Parameter spaces of separating hyperplanes

Every pair of relatively disjoint polytopes is dual to the parameter space of all their separating hyperplanes, which is also a polytope. For a polytope whose interior is disjoint from the relative interior of another polytope, the parameter space of all separating hyperplanes is a polytope of the same dimension. One face of this parameter space parametrizes the separating hyperplanes that also simultaneously support both polytopes. A separating hyperplane corresponds to a vertex of this face if and only if no other hyperplanes support the polytopes at the same intersection points. If all the vertices of the polytopes have all their coordinates in an ordered field, then the same results and their proofs hold with the same ordered field.     

A Lefschetz hyperplane theorem for Mori dream spaces

Let X be a smooth Mori dream space of dimension ?? 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron?CSeveri spaces of X and Y, and under this identification every Mori chamber of Y is a union of some Mori chambers of X, and the nef cone of Y is the same as the nef cone of X. This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking ??Mori dream hypersurfaces?? of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.     

On Passive and Conservative State/Signal Systems

This article is devoted to a study of continuous-time passive and conservative systems within the state/signal framework. The main idea of the state/signal approach is to not a priori distinguish between inputs and outputs, but rather to combine these two into a single external signal. The so-called node space is introduced as the product of two copies of the state space of the system and one copy of the space where the external signals of the system live. This node space is equipped with a sesquilinear product that makes it a Kre?n space. A generating subspace is defined as a closed subspace of the node space, and the node space determines the trajectories of a state/signal system. One of the main results of this article is that a subspace of the node space generates a passive state/signal system if and only if it is a maximally nonnegative subspace of the node space and it satisfies a certain nondegeneracy condition. In this case the generating subspace can be interpreted as the graph of a scattering-passive input/state/output system node.     

可调激活函数递进提升输出维的选参方法

针对一类变参数 Sigmoid可调激活函数构成三层前向神经网络 ,分析其可调激活函数中参数所表示意义 ;给出了递进提升输出向量空间维数的可调变参数激活函数中参数选取的方法 ,解决了隐含神经元采用相同激活函数限制了神经网络逼近能力这一问题 .其目的给人们在采用变参数可调激活函数神经网络解决问题时 ,如何选取激活函数中的参数提供了一种数学依据和方法 .     

Prevalence of locally parameter identifiable systems

The problem of local parameter identifiability of an input–output system is considered. A set lf of systems is studied for which the property of local parameter identifiability holds for almost all values of input signals and parameters in both topological and metric senses. Sufficient conditions are pointed out under which the set LI contains a prevalent subset. The proof is based on the prevalent transversality theorem proved by Kaloshin. Systems are considered that are characterized by a family of (structural) parameters a and a control block. It is shown that if the dimension of the set of parameters a is large enough (the structure of the system is rich enough), then, generically, a system f _a belongs to the class lf for a set of parameters a having full measure.     

On the complexity of some Euclidean optimal summing problems

The complexity status of several discrete optimization problems concerning the search for a subset of a finite set of Euclidean points (vectors) is analyzed. In the considered problems, the aim is to minimize objective functions depending either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is part of the input, then all analyzed problems are strongly NP-hard and, if the space dimension is fixed, then these problems are NP-hard even for dimension 2 (on a plane). It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.     

A note on the Picard bundle over a moduli space of vector bundles

Let ??(n , d ) be a coprime moduli space of stable vector bundles of rank n ≥ 2 and degree d over a complex irreducible smooth projective curve X of genus g ≥ 2 and ??_ξ ? ??(n , d ) a fixed determinant moduli space. Assuming that the degree d is sufficiently large, denote by ?? the vector bundle over X ×??(n , d ) defined by the kernel of the evaluation map H ⁰(X , E ) → E_x , where E ∈??(n , d ) and x ∈ X . We prove that ?? and its restriction ??_ξ to X × ??ξ are stable. The space of all infinitesimal deformations of ?? over X ×??(n , d ) is proved to be of dimension 3g and that of ??ξ over X × ??ξ of dimension 2g , assuming that g ≥ 3 and if g = 3 then n ≥ 4 and if g = 4 then n ≥ 3. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariant measures for parabolic IFS with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.

     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

Functionality defense through diversity: a design framework to multitier systems

Diversification is one of the most effective approaches to defend multitier systems against attacks, failure, and accidents. However, designing such a system with effective diversification is a challenging task because of stochastic user and attacker behaviors, combinatorial-explosive solution space, and multiple conflicting design objectives. In this study, we present a systematic framework for exploring the solution space, and consequently help the designer select a satisfactory system solution. A simulation model is employed to evaluate design solutions, and an artificial neural network is trained to approximate the behavior of the system based on simulation output. Guided by a trained neural network, a multi-objective evolutionary algorithm (MOEA) is proposed to search the solution space and identify potentially good solutions. Our MOEA incorporates the concept of Herbert Simon??s satisficing. It uses the decision maker??s aspiration levels for system performance metrics as its search direction to identity potentially good solutions. Such solutions are then evaluated via simulation. The newly-obtained simulation results are used to refine the neural network. The exploration process stops when the result converges or a satisfactory solution is found. We demonstrate and validate our framework using a design case of a three-tier web system.     

