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A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $\mu\vdash d$ and a $\mu$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{\mu,\mathbb{R}}$ of $\mu$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_\mu$ the complexification of $X_{\mu,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ of the dual variety of $X_\mu$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ is the square of the $\mu$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $\mu$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ for tensors of format $2\times 2\times 2$., Comment: 21 pages, 5 figures

The polarization constant of a Banach space $X$ is defined as $$\mathbf c(X):= \limsup\limits_{k\rightarrow \infty} \mathbf c(k, X)^\frac{1}{k},$$ where $\mathbf c(k, X)$ stands for the best constant $C>0$ such that $ \Vert \overset{\vee}{P} \Vert \leq C \Vert P \Vert$ for every $k$-homogeneous polynomial $P \in \mathcal P(^kX)$. We show that if $X$ is a finite dimensional complex space then $\mathbf c(X)=1$. We derive some consequences of this fact regarding the convergence of analytic functions on such spaces.The result is no longer true in the real setting. Here we relate this constant with the so-called Bochnak's complexification procedure. We also study some other properties connected with polarization. Namely, we provide necessary conditions related with the geometry of $X$ for $\mathbf c(2,X)=1$ to hold. Additionally we link polarization's constants with certain estimates of the nuclear norm of the product of polynomials., Math. Proc. Cambridge Philos. Soc., accepted

Complex systems such as ecosystems, electronic circuits, lasers or chemical reactions can be modelled by dynamical systems which typically experience bifurcations. Transients typically suffer extremely long delays at the vicinity of bifurcations and it is also known that these transients follow scaling laws as the bifurcation parameter gets closer the bifurcation value in deterministic systems. The mechanisms involved in local bifurca- tions are well-known. However, for saddle-node bifurcations, the relevant dynamics after the bifurcation occur in the complex phase space. Hence, the mechanism responsible for the delays and the associated inverse-square root scaling law for this bifurcation can be better understood by looking at the dynamics in the complex space. We follow this approach and complexify a simple ecological system undergoing a saddle-node bifurcation. The discrete model describes a biological system with facilitation (cooperation) under habitat destruction for species with non-overlapping generations. We study the complex (as opposed to real) dynamics once the bifurcation has occurred. We identify the fundamental mechanism causing these long delays (called ghosts), given by two repellers in the complex space. Such repellers appear to be extremely close to the real line, thus forming a nar- row channel close to the two new fixed points and responsible for the slow passage of the orbits, which remains tangible in the real numbers phase space. We analytically provide the relation between the inverse square-root scaling law and the multipliers of these repellers. We finally prove that the same phenomenon occurs for more general i.e., non-necessarily polynomial, models.

We propose topological Hochschild homology as a tool for measuring ramification of maps of structured ring spectra. We determine second order topological Hochschild homology of the $p$-local integers. For the tamely ramified extension of the map from the connective Adams summand to $p$-local complex topological K-theory we determine the relative topological Hochschild homology and show that it detects the tame ramification of this extension. We also determine relative topological Hochschild homology for the complexification map from connective real to complex topological K-theory and for some quotient maps with commutative quotients., Comment: Unfortunately, the published version of this paper contains a mistake. We are grateful to Eva H\"oning who noticed the error. An erratum is about to be published. In this version we correct the mistake and state Lemma 5.1 and Theorem 5.2 in their corrected form

We report the discovery of wide nested sequences of period-adding complexification routes spawning the whole control parameter space of the Henon map, a discrete-time proxy of single-mode loss-modulated CO 2 lasers. For a realistic map, the new adding routes found reproduce analogous phenomena previously observed in continuous-time differential equations describing, for instance, oscillatory phases in an enzyme reaction model. In contrast to differential equations, the map is analytically tractable to a large extent, and provides a prototypic framework to investigate intricate features of generic dissipative flows.

Let$M$be an$n$-dimensional closed oriented smooth manifold with$n\equiv 4\;\text{mod}\;8$, and$\unicode[STIX]{x1D702}$be a complex vector bundle over$M$. We determine the final obstruction for$\unicode[STIX]{x1D702}$to admit a stable real form in terms of the characteristic classes of$M$and$\unicode[STIX]{x1D702}$. As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.

