Easy Examples Research Articles

Perturbations of differentiable semigroups

If (A, D(A)) generates a C 0-semigroup T on a Banach space X and $$B \in {\mathcal{L}}(X)$$ then (A + B, D(A)) is also the generator of a C 0-semigroup, S B . There are easy examples to show that if T is eventually differentiable then S B need not be eventually differentiable. In 1995 an example was constructed to show that if T is immediately differentiable then S B need not be immediately differentiable. In this paper we establish necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S B for all $$B \in {\mathcal{L}}(X)$$ . We are therefore able to give a characterization of the immediately and eventually differentiable C 0-semigroups for which differentiability is a stable property under bounded perturbations of the generator. We also prove a characterization of the C 0-semigroups for which the norm of the resolvent of the generator decays on vertical lines and a new characterization of the Crandall-Pazy class of semigroups.

Satellite communication over quantum channel

Quantum computing offers revolutionary solutions in the field of computer sciences, applying the opportunities of quantum physics which are incomparably richer than classical physics. Although quantum computers are going to be the tools of the far future, there are already a few algorithms to solve problems which are very difficult to handle with traditional computers. Perhaps the easiest example of a structure of a quantum system is a quantum channel. Typically, one is interested in some basis in the Hilbert space representing the input of a channel, which is entangled with a second Hilbert space representing the environment, and then another (possibly the same) basis for the first space at a later time. Free-space quantum key distribution (QKD)—over an optical path of about 30 cm—was first introduced in 1991, and recent advances have led to demonstrations. Indeed there are certain key distribution problems in this category for which free-space QKD has practical advantages (for example it is not practical to send a courier to a satellite). Quantum computing algorithms can be used to affirm our communication in several ways (open-air communication, satellite communications, satellite broadcast, satellite-satellite communication). We set up a free-space quantum-channel-model at the university and made several simulations. The main aim is to trace some adoptable algorithms in the communication between Earth and the satellite and also between satellites. This paper is a theoretical study to compare the simulation results of the three models.

On the Construction of Affine Extractors

In this paper, we address the problem of explicit construction of so-called ‘affine extractors’ of δ-entropy ratio, when δ ≤ 1/2 (for δ > 1/2, there is a well-known and easy example based on the Hadamard function). We first give examples for δ slightly below 1/2 and produce then a construction for arbitrary given δ > 0. All these examples are again of a simple algebraic nature. The mathematics involved includes recent developments in the sum-product theory in finite fields and certain new bounds on multilinear exponential sums.

Diametrically Contractive Multivalued Mappings

AbstractDiametrically contractive mappings on a complete metric space are introduced by V. I. Istratescu. We extend and generalize this idea to multivalued mappings. An easy example shows that our fixed point theorem is more applicable than a former one obtained by H. K. Xu. A convergence theorem of Picard iteratives is also provided for multivalued mappings on hyperconvex spaces, thereby extending a Proinov's result.

Derivations of identities by symbolic computation

We have made the implementation of symbolic computation programs which can derive identity relations for arbitrarily given mathematical expressions. Simulations of relatively easy several concrete examples have been shown to run within practical speeds.

Characterization of scaling functions in a multiresolution analysis

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map A : R n → R n A: \mathbb {R}^n\rightarrow \mathbb {R}^n such that A ( Z n ) ⊂ Z n A(\mathbb {Z}^n) \subset \mathbb {Z}^n and all (complex) eigenvalues of A A have absolute value greater than 1. 1. In the general case the conditions depend on the map A . A. We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when A A is a diagonal matrix with all numbers in the diagonal equal to 2. 2. There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.

The LMDI approach to decomposition analysis: a practical guide

In a recent study, Ang (Energy Policy 32 (2004)) compared various index decomposition analysis methods and concluded that the logarithmic mean Divisia index method is the preferred method. Since the literature on the method tends to be either too technical or specific for most potential users, this paper provides a practical guide that includes the general formulation process, summary tables for easy reference and examples.

Hilbert series, Howe duality, and branching rules.

Let lambda be a partition, with l parts, and let F(lambda) be the irreducible finite dimensional representation of GL(m) associated to lambda when l < or = m. The Littlewood Restriction Rule describes how F(lambda) decomposes when restricted to the orthogonal group O(m) or to the symplectic group Sp(m2) under the condition that l < or = m2. In this paper, this result is extended to all partitions lambda. Our method combines resolutions of unitary highest weight modules by generalized Verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory. Corollaries include determination of the Gelfand-Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, and the determination of their Hilbert series (as a graded module for p(-)). Let L be a unitary highest weight representation of sp(n, R), so*(2n), or u(p, q). When the highest weight of L plus rho satisfies a partial dominance condition called quasi-dominance, we associate to L a reductive Lie algebra g(L) and a graded finite dimensional representation B(L) of g(L). The representation B(L) will have a Hilbert series P(q) that is a polynomial in q with positive integer coefficients. Let delta(L) = delta be the Gelfand-Kirillov dimension of L and set c(L) equal to the ratio of the dimensions of the zeroth levels in the gradings of L and B(L). Then the Hilbert series of L may be expressed in the form H(L)(q)=c(L) P(q)(1-q)(delta). In the easiest example of the correspondence L --> B(L), the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group.

