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The issue of testing invertibility of cellular automata has been often discussed. Constructing invertible automata is very useful for simulating invertible dynamical systems, based on local rules. The computation universality of cellular automata has long been positively resolved, and by showing that any cellular automaton could be simulated by an invertible one having a superior dimension, Toffoli proved that invertible cellular automaton of dimension d≥2 were computation-universal. Morita proved that any invertible Turing Machine could be simulated by a one-dimensional invertible cellular automaton, which proved computation-universality of invertible cellular automata. This article shows how to simulate any Turing Machine by an invertible cellular automaton with no loss of time and gives, as a corollary, an easier proof of this result.

The Klein–Gordon equation, one of the most fundamental equations in field theory, is known to be not invariant under conformal transformation. However, its massless limit exhibits symmetry under Bekenstein’s disformal transformation, subject to some conditions on the disformal part of the metric variation. In this study, we explore the symmetry of the Klein–Gordon equation under the general disformal transformation encompassing that of Bekenstein and a hierarchy of “sub-generalizations” explored in the literature (within the context of inflationary cosmology and scalar–tensor theories). We find that the symmetry in the massless limit can be extended under this generalization provided that the disformal factors take a special form in relation to the conformal factor. Upon settling the effective extension of symmetry, we investigate the invertibility of the general disformal transformation to avoid propagating nonphysical degrees of freedom upon changing the metric. We derive the inverse transformation and the accompanying restrictions that make this inverse possible.

Cellular automata are transformations of configuration spaces over finitely generated groups, such that the next state in a point only depends on the current state of a finite neighborhood of the point itself. Many questions arise about retrieving global properties from such local descriptions, and finding algorithms to perform these tasks.

We consider the case when the group is a semi-direct product of two finitely generated groups, and show that a finite factor (whatever it is) can be thought of as part of the alphabet instead of the group, preserving both the dynamics and some "finiteness" properties. We also show that, under reasonable hypotheses, this reduction is computable: this leads to some reduction theorems related to the invertibility problem.

Fix a group G and let X be an algebraic variety over an algebraically closed field k of characteristic zero. We investigate the invertibility of algebraic cellular automata, namely, G-equivariant uniformly continuous self-maps whose local defining maps are induced by morphisms of algebraic varieties $X^M\to X$ where $M\subset G$ is a finite memory set. When G is locally embeddable into finite groups (LEF), we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata and thus computable in polynomial time. Generalizations are also obtained for finite product Hopfian pointed object alphabets in concrete categories. Moreover, we prove that for algebraic cellular automata, the notions of reversibility and injectivity are equivalent whenever G is surjunctive and the field k is, additionally, uncountable of arbitrary characteristic.

We introduce a new approach to Vassiliev invariants. This approach deals with Vassiliev invariants directly on knots and does not make use of diagrams. We give a series of applications of this approach, (re)proving some new and known facts on Vassiliev invariants.

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

We introduce a polynomial invariant with two variables for an oriented virtual knot, which refines the odd writhe polynomial with one variable due to Cheng by using a modified version of the warping degree. Our invariant is a Vassiliev invariant of degree one, reduces to one variable for a checkerboard colorable virtual knot, vanishes for classical knots, and detects non-invertibility and non-amphicheirality for some cases. We raise some examples to show effectiveness of our invariant. Moreover we define a similar invariant for a flat virtual knot.

We introduce a parity of classical crossings of virtual link diagrams which extends the Gaussian parity of virtual knot diagrams and the odd writhe of virtual links that extends that of virtual knots introduced by Kauffman [A self-linking invariants of virtual knots, Fund. Math.184 (2004) 135–158]. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial introduced in [Index polynomial invariants of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. As consequences, we give some properties of our invariant, and raise some examples.

We introduce some polynomial invariants for flat virtual links which are similar to the Jones–Kauffman polynomial, the Miyazawa polynomial and the arrow polynomial for virtual link diagrams, and we give several properties and examples.

Canonical triangulations provide an algorithm to compute the symmetry groups of hyperbolic links and their complements, along with information on how each symmetry acts on the link components. The latter information determines the chirality and invertibility of the link. Symmetry groups of knots to ten crossings and multicomponent hyperbolic links to nine crossing are listed, with some exceptions. A Macintosh computer program, available free of charge, computes symmetry groups interactively.

