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Kleisli categories have long been recognised as a setting for modelling the linear behaviour of various types of systems. However, the final coalgebra in such settings does not, in general, correspond to a fixed notion of linear semantics. While there are well-understood conditions under which final coalgebras capture finite trace semantics, a general account of infinite trace semantics via finality has remained elusive. In this work, we present a sheaf-theoretic framework for infinite trace semantics in Kleisli categories that systematically constructs final coalgebras capturing infinite traces. Our approach combines Kleisli categories, sheaves over ordinals, and guarded (co)recursion, enabling infinite behaviours to emerge from coherent families of finite approximations via amalgamation. We introduce the notion of guarded behavioural functor and show that, under mild conditions, their final coalgebras directly characterise infinite traces.

Proponents of Condorcet voting face the question of what to do in the rare case when no Condorcet winner exists. Recent work provides compelling arguments for the rule that should be applied in three-candidate elections, but already with four candidates, many rules appear reasonable. In this paper, we consider a recent proposal of a simple Condorcet voting method for Final Four political elections. Our question is what normative principles could support this simple form of Condorcet voting. When there is no Condorcet winner, one natural principle is to pick the candidate who is closest to being a Condorcet winner. Yet there are multiple plausible ways to define closeness, leading to different results. Here we take the following approach: identify a relatively uncontroversial sufficient condition for one candidate to be closer than another to being a Condorcet winner; then use other principles to help settle who wins in cases when that condition alone does not. We prove that our principles uniquely characterize the simple Condorcet voting method for Final Four elections. This analysis also points to a new way of extending the method to elections with five or more candidates that is

We consider a generalized Heitler model for QED cascade. An exact formula for the final number of leptons is obtained by solving the kinetic equations. We demonstrate that in such a model the final number of leptons does not depend on photon and lepton free paths. We derive approximate formulas for the main characteristics of cascades at high energy, including the final number of leptons and the cascade depth. We show that in general the final number of leptons is asymptotically proportional to the energy of seed particle. It is also demonstrated how the original Heitler model is reproduced as a special case.

Let G be a context-free grammar with a total alphabet V, and let F be a final language over an alphabet W such that W is a subset of V. A final sentential form is any sentential form of G that, after omitting symbols from V - W, it belongs to F. The string resulting from the elimination of all nonterminals from W in a final sentential form is in the language of G finalized by F if and only if it contains only terminals. The language of any context-free grammar finalized by a regular language is context-free. On the other hand, it is demonstrated that L is a recursively enumerable language if and only if there exists a propagating context-free grammar G such that L equals the language of G finalized by {w#w^R | w is a string over a binary alphabet}, where w^R is the reversal of w.

In [4] Sturmfels linked the Hilbert Nullstellensatz to Gröbner bases through final polynomials. In (loc. cit.) it was claimed that final polynomials always appear in a lexicographic Gröbner basis of a certain ideal. In this paper, we give a counterexample to this claim. We also show how the introduction of an extra variable restores the claim in a deformed setup, which we call extended final polynomials.

We acuminate the idea of a final theory of physics in order to analyze its logical implications and consequences. It is argued that the rationale of a final theory is the principle of sufficient reason. This implies that a final theory of physics, presumed such a theory is possible, does not allow to incorporate substantial (non-trivial) propositions unless they are logically or mathematically deduced. Differences between physics and mathematics are discussed with emphasis on the role of physical constants. It is shown that it is logically impossible to introduce constants on the fundamental level of a final theory. The most fundamental constants emerging within a final theory are constants of motion. It is argued that the only possibility to formulate a final theory is necessarily a tautology: A final theory of physics can only be derived from those presumptions about reality that are inherent in the idea and practice of physics itself. It is argued that a final theory is based on the notion of objectivity, but it is logically impossible that an ideal final theory supports realism.

The semitauonic $B_c^- \to J/ψτ^-\barν_τ$ decay is optimal to scrutinize possible new physics effects in $b \to c τ^- \barν_τ$ transitions as indicated by the current data on $R(D^{(*)})$ anomalies. In this work, we study the $B_c^- \to J/ψτ^-\barν_τ$ decay with both polarized $τ$ and $J/ψ$. Their subsequent decays, with $J/ψ\to μ^+ μ^-$ as well as $τ^- \to π^- ν_τ$, $τ^- \to ρ^- ν_τ$ and $τ^- \to \ell^-\barν_\ellν_τ$, are exploited to extract the energy and angular distributions of the charged final-state particles in the processes. Starting with the most general effective Hamiltonian relevant for the $b \to c τ^- \barν_τ$ transitions, including all possible Lorentz structures of the dimension-six operators with both left- and right-handed neutrinos, we first derive the five-fold differential decay rate in terms of the visible final-state kinematics. From this distribution, we then construct in total 34 normalized observables, among which nine refer to the CP-violating triple product asymmetries that vanish within the Standard Model. We also construct five new observables based on the combinations of these normalized observables that can only be attributed to the right-handed neut

