We consider the stationary nonlinear magnetic Choquard equation 

$$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, ${\alpha \in (0, N)}$ and 2 − (α/N) < p < (2N − α)/(N−2). We assume that both A and V are compatible with the action of some group G of linear isometries of ${\mathbb{R}^{N}}$ . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$$

where ${\tau : G \rightarrow \mathbb{S}^{1}}$ is a given group homomorphism into the unit complex numbers.

Multiple solutions to a magnetic nonlinear Choquard equation

Nilpotence and descent in equivariant stable homotopy theory

In the renaming task, n+1 processes start with unique input names from a large space and must choose unique output names taken from a smaller name space, 0,1,…, K. To rule out trivial solutions, a protocol must be anonymous: the value chosen by a process can depend on its input name and on the execution, but not on the specific process ID.Attiya et al. [1990] showed that renaming has a wait-free solution when K≥ 2n. Several algebraic topology proofs of a lower bound stating that no such protocol exists when K

New combinatorial topology bounds for renaming: The upper bound

Motivated by topological Tverberg-type problems, we consider multiple (double, triple, and higher multiplicity) self-intersection points of maps from finite simplicial complexes (compact polyhedra) into Rd, and study conditions under which such multiple points can be eliminated. The most classical case is that of embeddings (i.e., maps without double points) of a k-dimensional complex K into R2k. For this problem, the work of van Kampen, Shapiro, and Wu provides an efficiently testable necessary condition for embeddability (namely, vanishing of the van Kampen obstruction). For k ≥ 3, the condition is also sufficient, and yields a polynomial-time algorithm for deciding embeddability: One starts with an arbitrary map f: K → R2k, which generically has finitely many double points; if k ≥ 3 and if the obstruction vanishes then one can successively remove these double points by local modifications of the map f. One of the main tools is the famous Whitney trick that permits eliminating pairs of double points of opposite intersection sign. We are interested in generalizing this approach to intersection points of higher multiplicity. We call a point y ∈ Rd an r-fold Tverberg point of a map f: Kk → Rd if y lies in the intersection f(σ1)∩…∩ f(σr) of the images of r pairwise disjoint simplices of K. The analogue of (non-)embeddability that we study is the problem Tverberg r k→d: Given a k-dimensional complex K, does it satisfy a Tverberg-type theorem with parameters r and d, i.e., does every map f:Kk → Rd have an r-fold Tverberg point? Here, we show that for fixed r, k and d of the form d = rm and k = (r−1)m, m ≥ 3, there is a polynomial-time algorithm for deciding this (based on the vanishing of a cohomological obstruction, as in the case of embeddings). Our main tool is an r-fold analogue of the Whitney trick: Given r pairwise disjoint simplices of K such that the intersection of their images contains two r-fold Tverberg points y+ and y− of opposite intersection sign, we can eliminate y+ and y− by a local isotopy of f. In a subsequent paper, we plan to develop this further and present a generalization of the classical Haefliger--Weber Theorem (which yields a necessary and sufficient condition for embeddability of k-complexes into R, for a wider range of dimensions) to intersection points of higher multiplicity.

Eliminating Tverberg Points, I. An Analogue of the Whitney Trick

Introduction.- Simplicial (co)homology.- Singular and cellular (co)homologies.- Products.- Poincar'e Duality.- Projective spaces.- Equivariant cohomology.- Steenrod squares.- Stiefel-Whitney classes.- Miscellaneous applications and developments.- Hints and answers for some exercises.

Mod Two Homology and Cohomology

In this work, we have proved a number of purely geometric statements by algebraic methods. Also we have proved Sylvester’s law of Nullity and Exercise: the nullity of the product BA never exceeds the sum of the nullities of the factor and is never less than the nullity of A. Keywords: Transformation of Groups, Nullity, Kernel, Image, Non-Singular, Symmetry Group, Shear, Compression, Elongation Reflection Journal of the Nigerian Association of Mathematical Physics , Volume 20 (March, 2012), pp 27 – 30

Transformation of Groups

: From the studies of HIV/AIDS transmission and treatment dynamics using mathematical modeling, literature reviews have shown that attention had not been given to the behavioral attitude of screen-aware infectives not ready to receive treatment, HIV-aware infectives that initiated treatment but truncated only to resume treatment later (therapy abuse) and those on consistent treatment protocols. Moreso, following the non-outright eradication of the deadly HI-virus, recommendations have been geared towards exploring optimal control theory for the maximization of healthy uninfected CD4 + T-cells. Therefore, this present investigation seeks and formulated an optimal control 6-Dimensional deterministic mathematical dynamic model, which accounted for the Role of Antiretroviral Therapy (ART) abuse in the treatment dynamics of the HIV/AIDS epidemic. The materials and methods for this model are constituted by a set of 6-Dimensional varying subpopulations interacting with concentrated HI-viral load. Interactions are investigated using bilinear control functions (condom use and ART) with empirically generated data. The model assumed a deterministic approach and was formulated using the fundamental theory of differential equations. Theoretical optimal predictions explored classical numerical methods with optimal control techniques (Pontryagin's maximum principle in conjunction with Hessian matrix) as a basis. Numerical simulations were conducted using in-built Runge-Kutta of the order of precision 4 in a Mathcad surface. Following the derived model for both off-optimal control and onset-optimal control functions and model optimal control pair as well as model optimality system, results of simulations indicated that at off-optimal control function, near zero population extinction was observed. From the application of optimal control functions under optimal control techniques, there exists tremendous rejuvenation of susceptible populations vindicated by a reduction in the rate of ART abuse under a minimal proportion of bilinear control functions. The study concluded that adopting optimal control techniques for the investigation of the role of ART abuse in HIV/AIDS treatment yield highly significant recovery of healthy CD4 + T-Cells at minimal systemic cost when compared with off-optimal control outcome. Therefore, the study not only affirmed the vital concept of optimal control strategy but also, instituted the viability of the model. Thus, this model can be extensively used in Bio-system and applied mathematics.

Adagba O Henry

Papers

Transformation of Groups

Optimal Control Techniques for the Role of Antiretroviral Therapy (ART) Abuse in HIV/AIDS Treatment Dynamics