Many of the primal ingredients of convex optimization extend naturally from Euclidean to Hadamard spaces $\unicode{x2014}$ nonpositively curved metric spaces like Euclidean, Hilbert, and hyperbolic spaces, metric trees, and more general CAT(0) cubical complexes. Linear structure, however, and the duality theory it supports are absent. Nonetheless, we introduce a new type of subgradient for convex functions on Hadamard spaces, based on Busemann functions. This notion supports a splitting subgradient method with guaranteed complexity bounds. In particular, the algorithm solves $p$-mean problems in general Hadamard spaces: we illustrate by computing medians in BHV tree space.

Recent work has shown the capability of Large Language Models (LLMs) to solve tasks related to Knowledge Graphs, such as Knowledge Graph Completion, even in Zero- or Few-Shot paradigms. However, they are known to hallucinate answers, or output results in a non-deterministic manner, thus leading to wrongly reasoned responses, even if they satisfy the user's demands. To highlight opportunities and challenges in knowledge graphs-related tasks, we experiment with three distinguished LLMs, namely Mixtral-8x7b-Instruct-v0.1, GPT-3.5-Turbo-0125 and GPT-4o, on Knowledge Graph Completion for static knowledge graphs, using prompts constructed following the TELeR taxonomy, in Zero- and One-Shot contexts, on a Task-Oriented Dialogue system use case. When evaluated using both strict and flexible metrics measurement manners, our results show that LLMs could be fit for such a task if prompts encapsulate sufficient information and relevant examples.

``Do Carroll particles move?'' The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from Carroll boost symmetry which implies dipole conservation, but not conversely. A massless Carroll particle may propagate by following the Hall law, consistently with the partial breaking of the Carroll boost symmetry. The framework is extended to Carroll field theory. In $d=2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parameterized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized Hall law which includes a Zeeman force. This theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure we call exotic photons, which move on the horizon of a Black Hole, giving rise to an anyonic spin-Hall Effect.

The aim of this paper is to give strict fixed point principles for multivalued operators $T:X\rightarrow P(X)$ satisfying some contraction conditions of \'Ciri\' c and of \'Ciri\' c-Reich-Rus type. We are interested, under which conditions, the multi-valued operator has a unique strict fixed point and, additionally, when the sequence of its multi-valued iterates $(T^n(x))_{n\in \mathbb{N}}$ converges to this unique strict fixed point. Moreover, some stability properties, such as data dependence on operator perturbation, Ulam-Hyers stability, well-posedness in the sense of Reich and Zaslavski and Ostrowski property of the strict fixed point problem are established.

This note proves a result on the existence of barycenters in a class of uniformly convex geodesic spaces.

Recent work has shown the capability of Large Language Models (LLMs) to solve tasks related to Knowledge Graphs, such as Knowledge Graph Completion, even in Zero- or Few-Shot paradigms. However, they are known to hallucinate answers, or output results in a non-deterministic manner, thus leading to wrongly reasoned responses, even if they satisfy the user's demands. To highlight opportunities and challenges in knowledge graphs-related tasks, we experiment with three distinguished LLMs, namely Mixtral-8x7b-Instruct-v0.1, GPT-3.5-Turbo-0125 and GPT-4o, on Knowledge Graph Completion for static knowledge graphs, using prompts constructed following the TELeR taxonomy, in Zero- and One-Shot contexts, on a Task-Oriented Dialogue system use case. When evaluated using both strict and flexible metrics measurement manners, our results show that LLMs could be fit for such a task if prompts encapsulate sufficient information and relevant examples.