To every realizable oriented matroid there corresponds an arrangement of real hyperplanes. The homeomorphism type of the complexified complement of such an arrangement is completely determined by the oriented matroid. In this paper we study arrangements of pseudohyperplanes; they correspond to non-realizable oriented matroids. These arrangements arise as a consequence of the Folkman-Lawrence topological representation theorem. We propose a generalization of the complexification process in this context. In particular we construct a space naturally associated with these pseudo-arrangements which is homeomorphic to the complexified complement in the realizable case. Further we generalize the classical theorem of Salvetti and show that this space has the homotopy type of a cell complex defined in terms of the oriented matroid., Comment: 19 pages, 3 figures. Third version: some more typos fixed, minor changes, final version. To appear in Proceedings Mathematical Sciences. In the second version: typos fixed, exposition improved, contact information updated. arXiv admin note: text overlap with arXiv:1110.1520

We show that a globally subanalytic function that is real analytic can be extended in a globally subanalytic way to a holomorphic function. We also establish a parametric version of this result. For other important ominimal structures, we show that unary definable real analytic functions can be extended definably to a holomorphic function.

We introduce tensor products in the category of lattice seminormed spaces. We show that the reasonable cross vector seminorms on the complexification of a lattice seminormed space are the same as the admissible vector seminorms. We then specialize these results to complexifications of Archimedean Riesz spaces.

We will give a characterization of stable real C*-algebras analogous to the one given for complex C*-algebras by Hjelmborg and Rørdam. Using this result, we will prove that any real C*-algebra satisfying the corona factorization property is stable if and only if its complexification is stable. Real C*-algebras satisfying the corona factorization property include AF-algebras and purely infinite C*-algebras. We will also provide an example of a simple unstable C*-algebra, the complexification of which is stable.

We first study the generalized Fourier-Gauss transforms of functionals defined on the complexification of an abstract Wiener space (, , ). Secondly, we introduce a new class of convolution products of functionals defined on and study several properties of the convolutions. Then we study various relations among the first variation the convolutions, and the generalized Fourier-Gauss transforms.

We define a notion of Picard-Vessiot extension for a homogeneous linear differential equation L = 0 defined over a real differential field K with a real closed field of constants C K . When L has differential Galois group GL n over the complexification of K, we prove that a Picard-Vessiot extension for L exists over K.

Multidimensional fractional actionlike variational problem with time-dependent dynamical fractional exponents is constructed. Fractional Euler–Lagrange equations are derived and discussed in some details. The results obtained are used to explore some novel aspects of fractional quantum field theory where many interesting consequences are revealed, in particular the complexification of quantum field theory, in particular Dirac operators and the novel notion of “mass without mass”.

We also show that, if the standard Feynman‐Kac formula is assumed to be known, then the classical Feynman‐Kac‐Ito theorem turns out to be an immediate corollary of the generalized Girsanov‐Maruyama formula. This result is an alternative interpretation of the remark on p. 172 of Simon’s book [4], which is incorrect in its original form. An important role in obtaining the results which we discuss is played by the notion of the logarithmic derivative of a function of a real argument taking values in the space of real (alternating) measures, which was introduced in [2]. We apply this notion to define the logarithmic derivative of such a measure along a vector field taking values in the complexification of the space of differentiability of this measure. In fact, we define the logarithmic derivative of a function of a real argument whose values are analytic continuations (to a suitable domain in the complex plane) of some functions of another real argument, taking values in the space of real measures. Such analytic extensions take values in the complexification of the space of real measures, i.e., in the space of complex-valued measures. In turn, the logarithmic derivative thus defined is an analytic continuation with respect to the same parameter of the logarithmic derivative of the just mentioned function taking values in the space of real measures. The logarithmic derivative is used to determine the corresponding trans

Given a hyperplane arrangement A \mathcal {A} in a real vector space V V , we introduce a real algebraic prevariety Z ( A ) \mathcal {Z}(\mathcal {A}) , and exhibit the complement of A \mathcal {A} in the complexification of V V as the total space of an affine bundle over Z ( A ) \mathcal {Z}(\mathcal {A}) with fibers modeled on the dual vector space V ∨ V^{\vee } .

