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OBJECTIVES To evaluate the diagnostic performance of N-terminal pro-B-type natriuretic peptide (NT-proBNP) thresholds for acute heart failure and to develop and validate a decision support tool that combines NT-proBNP concentrations with clinical characteristics. DESIGN Individual patient level data meta-analysis and modelling study. SETTING Fourteen studies from 13 countries, including randomised controlled trials and prospective observational studies. PARTICIPANTS Individual patient level data for 10 369 patients with suspected acute heart failure were pooled for the meta-analysis to evaluate NT-proBNP thresholds. A decision support tool (Collaboration for the Diagnosis and Evaluation of Heart Failure (CoDE-HF)) that combines NT-proBNP with clinical variables to report the probability of acute heart failure for an individual patient was developed and validated. MAIN OUTCOME MEASURE Adjudicated diagnosis of acute heart failure. RESULTS Overall, 43.9% (4549/10 369) of patients had an adjudicated diagnosis of acute heart failure (73.3% (2286/3119) and 29.0% (1802/6208) in those with and without previous heart failure, respectively). The negative predictive value of the guideline recommended rule-out threshold of 300 pg/mL was 94.6% (95% confidence interval 91.9% to 96.4%); despite use of age specific rule-in thresholds, the positive predictive value varied at 61.0% (55.3% to 66.4%), 73.5% (62.3% to 82.3%), and 80.2% (70.9% to 87.1%), in patients aged 75 years, respectively. Performance varied in most subgroups, particularly patients with obesity, renal impairment, or previous heart failure. CoDE-HF was well calibrated, with excellent discrimination in patients with and without previous heart failure (area under the receiver operator curve 0.846 (0.830 to 0.862) and 0.925 (0.919 to 0.932) and Brier scores of 0.130 and 0.099, respectively). In patients without previous heart failure, the diagnostic performance was consistent across all subgroups, with 40.3% (2502/6208) identified at low probability (negative predictive value of 98.6%, 97.8% to 99.1%) and 28.0% (1737/6208) at high probability (positive predictive value of 75.0%, 65.7% to 82.5%) of having acute heart failure. CONCLUSIONS In an international, collaborative evaluation of the diagnostic performance of NT-proBNP, guideline recommended thresholds to diagnose acute heart failure varied substantially in important patient subgroups. The CoDE-HF decision support tool incorporating NT-proBNP as a continuous measure and other clinical variables provides a more consistent, accurate, and individualised approach. STUDY REGISTRATION PROSPERO CRD42019159407.

We compute the “norm” of irreducible uniformly bounded representations of SL(2,R). We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in its similarity class of uniformly bounded representations.

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal

We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.

A distribution on a Heisenberg type group of homogeneous dimension Q a biradial kernel of type _it coincides with a biradial function, homogeneous of degree _ a － Q, and smooth away from the identity. We prove that a distribution is a biradial kernel of type a_, 0 _< a _ < Q, if and only if its Gelfand transform, defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous of degree -a/2. A similar result holds for radial kernels on the Heisenberg group.

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ObjectivesTo evaluate the diagnostic performance of N-terminal pro-B-type natriuretic peptide (NT-proBNP) thresholds for acute heart failure and to develop and validate a decision support tool that combines NT-proBNP concentrations with clinical characteristics.DesignIndividual patient level data meta-analysis and modelling study.SettingFourteen studies from 13 countries, including randomised controlled trials and prospective observational studies.ParticipantsIndividual patient level data for 10 369 patients with suspected acute heart failure were pooled for the meta-analysis to evaluate NT-proBNP thresholds. A decision support tool (Collaboration for the Diagnosis and Evaluation of Heart Failure (CoDE-HF)) that combines NT-proBNP with clinical variables to report the probability of acute heart failure for an individual patient was developed and validated.Main outcome measureAdjudicated diagnosis of acute heart failure.ResultsOverall, 43.9% (4549/10 369) of patients had an adjudicated diagnosis of acute heart failure (73.3% (2286/3119) and 29.0% (1802/6208) in those with and without previous heart failure, respectively). The negative predictive value of the guideline recommended rule-out threshold of 300 pg/mL was 94.6% (95% confidence interval 91.9% to 96.4%); despite use of age specific rule-in thresholds, the positive predictive value varied at 61.0% (55.3% to 66.4%), 73.5% (62.3% to 82.3%), and 80.2% (70.9% to 87.1%), in patients aged 75 years, respectively. Performance varied in most subgroups, particularly patients with obesity, renal impairment, or previous heart failure. CoDE-HF was well calibrated, with excellent discrimination in patients with and without previous heart failure (area under the receiver operator curve 0.846 (0.830 to 0.862) and 0.925 (0.919 to 0.932) and Brier scores of 0.130 and 0.099, respectively). In patients without previous heart failure, the diagnostic performance was consistent across all subgroups, with 40.3% (2502/6208) identified at low probability (negative predictive value of 98.6%, 97.8% to 99.1%) and 28.0% (1737/6208) at high probability (positive predictive value of 75.0%, 65.7% to 82.5%) of having acute heart failure.ConclusionsIn an international, collaborative evaluation of the diagnostic performance of NT-proBNP, guideline recommended thresholds to diagnose acute heart failure varied substantially in important patient subgroups. The CoDE-HF decision support tool incorporating NT-proBNP as a continuous measure and other clinical variables provides a more consistent, accurate, and individualised approach.Study registrationPROSPERO CRD42019159407.

