Differentiability of Spectral Functions for Relativistic <Emphasis Type="Italic">α</Emphasis>-Stable Processes with Application to Large Deviations

We consider the relativistic α-stable process, a pure jump Markov process generated by \(\mathcal{H}^{\alpha} = (-\Delta + m^{2/\alpha})^{\alpha /2}-m\). Let ?C(λ) be the bottom of spectrum of Schrödinger type operator \(\mathcal{H}^{\lambda \mu} = \mathcal{H}^{\alpha} - \lambda \mu\), where μ is a signed Kato measure. We prove the differentiability of C(λ). As an application of it, we establish a large deviation principle for the additive functional \(A_t^{\mu}\) corresponding to the measure μ.     

Dynamics of a creep-slip model of earthquake faults

Starting off from the relationship between time-dependent friction and velocity softening we present a generalization of the continuous, one-dimensional homogeneous Burridge–Knopoff (BK) model by allowing for displacements by plastic creep and rigid sliding. The evolution equations describe the coupled dynamics of an order parameter-like field variable (the sliding rate) and a control parameter field (the driving force). In addition to the velocity-softening instability and deterministic chaos known from the BK model, the model exhibits a velocity-strengthening regime at low displacement rates which is characterized by anomalous diffusion and which may be interpreted as a continuum analogue of self-organized criticality (SOC). The governing evolution equations for both regimes (a generalized time-dependent Ginzburg–Landau equation and a non-linear diffusion equation, respectively) are derived and implications with regard to fault dynamics and power-law scaling of event-size distributions are discussed. Since the model accounts for memory friction and since it combines features of deterministic chaos and SOC it displays interesting implications as to (i) material aspects of fault friction, (ii) the origin of scaling, (iii) questions related to precursor events, aftershocks and afterslip, and (iv) the problem of earthquake predictability. Moreover, by appropriate re-interpretation of the dynamical variables the model applies to other SOC systems, e.g. sandpiles.     

Application of the HN method to the critical slab problem for reflecting boundary conditions

The recently developed H_N method is used to solve the critical slab problem for a slab which is surrounded by a reflector. In the special case for R=0 (the reflection coefficient) the problem reduces to the one under vacuum boundary conditions. It is shown that the method is concise and leads to fast converging numerical results. The presented numerical results are compared with the data available in literature.     

Virtual-site correlation mean field approach to criticality in spin systems

We propose a virtual-site correlation mean field theory for dealing with interacting many-body systems. It involves a coarse-graining technique that terminates a step before the mean field theory: While mean field theory deals with only single-body physical parameters, the virtual-site correlation mean field theory deals with single- as well as two-body ones, and involves a virtual site for every interaction term in the Hamiltonian. We generalize the theory to a cluster virtual-site correlation mean field, that works with a fundamental unit of the lattice of the many-body system. We apply these methods to interacting Ising spin systems in several lattice geometries and dimensions, and show that the predictions of the onset of criticality of these models are generally much better in the proposed theories as compared to the corresponding ones in mean field theories.     

A criticality result for polycycles in a family of quadratic reversible centers

We consider the family of dehomogenized Loud's centers $X_{\mu }=y(x?1){?}_x+(x+D{x}^2+F{y}^2){?}_y$, where $\mu =(D,F)\in {\mathbb{R}}^2$, and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu },\mu \in {\mathbb{R}}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ${\mathrm{\Gamma }}_B$ of codimension 1 in ${\mathbb{R}}^2$. In the present paper we succeed in proving that a subset of ${\mathrm{\Gamma }}_B$ has criticality equal to one.     

Schrödinger equations with critical nonlinearity, singular potential and a ground state

We study semilinear elliptic equations in a generally unbounded domain Ω⊂RN when the pertinent quadratic form is nonnegative and the potential is generally singular, typically a homogeneous function of degree −2. We prove solvability results based on the asymptotic behavior of the potential with respect to unbounded translations and dilations, while the nonlinearity is a perturbation of a self-similar, possibly oscillating, term f_∞ of critical growth satisfying , j∈Z, s∈R. This paper focuses on two qualitatively different cases of this problem, one when the quadratic form has a generalized ground state and another where the presence of potential does not change the energy space. In the latter case we allow nonlinearities with oscillatory critical growth. An important example of such quadratic form is the one on RN with the radial Hardy potential −μ|x|⁻² with μ=μ_∗ in the first case, μ<μ_∗ in the second case, where is the largest constant for which the energy form remains nonnegative.     

Time to consensus

The dynamics of a two-state decision-making model (DMM) with a tunable control parameter K$K$ is described. On all-to-all (ATA) networks, the system undergoes a phase transition at a critical value of _Kc=1$K_c=1$. Scale-free networks were also found to undergo phase transitions, but the value of _Kc$K_c$ increases as the scale-free exponent increases. Time to consensus is defined as a first passage time to a specified threshold of the global mean field, which represents a level of majority for the network. At criticality, the time to consensus for sparse networks with broad degree distributions (e.g., scale-free networks) approaches that of the ATA networks, although somewhat more cooperation (higher _Kc$K_c$) is required.     

Existence and stability of a synchronous oscillation in a neural system with delayed coupling

The article deals with a particular case of a two-neuron network with two discrete time delays. We analyse the linear stability for the particular equilibrium (0,0)$(0,0)$, and address the Hopf bifurcation which may occur as one of the delays crosses some critical values. The network under consideration has no self-connections, and thus only one time delay appears in the characteristic equation.     

Franco-Russian comparison of mixed neutron and gamma radiation field dosimeters at the Silène reactor

This paper gives the results of dosimetry measurements carried out in the Silène reactor at Valduc (France) with neutron and photon dosimeters in mixed neutron and gamma radiation fields, in the frame of a Franco-Russian comparison of dosimeters. Neutron dosimetry was supplied by passive semiconductors, activation detectors and nuclear track detectors. For photon dosimetry, thermoluminescent and passive semiconductor detectors were used. The experiments were located at 3 m from the reactor core, in free air and also at the front and back of a tissue-equivalent phantom. The pulse operating mode of the reactor was used to simulate a criticality accident with solid fissile material, while the free evolution mode simulated a criticality accident in a fissile solution. The photon absorbed dose showed a slight increase on entering the phantom compared to measurements in free air, probably due to backscattering by the phantom. At the rear of the phantom, the neutron kerma was four times lower than on the front, whereas the photon dose was only two times lower. The heterogeneity of dose inside the phantom was far greater for neutrons than for photons.