We provide a concrete example exhibiting marked deviation from the PPN approximation in a modified theory of gravity. Specifically, we derive the exact formula for the Robertson parameter $\gamma$ in Brans-Dicke gravity for compact mass sources, explicitly incorporating the pressure content of these sources. We achieve this by exploiting the $\textit integrability$ of the 00-component of the Brans-Dicke field equation. In place of the conventional PPN result $\gamma_{PPN}=\frac{\omega+1}{\omega+2}$, we obtain the analytical expression $\gamma_{\,exact}=\frac{\omega+1+(\omega+2)\varTheta}{\omega+2+(\omega+1)\varTheta}$ where $\varTheta$ is the ratio of the total pressure $P_\parallel^*+2P_\perp^*$ and total energy $E^*$ contained within the mass source. Our $\textit non\text{-}perturbative$ formula is valid for all field strengths and types of matter comprising the mass source. We draw four key conclusions: (1) The usual $\gamma_{PPN}$ formula is violated in the presence of pressure, viz. when $\varTheta\neq0$, revealing a limitation of the PPN approximation in Brans-Dicke gravity. (2) The PPN result mainly stems from the assumption of pressureless matter. Even in the weak-field star case, non-zero pressure leads to a violation of the PPN $\gamma$ formula. Conversely, the PPN result is a good approximation for low-pressure matter, i.e. when $\varTheta\approx0$, for all field strengths. (3) Observational constraints on $\gamma$ set $\textit joint$ bounds on $\omega$ and $\varTheta$, with the latter representing a global characteristic of a mass source. If the equation of state of matter in the mass source approaches the ultra-relativistic form, entailing $\varTheta\simeq1$, $\gamma_{\,exact}$ converges to 1 $\textit irrespective$ of $\omega$. (4) In a broader context, our findings indicate the latent significance of considering the interior structure of stars in observational astronomy.

In vivo, neurons establish functional connections and preserve information along their synaptic pathways from one information processing stage to the next in a very efficient manner. Paired spiking (PS) enhancement plays a key role by acting as a temporal filter that deletes less informative spikes. We analyzed the spontaneous neural activity evolution in a hippocampal and a cortical network over several weeks exploring whether the same PS coding mechanism appears in neuronal cultures as well. We show that self-organized neural in vitro networks not only develop characteristic bursting activity, but feature robust in vivo-like PS activity. PS activity formed spatiotemporal patterns that started at early days in vitro (DIVs) and lasted until the end of the recording sessions. Initially random-like and sparse PS patterns became robust after three weeks in vitro (WIVs). They were characterized by a high number of occurrences and short inter-paired spike intervals (IPSIs). Spatially, the degree of complexity increased by recruiting new neighboring sites in PS as a culture matured. Moreover, PS activity participated in establishing functional connectivity between different sites within the developing network. Employing transfer entropy (TE) as an information transfer measure, we show that PS activity is robustly involved in establishing effective connectivities. Spiking activity at both individual sites and network level robustly followed each PS within a short time interval. PS may thus be considered a spiking predictor. These findings suggest that PS activity is preserved in spontaneously active in vitro networks as part of a robust coding mechanism as encountered in vivo. We suggest that, presumably in lack of any external sensory stimuli, PS may act as an internal surrogate stimulus to drive neural activity at different developmental stages.

Studying structural brain networks has witnessed significant advancement in recent decades. Findings have revealed a geometric principle, the exponential distance rule (EDR) showing that the number of neurons decreases exponentially with the length of their axons. An EDR based network model explained various characteristics of inter-areal cortical networks in macaques, mice, and rats. The complete connectome of the Drosophila fruit fly has recently been mapped at the neuronal level. Our study demonstrates that the EDR holds true in Drosophila, and the EDR model effectively accounts for numerous binary and weighted properties of neuropil networks, also called projectome. Our study illustrates that the EDR model is a suitable null model for analyzing networks of brain regions, as it captures geometric and physical constraints in very different species. The importance of the null model lies in its ability to facilitate the identification of functionally significant features that are not caused by inevitable geometric constraints, as we illustrate with the pronounced asymmetry of connection weights important for functional hierarchy.

In A&A 412, 35 (2003) Blanchard, Douspis, Rowan-Robinson, and Sarkar (BDRS) slightly modified the primordial fluctuation spectrum and produced an excellent fit to WMAP's CMB power spectrum for an Einstein-de Sitter (EdS) universe, bypassing dark energy. Curiously, they obtained a Hubble value of $H_0\approx46$, in sharp conflict with the canonical range $H_0\sim67-73$. However, we will demonstrate that the reduced value of $H_0\approx46$ achieved by BDRS is fully compatible with the use of variable speed of light in analyzing the late-time cosmic acceleration observed in Type Ia supernovae (SNeIa). In Phys. Lett. B 862, 139357 (2025) we uncovered a hidden aspect in a generic class of scale-invariant actions: the dynamics of the dilaton can induce a variation in the speed of light as $c\propto\chi^{1/2}$, causing $c$ to vary alongside $\chi$ across spacetime. For an EdS universe with varying $c$, besides the effects of cosmic expansion, light waves emitted from distant SNeIa are further subject to a refraction effect, which alters the Lemaitre redshift relation to $1+z=a^{-3/2}$. Based on this new formula, we achieve a fit to the SNeIa Pantheon Catalog exceeding the quality of the $\Lambda$CDM model. Crucially, our approach does not require dark energy and produces $H_0=47.2$ in strong alignment with the BDRS finding of $H_0\approx46$. Hence, BDRS's analysis of the (early-time) CMB power spectrum and our variable-$c$ analysis of the (late-time) Hubble diagram of SNeIa fully agree on two counts: (i) the dark energy hypothesis is avoided, and (ii) $H_0$ is reduced to $\sim47$, which also yields an age $t_0=2/(3H_0)=13.8$ Gy for an EdS universe, without requiring dark energy. Most importantly, we will demonstrate that the late-time acceleration can be attributed to the declining speed of light in an expanding EdS universe, rather than to a dark energy component.

