The mixed mode brittle fracture criteria in siliding mode fracture

It is well-known that the present mixed mode brittle fracture criteria are all theopening mode fracture criterion.We consider that mixed mode brittle fracture of slidingmode fracture exists too.Hence we propose three criteria of mixed mode brittle fracture ofsliding mode fracture;:the radial shearing stress criterion,the maximum shearing stresscriterion and the distortional strain-energy-density criterion.Thus,we can overall explainthe phenomena of brittle fracture in the structural elements with cracks.     

On the Consistency of a Positive Theory

In positive theories, we have an axiom scheme of comprehension for positive formulas. We study here the “generalized positive” theory GPK_∞⁺. Natural models of this theory are hyperuniverses. The author has shown in [2] that GPK_∞⁺ interprets the Kelley Morse class theory. Here we prove that GPK_∞⁺ + AC_WF (AC_WF being a form of the axiom of choice allowing to choose elements in well-founded sets) and the Kelley-Morse class theory with the axiom of global choice and the axiom “On is ramifiable” are mutually interpretable. This shows that GPK_∞⁺ + AC_WF is a “strong” theory since “On is ramifiable” implies the existence of a proper class of inaccessible cardinals.     

Adjoining Almost Disjoint Permutations

We show that it is consistent with ZFC + ¬CH that there is a maxima a most disjoint permutation family A ⊆ Sym(ℕ)such that A is a proper subset of an eventually different family E ⊆ ^ℕℕ and |A| < |E|. We also ask severa questions in this area.     

Indestructible Weakly Compact Cardinals and the Necessity of Supercompactness for Certain Proof Schemata

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.     

Revisiting the inhomogeneously driven sine–Gordon equation

The sine–Gordonequation is a semilinear wave equation used to model many physical phenomenon like seismic events that includes earthquakes, slow slip and after-slip processes, dislocation in solids etc. Solution of homogeneous sine–Gordon equation exhibit soliton like structure that propagates without change in its shape and structure. The question whether solution of sine–Gordon equation still exhibit soliton like behavior under an external forcing has been challenging as it is extremely difficult to obtain an exact solution even under simple forcing like constant. In this study solution to an inhomogeneous sine–Gordon equation with Heaviside forcing function is analyzed. Various one-dimensional test cases like kink and breather with no flux and non-reflecting boundary conditions are studied.     

Evidence of Dispersion Relations for the Nonlinear Response of the Lorenz 63 System

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.     

Mach-Zehnder interferometer-based tree architecture for all-optical logic and arithmetic operations

Interferometric devices for optical processing have been of great interest in recent years. Semiconductor optical amplifier (SOA)-based Mach-Zehnder interferometer (MZI) has already taken a significant role in the field of ultra-fast all-optical signal processing. Optical tree architecture (OTA) provides important contributions in optical interconnecting networks. In this communication, we have tried to exploit the advantages of both OTA and SOA-based MZI switches. We have proposed SOA-MZI-based tree architecture, a new and alternative scheme, for integrated all-optical logic and arithmetic operations. This architecture can enable one to perform all-optical processing of signals, including two input logic operations, half-adder, full-adder, full-subtractor, one-bit data comparator, etc.     

On the Laplacian spectral radii of trees with perfect matchings

Denote by the set of trees of order 2k with perfect matchings. GUO [Guo, Linear Algebra Appl. 368:379–385, 2003.] determined the largest value of Laplacian spectral radii μ(T) of the trees T in and gave the corresponding tree T in whose μ(T) reaches this largest value. In this paper, we determine the second to the sixth largest values of μ(T) of the trees T in and also give the corresponding trees T in whose μ(T) reach these values.     

带强迫项的高阶中立型时滞差分方程的振动定理

本文研究带有强迫项的高阶中立型时滞差分方程的振动性.我们建立了方程(E)的新的振动准则且给出了说明定理应用的例子.