Bell's Inequality for Two-Particle Mixed Spin States

We derive the Bell–Clauser–Horne–Shimony–Holt inequalities for two-particle mixed spin states both in the conventional quantum mechanics and in the hidden-variables theory. We consider two cases for the vectors , and specifying the axes onto which the particle spins of a correlated pair are projected. In the first case, all four vectors lie in the same plane, and in the second case, they are oriented arbitrarily. We compare the obtained inequalities and show that the difference between the predictions of the two theories is less for mixed states than for pure states. We find that the inequalities obtained in quantum mechanics and the hidden-variables theory coincide for some special states, in particular, for the mixed states formed by pure factorable states. We discuss the points of similarity and difference between the uncertainty relations and Bell's inequalities. We list all the states for which the right-hand side of the Bell–Clauser–Horne–Shimony–Holt inequality is identically equal to zero.     

Matroid representation over GF(3)

We show that a matroid is representable over GF(3) if and only if no minor is the five-point line or the Fano matroid, or their duals. Tutte's famous characterization of the regular matroids is a corollary. A key lemma states that two representations of the same matroid in the same vector space over GF(3) may be transformed one into the other by inverting some points through the origin and taking a linear transformation; no result of this kind holds in larger fields.     

Vector coherent states on Clifford algebras

The well-known canonical coherent states are expressed as infinite series in powers of a complex number z and a positive integer (m) = m!. In analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable z with a real Clifford matrix. We also present another class of vector coherent states by simultaneously replacing z with a real Clifford matrix and (m) with a real matrix. As examples, we present vector coherent states labeled by quaternions and octonions with their real matrix representations. We also present a physical example.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 9–21, April, 2005.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Dynamic effect of Bloch electrons in time-periodic electric fields

The dynamic effect of Bloch electrons under the action of temporally periodic electric fields is studied. The explicit solutions for the quasienergies and the Floquet states are obtained exactly and generally, from which the localized or extended nature of Floquet states is demonstrated to be controlled by a reduced vector potential in one period of electric fields. When this reduced vector potential is an irrational, all the Floquet states are localized except the cases where the reduced vector potential is extremely well approximated by rationals (i.e., a Liouville number) and simultaneously the instersite hopping does not decay fast enough, for which the Floquet states are found to be more and more extended at large distances. When this reduced vector potential approaches an ordinary rational, all the Floquet states are extended. However, if the reduced vector potential has typical Diophantine properties, and if the intersite hopping decreases fast enough, the Floquet states cannot be extended. Project supported in part by the National Natural Science Foundation of China and the Chinese Academy of Engineering and Physics.     

A Note on Lights-Out-Puzzle: Parity-State Graphs

A state of a graph G is an assignment of 0 or 1 to each vertex of G. A move of a state consists of choosing a vertex and then switching the value of the vertex as well as those of its neighbors. Two states are said to be equivalent if one state can be changed to the other by a series of moves. A parity-state graph is defined to be a graph in which two states are equivalent if and only if the numbers of 1’s in the two states have the same parity. We characterize parity-state graphs and present some constructions of parity-state graphs together with applications. Among other things, it is proved that the one-skeleton of the 3-polytope obtained from a simple 3-polytope by cutting off all vertices is a parity-state graph.     

Blowup criterion for viscous, compressible micropolar fluids with vacuum

We prove that the maximum norm of velocity gradients controls the possible breakdown of smooth (strong) solutions for the 3-dimensional viscous, compressible micropolar fluids. More precisely, if a solution of the system is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without the bound of the velocity gradients as the critical time approaches. Our result is a generalization of Huang et al. (2011) [13] from viscous barotropic flows to the viscous, compressible micropolar fluids. In addition, initial vacuum states are also allowed in our result.     

Weak continuous states on Banach algebras

We prove that if a unital Banach algebra A is the dual of a Banach space A_? then the set of normal states is weak^∗ dense in the set of all states on A. Further, normal states linearly span A_?.     

Analytic Representations Based on SU(1,1) Lie Algebra Coherent States for Squeezed Displaced Fock States

The coherent states (CSs) of the SU(1,1) group can be divided into two broad categories: (a) the Barut-Girardello coherent states (BGCSs) and (b) the Perelomov coherent states (PCSs). Some definitions for the squeezed displaced Fock states (SDFSs) are given. The hyperbolic analytic representation in the complex plane is considered. An analytic representation of the SU(1,1) Lie group is given and the representation in the unit disk based on the SU(1,1) PCSs for SDFSs is considered.      

Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

Earthquake曲线切向量上的范数

本文研究了定义在earthquake曲线切向量上的范数，首先证明了一条earthquake曲线ht上初始切向量的范数等价于earthquake测度σ的Thurston范数．其次证明了当t→∞时，ht的切向量Vt的范数增长渐近等于O(||→||The^Ct||σ||Th)，其中C是正的万有常数，||σ||Th是σ的Thurston范数，而O所代表的常数是渐近万有的，也即当t||σ||Th充分大时它是万有的．此外，附带证明了定义在Zygmund有界函数上的两种交比范数是等价的．     

Students’ perceived sociomathematical norms: The missing paradigm

This study proposes a framework for research which takes into account three aspects of sociomathematical norms: teachers’ endorsed norms, teachers’ and students’ enacted norms, and students’ perceived norms. We investigate these aspects of sociomathematical norms in two elementary school classrooms in relation to mathematically based and practically based explanations. Results indicate that even when the observed enacted norms are in agreement with the teachers’ endorsed norms, the students may not perceive these same norms. These results highlight the need to consider the students’ perspective when investigating sociomathematical norms.     

Characterization of the closedness of the sum of two shift-invariant spaces

We first present a formula for the supremum cosine angle between two closed subspaces of a separable Hilbert space under the assumption that the ‘generators’ form frames for the subspaces. We then characterize the conditions that the sum of two, not necessarily finitely generated, shift-invariant subspaces of L²(Rd) be closed. If the fibers of the generating sets of the shift-invariant subspaces form frames for the fiber spaces a.e., which is satisfied if the shift-invariant subspaces are finitely generated or if the shifts of the generating sets form frames for the respective subspaces, then the characterization is given in terms of the norms of possibly infinite matrices. In particular, if the shift-invariant subspaces are finitely generated, then the characterization is given wholly in terms of the norms of finite matrices.     

Equivalent complete norms and positivity

In the first part of the article we characterize automatic continuity of positive operators. As a corollary we consider complete norms for which a given cone E₊ in an infinite dimensional Banach space E is closed and we obtain the following result: every two such norms are equivalent if and only if has finite codimension. Without preservation of an order structure, on an infinite dimensional Banach space one can always construct infinitely many mutually non-equivalent complete norms. We use different techniques to prove this. The most striking is a set theoretic approach which allows us to construct infinitely many complete norms such that the resulting Banach spaces are mutually non-isomorphic. Received: 28 January 2009     

On trivialities of Stiefel-Whitney classes of vector bundles over highly connected complexes

It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel-Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel-Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces.     

Compatible Dubrovin–Novikov Hamiltonian Operators,Lie Derivative,and Integrable Systems of Hydrodynamic Type

We prove that two Dubrovin–Novikov Hamiltonian operators are compatible if and only if one of these operators is the Lie derivative of the other operator along a certain vector field. We consider the class of flat manifolds, which correspond to arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators. Locally, these manifolds are defined by solutions of a system of nonlinear equations, which is integrable by the method of the inverse scattering problem. We construct the integrable hierarchies generated by arbitrary pairs of compatible Dubrovin–Novikov Hamiltonian operators.     

On Existence Theorems for Solutions of Non‐Linear Systems

An equivalent formulation of a recent result in [1] states that if the conditions of the Theorem of Newton‐Kantorovich are satisfied in a slightly modified (but equivalent) form in the maximum norm for a function g : G → ℝⁿ, G ⊆ ℝⁿ, guaranteeing the existence of a zero in D, then the conditions of Miranda's Theorem are automatically satisfied. We prove that this result holds for arbitrary norms if the conditions of the Theorem of Newton‐Kantorovich are suitable strengthened and Miranda's Theorem is suitably generalized.     

可压Navier-Stokes方程真空状态的动力学行为

本文主要考虑了一维可压Navier-Stokes方程真空状态的动力学行为.对于任意的熵弱解,如果初始状态不存在真空,我们证明了密度函数关于时间和空间变量是连续的且对于任意时间它是处处为正的.同时,我们还得到了含有间断连接的真空状态的整体熵弱解的存在性,结果显示其真空区域以代数速率被压缩,并在有限时间内消失.     

On cooperative games,inseparable by semivalues

Semivalues like the Shapley value and the Banzhaf value may assign the same payoff vector to different games. It is even possible that two games attain the same outcome for all semivalues. Due to the linearity of the semivalues, this exactly occurs in case the difference of the two games is an element of the kernel of each semivalue. The intersection of these kernels is called the shared kernel, and its game theoretic importance is that two games can be evaluated differently by semivalues if and only if their difference is not a shared kernel element. The shared kernel is a linear subspace of games. The corresponding linear equality system is provided so that one is able to check membership. The shared kernel is spanned by specific {–1,0,1}-valued games, referred to as shuffle games. We provide a basis with shuffle games, based on an a-priori given ordering of the players.