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This work proposes a unified theory of regularity in one hypercomplex variable: the theory of $T$-regular functions. In the special case of quaternion-valued functions of one quaternionic variable, this unified theory comprises Fueter-regular functions, slice-regular functions and a recently-discovered function class. In the special case of Clifford-valued functions of one paravector variable, it encompasses monogenic functions, slice-monogenic functions, generalized partial-slice monogenic functions, and a variety of function classes not yet considered in literature. For $T$-regular functions over an associative $*$-algebra, this work provides integral formulas, series expansions, an Identity Principle, a Maximum Modulus Principle and a Representation Formula. It also proves some foundational results about $T$-regular functions over an alternative but nonassociative $*$-algebra, such as the real algebra of octonions., Comment: 75 pages

In the present paper, we prove a resolvent equation for the $\mathcal{S}$-resolvent operator in the quaternionic framework. Exploiting this resolvent equation, we find a series expansion for the $\mathcal{S}$-resolvent operator in an open neighborhood of any given quaternion belonging to the $\mathcal{S}$-resolvent set. Some consequences of the series expansion are deduced. In particular, we describe a property of the geometry of the $\mathcal{S}$-resolvent set in terms of the Cassini pseudo-metric on quaternions. The concept of vector-valued real analytic function of several variables plays a crucial role in the proof of the mentioned series expansion for the $\mathcal{S}$-resolvent operator.

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular function, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function $f:\Omega\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $\Omega$ (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent $f$ better and better and allow to prove increasingly stronger results., Comment: 42 pages, 12 figures, 1 table. To appear in Mathematische Zeitschrift

We define a very general notion of regularity for functions taking values in an alternative real $*$-algebra. Over Clifford numbers, this notion subsumes the well-established notions of monogenic function and slice-monogenic function. Over quaternions, in addition to subsuming the notions of Fueter-regular function and of slice-regular function, it gives rise to an entirely new theory, which we develop in some detail., Comment: 16 pages

We prove that every real algebraic set $V\subset\mathbb{R}^n$ with isolated singularities is homeomorphic to a set $V'\subset\mathbb{R}^m$ that is $\mathbb{Q}$-algebraic in the sense that $V'$ is defined in $\mathbb{R}^m$ by polynomial equations with rational coefficients. The homeomorphism $\phi:V\to V'$ we construct is semialgebraic, preserves nonsingular points and restricts to a Nash diffeomorphism between the nonsingular loci. In addition, we can assume that $V'$ has a codimension one subset of rational points. If $m$ is sufficiently large, we can also assume that $V'\subset\mathbb{R}^m$ is arbitrarily close to $V\subset\mathbb{R}^n\subset\mathbb{R}^m$, and $\phi$ extends to a semialgebraic homeomorphism from $\mathbb{R}^m$ to $\mathbb{R}^m$. A first consequence of this result is a $\mathbb{Q}$-version of the Nash-Tognoli theorem: Every compact smooth manifold admits a $\mathbb{Q}$-algebraic model. Another consequence concerns the open problem of making Nash germs $\mathbb{Q}$-algebraic: Every Nash set germ with an isolated singularity is semialgebraically equivalent to a $\mathbb{Q}$-algebraic set germ., Comment: 37 pages

We present a new mimetic finite difference method for diffusion problems that converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of $P_{0}$-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries., Comment: Accepted manuscript

We provide a novel framework to compute a discrete vector potential of a given discrete vector field on arbitrary polyhedral meshes. The framework exploits the concept of acyclic matching, a combinatorial tool at the core of discrete Morse theory. We introduce the new concept of complete acyclic matchings and we show that they give the same end result of Gaussian elimination. Basically, instead of doing costly row and column operations on a sparse matrix, we compute equivalent cheap combinatorial operations that preserve the underlying sparsity structure. Currently, the most efficient algorithms proposed in literature to find discrete vector potentials make use of tree-cotree techniques. We show that they compute a special type of complete acyclic matchings. Moreover, we show that the problem of computing them is equivalent to the problem of deciding whether a given mesh has a topological property called collapsibility. This fact gives a topological characterization of well-known termination problems of tree-cotree techniques. We propose a new recursive algorithm to compute discrete vector potentials. It works directly on basis elements of $1$- and $2$-chains by performing elementary Gaussian operations on them associated with acyclic matchings. However, the main novelty is that it can be applied recursively. Indeed, the recursion process allows us to sidetrack termination problems of the standard tree-cotree techniques. We tested the algorithm on pathological triangulations with known topological obstructions. In all tested problems we observe linear computational complexity as a function of mesh size. Moreover, the algorithm is purely graph-based so it is straightforward to implement and does not require specialized external procedures. We believe that our framework could offer new perspectives to sparse matrix computations.

