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We propose a modular framework for temporal disaggregation of quarterly GDP into monthly frequency, in which the regression step accommodates any supervised learning model while Mariano-Murasawa reconciliation enforces quarterly consistency. Comparing Chow-Lin, Elastic Net, XGBoost, and a Multi-Layer Perceptron across four countries, we find that regularization, not nonlinearity, drives the gains: Elastic Net achieves $R^2 = 0.87$ for the United States when lagged indicators are included, while nonlinear models cannot overcome the variance cost of small quarterly samples. We formalize this tradeoff through regime-switching bias and ridge-regularization results.

This paper presents an introduction and expository account of a beautiful, current, and active application of recursions to the computation of resistance distance. Resistance distance, also referred to as effective resistance, is a well-known graph metric that arises naturally by considering a graph as an electrical circuit; heuristically resistance distance measures both the number of paths between two vertices in a graph and the cost of each path. This topic finds applications in a rich array of fields including social, biological, ecological, and transportation networks, chemistry, graph theory, numerical linear algebra, and engineering. A variety of methods are used in the field to determine resistance distance including recursive, mathematical, and graphical techniques. Sequences familiar to the readers of the Fibonacci Quarterly such as the Fibonacci and Lucas sequences appear quite often in results in the literature. Twenty five to forty years ago there were a handful of papers on resistance that appeared in the Fibonacci Quarterly and the Proceedings and recently papers on the subject have appeared again. It is hoped that this introductory expository account will interest r

The use of drones offers police forces potential gains in efficiency and safety. However, their use may also harm public perception of the police if drones are refused. Therefore, police forces should consider the perception of bystanders and broader society to maximize drones' potential. This article examines the concerns expressed by members of the public during a field trial involving 52 test participants. Analysis of the group interviews suggests that their worries go beyond airspace safety and privacy, broadly discussed in existing literature and regulations. The interpretation of the results indicates that the perceived justice of drone use is a significant factor in acceptance. Leveraging the concept of organizational justice and data collected, we propose a catalogue of guidelines for just operation of drones to supplement the existing policy. We present the organizational justice perspective as a framework to integrate the concerns of the public and bystanders into legal work. Finally, we discuss the relevance of justice for the legitimacy of the police's actions and provide implications for research and practice.

We consider the minimum distance projection in the $L_2$-norm from an arbitrary point in an $n$-dimensional, Euclidian space onto the canonical simplex. It is shown that this problem reduces to a univariate problem that can be solved by a simple algorithm. This optimization problem occurs in the setting of credit risk, where one has stochastic matrices that describe transition probabilities between different credit ratings, and one wants to determine the roots of these matrices, or close approximations to them.

We formulate and study the negative gradient flow of an energy functional of Spin(7)-structures on compact $8$-manifolds. The energy functional is the $L^2$-norm of the torsion of the Spin(7)-structure. Our main result is the short-time existence and uniqueness of solutions to the flow. We also explain how this negative gradient flow contains, as the highest order terms, all independent second order differential invariants of Spin(7)-structures which can be made into an admissible $4$-form. We also study solitons of the flow and prove a non-existence result for compact expanding solitons.

We discuss general properties of strong G$_2$-structures with torsion and we investigate the twisted G$_2$ equation, which represents the G$_2$-analogue of the twisted Calabi-Yau equation for SU$(n)$-structures introduced by Garcia-Fernández - Rubio - Shahbazi - Tipler. In particular, we show that invariant strong G$_2$-structures with torsion do not occur on compact non-flat solvmanifolds. This implies the non-existence of non-trivial solutions to the twisted Calabi-Yau equation on compact solvmanifolds of dimensions $4$ and $6$. More generally, we prove that a compact, connected homogeneous space admitting invariant strong G$_2$-structures with torsion is diffeomorphic either to $S^3 \times T^4$ or to $S^3 \times S^3 \times S^1$, up to a covering, and that in both cases solutions to the twisted G$_2$ equation exist. Finally, we discuss the behavior of the homogeneous Laplacian coflow for strong G$_2$-structures with torsion on these spaces.

