Counting Arithmetical Structures on Paths and Cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles

Counting Arithmetical Structures on Paths and Cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag (d) - A) r=0 , where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag (d) - A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients ((2n-1)/(n-1)) , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles

More on GG-flux and General Hodge Cycles on the Fermat Sextic

We study M-Theory solutions with $G$-flux on the Fermat sextic Calabi-Yau fourfold, focussing on the relationship between the number of stabilized complex structure moduli and the tadpole contribution of the flux. We use two alternative approaches to define the fluxes: algebraic cycles and (appropriately quantized) Griffiths residues. In both cases, we collect evidence for the non-existence of solutions which stabilize all moduli and stay within the tadpole boundComment: v2: typos corrected and references adde

The calcium goes meow : effects of ions and glycosylation on Fel d 1, the major cat allergen

The major cat allergen, Fel d 1, is a structurally complex protein with two N-glycosylation sites that may be filled by different glycoforms. In addition, the protein contains three putative Ca2+ binding sites. Since the impact of these Fel d 1 structure modifications on the protein dynamics, physiology and pathology are not well established, the present work employed computational biology techniques to tackle these issues. While conformational effects brought upon by glycosylation were identified, potentially involved in cavity volume regulation, our results indicate that only the central Ca2+ion remains coordinated to Fel d 1 in biological solutions, impairing its proposed role in modulating phospholipase A2 activity. As these results increase our understanding of Fel d 1 structural biology, they may offer new support for understanding its physiological role and impact into cat-promoted allergy

Shock-Boundary Layer Interactions in Supersonic Turbine Cascades

The physics of shock-boundary layer interactions in a supersonic turbine cascade is investigated through a wall-resolved large eddy simulation. Special attention is given to the characterization of the low-frequency dynamics of the separation bubbles using flow visualization, spectral analysis, space-time cross correlations, and flow modal decomposition. The mean flowfield shows different shock structures formed on both sides of the airfoil. On the suction side, an oblique shock impinges on the turbulent boundary layer, whereas a Mach reflection interacts with the pressure side boundary layer. Instantaneous flow visualizations illustrate elongated streamwise structures on the incoming boundary layers and their interactions with the shocks and separation bubbles. The passage of high-speed (low-speed) streaks through the recirculation bubbles leads to the downstream (upstream) motion of the separation point on both suction and pressure sides, resulting in spanwise modulation of the bubbles. Space-time cross-correlations reveal that the near-wall streaks drive the suction side separation bubble motion, which in turn promotes the oscillations of the reattachment shock and shear layer flapping. Space-time correlations also indicate the existence of a $\pi$ phase jump in the pressure fluctuations along the separation bubble on the suction side. After this phase jump, a downstream propagating pressure disturbance is observed, while prior to this point, the pressure disturbances dominantly propagate in the upstream direction. Finally, the organized motions in the shock-boundary layer interactions and their corresponding characteristic frequencies are identified using proper orthogonal decomposition.Comment: 40 pages, 19 figures. Submitted to Physical Review Fluid

UPPERCARE: a community aware environment for post-surgical musculoskeletal recovery of elderly patients

Trabalho apresentado na Conferência Internacional realizada em Wellington, Nova Zelândia, de 26-28 de abril de 2017Disability from musculoskeletal diseases and comorbidities may lead to the worsening of social and economic well-being through a multitude of paths. Moreover since in European Union (EU) Member States it is projected that those aged 65 and over will become a much larger share (rising from 17% to 30% of the population), and those aged 80 and over (rising from 5% to 12%) will almost become as numerous as the young population in 2060, there is a great potential for Information and Communication Technologies (ICT) solutions for addressing the present and future living arrangements in older people. The UPPERCARE system is meant to affect positively both the intergenerational and partners care since it contributes to decrease usability barriers and promote collaborative environments for informal and self-care. UPPERCARE is a new approach for integrated care supported by ICT systems and services, focusing on post-operative rehabilitation of musculoskeletal pathologies, having as a case study the knee post-operative scenarios of prosthetic care. This paper presents the UPPERCARE system, that provides an integrated care solution, supported ICT, for empowering self-care and adherence to rehabilitation plans through natural interfaces, gamification and cross-modal paths for community care collaboration. The system addresses current barriers from technological, clinical, social and organisational perspectives in a multidisciplinary environment. Special attention is given to the patients’ needs and behaviours entailing the participation of a wide care community, including clinical and non-clinical people, associations, institutions and authorities) through an user driven interaction within the system.This work was supported by Project ”NanoSTIMA: Macroto-Nano Human Sensing: Towards Integrated Multimodal Health Monitoring and Analytics/NORTE-01-0145-FEDER- 000016” financed by the North Portugal Regional Operational Programme (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement, and through the European Regional Development Fund (ERDF).info:eu-repo/semantics/publishedVersio

Counting arithmetical structures on paths and cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d)-A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)-A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1,n-1), and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles