A variation of the prime k-tuples conjecture with applications to quantum limits

Let $$\mathcal {H}^{*}=\{h_1,h_2,\ldots \}$$ H ∗ = { h 1 , h 2 , … } be an ordered set of integers. We give sufficient conditions for the existence of increasing sequences of natural numbers $$a_j$$ a j and $$n_k$$ n k such that $$n_k+h_{a_j}$$ n k + h a j is a sum of two squares for every $$k\ge 1$$ k ≥ 1 and $$1\le j\le k.$$ 1 ≤ j ≤ k . Our method uses a novel modification of the Maynard–Tao sieve together with a second moment estimate. As a special case of our result, we deduce a conjecture due to D. Jakobson which has several implications for quantum limits on flat tori.

Transverse J-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$

J-holomorphic curves in nearly Kähler $$\mathbb {CP}^3$$ CP 3 are related to minimal surfaces in $$S^4$$ S 4 as well as associative submanifolds in $$\Lambda ^2_-(S^4)$$ Λ - 2 ( S 4 ) . We introduce the class of transverse J-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $$S^4$$ S 4 and construct moment-type maps from $$\mathbb {CP}^3$$ CP 3 to relate them to the theory of $$\mathrm {U}(1)$$ U ( 1 ) -invariant minimal surfaces on $$S^4$$ S 4 .

Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori

In this work, we show that an n-dimensional sublattice Λ′=mΛ of an n-dimensional lattice Λ induces a G=Zmn tessellation in the flat torus Tβ′=Rn/Λ′, where the group G is isomorphic to the lattice partition Λ/Λ′. As a consequence, we obtain, via this technique, toric codes of parameters [[2m2,2,m]], [[3m3,3,m]] and [[6m4,6,m2]] from the lattices Z2, Z3 and Z4, respectively. In particular, for n=2, if Λ1 is either the lattice Z2 or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell P0′ of each hexagonal sublattice Λ′ of hexagonal lattices Λ, using either the fundamental cell P0 or the Voronoi cell V0. These partitions allow us to present new classes of toric codes with parameters [[3m2,2,m]] and color codes with parameters [[18m2,4,4m]] in the flat torus from families of hexagonal lattices in R2.

Manifolds with Many Rarita–Schwinger Fields

The Rarita–Schwinger operator is the twisted Dirac operator restricted to $$\nicefrac 32$$ 3 2 -spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

Motivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

PREVALENCE OF ALVEOLAR EXOSTOSIS AMONG EDENTULOUS PATIENTS IN KASHMIRI POPULATION

Background: The most remarkable exostoses of the human jaws are torus palatinus (TP) and torus mandibularis (TM). The aim of the present study was to determine the prevalence of alveolar exostosis among the edentulous patients of Kashmir. Methods: The present study included 200 edentulous subjects aged between 55 and 75 years. The subjects were examined for the existence of tori by clini­cal inspection and palpation. Statistical analysis was performed using SPSS for Windows version. Results: Among the 200 subjects, 41 (20.5%) had prevalence of tori. Out of 41 patients, four subjects (9.75%) had TP and thirty-seven (18.5%) had TM. Prevalence of TP is more in female subjects than males. Out of the 41 subjects, 27 (65.85%) presented with flat tori, nine (21.95%) with spindle-shaped tori, and five (12.19%) with nodular-shaped tori. Most of the tori observed were small sized (92.68%). Conclusion: A comparatively increased prevalence of TM was observed with flat tori being the most common type.