Journal of Combinatorics and Number Theory

The Journal of Combinatorics and Number Theory (JCNT) is devoted to publishing peer-refereed original research papers on topics in combinatorics (including graph theory) or number theory. Papers involving both combinatorics and number theory are particularly welcome. The subject areas of JCNT correspond to the MSC (Mathematics Subject Classification) codes 05, 11, 20D60, and 52.

Journal Editor: Zhi-Wei Sun, Editor-in-Chief
ISSN: 1942-5600
Frequency: 3 x per year



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Volume 12

Volume 10, Issue 1

Volume 12

Maxillary Canine Internal Root Resorption Concomitant to Orthodontic Retraction; Cone Beam Evaluation Hussam M. Abdel-Kader and Ahmed S. M. Ammar

TMD-Related Headache and Facial Pain: A Review Ibrahim Mortada, Rola Mortada, Mohamad Al Bazzal, Angelo Leone, and Abdo Jurjus

Effect of Chemical and Thermal Factors on Ni-Ti Endodontic Instruments: A Review Zahed Mohammadi and Sousan Shalavi

Co-Relation of the Disease Severity between Rheumatoid Arthritis and Chronic Periodontitis in South Indian Population T. Sivasankari, T. Ramakrishnan, B. Aruna, Kamal Kanathasan, Uma Sudhakar, Shankar Ram, Danny Mathew, and S. Parthiban

Prevalence of Maxillofacial Fractures Reported at Sandeman Provincial Hospital, Quetta, Pakistan Ambreen Mengal, Mirza Khan Tareen, and Jahangir Hamad

Resolution of Buccal Dehiscence Defect after Endodontic Treatment: A Case Report Jefferson J. C. Marion, Maíra Prado, Nicole A. Tomiazzi, Cleverson O. Silva, Emmanuel José Silva, and Thais M. Duque

Prevalence of ROM Levels (in GCF / Saliva, Plasma) in Smokers and Non-Smokers with Chronic Periodontitis Uma Sudhakar, T. Ramakrishnan, Vikram Buddhanesan, Himanshu Shekhar, Dr. Parthiban, and Dr. Karthikaivelan

Structuring, Logistics and Legal Aspects of Human Teeth Bank on the Northern Brazil Natácya Menezes de Souza Lopes, Glenda Araújo Pinheiro, Camila Sagica dos Anjos, Bárbara da Silva Neves, Arielle Monteiro Cristo, Silvio Augusto Fernandes de Menezes, Patricia de Almeida Rodrigues Silva e Souza, Ricardo Roberto de Souza Fonseca

Cavity Disinfectants in Restorative Dentistry- Journey til Date Karuna Yarmunja Mahabala, Arathi Rao, and P Anupama Nayak

Introduction: Dental pulp when exposed to iritants will endanger its life. The iatrogenic injury that happened during cavity preparation is the common cause of the pulp irritation. The response of the pulp to irritation is inflammation and, if left untreated, this will eventually progress to pulp necrosis. The aim of this study was to evaluate the anti-inflammatory effect of mangosteen rind extract in rabbit teeth with reversible pulpitis. Material and Methods: This study was conducted on 9 rabbits. 4 central incisors were used from each rabbit,. The teeth were perforated using small bur until bleeding, then after cleansing, materials will be applied. The materials used were the mangosteen rind of concentration 5% and 10%, and biodentine. All the teeth prepared and materials were applied to the 3 central incisors with one tooth left as control. The rabbits randomly assigned into 3 groups: Group 1: 3 rabbits decapitated 1 day post-injury, Group 2:

This research work is carried out to assess the antimicrobial efficacy of an anti-plaque agent, Hybnex® against the leading periodontal pathogens which are known to cause periodontal disease. Complete exclusion of these infective microbes and thereby impeding disease progression is the prime focus of periodontal therapy, which is virtually achieved by scaling and root planing. However, for maintenance of such perceptible conditions are carried out by the adjunctive use of antimicrobial agents. A total of fifteen intact human single rooted teeth were inoculated with periodontal pathogens to form a biofilm. The biofilm formed was confirmed using scanning electron microscope (SEM). The specimens were then divided into 3 groups. Group I: specimens were treated with test agent. Group II: specimens were treated with 0.2% Chlorhexidine (CHX) AND Group III: Specimens were treated with deionized water. The results showed comparable antimicrobial activity between test agent (Hybnex®) and CHX (0.2%) when compared to the deionized water, which was statistically significant (p < 0.001).

