Compression Theorem Research Articles

Quantum data compression under localized features

In this paper, we first introduce the local quantum information processing task by compressing the density operator based on local quantum Bernoulli noises. Then, the local quantum compression theorem is given, that is, the local quantum entropy is the minimum achievable rate of local compression. Finally, we prove the theorem with the proof of the direct coding theorem and the inverse theorem. The direct coding theorem shows that a scheme with such a local compression rate exists, and that this local compression rate converges to the local quantum entropy. The inverse theorem shows that the compression scheme with the rate below the local entropy is unachievable. With the continuous development of quantum technology, local quantum data compression technology will have broad application prospects and development space.

Monotone positive solution of fourth order boundary value problem with mixed integral and multi-point boundary conditions

In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results.

Positive solutions of conformable fractional differential equations with integral boundary conditions

In this paper, we discuss the existence of positive solutions of the conformable fractional differential equation T_{alpha }x(t)+f(t,x(t))=0, tin [0,1], subject to the boundary conditions x(0)=0 and x(1)= lambda int_{0}^{1}x(t),mathrm{d}t, where the order α belongs to (1,2], T_{alpha }x(t) denotes the conformable fractional derivative of a function x(t) of order α, and f:[0,1]times [0,infty)mapsto [0,infty) is continuous. By use of the fixed point theorem in a cone, some criteria for the existence of at least one positive solution are established. The obtained conditions are generally weaker than those derived by using the classical norm-type expansion and compression theorem. A concrete example is given to illustrate the possible application of the obtained results.

Multiple Positive Solutions for a Boundary Value Problem with Nonlinear Nonlocal Riemann-Stieltjes Integral Boundary Conditions

Abstract In this paper, we study the existence of positive solutions to the fractional boundary value problem D 0 + α x ( t ) + q ( t ) f ( t , x ( t ) ) = 0 , 0 < t < 1 , $$\begin{array}{} \displaystyle D^{\alpha }_{0+}x(t)+q(t)f(t,x(t))=0, \,\, 0\lt t \lt1, \end{array}$$ together with the boundary conditions x ( 0 ) = x ′ ( 0 ) = ⋯ = x ( n − 2 ) ( 0 ) = 0 , D 0 + β x ( 1 ) = ∫ 0 1 h ( s , x ( s ) ) d A ( s ) , $$\begin{array}{} \displaystyle x(0)=x^{\prime}(0)= \cdots = x^{(n-2)}(0)=0, D_{0+}^{\beta }x(1)= \int^{1}_{0}h(s,x(s))\,dA(s), \end{array}$$ where n > 2, n – 1 < α ≤ n, β ∈ [1,α – 1], and D 0 + α $\begin{array}{} \displaystyle D^{\alpha }_{0+} \end{array}$ and D 0 + β $\begin{array}{} \displaystyle D^{\beta }_{0+} \end{array}$ are the standard Riemann-Liouville fractional derivatives of order α and β, respectively. We consider two different cases: f, h : [0, 1] × R → R, and f, h : [0, 1] × [0, ∞) → [0, ∞). In the first case, we prove the existence and uniqueness of the solutions of the above problem, and in the second case, we obtain sufficient conditions for the existence of positive solutions of the above problem. We apply a number of different techniques to obtain our results including Schauder’s fixed point theorem, the Leray-Schauder alternative, Krasnosel’skii’s cone expansion and compression theorem, and the Avery-Peterson fixed point theorem. The generality of the Riemann-Stieltjes boundary condition includes many problems studied in the literature. Examples are included to illustrate our findings.

Shannon entropy: a rigorous notion at the crossroads between probability, information theory, dynamical systems and statistical physics

Statistical entropy was introduced by Shannon as a basic concept in information theory measuring the average missing information in a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. In this paper, I describe how statistical entropy and entropy rate relate to other notions of entropy that are relevant to probability theory (entropy of a discrete probability distribution measuring its unevenness), computer sciences (algorithmic complexity), the ergodic theory of dynamical systems (Kolmogorov–Sinai or metric entropy) and statistical physics (Boltzmann entropy). Their mathematical foundations and correlates (the entropy concentration, Sanov, Shannon–McMillan–Breiman, Lempel–Ziv and Pesin theorems) clarify their interpretation and offer a rigorous basis for maximum entropy principles. Although often ignored, these mathematical perspectives give a central position to entropy and relative entropy in statistical laws describing generic collective behaviours, and provide insights into the notions of randomness, typicality and disorder. The relevance of entropy beyond the realm of physics, in particular for living systems and ecosystems, is yet to be demonstrated.

