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## Über dieses Buch

This contributed volume features invited papers on current research and applications in mathematical structures. Featuring various disciplines in the mathematical sciences and physics, articles in this volume discuss fundamental scientific and mathematical concepts as well as their applications to topical problems. Special emphasis is placed on important methods, research directions and applications of analysis within and beyond each field. Covered topics include Metric operators and generalized hermiticity, Semi-frames, Hilbert-Schmidt operator, Symplectic affine action, Fractional Brownian motion, Walker Osserman metric, Nonlinear Maxwell equations, The Yukawa model, Heisenberg observables, Nonholonomic systems, neural networks, Seiberg-Witten invariants, photon-added coherent state, electrostatic double layers, and star products and functions.
All contributions are from the participants of the conference held October 2016 in Cotonou, Benin in honor of Professor Mahouton Norbert Hounkonnou for his outstanding contributions to the mathematical and physical sciences and education. Accessible to graduate students and postdoctoral researchers, this volume is a useful resource to applied scientists, applied and pure mathematicians, and mathematical and theoretical physicists.

## Inhaltsverzeichnis

### Metric Operators, Generalized Hermiticity, and Partial Inner Product Spaces

A quasi-Hermitian operator is an operator in a Hilbert space that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of metric operators, bounded or unbounded. We introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties.
Next we consider canonical lattices of Hilbert space s generated by unbounded metric operators. Since such lattices constitute the simplest case of a partial inner product space (pip-space), we can exploit the technique of pip-space operators. Thus we apply some of the previous results to operators on a particular pip-space, namely, the scale of Hilbert space s generated by a single metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.
Jean-Pierre Antoine, Camillo Trapani

### Beyond Frames: Semi-frames and Reproducing Pairs

Frames are nowadays a standard tool in many areas of mathematics, physics, and engineering. However, there are situations where it is difficult, even impossible, to design an appropriate frame. Thus there is room for generalizations, obtained by relaxing the constraints. A first case is that of semi-frames, in which one frame bound only is satisfied. Accordingly, one has to distinguish between upper and lower semi-frames. We will summarize this construction. Even more, one may get rid of both bounds, but then one needs two basic functions and one is led to the notion of reproducing pair. It turns out that every reproducing pair generates two Hilbert spaces, conjugate dual of each other. We will discuss in detail their construction and provide a number of examples, both discrete and continuous. Next, we notice that, by their very definition, the natural environment of a reproducing pair is a partial inner product space (pip-space) with an L 2 central Hilbert space. A first possibility is to work in a rigged Hilbert space. Then, after describing the general construction, we will discuss two characteristic examples, namely, we take for the partial inner product space a Hilbert scale or a lattice of L p spaces.
Jean-Pierre Antoine, Camillo Trapani

### On Hilbert-Schmidt Operator Formulation of Noncommutative Quantum Mechanics

This work gives value to the importance of Hilbert-Schmidt operators in the formulation of noncommutative quantum theory. A system of charged particle in a constant magnetic field is investigated in this framework.
Isiaka Aremua, Ezinvi Baloïtcha, Mahouton Norbert Hounkonnou, Komi Sodoga

### Symplectic Affine Action and Momentum with Cocycle

Let G be a Lie group, $$\mathfrak {g}$$ its Lie algebra, and $$\mathfrak {g^{*}}$$ the dual of $$\mathfrak {g}$$. Let Φ be the symplectic action of G on a symplectic manifold (M, ω). If the momentum mapping $$\mu :M\rightarrow \mathfrak {g^{*}}$$ is not Ad -equivariant, it is a fact that one can modify the coadjoint action of G on $$\mathfrak {g^{*}}$$ in order to make the momentum mapping equivariant with respect to the new G-structure in $$\mathfrak {g^{*}}$$, and the orbit of the coadjoint action is a symplectic manifold. With the help of a two cocycle $$\sum :\mathfrak {g}\times \mathfrak {g}\rightarrow \mathbf {R}$$, $$(\xi ,\eta )\mapsto \sum (\xi ,\eta )=d\hat {\sigma }_{\eta }(e)\cdot \xi$$ associated with one cocycle $$\sigma :G\rightarrow \mathfrak {g^{*}};~~\sigma (g)=\mu (\phi _g(m))-Ad^*_g\mu (m)$$, we show that a symplectic structure can be defined on the orbit of the affine action $$\Psi (g,\beta ):=Ad_{g}^{*}\beta +\sigma (g)$$ of G on $$\mathfrak {g^{*}}$$, the orbit of which is a symplectic manifold with the symplectic structure $$\omega _{\beta }(\xi _{\mathfrak {g^{*}}}(v),\eta _{\mathfrak {g^{*}}}(v))=-\beta ([\xi ,\eta ])+\sum (\eta ,\xi )$$.
Furthermore, we introduce a deformed Poisson bracket on (M, ω) with which some classical results of conservative mechanics still hold true in a new setting.
Augustin Batubenge, Wallace Haziyu