Purpose: To evaluate the efficacy of a 37% phosphoric acid solution delivered by conventional irrigation, Endovac®, sonic and ultrasonic activated tips and plastic file on Ca(OH)2 removal using micro-CT. Methods: Fifty mesial roots of mandibular molars were manually instrumented. Ca(OH)2 paste was inserted into the root canal system with a 25 K-file. All teeth were scanned by micro-CT to determine the initial dressing volume. After 7 days, the Ca(OH)2 was removed with a 25 K-file and 10 mL of NaOCl. The specimens were allocated in 5 groups irrigated with phosphoric acid (n = 10): conventional irrigation (CI); Endovac (EV); Sonic irrigation (SI); Passive ultrasonic irrigation (PUI) and Plastic file (PF). The teeth were re-scanned by micro-CT. Results: The percentage volume of Ca(OH)2 removal was calculated. Data was evaluated by Kruskal-Wallis, Mann-Whitney and Wilcoxon tests (p < 0.05). PUI removed significantly more Ca(OH)2 than CI, EV and PF.

Purpose: To investigate the pulpal response to EPT before and after pulpotomy procedure. Methods: Sixty teeth (premolar and molar), belonged to patients aged 20-60 years, were included in this survey. Teeth had a history of spontaneous pain and were candidate for root canal treatment. Before anesthesia and emergency pulpotomy treatment, the response to EPT was recorded. Pulpotomy were accomplished and the access cavity were sealed using a temporary restoration 2-5 days after pulpotomy, the responses to EPT were recorded again and the root canal treatment were completed. Statistical analysis was performed with paired t-test. The significance level was set as p < 0.05. Results: According to the results of this study, the mean value of response to EPT was 3.08 ± 0.8 before and 4.5 ± 1.6 after pulpotomy. The pulpal response to EPT before and after pulpotomy had significant difference (p < .001).

Purpose: This study aimed to correlate the two quantitative radiomorphometric indices—MI and PMI—evaluated by panoramic radiographs, to the BMD obtained by bone densitometry in postmenopausal women in the Amazon region and demonstrate the importance of panoramic radiography as a screening tool for patients with low BMD. Methods: The indices measured on panoramic radiographs of 91 postmenopausal women were evaluated and compared with the diagnosis of bone densitometry, using the chi-square and Pearson correlation test for statistical evaluation. Results: Patients with osteoporosis/osteopenia presented lower index values than normal patients. The correlation test showed that patients classified by densitometry as having normal bone density or osteoporosis, are more likely to be classified similarly by PMI. Moreover, cases classified as normal or with osteopenia and osteoporosis by densitometry are more likely to be classified similarly by the MI. The evaluated indices were able to identify women with low bone mass, showing that panoramic radiography can be a useful tool for screening such

Purpose: Revascularization has emerged as a favorable treatment option, in particular for teeth with pulp necrosis in the early stages of root formation. However, internal tooth discoloration has been observed by triple antibiotic paste, as well as by mineral trioxide aggregate (MTA) used in the revascularization procedure. Case Report: In this report, a case of traumatized immature teeth was treated with pulp revascularization using two intracanal medicaments - triple antibiotic paste and the combination of calcium hydroxide and 2% chlorhexidine gel, and its effect on darkening was discussed. In addition, internal bleaching was proposed to solve the drawback of tooth discoloration after revascularization. Conclusion: At 15 months follow-up, clinical findings showed no tooth discoloration and radiographic examinations showed continuation of root formation and apical closure.

Purpose: The aim of the present study was to evaluate the surface tension of established and potential endodontic irrigants used during endodontic practice. Methods: The substances evaluated were: Distilled water; natrosol; sodium hypochlorite (1%, 2.5%, 5.25%, 6% - solution and gel); chlorhexidine (0.12%, 0.2%, 2% - solution and gel); 17% EDTA solution and gel; 37% phosphoric acid solution; citric acid (1%, 10%, 50% - solution and gel); sodium thiosulfate solution (0.5% and 5%); 10% sodium ascorbate solution; 100% Ethanol; absolute alcohol; and propyleneglycol. The surface tension was measured using the Du Noüy ring method. Ten measurements were taken for each substance (n = 10). Data were statistically analyzed using Anova and Tukey tests. Results: Comparing all substances evaluated, ethanol and absolute alcohol showed the lowest and 1% NaOCl the highest values of surface tension. Regarding the substances used during preparation, CHX showed the lowest surface tension values. Regarding NaOCl, in concentration of 6% this substance showed the lower surface tension. For smear layer removal, 17% EDTA showed the lowest values of surface tension. As final irrigant, ethanol and absolute alcohol showed the lowest values of surface tension followed by 2% CHX solution. Conclusion: It was concluded that the surface tension of the tested substances was varied, having the absolute alcohol the lowest and the 1% NaOCl the highest.

Purpose: The latest epidemiologic orofacial pain survey in Indonesia showed that orofacial pain remains as one of the main dental problems in Indonesia. Therefore, development of an assessment tool that evaluates the effect of orofacial pain on the earlier period of life is considered to be of important. The current study aimed on evaluating the validity as well as reliability of the Children Orofacial Pain Questionnaire (COPQ) that will be used as one of the assessment tools in the intensity, duration, frequency, and impact of orofacial pain on children. Method: One hundred and sixty-five schooled aged children (aged 6-12 years old) participated in the current study. Researcher interviewed all participants based on a questionnaire that evaluates the type of orofacial pain, intensity, duration, frequency, and the impact of orofacial pain on daily activity. Pearson’s correlation was used to test the construct validity of the questionnaire. Results: All questions tested were significantly correlated (p-value < 0.05) with a positive r values (ranged from 0.67 until 0.75). The reliability evaluation using the test-retest method showed a Cronbach’s alpha value of 0.74. Conclusion: It is concluded that the Indonesian COPQ is proven to be valid and reliable to be used as an assessment tool of orofacial pain on children.

