Compressible effect algebras

We define a special type of additive map J on an effect algebra E called a compression. We call J(1) the focus of J and if p is the focus of a compression then p is called a projection. The set of projections in E is denoted by P(E). A compression J is direct if J(a) ≤ a for all a ε E. We show that direct compressions are equivalent to projections onto components of cartesian products. An effect algebra E is said to be compressible if every compression on E is uniquely determined by its focus and every compression on E has a supplement. We define and characterize the commutant C(p) of a projection p and show that a compression with focus p is direct if and only if C(p) = E. We show that P(E) is an orthomodular poset. It is proved that the cartesian product of effect algebras is compressible if and only if each component is compressible. We then consider compressible sequential effect algebras, Lüders maps and conditional probabilities.     

Effect Algebras Are Not Adequate Models for Quantum Mechanics

We show that an effect algebra E possess an order-determining set of states if and only if E is semiclassical; that is, E is essentially a classical effect algebra. We also show that if E possesses at least one state, then E admits hidden variables in the sense that E is homomorphic to an MV-algebra that reproduces the states of E. Both of these results indicate that we cannot distinguish between a quantum mechanical effect algebra and a classical one. Hereditary properties of sharpness and coexistence are discussed and the existence of {0,1} and dispersion-free states are considered. We then discuss a stronger structure called a sequential effect algebra (SEA) that we believe overcomes some of the inadequacies of an effect algebra. We show that a SEA is semiclassical if and only if it possesses an order-determining set of dispersion-free states.     

Aplanatic hologram lenses on spherical surfaces

It is shown that if an off-axis Fresnel hologram is formed on a spherical surface the hologram as a lens will be aplanatic if the position of the reference and object points satisfy the condition (l′)^?1 + l^?1 = R^?1.     

Positivity and monotonicity properties ofC 0-semigroups. II

IfS _t =exp{?tH},T _t =exp{?tK}, are self-adjoint positivity preserving semigroups on a Hilbert space ?=L ²(X; dμ) we write (*) $$T_t \succ 0$$ ifT _t is positivity improving and (**) $$S_t \succ T_t $$ if the differenceS _t ?T _t is positivity improving. We derive a variety of characterizations of (*) and (**). In particular (*) is valid for allt>0 if, and only if,T _t ∪L ^∞ (X; dμ) is irreducible for somet>0. Similarly if the semigroups are ordered the strict order (**) is valid if, and only if, {S _t ?T _t }∪L ^∞(X; dμ) is irreducible for somet>0. These criteria are used to prove that if (*) is valid for allt>0 then $$e^{ - tf(K)} \succ 0,t > 0,$$ and if (**) is valid for allt>0 then $$e^{ - tf(H)} \succ e^{ - tf(K)} ,t > 0$$ for each non-constantf in the class characterized in the preceding paper. We discuss the decomposition of positivity preserving semigroups in terms of positivity improving semigroups on subspaces. Various applications to monotonicity properties of Green's functions are given.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

Continuous opinion model in small-world directed networks

In the compromise model of continuous opinions proposed by Deffuant et al., the states of two agents in a network can start to converge if they are neighbors and if their opinions are sufficiently close to each other, below a given threshold of tolerance ?. In directed networks, if agent i is a neighbor of agent j,j need not be a neighbor of i. In Watts-Strogatz networks we performed simulations to find the averaged number of final opinions 〈F〉 and their distribution as a function of ? and of the network structural disorder. In directed networks 〈F〉 exhibits a rich structure, being larger than in undirected networks for higher values of ?, and smaller for lower values of ?.     

On invariant states and the commutant of a group of quasi-free automorphisms of the CAR algebra. III

It is shown that a state of the CAR algebra is an extreme invariant state under the group of quasi-free automorphisms α_U with unitaries u in a von Neumann algebra M on the one- particle Hilbert space if and only if it is a gauge-invariant quasi-free state Ø _A corresponding to A in M' with 0 ≤ A ≤ 1, under the assumption that M does not contain any finite type I factor direct summands.     

Orthomodular Lattices of Subspaces Obtained from Quadratic Forms

Being given a field K of characteristic different from 2 and 3, a 3-dimensional vector space E over K, and a nonsingular symmetric bilinear form φ over E, we define a structure of orthomodular lattice T(E,φ) on the set of all nonisotropic subspaces of E. We give a structure Theorem about the irreducible and 3-homogeneous subalgebras of T(E,φ). In particular, these subalgebras are all of the form T(E',φ ') where E' is a 3-dimensional subspace of E, if E is regarded as a vector space over a subfield K' of K, and φ ' is induced by φ. This structure Theorem allows us to achieve an old project, concerning minimal orthomodular lattices (an orthomodular lattice L is called minimal if it is nonmodular and if all its proper subalgebras are either modular, or isomorphic to L).     

Function groups associated with constraint submanifolds

Let M be a submanifold of a manifold Q which has a generalized symplectic form ω. The submanifold M of Q is locally Hamiltonian if its vector fields are locally generated as a C^∞_ω- module by Hamiltonian vector fields of Q. If M is a first class submanifold, i.e., M is defined by first class constraints, then it is shown that M is locally Hamiltonian. It follows that if M is first class then M is a leaf of the singular foliation associated with the function group of all first class functions.     

The robustness of contextuality and the contextuality cost of empirical models

In this paper, we introduce and discuss the robustness of contextuality (RoC) R_C(e) and the contextuality cost C(e) of an empirical model e. The following properties of them are proved. (i) An empirical model e is contextual if and only if R_C(e) > 0; (ii) the RoC function R_C is convex, lower semi-continuous and un-increasing under an affine mapping on the set EM of all empirical models; (iii) e is non-contextual if and only if C(e) = 0; (iv) e is contextual if and only if C(e) > 0; (v) e is strongly contextual if and only if C(e) = 1. Also, a relationship between R_C(e) and C(e) is obtained. Lastly, the RoC of three empirical models is computed and compared. Especially, the RoC of the PR boxes is obtained and the supremum 0.5 is found for the RoC of all no-signaling type (2, 2, 2) empirical models.     

