
Quaternionic de Branges Spaces and Characteristic Operator Function
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini
This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the socalled Sspectrum and Sresolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed.

Commutative DoublyIdempotent Semirings Determined by Chains and by Preorder Forests
Natanael Alpay and Peter Jipsen
A commutative doublyidempotent semiring (cdisemiring) (S,V,·,0,1) is a semilattice (S,V,0) with x V 0 = x and a semilattices (S,·,1) with identity 1 such that x0 = 0, and x(y V z) = xy V xz holds for all x, y, z ϵ S. Bounded distributive lattices are cdisemirings that satisfy xy = x ^ y, and the variety of cdisemirings covers the variety of bounded distributive lattices. Chajda and Länger showed in 2017 that the variety of all cdisemirings is generated by a 3element cdisemiring. We show that there are seven cdisemirings with a Vsemilattice of height less than or equal to 2. We construct all cdisemirings for which their multiplicative semilattice is a chain with n + 1 elements, and we show that up to isomorphism the number of such algebras is the nth Catalan number C_{n} = (1/(n+1)) (2n/n ) . We also show that cdisemirings with a complete atomic Boolean Vsemilattice on the set of atoms A are determined by singletonrooted preorder forests on the set A. From these results we obtain efficient algorithms to construct all multiplicatively linear cdisemirings of size n and all Boolean cdisemirings of size 2n.

Weakening Relation Algebras and FL^{2}algebras
Nikolaos Galatos and Peter Jipsen
FL^{2}algebras are latticeordered algebras with two sets of residuated operators. The classes RA of relation algebras and GBI of generalized bunched implication algebras are subvarieties of FL^{2}algebras. We prove that the congruences of FL^{2}algebras are determined by the congruence class of the respective identity elements, and we characterize the subsets that correspond to this congruence class. For involutive GBIalgebras the characterization simplifies to a form similar to relation algebras.
For a positive idempotent element p in a relation algebra A, the double division conucleus image p/A/p is an (abstract) weakening relation algebra, and all representable weakening relation algebras (RWkRAs) are obtained in this way from representable relation algebras (RRAs). The class S(dRA) of subalgebras of {p/A/p∶ A ϵ RA; 1 ≤ p^{2} = p ϵ A} is a discriminator variety of cyclic involutive GBIalgebras that includes RA. We investigate S(dRA) to find additional identities that are valid in all RWkRAs. A representable weakening relation algebra is determined by a chain if and only if it satisfies 0 ≤ 1, and we prove that the identity 1 ≤ 0 holds only in trivial members of S(dRA).

Interrogating Regulatory Mechanisms in Signaling Proteins by Allosteric Inhibitors and Activators: A Dynamic View Through the Lens of Residue Interaction Networks
Lindy Astl, Amanda Tse, and Gennady M. Verkhivker
Computational studies of allosteric interactions have witnessed a recent renaissance fueled by the growing interest in modeling of the complex molecular assemblies and biological networks. Allosteric interactions in protein structures allow for molecular communication in signal transduction networks. In this chapter, we discuss recent developments in understanding of allosteric mechanisms and interactions of protein systems, particularly in the context of structural, functional, and computational studies of allosteric inhibitors and activators. Computational and experimental approaches and advances in understanding allosteric regulatory mechanisms are reviewed to provide a systematic and critical view of the current progress in the development of allosteric modulators and highlight most challenging questions in the field. The abundance and diversity of genetic, structural, and biochemical data underlies the complexity of mechanisms by which targeted and personalized drugs can combat mutational profiles in protein kinases. Structural and computational studies of protein kinases have generated in recent decade significant insights that allowed leveraging knowledge about conformational diversity and allosteric regulation of protein kinases in the design and discovery of novel kinase drugs. We discuss recent developments in understanding multilayered allosteric regulatory machinery of protein kinases and provide a systematic view of the current state in understanding molecular basis of allostery mediated by kinase inhibitors and activators. In conclusion, we highlight the current status and future prospects of computational biology approaches in bridging the basic science of protein kinases with the discovery of anticancer therapies.

On the Structure of Generalized Effect Algebras and Separation Algebras
Sarah Alexander, Peter Jipsen, and Nadiya Upegui
Separation algebras are models of separation logic and effect algebras are models of unsharp quantum logics. We investigate these closely related classes of partial algebras as well as their noncommutative versions and the subclasses of (generalized) (pseudo)orthoalgebras. We present an orderly algorithm for constructing all nonisomorphic generalized pseudoeffect algebras with n elements and use it to compute these algebras with up to 10 elements.

