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We explore functors between operator space categories, some properties of these functors, and establish relations between objects in these categories and their images under these functors, in particular regarding injectivity and injective envelopes. We also compare the purely categorical definition of injectivity with the ‘standard’ operator theoretical definition. An appendix by D. P. Blecher discusses the unitization of an operator space and its injective envelope. [ABSTRACT FROM AUTHOR]

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and C ∗ -envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group. [ABSTRACT FROM AUTHOR]

A recent publication in Nature has generated much heated discussion about evolution, its tendency towards increasing diversity and complexity, and its potential status above and beyond the known laws of fundamental physics. The argument at the heart of this controversy concerns assembly theory, a method to detect and quantify the influence of higher-level emergent causal constraints in computational worlds made of basic objects and their combinations. In this short essay, I briefly review the theory, its basic principles and potential applications. I then go on to critically examine its authors' assertions, concluding that assembly theory has merit but is not nearly as novel or revolutionary as claimed. It certainly does not provide any new explanation of biological evolution or natural selection, or a new grounding of biology in physics. In this regard, the presentation of the paper is starkly distorted by hype, which may explain some of the outrage it created. [ABSTRACT FROM AUTHOR]

This paper focuses on the investigation of the dynamics of novel 2-DOF coupled oscillators. The system consists of a linear oscillator (main structure) and an attached lightweight nonlinear oscillator, called a nonlinear energy sink (NES), under harmonic forcing in the regime of 1:1:1 resonance. The studied NES has geometrically nonlinear stiffness and damping. Due to the degeneracies that the NES brings to the system, diverse bifurcation structures and rich dynamical phenomena such as nonlinear beating and strongly modulated response occur. The latter two phenomena represent different patterns of energy transfer. To capture the bifurcation structure, the slow flow of the system can be acquired with the use of the complex-averaging method. Furthermore, by applying the bifurcation analysis technique, we get curve boundaries of several bifurcation points in the parameter space. These boundaries will induce different types of folding structures, which can lead to complicated patterns of strongly modulated responses, in which intense energy transfer from the main structure to NES occurs. To study the necessary parameter conditions of strongly modulated responses, we analyzed the dynamics of different time scales of the slow flow in detail and determined the corresponding parameter ranges finally. It is worth noting that the small parameter ε may have a qualitative impact on the dynamics of the system. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the problem of Schwarzschild space time in which the notion of density is substituted by that of a "fractal mass measure" defined over sets in the framework of the Fractional Action Like Variational Approach with time-dependent exponent. On a hypersphere of radius R centered at a point P, we assume that the mass enclosed behaves like: M (R) ∝ R β (β is a fractal dimension) if P is fractal and is equal to zero otherwise. Furthermore, we conjecture that the Newtonian's gravitational coupling constant increase with R like G N (R) ∝ R ζ , ζ ∈ R ≪ 1 for short distances which is one of the main consequences of the exhaustive examination of non-perturbatively renormalizable or asymptotically safe quantum gravity which envisages a fractal space time structure at sub-Planckian distances whose effective dimensionality is 2. Many interesting features are revealed and discussed in some details, in particular the emergence of complexified space-time, complexified gravity and Schwarzschild space-time metric deformation. [ABSTRACT FROM AUTHOR]

We classify derived-discrete algebras over the real numbers up to Morita equivalence, using the classification of complex derived-discrete algebras in D. Vossieck, (J. Algebra, 243, 168–176 2001). To this end, we investigate the quiver presentation of the complexified algebra of a real algebra given by a modulated quiver and an admissible ideal. [ABSTRACT FROM AUTHOR]

We study glued tensor and free products of compact matrix quantum groups with cyclic groups – so-called tensor and free complexifications. We characterize them by studying their representation categories and algebraic relations. In addition, we generalize the concepts of global colourization and alternating colourings from easy quantum groups to arbitrary compact matrix quantum groups. Those concepts are closely related to tensor and free complexification procedures. Finally, we also study a more general procedure of gluing and ungluing. [ABSTRACT FROM AUTHOR]

