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Quaternion (Q-) mathematics formally contains many fragments of physical laws; in particular, the Hamiltonian for the Pauli equation automatically emerges in a space with Q-metric. The eigenfunction method shows that any Q-unit has an interior structure consisting of spinor functions; this helps us to represent any complex number in an orthogonal form associated with a novel geometric image (the conic-gearing picture). Fundamental Q-unit-spinor relations are found, revealing the geometric meaning of spinors as Lam\'e coefficients (dyads) locally coupling the base and tangent surfaces., Comment: 7 pages, 1 figure

Based upon a new general vector-valued vector product, generalized complex numbers with respect to certain positively homogeneous functionals including norms and antinorms are introduced and a vector-valued Euler type formula for them is derived using a vector valued exponential function. Furthermore, generalized Cauchy–Riemann differential equations for generalized complex differentiable functions are derived. For random versions of the considered new type of generalized complex numbers, moments are introduced and uniform distributions on discs with respect to functionals of the considered type are analyzed. Moreover, generalized uniform distributions on corresponding circles are studied and a connection with generalized circle numbers, which are natural relatives of π, is established. Finally, random generalized complex numbers are considered which are star-shaped distributed.

Let η ( s ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m ( τ ) and consider finite Dirichlet series υ M ( τ , s ) = ∑ m = 1 M b M , m ( τ ) m − s and η N ( τ , s ) = ∑ M = 1 N υ M ( τ , s ) . Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N ( τ , s ) approximates η ( s ) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η ( s ) = ∑ M = 1 ∞ υ M ( τ , s ) . Moreover, it looks that lim M → ∞ ⁡ m υ M ( τ , 1 − σ + i t ) ‾ m υ M ( τ , σ + i t ) = η ( σ + i t ) m η ( 1 − σ + i t ) ‾ m ; in other words, the individual summands in expected expansion ( ⁎ ) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M ( τ , σ + i t ) = υ M ( τ , σ + i t ) / η ( σ + i t ) . When M, τ, and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M ( τ , σ + i t ) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function.

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number, the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains, however, it can be used as a substitute derivative in specific domains.

It is well known that Galilean conformal algebras play important roles in the nonrelativistic anti-de Sitter/conformal field theory correspondence. The finite Lie conformal algebras P G ( a , b ) of planar Galilean type can be viewed as Lie conformal analogues of certain planar Galilean conformal algebras. In this paper, we classify finite irreducible conformal modules over P G ( a , b ) for all complex numbers a and b.

The Hyers-Ulam stability of ($\alpha, \beta, \gamma$)-derivations on Lie $C^*$-algebras is discussed by following functional inequality $ \begin{eqnarray*} f(ax+by)+f(ax-by) = 2f(ax)+bf(y)+bf(-y), \end{eqnarray*} $ where $a, b$ are nonzero fixed complex numbers.

It is a known uniqueness result that if f is a transcendental meromorphic function of finite order with two Borel excetional values $$a \ne \infty , b$$ and its first order difference operator $${\triangle }_c f\not \equiv 0$$ , for some complex number c, and if f and $${\triangle }_c f$$ share a, b CM, then $$a=0, b=\infty $$ and $$f=\exp (Az+B)$$ , where $$A\ne 0, B \in {\mathbb {C}} $$ . This type of results has its origin dating back to Csillag-Tumura’s uniqueness theorems. In this paper, by using completely different methods, we shall show that the result holds for arbitrary higher order difference operators. Examples are provided to show that this result is not valid for meromorphic functions with infinite order, which also shows a distinction between the derivatives and the difference operators of meromorphic functions, in view of Csillag-Tumura type uniqueness theorems.

We construct a mass dimension one fermionic field associated with flag-dipole spinors. These spinors are related to Elko (flag-pole spinors) by a one-parameter matrix transformation $\mathcal{Z}(z)$ where $z$ is a complex number. The theory is non-local and non-covariant. While it is possible to obtain a Lorentz-invariant theory via $\tau$-deformation, we choose to study the effects of non-locality and non-covariance. Our motivation for doing so is explained. We show that a fermionic field with $|z|\neq1$ and $|z|=1$ are physically equivalent. But for fermionic fields with more than one value of $z$, their interactions are $z$-dependent thus introducing an additional fermionic degeneracy that is absent in the Lorentz-invariant theory. We study the fermionic self-interaction and the local $U(1)$ interaction. In the process, we obtained non-local contributions for fermionic self-interaction that have previously been neglected. For the local $U(1)$ theory, the interactions contain time derivatives that renders the interacting density non-commutative at space-like separation. We show that this problem can be resolved by working in the temporal gauge. This issue is also discussed in the context of gravity., Comment: 8 pages. Published in EPJC