Repeat Victimisation and the Policing of Communities

Three types of police service can be distinguished. • Reaction to relevant emergencies; • Identifying and processing crime perpetrators; • Proaction to prevent crime and disorder (referred to below as ‘unity policing’). The central problem for the organisation of policing is arguably the integration of the first two elements of work with the third. In this paper, the writers contend that focusing on identifying and providing relevant help to those previously victimised by crime offers a platform for such integration. The advantages conferred by this approach will be identified. The Huddersfield Biting Back project is described as an example of how such an approach can be implemented. The implications of risk-based targeting, of which repeat victimisation provides the easiest example to implement, are spelled out.

Bethe ansatz solvable multi-chain quantum systems

In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (‘basic’ model) of this class of models: the antiferromagnetic two-chain spin- 1 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate–incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the ‘quantum transfer matrix’) for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.

Rotation rate of a trajectory of an algebraic vector field around an algebraic curve

For a Lipschitzian vector field in $\R^n$, angular velocity of its trajectories with respect to any stationary point is bounded by the Lipschitz constant. The same is true for a rotation speed around any integral submanifold of the field. However, easy examples show that a trajectory of a $C^{\infty}$-vector field in $\R^3$ can make in finite time an infinite number of turns around a straight line. We show that for a trajectory of a polynomial vector field in $\R^3$, its rotation rate around any algebraic curve is bounded in terms of the degree of the curve and the degree and size of the vector field. As a consequence, we obtain a linear in time bound on the number of intersections of the trajectory with any algebraic surface.

Robust Stability of Discrete Polynomials with Polynomic Structure of Coefficients

A method for Schur robust stability analysis of polynomials with polynomic structure of their coefficients is presented in this paper. The presented method tests the stability in the coefficient space and is based on the Modified Jury criterion. Coefficients of this table are multivariate polynomial functions, whose positivity is tested by Sign-decomposition. This is a method which makes it possible to test positivity of some set of multivariate functions on a convex set. Another approach consists in using linear fractional transformation of discrete polynomials into continuous polynomials and then in using the Modified Routh criterion or the Hurwitz criterion together with Sign-decomposition for robust stability analysis. To determine elements of any table a new procedure based on fast Fourier transform for multivariate polynomial multiplication is used. Both methods are resulting in a necessary and sufficient condition. They are demonstrated on an easy example and compared from computational point of view.

On some examples of simple quantum groups

The aim of this note is to construct noncocommutative finite dimensional simple Hopf algebras by twisting the group algebras of finite simple groups. In [N1] Nikshych described those constructions and proved that if the twist is possible then the resulting Hopf algebra will remain simple and the Grothendieck rings of the two Hopf algebras will be the isomorphic as ordered rings with involution. We show that the twisting is always possible and describe all the cases in which his construction can be done. Recently in [N2] these methods have been used to describe all simple Hopf algebras of dimension at most 60. To fix the notation let k be an algebraically closed field of characteristic zero. We will only be interested in finite dimensional Hopf algebras over k. The easiest example is provided by the group algebra kG for G a finite group and with comultiplication ∆(g) = g⊗g, counit e(g) = 1 and antipode S(g) = g−1. We will call such an example a trivial Hopf algebra. A Hopf algebra H is simple if it has no Hopf quotients. In particular in the case of a group algebra kG the normal Hopf subalgebras are in 11 correspondence with the normal subgroups of G and in particular kG is simple if and only if G is simple. Also A is cocommutative (respectively commutative) iff A is a trivial (respectively the dual of a trivial algebra). We will prove the following :

Is There a Predictable Criterion for Mutual Singularity of Two Probability Measures on a Filtered Space?

The theme of providing criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open [J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987, p.~210]: We do not know whether it is possible to derive a criterion (necessary and sufficient condition) for $P_T'\perp P_T,\ldots$. It turns out that there are two answers to this question raised in the monograph of J.~Jacod and A.~N.~Shiryaev: On the negative side, we give an easy example showing that in general the answer is no, even when we use a rather wide interpretation of the concept of predictable criterion. The difficulty comes from...