This work looks at the theory of octonionic slice regular functions through the lens of differential topology. It proves a full-fledged version of the Open Mapping Theorem for octonionic slice regular functions. Moreover, it opens the path for a possible use of slice regular functions in the study of almost-complex structures in eight dimensions., Comment: 35 pages, published online in the Journal of Noncommutative Geometry

This paper deals with generators $\mathsf{A}$ of strongly continuous right linear semigroups in Banach two-sided spaces whose set of scalars is an arbitrary Clifford algebra $\mathit{C}\ell(0,n)$. We study the invertibility of operators of the form $P(\mathsf{A})$, where $P(x)\in\mathbb{R}[x]$ is any real polynomial, and we give an integral representation for $P(\mathsf{A})^{-1}$ by means of a Laplace-type transform of the semigroup $\mathsf{T}(t)$ generated by $\mathsf{A}$. In particular, we deduce a new integral representation for the operator $(\mathsf{A}^2 - 2\mathrm{Re}(q) \,\mathsf{A} + |q|^2)^{-1}$. As an immediate consequence, we also obtain a new proof of the well-known integral representation for the $S$-resolvent operator of $\mathsf{A}$ (also called spherical resolvent operator of $\mathsf{A}$).

The concept of slice regular function over the real algebra $\mathbb{H}$ of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let $\Omega$ be an open subset of $\mathbb{H}$, which intersects $\mathbb{R}$ and is invariant under rotations of $\mathbb{H}$ around $\mathbb{R}$. A function $f:\Omega\to\mathbb{H}$ is slice regular if it is of class $\mathscr{C}^1$ and, for all complex planes $\mathbb{C}_I$ spanned by $1$ and a quaternionic imaginary unit $I$, the restriction $f_I$ of $f$ to $\Omega_I=\Omega\cap\mathbb{C}_I$ satisfies the Cauchy-Riemann equations associated to $I$, i.e., $\overline{\partial}_I f_I=0$ on $\Omega_I$, where $\overline{\partial}_I=\frac{1}{2}\big(\frac{\partial}{\partial\alpha}+I\frac{\partial}{\partial\beta}\big)$. Given any positive natural number $n$, a function $f:\Omega\to\mathbb{H}$ is called slice polyanalytic of order $n$ if it is of class $\mathscr{C}^n$ and $\overline{\partial}_I^{\,n} f_I=0$ on $\Omega_I$ for all $I$. We define global slice polyanalytic functions of order $n$ as the functions $f:\Omega\to\mathbb{H}$, which admit a decomposition of the form $f(x)=\sum_{h=0}^{n-1}\overline{x}^hf_h(x)$ for some slice regular functions $f_0,\ldots,f_{n-1}$. Global slice polyanalytic functions of any order $n$ are slice polyanalytic of the same order $n$. The converse is not true: for each $n\geq2$, we give examples of slice polyanalytic functions of order $n$, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations $\overline{\partial}_I^{\,n} f_I=0$ on $\Omega_I$ and as solutions of their global version $\overline{\vartheta}^nf=0$ on $\Omega\setminus\mathbb{R}$. Our quaternionic results extend to the monogenic case., Comment: 20 pages

In this paper, we lay the foundations of the theory of slice regular functions in several variables ranging in any real alternative $^*$-algebra, including quaternions, octonions and Clifford algebras. This theory is an extension of the classical theory of holomorphic functions in several complex variables.

An efficient numerical algorithm for the computation of linking number is presented. The algorithm keep tracks or rounding error so that it can ensure the correctness of the results., Comment: 19 pages 1 figure