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\phi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\phi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space., Comment: 27 pages

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

We study a semilinear wave equation whose linear part corresponds to the linear Klein–Gordon equation in the non-relativistic limit, augmented with a nonlinearity that is Frechet-differentiable over the complex numbers. We show that this equation possesses an almost invariant manifold in phase space that generalizes the slow manifold which is known to exist for finite-dimensional Galerkin truncations of the system. This manifold is shown to be almost invariant to any algebraic order and can be constructed in the H s − 1 × H s phase space of the equation uniformly in the order of the approximation. In particular, we prove that the dynamics on this “slow manifold” shadows orbits of the full system over a finite interval of time.

We consider Berry's random planar wave model (1977) for a positive Laplace eigenvalue $E>0$, both in the real and complex case, and prove limit theorems for the nodal statistics associated with a smooth compact domain, in the high-energy limit ($E\to \infty$). Our main result is that both the nodal length (real case) and the number of nodal intersections (complex case) verify a Central Limit Theorem, which is in sharp contrast with the non-Gaussian behaviour observed for real and complex arithmetic random waves on the flat $2$-torus, see Marinucci et al. (2016) and Dalmao et al. (2016). Our findings can be naturally reformulated in terms of the nodal statistics of a single random wave restricted to a compact domain diverging to the whole plane. As such, they can be fruitfully combined with the recent results by Canzani and Hanin (2016), in order to show that, at any point of isotropic scaling and for energy levels diverging sufficently fast, the nodal length of any Gaussian pullback monochromatic wave verifies a central limit theorem with the same scaling as Berry's model. As a remarkable byproduct of our analysis, we rigorously confirm the asymptotic behaviour for the variances of the nodal length and of the number of nodal intersections of isotropic random waves, as derived in Berry (2002)., 51 pages

In this paper, we construct and study a new class of modules over the Block algebra B ( q ) , where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U ( h ) -free B ( q ) -modules of rank-1.

We initiate the study of the conformal bootstrap using Sturm-Liouville theory, specializing to four-point functions in one-dimensional CFTs. We do so by decomposing conformal correlators using a basis of eigenfunctions of the Casimir which are labeled by a complex number alpha. This leads to a systematic method for computing conformal block decompositions. Analyzing bootstrap equations in alpha space turns crossing symmetry into an eigenvalue problem for an integral operator K. The operator K is closely related to the Wilson transform, and some of its eigenfunctions can be found in closed form., 34 pages, 3 figures. v2: some minor changes and added subsection 2.3 on convergence

We discuss the notion about physical quantities as having values represented by real numbers, and its limiting to describe nature to be understood in relation to our appreciation that the quantum theory is a better theory of natural phenomena than its classical analog. Getting from the algebra of physical observables to their values on a fixed state is, at least for classical physics, really a homomorphic map from the algebra into the real number algebra. The limitation of the latter to represent the values of quantum observables with noncommutating algebraic relation is obvious. We introduce and discuss the idea of the noncommutative values of quantum observables and its feasibility, arguing that at least in terms of the representation of such a value as an infinite set of complex number, the idea makes reasonable sense theoretically as well as practically., 16 pages in latex, no figure; proof-read version

The flows considered so far were mostly stationary, that is, did not depend on time. As long as individual fluid parcels are not followed, the fluid could equally well be frozen. We turn now to flows that show mild modes of excitation in the form of wave propagation or unstable growth. Often, both phenomena are intimately linked: time-periodic, harmonic waves with angular speed \(\omega \) and amplitude \(\sim e^{i\,\omega t}\) can start to grow or decay exponentially, \(\sim e^{\pm |\omega |t}\), once \(\omega \) turns to be a complex number. Since the dispersion relation that describes waves as well as instabilities is often a nonlinear algebraic equation in \(\omega \) and the wavenumber k, such switching from real to imaginary angular speeds (or wavenumbers) is quite common.

