Witten–Dijkgraaf–Verlinde–Verlinde Equation and its Application to Relative Gromov–Witten Theory

We derive a recursive formula for certain relative Gromov–Witten invariants with a maximal tangency condition via the Witten–Dijkgraaf–Verlinde–Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.

From dawn to dusk

It used to be the case in some jurisdictions that a rhythmic knell of the Angelus bell would mark the onset of dawn, noon and the passing of a day at dusk. Between these times there is daylight whose duration varies from place to place and from day to day and which can be predicted either exactly or approximately. An illuminating problem concerning the number of daylight hours at a winter solstice in London was posed recently and answered in the Problem Corner of The Mathematical Gazette [1]. It was shown, for example, that a calculation of daylight hours rested strictly upon the numerical solution to a transcendental trigonometric equation. Related references to earlier works in the Gazette involving a point source Sun were also given.The purpose of this note is multifold. First, it is to point out that the above-cited equation might be viewed also as a “Sun-ray-to-Earth tangency condition”. Such a condition was developed earlier by the author in a publication that is now defunct [2], and so for completeness the steps necessary to establish the condition will be produced again here. Second, it will be evident from the manner of its derivation that the governing equation is valid at all orbit points, not just at a given solstice.

Near viability for fully nonlinear differential inclusions

We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.

Transpiration Boundary Conditions for a Steady Inverse Method

Computational fluid dynamics has been widely used in the analysis of turbomachinery blades, however, its use as a design tool is far from sophisticated. The inverse method is such a design approach, which lends it self to the latter category. One application of the inverse method is the so called “pure inverse methd”, which differs from common analysis solver mainly in the boundary conditions on the blade surfaces. For this application, the usual non-penetration boundary conditions on the blade surfaces are aborted, instead, some aerodynamic constraints are imposed, and the flow is allowed to transpire through the actual solid wall. A camber line generation equation is added to periodically re-generate the blade camber line and drive the normal velocities on the blade surfaces to zero. When converged, the inverse method should obtain the blade shapes which satisfy the specified aerodynamic performance. In the present paper, three transpiration boundary conditions for turbomachinery blades design are compared in terms of time cost, robustness, capability of coping separation flow etc. The first inverse boundary condition is based on the flow-tangency condition on the blade surfaces, the second relys on the propagating characteristics in the flow field, and the third is a hybrid version of the first and the second. The computation is validated for 2D Navier-Stokes equation. Two compressor cascades are taken as examples to compare the performance of the three transpiration boundary conditions. Finally some conclusions are drawn.

On the Analysis of Minimum Thickness in Circular Masonry Arches

In this paper, the so-called Couplet–Heyman problem of finding the minimum thickness necessary for equilibrium of a circular masonry arch, with general opening angle, subjected only to its own weight is reexamined. Classical analytical solutions provided by J. Heyman are first rederived and explored in details. Such derivations make obviously use of equilibrium relations. These are complemented by a tangency condition of the resultant thrust force at the haunches' intrados. Later, given the same basic equilibrium conditions, the tangency condition is more correctly restated explicitly in terms of the true line of thrust, i.e., the locus of the centers of pressure of the resultant internal forces at each theoretical joint of the arch. Explicit solutions are obtained for the unknown position of the intrados hinge at the haunches, the minimum thickness to radius ratio and the nondimensional horizontal thrust. As expected from quoted Coulomb's observations, only the first of these three characteristics is perceptibly influenced, in engineering terms, by the analysis. This occurs more evidently at increasing opening angle of the arch, especially for over-complete arches. On the other hand, the systematic treatment presented here reveals the implications of an important conceptual difference, which appears to be relevant in the statics of masonry arches. Finally, similar trends are confirmed as well for a Milankovitch-type solution that accounts for the true self-weight distribution along the arch.