On the complexity and approximability of some Euclidean optimal summing problems

The complexity status of several well-known discrete optimization problems with the direction of optimization switching from maximum to minimum is analyzed. The task is to find a subset of a finite set of Euclidean points (vectors). In these problems, the objective functions depend either only on the norm of the sum of the elements from the subset or on this norm and the cardinality of the subset. It is proved that, if the dimension of the space is a part of the input, then all these problems are strongly NP-hard. Additionally, it is shown that, if the space dimension is fixed, then all the problems are NP-hard even for dimension 2 (on a plane) and there are no approximation algorithms with a guaranteed accuracy bound for them unless P = NP. It is shown that, if the coordinates of the input points are integer, then all the problems can be solved in pseudopolynomial time in the case of a fixed space dimension.     

On the Order Reduction Approach for Singularly Perturbed Optimal Control Systems

The order reduction method for singularly perturbed optimal control systems consists of employing the system obtained while setting the small parameter to be zero. In many situations the differential-algebraic system thus obtained indeed provides an appropriate approximation to the singularly perturbed problem with a small parameter. In this paper we establish that if relaxed controls are allowed then the answer to the question whether or not this method is valid depends essentially on one simple parameter: the dimension of the fast variable, denoted n. More specifically, if n=1 then the order reduction method is indeed applicable, while if n>1 then the set of singularly perturbed optimal control systems for which it is not applicable is dense (in the L norm).     

Two-Sided Closed Ideals of Certain Algebras of Unbounded Operators

This paper investigates the two-sided uniformly closed ideals of the maximal Op*-algebra L⁺(D) of (bounded or unbounded) operators on a dense domain D in a HILBERT space. It is assumed that D is a FRECHET space with respect to the graph topology. The set of all non-trivial two-sided closed ideals of L⁺(D) is well-ordered by inclusion and the α-th closed ideal ??_α is generated by the orthogonal projections onto HILBERTian subspaces of D of dimension less then ??_α. An element A in L⁺(D) belongs to the minimal closed ideal ??₀ if and only if the following two equivalent conditions are satisfied: a) A maps bounded subsets of D into relatively compact sets. b) A maps weakly convergent sequences in D into convergent sequences.     

Poisson polyhedra in high dimensions

The zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated, as the space dimension tends to infinity. The model includes the zero cell of stationary and isotropic Poisson hyperplane tessellations as well as the typical cell of a stationary Poisson Voronoi tessellation as special cases. It is shown that asymptotically in the space dimension, with overwhelming probability these cells satisfy the hyperplane conjecture, if the distance exponent and the intensity parameter are suitably chosen dimension-dependent functions. Also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed. In the background are new identities linking the f-vector of the zero cell to certain dual intrinsic volumes.     

Detecting codimension one manifold factors with 0-stitched disks

The 0-stitched disks property is introduced and shown to detect codimension one manifold factors of dimension n?4. It is shown that if a space X is an ANR and has the 0-stitched disks property, then X has the disjoint homotopies property. It follows that if a space X is a resolvable generalized manifold of dimension n?4 with the 0-stitched disks property, then X is a codimension one manifold factor. Whether or not the 0-stitched disks property is equivalent to the disjoint homotopies property remains an open question.     

Adaptive regulation for deterministic systems

For the linear deterministic system with unknown orders and coefficients adaptive controls are given so that the closed-loop system is stabilized and the unknown parameters are consistently estimated. Moreover, if the parameter estimation is ignored, then the system input and output can be reduced to zero with an exponential rate.This project is supported by the National Natural Science Foundation of China.In Commemoration of the 15th Anniversary of the Acta Mathematicae Applicatae Sinica.     

On Semitransitive Collections of Operators

A collection F of operators on a vector space V is said to be semitransitive if for every pair of nonzero vectors x and y in V there exists a member T of F such that either Tx = y or Ty = x (or both). We study semitransitive algebras and semigroups of operators. One of the main results is that if the underlying field is algebraically closed, then every semitransitive algebra of operators on a space of dimension n contains a nilpotent element of index n. Among other results on semitransitive semigroups, we show that if the rank of nonzero members of such a semigroup acting on an n-dimensional space is a constant k, then k divides n.     

Rotation number quantization effect

We study a class of dynamical systems on a torus that includes dynamical systems modeling the dynamics of the Josephson transition. For systems in this class, we introduce certain characteristics including a sequence of functions depending on the system parameters. We prove that if this sequence converges at a given point in the parameter space, then its limit is equal to the classical rotation number, and we then call this point a quantization point for the rotation number. We prove that the rotation number of such a system takes only integer values at a quantization point. Quantization areas are thus defined in the parameter space, and the problem of effectively describing them becomes an important part of characterizing the systems under study. We present graphs of the rotation number at quantization points and under conditions when it is not quantized (an example of a half-integer rotation number) and diagrams for quantization areas.     

On output feedback control of singularly perturbed systems

We treat the problem of robustness of output feedback controllers with respect to singular perturbations. Given a singularly perturbed control system whose boundary layer system is exponentially stable and whose reduced order system is exponentially stabilizable via a (possibly dynamical) output feedback controller, we present a sufficient condition which ensures that the system obtained by applying the same controller to the original full order singularly perturbed control system is exponentially stable for sufficiently small values of the perturbation parameter. This condition, which is less restrictive than those previously given in the literature, is shown to be always satisfied when the singular perturbation is due to the presence of fast actuators and/or sensors. Furthermore, we show explicitly that, in the linear time-invariant case, if this condition is not satisfied then there exists an output feedback controller which stabilizes the reduced order system but destabilizes the full order system.