Odd Rank Research Articles

A Gross-Kohnen-Zagier theorem for non-split Cartan curves

Let p be a prime number and let E be rational an elliptic curve of conductor p2 and odd analytic rank. We prove that the positions of its special points arising from non-split Cartan curves and imaginary quadratic fields where p is inert are encoded in the Fourier coefficients of a Jacobi form of weight 6 and lattice index of rank 9, obtaining a result analogous to that of Gross, Kohnen and Zagier.

Maximal commutative subalgebras of a Grassmann algebra

We provide a new approach to the investigation of maximal commutative subalgebras (with respect to inclusion) of Grassmann algebras. We show that finding a maximal commutative subalgebra in Grassmann algebras is equivalent to constructing an intersecting family of subsets of various odd sizes in [Formula: see text] which satisfies certain combinatorial conditions. Then we find new maximal commutative subalgebras in the Grassmann algebra of odd rank [Formula: see text] by constructing such combinatorial systems for odd [Formula: see text]. These constructions provide counterexamples to conjectures made by Domoskos and Zubor.

On quantum numbers for Rarita–Schwinger fields

We consider first order linear operators commuting with the operator appearing in the linearized equation of motion of Rarita–Schwinger fields which comes directly from the action. First we consider a simplified operator giving an equation equivalent to the original equation, and classify first order operators commuting with it in four dimensions. In general such operators are symmetry operators of the original operator, but we find that some of them commute with it. We extend this result in four dimensions to arbitrary dimensions and give first order commuting operators constructed of odd rank Killing–Yano and even rank closed conformal Killing–Yano tensors with additional conditions.

Remarks on higher-rank ACM bundles on hypersurfaces

In terms of the number of generators, one of the simplest non-split rank-3 arithmetically Cohen–Macaulay bundles on a smooth hypersurface in P5 is 6-generated. We prove that a general hypersurface in P5 of degree d≥3 does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form OX(c) for some integer c. This verifies some cases of conjectures we discuss here.

A resolution of non-uniqueness puzzle of periodic orbits in the 2-dim anisotropic Kepler problem: bifurcation U → S + U′

Using binary coding of orbit we introduce a finite level (N) surface over the initial value domain D of 2-dim AKP. It gives a tiling of D by base ribbons. The scheme of the one-time map is studied and the properness of the tiling is proved. This analysis in turn resolves the long standing puzzle in AKP—the non-uniqueness issue of a PO for a given code. We argue that the unique existence of a periodic orbit (PO) for a given binary code generally holds (for inverse anisotropy parameter γ < 8/9) but there is a remarkable exception in which a ribbon with a certain code escapes from shrinking at large N and embodies the Broucke-type stable PO (S). It comes along the bifurcation of an unstable PO (U): U(R) → S(R) + U′(NR) (R for retracing and NR for non-retracing). An analysis based on orbit topology clarifies the pattern of the bifurcation; we give a conjecture that it occurs among odd rank Y-symmetric POs.

ODD SYMMETRIC TENSORS, AND AN ANALOGUE OF THE LEVI-CIVITA CONNECTION FOR ODD SYMPLECTIC STRUCTURE

We consider odd Poisson (odd symplectic) structure on supermanifolds induced by an odd symmetric rank 2 (non-degenerate) contravariant tensor field. We describe the difference between odd Riemannian and odd symplectic structure in terms of the Cartan prolongation of the corresponding Lie algebras, and formulate an analogue of the Levi- Civita theorem for an odd symplectic supermanifold.

Effects of a progressive muscle relaxation intervention on dementia symptoms, activities of daily living, and immune function in group home residents with dementia in Japan.