Odontomas are a common type of odontogenic tumor and are considered to be hamartomas, containing both epithelial and mesenchymal elements with unknown etiology. Odontomas have been classified into two main types – compound and complex. Compound odontomas consist of tooth-like structure, organized in an orderly pattern, but presents altered size and conformation. Complex odontomas are malformations in which all dental tissues are present as a disorganized amorphous mass and they are frequently detected around the second decade of life. This report attempts to describe a rare event of an erupting odontoma in a 25 year-old male patient, Caucasian, who sought care at the service of CESUPA School of Dentistry, complaining about slight discomfort at lower incisors' area. First was realized a clinical and radiographic examination and was found 3 teeth-like structures which one of them was exposed at oral cavity and the treatment of choice was cirurgical excision. Therefore, the professional must be able to diagnose and treat properly this uncommon condition.

Volume 12

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Volume 12, Issue 1

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Volume 10

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Volume 18, Issue 1

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Volume 13

Volume 13, Issue 1

Volume 32

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Volume 32, Issue 1-2

Volume 11

Special issue: Palliative care: Experience from Hong Kong

Special issue: Current opinion in cancer and exercise

Volume 12

Volume 11

Volume 9

Please purchase each Issue and Article separately. Thank you.

Volume 25

Volume 11

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Special issue: Tribute to the pediatrician Emanuel Chigier, 1928-2017
Edited by Joav Merrick

Larger than life ambassador: Emanuel Chigier, pediatrician, 1928-2017 Daniel Hardoff and Joav Merrick

The quest for an alternative paradigm of STEM education for young people Daniel TL Shek and Po Chung

Assessment and intervention for perinatal bonding disorder Miki Mizumoto and Yoshiyuki Tachibana

Measuring child survival in Trinidad: Preparation for Sustainable Development Goal 3 Kameel Mungrue, Katavia Balfour, Binta Baptiste, Asa Badall, and Analiese Bates

Training of potential program implementers for the Tin Ka Ping P.A.T.H.S. Project in China: Subjective outcome evaluation findings Daniel TL Shek, Janet TY Leung, Lawrence K Ma, Li Lin, and Florence KY Wu

A qualitative study on the role of gender in the health and development of adolescents in disadvantaged communities in Ibadan, Nigeria Adesola O Olumide and Oladosu A Ojengbede

Overparenting from the perspectives of Chinese parents and youths Janet TY Leung, Daniel TL Shek, and Lorrie SL Ng

Mothers’ attributions in the relationship between young children’s temperament and parenting stress Melissa Middleton and Kimberly Renk

Does head and neck cancer (HNC) education have impact on adolescents’ knowledge and attitude towards HNC and HNC peer education? An example from Nigeria Kehinde K Kanmodi and Omotayo F Fagbule

Secondhand tobacco smoke exposure and attitudes towards tobacco ban: A pilot survey of secondary school students in Ibokun Town, Nigeria Omotayo F Fagbule, Kehinde K Kanmodi, and Timothy O Aladelusi

Anaemia in infants: Is it always an 'iron clad' issue? An observational study Lutfi Jaber and Gary Diamond

Child abuse and alcohol misuse in a victim: The role of ethnic and societal factors, part 2 Sergei V Jargin

Incidental oral hairy leucoplakia (OHL) in a healthy asymptomatic individual Lyubov Matytsina-Quinlan, Mrinalini Mahto, Donald E Greydanus, and Laura Gwatkin

Volume 12, Issue 1

Synchronization between two Lorenz systems is analyzed through various forms of coupling between them. We have used direct, conjugate, delayed direct and delayed conjugate coupling mechanism. In each case Lyapunov exponent is computed for ascertaining the synchronized situation, which is then used to encrypt a signal and then retrieve the same by the second system. The situation is demonstrated with the help of electronic circuit. It is observed that both the numerical simulation and experimental results corroborates each other. Lastly the chaotic data from the circuit is also used to encrypt an image and retrieve by the second set. In both the cases the system security is discussed.