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra $\mathbb{O}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:\Omega_D\subset\mathbb{O}\to\mathbb{O}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra $\mathbb{H}_\mathbb{I}$ of $\mathbb{O}$ generated by a pair $\mathbb{I}=(I,J)$ of orthogonal imaginary units $I$ and $J$ ($\mathbb{H}_\mathbb{I}$ is a `quaternionic slice' of $\mathbb{O}$), the restriction of $f$ to $\Omega_D\cap\mathbb{H}_\mathbb{I}$ belongs to the kernel of the corresponding Cauchy-Riemann-Fueter operator $\frac{\partial}{\partial x_0}+I\frac{\partial}{\partial x_1}+J\frac{\partial}{\partial x_2}+(IJ)\frac{\partial}{\partial x_3}$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their `holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine $8$-dimesional domains of $\mathbb{O}$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $\Gamma$ and of slice Fueter operator $\bar{\vartheta}_F$ over octonions, which allow to characterize slice Fueter-regular functions as the $\mathscr{C}^2$-functions in the kernel of $\bar{\vartheta}_F$ satisfying a second order differential system associated with $\Gamma$. The paper contains eight open problems., Comment: 33 pages

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian $J_f$ of a slice regular function $f$ proving in particular that $\det(J_f)\geq0$, i.e. $f$ is orientation-preserving. We give a complete characterization of the fibers of $f$ making use of a new notion we introduce here, the one of wing of $f$. We investigate the singular set $N_f$ of $f$, i.e. the set in which $J_f$ is singular. The singular set $N_f$ turns out to be equal to the branch set of $f$, i.e. the set of points $y$ such that $f$ is not a homeomorphism locally at $y$. We establish the quasi-openness properties of $f$. As a consequence we deduce the validity of the Maximum Modulus Principle for $f$ in its full generality. Our results are sharp as we show by explicit examples., Comment: 27 pages. To appear in the Journal of Geometric Analysis

Let $Y\subset{\mathbb R}^n$ be a triangulable set and let $r$ be either a positive integer or $r=\infty$. We say that $Y$ is a $\mathscr{C}^r$-approximation target space, or a $\mathscr{C}^r\text{-}\mathtt{ats}$ for short, if it has the following universal approximation property: For each $m\in{\mathbb N}$ and each locally compact subset $X$ of~${\mathbb R}^m$, any continuous map $f:X\to Y$ can be approximated by $\mathscr{C}^r$ maps $g:X\to Y$ with respect to the strong $\mathscr{C}^0$ Whitney topology. Taking advantage of new approximation techniques we prove: if $Y$ is weakly $\mathscr{C}^r$ triangulable, then $Y$ is a $\mathscr{C}^r\text{-}\mathtt{ats}$. This result applies to relevant classes of triangulable sets, namely: (1) every locally compact polyhedron is a $\mathscr{C}^\infty\text{-}\mathtt{ats}$, (2) every set that is locally $\mathscr{C}^r$ equivalent to a polyhedron is a $\mathscr{C}^r\text{-}\mathtt{ats}$, and (3) every locally compact locally definable set of an arbitrary o-minimal structure is a $\mathscr{C}^1\text{-}\mathtt{ats}$ (this includes locally compact locally semialgebraic sets and locally compact subanalytic sets). In addition, we prove: if $Y$ is a global analytic set, then each proper continuous map $f:X\to Y$ can be approximated by proper $\mathscr{C}^\infty$ maps $g:X\to Y$. Explicit examples show the sharpness of our results., Comment: 29 pages

In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are differentiable of class~${\mathcal C}^\nu$ for a fixed integer $\nu\geq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $\nu=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $\nu \geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${\mathcal C}^\nu$ semialgebraically equivalent to a polyhedron, for instance when $T$ is a compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for instance when $T$ is a Nash set. Our density results are based on a recent ${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples., Comment: 26 pages

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge2$ has a natural reformulation over quaternions, obtained via rotation around the real axis. We prove that such a reformulation is true only for $d=2$. We present a new quaternionic version of the Gauss-Lucas Theorem valid for all $d\geq2$, together with some consequences., Comment: 7 pages, 1 figure. Remarks added in section 3. Proposition 14 added with complete proof

This work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem., Comment: 24 pages, published online in Proc. Roy. Soc. Edinburgh Sect. A

This paper deals with the existence of algebraic structures on compact Nash sets. We introduce the algebraic-topological notion of asymmetric Nash cobordism between compact Nash sets, and we prove that a compact Nash set is semialgebraically homeomorphic to a real algebraic set if and only if it is asymmetric Nash cobordant to a point or, equivalently, if it is strongly asymmetric Nash cobordant to a real algebraic set. As a consequence, we obtain new large classes of compact Nash sets semialgebraically homeomorphic to real algebraic sets. To prove our results, we need to develop new algebraic-topological approximation procedures. We conjecture that every compact Nash set is asymmetric Nash cobordant to a point, and hence semialgebraically homeomorphic to a real algebraic set., Comment: 19 pages, to appear in JEMS

We present an alternate proof, much quicker and more straightforward than the original one, of a celebrated Fulton's conjecture on the ample cone of the moduli space of stable rational curves with n marked points in the case n=7.

