Here, we describe the bicomplex (p,q)- Fibonacci numbers and the bicomplex (p,q)- Fibonacci quaternions that are based on these numbers and give some of their equations, including the Binet formula, generating function, Catalan, Cassini, d’Ocagne’s identities, and some summation formulas for both of them. Finally, we create a matrix for bicomplex (p,q)- Fibonacci quaternions, and we obtain a determinant of a special matrix that gives the terms of that quaternion.

We evaluate the nested sum $\sum_{a_{n - 1} = c}^{a_n } {\sum_{a_{n - 2} = c}^{a_{n - 1} } { \cdots \sum_{a_0 = c}^{a_1 } {x^{a_0 } } } }$ where $a_n$ and $c$ are any integers and $x$ is a real or complex variable. Consequently, we evaluate multiple sums involving the terms of a general second order sequence, the Horadam sequence (W_j (a, b; p, q)), defined for all non-negative integers j by the recurrence relation W₀ = a, W₁ = b; W_j = pW_j−1 − qW_j−2 (j≥2); where a, b, p and q are arbitrary complex numbers, with p ≠ 0, q ≠ 0.

It is not understood how the numbers and identities of vertebrae are controlled during mammalian development. The remarkable robustness and conservation of segmental numbers may suggest a digital nature of the underlying process. Here I propose a mechanism that allows cells to obtain and store the segmental information in digital form, and to produce a pattern of chromatin accessibility that in turn regulates Hox gene expression specific to the metameric segment. The model requires that a regulatory element be present such that the number of occurrences of the motif between two consecutive Hox genes equals the number of segments under the control of the anterior gene. This is true for the recently discovered HRC3 motif, associated with histone modifications and developmental genes. The finding not only allows correctly predicting the numbers of segments using only sequence information, but also resolves the 40-year-old enigma of the function of temporal and spatial collinearity of Hox genes. The logic of the mechanism is illustrated in an animated video: https://youtu.be/4a3XOQ7Lz28. I also discuss how different aspects of the proposed mechanism can be tested experimentally.

In the paper, the authors discover several series identities involving the Catalan numbers, the Catalan function, the Riemanian zeta function, and the alternative Hurwitz zeta function.

New technologies have resulted in a better understanding of blood and lymphatic vascular heterogeneity at the cellular and molecular levels. Meanwhile, we still need to learn more about the heterogeneity of the cardiovascular and lymphatic systems between different species at the anatomical and functional levels. Even the deceptively simple question of the functions of fish lymphatic vessels has yet to be conclusively answered. The most common interpretation assumes a similar dual setup of the vasculature in zebrafish and mammals: on the one hand, there is the cardiovascular, circulatory system, and on the other hand, the lymphatic vascular system, in which the unidirectional flow is derived from surplus interstitial fluid and returned into the cardiovascular system. A competing interpretation questions the identity of the lymphatic vessels in fish as at least some of them receive their flow from arteries via specialised anastomoses, neither requiring an interstitial source for the lymphatic flow nor stipulating unidirectionality. In this alternative view, the “fish lymphatics” are a specialised subcompartment of the cardiovascular system, called the “secondary vascular system” (SVS). Many of the contradictions found in the literature appear to stem from the fact that the SVS develops in part or its entirety from an embryonic lymphatic vascular system by transdifferentiation. Future research needs to establish the extent of embryonic transdifferentiation of lymphatics into SVS blood vessels. Similarly, more insight is needed into the molecular regulation of vascular development in fish. Most fish do feature more than the five VEGF genes and three VEGF receptor genes that we know from mice or humans, and the relative tolerance of fish to whole genome and gene duplications could underlie the evolutionary diversification of the vasculature. This review discusses the key elements of the fish lymphatics versus the SVS and attempts to draw a picture coherent with the existing data, including the phylogenetic knowledge.

