Vanishing Conditions Research Articles

Separation of flow over a diaphragm in a two-dimensional channel

A study is made of the hydrodynamic problem of the flow with separation over a diaphragm with sharp edges. The vorticity of the fluid in The separation region is determined by means of the Zhukovskii—Chaplygin condition of finiteness of the velocity at the sharp edge, and the extension of the separation region is found from the condition of vanishing of the velocity of the fluid at the point of closure of the separation region.

Soliton Solutions of the Derivative Nonlinear Schrödinger Equation

By using the inverse scattering method the soliton solutions are examined analytically and numerically under both (I) vanishing and (II) nonvanishing conditions. The two-soliton solution for (I) shows that two solitons collide as if they were particles. For the case (II) there appears a “paired soliton” which generally pulsates with a period (“pulsative soliton”) but degenerates to a stationary one (“pure soliton”) in a limited case. There exist two types of pure solitons, envelope bright and dark solitons, between which some cases of collisions are examined. The collision between solitons of same types is similar to that of (I), while the other collisions are different from the previous case.

Pomeranchuk-type theorems for total and elastic cross sections

In the framework of local field theory the asymptotic properties of the crossing-odd forward scattering amplitude $F(E)$ are investigated, $E$ being the laboratory energy. Confronting the analyticity properties of the logarithm of the amplitude with the Froissart-Martin bound, we obtain a series of sufficient conditions for a fast asymptotic vanishing of $\mathrm{Im}\frac{F(E)}{F}$ and $\mathrm{Re}\frac{F(E)}{E}$ in the mean. Analogous conditions for the vanishing of the total-cross-section difference $\ensuremath{\Delta}\ensuremath{\sigma}(E)$ follow via the optical theorem. We also find conditions for the existence of Meiman's generalized high-energy limit of $\ensuremath{\Delta}\ensuremath{\sigma}(E)$. Finally, two independent asymptotic bounds on $\ensuremath{\Delta}\ensuremath{\sigma}(E)$ are derived. The method is extended to nonforward scattering to give asymptotic bounds on differential cross sections.

A vanishing theorem for second degree cohomology

In this paper we give conditions for the cohomology group H2(G, V) to vanish, in the case where G is a finite group, V is a KG-module for some field K of characteristic p and G has a normal p-Sylow subgroup acting trivially on V. Interpreting these conditions when G is a Bore1 subgroup of L(K), a group of Lie type over K, we obtain a vanishing condition for P@(K), V) depending only on relations between roots and weights. The main theorem is proved in Section 1 and is applied to groups of Lie type in Section 2. In Section 3 we list the cases in which the vanishing conditions have been shown to hold and briefly describe what is known in some of the cases where the conditions fail.

The generalized harmonic balance method for determining the combination resonance in the parametric dynamic systems

For a multi-degree-of-freedom system under parametric excitation an attempt is made to generalize the harmonic balance method to the case of the combination resonance. The two harmonic components solution with uncommensurable frequencies has been assumed on the stability limits. It is found that besides the condition of zero value of the characteristic determinant of the algebraic system derived by the harmonic balance procedure, the additional condition of vanishing of all minors must be satisfied. The method has been applied to a two-degree-of-freedom system. The boundaries of the principal periodic and combination resonances have been calculated theoretically and then the results have been checked by an analog computer analysis. New essential features and peculiarities of the combination resonance have been found.

On born‐huang invariance conditions

AbstractThe Born‐Huang (B‐H) invariance conditions have been criticised on the ground of insufficiency of rotational invariance by several workers who have obtained additional conditions. This point is examined and it is shown that the B‐H invariance conditions are complete. The equilibrium conditions for vanishing of stress should be separately considered. Then the additional conditions mentioned above are nothing new.

A spatial restriction method in the theory of many-electron systems. I

An efficient method is suggested for calculating the behavior of many-electron systems in which the electron motion is spatially restricted by the wave function permutation symmetry. The method is based on introducing a trial function, satisfying the permutation rules and accounting explicitly for restriction in the one-electron configuration space. Applied to outer shell electrons of an atom, the method allows the inner shells to be taken into account through a vanishing boundary condition on a spherical surface of radius r0. A table of r0 values is constructed for atomic cores of the principal group elements. The energy levels of atoms and ions calculated with the aid of this table are in good agreement with a wide range of spectroscopic data. The results form a basis for ground state calculations of dielectric crystals (see Part II).

