Continuous Linear Map Research Articles

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.

Linearly topologized modules over a discrete valuation ring

In this paper certain aspects of the theory of linearly topologized modules over an arbitrary discrete valuation ring are studied. Among the main topics discussed in this work we may mention: the class of linearly topologized spaces, linearly topologized topological modules of continuous linear mappings, equicontinuity of sets of linear mappings, dual systems of modules and orthogonality, weakly continuous linear mappings.

A pair of pseudo-differential operators involving Hankel-type integral transformations

The main objective of this paper is to investigate a pair of pseudo-differential operators (p.d.o.) involving the Hankel-type integral transformations. The p.d.o. associated with the symbols belonging to the class \(H^{m}\) (or \(H_{0}^{m}\)) are the continuous linear mapping of the Zemanian type spaces into itself. Integral representation of p.d.o. involving first Hankel-type integral transformation is obtained.

Bounded elements of C*-inductive locally convex spaces

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: The first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to the natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.

On Centralizers of Banach Algebras

Let \(\mathcal {A}\) be a unital Banach algebra and \(\mathcal {M}\) be a unital Banach \(\mathcal {A}\)-bimodule. The main results characterize a continuous linear map \(\varphi :\mathcal {A}\rightarrow \mathcal {M}\) that satisfies \(a\varphi (a^{-1})=\varphi (1)\) or \(a\varphi (a^{-1})+ \varphi (a^{-1}) a=2\varphi (1)\) for all \(a\) in principal component of invertible elements of \(\mathcal {A}\). The proof is based on the consideration of a continuous bilinear map satisfying a related condition.

Zero products preserving maps from the Fourier algebra of amenable groups

Let $G$ be a locally compact amenable group. The goal of this paper is to investigate the problem of surjective zero products preserving maps from the Fourier algebra of $G$ into a completely contractive Banach algebra. We show that if $B$ is a completely contractive Banach algebra which is faithful and factors weakly, then every surjective completely bounded linear map from $A(G)$ into $B$ which preserves zero products is a weighted homomorphism. Moreover an equivalent condition is given for such a map to be a homomorphism. In particular, this result implies that if $B$ is a commutative C$^*$-algebra or a matrix space and $T:A(G)\rightarrow B$ is a continuous surjective linear map which preserves zero products, then $T$ is a weighted homomorphism and there is an equivalent condition for $T$ to be a homomorphism.

Linear maps preserving G-unitary operators in Hilbert space

Let H be a complex Hilbert space and B(H) the algebra of all bounded linear operators on H. We give the concrete forms of surjective continuous unital linear maps from B(H) onto itself that preserve G-unitary operators.

Orthogonally additive mappings on Hilbert modules

In this paper, we study the representation of orthogonally additive mappings acting on Hilbert C*-modules and Hilbert H*-modules. One of our main results shows that every continuous orthogonally additive mapping f from a Hilbert module W over K(H) or HS(H) to a complex normed space is of the form f(x)=T(x)+S( ) for all x in W, where T is a continuous additive mapping, and S is a continuous linear mapping.

Diagonally Canonical and Related Gaussian Random Elements

We call a Gaussian random element $\eta$ in a Banach space $X$ with a Schauder basis ${\bf e}=(e_n)$ diagonally canonical (for short, $D$-canonical) with respect to ${\bf e}$ if the distribution of $\eta$ coincides with the distribution of a random element having the form $B\xi$, where $\xi$ is a Gaussian random element in $X$, whose ${\bf e}$-components are stochastically independent and $B: X\to X$ is a continuous linear mapping. In this paper we show that if $X=l_p$, $1\le p<\infty$ and $p\neq2,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X,$ which is not $D$-canonical with respect to the natural basis of $X$. We derive this result in the case when $X=l_p$, $2< p<\infty$, or $X=c_0$ from the following statement, an analogue which was known earlier only for Banach spaces without an unconditional Schauder basis: if $X=l_p,$ $2<p<\infty,$ or $X=c_0,$ then there exists a Gaussian random element $\eta$ in $X$ such that the distribution of $\eta$ does not coincide with the distribution of the sum of almost surely convergent in $X$ series $\sum_{n=1}^\infty{}x_ng_n$, where $(x_n)$ is an unconditionally summable sequence of elements of $X$ and $(g_n)$ is a sequence of stochastically independent standard Gaussian random variables.

Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model

We study the bifurcation structure of the parameter space of a 1D continuous piecewise linear bimodal map which describes dynamics of a business cycle model introduced by Day-Shafer. In particular, we obtain the analytical expression of the boundaries of several periodicity regions associated with attracting cycles of the map (principal cycles and related fin structure). By crossing these boundaries the map displays robust chaos.

(m,n)-Jordan derivations

A subspace lattice L on H is called commutative subspace lattice if all projections in L commute pairwise. It is denoted by CSL. If L is a CSL, then algL is called a CSL algebra. Under the assumption m + n ? 0 where m,n are fixed integers, if ? is a mapping from L into itself satisfying the condition (m + n)?(A2) = 2m?(A)A + 2nA?(A) for all A?A, we call ? an (m,n) Jordan derivation. We show that if ? is a norm continuous linear (m,n) mapping from A into it self then ? is a (m,n)-Jordan derivation.