Let $\mathcal{𝒞}$ be a category with an involution ∗. Suppose that $\phi :X\to X$ is a morphism and $({\phi }_1,Z,{\phi }_2)$ is an (epic, monic) factorization of $\phi$ through $Z$, then $\phi$ is core invertible if and only if ${({\phi }^{\ast })}^2{\phi }_1$ and ${\phi }_2{\phi }_1$ are both left invertible if and only if $({({\phi }^{\ast })}^2{\phi }_1,Z,{\phi }_2)$, $({\phi }_2^{\ast },Z,{\phi }_1^{\ast }{\phi }^{\ast }\phi )$ and $({\phi }^{\ast }{\phi }_2^{\ast },Z,{\phi }_1^{\ast }\phi )$ are all essentially unique (epic, monic) factorizations of ${({\phi }^{\ast })}^2\phi$ through $Z$. We also give the corresponding result about dual core inverse. In addition, we give some characterizations about the coexistence of core inverse and dual core inverse of an $R$-morphism in the category of $R$-modules of a given ring $R$.

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v₁,…,v_n} and where each vertex v_i has a state x_vi in a finite field K, (b) a sequence of functions (F_vi,Y)_i, and (c) a word w = (w₁,…,w_k), where each w_i is a vertex in Y. The function F_vi,Y updates the state of vertex v_i as a function of the state of v_i and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map . In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ_n), and all the maps F_vi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ₄. Finally, we analyze invertible ACA and offer an interpretation of S₃₅ as the group of an SDS over the three-dimensional cube with local functions induced by nor₃ + nand₃.

The low accuracy near the boundary or the interface in SPH method has been paid extensive attention. The Finite Particle Method (FPM) is a significant improvement to the traditional SPH method, which can greatly improve the computational accuracy for boundary particles. However, there are still some inherent defects for FPM, such as the long computing time and the potential numerical instability. By conducting matrix decomposition on the basic equations of FPM, an improved FPM method (IFPM) is proposed, which can not only maintain the high accuracy of FPM for boundary particles, but also keep the invertibility of the coefficient matrix in FPM. The numerical results show that the IFPM is really an effective improvement to traditional FPM, which could greatly reduce the computing time. Finally, the modified method is applied to two transient problems.

Let S_p be the ideal of the Schatten-von Neumann operators with the norm N_p. For an integer p ≥ 3 and an A ∈ S_p, the inequalities of the type

We study properties of solutions of the operator equation , , where a closable linear operator on a Hilbert space , such that there exists a self-adjoint operator on , with the resolution of identity E(·), which commutes with . We are interested in the question of regular admissibility of the subspace , i.e. when for every there exists a unique (mild) solution u in of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ ℝ is a compact subset such that Λ ⋂ Σ = ∅, then is regularly admissible. If Λ ⊂ ℝ is an arbitrary Borel subset such that Λ ⋂ Σ = ∅, then, in general, needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of . Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12].

Linear preserver problems are questions about characterising linear maps on spaces of matrices or spaces of operators (or more generally on rings or algebras) that preserve certain properties. We present an exposition of three such problems on preserving invertibility or commutativity or rank one.

We will present a criterion for left, right and both-sided invertibility of matrix Wiener-Hopf plus Hankel operators with the same Fourier symbol in the Wiener subclass of the almost periodic algebra. The criterion is based on the value of a certain mean motion constructed from a particular Hausdorff set which is bounded away from zero.

Let be a C*-algebra of functional operators on the space L²(𝕋) generated by multiplication operators by piecewise slowly oscillating functions and by a group of unitary operators associated to an amenable group G of orientation-preserving homeomorphisms of the unit circle 𝕋 onto itself such that all g ∈ G\{e} have piecewise continuous derivatives and the same nonempty set Λ of periodic points, and there is a point t ∈ Λ with a finite G-orbit. An invertibility criterion for the operators is established.

In this chapter, we study the Box-Jenkins approach to the modelling of stochastic process. It is a versatile and a practically useful method for modelling stationary processes as the fundamental building blocks of most stochastic processes. The approach is systematic, involving a recipe of careful steps to arrive at the appropriate process for modelling and forecasting purposes. The steps involve identification of the process, making it stationary or making it as some differencings of an underlying nonstationary process, estimating the identified model, then validating it. We shall also study an important application to the modelling of inflation rates in the economy.