Rm End Research Articles

Noninvasive Evaluation of Central Haemodynamic Parameters

Ageing is one of the relevant cardiovascular risk factors. Non-invasive assessment of cardiac function is an important approach in preventive medicine. The aim of our study was to evaluate the effect of age and gender on central haemodynamics. Seventy-six clinically healthy individuals were divided into four groups according to their age and gender. Applanation tonometry was applied using Sphygmocor device (AtCor, Australia). Pulse wave analysis was utilized for non-invasive assessment of central pressures (systolic, diastolic, mean, pulse, augmentation and end-systolic). Several important derivative indices such as augmentation index AIx75 $$(\frac{\rm Augmentation\ pressure}{\rm Pulse\ pressure}\times 100$$, corrected for heart rate 75 beats/minute), arterial elastance (Ea,$$\frac{\rm End - systolic\ pressure}{\rm Stroke\ volume})$$ and wasted effort (Ew) were computed. Ew is the ventricular force that is generated to overcome the late systolic peak and does not contribute to an increased stroke volume. Arterial stiffening was confirmed by the significantly higher central systolic and pulse pressures, AIx75 and arterial elastance in the middle-aged groups ($$P<0.01$$). Those alterations were most pronounced in middle-aged females in support to the concept for a postmenopausal effect of female gender beyond ageing. We have shown a positive correlation between Ew and Ea valid only for the middle-aged individuals ($$R=0.70$$, $$P<0.001$$) that coincided with similar correlation between Ea and AIx75 for the same age groups ($$R=0.53$$, $$P<0.001$$). Our supposition is that arterial stiffening contributes to the elevation of the `pulsatile' afterload thus impairing vascular/ventricular coupling and increasing myocardial load and oxygen demands. In conclusion, applanation tonometry and PWA provide reliable measures of central haemodynamics and myocardial afterload.

On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

Let G be an additive finite abelian group and \(\Gamma \subset {\rm End} (G)\) be a subset of the endomorphism group of G. A sequence \(S = g_1 \cdot \ldots \cdot g_{\ell}\) over G is a (\(\Gamma\)-)weighted zero-sum sequence if there are \(\gamma_1, \ldots, \gamma_{\ell} \in \Gamma\) such that \(\gamma_1 (g_1) + \ldots + \gamma_{\ell} (g_{\ell})=0\). We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group \(\Gamma\)) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.

On the entropy and index of the winding endomorphisms of $p$-adic ring $C^*$-algebras

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $\chi_k\in{\rm End}(\mathcal{Q}_p)$ by setting $\chi_k(U):=U^k$ and $\chi_k(S_p):=S_p$. We then compute the entropy of $\chi_k$, which turns out to be $\log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $\chi_k$ showing that the entropy is the natural logarithm of the index.

Independence algebras, basis algebras and the distributivity condition

Stable basis algebras were introduced by Fountain and Gould and developed in a series of articles. They form a class of universal algebras, extending that of independence algebras, and reflecting the way in which free modules over well-behaved domains generalise vector spaces. If a stable basis algebra $$\mathbb{B}$$ satisfies the distributivity condition (a condition satisfied by all the previously known examples), it is a reduct of an independence algebra $$\mathbb{A}$$ . Our first aim is to give an example of an independence algebra not satisfying the distributivity condition. Gould showed that if a stable basis algebra $$\mathbb{B}$$ with the distributivity condition has finite rank, then so does the independence algebra $$\mathbb{A}$$ of which it is a reduct, and that in this case the endomorphism monoid $${\rm End}(\mathbb{B})$$ of $$\mathbb{B}$$ is a left order in the endomorphism monoid $${\rm End}(\mathbb{A})$$ of $$\mathbb{A}$$ . We complete the picture by determining when $${\rm End}(\mathbb{B})$$ is a right, and hence a two-sided, order in $${\rm End}(\mathbb{A})$$ . In fact (for rank at least 2), this happens precisely when every element of $${\rm End}(\mathbb{A})$$ can be written as $${\alpha}^{\sharp} \beta$$ where $$\alpha,\beta\in{\rm End}(\mathbb{B})$$ , $${\alpha}^{\sharp}$$ is the inverse of $$\alpha$$ in a subgroup of $${\rm End}(\mathbb{A})$$ and $$\alpha$$ and $$\beta$$ have the same kernel. This is equivalent to $${\rm End}(\mathbb{B})$$ being a special kind of left order in $${\rm End}(\mathbb{A})$$ known as straight.

