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We model the evolution of tokens within a deep stack of Transformer layers as a continuous-time flow on the unit sphere, governed by a mean-field interacting particle system, building on the framework introduced in (Geshkovski et al., 2023). Studying the corresponding mean-field Partial Differential Equation (PDE), which can be interpreted as a Wasserstein gradient flow, in this paper we provide a mathematical investigation of the long-term behavior of this system, with a particular focus on the emergence and persistence of meta-stable phases and clustering phenomena, key elements in applications like next-token prediction. More specifically, we perform a perturbative analysis of the mean-field PDE around the iid uniform initialization and prove that, in the limit of large number of tokens, the model remains close to a meta-stable manifold of solutions with a given structure (e.g., periodicity). Further, the structure characterizing the meta-stable manifold is explicitly identified, as a function of the inverse temperature parameter of the model, by the index maximizing a certain rescaling of Gegenbauer polynomials., Comment: 37 Pages, 6 figures

We develop a computer-assisted symbolic method to show that a linearized Boussinesq flow in self-similar coordinates gives rise to an invertible operator., Comment: The argument involves 7 Matlab files, which are attached to this submission

We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. This is done by exploiting a second-order effect, related to the classical Rayleigh--B\'enard instability, that overcomes the regularizing effect of transport. A similar result is established for the 3d Euler system based on the Taylor--Couette instability.

The SPIDER experiment, part of the Neutral Beam Test Facility (NBTF) at Consorzio RFX (Padua, Italy), is the prototype of the negative ion source for the ITER neutral beam injectors; the source is coupled to a 100 kV three-grid acceleration system. A Beam Emission Spectroscopy (BES) diagnostic was installed in SPIDER to study and optimize energy distribution, aiming, uniformity and divergence of the H-/D- beam extracted from the source. The diagnostic is based on the analysis of the Doppler shifted H{\alpha}/D{\alpha} light emitted in the interaction between the beam particles and the H2/D2 molecules of the background. In 2019 the BES diagnostic in SPIDER was installed and calibrated, and it allowed to characterize the first hydrogen beams extracted from the SPIDER source, in cesium free conditions. The number of active beamlets composing the beam was reduced from 1280 to 80, affecting the BES diagnostic capabilities. This paper presents the BES diagnostic setup and discusses the first collected results. Under limited extracted current density ($\sim$ 10 A/m^2) and ion energy ($\leq$35 keV), no significant vertical beam deflection caused by the magnetic filter field in the source was detected. In some cases the beamlets were observed to be elongated in horizontal direction; beamlet divergence values down to 20 mrad and 30 mrad e-folding were measured in vertical and horizontal direction, respectively; the intensity of the Doppler shifted radiation was found to be strongly correlated to the beam current and to the beam divergence. The progressive compensation of beamlet deflections (caused by electron suppression filter fields) with increasing voltage in the extraction gap was studied., Comment: 21 pages, 8 figures. Accepted manuscript of a published paper

ITER envisages the use of two heating neutral beam injectors plus an optional one as part of the auxiliary heating and current drive system. The 16.5 MW expected neutral beam power per injector is several notches higher than worldwide existing facilities. A Neutral Beam Test Facility (NBTF) was established at Consorzio RFX, exploiting the synergy of two test beds, SPIDER and MITICA. SPIDER is dedicated to developing and characterizing large efficient negative ion sources at relevant parameters in ITER-like conditions: source and accelerator located in the same vacuum where the beam propagates, immunity to electromagnetic interferences of multiple radio-frequency (RF) antennas, avoidance of RF-induced discharges on the outside of the source. Three years of experiments on SPIDER have addressed to the necessary design modifications to enable full performances. The source is presently under a long shut-down phase to incorporate learnings from the experimental campaign. Parallelly, developments on MITICA, the full-scale prototype of the ITER NBI featuring a 1 MV accelerator and ion neutralization, are underway including manufacturing of in-vessel components, while power supplies and auxiliary plants are already under final testing and commissioning. Integration, commissioning and tests of the 1MV power supplies are essential for this first-of-kind system, unparalleled both in research and industry field. The integrated test to confirm 1MV output by combining invertor systems, DC generators and transmission lines extracted errors/accidents in some components. To realize a concrete system for ITER, solutions for the repair and the improvement of the system were developed. Hence, NBTF is emerging as a necessary facility, due to the large gap with existing injectors, effectively dedicated to identify issues and find solutions to enable successful ITER NBI operations in a time bound fashion.