It is well-known that the Schrodinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal E ( ∞ ) space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.

We embed the space of totally real r-cycles of a totally real projective variety into the space of complex r-cycles by complexification. We provide a proof of the holomorphic taffy argument in the proof of Lawson suspension theorem by using Chow forms and this proof gives us an analogous result for totally real cycle spaces. We use Sturm theorem to derive a criterion for a real polynomial of degree d to have d distinct real roots and use it to prove the openness of some subsets of real divisors. This enables us to prove that the suspension map induces a weak homotopy equivalence between two enlarged spaces of totally real cycle spaces.

We study the connections between the norms of bounded operators on real and complex normed spaces, and with range in quaternionic spaces, and of their quaternionic extensions. We study some aspects of the relations between the quaternionization and the complexification of an inner product real space. The problems that arise when k-linear mappings are defined for the quaternionic case, are considered too. Copyright © 2006 John Wiley & Sons, Ltd.

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.

Though the Kerr–Newman metric which represents the field of a charged spinning black hole was deduced by Newman decades ago by introducing complex coordinates, the physical motivation has remained a mystery, as also the fact that this metric infact represents also the field of the electron including the purely Quantum Mechanical gyromagnetic ratio g=2. These two circumstances were characterized by Newman as an accident. In this note, we argue on the other hand that Newman's complexification is infact symptomatic of the fuzzy nature of the space time at the micro scale which can be described by a non-commutative geometry, which would also explain why the Kerr–Newman metric describes the field of the Quantum Mechanical electron. From these considerations, we argue that a time varying gravitational constant is implied, which apart from other things, explains the otherwise inexplicable anomalous accelerations of the Pioneer spacecrafts.

We consider real AW*-algebras, that is, Kaplansky algebras over the field of real numbers. As in the case of complex von Neumann algebras and complex AW*-algebras, real AW*-algebras are classified in terms of types , , , , and . We prove that if the complexification of a real AW*-algebra A also is an AW*-algebra, then the types of A and M coincide.

We define united K-theory for real C*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors -- real K-theory, complex K-theory, and self-conjugate K-theory -- and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Kunneth-type, non-splitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C*-algebras A and B which holds as long as the complexification of A is in the bootstrap category. Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product O_{k+1} otimes O_{l+1} is not determined solely by the greatest common divisor of k and l. Hence we have examples of non-isomorphic, simple, purely infinite, real C*-algebras whose complexifications are isomorphic., 67 pages, 13 tables, uses xy-pic for commutative diagrams, to appear in "K-theory"

A Dirac picture perturbation theory is developed for the time evolution operator in classical dynamics in the spirit of the Schwinger–Feynman–Dyson perturbation expansion and detailed rules are derived for computations. Complexification formalisms are given for the time evolution operator suitable for phase space analyses, and then extended to a two-dimensional setting for a study of the geometrical Berry phase as an example. Finally a direct integration of Hamilton's equations is shown to lead naturally to a path integral expression, as a resolution of the identity, as applied to arbitrary functions of generalized coordinates and momenta.

Which differential equations can be integrated using functions that appear in the Differential Calculus? This is an ancient problem that has been considered by I. Newton, H. Poincare, H. Dulac and P. Painleve ([7],[8]). We address this question for differential equations of the form (1) $ {dy_1\over dx_1} = {P_1(x_1,y_1)\over Q_1(x_1,y_1)} $ where $ P_1,Q_1\in {\Bbb R}[x_1,y_1] $ are polynomials in two real variables. The classification of such an integrable equation is strongly related to the study of its complexification, mainly its complex singularities. For instance, we prove that if the complexification has only generic singularities then a polynomial differential equation (1) as above, which is integrable in the sense of Liouville (see §2 for the definition), must come from a linear differential equation after introduction of complex coefficients. Relaxing slightly the hypothesis on the singularities we obtain also Bernoulli differential equations (see §3 for these main results). These conclusions enforce original ideas of P. Painleve (see Remark 3 in §3). Based on this classification, an interesting application to certain polynomial vector fields whose orbits have an algebraic curve as limit set in $ {\Bbb R}^2 $ is given in §4. It is proved that under appropriate hypothesis these equations also come from linear or Bernoulli equations.