Truesdell transport in general relativity

The Truesdell transport of a contravariant tensor fieldX ^ab is defined with respect to a time-like vector fieldu ^a analogous to the classical definition in continuum mechanics. Truesdell stress rate of the shear tensor (σ ^ab) and the metric tensor (g ^ab ) is Truesdell transported alongl _a if and only if the medium is rigid. In case of two-dimensional projection operatorsγ ^ab andγ ^*ab are Truesdell transported along timelike vector fieldu ^a if and only if they are expansion free (alongu ^a ) with \(\tau = \lambda - \bar \sigma + \mu - \rho = 0\) . A collapsing perfect fluid is Truesdell transported with respect tou ^a if and only if it is expansion free (alongu ^a ), energy density (D), pressure (P), and density of neutrino radiation (q) are conserved alongu ^a withσt=λ, \(\alpha + \bar \beta = \tau = 0\) . In all the above cases, analogous results are found alongl ^a ,n ^a , andm ^a using freedom conditions (k=ε=π=γ+γ ^?=0) as null rays are always restricted to be geodesic (Carter [1]).     

Thermodynamics of hadrons: Delimiting the temperature

We assume an exponentially rising hadronic mass spectrum ～exp (m/T₀) and consider means for avoiding a limiting temperature at T = T₀. It is possible to do so if the single-particle potential rises at least as fast as m, or if the finite size of hadrons is taken into account, or both. Applications to heavy ion collisions, the early universe and Monte Carlo simulations of the thermodynamics of lattice QCD are mentioned.     

Dynamics of a neural network model with finite connectivity and cycle stored patterns

The spatiotemporal evolution and memory retrieval properties of a Hopfield-like neural network with cycle-stored patterns and finite connectivity are studied. The analytical studies on a mean-field version show that, given the number of stored patterns p, there is a critical connectivity k_c such that the retrieval states are stable fixed points if and only if k > k_c. The dependence of k_c on the number of stored patterns is also present. The numerical simulations are applied to the short-ranged model with local interaction. It is revealed that, given p, the memory retrieval function is kept if the connectivity is high enough while the dynamics of the system is in the frozen phase. However when the connectivity k is less than a critical value k_c the system is in the chaotic phase and loses its memory retrieval ability. The critical points of both the dynamical phase transition and memory-loss phase transition are obtained by simulation data.     

On locally continuous cocycles

Cohomology groups of locally continuous cocycles of a topological group G with values in a topological G-module A_Ψ are considered and the role of those of degree 2 in the theory of topological extensions of G by A_Ψ is analyzed. The exactness of long sequence is proven. It is shown that, if G and A_Ψ are Polish and if all topological extensions of G by A_Ψ are fibered, the cohomology groups of Borel cocycles and those of locally continuous Borel cocycles are isomorphic for degree 0,1,2.     

Scaling limits for the Lego discrepancy

For the Lego discrepancy with M bins, which is equivalent with a χ²-statistic with M bins, we present a procedure to calculate the moment generating function of the probability distribution perturbatively if M and N, the number of uniformly and randomly distributed data points, become large. Furthermore, we present a phase diagram for various limits of the probability distribution in terms of the standardized variable if M and N become infinite.     

Groups of automorphisms with spectrum condition and the lifting problem

We show that a continuous one-parameter group {α _t } of automorphisms of a separableC*-algebraA satisfies a spectrum condition if and only if it is the limit, pointwise onA, uniformly on compact subsets of IR of a sequence of inner automorphism groups whose positive generators do not increase too fast. Moreover we prove in this case that if π is a surjective morphism of a separableC*-algebraB onA then there is a similar group \(\{ \bar \alpha _t \} \) of automorphisms ofB such that \(\pi \circ \bar \alpha _t = \alpha _t \circ \pi \) for allt.     

On the use of the mean-field theory in optical bistability

The validity of the mean-field approximation in the theory of optical bistability is discussed. We show that major deviations from its predictions may occur if the mirrorreflectivity is decreased below a value which is strongly dependent on the bistability parameter C, and if the absorption αL is larger than a value weakly dependent on C (αL?4).     

C*-algebras and automorphism groups

Let (A,G, α) be aC*-dynamical system withG a topological group. Let π be a representation ofA. We will show that there exists a quasiequivalent representation \(\hat \pi \) to π which is a covariant representation, if and only if the folium of π is invariant under the action ofG and this action is strongly continuous.     

Almost Orthogonality and Hausdorff Interval Topologies of de Morgan Lattices and Lattice Effect Algebras

The topologies on ordered structures have been intensively studied by mathematicians and computer scientists. Various types of topologies may be introduced, depending on the nature of the ordered sets considered. Our purpose here is to study the interval topology τ _i , the order topology τ _o and the topology τ _Φ induced by a canonical intrinsic uniformity generated by a certain family of pseudometrics on de Morgan lattices. This uniformity and topology may be regarded as a “two-sided symmetrization” of a similar intrinsic uniformity introduced by Erné and Palko for an order-theoretical construction of certain uniform completions. We prove that on a de Morgan lattice L with a join-dense set $\mathcal{U}$ the interval topology τ _i is Hausdorff and L is compactly generated by the elements of $\mathcal{U}$ if and only if L is $\mathcal{U}$ -almost orthogonal if and only if any element of $\mathcal{U}$ is hypercompact.