The Socle of the Last Term in the Minimal Injective Resolution of a Gorenstein Module

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and $_RU$ a Gorenstein module with $S={\rm End}(_RU)$. If the injective dimensions of $_RU$ and $U_S$ are finite, then the last term in the minimal injective resolution of $_{R}U$ has an essential socle.

The spin-Brauer diagram algebra

We investigate the spin-Brauer diagram algebra, denoted ${\bf SB}_n(\delta)$, that arises from studying an analogous form of Schur-Weyl duality for the action of the pin group on ${\bf V}^{\otimes n} \otimes \Delta$. Here ${\bf V}$ is the standard $N$-dimensional complex representation of ${\bf Pin}(N)$ and $\Delta$ is the spin representation. When $\delta = N$ is a positive integer, we define a surjective map ${\bf SB}_n(N) \twoheadrightarrow {\rm End}_{{\bf Pin}(N)}({\bf V}^{\otimes n} \otimes \Delta)$ and show it is an isomorphism for $N \geq 2n$. We show ${\bf SB}_n(\delta)$ is a cellular algebra and use cellularity to characterize its irreducible representations.

Relative cluster tilting objects in triangulated categories

Assume that $\D$ is a Krull-Schmidt, Hom-finite triangulated category with a Serre functor and a cluster-tilting object $T$. We introduce the notion of relative cluster tilting objects, and $T[1]$-cluster tilting objects in $\D$, which are a generalization of cluster-tilting objects. When $\D$ is $2$-Calabi-Yau, the relative cluster tilting objects are cluster-tilting. Let $\la={\rm End}^{op}_{\D}(T)$ be the opposite algebra of the endomorphism algebra of $T$. We show that there exists a bijection between $T[1]$-cluster tilting objects in $\D$ and support $\tau$-tilting $\la$-modules, which generalizes a result of Adachi-Iyama-Reiten \cite{AIR}. We develop a basic theory on $T[1]$-cluster tilting objects. In particular, we introduce a partial order on the set of $T[1]$-cluster tilting objects and mutation of $T[1]$-cluster tilting objects, which can be regarded as a generalization of `cluster-tilting mutation'. As an application, we give a partial answer to a question posed in \cite{AIR}.

The Infinitesimal Moduli Space of Heterotic G2 Systems

Heterotic string compactifications on integrable G2 structure manifolds Y with instanton bundles {(V,A), (TY,tilde{theta})} yield supersymmetric three-dimensional vacua that are of interest in physics. In this paper, we define a covariant exterior derivative {mathcal{D}} and show that it is equivalent to a heterotic G2 system encoding the geometry of the heterotic string compactifications. This operator {mathcal{D}} acts on a bundle {mathcal{Q}=T^*Y oplus {rm End}(V) oplus {rm End}(TY)} and satisfies a nilpotency condition {check{{mathcal{D}}}^2=0} , for an appropriate projection of {mathcal D}. Furthermore, we determine the infinitesimal moduli space of these systems and show that it corresponds to the finite-dimensional cohomology group {check H^1_{check{{mathcal{D}}}}(mathcal{Q})}. We comment on the similarities and differences of our result with Atiyah’s well-known analysis of deformations of holomorphic vector bundles over complex manifolds. Our analysis leads to results that are of relevance to all orders in the {alpha'} expansion.