We reveal a novel aspect of scale-invariant actions that allow matter to couple with a dilaton field: $\,$The dynamics of the dilaton can induce variations in the Planck constant $\hbar$ and speed of light $c$. $\,$Our mechanism for generating variable $\hbar$ and $c$ in $\textit{curved}$ spacetimes via the dilaton offers a viable alternative account for late-time cosmic acceleration, bypassing the need for dark energy.

The complex software systems developed nowadays require assessing their quality and proneness to errors. Reducing code complexity is a never-ending problem, especially in today's fast pace of software systems development. Therefore, the industry needs to find a method to determine the qualities of a software system, the degree of difficulty in developing new functionalities, or the system's proneness to errors. One way of measuring and predicting the quality attributes of a software system is to analyse the software metrics values for it and the relationships between them. More precisely, we should study the metrics that measure and determine the degree of complexity of the code. This paper aims to analyse a novel complexity metric, Hybrid Cyclomatic Complexity (HCC) and its efficiency as a feature in a defect prediction model. The main idea behind this new metric is that inherited complexity should play a role in the complexity of a class, hence the need for a metric that calculates the total complexity of a class, taking into account the complexities of its descendants. Moreover, we will present a comparative study between the HCC metric and its two components, the inherited complexity and the actual complexity of a class in the object-oriented context. Since we want this metric to be as valuable as possible, the experiments will use data from open-source projects. One of the conclusions that can be drawn from these experiments is that inherited complexity is not correlated with class complexity. Therefore, HCC can be considered a valid metric from this point of view. Moreover, the evaluation of the efficiency of the prediction models shows us a similar efficiency for HCC and the inherited complexity. Additionally, there is a need for a clear distinction between a class's complexity and its inherited complexity when defining complexity metrics.

We present an exact static spherisymmetric solution for the Brans-Dicke action sourced by a self-gravitating massless Klein-Gordon helicity-0 field. In contrast to the Maxwell electromagnetic field, a Klein-Gordon field possesses an energy-momentum tensor with $\textit{non-vanishing trace}$. Upon a Weyl mapping into the Einstein frame, the transformed Brans-Dicke scalar field takes on the role of a "dilaton" coupled with the Klein-Gordon field. Despite this dilatonic coupling, the field equations of the resulting Einstein-Klein-Gordon-dilaton action are fully soluble when employing the harmonic radial coordinate. The exact solution derived herein can serve as a prototype for future Brans-Dicke gravity studies involving trace-carrying matter fields. Notably, in the limit of infinite $\omega$, the Brans-Dicke scalar field exhibits an anomalous behavior of ${\cal O}\,(1/\sqrt\omega)$ as opposed to ${\cal O}\,(1/\omega)$. As a consequence, the solution converges to a spacetime configuration of General Relativity sourced by the original Klein-Gordon field and a free scalar field, the latter of which is the ${\cal O}\,(1/\sqrt\omega)$ "remnant" of the Brans-Dicke scalar field. Furthermore, we provide a formal mathematical proof substantiating these two conclusions. Although the ${\cal O}\,(1/\sqrt\omega)$ anomaly has been previously discovered for Brans-Dicke vacuum and Brans-Dicke-Maxwell electrovacuum, our findings establish its prevalence in Brans-Dicke gravity $\textit{regardless}$ of the trace of the energy-momentum tensor of the source. Taken together, the ${\cal O}\,(1/\sqrt\omega)$ anomaly challenges the conventional belief in the ${\cal O}\,(1/\omega)$ signature commonly associated with Brans-Dicke gravity. In particular, it may have implications in improving the relativistic corrections to Newtonian gravity beyond the weak-field parametrized post-Newtonian formalism.