Identifying the Information Gain of a Quantum Measurement

We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs [Winter, CMP 244 (157), 2004] to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for "classically coherent states" as well as the post-selection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579), 2011].

Positivity of matrices with generalized matrix functions

Using an elementary fact on matrices we show by a unified approach the positivity of a partitioned positive semidefinite matrix with each square block replaced by a compound matrix, an elementary symmetric function or a generalized matrix function. In addition, we present a refined version of the Thompson determinant compression theorem.

A Nearly One-to-One Method to Convert Analog Signals into a Small Volume of Data

The actual quantizing and sampling procedure leads to irreducible errors that do not permit to reconstruct correctly the initial analog signal. To diminish the reconstruction errors to an acceptable level one has to increase the volume of transmitted data. Then, it results in big (or even very big) data volumes that are difficult to store and/or transmit. A deal has been made sometimes between the quality and the resulted data volume in order to have small enough volumes. The paper presents a new (theoretically) one-to-one method to convert physical realizable analog signals into a small volume of data. This method uses the (Shannon) Sampling Theorem, the Bandwidth Compression Theorem and a new Quantizing and Sampling Theorem described in this paper. The Quantizing and Sampling Theorem is based on the properties of phase/frequency modulation and on the properties of a Phase Lock Loop (PLL) to decode also amplitude-modulated signals. Practically, this new method may realize a (theoretically) nearly one-to-one conversion of a physically realizable analog signal into a small volume of data. Adequate software and hardware have to be realized to achieve technically this goal. To diminish the errors of calculus, error- free calculus methods have to be used.

The geometry of Whitney’s critical sets

In this paper, the complete geometric characterizations, including decomposition and compression theorems, are obtained for a connected and compact set to be a critical set in Whitney’s sense, i.e., a set such that there exists a differentiable function critical but not constant on it. The problem to characterize these sets geometrically was posed by H. Whitney [21] in 1935. We also provide a complete geometrical characterization for monotone Whitney arc, i.e., there exists a differentiable function critical but also increasing along the arc. All examples appearing in the literature are monotone Whitney arcs, for example, the examples by Whitney [21] and Besicovitch [2], Norton’s t-quasi-arcs with Hausdorff dimension > t [14], and self-similar arcs [19]. Furthermore, after introducing the notion of homogeneous Moran arc, we can completely characterize all the monotone Whitney arcs of criticality > 1, which include t-quasi arcs and self-conformal arcs. Some applications to arcs which are attractors of Iterated Function Systems are discussed, including self-conformal arcs, self-similar arcs and self-affine arcs. Finally, we give an example of critical arc such that any of its subarcs fails to be a t-quasi-arc for any t, providing an affirmative answer to an open question by Norton.

Compression theorems for surfaces and their applications

Let M be a complete glued surface whose sectional curvature is greater than or equal to k and ▵ p q r a geodesic triangle domain with vertices p , q , r in M . We prove a compression theorem that there exists a distance nonincreasing map from ▵ p q r onto the comparison triangle domain ▵ ˜ p q r in the two-dimensional space form with sectional curvature k . Using the theorem, we also have some compression theorems and an application to a circular billiard ball problem on a surface.

Leray–Schauder and Krasnoselskii results for multivalued maps defined on pseudo-open subsets of a Fréchet space

New Leray–Schauder alternatives and Krasnoselskii expansion and compression theorems are presented for multivalued maps defined on subsets of a Fréchet space E . The proof relies on the notion of a pseudo-open set and on viewing E as the projective limit of a sequence of Banach spaces.

The Data Compression Theorem for Ergodic Quantum Information Sources

We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate.

A Krasnoselskii cone compression result for multimaps in the S-KKM class

The aim of this article is obtain a Krasnoselskii cone compression theorem for multimaps in the class S-KKM.

Computation of the equation of state of the quantum hard-sphere fluid utilizing several path-integral strategies.