### Some Difference Integral Inequalities

We establish difference versions of the classical integral inequalities of Hölder, Cauchy-Schwartz, Minkowski and integral inequalities of Grönwall, Bernoulli and Lyapunov based on the Lagrange method of linear difference equation of first order.
G. Bangerezako, J. P. Nuwacu

### Theoretical and Numerical Comparisons of the Parameter Estimator of the Fractional Brownian Motion

The fractional Brownian motion which has been defined by Kolmogorov (CR (Doklady) Acad Sci URSS (N.S.) 26:115–118) and numerous papers was devoted to its study since its study in Mandelbrot and Van Ness (SIAM Rev 10:422–437, 1968) [19] present it as a paradigm of self-similar processes. The self-similarity parameter, also called the Hurst parameter, commands the dynamic of this process and the accuracy of its estimation is often crucial. We present here the main and used methods of estimation, with the limit theorems satisfied by the estimators. A numerical comparison is also provided allowing to distinguish between the estimators.
Jean-Marc Bardet

### Minimal Lethal Disturbance for Finite Dimensional Linear Systems

In this work we consider the problem of robust viability and viability radius for finite dimensional disturbed linear systems. The problem consists in the determination of the smallest disturbance f (in some disturbance set $${\mathcal {F}}$$), for which a given viable state z 0 does not remains viable. We also consider the problem of the determination of the smallest disturbance f for which the viability set $$Viab_{{\mathcal {K}}}^{f}$$ becomes empty; the smallest disturbance that makes all the $${\mathcal {K}}$$-viable states non viable, which we call the Minimal Lethal Disturbance (MLD). We give some characterizations of the viability radius and an illustration through some examples and connection with toxicity in biology.
Abdes Samed Bernoussi, Mina Amharref, Mustapha Ouardouz

### Walker Osserman Metric of Signature (3, 3)

A Walker m-manifold is a pseudo-Riemannian manifold, which admits a field of parallel null r-planes, with $$r{\leqslant } \frac {m}{2}$$. The Riemann extension is an important method to produce Walker metric on the cotangent bundle T M of any affine manifold (M, ∇). In this paper, we investigate the torsion-free affine manifold (M, ∇) and their Riemann extension $$(T^* M,\bar {g})$$ as concerns heredity of the Osserman condition.
Abdoul Salam Diallo, Mouhamadou Hassirou, Ousmane Toudou Issa

### Conformal Symmetry Transformations and Nonlinear Maxwell Equations

We make use of the conformal compactification of Minkowski spacetime M # to explore a way of describing general, nonlinear Maxwell fields with conformal symmetry. We distinguish the inverse Minkowski spacetime [M #]−1 obtained via conformal inversion, so as to discuss a doubled compactified spacetime on which Maxwell fields may be defined. Identifying M # with the projective light cone in (4 + 2)-dimensional spacetime, we write two independent conformal-invariant functionals of the 6-dimensional Maxwellian field strength tensors—one bilinear, the other trilinear in the field strengths—which are to enter general nonlinear constitutive equations. We also make some remarks regarding the dimensional reduction procedure as we consider its generalization from linear to general nonlinear theories.
Gerald A. Goldin, Vladimir M. Shtelen, Steven Duplij

### The Yukawa Model in One Space - One Time Dimensions

The Yukawa Model is revisited in one space - one time dimensions in an approach completely different to those available in the literature. We show that at the classical level it is a constrained system. We apply the Dirac method of quantization of constrained systems. Then by means of the bosonization procedure we uniformize the Hamiltonian at the quantum level in terms of a pseudo-scalar field and the chiral components of a real scalar field.
Laure Gouba

### Towards the Quantum Geometry of Saturated Quantum Uncertainty Relations: The Case of the (Q, P) Heisenberg Observables

This contribution to the present Workshop Proceedings outlines a general programme for identifying geometric structures—out of which to possibly recover quantum dynamics as well—associated with the manifold in Hilbert space of the quantum states that saturate the Schrödinger–Robertson uncertainty relation associated with a specific set of quantum observables which characterise a given quantum system and its dynamics. The first step in such an exploration is addressed herein in the case of the observables Q and P of the Heisenberg algebra for a single degree of freedom system. The corresponding saturating states are the well-known general squeezed states, whose properties are reviewed and discussed in detail together with some original results, in preparation of a study deferred to a separated analysis of their quantum geometry and of the corresponding path integral representation over such states.
Jan Govaerts

### The Role of the Jacobi Last Multiplier in Nonholonomic Systems and Locally Conformal Symplectic Structure

In this pedagogic article we study the geometrical structure of nonholonomic system and elucidate the relationship between Jacobi’s last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers (Rep Math Phys 42(1–2):231–247, 1998; Rep Math Phys 49(2–3):143–149, 2002; What is a completely integrable nonholonomic dynamical system, in Proceedings of the XXX symposium on mathematical physics, Toruń, 1998; Rep Math Phys 32:99–115, 1993), can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system.
Partha Guha

