Odd dimensional analogue of the Euler characteristic

When compact manifolds X and Y are both even dimensional, their Euler characteristics obey the Künneth formula χ(X × Y) = χ(X)χ(Y). In terms of the Betti numbers bp(X), χ(X) = Σp(−1)pbp(X), implying that χ(X) = 0 when X is odd dimensional. We seek a linear combination of Betti numbers, called ρ, that obeys an analogous formula ρ(X × Y) = χ(X)ρ(Y) when Y is odd dimensional. The unique solution is ρ(Y) = − Σp(−1)ppbp(Y). Physical applications include: (1) ρ → (−1)mρ under a generalized mirror map in d = 2m + 1 dimensions, in analogy with χ → (−1)mχ in d = 2m; (2) ρ appears naturally in compactifications of M-theory. For example, the 4-dimensional Weyl anomaly for M-theory on X4× Y7 is given by χ(X4)ρ(Y7) = ρ(X4× Y7) and hence vanishes when Y7 is self-mirror. Since, in particular, ρ(Y × S1) = χ(Y), this is consistent with the corresponding anomaly for Type IIA on X4× Y6, given by χ(X4)χ(Y6) = χ(X4× Y6), which vanishes when Y6 is self-mirror; (3) In the partition function of p-form gauge fields, ρ appears in odd dimensions as χ does in even.

ADHD and ODD Dimensions: Time Varying Prediction of Internalizing Problems from Childhood to Adolescence

Objective Although childhood ADHD is a risk factor for internalizing problems, it consists of separable inattention and hyperactivity dimensions that differentially predict outcomes. Oppositional defiant disorder also consists of separable dimensions (i.e., irritable, oppositional), co-occurs with ADHD, and predicts internalizing outcomes. To discern independent associations with internalizing problems, dimensions must be considered simultaneously. Methods Controlling for age, sex, and race, we tested inattention, hyperactivity, irritability, and oppositionality as time-varying predictors of 6 to 7-year prospective change in parent- and teacher-rated internalizing problems in 230 ethnically- diverse (50% Caucasian) 5 to 10 year old youth ( M = 7.4 years, 68% male) with ( n = 120) and without ADHD ( n = 110). Results Escalating inattention and irritability, but not hyperactivity and oppositionality, uniquely predicted internalizing problems. Conclusion These findings suggest that inattention and irritability are unique risk factors for later internalizing problems. These dimensions may catalyze internalizing problems across development and constitute important intervention targets.

Entangled symmetric states and copositive matrices

Entanglement in symmetric quantum states and the theory of copositive matrices are intimately related concepts. For the simplest symmetric states, i.e., the diagonal symmetric (DS) states, it has been shown that there exists a correspondence between exceptional (non-exceptional) copositive matrices and non-decomposable (decomposable) Entanglement Witnesses (EWs). Here we show that EWs of symmetric, but not DS, states can also be constructed from extended copositive matrices, providing new examples of bound entangled symmetric states, together with their corresponding EWs, in arbitrary odd dimensions.

Cusp cobordism group of Morse functions

By a Morse function on a compact manifold with boundary we mean a real-valued function without critical points near the boundary such that its critical points as well as the critical points of its restriction to the boundary are all nondegenerate. For such Morse functions, Saeki and Yamamoto have previously defined a certain notion of cusp cobordism, and computed the unoriented cusp cobordism group of Morse functions on surfaces. In this paper, we compute unoriented and oriented cusp cobordism groups of Morse functions on manifolds of any dimension by employing Levine’s cusp elimination technique as well as the complementary process of creating pairs of cusps along fold lines. We show that both groups are cyclic of order two in even dimensions, and cyclic of infinite order in odd dimensions. For Morse functions on surfaces our result yields an explicit description of Saeki–Yamamoto’s cobordism invariant which they constructed by means of the cohomology of the universal complex of singular fibers.

Parrondo's paradox for homoeomorphisms

We construct two planar homoeomorphisms $f$ and $g$ for which the origin is a globally asymptotically stable fixed point whereas for $f \circ g$ and $g \circ f$ the origin is a global repeller. Furthermore, the origin remains a global repeller for the iterated function system generated by $f$ and $g$ where each of the maps appears with a certain probability. This planar construction is also extended to any dimension $>$ 2 and proves for first time the appearance of the dynamical Parrondo's paradox in odd dimensions.

Charge algebra in Al(A)dSn spacetimes

The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

A correspondence between Ricci-flat Kerr and Kaluza-Klein AdS black hole

We establish an explicit correspondence of Einstein gravity on the squashed spheres that are the U(1) bundles over ℂℙm to the Kaluza-Klein AdS gravity on the tori. This allows us to map the Ricci-flat Kerr metrics in odd dimensions with all equal angular momenta to charged Kaluza-Klein AdS black holes that can be lifted to become singly rotating M-branes and D3-branes. Furthermore, we find maps between Ricci-flat gravitational instantons to the AdS domain walls. In particular the supersymmetric bolt instantons correspond to domain walls that can be interpreted as distributed M-branes and D3-branes, whilst the non-supersymmetric Taub-NUT solutions yield new domain walls that can be lifted to become solutions in M-theory or type IIB supergravity. The correspondence also inspires us to obtain a new superpotential in the Kaluza-Klein AdS gravity in four dimensions.