Modular Curvature and Morita Equivalence

The curvature of the noncommutative torus $T^2_\theta$ ($\theta$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_\theta = C(T^2_\theta)$ with other toric algebras $A_{\theta'}$. In the enlarged context the conformal metric on $T^2_\theta$ is exchanged with an arbitrary Hermitian metric on the Heisenberg $(A_\theta, A_{\theta'})$-bimodule $E'$ for which ${\rm End}_{A_{\theta'}}(E') = A_\theta $. We prove that the Ray-Singer log-determinant of the corresponding Laplacian, viewed as a functional on the space of all Hermitian metrics on $E'$, attains its extremum at the unique Hermitian metric whose corresponding connection has constant curvature. The gradient of the log-determinant functional gives rise to a noncommutative analogue of the Gaussian curvature. The genuinely new outcome of this paper is that the latter is shown to be independent of any Heisenberg bimodule $E'$ such that $A_\theta = {\rm End}_{A_{\theta'}}(E')$, and in this sense it is Morita invariant. To prove the above results we extend Connes' pseudodifferential calculus to Heisenberg modules. The twisted version, which offers more flexibility even in the case of trivial coefficients, could potentially be applied to other problems in the elliptic theory on noncommutative tori. A noteworthy technical feature is that we systematize the computation of the resolvent expansion for elliptic differential operators on noncommutative tori to an extent which makes the (previously employed) computer assistance unnecessary.

Endomorphism rings of some Young modules

Let $\Sigma_r$ be the symmetric group acting on $r$ letters, $K$ be a field of characteristic 2 and $\lambda$ and $\mu$ be partitions of $r$ in at most two parts. Denote the permutation module corresponding to the Young subgroup $\Sigma_\lambda$ in $\Sigma_r$ by $M^\lambda$, and the indecomposable Young module by $Y^\mu$. We give an explicit presentation of the endomorphism algebra ${\rm End}_{K[\Sigma_r]}(Y^\mu)$, using the idempotents found by Doty, Erdmann and Henke in [1].

Investigating the global collapse of filaments using smoothed particle hydrodynamics

We use Smoothed Particle Hydrodynamic simulations of cold, uniform density, self-gravitating filaments, to investigate their longitudinal collapse timescales; these timescales are important because they determine the time available for a filament to fragment into cores. A filament is initially characterised by its line-mass, $\mu$, its radius, $R$ (or equivalently its density $\rho\!=\!\mu/\pi R^2$), and its aspect ratio, $A\;\,(\equiv Z/R$, where $Z$ is its half-length). The gas is only allowed to contract longitudinally, i.e. parallel to the symmetry axis of the filament (the $z$-axis). Pon et al. (2012) have considered the global dynamics of such filaments analytically. They conclude that short filaments ($A\! < \!5$) collapse along the $z$-axis more-or-less homologously, on a time-scale $t_{_{\rm HOM}} \sim 0.44\,A\,(G\rho)^{-1/2}$; in contrast, longer filaments ($A\! > \!5$) undergo end-dominated collapse, i.e. two dense clumps form at the ends of the filament and converge on the centre sweeping up mass as they go, on a time-scale $t_{_{\rm END}} \sim 0.98\,A^{1/2}\,(G\rho)^{-1/2}$. Our simulations do not corroborate these predictions. First, for all $A\! > \!2$, the collapse time satisfies a single equation \[t_{_{\rm COL}}\;\sim\;(0.49+0.26A)(G\rho)^{-1/2}\,,\] which for large $A$ is much longer than the Pon et al. prediction. Second, for all $A\! > \!2$, the collapse is end-dominated. Third, before being swept up, the gas immediately ahead of an end-clump is actually accelerated outwards by the gravitational attraction of the approaching clump, resulting in a significant ram pressure. For high aspect ratio filaments the end-clumps approach an asymptotic inward speed, due to the fact that they are doing work both accelerating and compressing the gas they sweep up. Pon et al. appear to have neglected the outward acceleration and its consequences.