We study the massive scalar field equation $\Box_g \phi = m^2 \phi$ on a stationary and spherically symmetric black hole $g$ (including in particular the Schwarzschild and Reissner--Nordstr\"om black holes in the full sub-extremal range) for solutions $\phi$ projected on a fixed spherical harmonic. Our problem involves the scattering of an attractive long-range potential (Coulomb-like) and thus cannot be treated perturbatively. We prove precise (point-wise) asymptotic tails of the form $t^{-5/6} f(t)+ O(t^{-1+\delta})$, where $f(t)$ is an explicit oscillating profile. Our asymptotics appear to be the first rigorous decay result for a massive scalar field on a black hole. Establishing these asymptotics is also an important step in retrieving the assumptions used in work of the third author regarding the interior of dynamical black holes and Strong Cosmic Censorship., Comment: Version 2, 41 pages. Minor modifications

The electron density close to the extraction grids and the co-extracted electrons represent a crucial issue when operating negative ion sources for fusion reactors. An excessive electron density in the plasma expansion region can indeed inhibit the negative ion production and introduce potentially harmful electrons in the accelerator. Among the set of plasma and beam diagnostics proposed for SPIDER upgrade, a heterodyne microwave (mw) interferometer at 100 GHz is being explored as a possibility to measure electron density in the plasma extraction region. The major issue in applying this technique in SPIDER is the poor accessibility of the probing microwave beam through the source metal walls and the long distance of 4 m at which mw modules should be located outside the vacuum vessel. Numerical investigations in a full-scale geometry showed that the power transmitted through the plasma source apertures was sufficient for the microwave module sensitivity. An experimental proof-of-principle of the setup was then performed. The microwave system was tested on an experimental full-scale test-bench mimicking SPIDER viewports accessibility constraints, including the presence of a SPIDER-like plasma. The outcome of first tests revealed that, despite the geometrical constraints, in certain conditions, the electron density measurements are possible. The main issue arises from decoupling the one-pass signal from spurious multipaths generated by mw beam reflections, requiring signal cross correlation analysis. These preliminary tests demonstrate that despite the 4 m distance between the mw modules and the presence of metal walls, plasma density measurement is possible when the 80-mm diameter ports are available. In this contribution, we discuss the numerical simulations, the preliminary experimental tests and suggest design upgrades of the interferometric setup to enhance signal transmission.

Neutral Beam Injectors (NBIs) based on negative ions are the workhorses of future fusion reactors, such as ITER, which they are expected to provide with up to \SI{33}{MW} of power to heat the fusion plasma. The negative hydrogen ions are extracted from a RF plasma, in which a magnetic filter field cools down the electrons reaching the so-called expansion region and allows the formation of negative ions near the apertures in the plasma grid. To further improve the production of negative ions, cesium is usually evaporated inside the source and deposited onto the plasma walls, reducing the work function of the surfaces. This dramatically increases the density of negative hydrogen ions near the surfaces, causing the transition to an electronegative plasma in the vicinity of the plasma grid. This condition can be observed with Langmuir probes, which can then be used to provide a local meaurement of negative ion density in the ion source. In this paper we use the measurements provided by the Langmuir probe sensors embedded in the plasma grid of SPIDER, the prototype ion source of ITER NBIs, to determine the density of negative ions. A fitting method based on the determination of the collection area of the different plasma species is proposed and adapted to SPIDER experimental condition, taking into account the shape of the probes and the local topology of the magnetic field. The method is then applied to the experimental data, determining the densities of the positive and negative ions and of the electrons during a plasma pulse. Finally, a vertical array of four probes in the plasma grid is used to assess the vertical profile of plasma parameters., Comment: 12 pages, 7 figures, contributed paper to the NIBS 22 conference. Submitted to Journal of Instrumentation