Let X ⊂ R n be a (global) real analytic surface. Then every positive semidefinite meromorphic function on X is a sum of 10 squares of meromorphic functions on X. As a consequence, we provide a real Nullstellensatz for (global) real analytic surfaces. [ABSTRACT FROM AUTHOR]

In this work, we study some algebraic and topological properties of the ring of global analytic functions on the normalization of a reduced complex analytic space. If is a Stein space, we characterize in terms of the (topological) completion of the integral closure of the ring of global holomorphic functions on X (inside its total ring of fractions) with respect to the usual Fréchet topology of. This shows that not only the Stein space but also its normalization is completely determined by the ring of global analytic functions on X. This result was already proved in 1988 by Hayes–Pourcin when is an irreducible Stein space, whereas in this paper we afford the general case. We also analyze the real underlying structures and of a reduced complex analytic space and its normalization. We prove that the complexification of provides the normalization of the complexification of if and only if is a coherent real analytic space. Roughly speaking, coherence of the real underlying structure is equivalent to the equality of the following two combined operations: (1) normalization + real underlying structure + complexification, and (2) real underlying structure + complexification + normalization. [ABSTRACT FROM AUTHOR]

We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in Kusraeva (Vladikavkaz Math J 16(4):49-53, 2014) actually characterize them. Secondly, by employing complexifications of the unique symmetric multilinear maps associated with orthogonally additive polynomials, we derive new characterizing formulas. [ABSTRACT FROM AUTHOR]

Considering tensor products of special commutative algebras and general real Clifford algebras, we arrive at extended Clifford algebras. We have found that there are five types of extended Clifford algebras. The class of extended Clifford algebras is closed with respect to the tensor product. [ABSTRACT FROM AUTHOR]

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485-496, , Illinois J Math 47(4):1047-1062, ), 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. [ABSTRACT FROM AUTHOR]

In classical linear algebra, extending the ring of scalars of a free module gives rise to a new free module containing an isomorphic copy of the former and satisfying a certain universal property. Also, given two free modules on the same ring of scalars and a morphism between them, enlarging the ring of scalars results in obtaining a new morphism having the nice property that it coincides with the initial map on the isomorphic copy of the initial free module in the new one. We investigate these problems in the category of free $${\mathcal{A}}$$-modules, where $${\mathcal{A}}$$ is an $${\mathbb{R}}$$- algebra sheaf. Complexification of free $${\mathcal{A}}$$-modules, which is defined to be the process of obtaining new free $${\mathcal{A}}$$-modules by enlarging the $${\mathbb{R}}$$-algebra sheaf $${\mathcal{A}}$$ to a $${\mathbb{C}}$$- algebra sheaf, denoted $${\mathcal{A}_\mathbb{C}}$$, is an important particular case (see Proposition 2.1, Proposition 3.1). Attention, on the one hand, is drawn on the sub- $${_{\mathbb{R}}\mathcal{A}}$$- sheaf of almost complex structures on the sheaf $${{_\mathbb{R}}\mathcal{A}^{2n}}$$, the underlying $${\mathbb{R}}$$-algebra sheaf of a $${\mathbb{C}}$$- algebra sheaf $${\mathcal{A}}$$, and on the other hand, on the complexification of the functor $${\mathcal{H}om_\mathcal {A}}$$, with $${\mathcal{A}}$$ an $${\mathbb{R}}$$-algebra sheaf. [ABSTRACT FROM AUTHOR]