In potato, high levels of nitrogen (N) can lead to excessive vegetative growth at the expense of tuber development, resulting in lower yield and poor quality tubers. We found that Solanum tuberosum CLE4 (StCLE4) is expressed most strongly in the roots on N-rich media and positively regulates root growth of potato in N-deficient conditions. We noted that StCLE4 functions as a negative regulator of normal shoot apex development similar to CLV3 in Arabidopsis. Transcriptomic analysis revealed that overexpression of StCLE4 results in repression of the StIT1 gene, a regulator of potato tuber initiation. Overexpressing StCLE4 stolons were transformed into branches that were similar to milder it1 phenotype. We also found that NIN-like proteins, key regulators of nitrate signaling, bind to the regulatory sequence of StIT1 in yeast one-hybrid assay. Taken together, our findings suggest that StCLE4 regulates shoot, root and stolon growth in potato.

Conditional Privacy Preserving Authentication (CPPA) schemes are an effective way of securing communications in Vehicular Adhoc Networks (VANETs), as well as ensuring user privacy and accountability. Cryptanalysis plays a crucial role in pointing out the vulnerabilities in existing schemes to enable the development of more resilient ones. In 2019, Zhang proposed a CPPA scheme for VANET security (PA-CRT), based on identity batch verification (IBV) and Chinese remainder theorem (CRT). In 2023, this scheme was crypt-analyzed by Tao. In this paper we provide our own crypt analysis of Zhang’s scheme and propose counter measures to mitigate Tao’s attacks. Furthermore, in 2021, Xiong proposed a Certificateless Aggregate Signature (CLAS) scheme which was cryptanalyzed by Shim in 2023. Xiong subsequently proposed a modified CLAS scheme which is also cryptanalyzed in this paper.

Within the framework of theoretical developments in so-called participatory culture and the context of funds of identity, incorporated within what is known as the funds of knowledge approach, an innovative teaching methodology is implemented that allows students to actively participate in knowledge construction. The project translates into a proposal for educational contextualization and personalization based on the students’ funds of identity, that is, those resources (people, artifacts, places, activities, institutions) they consider to be most relevant and significant to define themselves. Once these have been identified through identity artifacts, such as collages or self-drawings, students link some of these funds of identity to curricular content of the subject and produce a video that shows the results of this academic work. The final product is shared on a YouTube channel containing the videos of all of the students in the class. The phases of the project are described and illustrated. We argue that the proposed teaching and learning project, which is cross-disciplinary in nature, allows for principles such as educational contextualization, funds of identity, and participatory culture to be incorporated into what we call here the funds of identity 2.0 approach, putting it into educational practice.

This article explores the enduring influence of colonial legacies on the economic development of the Americas, tracing their historical roots and contemporary repercussions. By examining a comprehensive range of historical and modern research, the article analyzes how colonial institutions, policies, and labour systems have profoundly shaped the economic trajectories of countries across the region. Central to the discussion is the lasting impact of colonial-era practices, such as the mining mita in Peru, and how factor endowments have influenced inequality and development patterns. The article further investigates the broader implications of these findings for understanding the deep-seated origins of economic disparities and the formidable challenge of overcoming colonial legacies in the present day. Special attention is given to the role of inclusive institutions, regional cooperation, and the intersection of culture and identity in fostering more equitable economic and social progress. Through this analysis, the article provides valuable insights into the historical foundations of contemporary economic imbalances and offers policy recommendations for addressing these persistent legacies.

Perceived peer support is a critical factor in the academic adjustment of higher vocational college students. While previous studies have explored this relationship, few studies have delved into the underlying mechanisms. This study surveyed 9782 students from 35 higher vocational colleges in 13 provinces and cities in China to explore the relationship and mechanisms between perceived peer support and academic adjustment. Drawing on Conservation of Resources theory and Social Support Resource theory, the findings revealed that perceived peer support not only directly predicted the academic adjustment of higher vocational college students, but also indirectly through the independent mediating effect of academic hope and professional identity, as well as the chain-mediating effect of academic hope to professional identity. Additionally, students from vocational high schools showed a more significant increase in academic hope due to perceived peer support compared to those from general high schools, highlighting the moderating effect of educational background. The study provided a new theoretical perspective for understanding the complex mechanisms of academic adjustment in students and offers valuable guidance for educational practice, emphasizing the importance of fostering perceived peer support, academic hope, and professional identity in higher vocational education.