Clinical microbiologists are routinely engaged in the identification and taxonomic categorization of microorganisms having ‘morphological and genetic plasticity’ such as Enterobacter, Escherichia and Shigella responsible for urinary tract infections (UTI) in immunocompromised nosocomial patients. Nosocomial patients are amiable host for

The so-called experiencing of “anomalous experiences” has been reported by people in various cultures since ancient times. However, they only received systematic attention from science at the eighteenth century. Understood as human experiences that appear to have no known psychological or physical explanation, they suggest the possibility of direct interactions between mind and matter. This paper reports the results of a qualitative research that aimed to describe and characterize “anomalous” phenomena experienced by doctoral students of the Program of Engineering and Knowledge Management, at the Federal University of Santa Catarina, Brazil. Ten respondents, seven women and three men, aged between thirty-four and fifty-five years old, who volunteered for the survey and were interviewed. The data was collected and discussed according to five emerging categories of analysis: (1) types, (2) feelings and emotions, (3) time-space, (4) empowerment and interest, and (5) knowledge. Despite the impossibility of generalization of the findings, they indicate that people, including those with scientific and technological training, may be living much more with “anomalous” phenomena than one might expect. More qualitative research is suggested, analyzing new relationships and processes present in the context of these occurrences.

The Mandelbrot sets for complex polynomials were obtained using one-step feedback process since their advent. From 2004 onwards, two-step feedback process was used via superior iterates to obtain Mandelbrot sets for polynomials. In this paper, pattern in quadratic Mandelbrot sets called as superior Mandelbrot sets has been studied.

This paper deals with the connection between longitudinal and transversal sections of the state space of chaotic events in the nonlinear damped and forced oscillator (NLDFO), which may be considered the drosophila of physical chaos. In order to extract the above mentioned sections, which are nothing but tomographies of state space, the Poincaré techniques of slicing the cited space have been applied. The longitudinal sections of state space display the period bifurcation cascade of the chaotic event and, its transversal sections show the attractors of the dynamics; these display evidence that even during the most furious stages of chaos, there is order and structure in the chaotic behavior of the oscillator. It has been found that the attractor of a complete chaotic event is made up of a series of partial strange attractors, one after another and, that the transition from one to the next is smooth, i.e., the end of an attractor gradually becomes the beginning of the next. This report includes also some conclusions of investigations carried out by this author in the NLDFO; since this system transitions to chaos through a cascade of period bifurcations, these conclusions may be generalized to those systems transitioning to chaos following the same route.

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Volume 11

Special issue: Development and validation of assessment tools on service leadership knowledge, attitude and behavior

Edited by Daniel TL Shek, Xiaoqin Zhu, and Joav Merrick

Development and validation of assessment tools on service leadership knowledge, attitude, and behavior Daniel TL Shek, Xiaoqin Zhu, and Joav Merrick

Leadership assessment tools in different Chinese contexts Li Lin and Daniel TL Shek

Conceptual background and the development of service leadership knowledge scale Daniel TL Shek, Xiaoqin Zhu, and Alex YF Zhu

Development of the attitude to Service Leadership Scale in Hong Kong Cecilia MS Ma, Daniel TL Shek, and Yanto Chandra

Development of Service Leadership Behavior Scale: Background and conceptual model Daniel TL Shek, Xiaoqin Zhu, and Kin-Man Chan

Psychometric properties of the Service Leadership Behavior Scale: Preliminary findings Daniel TL Shek, Lawrence K Ma, Li Lin, and Hildie Leung

Content validation of a Service Leadership Behavior Scale in Hong Kong Daniel TL Shek, Jing Wu, Li Lin, and Xiang Li

Validation of the Service Leadership Knowledge Scale: Factorial and convergent validity Daniel TL Shek, Lawrence K Ma, Lu Yu, and Loretta MK Leung

Convergent and factorial validation of the Service Leadership Behavior Scale Daniel TL Shek, Lawrence K Ma, Cecilia MS Ma, and A Reza Hoshmand YF Zhu, Lawrence K Ma, and Li Lin

The Service Leadership Knowledge Scale: Norms and psychological correlates Daniel TL Shek, Xiaoqin Zhu, and Samson Tse

The Service Leadership Attitude Scale: Normative data and personal correlates in Chinese university students Daniel TL Shek, Jing Wu, and Ben YB Chan

The Service Leadership Behavior Scale: Norms and personal correlates Daniel TL Shek, Diya Dou, and Robin S Snell

Volume 11

Volume 9, Issue 2

Volume 25, Issue 3

Volume 12

Volume 11

Volume 11, Issue 3

All the contributions collected here have a unitary inspiration in the Mind Force conference and think-tank held in Siena on September 2016 at the Pontignano Charterhouse. The cloisters and gardens have inspired again an atmosphere of friendly, free and deep interdisciplinary reflections. All participants shared a sense of being a community based on Complexity Science and of being on a historical mission in laying the foundations of the shape of a new science to come. This special issue of Chaos and Complexity Letters contains some of the most interesting contributions shared and discussed during the conference. Lewis Carroll wrote in his Alice, “Sometimes I’ve believed as many as six impossible things before breakfast.” We kept believing and realised that Mind Force, as an integrated theory, is not only possible but already a shared common ground between scientists from different domains.

