Some combinatorial identities related to commuting varieties and Hilbert schemes

In this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane

The interaction of a gap with a free boundary in a two dimensional dimer system

Let $\ell$ be a fixed vertical lattice line of the unit triangular lattice in the plane, and let \Cal H be the half plane to the left of $\ell$. We consider lozenge tilings of \Cal H that have a triangular gap of side-length two and in which $\ell$ is a free boundary - i.e., tiles are allowed to protrude out half-way across $\ell$. We prove that the correlation function of this gap near the free boundary has asymptotics $\frac{1}{4\pi r}$, $r\to\infty$, where $r$ is the distance from the gap to the free boundary. This parallels the electrostatic phenomenon by which the field of an electric charge near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te

The influence of increased venous return on right ventricular dyssynchrony during acute and sustained hypoxaemia.

NEW FINDINGS: What is the central question of this study? Right ventricular dyssynchrony is a marker of function that is elevated in healthy individuals exposed to acute hypoxia, but does it remain elevated during sustained exposure to high altitude hypoxia, and can it be normalised by augmenting venous return? What is the main finding and its importance? For the first time it is demonstrated that (i) increasing venous return in acute hypoxia restores the synchrony of right ventricular contraction and (ii) dyssynchrony is evident after acclimatisation to high altitude, and remains sensitive to changes in venous return. Therefore, the interpretation of right ventricular dyssynchrony requires consideration the prevailing haemodynamic state. ABSTRACT: Regional heterogeneity in timing of right ventricular (RV) contraction (RV dyssynchrony; RVD) occurs when pulmonary artery systolic pressure (PASP) is increased during acute hypoxia. Interestingly, RVD is not observed during exercise, a stimulus that increases both PASP and venous return. Therefore, we hypothesised that RVD in healthy humans is sensitive to changes in venous return, and examined whether (i) increasing venous return in acute hypoxia lowers RVD and (ii) if RVD is further exaggerated in sustained hypoxia, given increased PASP is accompanied by decreased ventricular filling at high altitude. RVD, PASP and right ventricular end-diastolic area (RVEDA) were assessed using transthoracic two-dimensional and speckle-tracking echocardiography during acute normobaric hypoxia ( F i O 2  = 0.12) and sustained exposure (5-10 days) to hypobaric hypoxia (3800 m). Venous return was augmented with lower body positive pressure at sea level (LBPP; +10 mmHg) and saline infusion at high altitude. PASP was increased in acute hypoxia (20 ± 6 vs. 28 ± 7, P < 0.001) concomitant to an increase in RVD (18 ± 7 vs. 38 ± 10, P < 0.001); however, the addition of LBPP during hypoxia decreased RVD (38 ± 0 vs. 26 ± 10, P < 0.001). Sustained hypoxia increased PASP (20 ± 4 vs. 26 ± 5, P = 0.008) and decreased RVEDA (24 ± 4 vs. 21 ± 2, P = 0.042), with RVD augmented (14 ± 5 vs. 31 ± 12, P = 0.001). Saline infusion increased RVEDA (21 ± 2 vs. 23 ± 3, P = 0.008) and reduced RVD (31 ± 12 vs. 20 ± 9, P = 0.001). In summary, an increase in PASP secondary to acute and sustained exposure to hypoxia augments RVD, which can be at least partly reduced via increased venous return

Coloured peak algebras and Hopf algebras

For $G$ a finite abelian group, we study the properties of general equivalence relations on G_n=G^n\rtimes \SG_n, the wreath product of $G$ with the symmetric group \SG_n, also known as the $G$-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of \k G_n as well as graded connected Hopf subalgebras of \bigoplus_{n\ge o} \k G_n. In particular we construct a $G$-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or $G$-coloured descent algebra). We show that the direct sum of the $G$-coloured peak algebras is a Hopf algebra. We also have similar results for a $G$-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the $G$-coloured descent Hopf algebra whose image is the $G$-coloured peak Hopf algebra. We outline a theory of combinatorial $G$-coloured Hopf algebra for which the $G$-coloured quasi-symmetric Hopf algebra and the graded dual to the $G$-coloured peak Hopf algebra are central objects.Comment: 26 pages latex2

No heartbreak at high altitude; preserved cardiac function in chronic hypoxia

High altitude hypoxia presents a series of challenges to the human heart due to concomitant changes in preload, afterload and contractility. This challenge is characterised by a decrease in blood volume due to plasma volume constriction, an increase in right ventricular afterload via hypoxic pulmonary vasoconstriction, and an increase in sympathetic nerve activity . As such, understanding how the heart adapts to this multifaceted challenge has been a topic of interest to physiologists and clinicians for decades. In the current issue of Experimental Physiology, Maufrais et al. (2019) use modern speckle tracking technology to investigate region-specific cardiac performance in chronic hypoxia

On the uniqueness of promotion operators on tensor products of type A crystals

The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.Comment: 31 pages; 8 figure

The independent effects of hypovolemia and pulmonary vasoconstriction on ventricular function and exercise capacity during acclimatisation to 3800 m