We present an efficient algorithm for the construction of a basis of $H_2(\overline{\Omega},\partial\Omega;\mathbb Z)$ via the Poincar\'e--Lefschetz duality theorem. Denoting by $g$ the first Betti number of $\overline \Omega$ the idea is to find, first $g$ different $1$-boundaries of $\overline{\Omega}$ with supports contained in $\partial\Omega$ whose homology classes in $\mathbb R^3 \setminus \Omega$ form a basis of $H_1(\mathbb R^3 \setminus \Omega;\mathbb Z)$, and then to construct in $\overline{\Omega}$ a homological Seifert surface of each one of these $1$-boundaries. The Poincar\'e--Lefschetz duality theorem ensures that the relative homology classes of these homological Seifert surfaces in $\overline\Omega$ modulo $\partial\Omega$ form a basis of $H_2(\overline\Omega,\partial\Omega;\mathbb Z)$. We devise a simply procedure for the construction of the required set of $1$-boundaries of $\overline{\Omega}$ that, combined with a fast algorithm for the construction of homological Seifert surfaces, allows the efficient computation of a basis of $H_2(\overline{\Omega},\partial\Omega;\mathbb Z)$ via this very natural geometrical approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms., Comment: arXiv admin note: text overlap with arXiv:1409.5487

The main goal of this work is classifying the singularities of slice regular functions over a real alternative *-algebra A. This function theory has been introduced in 2011 as a higher-dimensional generalization of the classical theory of holomorphic complex functions, of the theory of slice regular quaternionic functions launched by Gentili and Struppa in 2006 and of the theory of slice monogenic functions constructed by Colombo, Sabadini and Struppa since 2009. Along with this generalization step, the larger class of slice functions over A has been defined. We introduce here a new type of series expansion near each singularity of a slice regular function. This instrument, which is new even in the quaternionic case, leads to a complete classification of singularities. This classification also relies on some recent developments of the theory, concerning the algebraic structure and the zero sets of slice functions. Peculiar phenomena arise, which were not present in the complex or quaternionic case, and they are studied by means of new results on the topology of the zero sets of slice functions. The analogs of meromorphic functions, called (slice) semiregular functions, are introduced and studied., Comment: 36 pages, to appear in Advances in Mathematics

In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy integral formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included., Comment: A misprint in the second displayed formula of Definition 6.11 (p. 28) has been corrected

The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann's foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has been started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of quaternionic formulation is the description of composite quantum systems in absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover in particular the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator., Comment: 62 pages, no figures. References added and updated. Accepted for publication in Reviews in Mathematical Physics

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative $^*$-algebra $A$ over $\mathbb{R}$. These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over $A$, which comprise all polynomials over $A$, form an alternative $^*$-algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront with questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest., Comment: 38 pages, to appear in Transactions of the American Mathematical Society

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for C^1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions., Comment: 13 pages, an example has been added at the end of Section 4. To appear in "Complex Analysis and Operator Theory"

Let $\Omega$ be a bounded domain of $\mathbb{R}^3$ whose closure $\overline{\Omega}$ is polyhedral, and let $\mathcal{T}$ be a triangulation of $\overline{\Omega}$. Assuming that the boundary of $\Omega$ is sufficiently regular, we provide an explicit formula for the computation of homological Seifert surfaces of any $1$-boundary $\gamma$ of $\mathcal{T}$; namely, $2$-chains of $\mathcal{T}$ whose boundary is $\gamma$. It is based on the existence of special spanning trees of the complete dual graph of $\mathcal{T}$, and on the computation of certain linking numbers associated with those spanning trees. If the triangulation $\mathcal{T}$ is fine, the explicit formula is too expensive to be used directly. For this reason, making also use of a simple elimination procedure, we devise a fast algorithm for the computation of homological Seifert surfaces. Some numerical experiments illustrate the efficiency of this algorithm., Comment: 30 pages, 14 figures, 2 tables