The Nambu--Jona-Lasinio model is widely used to study strong-interaction phenomena in vacuum and quark matter. Since the model is nonrenormalizable, one needs to work within a specific regularization scheme to obtain finite results. Here we show that a commonly used cutoff regularization scheme leads to unphysical results, such as superluminal speed of sound and wrong high-temperature behavior of the specific heat and other thermodynamical quantities. Such a troublesome feature of the cutoff regularization invalidates the model for temperature and baryon density values relevant to the phenomenology of heavy-ion collisions and compact stars. We show that the source of the problems stems from cutting off momentum modes in finite integrals depending on thermal distribution functions in the grand canonical potential. The problems go away when taking into account the full momentum range of those integrals. Explicit examples are worked out in the SU(2)-flavor version of the model., Comment: Typos corrected, comments added. Matches published version

We construct a class of time-symmetric initial data sets for the Einstein vacuum equation modeling elementary configurations of multiple ``almost isolated" systems. Each such initial data set consists of a collection of several localized sources of gravitational radiation, and lies in a family of data sets which is closed under scaling out the distances between the systems by arbitrarily large amounts. This class contains data sets which are not asymptotically flat, but to which nonetheless a finite ADM mass can be ascribed. The construction proceeds by a gluing scheme using the Brill--Lindquist metric as a template. Such initial data are motivated in part by a desire to understand the dynamical interaction of distant systems in the context of general relativity. As a by-product of the construction, we produce complete, scalar-flat initial data with trivial topology and infinitely many minimal spheres, as well as initial data with infinitely many Einstein--Rosen bridges., Comment: 40 pages

We investigate the threshold $p_{\vec H}=p_{\vec H}(n)$ for the Ramsey-type property $G(n,p)\to \vec H$, where $G(n,p)$ is the binomial random graph and $G\to\vec H$ indicates that every orientation of the graph $G$ contains the oriented graph $\vec H$ as a subdigraph. Similarly to the classical Ramsey setting, the upper bound $p_{\vec H}\leq Cn^{-1/m_2(\vec H)}$ is known to hold for some constant $C=C(\vec H)$, where $m_2(\vec H)$ denotes the maximum $2$-density of the underlying graph $H$ of $\vec H$. While this upper bound is indeed the threshold for some $\vec H$, this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic $\{K_3,K_{3,3}\}$-free graphs) and arbitrarily rooted transitive triangles., Comment: 12 pages, 1 figure. To appear in RAIRO-Operations Research

The SPIDER experiment, operated at the Neutral Beam Test Facility of Consorzio RFX, Padua, hosts the prototype of the H-/D- ion source for the ITER neutral beam injectors. The maximization of the ion current extracted from the source and the minimization of the amount of co-extracted electrons are among the most relevant targets to accomplish. The Cavity Ring-Down Spectroscopy diagnostic measures the negative ion density in the source close to the plasma grid (the plasma-facing grid of the ion acceleration system), so to identify the source operational parameters that maximize the amount of negative ions which can be extracted. In this study SPIDER was operated in hydrogen and deuterium in Cs-free conditions, therefore negative ions were mostly produced by reactions in the plasma volume. This work shows how the magnetic filter field and the bias currents, present in SPIDER to limit the amount of co-extracted electrons, affect the density of negative ions available for extraction. The results indicate that the magnetic filter field in front of the acceleration system should be set between about 1.6 mT, condition that maximizes the density of available negative ions, and about 3.2 mT, condition that minimizes the ratio of electron current to ion current. The negative ion density also resulted to be maximized when the plasma grid and its surrounding bias plate was positively biased against the source body with a total current in the range 0 A{\div}100 A. The paper shows also how much, in Cs-free conditions, the electric fields in the acceleration system can affect the density of negative ions in the source, close to the plasma grid apertures., Comment: 19 pages, 10 figures. Accepted manuscript of a published paper