### Non-perturbative Renormalization Group of a U(1) Tensor Model

This paper aims at giving some comment on our new development on the functional renormalization group applied to the U(1) tensor model previously studied in [Phys. Rev. D 95, 045013 (2017)]. Using the Wetterich non-perturbative equation, the flow of the couplings and mass parameter are discussed and the physical implication such as the asymptotically safety of the model is provided.
Vincent Lahoche, Dine Ousmane Samary

### Ternary Z2 and Z3 Graded Algebras and Generalized Color Dynamics

We discuss cubic and ternary algebras which are a direct generalization of Grassmann and Clifford algebras, but with Z 3-grading replacing the usual Z 2-grading. Combining Z 2 and Z 3 gradings results in algebras with Z 6 grading, which are also investigated. We introduce a universal constitutive equation combining binary and ternary cases.
Elementary properties and structures of such algebras are discussed, with special interest in low-dimensional ones, with two or three generators.
Invariant anti-symmetric quadratic and cubic forms on such algebras are introduced, and it is shown how the SL(2, C) group arises naturally in the case of lowest dimension, with two generators only, as the symmetry group preserving these forms.
In the case of lowest dimension, with two generators only, it is shown how the cubic combinations of Z 3-graded elements behave like Lorentz spinors, and the binary product of elements of this algebra with an element of the conjugate algebra behave like Lorentz vectors.
Then Pauli’s principle is generalized for the case of the Z 3 graded ternary algebras leading to cubic commutation relations. A generalized Dirac equation is introduced.
The model displays the color SU(3) symmetry of strong interactions, as well as the SU(2) and U(1) symmetries giving rise to the Standard Model gauge fields.
Richard Kerner

### Pseudo-Solution of Weight Equations in Neural Networks: Application for Statistical Parameters Estimation

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is implemented for the approximation of smooth batch data containing input–output of the hidden neurons and the final neural output of the network. The training set is associated with the adjustable parameters of the network by weight equations that may be compatible or incompatible.Then we have obtained the exact input weight of the nonlinear equations and the approximated output weight of the linear equations using the conjugate gradient method with an adaptive learning rate. Using the multi-agent system as the different rates of traders of five regions in the Republic of Benin smuggling the fuel from the Federal Republic of Nigeria and the computational neural networks, one can predict the average rates of fuel smuggling traders thinking of this activity in terms of its dangerous character and those susceptible to give up this activity, respectively. This information enables the planner or the decision-maker to compare alternative actions, to select the best one for ensuring the retraining of these traders.
Vincent J. M. Kiki, Villévo Adanhounme, Mahouton Norbert Hounkonnou

### A Note on Curvatures and Rank 2 Seiberg–Witten Invariants

In this paper, we investigate rank 2 Seiberg–Witten equations which were introduced and studied in Massamba and Thompson (J Geom Phys 56:643–665, 2006). We derive some lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with non-trivial rank 2 Seiberg–Witten invariants. Existence of Einstein and anti-self-dual metrics on some compact oriented 4-manifolds is also discussed.
Fortuné Massamba

### Shape Invariant Potential Formalism for Photon-Added Coherent State Construction

An algebro-operator approach, called shape invariant potential method, of constructing generalized coherent states for photon-added particle system is presented. Illustration is given on Pöschl–Teller potentials.
Komi Sodoga, Isiaka Aremua, Mahouton Norbert Hounkonnou

### On the Fourier Analysis for L2 Operator-Valued Functions

We endow the set of square integrable operator-valued functions on a locally compact group with a pre-Hilbert module structure and define the ρ-Fourier transform for such functions. We also describe the Fourier transform of Hilbert-Schmidt operator-valued function on compact groups.
Mawoussi Todjro, Yaogan Mensah

### Electrostatic Double Layers in a Magnetized Isothermal Plasma with two Maxwellian Electrons

Finite amplitude nonlinear ion-acoustic double layers are discussed in a magnetized plasma consisting of warm isothermal ion fluid and two Boltzmann distributed electron species by assuming the charge neutrality condition at equilibrium. The model is compatible with the evolution of negative potential double layer structures in the auroral acceleration region. The model predicts maximum electric field amplitude of about ∼ 30 mV/m, which is within the satellite measurements in the auroral acceleration region of the Earth’s magnetosphere.
Odutayo Raji Rufai

### Star Products, Star Exponentials, and Star Functions

We give a brief review on non-formal star products and star exponentials and star functions (Omori et al., Deformation of expressions for elements of an algebra, in Symplectic, Poisson, and Noncommutative Geometry. Mathematical Sciences Research Institute Publications, vol. 62 (Cambridge University Press, Cambridge, 2014), pp. 171–209; Deformation of expressions for elements of algebra, arXiv:1104.1708v1[math.ph]; Deformation of expressions for elements of algebras (II), arXiv:1105.1218v2[math.ph]). We introduce a star product on polynomials with a deformation parameter ħ > 0. Extending to functions on complex space enables us to consider exponential element in the star product algebra, called a star exponential. By means of the star exponentials we can define several functions called star functions in the algebra, with some noncommutative identities. We show certain examples.
Akira Yoshioka

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