Let $(G,K)$ be a Gelfand pair, with $G$ a Lie group of polynomial growth, and let $\Sigma\subset{\mathbb R}^\ell$ be a homeomorphic image of the Gelfand spectrum, obtained by choosing a generating system $D_1,\dots,D_\ell$ of $G$-invariant differential operators on $G/K$ and associating to a bounded spherical function $\varphi$ the $\ell$-tuple of its eigenvalues under the action of the $D_j$'s. We say that property (S) holds for $(G,K)$ if the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto ${\mathcal S}(\Sigma)$, the space of restrictions to $\Sigma$ of the Schwartz functions on ${\mathbb R}^\ell$. This property is known to hold for many nilpotent pairs, i.e., Gelfand pairs where $G=K\ltimes N$, with $N$ nilpotent. In this paper we enlarge the scope of this analysis outside the range of nilpotent pairs, stating the basic setting for general pairs of polynomial growth and then focussing on strong Gelfand pairs.

Let δ be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of δ with a multiple of the identity on L p spaces. © 2012 Elsevier Inc.

We prove that Huygens’ principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek–Ricci spaces. We also prove a Paley–Wiener type theorem for the inverse of the Helgason Fourier transform on Damek–Ricci spaces.

We prove that a Damek–Ricci space is symmetric if and only if the geodesic inversion preserves the set of horocycles.

We prove several Paley–Wiener-type theorems related to the spherical transform on the Gelfand pair $$\big ({H_n}\rtimes {\text {U}(n)},{\text {U}(n)}\big )$$(Hn⋊U(n),U(n)), where $${H_n}$$Hn is the $$2n+1$$2n+1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in $$\mathbb {R}^2$$R2, we prove that spherical transforms of $${\text {U}(n)}$$U(n)-invariant functions and distributions with compact support in $${H_n}$$Hn admit unique entire extensions to $$\mathbb {C}^2$$C2, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.

Let N be an H-type group of homogeneous dimension e. We study the space of biradial Schwartz functions on N by means of the Gelfand transform. This enables us to characterize the class of biradial homogeneous distributions on N of degree alpha, with 0 less than or equal to alpha < Q, which are C-infinity away from the identity, via the Gelfand transform

Let H1 be the 3-dimensional Heisenberg group. We prove that a modi-fied version of the spherical transform is an isomorphism between the space Sm(H1) of Schwartz functions of type m and the space S(Σm) consisting of restrictions of Schwartz functions on R2 to a subset Σm of the Heisenberg fan with jmj of the half-lines removed. This result is then applied to study the case of general Schwartz functions on H1.

The generalised Cayley transform C \mathcal {C} from an Iwasawa N N -group into the corresponding real unit sphere S \mathbb {S} induces isomorphisms between suitable Sobolev spaces H α ( S ) \mathcal {H}^\alpha (\mathbb {S}) and H α ( N ) \mathcal {H}^\alpha (N) . We study the differential of C \mathcal {C} , and we obtain a criterion for a function to be in H α ( S ) \mathcal {H}^\alpha (\mathbb {S}) .

Let σ be the geodesic inversion on a Heisenberg type group N with homogeneous dimension Q, and denote by S the jacobian of σ. We prove that, for $ - \frac{1}{2}Q< \alpha< \frac{1}{2}Q$ , the operators $T_\alpha :f \mapsto S^{1/2 - \alpha /Q} (f \circ \sigma )$ are bounded on certain homogeneous Sobolev spaces $\mathcal{H}^\alpha (N)$ if and only if N is an Iwasawa N-group.

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.