LetM be a cgoherentD-module (e.g., an overdetermined system of partial dierential equations) on the complexification of a real analytic manifold M. Assume that the characteristic variety ofM is hyperbolic with respect to a submanifold N M. Then, it is well-known that the Cauchy problem forM with data on N is well posed in the space of hyperfunctions. In this paper, under the additional assumption thatM has regular singularities along a regular involutive submanifold of real type, we prove that the Cauchy problem is well posed in the space of distributions. When M is induced by a single dierential operator (or by a normal square system) with characteristics of constant multiplicities, our hypotheses correspond to Levi conditions, and we recover a classical result.

New type Paley-Wiener theorems for the modified multidimensional Mellin and inverse Mellin transforms are established. The supports of functions are described in terms of their modified Mellin (or inverse Mellin) transform without passing to the complexification.

International audience; Given a polynomial with real coefficients, we produce a motivic analog of a simple identity that relates the complex conjugation and the monodromy of the Milnor fibre of its complexification. To that purpose, we introduce motivic Zeta functions that take into account complex conjugation and monodromy.

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485–496, 2003, Illinois J Math 47(4):1047–1062, 2003), of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, \(C^*\)-envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided \(M\)-ideal theory and characterize one-sided \(M\)-ideals in real \(C^*\)-algebras and real operator algebras with contractive approximate identity.

Relations between the norms of an operator and its complexification as a mapping from Lp to Lq has been recognized as a serious problem in analysis after the publication of Marcel Riesz's work on convexity and bilinear forms in 1926. We summarize here what it is known about these relations in the case of normed Legesgue spaces and investigate the quasinormed case, i. e. we consider all 0 < p,q ≤ 8 . In particular, in the lower triangle, that is, for 0< p≤q≤∞ these norms are the same. In the upper triangle and the normed case, that is, when 1 ≤ q < p ≤ ∞ the norm of the complexification of a real operator is obviously not bigger than 2 times its real norm. In 1977 Krivine proved that the constant 2 can be replaced by √2 . On the other hand, it was suspected that in the case of quasi-normed Lebesgue spaces (0 < q < p ≤ ∞ ) the corresponding constant could be arbitrarily large, but as we will see this is not the case. More precisely, we prove that this constant for quasi-normed Lebesgue spaces is between 1 and 2. Some additional properties and estimates of this constant with some results about the relation between complex and real norms of operators, including those between two-dimensional Orlicz spaces are presented in the first four chapters. Finally, in Chapter 5, we use the results on the estimates of the norms in the proof of the real Riesz-Thorin interpolation theorem valid in the first quadrant Validerad; 2011; 20110317 (andbra)

We investigate the homogeneity of certain kind of slices of the complete complexification of a proper complex equifocal submanifold in a symmetric space of non-compact type., Comment: 30pages

The purpose of this paper is to construct the "sheaf" of 2-hyperfunctions at the boundary along an involutive submanifold and to generalize the notion of microhyperbolicity at the boundary. Let M be a real analytic manifold, X a complexification of M, and let Q be an open subset of M with Cw-boundary N. Let Fbe a conic involutive submanifold of T^X which intersects transversally to N x T%f X with regular involutive intersection. Then we define the complex of ^j-Modules $2$x °f 2-hyperfunctions at the boundary along V, which appears to be a useful tool in studying noncharacteristic boundary value problems. Remark that the complex ^Q\X was first introduced by P. Schapira [S 3] for the microlocal study of boundary value problems. Next we introduce the notion of O-F-hyperboli city of a system M of microdifferential equations and prove that it implies "propagation of zeros up to the boundary" of cohomology groups of the complex Rjf#mSx(Jt, ^2Q\x}- This implies in particular "D-regularity" of M in the sense of [S 3].