Constructing supersingular elliptic curves with a given endomorphism ring

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathcal{O}$ be a maximal order in the quaternion algebra $B_p$ over $\mathbb{Q}$ ramified at $p$ and $\infty $. The paper is about the computational problem: construct a supersingular elliptic curve $E$ over $\mathbb{F}_p$ such that ${\rm End}(E) \cong \mathcal{O}$. We present an algorithm that solves this problem by taking gcds of the reductions modulo $p$ of Hilbert class polynomials.New theoretical results are required to determine the complexity of our algorithm. Our main result is that, under certain conditions on a rank three sublattice $\mathcal{O}^T$ of $\mathcal{O}$, the order $\mathcal{O}$ is effectively characterized by the three successive minima and two other short vectors of $\mathcal{O}^T\! .$ The desired conditions turn out to hold whenever the $j$-invariant $j(E)$, of the elliptic curve with ${\rm End}(E) \cong \mathcal{O}$, lies in $\mathbb{F}_p$. We can then prove that our algorithm terminates with running time $O(p^{1+\varepsilon })$ under the aforementioned conditions.As a further application we present an algorithm to simultaneously match all maximal order types with their associated $j$-invariants. Our algorithm has running time $O(p^{2.5 + \varepsilon })$ operations and is more efficient than Cerviño’s algorithm for the same problem.

Braidings of Tensor Spaces

Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V.

On the arithmetic of the endomorphisms ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$

For a prime number p, Bergman (Israel J Math 18:257–277, 1974) established that $${{\rm End}(\mathbb{Z}_{p}\times \mathbb{Z}_{p^{2}})}$$ is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$ with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the first row belong to $${\mathbb{Z}_{p}}$$ and the elements in the second row belong to $${\mathbb{Z}_{p^{2}}}$$; also, using the arithmetic in $${\mathbb{Z}_{p}}$$ and $${\mathbb{Z}_{p^{2}}}$$, we introduce the arithmetic in that ring and prove that the ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$ is isomorphic to the ring E p . Finally, we present a Diffie-Hellman key interchange protocol using some polynomial functions over E p defined by polynomials in $${\mathbb{Z}[X]}$$.

Strongly tilting truncated path algebras

For any truncated path algebra Λ, we give a structural description of the modules in the categories \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) and \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) , consisting of the finitely generated (resp. arbitrary) Λ-modules of finite projective dimension. We deduce that these categories are contravariantly finite in Λ−mod and Λ-Mod, respectively, and determine the corresponding minimal \({\mathcal{P}^{<\infty}}\) -approximation of an arbitrary Λ-module from a projective presentation. In particular, we explicitly construct—based on the Gabriel quiver Q and the Loewy length of Λ—the basic strong tilting module ΛT (in the sense of Auslander and Reiten) which is coupled with \({\mathcal{P}^{<\infty}(\Lambda\text{-}{\rm mod})}\) in the contravariantly finite case. A main topic is the study of the homological properties of the corresponding tilted algebra \({\tilde{\Lambda} = {\rm End}_\Lambda(T)^{\rm op}}\) , such as its finitistic dimensions and the structure of its modules of finite projective dimension. In particular, we characterize, in terms of a straightforward condition on Q, the situation where the tilting module \({T_{\tilde{\Lambda}}}\) is strong over \({\tilde{\Lambda}}\) as well. In this Λ-\({\tilde{\Lambda}}\)-symmetric situation, we obtain sharp results on the submodule lattices of the objects in \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) , among them a certain heredity property; it entails that any module in \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) is an extension of a projective module by a module all of whose simple composition factors belong to \({\mathcal{P}^{<\infty}({\rm Mod}\text{-}\tilde{\Lambda})}\) .

Gradings and Derived Categories

Let \({{\mathcal G}}\) be a group, Λ a \({{\mathcal G}}\)-graded Artin algebra and gr(Λ) denote the category of finitely generated \({{\mathcal G}}\)-graded Λ-modules. This paper provides a framework that allows an extension of tilting theory to \({{\mathcal D}}^b(\rm gr(\Lambda))\) and to study connections between the tilting theories of \({{\mathcal D}}^b(\Lambda)\) and \({{\mathcal D}}^b(\rm gr(\Lambda))\). In particular, using that if T is a gradable Λ-module, then a grading of T induces a \({{\mathcal G}}\)-grading on EndΛ(T), we obtain conditions under which a derived equivalence induced by a gradable Λ-tilting module T can be lifted to a derived equivalence between the derived categories \({{\mathcal D}}^b(\rm gr(\Lambda))\) and \({{\mathcal D}}^b(\rm gr(\rm End_{\Lambda}(\textit T)))\).