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions $H\simeq R_{0,2}$, or to the real Clifford algebra $R_{0,3}$. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of $R_{0,3}$, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases $R_{0,0}\simeq R$, $R_{0,1}\simeq C$ and the trivial case $R_{1,0}\simeq R\oplus R$, the interpolation problem on Clifford algebras $R_{p,q}$ with $(p,q)\neq(0,2),(0,3)$ seems to have some intrinsic difficulties., Comment: Two examples added. Accepted by the Journal of Algebra and Its Applications

The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy integral formula. It follows that such a semigroup is analytic in time., Comment: 29 pages; to appear in Transactions of the American Mathematical Society; formula (1.1) has been corrected

General, especially spectral, features of compact normal operators in quaternionic Hilbert spaces are studied and some results are established which generalize well-known properties of compact normal operators in complex Hilbert spaces. More precisely, it is proved that the norm of such an operator always coincides with the maximum of the set of absolute values of the eigenvalues (exploiting the notion of spherical eigenvalue). Moreover the structure of the spectral decomposition of a generic compact normal operator $T$ is discussed also proving a spectral characterization theorem for compact normal operators., Comment: 11 pages, no figures. arXiv admin note: text overlap with arXiv:1207.0666

We study two types of series over a real alternative $^*$-algebra $A$. The first type are series of the form $\sum_{n} (x-y)^{\punto n}a_n$, where $a_n$ and $y$ belong to $A$ and $(x-y)^{\punto n}$ denotes the $n$--th power of $x-y$ w.r.t.\ the usual product obtained by requiring commutativity of the indeterminate $x$ with the elements of $A$. In the real and in the complex cases, the sums of power series define, respectively, the real analytic and the holomorphic functions. In the quaternionic case, a series of this type produces, in the interior of its set of convergence, a function belonging to the recently introduced class of slice regular functions. We show that also in the general setting of an alternative algebra $A$, the sum of a power series is a slice regular function. We consider also a second type of series, the spherical series, where the powers are replaced by a different sequence of slice regular polynomials. It is known that on the quaternions, the set of convergence of these series is an open set, a property not always valid in the case of power series. We characterize the sets of convergence of this type of series for an arbitrary alternative $^*$-algebra $A$. In particular, we prove that these sets are always open in the quadratic cone of $A$. Moreover, we show that every slice regular function has a spherical series expansion at every point., Comment: To appear in Indiana University Mathematics Journal

The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic $C^*$--algebras and to define, on each of these $C^*$--algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included., Comment: 71 pages, some references added. Accepted for publication in Reviews in Mathematical Physics

We introduce a family of Cauchy integral formulas for slice and slice regular functions on a real associative *-algebra. For each suitable choice of a real vector subspace of the algebra, a different formula is given, in which the domains of integration are subsets of the subspace. In particular, in the quaternionic case, we get a volume Cauchy formula. In the Clifford algebra case, the choice of the paravector subspace R^(n+1) gives a volume Cauchy formula for slice monogenic functions.

In this paper we develop a theory of slice regular functions on a real alternative algebra $A$. Our approach is based on a well--known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in $A$ and we prove a Cauchy integral formula for slice functions of class $C^1$., Comment: 32 pages - accepted for publication in Advances in Mathematics

The goal of this paper is to describe and clarify as much as possible the 3-dimensional topology underlying the Helmholtz cuts method, which occurs in a wide theoretic and applied literature about Electromagnetism, Fluid dynamics and Elasticity on domains of the ordinary space. We consider two classes of bounded domains that satisfy mild boundary conditions and that become "simple" after a finite number of disjoint cuts along properly embedded surfaces. For the first class (Helmholtz), "simple" means that every curl-free smooth vector field admits a potential. For the second (weakly-Helmholtz), we only require that a potential exists for the restriction of every curl-free smooth vector field defined on the whole initial domain. By means of classical and rather elementary facts of 3-dimensional geometric and algebraic topology, we give an exhaustive description of Helmholtz domains, realizing that their topology is forced to be quite elementary (in particular, Helmholtz domains with connected boundary are just possibly knotted handlebodies, and the complement of any non-trivial link is not Helmholtz). The discussion about weakly-Helmholtz domains is a bit more advanced, and their classification appears to be a quite difficult issue. Nevertheless, we provide several interesting characterizations of them and, in particular, we point out that the class of links with weakly-Helmholtz complements eventually coincides with the class of the so-called homology boundary links, that have been widely studied in Knot Theory., Comment: 39 pages, 14 figures

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials., Comment: Proof of Lemma 7 rewritten (thanks to an anonymous reviewer)