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A uniform mathematical description of quantum automata as quantum finite automata, quantum linear bounded automata, and quantum Turing machines is introduced, and the existence of such automata under certain circumstances is investigated. Such quantum automata are modelled by isometric rather than unitary operators, reflecting the fact that computation is primarily irreversible in contrast to physical phenomena in the microworld. The global conditions determine the local conditions of the operators.

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τ_i, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τ_i is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τ_i to operators on the Hilbert space L²(μ). Instead, we show that it is possible to realize the maps τ_i quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫_X|φ|² dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and

We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

We describe all the holomorphic Hermitian vector bundles (E, h) over the upper half plane in ℂ with the property that (f* E, f* h) is holomorphically isometric to (E, h) for any holomorphic automorphism f of the upper half plane. We give an explicit construction of all such holomorphic Hermitian vector bundles using some linear algebraic data.

Entangled quantum states are an important component of quantum computing techniques such as quantum error-correction, dense coding and quantum teleportation. We determine the requirements for a state in the Hilbert space C⁴ to be entangled and a solution to the corresponding "factorization" problem if this is not the case. The factorization of nonentangled states is implemented in C++.

We show how SymbolicC++ can be used to simulate quantum circuits.

We consider finite-dimensional Hilbert spaces with with n ≥ 2 and unitary operators. In particular, we consider the case n = 2^m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU^* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.

This paper presents a low-order and accurate method for the design of FIR fractional delay (FD) filters with complex coefficients. This method employs least square technique in Hilbert space to approximate the ideal FD transfer function with a FIR filter and to calculate its coefficients. The main advantages of the resulting filter are: very good response at all frequencies compared to other FD filter design methods and a good method to create very small delay. Design examples are presented to illustrate the effectiveness of this new design approach.

It is widely understood that quantum computing — quantum gates upon qubits — is the general case, encompassing computing by classical means, viz. Boolean logic upon classical bits. It also seems reasonable that Quantum Software should encompass Classical Software. However, to accept such a statement regarding software, the feeling that it seems reasonable is not enough. One needs clear-cut definitions and formal conclusions. This is exactly the purpose of this paper. Previously, we have represented Classical Software by the Laplacian Matrix. More recently, we have shown that Quantum Software is faithfully represented by Density Matrices. It turns out that a Laplacian Matrix normalized by the Laplacian Trace easily obtains a Density Matrix. This opens the horizons for Quantum Software operations — such as unitary and reversible evolution — not naturally available with the classical Laplacian. This paper provides the necessary definitions and conclusions, illustrating the more general Quantum operations with a relevant case study, playing the double role of both classical and quantum software.

Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.

We construct a bounded function H : l₂ × l₂ → ℝ with continuous Fréchet derivative such that for any q₀ ∈ l₂ the Cauchy problem

This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

In this work stochastic integration with respect to cylindrical Lévy processes with weak second moments is introduced. It is well known that a deterministic Hilbert–Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Lévy process by random Hilbert–Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Lévy process. We finish this work by deriving an Itô isometry and by considering shortly stochastic partial differential equations driven by cylindrical Lévy processes.

A parabolic partial differential equation $u_t^{\prime }(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent (but dependent on $x$) coefficients. We assume that the spatial coordinate $x$ belongs to a finite- or infinite-dimensional real separable Hilbert space $H$. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator $L$. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over $H$ as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on $H$. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in $L$ is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation.

Let $A$ be a compact operator in a separable Hilbert space and ${\lambda }_k(A)(k=1,2,\dots )$ be the eigenvalues of $A$ with their multiplicities enumerated in the non-increasing order of their absolute values. We prove the inequality

In this paper, we study long memory phenomenon of functional time series. We consider an operator fractional Brownian motion with values in a Hilbert space defined via operator-valued Hurst coefficient. We prove that this process is a limiting one for polygonal lines constructed from partial sums of time series having space varying long memory.

The Central Limit Theorem is studied for stationary sequences that are sums of countable collections of linear processes. Two sets of sufficient conditions are obtained. One restricts only the coefficients and is shown to be best possible among such conditions. The other involves an interplay between the coefficients and the distribution functions of the innovations and is shown to be necessary for the Conditional Central Limit Theorem in the case of a causal process with independent innovations.

We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

We study the connection between *-representations of algebras associated with graphs, locally-scalar graph representations and the problem about the spectrum of a sum of two Hermitian operators. For algebras associated with Dynkin graphs we give an explicit description of the parameters for which there are irreducible representations and an algorithm for constructing these representations.

A frame is a system of vectors $S$ in Hilbert space $\mathcal{\mathscr{H}}$ with properties which allow one to write algorithms for the two operations, analysis and synthesis, relative to $S$, for all vectors in $\mathcal{\mathscr{H}}$; expressed in norm-convergent series. Traditionally, frame properties are expressed in terms of an $S$-Gramian, $G_S$ (an infinite matrix with entries equal to the inner product of pairs of vectors in $S$); but still with strong restrictions on the given system of vectors in $S$, in order to guarantee frame-bounds. In this paper, we remove these restrictions on $G_S$, and we obtain instead direct-integral analysis/synthesis formulas. Applications are given to reproducing kernel Hilbert spaces, and to random fields.

We study reduction schemes for functions of “many” variables into system of functions in one variable. Our setting includes infinite dimensions. Following Cybenko–Kolmogorov, the outline for our results is as follows: We present explicit reduction schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in “many” variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.