In contemporary debate regarding neuroscience, psychology, cognitive science, philosophy or anthropology it is a commonplace to analyse the problem of the mind focusing on its emotional or, alternatively, cognitive and, lastly its symbolic character. Therefore, it is frequently spoken of as an emotional, a cognitive, or a symbolic mind. In the context of an increasingly branched dialogue and convergence of scientific and philosophical investigation with respect to the mind-body problem, philosophers, psychologists, neuroscientists, anthropologists and physicists have focused, in recent years, on the aesthetic dimension of the mind.

The epistemological characteristics of integration in psychotherapy, as part of life and health sciences, are presented and discussed from an analysis of the definitions of multidisciplinarity, interdisciplinarity, consilience and transdisciplinarity, as models of scientific work.

Neuroscience and experimental psychology are recently making significant advances in the nature of the human mind, particularly in consciousness and the ability to process information unconsciously. Aside from their interest in terms of basic science, these advances have provided some new ideas for the interpretation of different psychopathological disorders, and for their treatment. In the first part of this paper, we analyze the psychological characteristics of consciousness and unconscious cognition (UCo), mainly unconscious perception (UPe), as well as their neurological bases, structured around the theory of the global neuronal workspace (GNW). In the second part, we analyze the consequences of these considerations for psychopathology with a dual mind model (DMM), made up of controlled and conscious cognitive processes and automatic and unconscious cognitive processes. Thereby, we highlight the incidence of the alterations of consciousness in some psychopathologies, such as schizophrenia, and of UCo, through automatic cognitive bias, in others, such as depression, phobias, or anxiety and consumption disorders (drug addiction and bulimia). Finally, we analyze different psychotherapeutic intervention procedures developed from previous approaches, especially the techniques of modification of cognitive biases.

In cognitive constructivist psychotherapy therapist-patient relationship represent a particular interaction with a high level of intersubjective confidence and emotional involvement [1, 2]. Both subjects inside the relationship are active subjects and they are emotionally involved in the interaction. The focus of this study is to analyze psychophysiological correlation in therapeutic relationship, in particular way, we will check the presence of emotional reciprocity. We used two synchronized physiopolygraphs, to measure activation and regulatory responses of fast and slow autonomic physiological indices. The entire sample consisted of ten dyads, patient-therapist, unique and homogeneous. During psychotherapeutic session we observed that there is an emotional dynamic between patient and therapist (positive or negative). This synchronization, or empathy, enables the therapist to induce emotional disturbances favoring processes that can help the patient to explore their ways of knowing and to re-organize their own personal meanings.

Innovations in information technology opened the way to monitor the nonlinear features of human change dynamics in real time. Especially the internet-based Synergetic Navigation System (SNS) was optimized for high-frequency assessment in real-world settings and for the nonlinear analysis of the collected time series data. The technology also has an impact on the conceptualization of psychotherapy feedback, e.g., concerning measurement frequencies and sampling rates, the variables to be assessed, the methods of time series analysis, the way how to practically use the technology, and how to do feedback-based interviews. One important aim is to identify order transitions and their precursors in psychotherapy and counseling. The options available in the SNS for analyzing and visualizing non-stationarities and related precursors are described and illustrated by Figures. The paper is completed by two perspectives on practice and theory – one on the individualization of measurement procedures and process-sensitive treatment designs, the other on the mathematization of models for understanding the complexity of change processes (computational systems psychology).

Volume 12, Issue 2

Volume 12

Special issue: Promotion of holistic development of university students in Hong Kong

The impact of a new 4-year undergraduate program in Hong Kong Daniel TL Shek, Yu Lu, and Joav Merrick

Leadership qualities as a foundation of serivce learning: The Hong Kong experience Daniel TL Shek

Service learning as a mandatory credit-bearing subject at The Hong Kong Polytechnic University Stephen CF Chan, Grace Ngai, and Daniel TL Shek

University social responsibility and promotion of the quality of life Daniel TL Shek

The quest for holistic youth leadership development: What should be the desired attributes of youth leaders? Daniel TL Shek and Angelina Tsang-Yuen

General education at The Hong Kong Polytechnic University: A comprehensive evaluation study Daniel TL Shek, Lu Yu, Wen Yu Chai, Kevin Chan, and Joe Ngai

Achievement of desired university graduate attributes through a new 4-year undergraduate program Daniel TL Shek, Lu Yu, Wen Yu Chai, Florence KY Wu, and Wynants WL Ho

Perceptions of teachers of the new general education curriculum in Hong Kong: A focus group study Daniel TL Shek, Catherine WY Chai, and Florence KY Wu

Wellbeing of university students in Hong Kong: A longitudinal case study Florence KY Wu, Daniel TL Shek, Catherine WY Chai, and Julie XQ Zhu

Development of undergraduate university students: A 4-year longitudinal case study in Hong Kong Daniel TL Shek, Wen Yu Chai, Huijing Lu, Lu Yu, Kevin Chan, Wynants Ho, Cecilia MS Ma, Florence KY Wu, Hildie Leung, Li Lin, and Moon Law

