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We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman--Vaught--Mostowski (FVM) type theorems, which characterize how logical equivalence behaves under composition and transformation of models. Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the existential and positive existential fragments, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads. We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments. This is an extended version of the LiCS 2023 conference paper of the same name., Comment: This is an extended version of arXiv:2304.10196 which, apart from providing full proofs of all statements, takes a more categorical point of view to tell the whole story. In particular, we highlight and explain the underlying categorical constructions in detail

We build on a recent result stating that the frame $\mathsf{SE}(L)$ of strongly exact filters for a frame $L$ is anti-isomorphic to the coframe $\mathsf{S}_o(L)$ of fitted sublocales. The collection $\mathsf{E}(L)$ of exact filters of $L$ is known to be a sublocale of this frame. We consider several other subcollections of $\mathsf{SE}(L)$: the collections $\mathcal{J}(\mathsf{CP}(L))$ and $\mathcal{J}(\mathsf{SO}(L))$ of intersections of completely prime and Scott-open filters, respectively, and the collection $\mathsf{R}(L)$ of regular elements of the frame of filters. We show that all of these are sublocales of $\mathsf{SE}(L)$, and as such they correspond to subcolocales of $\mathsf{S}_o(L)$, which all turn out to have a concise description. By using the theory of polarities of Birkhoff, one can show that all of the structures mentioned above enjoy universal properties which are variations of that of the canonical extension. We also show how some of these subcollections can be described as polarities and give three new equivalent definitions of subfitness in terms of the lattice of filters.

A contactless control of fluctuations of phase space variables of a nanoobject belongs among the key methods needed for ultra-precise nanotechnology and the upcoming quantum technology of macroscopic systems. Here we utilize the experimental platform of a single levitating nanoparticle (NP) to demonstrate essential protocols providing linear amplification of the mechanical phase space variables together with squeezing of phase space probability distribution. The protocol combines a controlled fast switching between the parabolic trapping potential and either weak parabolic or inverted parabolic amplifying potential leading to amplification of mean value and variance (fluctuations) along an arbitrary phase space variable and squeezing along the complementary one. The protocol is completed with cold damping scheme to control the initial fluctuations of the NP phase space variables. We reached the amplification gain $|G|>2$, the squeezing coefficient above 4 dB, and the second-order energy correlation function approaching 3 which corresponds to a maximum for a stochastic non-equilibrium classical state. These experimental results will already allow pre-amplification and manipulation of nanomechanical NP motion for all quantum protocols if the NP cooling towards the ground state is applied.

Nanoparticles levitated in an optical trap provide a versatile platform to study mechanical oscillators in a controlled environment with tuneable parameters. Recently, it has become possible to couple two of these optomechanical oscillators. Here, we demonstrate the collective non-Hermitian dynamics of such a pair of non-conservatively coupled oscillators. We take advantage of the tunability of the optical interactions between the particles in our system and set the optical interaction between the particles to be purely non-reciprocal. By continuously varying the relative power of the trapping beams, we take the system through a transition, similar to a parity-time phase transition. A Hopf bifurcation at a critical point results in the formation of collective limit cycle oscillations, resembling those observed in phonon lasers. These coupled levitated oscillators provide a platform for exceptional point optomechanical sensing and can be extended to multi-particle systems, paving the way for the development of topological optomechanical media., Comment: Nat. Phys. (2024)

Methods for controlling the motion of single particles, optically levitated in vacuum, have developed rapidly in recent years. The technique of cold damping makes use of feedback-controlled, electrostatic forces to increase dissipation without introducing additional thermal fluctuations. This process has been instrumental in the ground-state cooling of individual electrically charged nanoparticles. Here we show that the same method can be applied to a pair of nanoparticles, coupled by optical binding forces. These optical binding forces are about three orders of magnitude stronger than typical Coulombic inter-particle force and result in a coupled motion of both nanoparticles characterized by a pair of normal modes. We demonstrate cold damping of these normal modes, either independently or simultaneously, to sub-Kelvin temperatures at pressures of 5x10^{-3} mbar. Experimental observations are captured by a theoretical model which we use to survey the parameter space more widely and to quantify the limits imposed by measurement noise and time delays. Our work paves the way for the study of quantum interactions between meso-scale particles and the exploration of multiparticle entanglement in levitated optomechanical systems.

We present a categorical theory of the composition methods in finite model theory -- a key technique enabling modular reasoning about complex structures by building them out of simpler components. The crucial results required by the composition methods are Feferman-Vaught-Mostowski (FVM) type theorems, which characterize how logical equivalence behaves under composition and transformation of models. Our results are developed by extending the recently introduced game comonad semantics for model comparison games. This level of abstraction allow us to give conditions yielding FVM type results in a uniform way. Our theorems are parametric in the classes of models, logics and operations involved. Furthermore, they naturally account for the positive existential fragment, and extensions with counting quantifiers of these logics. We also reveal surprising connections between FVM type theorems, and classical concepts in the theory of monads. We illustrate our methods by recovering many classical theorems of practical interest, including a refinement of a previous result by Dawar, Severini, and Zapata concerning the 3-variable counting logic and cospectrality. To highlight the importance of our techniques being parametric in the logic of interest, we prove a family of FVM theorems for products of structures, uniformly in the logic in question, which cannot be done using specific game arguments.

We explore, experimentally and theoretically, the emergence of coherent coupled oscillations and synchronization between a pair of non-Hermitian, stochastic, opto-mechanical oscillators, levitated in vacuum. Each oscillator consists of a polystyrene microsphere trapped in a circularly polarized, counter-propagating Gaussian laser beam. Non-conservative, azimuthal forces, deriving from inhomogeneous optical spin, push the micro-particles out of thermodynamic equilibrium. For modest optical powers each particle shows a tendency towards orbital circulation. Initially, their stochastic motion is weakly correlated. As the power is increased, the tendency towards orbital circulation strengthens and the motion of the particles becomes highly correlated. Eventually, centripetal forces overcome optical gradient forces and the oscillators undergo a collective Hopf bifurcation. For laser powers exceeding this threshold, a pair of limit cycles appear, which synchronize due to weak optical and hydrodynamic interactions. In principle, arrays of such Non-Hermitian elements can be arranged, paving the way for opto-mechanical topological materials or, possibly, classical time crystals. In addition, the preparation of synchronized states in levitated optomechanics could lead to new and robust sensors or alternative routes to the entanglement of macroscopic objects.

Game comonads have brought forth a new approach to studying finite model theory categorically. By representing model comparison games semantically as comonads, they allow important logical and combinatorial properties to be exressed in terms of their Eilenberg-Moore coalgebras. As a result, a number of results from finite model theory, such as preservation theorems and homomorphism counting theorems, have been formalised and parameterised by comonads, giving rise to new results simply by varying the comonad. In this paper we study the limits of the comonadic approach in the combinatorial and homomorphism-counting aspect of the theory, regardless of whether any model comparison games are involved. We show that any standard graph parameter has a corresponding comonad, classifying the same class. This comonad is constructed via a simple Kan extension formula, making it the initial solution to this problem and, furthermore, automatically admitting a homomorphism-counting theorem.

Bilinear maps and their classifying tensor products are well-known in the theory of linear algebra, and their generalization to algebras of commutative monads is a classical result of monad theory. Motivated by constructions needed in categorical approaches to finite model theory, we generalize the notion of bimorphism much further. To illustrate these maps are mathematically natural notions, we show that many common axioms in category theory can be phrased as certain morphisms being bimorphisms. We also show that much of the established theory of bimorphisms goes through in much greater generality. Our results carefully identify which assumptions are needed for the different components of the theory, including when good properties hold globally, or can at least be established locally. We include a brief string diagrammatic account of the bimorphism axiom, and conclude by recovering a simple proof of a classical theorem, emphasizing the efficacy of the bimorphism perspective.

Game comonads, introduced by Abramsky, Dawar and Wang, and developed by Abramsky and Shah, give a categorical semantics for model comparison games. We present an axiomatic account of Feferman-Vaught-Mostowski (FVM) composition theorems within the game comonad framework, parameterized by the model comparison game. In a uniform way, we produce compositionality results for the logic in question, and its positive existential and counting quantifier variants. Secondly, we extend game comonads to the second order setting, specifically in the case of Monadic Second Order (MSO) logic. We then generalize our FVM theorems to the second order case. We conclude with an abstract formulation of Courcelle's algorithmic meta-theorem, exploiting our earlier developments. This is instantiated to recover well-known bounded tree-width and bounded clique-width Courcelle theorems for MSO on graphs.