This article presents the use of data processing to apprehend mathematical questions such as the Riemann Hypothesis (RH) by numerical calculation. Calculations are performed alongside graphs of the argument of the complex numbers &zeta, (x+iy)=a+ib and &xi, (x+iy)=p+iq, in the critical strip. On the one hand, the two-dimensional surface angle tan&minus, 1(b/a) of the Riemann Zeta function &zeta, &nbsp, is related to the semi-angle of the fractional part of y2&pi, ln(y2&pi, ) and, on the other hand, the Ksi function &zeta, of the Riemann functional equation is analyzed with respect to the coordinates (x,1&minus, x, y). The computation of the power series expansion of the &xi, function with its symmetry analysis highlights the RH by the underlying ratio of Gamma functions inside the &zeta, formula. The &zeta, power series beside the angle of both surfaces of the &zeta, function enables to exhibit a B&eacute, zout identity au+bv&equiv, c between the components (a,b) of the &zeta, function, which illustrates the RH. The geometric transformations in complex space of the Zeta and Ksi functions, illustrated graphically, as well as series expansions, calculated by computer, make it possible to elucidate this mathematical problem numerically. A final theoretical outlook gives deeper insights on the functional equation&rsquo, s mechanisms, by adopting a computer&ndash, scientific perspective.

Let G = G L n ( C ) , the general linear group over the complex numbers, and let B be the set of invertible upper triangular matrices in G . Let b = Lie ( B ) . For μ : T ∗ ( b × C n ) → b ∗ , where b ∗ ≅ g ∕ u and u being strictly upper triangular matrices in g = Lie ( G ) , we prove that the Hamiltonian reduction μ − 1 ( 0 ) r s s ∕ ∕ B of the extended regular semisimple locus b r s s of the Borel subalgebra is smooth, affine, reduced, and scheme-theoretically isomorphic to a dense open locus of C 2 n . We also show that the B -invariant functions on the regular semisimple locus of the Hamiltonian reduction of b × C n arise as the trace of a certain product of matrices.

Throughout this paper we use the following notations. By Zp we denote the ring of p-adic rational integers, Q denotes the field of rational numbers, Qp denotes the field of p-adic rational numbers, C denotes the complex number field, and Cp denotes the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp with |p|p = p−νp(p) = p−1. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp. If q ∈ C one normally assume that |q| < 1. If q ∈ Cp, we normally assume that |q − 1|p < p 1 p−1

Motivated by the Lie structure of the planar Galilean conformal algebra, we construct a class of infinite rank Lie conformal algebras C P G ( a , b ) , where a, b are complex numbers. All their conformal derivations are shown to be inner. The rank-one conformal modules and ℤ-graded free intermediate series modules over C P G ( a , b ) are completely classified. The parallel results of the finite Lie conformal subalgebra C P G ( a , b ) of C P G ( a , b ) are also presented.

In this paper, we first study an expansion for the operators $$\begin{aligned} (\partial _{x}-\lambda )^{k}, \end{aligned}$$ where $$\partial _{x}$$ is the Dirac operator in superspace and $$\lambda $$ is a complex number. Then we investigate expansions for polynomial Dirac operators in superspace. These expansions are regarded as generalized Almansi expansions in superspace. As an application of the expansions, the modified Riquier problem in superspace is considered.

In this paper we define the so-called twisted Heisenberg superalgebras over the complex number field by adding derivations to Heisenberg superalgebras. We classify the fine gradings up to equivalence on twisted Heisenberg superalgebras and determine the Weyl groups of those gradings., 30 pages. arXiv admin note: text overlap with arXiv:1405.4093 by other authors

International audience; This paper deals with the study of the two-dimensional Dirac operatorwith infinite mass boundary condition in a sector. We investigate the question ofself-adjointness depending on the aperture of the sector: when the sector is convexit is self-adjoint on a usual Sobolev space whereas when the sector is non-convexit has a family of self-adjoint extensions parametrized by a complex number of theunit circle. As a byproduct of this analysis we are able to give self-adjointnessresults on polygones. We also discuss the question of distinguished self-adjointextensions and study basic spectral properties of the operator in the sector.