The Sharing Economy (SE) has dawn great attention from several stakeholders in society in the last five years. While business actors are interested in financial opportunities to meet consumer needs, new business models, the academia and governmental organizations are concerned with potential unintended effects on the society and environment. In the process of making a clearer comprehension of the SE phenomenon, researchers have identified that, despite its notable global growth, there still persists a lack of a more solid ground in understanding its origins and respective mechanisms under which it has been evolving over time as a category. In this research, we address the problematics of the origins and ascendency of the SE by examining the process by which the SE is arising as a new category, searching for conceptual clarification and pinpointing the legitimacy granted by key stakeholders. Our guiding research questions are: (1) how the SE was formed and evolved as a market category; and (2), as a market category, is the SE legitimate? Additionally, we attempt to identify the nature of the SE as a category. To answer these questions, we conducted an historical analysis of the expression SE and its equivalents. This paper deepens the discussion about the nature of the SE by providing evidence that (i) the SE has predominantly been formed by emergence processes, comprising social movement, similarity clustering and truce components. It is the combination of all these aforementioned processes that renders the SE a special case of market category formation, which, in turn, has been allowing communication, entrepreneurship, regulation and research about what really is the SE, and despite the evident lack of agreements regarding both the label and its content; (ii) there is a generalized legitimacy granted to the SE by a vast number of stakeholders, even though still lacking on the consolidation of socio-political legitimation, and (iii) the nature of the SE seems to fall in a metaphorical approach, particularly, the notion of radial categories.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

We propose a partial information decomposition based on the newly introduced framework of causal tensors, i.e., multilinear stochastic maps that transform source data into destination data. The innovation that causal tensors introduce is that the framework allows for an exact expression of an indirect association in terms of the constituting, direct associations. This is not possible when expressing associations only in measures like mutual information or transfer entropy. Instead of a priori expressing associations in terms of mutual information or transfer entropy, the a posteriori expression of associations in these terms results in an intuitive definition of a nonnegative and left monotonic redundancy, which also meets the identity property. Our proposed redundancy satisfies the three axioms introduced by Williams and Beer. Symmetry and self-redundancy axioms follow directly from our definition. The data processing inequality ensures that the monotonicity axiom is satisfied. Because causal tensors can describe both mutual information as transfer entropy, the partial information decomposition applies to both measures. Results show that the decomposition closely resembles the decomposition of other another approach that expresses associations in terms of mutual information a posteriori. A negative synergistic term could indicate that there is an unobserved common cause.

To handle the challenges and uncertainties of the actual labor market, the concept of employability becomes increasingly relevant, especially for graduates facing the transition from university to work. Graduate employability is a complex concept comprising different dimensions that may trigger effective job search and career management. This study introduces a new multifaceted tool, labeled the Orientamentis Questionnaire, to assess graduate employability from an agentic perspective, testing its psychometric properties. The Orientamentis questionnaire measures 14 dimensions merged into four 4 areas, namely graduate psychological capital, graduate agentic capabilities, graduate career perspective, and graduate proactive strategies. The tool was administered to a sample of 1238 graduate students (66% females) examining its factorial structure and reliability. To test its criterion validity, associations with three occupational outcomes collected one year later were explored on a subsample of 273 participants (66% females). Results attested the factorial structure of the Orientamentis questionnaire and the reliability of all dimensions. Moreover, the four areas and most of the dimensions were differently related to at least one subsequent occupational outcome. The Orientamentis Questionnaire represents a sound and reliable tool providing a multifaceted profile of graduate employability strengths and weaknesses, that could be used in academic placement programs to foster graduates’ self-awareness and proactive job search.

This paper looks at key elements of global culture that are driving a new paradigm shift in museums causing them to question their raison d'être, their design and physical space, recognizing the need to accommodate visitor interaction and participation, and to reprioritize institutional outcomes and goals reexamining their priorities. As heritage sharing in online spaces reaches across national, political, and social boundaries on platforms and networks, this has been driven by museum engagement with Internet life during the pandemic. Museum relationships and interactions with communities both local and global, continue to challenge core values and precepts, leading to radical changes in how museums define their roles and responsibilities. In this new cultural landscape, museums are responding to human digital identity in a tidal wave of human interactions on the Internet, from social media to online sharing of images and videos. This is revealing shared perspectives on cultural conflict as being tied to freedom of expression of one’s heritage embedded in digital identity.