Introduction to Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations: A Volume Dedicated to Heinz Langer
Daniel Alpay and Bernd Kirstein
"This volume is a tribute to Heinz Langer on the occasion of his eightieth birthday. Two earlier OT volumes (namely, volume 106 and volume 163; see [19] and [35] respectively) were dedicated to Heinz, the first one on the occasion of his sixtieth birthday, and the second one on the occasion of his retirement...The first named editor (DA) worked very hard on Heinz’ papers (and in particular on the abovementioned series of papers with M.G. Krein) during his doctoral studies [1] at the Weizmann Institute in Israel."

Multiparameter Assessment of Pre‐Earthquake Atmospheric Signals
Dimitar Ouzounov, Sergey Pulinets, JannYenq Liu, Katsumi Hattori, and Peng Han
We apply interdisciplinary observation to study earthquake processes, their physics, and the phenomena that precede their energy release. Our approach is based on multisensor observations of shortterm preearthquake phenomena preceding large earthquakes (M>6). The integrated satellite and terrestrial framework is our method for validation and is based on a sensor web of several physical and environmental parameters (satellite thermal infrared radiation (STIR), electron concentration in the ionosphere, air temperature, and relative humidity measurements) that were associated with earthquakes. The scientific rationale for multidisciplinary analysis is founded on the concept lithosphereatmosphereionosphere coupling. To check the predictive potential of preearthquake signals we validate in retrospective and prospective modes. Our validation processes consist of two steps: (a) a retrospective analysis preformed over three different regions with high seismic activity (M 6.0 Napa of 2014, M 6.0 Taiwan of 2016, and M 7.9 Kumamoto, Japan of 2016); (b) testing of Molchan's error diagram (MED) for STIR and differential total electron content anomalous events over Japan and Taiwan. Our findings suggest that: (a) preearthquake signals (with 130 days time lag) follow a general temporalspatial evolution pattern; (b) preearthquake atmospheric anomalies can provide shortterm predictive information for the occurrence of major earthquakes in the tested regions.

Lithosphere–Atmosphere–Ionosphere–Magnetosphere Coupling—A Concept for Pre‐Earthquake Signals Generation
Sergey Pulinets, Dimitar Ouzounov, Alexander Karelin, and Dmitry Davidenko
"The physics of earthquake genesis at the latest stage of a seismic cycle has always been the subject of scientific interest and discussion. In such discussion and in model development, however, only solid Earth processes have been contemplated. Development of satellite technologies brought new insight to the problem: new types of anomalies in the atmosphere and ionosphere were discovered that appear a few days/weeks before the main shock within the zone of earthquake preparation. These anomalies were thoroughly investigated, and results of the research have demonstrated a high statistical confidence warrants their identification as shortterm earthquake precursors. In this chapter we seek to clarify how information and energy are transported from underground to the upper and lower layers of the atmosphere, including nearEarth space. Owing to the interdisciplinary character of the model, it is very difficult to create a common code for all its physical and spatial domains. That is why the material presented here should be considered conceptual rather than modeling output. As an example, if the boundary layer plasmachemistry instrumentation is applied, for ionospheric anomalies the electromagnetism approach is adopted. This conceptual approach, however, proved to be valid not only for earthquakes but also for other natural and technological disasters where air ionization occurs."

Preface to Natural Hazards: Earthquakes, Volcanoes, and Landslides
Ramesh Singh and Darius Bartlett
A preface to Natural Hazards: Earthquakes, Volcanoes, and Landslides, edited by Ramesh Singh and Darius Bartlett, that gives an overview of the book and the contributions of each chapter.

Beyond Wavefunctions: A TimeSymmetric Nonlocal Ontology for Quantum Mechanics
Yakir Aharonov, Eliahu Cohen, and Avshalom C. Elitzur
"We take Agassi's attitude to QM as an invitation to present some insights we have gained during our research in this field. Following is a highly nontechnical account of a few works which we believe begins to merge into a novel and rich picture of physical reality."

Forcing Optimality and Brandt's Principle
Domenico Napoletani, Marco Panza, and Daniele C. Struppa
"In a series of previous papers...we described what we call the 'microarray paradigm' and we showed that there are scientific methodological motifs that structure the approach of data analysis to scientific problems. By 'microarray paradigm' we referred to the belief that sufficiently large data collected from a phenomenon allows answering any question about the phenomenon itself. Answers are then found through a process of automatic fitting of the data to models that do not carry any structural understanding beyond the actual solution of the problem. This is a process we suggested to label 'agnostic science'."