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincare superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

Given complex numbers $w_1, \ldots, w_n$, we define the weight $w(X)$ of a set $X$ of 0-1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors $(x_1, \ldots, x_n)$ in $X$. We present an algorithm, which for a set $X$ defined by a system of homogeneous linear equations with at most $r$ variables per equation and at most $c$ equations per variable, computes $w(X)$ within relative error $\epsilon >0$ in $(rc)^{O(\ln n-\ln \epsilon)}$ time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant $\beta >0$ and all $j=1, \ldots, n$. A similar algorithm is constructed for computing the weight of a linear code over ${\Bbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs., Comment: The exposition is improved, a couple of inaccuracies corrected

In a special representation of complex action theory that we call ``future-included'', we study a harmonic oscillator model defined with a non-normal Hamiltonian $\hat{H}$, in which a mass $m$ and an angular frequency $\omega$ are taken to be complex numbers. In order for the model to be sensible some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in the plane of the arguments of $m$ and $\omega$, according to which the model is classified into several types. In addition, we formulate two pairs of annihilation and creation operators, two series of eigenstates of the Hamiltonians $\hat{H}$ and $\hat{H}^\dag$, and coherent states. They are normalized in a modified inner product $I_Q$, with respect to which the Hamiltonian $\hat{H}$ becomes normal. Furthermore, applying to the model the maximization principle that we previously proposed, we obtain an effective theory described by a Hamiltonian that is $Q$-Hermitian, i.e. Hermitian with respect to the modified inner product $I_Q$. The generic solution to the model is found to be the ``ground'' state. Finally we discuss what the solution implies., Comment: Latex 42 pages, 3 figures, typos corrected, presentation improved, the final version to appear in Prog.Theor.Exp.Phys

The vector space of symmetric matrices of size $n$ has a natural map to a projective space of dimension $2^n-1$ given by the principal minors. This map extends to the Lagrangian Grassmannian ${\rm LG}(n,2n)$ and over the complex numbers the image is defined, as a set, by quartic equations. In case the characteristic of the field is two, it was observed that, for $n=3,4$, the image is defined by quadrics. In this paper we show that this is the case for any $n$ and that moreover the image is the spinor variety associated to ${\rm Spin}(2n+1)$. Since some of the motivating examples are of interest in supergravity and in the black-hole/qubit correspondence, we conclude with a brief examination of other cases related to integral Freudenthal triple systems over integral cubic Jordan algebras.

We compute orbi-disk invariants of compact Gorenstein semi-Fano toric orbifolds by extending the method used for toric Calabi-Yau orbifolds. As a consequence the orbi-disc potential is analytic over complex numbers., arXiv admin note: text overlap with arXiv:1306.0437

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-It\^o chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations., Comment: 50 pages. The use of combinatorial formulae for controlling non-singular pairs of squares has been clarified

In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular, we prove that for $n \geq 2$ the $\Pi$-projective space $\mathbb{P}^n_\Pi$ can be constructed as the non-projected supermanifold determined by three elements $(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n}, \lambda)$, where $\mathbb{P}^n$ is the ordinary complex projective space, $\Omega^1_{\mathbb{P}^n}$ is its cotangent sheaf and $\lambda $ is a non-zero complex number, representative of the fundamental obstruction class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^n} \otimes \bigwedge^2 \Omega^1_{\mathbb{P}^n}) \cong \mathbb{C}.$ Likewise, in the case $n=1$ the $\Pi$-projective line $\mathbb{P}^1_\Pi$ is the split supermanifold determined by the pair $(\mathbb{P}^1, \Omega^1_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} (-2)).$ Moreover we show that in any dimension $\Pi$-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also $\Pi$-Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of $\Pi$-geometry., Comment: 15 pages. Misprints fixed and exposition improved. Some of the main propositions of section 4 got rewritten in a more precise form. Main results are unaffected

These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this framework. In particular, classical mechanics can be obtained as a theory with noncommutative observables and a non-zero Planck constant if we replace complex numbers in quantum mechanics by dual numbers. Our consideration is based on induced representations which are build from complex-/dual-/double-valued characters. Dynamic equations, rules of additions of probabilities, ladder operators and uncertainty relations are discussed. Finally, we prove a Calder\'on--Vaillancourt-type norm estimation for relative convolutions., Comment: 55 pages, 9 figures, lectures read in Jun 2016 at Varna.XVIII Conference on Symmetries, Integrability, Quantisation

The discrete-time quantum walk (QW) is determined by a unitary matrix whose component is complex number. Konno (2015) extended the QW to a walk whose component is quaternion.We call this model quaternionic quantum walk (QQW). The probability distribution of a class of QQWs is the same as that of the QW. On the other hand, a numerical simulation suggests that the probability distribution of a QQW is different from the QW. In this paper, we clarify the difference between the QQW and the QW by weak limit theorems for a class of QQWs., Comment: 11 pages, 2 figures, Interdisciplinary Information Sciences (in press)