AimTo evaluate the effects of progressive muscle relaxation on the behavioral and psychological symptoms of dementia, activities of daily living, and immune function of elderly patients with dementia in group homes.MethodsThe participants were ranked by their group home unit. Odd ranks were assigned to the intervention group and even ranks to the control group. The intervention group participated in progressive muscle relaxation for 15 min each day for 90 days in the group environment; the control group members continued with their normal routine. All the participants’ secretory immunoglobulin A was measured and they were assessed with the Neuropsychiatric Inventory‐Nursing Home version, Nishimura Mental State Scale for the Elderly, and Nishimura Activities of Daily Living Scale.ResultsThe intervention group comprised 18 participants from six units and the control group comprised 19 participants from five units. After the intervention, the Neuropsychiatric Inventory scores were significantly better in the intervention group, particularly for Agitation and Anxiety. The intervention group also showed significantly lower Apathy and Irritability scores and significant improvement in the Interest, Volition, and Social relationships scores on the Mental State Scale, with improvement in the activities of daily living total. However, there was no difference in the secretory immunoglobulin A level between the groups.ConclusionThe results suggest that progressive muscle relaxation improves the behavioral and psychological symptoms of dementia and activities of daily living in group home residents with dementia, but does not affect their immune function.

Summation Paths in Clenshaw-Curtis Quadrature

Two topics concerning the use of Clenshaw-Curtis quadrature within the Bayesian automatic adaptive quadrature approach to the numerical solution of Riemann integrals are considered. First, it is found that the efficient floating point computation of the coefficients of the Chebyshev series expansion of the integrand is to be done within a mathematical structure consisting of the union of coefficient families ordered into complete binary trees. Second, the scrutiny of the decay rates of the involved even and odd rank Chebyshev expansion coefficients with the increase of their rank labels enables the definition of Bayesian decision paths for the advancement to the numerical output.

Singular moduli and the distribution of partition ranks modulo 2

AbstractIn this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve

An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions.

One-level density of families of elliptic curves and the Ratios Conjecture

Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then determine the underlying symmetry types of the families. The one-level scaling density for the first family corresponds to the orthogonal distribution as predicted by the conjectures of Katz and Sarnak, and the one-level scaling density for the second family is the sum of the Dirac distribution and the even orthogonal distribution. This is a new phenomenon for a family of curves with odd rank: the trivial zero at the central point accounts for the Dirac distribution, and also affects the remaining part of the scaling density which is then (maybe surprisingly) the even orthogonal distribution. The one-level density for this family was studied in the past for test functions with Fourier transforms of limited support, but since the Fourier transforms of the even orthogonal and odd orthogonal distributions are undistinguishable for small support, it was not possible to identify the distribution with those techniques. This can be done with the Ratios Conjecture, and it sheds more light on “independent” and “non-independent” zeroes, and the repulsion phenomenon.

Erratum to: On odd rank integral quadratic forms: canonical representatives of projective classes and explicit construction of integral classes with square-free determinant

Two rank \(n\), integral quadratic forms \(f\) and \(g\) are said projectively equivalent if there exist nonzero rational numbers \(r\) and \(s\) such that \(rf\) and \(sg\) are rationally equivalent. Two odd dimensional, integral quadratic forms \(f\) and \(g\) are projectivelly equivalent if and only if their adjoints are rationally equivalent. We prove that a canonical representative of each projective class of forms of odd rank, exists and is unique up to genus (integral equivalence for indefinite forms). We give a useful characterization of this canonical representative. An explicit construction of integral classes with square-free determinant is given. As a consequence, two tables of ternary and quinary integral quadratic forms of index \(1\) and with square-free determinant are presented.

Ranks of complex skew symmetric operators and applications to Toeplitz operators

We study the rank of complex skew symmetric operators on separable Hilbert spaces. We prove that a finite rank complex skew symmetric operator can't have an odd rank. As applications, we show that any finite rank commutator of two Toeplitz operators on the pluriharmonic Bergman space of the ball can't have an odd rank. We also show that for any positive even integer N, there are two Toeplitz operators whose commutator is exactly of rank N. Also we obtain the similar result for certain truncated Toeplitz operators.