Forcing Optimality and Brandt's Principle
Domenico Napoletani, Marco Panza, and Daniele C. Struppa
We argue that many optimization methods can be viewed as representatives of “forcing”, a methodological approach that attempts to bridge the gap between data and mathematics on the basis of an a priori trust in the power of a mathematical technique, even when detailed, credible models of a phenomenon are lacking or do not justify the use of this technique. In particular, we show that forcing is implied in particle swarms optimization methods, and in modeling image processing problems through optimization. From these considerations, we extrapolate a principle for general data analysis methods, what we call ‘Brandt’s principle’, namely the assumption that an algorithm that approaches a steady state in its output has found a solution to a problem, or needs to be replaced. We finally propose that biological systems, and other phenomena that respect general rules of morphogenesis, are a natural setting for the application of this principle

Bernsteintype Inequalities for Bicomplex Polynomials
Irene Sabadini, Adrian Vajiac, and Mihaela Vajiac
This paper considers the wellknown Bernstein and Erdős–Lax inequalities in the case of bicomplex polynomials. We shall prove that the validity of these inequalities depends on the norm in use and we consider the cases of the Euclidean, Lie, dual Lie and hyperbolicvalued norms. In particular, we show that in the case of the Euclidean norm the inequalities holds keeping the same relation with the degree of the polynomial that holds in the classical complex case, but we have to enlarge the radius of the ball. In the case of the dual Lie norm both the relation with the degree and the radius of the ball have to be changed. Finally, we prove that the exact analogs of the two inequalities hold when considering the Lie norm and the hyperbolicvalued norm. In the case of these two norms we also show the validity of the maximum modulus principle for bicomplex holomorphic functions.

A Complex Analysis Problem Book (Second Edition)
Daniel Alpay
This second edition presents a collection of exercises on the theory of analytic functions, including completed and detailed solutions. It introduces students to various applications and aspects of the theory of analytic functions not always touched on in a first course, while also addressing topics of interest to electrical engineering students (e.g., the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). It provides examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space), and also includes a section reviewing essential aspects of topology, functional analysis and Lebesgue integration.
Benefits of the 2nd edition
Rational functions are now covered in a separate chapter. Further, the section on conformal mappings has been expanded.

An Introduction to Superoscillatory Sequences
Fabrizio Colombo, Irene Sabadini, and Daniele C. Struppa
The notion of superoscillating functions, or more properly of superoscillatory sequences, is a byproduct of Aharonov's theory of weak measurements and weak values in quantum mechanics. Recently, many mathematicians and physicists have begun to pay attention to the mathematical significance of such objects, and have been able to begin a theory of superoscillatory behavior. Not surprisingly, this theory is based on some classical results in Fourier analysis, and it displays interesting connections with the theory of convolution equations. In this paper we will put these connections in a larger context, and show how to use this context to generate a large class of superoscillating sequences. As a concrete example we discuss the Cauchy problem with superoscillatory datum for the harmonic oscillator. Finally, we show how this theory can be generalized to the case of several variables.

Flipped Classroom Model: Effects on Performance, Attitudes and Perceptions in High School Algebra
Peter Esperanza, Khristin Fabian, and Criselda Toto
In this study, we evaluated student perceptions of the flipped classroom model and its effects to students’ performance and attitudes to mathematics. A randomized controlled trial with 91 high school algebra students was conducted. The experimental group participated in a yearlong intervention of the flipped classroom model while the control group followed the traditional lesson delivery. Results of the yearend evaluation of this model showed positive student perceptions. An analysis of covariance of the algebra posttest score with learning model as treatment factor and pretest as covariate resulted in a significant treatment effect at .05 level of significance. A pairedsample ttest by treatment group to compare pretest and posttest math attitude scores resulted in a significant decrease in the control groups’ value of mathematics while the experimental group had a significant positive change in their confidence and enjoyment of mathematics.

Mathematics Is Physics
Matthew S. Leifer
In this essay, I argue that mathematics is a natural sciencejust like physics, chemistry, or biologyand that this can explain the alleged "unreasonable" effectiveness of mathematics in the physical sciences. The main challenge for this view is to explain how mathematical theories can become increasingly abstract and develop their own internal structure, whilst still maintaining an appropriate empirical tether that can explain their later use in physics. In order to address this, I offer a theory of mathematical theorybuilding based on the idea that human knowledge has the structure of a scalefree network and that abstract mathematical theories arise from a repeated process of replacing strong analogies with new hubs in this network. This allows mathematics to be seen as the study of regularities, within regularities, within ..., within regularities of the natural world. Since mathematical theories are derived from the natural world, albeit at a much higher level of abstraction than most other scientific theories, it should come as no surprise that they so often show up in physics.
This version of the essay contains an addendum responding to Slyvia Wenmackers' essay and comments that were made on the FQXi website.