Parallelity in the real elliptic 3-space was defined by W. K. Clifford in 1873 and by F. Klein in 1890; we compare the two concepts. A Clifford parallelism consists of all regular spreads of the real projective 3-space $${{\rm PG}(3,\mathbb{R})}$$ whose (complex) focal lines (=directrices) form a regulus contained in an imaginary quadric (D1 = Klein's definition). Our new access to the topic 'Clifford parallelism' is free of complexification and involves Klein's correspondence λ of line geometry together with a bijective map γ from all regular spreads of $${{\rm PG}(3,\mathbb{R})}$$ onto those lines of $${{\rm PG}(5,\mathbb{R})}$$ having no common point with the Klein quadric; a regular parallelism P of $${{\rm PG}(3,\mathbb{R})}$$ is Clifford, if the spreads of P are mapped by γ onto a plane of lines (D2 = planarity definition). We prove the equivalence of (D1) and (D2). Associated with γ is a simple dimension concept for regular parallelisms which allows us to say instead of (D2): the 2-dimensional regular parallelisms of $${{\rm PG}(3,\mathbb{R})}$$ are Clifford (D3 = dimensionality definition). Submission of (D2) to λ yields a complexification free definition of a Clifford parallelism which uses only elements of $${{\rm PG}(3,\mathbb{R})}$$: A regular parallelism P is Clifford, if the union of any two distinct spreads of P is contained in a general linear complex of lines (D4 = line geometric definition). In order to see (D1) and (D2) simultaneously at work we discuss the following two examples using, at the one hand, complexification and (D1) and, at the other hand, (D2) under avoidance of complexification. Example 1. In the projectively extended real Euclidean 3-space a rotational regular spread with center o is submitted to the group of all rotations about o; we prove, that a Clifford parallelism is generated. Example 2. We determine the group $${Aut_e({\bf P}_{\bf C})}$$ of all automorphic collineations and dualities of the Clifford parallelism P and show $${Aut_e({\bf P}_{\bf C})\hspace{1.5mm} \cong ({\rm SO}_3\mathbb{R} \times {\rm SO}_3\mathbb{R})\rtimes \mathbb{Z}_2}$$. [ABSTRACT FROM AUTHOR]

In this paper, we study a Targeted Energy Transfer (TET) problem between a p degrees-of-freedom (dof) linear master structure and several coupled parallel slave Nonlinear Energy Sink (NES) systems. In detail, each lth dof l=1,2,..., p contains n parallel NES; so the linear structure has ( n+ n+⋅⋅⋅+ n+⋅⋅⋅+ n) NES. We are interested to study analytically the TET phenomenon during the first mode of the compound system. To this end, complexification, averaging, and multiple scales methods are used. The system is studied under 1:1 resonance for the transient regime and under harmonic excitation. The influence of the system parameters is observed through dimensionless variables. An analytical criterion is defined to tune NES parameters which lead to an efficient TET for the transient and the forced regimes. It will be demonstrated that analytical results are in good agreement with numerical ones. This paper will be followed by a companion paper which mainly deals with the governing equations for compound nonlinear systems with trees of NES devices at each dof; then experimental results of a four storey structure with two parallel NES at the top floor which are tuned by the mentioned technique in the current paper will be demonstrated and commented upon. [ABSTRACT FROM AUTHOR]

We investigate geometric properties of Grassmann manifolds and their complexifications in an infinite-dimensional setting. Specific structures of quaternionic type are constructed on these complexifications by a direct method that does not require any use of the cotangent bundles. [ABSTRACT FROM AUTHOR]

In this paper we discuss the problem of how to recognize a complex lattice homomorphism on the complexification $${L_\mathbb{C}}$$ of a real vector lattice L from its behavior on a small subset of $${L_\mathbb{C}}$$ . [ABSTRACT FROM AUTHOR]

The Doss trick is employed to find solutions of Schrüdinger equations on symmetric spaces of compact type. The potentials and initial conditions are taken from an algebra of functions which admit an holomorphic extension to the complexification of the considered symmetric spaces. [ABSTRACT FROM AUTHOR]

We consider a complex version of a Dirac-Kähler-type equation, the eight-component complex Dirac-Kähler equation with a nonvanishing mass, which can be decomposed into two Dirac equations by only a nonunitary transformation. We also write an analogue of the complex Dirac-Kähler equation in five dimensions. We show that the complex Dirac-Kähler equation is a special case of a Bhabha-type equation and prove that this equation is invariant under the algebra of purely matrix transformations of the Pauli-Gürsey type and under two different representations of the Poincaré group, the fermionic (for a two-fermion system) and bosonic-representations. The complex Dirac-Kähler equation is also written in a manifestly covariant bosonic form as an equation for the system (µ?, F,µ) of irreducible self-dual tensor, scalar, and vector fields. We illustrate the relation between the complex Dirac-Kähler equation and the known 16-component Dirac-Kähler equation. [ABSTRACT FROM AUTHOR]