This study experimentally examines the effect and changes in the delivered fields, using water-equivalent phantoms with and without titanium (Ti) dental implants positioned along the primary beam path. We measure in detail the composition and spectral-tracking characterization of particles generated in the entrance region of the Bragg curve using high-spatial resolution, spectral and time-sensitive imaging detectors with a pixelated array provided by the ASIC chip Timepix3. A 170 MeV proton beam was collimated and modulated in a polymethyl methacrylate (PMMA). Placing two dental implants at the end of the protons range in the phantom, the radiation was measured using two pixeled detectors with Si sensors. The Timepix3 (TPX3) detectors equipped with silicon sensors measure in detail particle fluxes, dose rates (DR) and linear energy transfer (LET) spectra for resolved particle types. Artificial intelligence (AI) based-trained neural networks (NN) calibrated in well-defined radiation fields were used to analyze and identify particles based on morphology and characteristic spectral-tracking response. The beam was characterized and single-particle tracks were registered and decomposed into particle-type groups. The resulting particle fluxes in both setups are resolved into three main classes of particles: i) protons, ii) electrons and photons iii) ions. Protons are the main particle component responsible for dose deposition. High-energy transfer particles (HETP), namely ions exhibited differences in both dosimetric aspects that were investigated: DR and particle fluxes, when the Ti implants were placed in the setup. The detailed multi-parametric information of the secondary radiation field provides a comprehensive understanding of the impact of Ti materials in proton therapy.

We investigate the transition to Self Organized Criticality in a two-dimensional model of a flux tube with a background flow. The magnetic induction equation, represented by a partial differential equation with a stochastic source term, is discretized and implemented on a two dimensional cellular automaton. The energy released by the automaton during one relaxation event is the magnetic energy. As a result of the simulations we obtain the time evolution of the energy release, of the system control parameter, of the event lifetime distribution and of the event size distribution, respectively, and we establish that a Self Organized Critical state is indeed reached by the system. Moreover, energetic initial impulses in the magnetohydrodynamic flow can lead to one dimensional signatures in the magnetic two dimensional system, once the Self Organized Critical regime is established. The applications of the model for the study of Gamma Ray Bursts is briefly considered, and it is shown that some astrophysical parameters of the bursts, like the light curves, the maximum released energy, and the number of peaks in the light curve can be reproduced and explained, at least on a qualitative level, by working in a framework in which the systems settles in a Self Organized Critical state via magnetic reconnection processes in the magnetized Gamma Ray Burst fireball.

Plasmonic nanoparticles embedded into a solid matrix could play crucial role in laser-matter interactions. In this study, excess energy creation was observed during the single-shot irradiation of a polymer matrix containing plasmonic gold nanorods, resonant to the laser wavelength, with a high intensity femtosecond laser pulse. This effect was manifested in a 7-fold rise in the crater volume for a 1.7-fold increase of the laser intensity, and was absent in the pure polymer without the gold doping. It occurred at laser intensities > 1.5 x 1017 W/cm2, being the vanishing threshold of plasma mirror formation, resulting in a more than 80% increase of the amount of laser light entering the target. This threshold was found to be critical for the plasmonic effect of gold nanoantennas tuned to the wavelength of the laser on the crater formation.

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.

Motivated by work of Haythorpe, Thomassen and the author showed that there exists a positive constant $c$ such that there is an infinite family of 4-regular 4-connected graphs, each containing exactly $c$ hamiltonian cycles. We complement this by proving that the same conclusion holds for planar 4-regular 3-connected graphs, although it does not hold for planar 4-regular 4-connected graphs by a result of Brinkmann and Van Cleemput, and that it holds for 4-regular graphs of connectivity 2 with the constant 144 < c, which we believe to be minimal among all hamiltonian 4-regular graphs of sufficiently large order. We then disprove a conjecture of Haythorpe by showing that for every non-negative integer $k$ there is a 5-regular graph on $26 + 6k$ vertices with $2^{k+10} \cdot 3^{k+3}$ hamiltonian cycles. We prove that for every $d \ge 3$ there is an infinite family of hamiltonian 3-connected graphs with minimum degree $d$, with a bounded number of hamiltonian cycles. It is shown that if a 3-regular graph $G$ has a unique longest cycle $C$, at least two components of $G - E(C)$ have an odd number of vertices on $C$, and that there exist 3-regular graphs with exactly two such components.

In this paper, we extend the concept of $b$-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the $b$-metric setting: fixed-point theorems, stability results, and a variant of Ekeland's variational principle. As a consequence, we also derive a variant of Caristi's fixed-point theorem