We propose a novel black hole model in which singular and regular black holes are combined as a whole and more precisely singular and regular black holes are regarded as different states of parameter evolution. We refer to them as singular and regular states, respectively. Furthermore, the regular state is depicted by the final state of parameter evolution in the model. We also present the sources that can generate such a black hole spacetime in the framework of $F(R)$ gravity. This theory of modified gravity is adopted because it offers a possible resolution to a tough issue in the thermodynamics of regular black holes, namely the discrepancy between the thermal entropy and Wald entropy. The dynamics and thermodynamics of the novel black hole model are also discussed when a singular state evolves into a regular state during the change of charge or horizon radius from its initial value to its extreme value.

We utilize the lepton number violation signal process $p\, e^- \to τ^+ jjj$ to search for heavy Majorana neutrinos at future proton-electron colliders. The LHeC (FCC-eh) is considered to run with an electron beam energy of 60 GeV, a proton beam energy of 7 (50) TeV and an integrated luminosity of 1 (3) ab$^{-1}$, and the electron beam is considered to be unpolarized. We apply detector configurations and simulate signal and related standard model background events for both hadronic $τ_h$ and leptonic $τ_\ell$ final states, $\ell$ being a muon. After preselection, multivariate analyses are performed to reject the background. The strategy to reconstruct the heavy neutrino mass is developed and distributions of reconstructed mass are presented. Discovery sensitivities on parameter $|V_{τN}|^2 |V_{eN}|^2 / ( |V_{τN}|^2 + |V_{eN}|^2 )$ for the heavy neutrino mass between 10 and 3000 GeV are predicted. At the 2-$σ$ significance, the best discovery sensitivity is $\sim 1.2 \times10^{-5} \,\,(5.0 \times 10^{-6})$ at the LHeC (FCC-eh) when $m_N \sim 100$ GeV for the hadronic $τ_h$ final state. Sensitivities for the leptonic $τ_\ell$ final state are found to be similar to those for the hadron

Generative Networks have shown great promise in generating photo-realistic images. Despite this, the theory surrounding them is still an active research area. Much of the useful work with Generative networks rely on heuristics that tend to produce good results. One of these heuristics is the advice not to use Batch Normalization in the final layer of the generator network. Many of the state-of-the-art generative network architectures use this heuristic, but the reasons for doing so are inconsistent. This paper will show that this is not necessarily a good heuristic and that Batch Normalization can be beneficial in the final layer of the generator network either by placing it before the final non-linear activation, usually a $tanh$ or replacing the final $tanh$ activation altogether with Batch Normalization and clipping. We show that this can lead to the faster training of Generator networks by matching the generator to the mean and standard deviation of the target distribution's image colour values.

The black hole singularity plays a crucial role in formulating Hawking's information paradox. The global spacetime analysis may be reconciled with unitarity by imposing a final state boundary condition on the spacelike singularity. Motivated by the final state proposal, we explore the effect of final state projection in two dimensional conformal field theories. We calculate the time evolution under postselection by employing the real part of pseudo-entropy to estimate the amount of quantum entanglement averaged over histories between the initial and final states. We find that this quantity possesses a Page curve-like behavior.

The Stanford Linear Collider (SLC) was the first prototype of a new type of accelerator, the electron-positron linear collider. Many years of dedicated effort were required to understand the physics of this new technology and to develop the techniques for maximizing performance. Key issues were emittance dilution, stability, final beam optimization and background control. Precision, non-invasive diagnostics were required to measure and monitor the beams throughout the machine. Beam-based feedback systems were needed to stabilize energy, trajectory, intensity and the final beam size at the interaction point. A variety of new tuning techniques were developed to correct for residual optical or alignment errors. The final focus system underwent a series of refinements in order to deliver sub-micron size beams. It also took many iterations to understand the sources of backgrounds and develop the methods to control them. The benefit from this accumulated experience was seen in the performance of the SLC during its final run in 1997-98. The luminosity increased by a factor of three to 3*10**30 and the 350,000 Z data sample delivered was nearly double that from all previous runs combined.