The Neutral Beam Injectors of the ITER experiment will rely on negative ion sources to produce 16.7 MW beams of H/D particles accelerated at 1 MeV. The prototype of these sources was built and is currently operated in the SPIDER experiment (Source for the Production of Ions o Deuterium Extracted from an RF plasma), part of the Neutral Beam Test Facility of Consorzio RFX, Padua. In SPIDER, the H-/D- ion source is coupled to a three grids 100 kV acceleration system. One of the main targets of the experimentation in SPIDER is to uniformly maximize the extracted current density; to achieve this it is important to study the density of negative ions available in proximity of the ion acceleration system. In SPIDER, line-integrated measurements of negative ion density are performed by a Cavity Ring Down Spectroscopy (CRDS) diagnostic. Its principle of operation is based on the absorption of the photons of a laser beam pulse by H-/D- photo-detachment; the absorption detection is enhanced by trapping the laser pulse in an optical cavity, containing the absorbing medium (i.e. negative ions). The paper presents and discusses the CRDS diagnostic setup in SPIDER, including the first measurements of negative ion density, correlated to the main source parameters., Comment: 5 pages, 7 figures. Contributer paper for the HTPD 2020 conference. Accepted manuscript

The NIO1 experiment hosts a flexible RF H- ion source, developed by INFN-LNL and Consorzio RFX to improve the present concepts for the production and acceleration of negative ions. The source is also used to benchmark the instrumentation dedicated to the ITER neutral beam test facility. Many diagnostics are installed in NIO1 to characterize the source and the extracted negative ion beam. Among them, Beam Emission Spectroscopy (BES) has been used in NIO1 to measure the divergence and the uniformity of the beam, together with the fraction of beam ions which was neutralized inside the acceleration system. The diagnostic method is based on the analysis of the Doppler shifted $H_\alpha$ photons emitted by the fast beam particles and collected along a line of sight. The article presents the experimental setup and the analysis algorithms of the BES diagnostic, together with a discussion of the first measurements and of their correlation with the operational parameters., Comment: 3 pages, 2 figures. Contributed paper for the ICIS 2017 conference. Accepted manuscript of a published paper

The NIO1 experiment, built and operated at Consorzio RFX, hosts an RF negative ion source, from which it is possible to produce a beam of maximum 130 mA in H- ions, accelerated up to 60 kV. For the preliminary tests of the extraction system the source has been operated in oxygen, whose high electronegativity allows to reach useful levels of extracted beam current. The efficiency of negative ions extraction is strongly influenced by the electron density and temperature close to the Plasma Grid, i.e. the grid of the acceleration system which faces the source. To support the tests, these parameters have been measured by means of the Optical Emission Spectroscopy diagnostic. This technique has involved the use of an oxygen-argon mixture to produce the plasma in the source. The intensities of specific Ar I and Ar II lines have been measured along lines of sight close to the Plasma Grid, and have been interpreted with the ADAS package to get the desired information. This work will describe the diagnostic hardware, the analysis method and the measured values of electron density and temperature, as function of the main source parameters (RF power, pressure, bias voltage and magnetic filter field). The main results show that not only electron density but also electron temperature increase with RF power; both decrease with increasing magnetic filter field. Variations of source pressure and plasma grid bias voltage appear to affect only electron temperature and electron density, respectively., Comment: 7 pages 4 figures. Contributed paper for the NIBS 2016 conference. Accepted manuscript

The SPIDER H-/D- ion source is currently in operation in the Neutral Beam Test facility (NBTF) at Consorzio RFX (Padova, Italy) to prove the possibility of generating up to 40 A of negative ions, with a maximum extracted current density of 350 A/m2 (H)/285 A/m2 (D) and a fraction of co-extracted electrons not greater than 0.5 (H)/1 (D). These performances are required for the realization of the ITER Neutral Beam Injector (NBI), which should deliver 16.7 MW to the plasma by means of negative ions accelerated up to 1 MeV and neutralized before being injected into the ITER tokamak. In order to obtain such high extracted current densities and low co-extracted electron fractions it is necessary to lower the work function of the surface of the acceleration system grid facing the source; this will be accomplished by coating the surfaces with Cs, routinely evaporated by three ovens. The functionality of the ovens has been tested at the CAesium oven Test Stand (CATS), hosted at NBTF. The test stand is equipped with several diagnostics, among which a Laser Absorption Spectroscopy (LAS) diagnostic. Using a tunable laser diode, the LAS diagnostic gets the high resolution absorption spectrum of the Cs 852 nm D2 line along a line of sight to measure Cs density at ground state. The paper describes the test stand and the LAS diagnostic, together with the characterization of the ovens to be installed in SPIDER. The paper will also study the systematic density underestimation effect caused by Cs ground state depopulation, as a function of laser intensity and of Cs density, in the perspective of correcting the density evaluation., Comment: 8 pages, 4 figures. Contribution to the LAPD 2019 conference. Accepted manuscript of a published paper (embargo expired)