We aimed to determine the isolated and combined contribution of hypovolemia and hypoxic pulmonary vasoconstriction in limiting left ventricular (LV) function and exercise capacity under chronic hypoxemia at high altitude. In a double‐blinded, randomized and placebo‐controlled design, twelve healthy participants underwent echocardiography at rest and during submaximal exercise before completing a maximal test to exhaustion at sea level (SL; 344 m) and after 5–10 days at 3800 m. Plasma volume was normalised to SL values, and hypoxic pulmonary vasoconstriction was reversed by administration of Sildenafil (50 mg) to create four unique experimental conditions that were compared with SL values; high altitude (HA), Plasma Volume Expansion (HA‐PVX), Sildenafil (HA‐SIL) and Plasma Volume Expansion with Sildenafil (HA‐PVX‐SIL). High altitude exposure reduced plasma volume by 11% (P < 0.01) and increased pulmonary artery systolic pressure (19.6 ± 4.3 vs. 26.0 ± 5.4, P < 0.001); these differences were abolished by PVX and SIL respectively. LV end‐diastolic volume (EDV) and stroke volume (SV) were decreased upon ascent to high altitude, but were comparable to sea level in the HA‐PVX. LV EDV and SV were also elevated in the HA‐SIL and HA‐PVX‐SIL trials compared to HA, but to a lesser extent. Neither PVX or SIL had a significant effect on the LV EDV and SV response to exercise, or the maximal oxygen consumption or peak power output. In summary, at 3800 m both hypovolemia and hypoxic pulmonary vasoconstriction contribute to the decrease in LV filling, however, restoring LV filling does not confer an improvement in maximal exercise performance

The Tchebyshev transforms of the first and second kind

We give an in-depth study of the Tchebyshev transforms of the first and second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform (of the first kind) preserves desirable combinatorial properties, including Eulerianess (due to Hetyei) and EL-shellability. It is also a linear transformation on flag vectors. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg and Readdy omega map of oriented matroids. One consequence is that nonnegativity of the cd-index is maintained. The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on the space of quasisymmetric functions QSym. It coincides with Stembridge's peak enumerator for Eulerian posets, but differs for general posets. The complete spectrum is determined, generalizing work of Billera, Hsiao and van Willigenburg. The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's classical quasisymmetric function of a poset, this map is a comodule morphism with respect to the quasisymmetric functions QSym. Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps. One such occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and Sottile's result on the terminal object in the category of combinatorial Hopf algebras. In contrast, the chain map of the first kind is both an algebra map and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page

The effect of an acute bout of resistance exercise on carotid artery strain and strain rate

Arterial wall mechanics likely play an integral role in arterial responses to acute physiological stress. Therefore, this study aimed to determine the impact of low and moderate intensity double-leg press exercise on common carotid artery (CCA) wall mechanics using 2D vascular strain imaging. Short-axis CCA ultrasound images were collected in 15 healthy men (age: 21 ± 3 years; stature: 176.5 ± 6.2 cm; body mass; 80.6 ± 15.3 kg) before, during, and immediately after short-duration isometric double-leg press exercise at 30% and 60% of participants’ one-repetition maximum (1RM: 317 ± 72 kg). Images were analyzed for peak circumferential strain (PCS), peak systolic and diastolic strain rate (S-SR and D-SR) and arterial diameter. Heart rate (HR), systolic and diastolic blood pressure (SBP and DBP) were simultaneously assessed and arterial stiffness indices were calculated post hoc. A two-way repeated measures ANOVA revealed that during isometric contraction, PCS and S-SR decreased significantly (P < 0.01) before increasing significantly above resting levels post-exercise (P < 0.05 and P < 0.01 respectively). Conversely, D-SR was unaltered throughout the protocol (P = 0.25). No significant differences were observed between the 30% and 60% 1RM trials. Multiple regression analysis highlighted that HR, BP and arterial diameter did not fully explain the total variance in PCS, S-SR and D-SR. Acute double-leg press exercise is therefore associated with similar transient changes in CCA wall mechanics at low and moderate intensities. CCA wall mechanics likely provide additional insight into localized intrinsic vascular wall properties beyond current measures of arterial stiffness

Macdonald polynomials in superspace: conjectural definition and positivity conjectures

We introduce a conjectural construction for an extension to superspace of the Macdonald polynomials. The construction, which depends on certain orthogonality and triangularity relations, is tested for high degrees. We conjecture a simple form for the norm of the Macdonald polynomials in superspace, and a rather non-trivial expression for their evaluation. We study the limiting cases q=0 and q=\infty, which lead to two families of Hall-Littlewood polynomials in superspace. We also find that the Macdonald polynomials in superspace evaluated at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In particular, their expansion coefficients in the corresponding Hall-Littlewood bases appear to be polynomials in t with nonnegative integer coefficients. More strikingly, we formulate a generalization of the Macdonald positivity conjecture to superspace: the expansion coefficients of the Macdonald superpolynomials expanded into a modified version of the Schur superpolynomial basis (the q=t=0 family) are polynomials in q and t with nonnegative integer coefficients.Comment: 18 page