We review recent progress in studies of nuclear final-state interactions in deep inelastic scattering (DIS) off the lightest nuclei tagged by a recoil nucleon. These processes hold a lot of potential for resolving the outstanding issues related to the dynamics of hadronization in QCD. Within the minimal Fock component framework, valid at large Bjorken $x$, the main features of the theoretical approach based on the virtual nucleon approximation are elaborated. In this approach, the strong final-state interaction of the DIS products with the nuclear fragments is described by an effective eikonal amplitude, whose parameters can be extracted from the analysis of semi-inclusive DIS off the deuteron target. The extraction of the $Q^2$ and $W$ mass dependences of these parameters gives a new observable in studying the QCD structure of DIS final states. Another important feature of tagged DIS off the lightest nuclei is the possibility of performing pole extrapolation with a high degree of accuracy. Such extrapolation allows an extraction of the neutron structure function in a model independent way due to suppression of the final-state interaction in the on-shell limit of the struck nucleon

During a galaxy merger, the supermassive black hole (SMBH) in each galaxy is thought to sink to the center of the potential and form a supermassive black hole binary; this binary can eject stars via 3-body scattering, bringing the SMBHs ever closer. In a static spherical galaxy model, the binary stalls at a separation of about a parsec after ejecting all the stars in its loss cone -- this is the well-known final parsec problem. Earlier work has shown that the centrophilic orbits in triaxial galaxy models are key in refilling the loss cone at a high enough rate to prevent the black holes from stalling. However, the evolution of binary SMBHs has never been explored in axisymmetric galaxies, so it is not clear if the final parsec problem persists in these systems. Here we use a suite of direct N-body simulations to follow SMBH binary evolution in galaxy models with a range of ellipticity. For the first time, we show that mere axisymmetry can solve the final parsec problem; we find the the SMBH evolution is independent of N for an axis ratio of c/a=0.8, and that the SMBH binary separation reaches the gravitational radiation regime for c/a=0.75.

The hadronic final state in electron-proton collisions at HERA has provided a rich testing ground for development of the theory of the strong force, QCD. In this review, over 200 publications from the H1 and ZEUS Collaborations are summarised. Short distance physics, the measurement of processes at high energy scales, has provided rigorous tests of perturbative QCD and constrained the structure of the proton as well as allowing precise measurements of the strong coupling constant to be made. Non-perturbative or low energy processes have also been investigated and results on hadronisation interpreted together with those from other experiments. Searches for exotic QCD objects, such as pentaquarks, glueballs and instantons have been performed. The subject of diffraction has been re-invigorated through its precise measurement, such that it can now be described by perturbative QCD. After discussion of HERA, the H1 and ZEUS detectors and the techniques used to reconstruct differing hadronic final states, the above subject areas are elaborated. The major achievements are then condensed further in a final section summarising what has been learned.

In a previous article (physics/9705004), the best 100m performances in track and field were determined by accounting for wind effects and atmospheric drag. In this report, I seek to ascertain what was the "fastest" 100m final through a statistical analysis of the competitors' times. Considered are the 100 finals from the 1984-1996 Olympic Games, the 1983-1997 World Championships, and the 1996 Grand Prix Final from Lausanne, SWI, in which Frank Fredericks (Namibia) recorded the only sub-9.90s time ever into a head-wind (9.86s, -0.4 m/s).

We give conditions on a finitary endofunctor of a finitely accessible category to admit a final coalgebra. Our conditions always apply to the case of a finitary endofunctor of a locally finitely presentable (l.f.p.) category and they bring an explicit construction of the final coalgebra in this case. On the other hand, there are interesting examples of final coalgebras beyond the realm of l.f.p. categories to which our results apply. We rely on ideas developed by Tom Leinster for the study of self-similar objects in topology.

This article summarises three searches for diboson resonances in the all-hadronic final state using data collected at a center-of-mass energy of $\sqrt{\rm{s}}=13$ TeV with the CMS experiment at the CERN LHC. The boson decay products are contained in one large-radius jet, resulting in dijet final states which are resolved using jet substructure techniques. The analyses presented use 2.3, 35.9 and 77.3 inverse femtobarns of data collected between 2015 and 2017. These include the first search for diboson resonances using data collected at a 13 TeV collision energy, the introduction of a new algorithm to tag vector bosons in the context of analyzing the data collected in 2016, and the development of a novel multidimensional fit improving on the sensitivity of the previous search method with up to $30\%$. The results presented here are the most sensitive to date of diboson resonance searches in the dijet final state. An emphasis on improvements in technique for vector boson tagging is made.

We present a criterion for $2$-final $(2,1)$-functors, analoguous to the classical one for final $1$-functor: a $(2,1)$-functor $F \colon A \to B$ is $2$-final if and only if, for any object $b$ of $B$, the slice $(2,1)$-category $b / F$ is nonempty, connected and simply connected. We also give a combinatorial presentation of paths and homotopies of paths in a $(2,1)$-category.