X-ray Natural Circular Dichroism

Natural Circular Dichroism was only recently discovered in the x-ray range[1]. This effect stems from the interference terms which mix multipole transition moments of opposite parity: the Electric Dipole-Electric Quadrupole (E1.E2) and the Electric Dipole-Magnetic Dipole (E1.M1) exist only in structures with broken space inversion symmetry. The scalar E1.M1 term known to be responsible for Circular Dichroism at optical wavelengths is usually considered to be vanishingly small for core level spectroscopies. The E1.E2 interference term, on the contrary, can be large in the X-ray region, but it is a parity odd second rank tensor and therefore observable only in 13 non-centrosymmetric crystal classes. X-ray Natural Circular Dichroism has now been detected in the XANES region for several uniaxial and biaxial crystals. It can give access to the absolute configuration of chiral absorbing centers. On the other hand, Chiral-EXAFS, i.e. the analog of Magnetic-EXAFS for Natural Circular Dichroism has also been measured recently using a uniaxial optically active crystal of paratellurite (TeO2). Chiral-EXAFS originates from symmetry allowed multiple scattering paths. In this presentation, we wish to report on recent advances in X-ray natural circular dichroism and its applications. Determination of absolute configuration is illustrated with measurements of both E1.E2 and E1.M1 terms in chiral alpha-Ni(H2O)6·SO4 single crystals. Manifestation of X-ray optical acitivity in magnetoelectric crystals will be illustrated with various dichroisms measured at the Fe K-edge in multiferroic GaFeO3 crystal. Finally, we will review briefly the perspectives open by our experiments.

On odd rank integral quadratic forms: canonical representatives of projective classes and explicit construction of integral classes with square-free determinant

On odd rank integral quadratic forms: canonical representatives of projective classes and explicit construction of integral classes with square-free determinant

Mesonic Differential Forms

When M is a differentiable manifold, the exterior differential k -forms on M are the alternate k -linear forms on the tangent bundle T(M) . The mesonic differential k -forms are the k -linear forms on T(M) that are alternate with respect to the variables of odd rank, and also alternate with respect to the variables of even rank. After a reminder about meson algebras, and after the presentation of elementary properties of mesonic forms, this article introduces the mesonic differentiation of mesonic forms, which can be partially compared to the exterior differentiation of exterior forms. Some applications to riemannian manifolds and flat manifolds follow.

Symmetric doubly dual hyperovals have an odd rank

We show that a symmetric, doubly dual hyperoval has an odd rank. This is a weak support for the conjecture that doubly dual hyperovals over $$\mathbb{F }_2$$ F 2 only exist, if the rank of the dual hyperoval is odd (see [2]).

On unitary subsectors of polycritical gravities

We study higher-derivative gravity theories in arbitrary space-time dimension d with a cosmological constant at their maximally critical points where the masses of all linearized perturbations vanish. These theories have been conjectured to be dual to logarithmic conformal field theories in the (d-1)-dimensional boundary of an AdS solution. We determine the structure of the linearized perturbations and their boundary fall-off behaviour. The linearized modes exhibit the expected Jordan block structure and their inner products are shown to be those of a non-unitary theory. We demonstrate the existence of consistent unitary truncations of the polycritical gravity theory at the linearized level for odd rank.

Ranks of commutators of Toeplitz operators on the harmonic Bergman space

We study the rank of commutators of two Toeplitz operators on the harmonic Bergman space of the unit disk. We first show that the commutator of any two Toeplitz operators with general symbols can’t have an odd rank. But, given any integer n ≥ 0, we also show that there are two symbols for which the corresponding Toeplitz operators induce the commutator with rank 2n exactly.

Unitary truncations and critical gravity: a toy model

Abstract We investigate a higher-derivative scalar field model in a fixed d + 1 dimensional AdS background as a toy model for a gravitational dual to a higher-rank logarithmic CFT. The holographic two-point correlation functions on the boundary agree with higher-rank LCFT correlation functions. For odd rank, the theory allows for a truncation to a nontrivial subspace with non-negative scalar product. We discuss possible implications for higherderivative critical gravity theories.