An Advanced Complex Analysis Problem Book: Topological Vector Spaces, Functional Analysis, and Hilbert Spaces of Analytic Functions
Daniel Alpay
This is an exercises book at the beginning graduate level, whose aim is to illustrate some of the connections between functional analysis and the theory of functions of one variable. A key role is played by the notions of positive definite kernel and of reproducing kernel Hilbert space. A number of facts from functional analysis and topological vector spaces are surveyed. Then, various Hilbert spaces of analytic functions are studied.

Preface to "Intertwingled: The Work and Influence of Ted Nelson"
Douglas R. Dechow and Daniele C. Struppa
This is the preface to "Intertwingled: The Work and Influence of Ted Nelson", which examines and honors the work and influence of the computer visionary and reimagines its meaning for the future. Emerging from a conference held in 2014 at Chapman University, it includes contributions from worldrenowned computer scientists and media figures.
The full text of this book is available on an open access basis at Springer.
The blog for the Intertwingled Conference can be read here.

"It from Bit" and the Quantum Probability Rule
Matthew S. Leifer
I argue that, on the subjective Bayesian interpretation of probability, "it from bit" requires a generalization of probability theory. This does not get us all the way to the quantum probability rule because an extra constraint, known as noncontextuality, is required. I outline the prospects for a derivation of noncontextuality within this approach and argue that it requires a realist approach to physics, or "bit from it". I then explain why this does not conflict with "it from bit". This version of the essay includes an addendum responding to the open discussion that occurred on the FQXi website. It is otherwise identical to the version submitted to the contest.

Generalized Quaternionic Schur Functions in the Ball and HalfSpace and KreinLanger Factorization
Daniel Alpay, Fabrizio Colombo, and Irene Sabadini
In this paper we prove a new version of KreinLanger factorization theorem in the slice hyperholomorphic setting which is more general than the one proved in [8]. We treat both the case of functions with κ negative squares defined on subsets of the quaternionic unit ball or on subsets of the half space of quaternions with positive real part. A crucial tool in the proof of our results is the SchauderTychonoff theorem and an invariant subspace theorem for contractions in a Pontryagin space.

The Fock Space in the Slice Hyperholomorphic Setting
Daniel Alpay, Fabrizio Colombo, Irene Sabadini, and Guy Salomon
In this paper we introduce and study some basic properties of the Fock space (also known as SegalBargmann space) in the slice hyperholomorphic setting. We discuss both the case of slice regular functions over quaternions and also the case of slice monogenic functions with values in a Clifford algebra. In the specific setting of quaternions, we also introduce the full Fock space. This paper can be seen as the beginning of the study of infinite dimensional analysis in the quaternionic setting.

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis
Daniel Alpay, M. E. LunaElizarrarás, Michael Shapiro, and Daniele C. Struppa
"With the goal of providing the foundations for a rigorous study of modules of bicomplex holomorphic functions, we develop here a general theory of functional analysis with bicomplex scalars."

Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis
D. Alpay, M. E. LunaElizarrarás, M. Shapiro, and Daniele C. Struppa
This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers and then defines the notion of bicomplex modules. After introducing a number of norms and inner products on such modules (some of which appear in this volume for the first time), the authors develop the theory of linear functionals and linear operators on bicomplex modules. All of this may serve for many different developments, just like the usual functional analysis with complex scalars and in this book it serves as the foundational material for the construction and study of a bicomplex version of the well known Schur analysis.

Dust Storms and Their Influence on Atmospheric Parameters Over the IndoGangetic plains
Ramesh P. Singh
Dust storms are very common in the northern parts of India, and every year people living in the IndoGangetic plains suffer greatly. Dust storms affect daytoday lives of people living in the IndoGangetic plains (IGP) and impact their health. The atmospheric and meteorological parameters are highly influenced by the dust storms and are found to affect the air quality that creates a big health threat and also affects the weather conditions. In this chapter, we discuss use of satellite remote sensing data in monitoring the dust events which occur every year during premonsoon season and their impacts on ocean, atmosphere, and meteorological parameters. Longterm effects of such dust storms on the climate of the northern parts of India are discussed. Such dust storms can be easily monitored using satellite data that can be used in issuing warning to the people so that they would not be exposed to such strong dust storms.
Below you may find selected books and book chapters from Mathematics, Physics, and Computer Science faculty in the Schmid College of Science and Technology.
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