Thermodynamic formalism for transient dynamics on the real line

We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic Z -extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the full dimension spectrum with respect to α-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.

Scaffolding layered control architectures through constraint closure: insights into brain evolution and development

The functional organization of the mammalian brain can be considered to form a layered control architecture, but how this complex system has emerged through evolution and is constructed during development remains a puzzle. Here we consider brain organization through the framework of constraint closure, viewed as a general characteristic of living systems, that they are composed of multiple sub-systems that constrain each other at different timescales. We do so by developing a new formalism for constraint closure, inspired by a previous model showing how within-lifetime dynamics can constrain between-lifetime dynamics, and we demonstrate how this interaction can be generalized to multi-layered systems. Through this model, we consider brain organization in the context of two major examples of constraint closure—physiological regulation and visual orienting. Our analysis draws attention to the capacity of layered brain architectures to scaffold themselves across multiple timescales, including the ability of cortical processes to constrain the evolution of sub-cortical processes, and of the latter to constrain the space in which cortical systems self-organize and refine themselves. This article is part of the theme issue ‘Systems neuroscience through the lens of evolutionary theory’.

Time-independent perturbation theory with Lagrange multipliers

We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Lagrange-multiplier regularization of eigenproblem for Jx

We solve the eigenproblem of the angular momentum $$J_x$$ J x by directly dealing with the non-diagonal matrix unlike the conventional approach rotating the trivial eigenstates of $$J_z$$ J z . Characteristic matrix is reduced into a tri-diagonal form following Narducci–Orszag rescaling of the eigenvectors. A systematic reduction formalism with recurrence relations for determinants of any dimension greatly simplifies the computation of tri-diagonal matrices. Thus the secular determinant is intrinsically factorized to find the eigenvalues immediately. The reduction formalism is employed to find the adjugate of the characteristic matrix. Improving the recently introduced Lagrange-multiplier regularization, we identify that every column of that adjugate matrix is indeed the eigenvector. It is remarkable that the approach presented in this work is completely new and unique in that any information of $$J_z$$ J z is not required and only algebraic operations are involved. Collapsing of the large amount of determinant calculation with the recurrence relation has a wide variety of applications to other tri-diagonal matrices appearing in various fields. This new formalism should be pedagogically useful for treating the angular momentum problem that is central to quantum mechanics course.

Nonclinical study and applicability of the absorbed dose conversion method with a single biodistribution measurement for targeted alpha-nuclide therapy

Background We recently reported a new absorbed dose conversion method, RAP (RAtio of Pharmacokinetics), for 211At-meta-astatobenzylguanidine (211At-MABG) using a single biodistribution measurement, the percent injected dose/g. However, there were some mathematical ambiguities in determining the optimal timing of a single measurement of the percent injected dose/g. Thus, we aimed to mathematically reconstruct the RAP method and to examine the optimal timing of a single measurement. Methods We derived a new formalism of the RAP dose conversion method at time t. In addition, we acquired a formula to determine the optimal timing of a single measurement of the percent injected dose/g, assuming the one-compartment model for biological clearance. Results We investigated the new formalism’s performance using a representative RAP coefficient with radioactive decay weighting. Dose conversions by representative RAP coefficients predicted the true [211At]MABG absorbed doses with an error of 10% or less. The inverses of the representative RAP coefficients plotted at 4 h post-injection, which was the optimal timing reported in the previous work, were very close to the new inverses of the RAP coefficients 4 h post-injection. Next, the behavior of the optimal timing was analyzed by radiolabeled compounds with physical half-lives of 7.2 h and 10 d on various biological clearance half-lives. Behavior maps of optimal timing showed a tendency to converge to a constant value as the biological clearance half-life of a target increased. The areas of optimal timing for both compounds within a 5% or 10% prediction error were distributed around the optimal timing when the biological clearance half-life of a target was equal to that of the reference. Finally, an example of RAP dose conversion was demonstrated for [211At]MABG. Conclusions The RAP dose conversion method renovated by the new formalism was able to estimate the [211At]MABG absorbed dose using a similar pharmacokinetics, such as [131I]MIBG. The present formalism revealed optimizing imaging time points on absorbed dose conversion between two radiopharmaceuticals. Further analysis and clinical data will be needed to elucidate the validity of a behavior map of the optimal timing of a single measurement for targeted alpha-nuclide therapy.

