Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies

The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α∈(0,1) and a Bolza-type cost functional. A relationship between the differential game and the Cauchy problem for the corresponding Hamilton–Jacobi–Bellman–Isaacs equation with fractional coinvariant derivatives of the order α and the natural boundary condition is established. An emphasis is given to construction of optimal positional (feedback) strategies of the players. First, a smooth case is studied when the considered Cauchy problem is assumed to have a sufficiently smooth solution. After that, to cope with a general non-smooth case, a generalized minimax solution of this problem is involved.

Entropy in Thermodynamics: from Foliation to Categorization

We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon’s relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds.

On a class of topological conjugacy with a homothety

We consider a class H(Rn) of orientation-preserving homeomorphisms of Euclidean space Rn such that for any homeomorphism h∈H(Rn) and for any point x∈Rn a condition limn→+∞hn(x)→O holds, were O is the origin. It is proved that for any n≥1 an arbitrary homeomorphism h∈H(Rn) is topologically conjugated with the homothety an:Rn→Rn, given by an(x1,…,an)=(12x1,…,12xn). For a smooth case under the condition that all eigenvalues of the differential of the mapping h have absolute values smaller than one, this fact follows from the classical theory of dynamical systems. In the topological case for n∉{4,5} this fact is proven in several works of 20th century, but authors do not know any papers where it would be proven for n∈{4,5}. This paper fills this gap.

Range-Kernel Orthogonality and Elementary Operators: The Nonsmoothness Case

The characterization of the points in C1ℋ, the trace class operators, that are orthogonal to the range of elementary operators has been carried out for certain kinds of elementary operators by many authors in the smooth case. In this note, we study that the characterization is a problem in nonsmoothness case for general elementary operators, and we give a counter example to S. Mecheri and M. Bounkhel results.

On kernel bundles over reducible curves with a node

Given a vector bundle [Formula: see text] on a complex reduced curve [Formula: see text] and a subspace [Formula: see text] of [Formula: see text] which generates [Formula: see text], one can consider the kernel of the evaluation map [Formula: see text], i.e. the kernel bundle [Formula: see text] associated to the pair [Formula: see text]. Motivated by a well-known conjecture of Butler about the semistability of [Formula: see text] and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a reducible curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known in the smooth case. For example, [Formula: see text] is actually quite never [Formula: see text]-semistable. Conditions which gives the [Formula: see text]-semistability of [Formula: see text] when [Formula: see text] or when [Formula: see text] is a line bundle are then given.

Fujiki relations and fibrations of irreducible symplectic varieties

This paper concerns different types of singular complex projective varieties generalizing irreducible symplectic manifolds. We deduce from known results that the generalized Beauville-Bogomolov form satisfies the Fujiki relations and has the same rank as in the smooth case. This enables us to study fibrations of these varieties; imposing the newer definition from [GKP16, Definition 8.16.2] we show that they behave much like irreducible symplectic manifolds. Comment: Published version

Effect of Blade Profile on Four-Passage Serpentine Configuration Designed to Negate Coriolis Effect on Heat and Fluid Flow

Gas turbine blades are equipped with serpentine internal cooling channels with 180-degree bends, through which relatively colder air is routed to cool the internal walls. It has been established that under the influence of rotation, pressure and suction side internal wall heat transfer characteristics are very different, which leads to non-uniform metal temperatures, and hence higher levels of thermal stresses. Present study addresses this non-uniformity in heat transfer using parallel rotation to negate Coriolis effect. Further, the blade curvature does not allow rectangular or trapezoidal passages, which are typically studied. In this paper, we have numerically investigated a realistic design for the four-passage channel, where the cooling design can actually be incorporated in a blade. Four-passage configuration also features 90-degree square shaped rib turbulators, and the corresponding baseline case is smooth channel. Numerical simulations have been carried out at Reynolds numbers of 5000, 10000 and 25000 and Rotation numbers were varied between 0 and 0.25. For smooth case, heat transfer enhancement was found to be higher on suction (leading) side compared to pressure (trailing) side under both stationary and rotating conditions. The enhancement levels between stationary and rotation conditions varied marginally in these designs, indicating that buoyancy effects were insignificant. For ribbed case, the effect of 90-degree rib turbulators on local heat transfer was more pronounced on the suction side when compared to smooth case. Under rotating conditions, it was found that the cooling levels were similar to the stationary condition for both pressure and suction side internal walls.