This article explores the multifaceted argument for Scottish independence, emphasizing Scotland's historical context, cultural identity, and the potential benefits of rejoining the European Union (EU). It begins with a historical overview, tracing Scotland's deep roots in independence and its eventual union with England in 1707, which, despite economic benefits, subordinated Scottish interests to those of England. The article argues that Scotland's distinct identity and ongoing desire for autonomy justify a renewed push for independence. Reuniting with Europe is presented as a key advantage, offering Scotland access to the EU's single market, financial support, and a platform for political influence. Additionally, independence would empower Scotland to harness its renewable energy resources, develop tailored economic policies, and uphold its tradition of social justice. However, the article also acknowledges the unique challenges Scotland faces, particularly in formulating citizenship policies amidst the legacy of English imperialism. Ultimately, the article posits that despite these challenges, independence offers Scotland a path to realizing its full potential as a sovereign nation committed to democracy, equality, and cooperation.

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n^2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials D_m(n,k) and U_m(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n^2m+1.

This paper addresses the growing concerns surrounding cloud computing security and highlights key strategies for ensuring the privacy and protection of data in information-critical industries. Cloud computing has revolutionized the field of information technology, providing convenience and efficiency in data processing. However, it has also brought to light significant challenges in terms of data security and privacy. Given the rise in cyber threats and the potential risks associated with storing sensitive information on cloud servers, critical industries such as healthcare and banking have been hesitant to adopt cloud computing as a trusted solution for their data needs. This work examines the specific security concerns that these industries face and proposes measures to mitigate them. Further, the paper explores the barriers that hinder the adoption of cloud computing in critical industries and offers solutions to address these concerns.

During the SARS-CoV-2 pandemic, many virus-binding monoclonal antibodies have been developed for clinical and diagnostic purposes. This underlines the importance of antibodies as universal bioanalytical reagents. However, little attention is given to the reproducibility crisis that scientific studies are still facing to date. In a recent study, not even half of all research antibodies mentioned in publications could be identified at all. This should spark more efforts in the search for practical solutions for the traceability of antibodies. For this purpose, we used thirty-five monoclonal antibodies against SARS-CoV-2 to demonstrate how sequence-independent antibody identification can be achieved by simple means applied onto the protein. First, we examined the intact and light chain masses of the antibodies relative to the reference material NIST-mAb 8671. Already half of the antibodies could be identified based solely on these two parameters. In addition, we developed two complementary peptide mass fingerprinting methods with MALDI-TOF-MS that can be performed in 45 minutes and had a combined sequence coverage of over 80%. One method is based on the partial acidic hydrolysis of the protein by 5 mM of sulfuric acid at 99 °C. Furthermore, we established a fast way for a tryptic digest without an alkylation step. We were able to show that the distinction of clones is possible simply by a brief visual comparison of the mass spectra. In this work, two clones originating from the same immunization gave the same fingerprints. Later, a hybridoma sequencing confirmed the sequence identity of these sister clones. In order to automate the spectral comparison for larger libraries of antibodies, we developed the online software ABID 2.0 (https://gets.shinyapps.io/ABID/). This open-source software determines the number of matching peptides in the fingerprint spectra. We propose that publications and other documents critically relying on monoclonal antibodies with unknown amino acid sequences should include at least one antibody fingerprint. By fingerprinting an antibody in question, its identity can be confirmed by comparison with a library spectrum at any time and context.

In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.