On duality in linearly topologized spaces

For a weakly continuous linear mapping u, a necessary and sufficient condition for the solubility of the equation u(x) = y0 is obtained. The equivalence between the category of dual systems and the category of linearly topologized spaces whose topologies are weak is also established. Mathematics Subject Classification: 46S10, 46A20, 46M15

The Tensor Product Representation of Polynomials of Weak Type in a DF-Space

LetEandFbe locally convex spaces overCand letP(nE;F)be the space of all continuousn-homogeneous polynomials fromEtoF. We denote by⨂n,s,πEthen-fold symmetric tensor product space ofEendowed with the projective topology. Then, it is well known that each polynomialp∈P(nE;F)is represented as an element in the spaceL(⨂n,s,πE;F)of all continuous linear mappings from⨂n,s,πEtoF. A polynomialp∈P(nE;F)is said to beof weak typeif, for every bounded setBofE,p|Bis weakly continuous onB. We denote byPw(nE;F)the space of alln-homogeneous polynomials of weak type fromEtoF. In this paper, in case thatEis a DF space, we will give the tensor product representation of the spacePw(nE;F).

Local asymptotic minimax estimation of nonregular parameters with translation-scale equivariant maps

When a parameter of interest is defined to be a nondifferentiable transform of a regular parameter, the parameter does not have an influence function, rendering the existing theory of semiparametric efficient estimation inapplicable. However, when the nondifferentiable transform is a known composite map of a continuous piecewise linear map with a single kink point and a translation-scale equivariant map, this paper demonstrates that it is possible to define a notion of asymptotic optimality of an estimator as an extension of the classical local asymptotic minimax estimation. This paper establishes a local asymptotic risk bound and proposes a general method to construct a local asymptotic minimax decision.

Hankel–Clifford transformations on some ultradifferentiable function spaces and pseudo-differential operators

Continuity of the first Hankel–Clifford transformation on the spaces of the type \(H_\mu \) are investigated. Pseudo-differential operator \(h_{1,\mu ,a}\) associated with Bessel type operator \(xD_{x}^{2}+(1-\mu )D_{x}\) involving the symbol \(a(x,y)\) whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain ultradifferentiable function spaces. It is shown that the operator \(h_{1,\mu ,a}\) is a continuous linear mapping of one ultradifferentiable function spaces into another spaces of same type.

Analytical expressions for power spectral density issued from one-dimensional continuous piecewise linear maps with three slopes

During the two last decades, chaotic signals have been increasingly considered in telecommunications, signal processing or secure transmissions. Despite the importance of frequency in the telecommunications and transmission security there are few works that explore the spectrum of chaotic signals by considering the bandwidths. Previous works have concerned skew tent maps. In this paper, we derive analytical expressions for the autocorrelation sequence (ACS) and power spectral density (PSD) of chaotic signals generated by one-dimensional continuous piecewise linear maps with three slopes. We obtain similar results to those obtained using the skew-tent map. Our results permit to conjecture that the bandwidth of chaotic signals obtained from piecewise linear continuous map can have the same behaviour whatever be the number of slopes in the map.

Dagger Categories of Tame Relations

Within the context of an involutive monoidal category the notion of a comparison relation \({\mathsf{cp} : \overline{X} \otimes X \rightarrow \Omega}\) is identified. Instances are equality = on sets, inequality \({\leq}\) on posets, orthogonality \({\perp}\) on orthomodular lattices, non-empty intersection on powersets, and inner product \({\langle {-}|{-} \rangle}\) on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame” relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation.

UNFOLDING NONSMOOTH BIFURCATION PATTERNS IN A 1-D PWL MAP AS A MODEL OF A SINGLE-INDUCTOR TWO-OUTPUT DC–DC SWITCHING CONVERTER

A piecewise linear (PWL) continuous map is used to analyze the nonsmooth bifurcation phenomena in a single-inductor two-output (SITO) DC–DC converter under a PWM interleaved control scheme. This map models the dynamical behavior of the converter when the waveforms of the inductor current can be approximated by straight lines during each switching subinterval. The parameter space is constrained by the interleaving control and the physical restriction on the values of some parameters. The main focus of this paper is on the existence and stability conditions of the rich variety of k-periodic orbits and the different bifurcation patterns that can be exhibited in this system. The analytical results in the form of 1-D and 2-D bifurcation diagrams are compared with numerical simulations obtained from the circuit-based switched model getting a good agreement between the two approaches.

Chain rule for conic derivatives

For all “nice” definitions of differentiability, the Chain Rule should be valid. We show that the Chain Rule remains true for some wide class of definitions of differentiability if one considers as approximative mappings (derivatives) not just continuous linear, but positively homogeneous mappings satisfying certain topological conditions (which are fulfilled for continuous linear mappings). For brevity, we call such derivatives conic. We will give corollaries for the case of conic differentiation of mappings between normed spaces, especially for the case of Frechet conic differentiation and compact conic differentiation.

The Fractional S-Transform on Spaces of Type

Certain properties of the S-transform are given and composition of S-transform is defined. We study the fractional S-transform on spaces of type . It is shown that the fractional S-transform is a continuous linear map of the space of type defined on into the space of type defined on .