Behavior of Two-Point Correlation Functions Near and on a Phase Boundary

The asymptotic decay of the two-point correlation function ${G}_{\mathrm{AB}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}})$ at and near a phase-separation point is discussed for $d$-dimensional, spin-\textonehalf{} Ising models at low temperature. The general behavior, even on the phase boundary ($H=0$), is in agreement with extended Ornstein-Zernike predictions. It is shown why the nearest-neighbor two-dimensional model in zero field is an exception. The decay of ${G}_{\mathrm{AB}}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{1},{\stackrel{\ensuremath{\rightarrow}}{\mathrm{R}}}_{2})$ near a free surface at high temperatures agrees with phenomenological predictions using a vanishing boundary condition. At low temperature, however, the decay of correlation near a surface is exponentially slower than in the bulk.

X‐ray Topographic Study of Defects in Quartz I. Brazil Twin Boundaries

AbstractA systematic investigation has been made of the diffraction contrast effects observed by X‐ray traverse topography at Brazil twin boundaries in natural quartz. Evidence is presented to show that the crystal axes are not coincident across the Brazil twin boundaries but are displaced by a vector R which depends upon the composition plane of the boundaries. Consequently the displacement vectors for intersecting Brazil twin boundaries are not identical. These displacement vectors have been determined from the observed vanishing conditions for the diffraction contrast fringes at the boundaries. The expected stair‐rod type dislocations at the intersections of the twin boundaries have been found. Probable atomic models for the Brazil twin boundaries are proposed and discussed.

Intersecting faults on basal and prismatic planes in aluminium nitride

Abstract Intersecting faults on basal (0001) and prismatic (1210) planes in A1N (hexagonal, wurtzite structure) have been studied by diffraction contrast with the electron microscope. The faults on (1210) had fault vectors 3/2 p + 1/2 c=½[1011], as deduced from the observed vanishing conditions. The expected stair-rod dislocations at the intersections of the faults have been found, and they had screw components to their Burgers vectors. An atomic model for the faults on (1210) is proposed, and possible mechanisms of formation during crystal growth are discussed. The determination of the nature of faults in hexagonal crystals by means of diffraction contrast is considered in detail, since the information obtainable is different for h.c.p. than for f.c.c.

Some Conditions for the Apparent Vanishing of the Energy Gap in Superconductors

Paramagnetic impurities in one of the two superconducting films causes the energy gap to disappear in tunneling bebetween two superconductors. An expression for the change in tunneling current due to impurities is derived using perturbation theory to deal with exchange coupling between conduction electrons and impurity spins. Effect of changes in spin states with quasi-particle occupation is also examined. (D.C.W.)

Some Conditions for the Apparent Vanishing of the Energy Gap in Superconductors

Paramagnetic impurities in one of the two superconducting films causes the energy gap to disappear in tunneling bebetween two superconductors. An expression for the change in tunneling current due to impurities is derived using perturbation theory to deal with exchange coupling between conduction electrons and impurity spins. Effect of changes in spin states with quasi-particle occupation is also examined. (D.C.W.)

Aspects of $(2,2)$ and $(0,2)$ hybrid models

In this work we study the topological rings of two dimensional (2,2) and (0,2) hybrid models. In particular, we use localization to derive a formula for the correlators in both cases, focusing on the B- and B/2-twists. Although our methods apply to a vast range of hybrid CFTs, we focus on hybrid models suitable for compactifications of the heterotic string. In this case, our formula provides unnormalized Yukawa couplings of the spacetime superpotential. We apply our techniques to hybrid phases of linear models, and we find complete agreement with known results in other phases. We also obtain a prediction for a certain class of correlators involving twisted operators in (2,2) Landau-Ginzburg orbifolds. For (0,2) theories, our argument does not rely on the existence of a (2,2) locus. Finally, we derive vanishing conditions concerning worldsheet instanton corrections in (0,2) B/2-twisted hybrid models.