A regional organisation (RO) is a group of states that, through the formation of mutual relationships, work together for shared regional goals such as economic growth, social welfare, security, and democracy. The Association of South East Asian Nations (ASEAN) is one such organisation, of interest because of its operative paradoxes. However, current theory to study ROs is both fragmented across the field of study, and inadequate. Taking ROs to be complex adaptive systems, diagnosing ROs requires theory that can to explore their hidden aspects. The metacybernetics framework enables the creation of a critical realism configurative a-priori inquiry approach to theory building, the configurations providing coordinative schemas able to respond to predefined complex modelling needs. To analyse ROs using metacybernetics, we offer a derivative schema, Cultural Agency Theory (CAT). Currently this is limited to exploring modes of information acquisition processes through cognitive style, and until now has had no inherent mechanism to explore social relationships as is required to explore ROs. To enable CAT to explore such relationships, we configure the Tönnies social organisation paradigm for CAT. Mindset Agency Theory (MAT) is a specific elaboration of CAT, and creates an empirical setting for the set of formative traits that de-fine an RO character. A related derivative of CAT is that of Multiple Identity Theory (MIT) which enables organisational pathologies to be explored in terms of identity conflicts. This multifaceted paradigm provides a relatively simple approach to the RO diagnosis, in particular in terms of stability/coherence. This paper is in two parts. In this part 1, the CAT theory will be further developed through the joint configuration of Tönnies social organisation paradigm allowing its pathologies to be diagnosed. In part 2 of the paper, the theory will be applied to ASEAN and the diagnosis will show that it is an inherently incoherent organisation.

A regional organisation (RO) is a group of states that, through the formation of mutual relation-ships, work together for shared regional goals such as economic growth, social welfare, security, and democracy. The Association of South East Asian Nations (ASEAN) is one such organisation, of interest because of its operative paradoxes. However, current theory to study ROs is both fragmented across the field of study, and inadequate. Taking ROs to be complex adaptive systems, diagnosing ROs requires theory that can to explore their hidden aspects. The metacybernetics framework enables the creation of a critical realism configurative a-priori inquiry approach to theory building, the configurations providing coordinative schemas able to respond to predefined complex modelling needs. To analyse ROs using metacybernetics, we offer a derivative schema, Cultural Agency Theory (CAT). Currently this is limited to exploring modes of information acquisition processes through cognitive style, and until now has had no inherent mechanism to explore social relationships as is required to explore ROs. To enable CAT to explore such relationships, we configure the Tönnies social organisation paradigm for CAT. Mindset Agency Theory (MAT) is a specific elaboration of CAT, and creates an empirical setting for the set of formative traits that de-fine an RO character. A related derivative of CAT is that of Multiple Identity Theory (MIT) which enables organisational pathologies to be explored in terms of identity conflicts. Thes multifaceted paradigm provides a relatively simple approach to the RO diagnosis, in particular in terms of stability/coherence. This paper is in two parts. In this part 1, the CAT theory will be further developed through the joint configuration of Tönnies social organisation paradigm allowing its pathologies to be diagnosed. In part 2 of the paper, the theory will be applied to ASEAN and the diagnosis will show that it is an inherently incoherent organisation

Motivated by a common Mathematical Finance topic, this paper surveys several results concerning windings of 2-dimensional processes, including planar Brownian motion, complex-valued Ornstein-Uhlenbeck processes and planar stable processes. In particular, we present Spitzer's asymptotic Theorem for each case. We also relate this study to the pricing of Asian options.

Role identity theory describes the purpose and meaning in life that comes, in part, from occupying social roles. While robustly linked to health and well-being, this may become, however, when an individual is unable to fulfil the perceived requirements of an especially salient role in the way that they believe they should. Amid high rates of mental illness among public safety personnel, we interviewed a purposely selected sample of 21 paramedics from a single service in Ontario, Canada to explore incongruence between an espoused and able-to-enact paramedic role identity. Situated in an interpretivist epistemology, and using successive rounds of thematic analysis, we developed a framework for role identity dissonance wherein chronic, identity-relevant disruptive events cause emotional and psychological distress. While some participants were able to recalibrate their sense of self and understanding of the role, for others, this dissonance was irreconcilable, contributing to disability and lost time from work. In addition to contributing a novel perspective on paramedic mental health and well-being, our work also offers a modest contribution to the theory in using the paramedic context as an example to consider identity disruption through chronic workplace stress.