Volume 11, Issue 2

Quantum probability fields are involved during processing of human perception and cognition with a wave function that mathematically is articulated in a proper Hilbert space and conceptually represents the space of all the potential alternatives at a semantic and semiotic level. Each subject has developed a unique mind within his/her correspondent context, hence his/her own wave function must be displayed as a complex-valued probability amplitude. Reaction time as well as the time of observation / decision / measurement also represent important variables contributing to our cognitive response and having also proper neurological correlates. Mind is a complex abstract entity so that we may assert that, generally speaking, the observer and the observed system are considered to be incompletely specified systems. An elementary wave function remains insufficient to represent a mental state, whereas a quantum wave function, which exhibits intrinsic indetermination and fluctuation of the basic probability amplitudes, may be implemented. In this paper, along with von Neumann’s postulate for non-existing dispersion-free ensembles, a new quantum statistical approach is elaborated to elucidate the peculiar quantum mechanical feature of the context dependent dynamics of human cognitive conceptual processing. In the framework of the present quantum statistical elaboration we introduce for the first time the possibility of using the Pareto distribution as a probability density function. This result links quantum statistical mechanics with the science of complexity.

The aim of this work is to describe a new formula relating the nontrivial zeros of the Riemann Zeta function to the energy levels of the harmonic oscillator, which we call the “Riemann wave,” whose nodes are located at the height of the non-trivial zeros (on the Riemann Hypothesis, RH). We illustrate the formula by means of various Figures, and we present a calculation up to relatively “high” heights. Then, we propose formally an “operator” in agreement to the Polya’s idea, which involves here the Lambert W function. We call it the “Quantum Riemann Wave.”

The aim of this work is to describe as fractal geometry, its property of self-similarity, and its processes of bifurcation can be appear in the arts and in architecture. These fractal features are common in different cultures and in different architectural styles.

The complexity of multimedia knowledge requires new methods of representation. This work addresses the problem by proposing the use of Petri nets, a powerful knowledge representation method suitable to manage complex concurrent events. An interactive interface allows the exploration of both the knowledge structure and the multimedia content. We have developed a web application that collects and systematizes Giovanni Degli Antoni's scientific and documentary work, mainly composed of web multimedia contributions, underlining his long-life research in the field of hypermedia world and Petri nets theory and applications.

A new nonlinear equation is proposed where the forcing is realised in a different manner and is aperiodic. It is observed that the system has a host of fixed points and the bifurcation pattern with respect to all the four parameters unravel a very rich structure of Chaotic dynamics. The Lyapunov exponents help to justify the range of values therein. Biparametric plots reveal the whole scenario in a very compact manner. Existence of attractor and various multi-periodic states are seen to occur for different choice of parameter values. They are actually projected very neatly in the biparametric plots.

Volume 11

Volume 25, Issue 1

Volume 12

Special issue: Awareness of Parkinson’s disease among young adults

Volume 11, Issue 1

This volume is dedicated, not just to Walter, but also to the cabal of scientists, scholars, and artist included in this volume, most of whom are one degree of authorship-connectivity to Walter. You could say they represent a community who have elected Walter as their honored colleague. Their scientific and philosophical exploits exemplify the standards he set. They recognize that the many arenas of his competency have made for an innovative and coherent corpus of research and conjecture. They also exemplify Köhler’s “The whole is different from the sum of its parts.”

Many have observed that Walter was ahead of his time and this is certainly true. But the question is, how did he get there? Was it just that he was more intelligent than others or was it simply a matter of luck and good timing? Perhaps. Had Walter been gifted with a duller mind or had he been born when Sherrington was born things might have turned out very differently. But I think the real explanation was to be found by looking around that strange library of his. Walter’s library attested to the fact that poetry and philosophy determined the fundamental mindset that he carried with him into the field of neuroscientific research. They were not a side line, not a diversion from what really mattered to him. And I believe that it was poetry more than anything else that set the parameters of his thinking and that shaped his unique perspective on everything, including the brain. For poetry holds things up in the light of being to reveal that they are more than what they seem to be individually because they are caught up in the dynamic of something much greater—the dynamic of being. Poetic thinking causes individual things to “resonate” in ways that make being itself palpable. And it is this kind of vision—a fundamentally poetic vision—that made Walter J. Freeman a truly great neuroscientist.

Remarkably, in our Cuban meeting Walter had referred to cell ensembles supported by reafference, a key concept for understanding the role of my cognits in the perception-action cycle, as well as the role of the frontal lobe in it. At the same time it was enlightening, already then, to hear him refer to the function of those ensembles in “biological intelligence,” with its basic attributes of autonomy and creative power. Today we know that those attributes of “biological intelligence” derive from the predictive and prospective cognitive functions of the prefrontal cortex.

My time in the lab was during the heyday of Walter’s writing on Chaos in brains and the accompanying backlash among US neuroscientists. It was also a time in which the use of dynamical systems to explain physical, social and biological processes was accelerating rapidly. I tried in my dissertation to sort out the metaphor. It is a difficult ask to tie the theoretical concepts of complex nonlinear dynamics and chaotic systems to the messy complexity and nonstationarity of a mammalian brain. The methods in use at the time to point at chaos in neural data, correlation dimensions, etc., were very sensitive to nonstationarities in our data. My question was and remains: once we obtain some measure of complexity, will it help us to understand what brains are doing, or do we just swap one set of complexities for another? Walter approached proving chaos by building computational models and showing that the regimes in which they best reproduced physiological states were chaotic ones [6].