The compressibility factor of the quantum hard-sphere fluid within the region (rho(N) (*)</=0.8,lambda(B) (*)</=0.9) is computed by following four distinct routes involving the three pair radial correlation functions that are significant in the path-integral context, namely, instantaneous, pair linear response, and centroids. These functions are calculated with path-integral Monte Carlo simulations involving the Cao-Berne propagator. The first route to the equation of state is the instantaneous standard one, i.e., the usual volume derivative of the partition function expressed in terms of the instantaneous pair radial correlations. The other three routes stem from the extended compressibility theorem, which associates the isothermal compressibility with the three pair radial structures mentioned above and involves the solving of appropriate Ornstein-Zernike equations. An analysis of the error bars in the quantities computed is reported, and it is proven the usefulness of the centroid pair correlations to fix quantum equations of state. Also, the regions where the fluid-solid changes of phase should take place are identified with the use of indicators sensitive to order in the sample. The consistency of the current results is assessed and comparison with data available in the literature is made wherever possible.

The compression theorem III: applications

This is the third of three papers about the Compression Theorem: if M^m is embedded in Q^q X R with a normal vector field and if q-m > 0, then the given vector field can be straightened (ie, made parallel to the given R direction) by an isotopy of M and normal field in Q X R. The theorem can be deduced from Gromov's theorem on directed embeddings [M Gromov, Partial differential relations, Springer--Verlag (1986) 2.4.5 C'] and the first two parts (math.GT/9712235 and math.GT/0003026) gave proofs. Here we are concerned with applications. We give short new (and constructive) proofs for immersion theory and for the loops--suspension theorem of James et al and a new approach to classifying embeddings of manifolds in codimension one or more, which leads to theoretical solutions. We also consider the general problem of controlling the singularities of a smooth projection up to C^0--small isotopy and give a theoretical solution in the codimension > 0 case.

The compressibility theorem for quantum simple fluids at equilibrium

An extension of the compressibility theorem for quantum simple fluids within the pathintegral approach is presented. First, it is demonstrated that in the absence of quantum exchange, the isothermal compressibility can be formulated in an exact manner with the use of the pair radial correlation function of the path-integral centroids corresponding to the particles of the fluid. This adds up to the two known formulations based on the pair correlations between true quantum particles, namely the instantaneous and the pair linear response correlations. To complement this extension, an exact Ornstein-Zernike equation for pair centroid correlations is derived, which permits accurate estimates for the isothermal compressibility to be obtained. Several fluids are studied, new numerical results for the latter quantity are reported to support the theoretical points, and some difficulties present in this sort of calculation are discussed. The systems studied are the following: the quantum hard sphere fluid with and without attractive Yukawa interaction, liquid helium-4 and liquid para-hydrogen. Finally, the possibilities of extending the theorem to deal with quantum exchange are considered, and it is shown that the extension and its computational Ornstein-Zernike scheme also hold for a Bose fluid.

A Krasnoselskii Cone Compression Theorem for $\scr{U}_{c}^{k}$ Maps

The main purpose of this paper is to prove a generalisation of the Krasnoselskii cone compression theorem for the $\scr{U}_{c}^{k}$ maps of Park. Our analysis is elementary and relies on the fact that the unit sphere (in a cone) is a retract of the unit ball (in a cone).

Poincare Duality Embeddings and Fibrewise Homotopy Theory, II

We prove a relative Poincarembedding theorem for maps of pairs into a Poincarpair. This result has applications: it is the engine of the compression theorem of the author's paper in Algebr. Geom. Topol. 2, (2002) 311-336, it yields a Poincarspace version of Hudson's embedding theorem, and it can be used to equip 2-connected Poincar´ e spaces with handle decompositions.

Statistical mechanics of the data compression theorem

We analyse the performance of a linear code used for data compression of a Slepian-Wolf type. In our framework, two correlated data are separately compressed into codewords employing Gallager-type codes and cast into a communication network through two independent input terminals. At the output terminal, the received codewords are jointly decoded by a practical algorithm based on the Thouless-Anderson-Palmer approach. Our analysis shows that the achievable rate region presented in the data compression theorem is described as first-order phase transitions among several phases. The typical performance of the practical decoder is also well evaluated by the replica method.

Stationary quantum source coding

In this article the quantum version of the source coding theorem is obtained for a completely ergodic source. This result extends Schumacher’s quantum noiseless coding theorem for memoryless sources. The control of the memory effects requires some earlier results of Hiai and Petz on high probability subspaces. Our result is equivalently considered as a compression theorem for noiseless stationary channels.