A Novel Framework for Modeling Weakly Lensing Shear Using Kinematics and Imaging at Moderate Redshift

Kinematic weak lensing describes the distortion of a galaxy’s projected velocity field due to lensing shear, an effect recently reported for the first time by Gurri et al. based on a sample of 18 galaxies at z ∼ 0.1. In this paper, we develop a new formalism that combines the shape information from imaging surveys with the kinematic information from resolved spectroscopy to better constrain the lensing distortion of source galaxies and to potentially address systematic errors that affect conventional weak-lensing analyses. Using a Bayesian forward model applied to mock galaxy observations, we model distortions in the source galaxy’s velocity field simultaneously with the apparent shear-induced offset between the kinematic and photometric major axes. We show that this combination dramatically reduces the statistical uncertainty on the inferred shear, yielding statistical error gains of a factor of 2–6 compared to kinematics alone. While we have not accounted for errors from intrinsic kinematic irregularities, our approach opens kinematic lensing studies to higher redshifts where resolved spectroscopy is more challenging. For example, we show that ground-based integral-field spectroscopy of background galaxies at z ∼ 0.7 can deliver gravitational shear measurements with signal-to-noise ratio of ∼1 per source galaxy at 1 arcminute separations from a galaxy cluster at z ∼ 0.3. This suggests that even modest samples observed with existing instruments could deliver improved galaxy cluster mass measurements and well-sampled probes of their halo mass profiles to large radii.

The field-theoretical methods in Lovelock gravity

The field-theoretical methods are used to construct conserved currents and related superpotentials for perturbations on arbitrary backgrounds in the Lovelock gravity. The perturbations are considered as a dynamic field configuration propagating in a given spacetime. The field-theoretical formalism is exact (without approximations) and equivalent to the original metric theory. As Lagrangian based formalism, it allows us to apply the Noether theorem. As a result, we construct conserved currents and superpotentials, where we use arbitrary displacement vectors, not only the Killing ones or other special vectors. The developed formalism is checked in calculating mass of the Schwarzschild-anti-de Sitter (AdS) black hole. The new formalism is adopted to the case of a so-called pure Lovelock gravity, where in the Lagrangian only a one polynomial in Riemannian tensor presents. We construct conserved charges and currents for static and dynamic black holes of the Vaidya type with AdS, dS and flat asymptotics. New properties of the solutions under consideration have been found. The more results are discussed. The first section in your paper

Lagrangian approach to super-sample effects on biased tracers at field level: galaxy density fields and intrinsic alignments

It has been recognized that the observables of large-scale structure (LSS) is susceptible to long-wavelength density and tidal fluctuations whose wavelengths exceed the accessible scale of a finite-volume observation, referred to as the super-sample modes. The super-sample modes modulate the growth and expansion rate of local structures, thus affecting the cosmological information encoded in the statistics of galaxy clustering data. In this paper, based on the Lagrangian perturbation theory, we develop a new formalism to systematically compute the response of a biased tracer of LSS, which is expressed perturbatively in terms of the matter density field of sub-survey modes, to the super-sample modes at the field level. The formalism presented here reproduces the power spectrum responses that have been previously derived, and provides an alternative way to compute statistical quantities with super-sample modes. As an application, we consider the statistics of the intrinsic alignments of galaxies and halos, and derive the field response of the galaxy/halo shape bias to the super-sample modes. Possible impacts of the long-mode contributions on the covariance of the three-dimensional power spectra of the intrinsic alignment are also discussed, and the signal-to-noise ratios are estimated.

Modeling spatial, developmental, physiological, and topological constraints on human brain connectivity

The complex connectivity of nervous systems is thought to have been shaped by competitive selection pressures to minimize wiring costs and support adaptive function. Accordingly, recent modeling work indicates that stochastic processes, shaped by putative trade-offs between the cost and value of each connection, can successfully reproduce many topological properties of macroscale human connectomes measured with diffusion magnetic resonance imaging. Here, we derive a new formalism with the aim to more accurately capture the competing pressures of wiring cost minimization and topological complexity. We further show that model performance can be improved by accounting for developmental changes in brain geometry and associated wiring costs, and by using inter-regional transcriptional or microstructural similarity rather than topological wiring-rules. However, all models struggled to capture topologies spatial embedding. Our findings highlight an important role for genetics in shaping macroscale brain connectivity and indicate that stochastic models offer an incomplete account of connectome organization.