Walter Freeman was not only a great neuroscientist and pioneer in the field of mesoscopic brain dynamics, he was also an excellent teacher and mentor to many, all over the world. In this paper, I give a personal account of what Walter Freeman has meant to me and my research, primarily on computational modeling of the olfactory cortex, but also as a friend and mentor. I briefly describe the path from my postdoc period at Caltech in the late 1980s, when I first met Walter Freeman, through our meetings at numerous conferences around the world, and finally to the TSC conference in Tucson 2016, which Walter missed by a couple of days. In more detail, I describe my modeling work on complex neurodynamics, and how oscillations, noise and chaos may play a role for perception and associative memory. I end with some personal remarks and memories from discussions with Freeman, including the relevance of the complex neurodynamics for cognitive and mental functions, in particular relating to intentionality and consciousness.

Walter Jackson Freeman III (January 30, 1927 – April 24, 2016) was a true explorer, a Renaissance Man, who transcended the boundaries of disciplines and scientific knowledge. He has revolutionized the field of neuroscience, by bringing into it many pioneering ideas on brain dynamics. The authors of this brief essay address the main legacy of Walter Freeman through their framework of Operational Architectonics of brain-mind functioning that encompasses Freeman’s mass action in the nervous system in the form of nested, dynamic neuronal assemblies and his cinematic model of cognitive dynamics, leading to emergence of consciousness. According to Operational Architectonics theory, the hierarchy of phenomenal world (features, patterns, objects, scenes) has its electrophysiological equivalent in an operational hierarchy of neuronal assemblies and nested spatial-temporal conglomerates of them in the form of operational modules (with different size and life-span), which correspond to the phenomenal entities of different complexity.

In this commemoration of Walter Freeman’s life and legacy, we review some his key contributions to the field of cognitive neuroscience. We elaborate on his groundbreaking contribution to neural aspects of intentionality, what he called intentional neurodynamics. We describe the conceptual framework of intentional neurodynamics and its neurophysiological manifestations. We conclude with the outline of Freeman K (Katchalsky) sets, a hierarchical model of brain structure, dynamics, and functions, which provide a suitable mathematical and computational framework to grasp essential aspects of intentionality.

It has been proposed in cognitive neuroscience that the operations of large-scale cortical networks are directly related to various aspects of cognitive function. However, the dynamic neural mechanisms responsible for these network operations are only poorly understood. This paper re-introduces Walter Freeman’s idea of the wave packet as both an experimentally and theoretically unifying concept in neural dynamics. We begin by discussing the history of the wave packet as empirically observed in the olfactory local field potential by Freeman and colleagues. Freeman claimed that the wave packet, a mesoscopically self-organized cooperative neuronal activity pattern, is the way in which sensory information is turned into percepts, and ultimately, by which brains create meaning. He surmised that the wave packet is the most fundamental functional unit in the brain that carries meaningful context-dependent information. We extend Freeman’s work to all cognitive function, which we propose is subserved by large-scale networks of linked neuronal populations in different neocortical areas, each generating a wave packet. We propose that the neocortical wave packet is comprised of locally-generated gamma oscillatory activity, as it is in paleocortex, and the linking mechanism consists of inter-areal phase coupling of beta oscillatory activity. We argue that neurocognitive function derives from wave packets in multiply linked neocortical areas reaching a functionally consistent consensus, and claim that the wave packet, discovered in the paleocortex, is key to the operation of the neocortex in cognition.

This article pursues Walter J. Freeman’s agenda to understand how we come to encode and represent the causes of our sensations. It attempts to derive the sentient aspects of any dynamical system (e.g., a brain), where the system’s internal (e.g., neuronal) states come to represent the outside world in a probabilistic fashion. The argument goes as follows: any self-organising dynamical system must be statistically distinguishable from the environment in which it is immersed. This leads to the notion of a Markov blanket; namely, a set of states that intervene between the internal states of a system and those external to the Markov blanket. This Markov blanket plays the role of sensory epithelia – that mediate the influence of the world on the system – and active states, such as our actuators and effectors – that mediate the influence of our internal states on the world. If a random dynamical system (e.g., you or me) persists over a sufficient amount of time, then we possess an attracting set (technically, a random dynamical attractor). Crucially, this means that there is always a mapping between the (expected) internal and external states, when conditioned upon the Markov blanket. This provides a probabilistic link between the internal states of a system and the external states of the world it has to navigate. By following some fairly straightforward maths, it is possible to show that the internal states perform approximate Bayesian inference about external states; thereby equipping internal dynamics with a representational or sentient aspect. These points are illustrated using simulations of loosely coupled dynamical systems to show that even (synthetic) virus-like particles behave like little Bayesian statisticians.