NIO1 is a compact and flexible radiofrequency H- ion source, developed by Consorzio RFX and INFN-LNL. Aim of the experimentation on NIO1 is the optimization of both the production of negative ions and their extraction and beam optics. In the initial phase of its commissioning, NIO1 was operated with nitrogen, but now the source is regularly operated also with hydrogen. To evaluate the source performances an optical emission spectroscopy diagnostic was installed. The system includes a low resolution spectrometer in the spectral range of 300-850 nm and a high resolution (50 pm) one, to study respectively the atomic and the molecular emissions in the visible range. The spectroscopic data have been interpreted also by means of a collisional-radiative model developed at IPP Garching. Besides the diagnostic hardware and the data analysis methods, the paper presents the first plasma measurements across a transition to the full H mode, in a hydrogen discharge. The characteristic signatures of this transition in the plasma parameters are described, in particular the sudden increase of the light emitted from the plasma above a certain power threshold., Comment: 3 pages, 2 figures. Contributed paper for the ICIS 2015 conference. Accepted manuscript

A test facility for the development of the Neutral Beam Injection system for ITER is under construction at Consorzio RFX. It will host two experiments: SPIDER, a 100 keV H-/D- ion RF source, and MITICA, a prototype of the full performance ITER injector (1 MV, 17 MW beam). A set of diagnostics will monitor the operation and allow to optimize the performance of the two prototypes. In particular, Beam Emission Spectroscopy will measure the uniformity and the divergence of the fast particles beam exiting the ion source and travelling through the beam line components. This type of measurement is based on the collection of the H{\alpha}/D{\alpha} emission resulting from the interaction of the energetic particles with the background gas. A numerical model has been developed to simulate the spectrum of the collected emissions in order to design this diagnostic and to study its performance. The paper describes the model at the base of the simulations and presents the modeled $H_\alpha$ spectra in the case of MITICA experiment., Comment: 4 pages, 5 figures. Contributed paper for the HTPD 2014 conference. Accepted manuscript

The Heating Neutral Beam Injectors (HNBs) for ITER will have to deliver 16.7 MW beams of H/D particles at 1 MeV energy. The beams will be produced from H-/D- ions, generated by a radiofrequency plasma source coupled to an ion acceleration system. A prototype of the ITER HNB ion source is being tested in the SPIDER experiment, part of the ITER Neutral Beam Test Facility at Consorzio RFX. Reaching the design targets for beam current density and fraction of coextracted electrons is only possible by evaporating cesium in the source, in particular on the plasma facing grid (PG) of the acceleration system. In this way the work function of the surfaces decreases, significantly increasing the amount of surface reactions that convert neutrals and positive ions into H-/D-. It is then of paramount importance to monitor the density of negative ions and the density of Cs in the proximity of the PG. Monitoring the Cs spatial distribution along the PG is also essential to guarantee the uniformity of the beam current. In SPIDER, this is possible thanks to the Cavity Ringdown Spectroscopy (CRDS) and the Laser absorption Spectroscopy diagnostics (LAS), which provide line-integrated measurements of negative ion density and neutral, ground state Cs density, respectively. The paper discusses the CRDS and LAS measurements as a function of input power and of the magnetic and electric fields used to reduce the coextraction of electrons. Negative ion density data are in qualitative agreement with the analogous measurements in Cs-free conditions. In agreement with simulations, Cs density is peaked in the center of the source; a top/bottom non uniformity is also present. Several effects of plasma on Cs deposition and negative ion production are presented., Comment: 21 pages, 12 figures. Paper (accepted manuscript) following the poster contribution at the SOFT 2022 conference. Published in Fusion Engineering and Design