This article constructs the Hilbert space for the algebra αβ − eiθβα = 1 that provides a continuous interpolation between the Clifford and Heisenberg algebras—this particular form is inspired by the properties of anyons. We study the eigenvalues of a generalized number operator (N=βα) and construct the Hilbert space, classified by values of a complex coordinate (λ0): the eigenvalues lie on a circle. For θ being an irrational multiple of 2π, we get an infinite-dimensional representation; however, for a rational multiple (MN) of 2π, it is finite-dimensional, parameterized by the complex coordinate λ0. The case for N = 2, θ = π is the usual Clifford algebra for fermions, while the case for N = ∞, θ = 0 is the Heisenberg algebra of bosons, albeit with two copies for positive and negative eigenvalues. We find a smooth transition from the fermion to the boson situation as N → ∞ from N = 2. After constructing the Hilbert space from the algebra, the cases for N = 2, 3 can be mapped to the “fuzzy sphere” of SU(2), while for general N, the “fuzzy pancake” is found to be the correct representation. Then, we motivate the study of coherent states, which are the eigenstates of α (the lowering operator), labeled by complex numbers for non-zero λ0. We specialize the study of coherent states to the very interesting case of λ0 = 0 and construct a calculus of the generalized Grassmann variables that result, applying it to compute a partition function for these particles. We then make some remarks about extending this study to that of anyons.

The purpose of this paper is to put the description of number scaling and its effects on physics and geometry on a firmer foundation, and to make it more understandable. A main point is that two different concepts, number and number value are combined in the usual representations of number structures. This is valid as long as just one structure of each number type is being considered. It is not valid when different structures of each number type are being considered. Elements of base sets of number structures, considered by themselves, have no meaning. They acquire meaning or value as elements of a number structure. Fiber bundles over a space or space time manifold, M, are described. The fiber consists of a collection of many real or complex number structures and vector space structures. The structures are parameterized by a real or complex scaling factor, s. A vector space at a fiber level, s, has, as scalars, real or complex number structures at the same level. Connections are described that relate scalar and vector space structures at both neighbor M locations and at neighbor scaling levels. Scalar and vector structure valued fields are described and covariant derivatives of these fields are obtained. Two complex vector fields, each with one real and one imaginary field, appear, with one complex field associated with positions in $M$ and the other with position dependent scaling factors. A derivation of the covariant derivative for scalar and vector valued fields gives the same vector fields. The derivation shows that the complex vector field associated with scaling fiber levels is the gradient of a complex scalar field. Use of these results in gauge theory shows that the imaginary part of the vector field associated with M positions acts like the electromagnetic field. The physical relevance of the other three fields, if any, is not known., 16 pages, 1 figure

We discuss the extension of the Lewis and Riesenfeld invariant method to cases where the quantum systems are modulated by time-dependent non-Hermitian Hamiltonians having $\mathcal{PT}$ symmetry. As an explicit example of this extension, we study the quantum motion of a particle submitted to action of a complex time-dependent linear potential with $\mathcal{PT}$ symmetry. We solve the time-dependent Schrodinger equation for this problem and construct a Gaussian wave packet solution. Afterwards, we use this Gaussian packet to calculate the expectation values of the position and the momentum and the uncertainty product. We find that these expectation values are complex numbers and consequently the position and momentum operators are not observables.

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I -statistical convergence, which is a recently introduced summability method. The names of our new methods are A I -lacunary statistical convergence and strongly A I -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S θ A I , F and N θ A I , F , respectively. We give some inclusion relations between S A I , F , S θ A I , F and N θ A I , F . We also investigate Cesáro summability for A I and we obtain some basic results between A I -Cesáro summability, strongly A I -Cesáro summability and the spaces mentioned above.