Let be a power function $f_{r,M}(s)$ defined for every $s$ within the finite set $M$ as follows $$f_{r,M}(s)= \begin{cases} s^r, \ &s\in M,\\ 0, \ &\mathrm{otherwise}. \end{cases} $$ Let a discrete convolution of $f_{r,M}(s)$ be denoted as follows $\mathrm{Conv}_{r,M}[n]=(f_{r,M}*f_{r,M})[n]$. Let a real coefficients $A_{m,j}$ be given by the following recurrence $$ A_{m,j} = \begin{cases} 0, & \mathrm{if } \ j<0 \ \mathrm{or } \ j>m, \\ (2j+1)\binom{2j}{j} \sum_{d=2j+1}^{m} A_{m,d} \binom{d}{2j+1} \frac{(-1)^{d-1}}{d-j} B_{2d-2j}, & \mathrm{if } \ 0 \leq j < m, \\ (2j+1)\binom{2j}{j}, & \mathrm{if } \ j=m. \end{cases} $$ In this paper we show that for every $n>0$ the following odd-power identities involving coefficients $A_{m,j}$ and convolution transform $\mathrm{Conv}_{r,M}[n]$ hold $$ \begin{split} n^{2m+1}+1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{N}}[n],\\ n^{2m+1}-1&=\sum_{r=0}^{m}A_{m,r}\mathrm{Conv}_{r,\mathbb{Z}_{>0}}[n],\\ n^{2m+1}&=\sum_{r=0}^{m}A_{m,r}\sum_{k=1}^{n} k^r(n-k)^r\\ &=\sum_{r=0}^{m}A_{m,r}\sum_{k=0}^{n-1} k^r(n-k)^r. \end{split} $$

Historically, the military and non-religious, sub-continent’s intellectual elite Muslim men crafted state ideologies before partition, by actively contributing to sociocultural constructs prevalent to this day, in service of colonial patriarchal concepts. The purpose of this ideological collaboration has been to mould a Pakistani cis-gendered female child primarily into the ideal “Pakistan-Muslim” bride for marriage, carried out through her family and society. This has shaped the lives of cis-gendered female children in Pakistan through coerced, gendered control, implemented as prescriptive societal norms. These norms are rooted in colonial, patriarchal ideologies and are functions of capital and coercion intentionally perpetuated to erase her pluralistic identity, reducing her to a central resource within a family unit. To the military concept of nationalism, her mind and her body symbolize the unification and centralization of a statist ideology. Thus, the military state plays an essential role in shaping her purpose around the concept of nationalism. The intellectual elite, via the state, uphold colonial, patriarchal ideologies, where the primary identity of a cis-gendered female child is actualized through her marriage. The cis-gendered female child’s life is a vehicle to unify the state’s two-fold “Pakistani-Muslim” agenda, by implementing a ‘singular statist narrative’. A narrative that attempts to resolve two major tensions: First, it seeks to wipe out the possibility of a nation connecting with and exploring its own immense, internal ethnocultural diversity. It does this by imposing an imagined “homogenous Muslim” identity — which doesn’t exist — and is instead imported from multiple countries to reinforce a ‘singular statist narrative’, Second, it aims to control the mixing of local, sociocultural narratives by bridging any disjuncture between them by labelling colonial institutional patriarchy concepts as inherently “Pakistani”; in service of its preferred ‘singular statist narrative’.

In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the first and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.

The notion of Critical Point Theory has extensive application in the filed of Partial Differential Equations, one of which is studying the existence and commenting on the uniqueness and multiplicity (in case when the solution is not unique) of weak solutions to Elliptic PDEs under certain boundary conditions.\par This article provides a brief survey about specific concepts like : differentiation on Banach Spaces, maxima and minima and its applications to PDEs. Furthermore, we've discussed the notion of defining a weak topology on Banach Spaces, before introducing the Variational Principle and its applications.\par The epilogue of the article primarily deals with results which studies the existence of weak solutions to Dirichlet Boundary Value Problems under specific conditions. The most notable aspect of this article is the proof of \textit{Rabinowitz's Saddle Point Theorem} using the method of \textit{Brouwer Degree}. Readers who are highly motivated in pursuing research in any of the topics relevent to the contents of this paper will surely find the \texttt{References} section to be extremely resourceful.