network, while the second objective is to minimize the carbon emission resulting from production, transportation, and disposal processes using carbon cap and carbon tax regularity policies. In the third objective, we maximize the service level of retailers by using maximum covering location as a measure of service level. To model the proposed problem, a physical programming approach is developed. This work contributes to the literature in designing an optimum CLSC network considering the service level objective and product substitution.

The purpose of this paper is to assess the application of graph coloring and domination to solve the airline-scheduling problem. Graph coloring and domination in graphs have plenty of applications in computer, communication, biological, social, air traffic flow network and airline scheduling.

The process of merging the concept of graph node coloring and domination is called the dominator coloring or the χ_d coloring of a graph, which is defined as a proper coloring of nodes in which each node of the graph dominates all nodes of at least one-color class.

The smallest number of colors used in dominator coloring of a graph is called the dominator coloring number of the graph. The dominator coloring of line graph, central graph, middle graph and total graph of some generalized Petersen graph P_(n ,1) is obtained and the relation between them is established.

The dominator coloring number of certain graph is obtained and the association between the dominator coloring number and domination number of it is established in this paper.

This paper provides an introduction to the Metadata Object Description Schema (MODS), a MARC21 compatible XML schema for descriptive metadata. It explains the requirements that the schema targets and the special features that differentiate it from MARC, such as user‐oriented tags, regrouped data elements, linking, recursion, and accommodations for electronic resources.

There continues to be a perception that online education is inferior to traditional education. In the U.S. online learning is more developed than in the U.K. This paper provides insights into a U.S. provision and takes a close look at what are per ceived as weak nesses of on line learn ing and ar gues that these are not necessarily inherent weaknesses of this form of educational delivery. Then, results of two major studies, undertaken in the U.S. are provided comparing the effectiveness of online education to traditional education as perceived by current MBA students and past graduates. Results of these studies suggest that students of MBA modules and MBA graduates perceive the quality and effectiveness of online education to be similar to, if not higher than, the quality and effectiveness of traditional modules and programmes.

The paper's aim is to evaluate the changing patterns of defence requirements and their implications on supply chains and relationships within the UK defence industry.

The paper builds a case study on the UK defence industry comprising 22 face‐to‐face interviews with senior management from the Ministry of Defence (MoD) and major first tier suppliers, as well as senior officers in the British armed forces.

The results suggest that there are major changes currently taking place that have major impacts on defence supply relationships. The authors find a consensus in the industry concerning a shift towards through‐life management (TLM), where major equipment platforms are kept in service for several decades. TLM is widely acknowledged as requiring much closer partnerships in the defence supply chain, in which suppliers assume much greater responsibilities in areas such as in‐service support and maintenance. Yet the findings with MoD and suppliers reveal different perceptions of the feasibility and practical implications of the proposed changes.

Product‐service specific capabilities need to be developed especially in areas such as accurate lifecycle costing. The development of integrated supply partnerships requires greater emphasis on openness, risk and reward sharing, trust and long‐term commitment in supplier relationships. There is also a need for early supplier involvement to ensure not only design for manufacture, but design for maintainability and logistics, instigated and managed by the customer (i.e. MoD).

The analysis demonstrates the importance of adopting a through‐life perspective when considering industrial contexts characterised by very long product lifecycles. This study shows that a through‐life perspective creates a blurring of the boundary between customers and suppliers, and increases long‐term supplier responsibility. This gives rise to new considerations, such as sophisticated risk and rewards sharing mechanisms, design for maintainability, and technology insertion.

The purpose of this paper is to highlight the international significance of multi-owned developments (MODs), present an MOD life cycle conceptual model and review the range of identified peer-reviewed empirical research papers.

The paper utilises an exploratory qualitative methodology to collate and analyse literature focusing on MODs. From the 403 research papers identified 96 peer-reviewed empirical research papers specific to MODs were examined. A MOD life cycle model has been conceptualised to facilitate a content analysis of the reviewed papers.

The findings of this paper highlights the gaps in knowledge pertinent to MODs and outlines avenues for future research that argues for the need to develop a more holistic and multi-disciplinary research approach.

The research is based on reviewing published papers as identified using specific search terms.

The paper makes an important and innovative contribution to the body of knowledge by developing a MOD life cycle model and identifying the range and scope of peer-reviewed empirical research literature published on MODs. Understanding the MOD life cycle phases and the gaps in the literature can enable academics from a multitude of disciplines to enhance this field of inquiry in order to better understand this emergent property type.

This paper describes the Metadata Object Description Schema (MODS), its accompanying documentation and some of its applications. It reviews the MODS user guidelines provided by the Library of Congress and how they enable a user of the schema to consistently apply MODS as a metadata scheme. Because the schema itself could not fully document appropriate usage, the guidelines provide element definitions, history, relationships with other elements, usage conventions, and examples. Short descriptions of some MODS applications are given and a more detailed discussion of its use in the Library of Congress’s Minerva project for Web archiving is given.

This case involves the March of Dimes (MOD) Foundation, the “leading nonprofit organization for pregnancy and baby health.” MOD’s mission was to support medical research, organize volunteer workers, and provide community services and education to save babies’ lives (www.marchofdimes.org). The strategic issue in the case involves creating awareness of both the mission and services of MOD and the critical issue driving that mission – premature births. The organization must create a desire for various target markets to take action in response to the problem. The main protagonist is Doug Staples, Senior Vice President for Marketing and Communications.

Data were collected via personal interviews with the primary protagonists, Doug Staples, and Mike Swenson of the Barkley agency. The MOD provided quantitative Gallup studies they commissioned, as well as documents unveiling the roll-out in the San Jose, CA region. The Barkley Agency provided qualitative data from a study which consisted of eight focus groups conducted in two markets and ten personal interviews. Secondary research was used to provide a support for industry and market data, to supplement organizational facts provided by the MOD, and to identify and link marketing theory to the situations provided in the case. The organization, facts and characters in this case were not disguised. MOD was consulted throughout the case development process.

This case study is recommended for marketing courses at the undergraduate level. It is most appropriate for marketing management, introductory marketing, or marketing strategy classes. Additionally, this case is a good fit for courses focused upon not-for-profit marketing issues.

The strongest opportunities to apply theory using this case relate to branding (see De Chernatony and Dall’Olmo Riley, 1998 for a content analysis of the brand literature). These theories include brand image and personality (Aaker, 1997; Belk, 1998; Grohmann, 2009), brand awareness (Aaker, 2002), brand involvement and customer loyalty (Brakus et al., 2009), brand engagement (Sprott et al., 2009), brand relationships (Breivik and Thorbjornsen, 2008), and brand equity (Aaker, 2002, 2008). Specifically, question 2 addresses brand personality, and questions 3 and 4 explore relationships with the brand such as the emotional power of the brand and brand association. Question 6 focuses on positioning strategy.

In this paper, the author defines the function B¯i,jδ,k(n), the number of singular overpartition pairs of n without multiples of k in which no part is divisible by δ and only parts congruent to ± i, ± j modulo δ may be overlined.

Andrews introduced to combinatorial objects, which he called singular overpartitions and proved that these singular overpartitions depend on two parameters δ and i can be enumerated by the function C¯δ,i(n), which gives the number of overpartitions of n in which no part divisible by δ and parts ≡ ± i(Mod δ) may be overlined.

Using classical spirit of q-series techniques, the author obtains congruences modulo 4 for B¯2,48,3(n), B¯2,48,5 and B¯2,412,3.

The results established in this work are extension to those proved in Andrews’ singular overpatition pairs of n.