We construct symplectic groupoids integrating log-canonical Poisson structures on cluster varieties of type $\mathcal{A}$ and $\mathcal{X}$ over both the real and complex numbers. Extensions of these groupoids to the completions of the cluster varieties where cluster variables are allowed to vanish are also considered. In the real case, we construct source-simply-connected groupoids for the cluster charts via the Poisson spray technique of Crainic and M\u{a}rcu\c{t}. These groupoid charts and their analogues for the symplectic double and blow-up groupoids are glued by lifting the cluster mutations to groupoid comorphisms whose formulas are motivated by the Hamiltonian perspective of cluster mutations introduced by Fock and Goncharov., Comment: 33 pages

Canonical coherent states can be written as infinite series in powers of a single complex number $z$ and a positive integer $\rho(m)$. The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers $\{\rho(m)\}_{m=0}^\infty$. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number $z$ and the moments $\rho(m)$ of the canonical coherent states by $n \times n$ matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes-Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator., Comment: 18 pages

We construct L-theory with complex coefficients from the geometry of 1|2-dimensional perturbative mechanics. Methods of perturbative quantization lead to wrong-way maps that we identify with those coming from the MSO-orientation of L-theory tensored with the complex numbers., Comment: 12 pages

In this paper we give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori $\mathbb{T}_{\theta}^2$ equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number $\tau$ in the upper half plane, representing the conformal class of a metric on $\mathbb{T}_{\theta}^2$, and a Weyl factor given by a positive invertible element $k \in C^{\infty}(\mathbb{T}_{\theta}^2)$, the value at the origin, $\zeta (0)$, of the spectral zeta function of the Laplacian $\triangle'$ attached to $(\mathbb{T}_{\theta}^2, \tau, k)$ is independent of $\tau$ and $k$.

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that $ C\ell \left (\Re^3 \right) $ provides a particularly versatile algebraic framework., 21 pages, 0 figures

The three Pauli matrices are normally given [1] as the 2 × 2 matrices:where ‘i’ is the usual complex number imaginary unit.These matrices obey the relations a2 = I = b2 = c2(where I is the 2 × 2 identity matrix), as well as the anticommutation relations:Within the quantities ia,ib and ic,i is a scalar multiplier of the 2 × 2 Pauli matrices and, of course, commutes with each of a, b, c.

The XXZ spin chain is an integrable generalization of the Heisenberg chain that accounts for a uni-axial anisotropy in the spin interaction. Its Bethe Ansatz solution is a “straightforward” generalization of the one employed in the previous chapter, but the classification of complex roots is more involved and the nature of the low energy excitations changes with the anisotropy. After previewing the phase diagram of the chain in Sect. 4.1, we recap the coordinate Bethe Ansatz solution in Sect. 4.2. We then analyze the different phases in Sects. 4.3, 4.4, and 4.5, by focusing on the physical properties and skipping some technical derivations.

A functional is a function from the space of functions to a number field. We constructed an integral theory for a functional, using double extensions of real, complex number field in nonstandard arguments. In this paper, we consider a good system of infinite harmonic oscillators. Our functional integral implies that the propagator of the system is represented as the Riemann zeta function. The result is a direct generalization of our previous work [T. Nitta, Complexification of the propagator for the harmonic oscillator, in: Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, World Scientific, 2006, pp. 261–268]. We remark that the variables are not only real numbers but also complex numbers. We will assume that the reader is familiar with Nonstandard Analysis.

In this paper, we study the existence of analytic solutions of a second-order differential equation $$\begin{aligned} \alpha z+\beta x'(z)+\gamma x''(z)=x(az+bx''(z)), \end{aligned}$$ in the complex field $$\mathbb C,$$ where $$\alpha , \beta , \gamma , a, b$$ are complex numbers. We discuss not only the constant $$\lambda $$ at resonance, i.e. at a root of the unity, but also those $$\lambda $$ near resonance (near a root of the unity) under the Brjuno condition.

For each prime p and each embedding of the multiplicative group of an algebraic closure of F_p as complex roots of unity, we construct a p-adic indecomposable representation of the integral BC-system as additive endomorphisms of the big Witt ring of an algebraic closure of F_p. The obtained representations are the p-adic analogues of the complex, extremal KMS states at zero temperature of the BC-system. The role of the Riemann zeta function, as partition function of the BC-system over complex numbers is replaced, in the p-adic case, by the p-adic L-functions and the polylogarithms whose values at roots of unity encode the KMS states. We use Iwasawa theory to extend the KMS theory to a covering of the completion of an algebraic closure of the p-adic field. We show that our previous work on the hyperring structure of the adeles class space, combines with p-adic analysis to refine the space of valuations on the cyclotomic extension of Q as a noncommutative space intimately related to the integral BC-system and whose arithmetic geometry comes close to fulfill the expectations of the "arithmetic site". Finally, we explain how the integral BC-system appears naturally also in de Smit and Lenstra construction of the standard model of an algebraic closure of F_p which singles out the subsystem associated to the Z^-extension of Q., Comment: 61 pages