Lov\'asz (1967) showed that two finite relational structures A and B are isomorphic if, and only if, the number of homomorphisms from C to A is the same as the number of homomorphisms from C to B for any finite structure C. Soon after, Pultr (1973) proved a categorical generalisation of this fact. We propose a new categorical formulation, which applies to any locally finite category with pushouts and a proper factorisation system. As special cases of this general theorem, we obtain two variants of Lov\'asz' theorem: the result by Dvo\v{r}\'ak (2010) that characterises equivalence of graphs in the k-dimensional Weisfeiler-Leman equivalence by homomorphism counts from graphs of tree-width at most k, and the result of Grohe (2020) characterising equivalence with respect to first-order logic with counting and quantifier depth k in terms of homomorphism counts from graphs of tree-depth at most k. The connection of our categorical formulation with these results is obtained by means of the game comonads of Abramsky et al. We also present a novel application to homomorphism counts in modal logic., Comment: 23 pages. To appear in the Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021)

A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of (finitely additive) measures arises -- via Stone-Priestley duality and the notion of types from model theory -- by enriching the expressive power of first-order logic with certain "probabilistic operators". We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively., Comment: arXiv admin note: text overlap with arXiv:1907.04036

We identify and highlight certain landmark results in Samson Abramsky's work which we believe are fundamental to current developments and future trends. In particular, we focus on the use of (i) topological duality methods to solve problems in logic and computer science; (ii) category theory and, more particularly, free (and co-free) constructions; (iii) these tools to unify the `power' and `structure' strands in computer science., Comment: 29 pages

Canonical extension of finitary ordered structures such as lattices, posets, proximity lattices, etc., is a certain completion which entirely describes the topological dual of the ordered structure and it does so in a purely algebraic and choice-free way. We adapt the general algebraic technique that constructs them to the theory of frames. As a result, we show that every locally compact frame embeds into a completely distributive lattice by a construction which generalises, among others, the canonical extensions for distributive lattices and proximity lattices. This construction also provides a new description of a construction by Marcel Ern\'e. Moreover, canonical extensions of frames enable us to frame-theoretically represent monotone maps with respect to the specialisation order.

A systematic theory of structural limits for finite models has been developed by Nesetril and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises --- via Stone-Priestley duality and the notion of types from model theory --- by enriching the expressive power of first-order logic with certain ``probabilistic operators''. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction. The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively., Comment: 19 pages

We provide a vivid demonstration of the mechanical effect of transverse spin momentum in an optical beam in free space. This component of the Poynting momentum was previously thought to be virtual, and unmeasurable. Here, its effect is revealed in the inertial motion of a probe particle in a circularly polarized Gaussian trap, in vacuum. Transverse spin forces combine with thermal fluctuations to induce a striking range of non-equilibrium phenomena. With increasing beam power we observe (i) growing departures from energy equipartition, (ii) the formation of coherent, thermally excited orbits and, ultimately, (iii) the ejection of the particle from the trap. Our results complement and corroborate recent measurements of spin momentum in evanescent waves, and extend them to a new geometry, in free space. In doing so, we exhibit fundamental, generic features of the mechanical interaction of circularly polarized light with matter. The work also shows how observations of the under-damped motion of probe particles can provide detailed information about the nature and morphology of momentum flows in arbitrarily structured light fields as well as providing a test bed for elementary non-equilibrium statistical mechanics.

Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard statistical analysis of unstable mechanical processes and their applications. A newly proposed approach takes advantage of the local characteristics of the most probable particle motion instead of the divergent averages. We experimentally verify its theoretical predictions for a Brownian particle moving near an inflection in a highly unstable cubic optical potential. The most-likely position of the particle atypically shifts against the force despite the trajectories diverge in the opposite direction. The local uncertainty around the most-likely position saturates even for strong diffusion and enables well-resolved position detection. Remarkably, the measured particle distribution quickly converges to the quasi-stationary one with the same atypical shift for different initial particle positions. The demonstrated experimental confirmation of the theoretical predictions approves the utility of local characteristics for highly unstable systems which can be exploited in thermodynamic processes to uncover energetics of unstable systems., Comment: SM as ancillary file