MODULES WITH FI-EXTENDING HULLS

AbstractIt is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension $H_{\frak{FI}}(P_R)$ (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))\cong {\rm End}(H_{\frak{FI}}(P_R))$, where $\widehat{Q}_{\frak{FI}}({\rm End}(P_R))$ is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

Frobenius fields for elliptic curves

Let $E/{\Bbb Q}$ be an elliptic curve over the field of rational numbers, with ${\mathop{\rm End}}_{\bar{{\Bbb Q}}}(E) = {\Bbb Z}$. Let $K$ be a fixed imaginary quadratic field over ${\Bbb Q}$, and $x$ a positive real number. For each prime $p$ of good reduction for $E$, let $\pi_p(E)$ be a root of the characteristic polynomial of the Frobenius endomorphism of $E$ over the finite field ${\Bbb F}_p$. Let $\Pi_{E}(K; x)$ be the number of primes $p \leq x$ such that the field extension ${\Bbb Q}(\pi_p(E))$ is the fixed imaginary quadratic field $K$. We present upper bounds for $\Pi_{E}(K; x)$ obtained using two different approaches. The first one, inspired from work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to $K$ and to the elliptic curve $E$. The second one, inspired from work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained using the first approach are better, $\Pi_E(K; x) \ll x^{4/5}/(\log x)^{1/5}$, and are the best known so far. The bounds obtained using the second approach are weaker, but are independent of the number field $K$, a property which is essential for other applications. All results are conditional upon GRH.

Projective toric varieties as fine moduli spaces of quiver representations

This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional algebra $\mathop{\rm End}\nolimits( \textstyle\bigoplus\nolimits_{i=0}^{r} L_i )$ where ${\cal L} := ({\scr O}_X,L_1, \ldots, L_r)$ is a list of line bundles on a projective toric variety $X$. The quiver $Q$ defines a smooth projective toric variety, called the multilinear series $|{\cal L}|$, and a map $X \longrightarrow |{\cal L}|$. We provide necessary and sufficient conditions for the induced map to be a closed embedding. As a consequence, we obtain a new geometric quotient construction of projective toric varieties. Under slightly stronger hypotheses on ${\cal L}$, the closed embedding identifies $X$ with the fine moduli space of stable representations for the bound quiver $(Q,R)$.

On Endomorphisms of Quantum Tensor Space

We give a presentation of the endomorphism algebra $${\rm End}_{\mathcal {U}_{q}(\mathfrak {sl}_{2})}(V^{\otimes r})$$ , where V is the three-dimensional irreducible module for quantum $${\mathfrak {sl}_2}$$ over the function field $${\mathbb {C}(q^{\frac{1}{2}})}$$ . This will be as a quotient of the Birman–Wenzl–Murakami algebra BMW r (q) : = BMW r (q −4, q 2 − q −2) by an ideal generated by a single idempotent Φ q . Our presentation is in analogy with the case where V is replaced by the two-dimensional irreducible $${\mathcal {U}_q(\mathfrak {sl}_{2})}$$ -module, the BMW algebra is replaced by the Hecke algebra H r (q) of type A r-1, Φ q is replaced by the quantum alternator in H 3(q), and the endomorphism algebra is the classical realisation of the Temperley–Lieb algebra on tensor space. In particular, we show that all relations among the endomorphisms defined by the R-matrices on $${V^{\otimes r}}$$ are consequences of relations among the three R-matrices acting on $${V^{\otimes 4}}$$ . The proof makes extensive use of the theory of cellular algebras. Potential applications include the decomposition of tensor powers when q is a root of unity.