Consorzio RFX and INFN-LNL are building a flexible small ion source (NIO1) capable of producing about 130 mA of H- ions accelerated at 60 KeV. Aim of the experiment is to test and develop the instrumentation for SPIDER and MITICA, the prototypes respectively of the negative ion sources and of the whole neutral beam injectors which will operate in the ITER experiment. As SPIDER and MITICA, NIO1 will be monitored with Beam Emission Spectroscopy (BES), a non-invasive diagnostic based on the analysis of the spectrum of the $H_\alpha$ emission produced by the interaction of the energetic ions with the background gas. Aim of BES is to monitor direction, divergence and uniformity of the ion beam. The precision of these measurements depends on a number of factors related to the physics of production and acceleration of the negative ions, to the geometry of the beam and to the collection optics. These elements were considered in a set of codes developed to identify the configuration of the diagnostic which minimizes the measurement errors. The model was already used to design the BES diagnostic for SPIDER and MITICA. The paper presents the model and describes its application to design the BES diagnostic in NIO1., Comment: 3 pages, 3 figures. Contributed paper for the ICIS 2013 conference. Accepted manuscript

The ITER Heating Neutral Beam (HNB) injector is required to deliver 16.7 MW power into the plasma from a neutralised beam of H-/D- negative ions, produced by an RF source and accelerated up to 1 MeV. To enhance the H-/D- production, the surface of the acceleration system grid facing the source (the plasma grid) will be coated with Cs because of its low work function. Cs will be routinely evaporated in the source by means of specific ovens. Monitoring the evaporation rate and the distribution of Cs inside the source is fundamental to get the desired performances on the ITER HNB. In order to proper design the source of the ITER HNB and to identify the best operation practices for it, the prototype RF negative ion source SPIDER has been developed and built in the Neutral Beam Test Facility at Consorzio RFX. A Laser Absorption Spectroscopy diagnostic will be installed in SPIDER for a quantitative estimation of Cs density. By using a wavelength tunable laser, the diagnostic will measure the absorption spectrum of the 852 nm line along 4 lines of sight, parallel to the plasma grid surface and close to it. From the absorption spectra the line-integrated density of Cs at ground state will be measured. The design of this diagnostic for SPIDER is presented, with details of the layout and of the key components. A preliminary installation of the diagnostic on the test stand for Cs ovens is also described, together with its first experimental results; the effect of ground state depopulation on collected measurements is discussed and partially corrected., Comment: 6 pages, 5 figures. Contributed paper for the SOFT 2018 conference. Accepted manuscript (embargo expired)

The Neutral beam Injectors of the ITER experiment will be based on negative ion sources for the generation of beams composed by 1 MeV H/D particles. The prototype of these sources is currently under testing in the SPIDER experiment, part of the Neutral Beam Test Facility of Consorzio RFX, Padua, Italy. Among the targets of the experimentation in SPIDER, it is of foremost importance to maximize the beam current density extracted from the source acceleration system. The SPIDER operating conditions can be optimized thanks to a Cavity Ring-down Spectroscopy diagnostic, which is able to give line-integrated measurements of negative ion density in proximity of the acceleration system apertures. Regarding the diagnostic technique, this work presents a phenomenon of drift in ring down time measurements, which develops in a time scale of few hours. This issue may significantly affect negative ion density measurements for plasma pulses of 1 h duration, as required by ITER. Causes and solutions are discussed. Regarding the source performance, this paper presents how negative ion density is influenced by the RF power used to sustain the plasma, and by the magnetic filter field present in SPIDER to limit the amount of co-extracted electrons. In this study, SPIDER was operated in hydrogen and deuterium, in Cs-free conditions., Comment: 9 pages, 5 figures. Accepted manuscript of a published article