We compute the stable cohomology of moduli spaces of hyperelliptic curves of a fixed genus embedded on a fixed Hirzebruch surface. We also describe these moduli spaces of embedded hyperelliptic curves in terms of moduli spaces of pointed non-embedded hyperelliptic curves.

Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness. These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.

We continue the investigation of general geometric flows of $G_2$-structures initiated by the third author in "Flows of $G_2$-structures, I." Specifically, we determine the possible geometric flows (up to lower order terms) of $G_2$-structures which are second order quasilinear, by explicitly computing all independent second order differential invariants of $G_2$-structures which are $3$-forms. There are four symmetric $2$-tensors and two vector fields. We do this by deriving explicit computational descriptions of the decompositions of the curvature and the covariant derivative of the torsion into irreducible $G_2$-representations, as well as the decomposition of the $G_2$-Bianchi identity into independent relations. We also show that these six tensors arise as leading order contributions to the Euler-Lagrange equations for the energy functionals of the four independent torsion components, and we establish a $G_2$-analogue of the classical block decomposition of the Riemann curvature operator on oriented $4$-dimensional Riemannian manifolds. Finally, we present a large class of geometric flows of $G_2$-structures which are directly amenable to a deTurck type trick to establish shor

In this survey article, we present two applications of surface curvatures in theoretical physics. The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy. In this formalism, the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics. We first show that there is an obstruction, arising from the spontaneous curvature, to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori. We then propose a scale-invariant anisotropic bending energy, which extends the Canham energy, and show that it possesses a unique toroidal energy minimizer, up to rescaling, in all parameter regime. Furthermore, we establish some genus-dependent topological lower and upper bounds, which are known to be lacking with the Helfrich energy, for the proposed energy. We also present the shape equation in our context, which extends the Helfrich shape equation. The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmi

Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples.

A bi-variant theory $\mathbb B(X,Y)$ defined for a pair $(X,Y)$ is a theory satisfying properties similar to those of Fulton--MacPherson's bivariant theory $\mathbb B(X \xrightarrow f Y)$ defined for a morphism $f:X \to Y$. In this paper, using correspondences we construct a bi-variant algebraic cobordism $Ω^{*,\sharp}(X, Y)$ such that $Ω^{*,\sharp}(X, pt)$ is isomorphic to Lee--Pandharipande's algebraic cobordism of vector bundles $Ω_{-*,\sharp}(X)$. In particular, $Ω^{*}(X, pt)=Ω^{*, 0}(X, pt)$ is isomorphic to Levine--Morel's algebraic cobordism $Ω_{-*}(X)$. Namely, $Ω^{*,\sharp}(X, Y)$ is \emph{a bi-variant vesion} of Lee--Pandharipande's algebraic cobordism of bundles $Ω_{*,\sharp}(X)$.

In this note we present the solution of Problem H-691 (The Fibonacci Quarterly, 50 (1) 2012) with some corrections and more details. The solution involves three nontrivial integrals whose evaluations are given here.

Google tries balancing AI data center emissions with clean energy efforts

Scientists have found that staple-shaped particles can tangle together to create a material that is both strong and flexible。 Unlike conventional materials, these particles can be locked into a sturdy structure or rapidly unraveled using vibrations。 The unusual behavior could open the door to recyclable buildings, reconfigurable structures, and eve

Astronomers have released the largest gravitational wave catalog ever, revealing 161 new black hole collisions and pushing the total number of detections to 390。 Among the highlights are the clearest gravitational wave signal ever recorded, the most accurate location of a black hole merger, and growing evidence that some black holes are the product

T-Mobile wants Broadcom to keep supporting its VMware perpetual licenses

A distant galaxy nicknamed Shadow Blaster may have revealed a surprising source of cosmic neutrinos: extreme star formation instead of a supermassive black hole。 The discovery suggests that hidden, dust-filled starburst galaxies could account for a significant fraction of the Universe’s high-energy neutrinos