Image filtering, mean curvature, Dirichlet problems

Numerical experiments of the model for image filtering proposed in [1] by Crandall, Lions and others show that, given the PDE, the differential problem which is correct to solve is the Dirichlet one, on a domain Ω convex with holes. The aim of this paper is to produce a result of existence and uniqueness of the viscosity solution for this problem. The result is stated for convex domains and for nonconvex ones, we propose an easy example in order to show that an existence theorem has not been expected.

Null ideals and spanning ranks of matrices, II

Suppose R is an integral domain and A ∈ M n × n(R) \\{O}. If D is a spanning rank partner of A, then precisely one of the following three relationships holds: N A = N D N A = XN D or XN A = N D. Here X is an indeterminate and N A(N D) denotes the null ideal of A(D) in R[X]. There are easy examples of A and D for which N A = N D and N A = XN D. In this paper, we give an example where XN A = N D. We give sharper versions of the theorem for n ≤ 4.

A fast guidance algorithm for an autonomous navigation system

In the present paper a fast and robust algorithm is presented for optimization of finite thrust orbit transfers. Generally optimal nominal trajectories are designed very accurately on ground solving two point boundary value problems with hard numerical work and computational time. Of course, an on board guidance system is needed to take into account the many perturbations occurring during the transfer trajectory. Two approaches are generally followed: in the first a steering program is actuated each time the navigation system detects sensible errors from the nominal trajectory. This guidance (perturbative guidance) is not optimal. The second approach (adaptive guidance) is to design a new optimal trajectory starting from the current state to the required final condition. This guidance is optimal but it requires the on board solution of two point boundary value problems any time it is necessary to update the trajectory. Therefore, complex algorithms and computational times, related to the convergence of the numerical solutions, are needed. In the present paper a method to reduce the complexity of adaptive guidance systems is described: the main point is to reduce the required two-point boundary value problems to differential problems with initial conditions only. This allows to design a fast and easy guidance law, suitable for on-board computers with a restricted storage capability, as in the case of small satellites and small launchers. The algorithm is robust, allowing rather large deviation from the nominal trajectory, and avoiding the use of uncontrolled iterative routines. This method is applied to an easy example: an in plane guidance law maximizing the final horizontal velocity.

Spatial Breakdown Point of Variogram Estimators

In the context of robust statistics, the breakdown point of an estimator is an important feature of reliability. It measures the highest fraction of contamination in the data that an estimator can support before being destroyed. In geostatistics, variogram estimators are based on measurements taken at various spatial locations. The classical notion of breakdown point needs to be extended to a spatial one, depending on the construction of most unfavorable configurations of perturbation. Explicit upper and lower bounds are available for the spatial breakdown point in the regular unidimensional case. The difficulties arising in the multidimensional case are presented on an easy example in IR2, as well as some simulations on irregular grids. In order to study the global effects of perturbations on variogram estimators, further simulations are carried out on data located on a regular or irregular bidimensional grid. Results show that if variogram estimation is performed with a 50% classical breakdown point scale estimator, the number of initial data likely to be contaminated before destruction of the estimator is roughly 30% on average. Theoretical results confirm the previous statement on data in IRd, d ≥ 1.

A Design Method of the Optimal PID Controller for Multiple Objective Optimization Using a Genetic Algorithms.

In this paper, we propose a method of Genetic Algorithms (GA) for multiple objective optimization problems. And, it is shown that the proposed GA improve the convergence of GA and the diversity of Pareto-optimal solutions, through two kinds of easy example problems of 2 objective optimization. Then, we apply the proposed GA to PID control design of 2 objective optimization problems. We solve 2 disk type mixed sensitivity problem and robust sensitivity minimization problem that include limiting control variable problem. The effectiveness of the PID controllers obtained by the proposed method is confirmed by the characteristics of the step response and gain diagrems of sensitivity function and complementary sensitivity function.

Bose-Mesner Algebras Related to Type II Matrices and Spin Models

A type II matrix is a square matrixW with non-zero complex entries such that the entrywise quotient of any two distinct rows of W sums to zero. Hadamard matrices and character tables of abelian groups are easy examples, and other examples called spin models and satisfying an additional condition can be used as basic data to construct invariants of links in 3-space. Our main result is the construction, for every type II matrix W, of a Bose-Mesner algebraN(W) , which is a commutative algebra of matrices containing the identity I, the all-one matrix J, closed under transposition and under Hadamard (i.e., entrywise) product. Moreover, ifW is a spin model, it belongs to N(W). The transposition of matrices W corresponds to a classical notion of duality for the corresponding Bose-Mesner algebrasN(W) . Every Bose-Mesner algebra encodes a highly regular combinatorial structure called an association scheme, and we give an explicit construction of this structure. This allows us to compute N(W) for a number of examples.