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.

Second-order derivative of domain-dependent functionals along Nehari manifold trajectories

Assume that a family of domain-dependent functionals EΩt possesses a corresponding family of least energy critical points ut which can be found as (possibly nonunique) minimizers of EΩt over the associated Nehari manifold N(Ωt). We obtain a formula for the second-order derivative of EΩt with respect to t along Nehari manifold trajectories of the form αt(u0(Φt−1(y)) + tv(Φt−1(y))), y ∈ Ωt, where Φt is a diffeomorphism such that Φt(Ω0) = Ωt, αt ∈ ℝ is a N(Ωt)-normalization coefficient, and v is a corrector function whose choice is fairly general. Since EΩt [ut] is not necessarily twice differentiable with respect to t due to the possible nonuniqueness of ut, the obtained formula represents an upper bound for the corresponding second superdifferential, thereby providing a convenient way to study various domain optimization problems related to EΩt. An analogous formula is also obtained for the first eigenvalue of the p-Laplacian. As an application of our results, we investigate the behaviour of the first eigenvalue of the Laplacian with respect to particular perturbations of rectangles.

On the formula of Cohen–Vogt relatively pointed topological semi-simplicial sets

In the papers [1] and [6], for an inverse sequence of pointed topological spaces and fibrations preserving the base points E=E_{1}\xleftarrow{p_{1}}E_{2}\xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}E_{m+1}, there exists an exact sequence *\rightarrow{\varprojlim}^{(1)}[X,\Omega E_{m}]\rightarrow[X,\varprojlim E]% \rightarrow{\varprojlim}^{(1)}[X,E_{m}]\rightarrow*. In the present paper, for an inverse sequence of pointed topological semi-simplicial sets and fibrations preserving base points \underline{E}=\underline{E}{}_{1}\xleftarrow{p_{1}}\underline{E}{}_{2}% \xleftarrow{p_{2}}\cdots\xleftarrow{p_{m}}\underline{E}{}_{m+1}\xleftarrow{% \hphantom{p_{1}}}\cdots, an analogous formula is proved.

Detailed study of the Λb → Λcτν̄τ decay

We examine in detail the semileptonic decay [Formula: see text], which may confirm previous hints, from the analogous [Formula: see text] decay, of a new physics beyond the Standard Model. First of all, starting from rather general assumptions, we predict the partial width of the decay. Then we analyze the effects of five possible new physics interactions, adopting in each case five different form factors. In particular, for each term beyond the Standard Model, we find some constraints on the strength and phase of the coupling, which we combine with those found by other authors in analyzing the analogous semileptonic decays of [Formula: see text]. Our analysis involves some dimensionless quantities, substantially independent of the form factor, but which, owing to the constraints, turn out to be strongly sensitive to the kind of nonstandard interaction. We also introduce a criterion thanks to which one can discriminate among the various new physics terms: the left-handed current and the two-Higgs-doublet model appear privileged, with a neat preference for the former interaction. Finally, we suggest a differential observable that could, in principle, help to distinguish between the two cases.

Gravitational microlensing of an elliptical source near a fold caustic

We consider the amplification factor for the luminosity of an extended source near the fold caustic of the gravitational lens. It is assumed that the source has elliptical shape, and the brightness distribution along the radial directions is Gaussian. During the microlensing event the total brightness of all microimages is observed, which changes when the source moves relative to the caustic. The main contribution to the variable component is given by the so-called critical images that arise/disappear at the intersection of the caustic by the source. In the present paper we obtained an analogous formula for elliptical Gaussian source. The formula involves a dependence on the coordinates of the source centre, its geometric dimensions, and its orientation relative to the caustic. We show that in the linear caustic approximation the amplification of the circular and elliptical sources is described by the same (rescaled) formula. However, in the next approximations the differences are significant. We compare analytical calculations of the amplification curves for different orientations of an elliptical source and for a circular source with the same luminosity for the model example.

Turán Numbers for 3-Uniform Linear Paths of Length 3

In this paper we confirm a special, remaining case of a conjecture of Füredi, Jiang, and Seiver, and determine an exact formula for the Turán number $\mathrm{ex}_3(n; P_3^3)$ of the 3-uniform linear path $P^3_3$ of length 3, valid for all $n$. It coincides with the analogous formula for the 3-uniform triangle $C^3_3$, obtained earlier by Frankl and Füredi for $n\ge 75$ and Csákány and Kahn for all $n$. In view of this coincidence, we also determine a `conditional' Turán number, defined as the maximum number of edges in a $P^3_3$-free 3-uniform hypergraph on $n$ vertices which is not $C^3_3$-free.

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [24] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an [Formula: see text] analog of the Masur–Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [16] give a strictly finer invariant in the analogous [Formula: see text] setting, we determine which of the 21 connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in [Formula: see text]. The rank 2 case is actually a direct consequence of the work of Masur and Smillie, as all elements of [Formula: see text] are induced by surface homeomorphisms and the index list and ideal Whitehead graph for a surface homeomorphism give precisely the same data.

Using Continued Fractions to Compute Iwasawa Lambda Invariants of Imaginary Quadratic Number Fields

Let l>3 be a prime such that l≡3 (mod 4) and Q(l) has class number 1. Then Hirzebruch and Zagier noticed that the class number of Q(-l) can be expressed as h(-l)=(1/3)(b1+b2+⋯+bm)-m where the bi are partial quotients in the “minus” continued fraction expansion l=[[b0;b1,b2,…,bm¯]]. For an odd prime p≠l, we prove an analogous formula using these bi which computes the sum of Iwasawa lambda invariants λp(-l)+λp(-4) of Q(-l) and Q(-1). In the case that p is inert in Q(-l), the formula pleasantly simplifies under some additional technical assumptions.

Product formula for p -adic epsilon factors

Let $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.