Let $E$ be a vector space in which some notion of boundedness is defined. Then $T:E\to E$ is said to have the boundedness stability property (BSP) if for each $x\in E$, the sequence $(T^n_x)^\infty_{n=1}$ is bounded whenever a subsequence $(T^{n_i}x)^\infty_{i=1}$ is bounded. It is shown that (1) every affine operator on a finite-dimensional Banach space has the (BSP); (2) every affine operator on an infinite-dimensional vector space has the functional (BSP); (3) when $E$ is an infinite-dimensional Banach space, an affine operator $T$ on $E$ has the (BSP) if its linear part $A_T=T-T(0)$ is a compact perturbation of a bounded linear operator with spectral radius less than one and (4) when $E$ is a Hilbert space, every normal or subnormal bounded linear operator has the (BSP). Some results on affine operators on a Hilbert space whose linear parts are normal or subnormal are also obtained. Finally, some problems are posed.

It is proved that a point $f$ of the complexification $E^C$ of a real Köthe space $E$ is a complex extreme point if and only if $|f|$ is a point of upper monotonicity in $E$. As a corollary it follows that $E$ is strictly monotone if and only if $E^C$ is complex rotund. It is also shown that $E$ is uniformly monotone if and only if $E^C$ is uniformly complex rotund. Next, the fact that $|x|\in S(E^+)$ is a ULUM-point of $E$ whenever $x$ is a $C$-LUR-point of $S(E^C)$ is proved, whence the relation that $E$ is a ULUM-space whenever $E^C$ is $C$-LUR is concluded. In the second part of this paper these general results are applied to characterize complex rotundity of properties Calderón-Lozanovskiĭ spaces, generalized Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces.

We will deal with two “hidden” real structures in the theory of models of subspace arrangements. Given a real subspace arrangement A and its complexification A3/8, the first structure is a real De Concini-Procesi model Y~A that can be seen as the manifold ReYA3/8 of (canonical) real points inside the complex De Concini-Procesi model YA3/8. We will study its combinatorial properties by describing it as a quotient of a real model with corners CYA introduced in 2003. A second structure arises, on the contrary, as an “extension” of CYA, when A is a Coxeter arrangement. We will “add faces” to CYA and obtain a convex body (or even a polytope); this gives rise to an interesting new family of “realized” posets which includes for instance Kapranov's permutoassociahedra.

Let V be a real vector space. An arrangement of hyperplanes in V is a finite set A of hyperplanes through the origin. A chamber of A is a connected component of V − (∪ H ∈ A H). The arrangement A is called simplicial if ∩ H ∈ A H = {0} and every chamber of A is a simplicial cone. For an arrangement A of hyperplanes in V, we set ... (formule)... where V C = C ⊗ V is the complexification of V, and, for H ∈ A, H C is the complex hyperplane of V C spanned by H. Let A be an arrangement of hyperplanes of V. Salvetti constructed a simplicial complex Sal(A) and proved that Sal(A) has the same homotopy type as M(A). In this paper we give a new short proof of this fact

CONTENTS Introduction Chapter I. Real theory of Fourier integral operators ??1. Densities, pseudodifferential operators, and asymptotic expansions ??2. Homogeneous Lagrangian immersions ??3. The canonical operator ??4. Fourier integral operators ??5. Examples and applications Chapter II. Complex theory of Fourier integral operators ??1. Introductory remarks ??2. Analysis on -analytic homogeneous manifolds ??3. Complexification of the phase space ??4. Definition of Fourier integral operators in the complex case Conclusion References

In this paper the cartesian product structure of complex analytic singularities is studied. A singularity is called indecomposable if it cannot be written as the cartesian product of two singularities of lower dimension. It is shown that there is an essentially unique way to write any reduced irreducible singularity as a cartesian product of indecomposable singularities. This result is applied to give an explicit description of the set of reduced irreducible complex singularities having a given underlying real analytic structure.

We study Yang-Mills connections on holomorphic bundles over complex K\"ahler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk\"ahler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk\"ahler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K\"ahler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk\"ahler manifold and the space of rational curves in the twistor space of the ``Mukai dual'' hyperk\"ahler manifold., Comment: 48 pages, LaTeX 2e