GEOMETRIC WEIGHT-SHIFTING OPERATORS ON HILBERT MODULAR FORMS IN CHARACTERISTIC p

We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic p, generalising the author’s prior work with Sasaki to the case where p is ramified in the totally real field. In particular, we use the partial Hasse invariants and Kodaira–Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren’s construction of partial $\Theta $ -operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures. Furthermore, we describe the kernels of partial $\Theta $ -operators in terms of images of geometrically constructed partial Frobenius operators. Finally, we apply our results to prove a partial positivity result for minimal weights of mod p Hilbert modular forms.

DEVELOPMENTAL DELAY IN 14 MONTH OLD BOY WITH GRADE I MALNUTRITION

When a child does not meet his or her developmental milestones within the predicted time or age, this is referred to as developmental delay. Developmental milestones are activities that certain children master or acquire at specific ages. Head balance, rolling, crawling, driving, and talking are a few examples. The mechanism of growth and the completion of such developmental milestones are referred to as child development. The patient was a 14 months old male child came to the hospital with complaint of inability to sit also he is unable to hold anything in his hand. As narrated by the mother the child was born to G2P1L1A1 mother at 9 months 5 days of gestation. The child has not yet attained gross motor milestone such as rolling over and no sit with support. The child attained immature pincer grasp at 12 months of age and has not attained mature pincer grasp. Social smile was attained at 6 months, stranger anxiety at 12 months and the child has not attained bye-bye. The child was alert to sounds by the age of 9 months .babbles at 10 months and says mama-dada at 12 months. The child is immunized till age. The marriage of the parents was non-consanguineous marriage. The child was born with a weight of 2.5 kg the mother had LSCS and breech delivery. The child did not cry immediately after birth. The child has history of fever after 3 months of delivery. The child is also interpreted as grade 1 mild malnutritious (71-80) %. Developmental delay and grade 1 mild malnutrition. The above study shows that Neck facilitation exercises in prone lying, Ice technique and proper brush technique for neck facilitation improves neck holding. Rolling facilitation in side lying position and proper handling positions helps in attaining milestones such as Rolling over. Facilitation of extension tone with vestibular rehabilitation helps to facilitate extensors. Family plays an important role in the management of such cases all the exercises should be properly explained to them for more improvement. Supporting the child while sitting, for example, might help him or her develop weight shifting, rotation, coordination, and balance.

Recursion relations for 5-point conformal blocks

We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.

The Cosmological Bootstrap: Spinning correlators from symmetries and factorization

We extend the cosmological bootstrap to correlators involving massless spinning particles, focusing on spin-1 and spin-2. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields with conformally coupled scalars. Four-point functions arising from tree-level exchange are singular in particular kinematic configurations, and the coefficients of these singularities satisfy certain factorization properties. We show that in many cases these factorization limits fix the structure of the correlators uniquely, without having to solve the conformal Ward identities. The additional constraint of locality for massless spinning particles manifests itself as current conservation on the boundary. We find that the four-point functions only satisfy current conservation if the s, t, and u-channels are related to each other, leading to nontrivial constraints on the couplings between the conserved currents and other operators in the theory. For spin-1 currents this implies charge conservation, while for spin-2 currents we recover the equivalence principle from a purely boundary perspective. For multiple spin-1 fields, we recover the structure of Yang--Mills theory. Finally, we apply our methods to slow-roll inflation and derive a few phenomenologically relevant scalar-tensor three-point functions.

Higher spin 3-point functions in 3d CFT using spinor-helicity variables

In this paper we use the spinor-helicity formalism to calculate 3-point functions involving scalar operators and spin-s conserved currents in general 3d CFTs. In spinor-helicity variables we notice that the parity-even and the parity-odd parts of a correlator are related. Upon converting spinor-helicity answers to momentum space, we show that correlators involving spin-s currents can be expressed in terms of some simple conformally invariant conserved structures. This in particular allows us to understand and separate out contact terms systematically, especially for the parity-odd case. We also reproduce some of the correlators using weight-shifting operators.

Momentum space parity-odd CFT 3-point functions

We study the parity-odd sector of 3-point functions comprising scalar operators and conserved currents in conformal field theories in momentum space. We use momentum space conformal Ward identities as well as spin-raising and weight-shifting operators to fix the form of some of these correlators. Wherever divergences appear we discuss their regularisation and renormalisation using appropriate counter-terms.

Compromised Brain Activity With Age During a Game-Like Dynamic Balance Task: Single- vs. Dual-Task Performance

Background: Postural control and cognition are affected by aging. We investigated whether cognitive distraction influenced neural activity differently in young and older adults during a game-like mediolateral weight-shifting task with a personalized task load.Methods: Seventeen healthy young and 17 older adults performed a balance game, involving hitting virtual wasps, serial subtractions and a combination of both (dual-task). A motion analysis system estimated each subject's center of mass position. Cortical activity in five regions was assessed by measuring oxygenated hemoglobin (HbO2) with a functional Near-Infrared Spectroscopy system.Results: When adding cognitive load to the game, weight-shifting speed decreased irrespective of age, but older adults reduced the wasp-hits more than young adults. Accompanying these changes, older adults decreased HbO2 in the left pre-frontal cortex (PFC) and frontal eye fields (FEF) compared to single-tasking, a finding not seen in young adults. Additionally, lower HbO2 levels were found during dual-tasking compared to the summed activation of the two single tasks in all regions except for the right PFC. These relative reductions were specific for the older age group in the left premotor cortex (PMC), the right supplementary motor area (SMA), and the left FEF.Conclusion: Older adults showed more compromised neural activity than young adults when adding a distraction to a challenging balance game. We interpret these changes as competitive downgrading of neural activity underpinning the age-related deterioration of game performance during dual-tasking. Future work needs to ascertain if older adults can train their neural flexibility to withstand balance challenges during daily life activities.