Metastability has been proposed as a new principle of behavioral and brain function and may point the way to a truly complementary neuroscience. From elementary coordination dynamics we show explicitly that metastability is a result of a symmetry breaking caused by the subtle interplay of two forces: the tendency of the components to couple together and the tendency of the components to express their intrinsic independent behavior. The metastable regime reconciles the well-known tendencies of specialized brain regions to express their autonomy (segregation) and the tendencies for those regions to work together as a synergy (integration). Integration ~ segregation is just one of the complementary pairs (denoted by the tilde (~) symbol) to emerge from the science of coordination dynamics. We discuss metastability in the brain by describing the favorable conditions existing for its emergence and by deriving some predictions for its empirical characterization in neurophysiological recordings.

The results of laboratory observations and of theoretical analysis carried on by Walter Freeman and other investigators suggest that the macroscopic functional stability and high efficiency of the brain depend on the coherent oscillations of assemblies of millions of neurons, not solely by the properties of single neurons. The approach focusing on the study of single neurons that dominates current neuroscientific research, although absolutely necessary, reveals itself insufficient for understanding how to fill the gap between the fluctuations in the biochemical cellular and molecular microscopic activity and the stable and efficient global functional performances of the brain. In this paper, it is discussed how some aspects of the dissipative quantum model of the brain may contribute to find a solution to such a problem. The “brain like an orchestra” metaphor proposed by Freeman is mentioned to emphasize the integration within and between these domains to which quantum concepts may contribute. Chaotic behavior and fractal-like self-similarity properties of the brain functional activity are briefly discussed.

Since 1960 at least, I have been sharing chaos thinking with my brother Fred, and reciprocally, learning about brain research from him. This is a brief reminiscence of those early years.

The exclusion of semantics, meaning, and interpretation from exact sciences is nothing but an antiquated prejudice. It is a consequence of a centuries-old approach toward science, and it seems inevitable to update scientific reasoning in a way capable of taking these issues into account (Atmanspacher, 1994; Atmanspacher et al. 1992). This does not at all mean that any separation of different semiotic approaches or any sort of operationalization criteria should be abandoned in general.

Across disciplines, cultures and millennia, transformative instability has been viewed as a potentiating force that is the essence of all creativity. It is given many descriptive forms but reveals itself only in the patterns it informs, arising out of turbulent gaps and at chaotic edges of liminal space. This essay looks at three depictions of such primal shaping energies: one, ancient philosophy of the Tao, a pathless path to enlightenment that, like processes in the natural world, proceeds spontaneously and mindlessly; two, complexity theory's strange attractors, activating unpredictable new order out of chaos; and three, the unconscious, the subterranean fount of much if not most of our behavior, presented in the context of Dada's anarchic anti-reasoning that found creative inspiration in vast realms of the unconscious. Strongly influenced by Walter Freeman's work, in particular his writing concerning the “I don't know” mode of neural activity, I weave these commonalities together into the argument offered here, that knowing by not-knowing is not only an essential phase of cognitive neurodynamics, it is also a powerful heuristic for heightened creativity. Several of my paintings serve to illustrate, in themes and process, how this paradoxical stance of unknowing draws strength from imperceptible attractions-interactions of poetic, noetic and somatic ways of knowing.

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Zhi-Wei Sun
Department of Mathematics
Nanjing University
P.R. China


Florian Luca
School of Mathematics
University of the Witwatersrand
Johannesburg, South Africa

Jiang Zeng
Camille Jordan Institute
University of Claude Bernard Lyon 1


Robin Chapman
Mathematics Research Institute
University of Exeter
United Kingdom

Zoltan Furedi
Department of Mathematics
University of Illinois at Urbana-Champaign

Ronald Graham
Department of Mathematics
University of California at San Diego
California, USA

Neal Koblitz
Department of Mathematics
University of Washington

Michael Krivelevich
School of Mathematical Sciences
Tel Aviv University

Wen-Ching Winnie Li
Department of Mathematics
Pennsylvania State University

Monika Ludwig
Department of Mathematics
Polytechnic University New York

Ken Ono
Department of Mathematics
Emory University
Atlanta, GA USA

Alberto Perelli
Department of Mathematics
Genova University

Janos Pintz
Alfred Renyi Institute of Mathematics
Hungarian Academy of Sciences
Budapest, Hungary

Carl Pomerance
Department of Mathematics
Dartmouth College
Hanover, USA

Imre Ruzsa
Alfred Renyi Institute of Mathematics
Hungarian Academy of Sciences
Budapest, Hungary

René Schoof
Dipartimento di Matematica
Università di Roma "Tor Vergata"
Roma, Italy

Ole Warnaar
School of Mathematics and Physics
The University of Queensland
Brisbane, Australia

Qing Xiang
Department of Mathematical Sciences
University of Delaware
Newark, USA

Catherine Yan
Department of Mathematics
Texas A&M University

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