A fully reconfigurable two-dimensional (2D) rocking ratchet system created with holographic optical micromanipulation is presented. We can generate optical potentials with the geometry of any Bravais lattice in 2D and introduce a spatial asymmetry with arbitrary orientation. Nontrivial directed transport of Brownian particles along different directions is demonstrated numerically and experimentally, including on-axis, perpendicular and oblique with respect to an unbiased ac driving. The most important aspect to define the current direction is shown to be the asymmetry and not the driving orientation, and yet we show a system in which the asymmetry orientation of each potential well does not coincide with the transport direction, suggesting an additional symmetry breaking as a result of a coupling with the lattice configuration. Our experimental device, due to its versatility, opens up a new range of possibilities in the study of nonequilibrium dynamics at the microscopic level., Comment: 5 pages and 4 figures

A general theory of presentations for d-frames does not yet exist. We review the difficulties and give sufficient conditions for when they can be overcome. As an application we prove that the category of d-frames is closed under coproducts.

We report on a measurement of the Upsilon(1S+2S+3S) -> e+e- cross section at midrapidity in p+p collisions at sqrt(s)=200 GeV. We find the cross section to be 114 +/- 38 (stat.) +23,-24 (syst.) pb. Perturbative QCD calculations at next-to-leading order in the Color Evaporation Model are in agreement with our measurement, while calculations in the Color Singlet Model underestimate it by 2 sigma. Our result is consistent with the trend seen in world data as a function of the center-of-mass energy of the collision and extends the availability of Upsilon data to RHIC energies. The dielectron continuum in the invariant mass range near the Upsilon is also studied to obtain a combined cross section of Drell-Yan plus (b b-bar) -> e+e-., Comment: 17 pages, 14 figures, STAR Collaboration

High Energy Nuclear Physics (HENP) collaborations' experience show that the computing resources available at a single site are often neither sufficient nor satisfy the need of remote collaborators. From latencies in the network connectivity to the lack of interactivity, work at distant computing centers is often inefficient. Having fully functional software stack on local resources is a strong enabler of science opportunities for any local group who can afford the time investment. Prague's heavy-ions group participating in STAR experiment at RHIC has been a strong advocate of local computing as the most efficient means of data processing and physics analyses. Tier 2 computing center was set up at a Regional Computing Center for Particle Physics called "Golias". We report on our experience in setting up a fully functional Tier 2 center and discuss the solutions chosen to address storage space and analysis issues and the impact on the farms overall functionality. This includes a locally built STAR analysis framework, integration with a local DPM system (a cost effective storage solution), the influence of the availability and quality of the network connection to Tier 0 via a dedicated CESNET/ESnet link and the development of light-weight yet fully automated data transfer tools allowing the movement of entire datasets from BNL (Tier 0) to Golias (Tier 2)., Comment: To appear in proceedings of Computing in High Energy and Nuclear Physics 2009

We present measurements of net charge fluctuations in $Au + Au$ collisions at $\sqrt{s_{NN}} = $ 19.6, 62.4, 130, and 200 GeV, $Cu + Cu$ collisions at $\sqrt{s_{NN}} = $ 62.4, 200 GeV, and $p + p$ collisions at $\sqrt{s} = $ 200 GeV using the dynamical net charge fluctuations measure $\nu_{+-{\rm,dyn}}$. We observe that the dynamical fluctuations are non-zero at all energies and exhibit a modest dependence on beam energy. A weak system size dependence is also observed. We examine the collision centrality dependence of the net charge fluctuations and find that dynamical net charge fluctuations violate $1/N_{ch}$ scaling, but display approximate $1/N_{part}$ scaling. We also study the azimuthal and rapidity dependence of the net charge correlation strength and observe strong dependence on the azimuthal angular range and pseudorapidity widths integrated to measure the correlation., Comment: 15 pages, 6 figures, STAR Collaboration

We report the measurements of $\Sigma (1385)$ and $\Lambda (1520)$ production in $p+p$ and $Au+Au$ collisions at $\sqrt{s_{NN}} = 200$ GeV from the STAR collaboration. The yields and the $p_{T}$ spectra are presented and discussed in terms of chemical and thermal freeze-out conditions and compared to model predictions. Thermal and microscopic models do not adequately describe the yields of all the resonances produced in central $Au+Au$ collisions. Our results indicate that there may be a time-span between chemical and thermal freeze-out during which elastic hadronic interactions occur., Comment: 6 pages, 3 figures, Phys. Rev. Lett. 97, 132301 (2006)