arXiv subjects
Publications and source records attributed to Marco Frasca.
We provide a study of quantum chromodynamics with the technique of Dyson-Schwinger equations in differential form. In this way, we are able to approach the non-perturbative limit and recover, with some approximations, the 't Hooft limit of the theory. Quark mass in the propagator term goes off-shell at low-energies signaling confinement. A condition for such occurrence in the theory is provided.
Using a technique devised by Bender, Milton and Savage, we derive the Dyson-Schwinger equations for quantum chromodynamics in differential form. We stop our analysis to the two-point functions. The 't~Hooft limit of color number going to infinity is derived showing how these equations can be cast into a treatable even if approximate form. It is seen how this limit gives a sound description of the low-energy behavior of quantum chromodynamics by discussing the dynamical breaking of chiral symmetry and confinement, providing a condition for the latter. This approach exploits a background field technique in quantum field theory.
Recent advances in deep learning have made available large, powerful convolutional neural networks (CNN) with state-of-the-art performance in several real-world applications. Unfortunately, these large-sized models have millions of parameters, thus they are not deployable on resource-limited platforms (e.g. where RAM is limited). Compression of CNNs thereby becomes a critical problem to achieve memory-efficient and possibly computationally faster model representations. In this paper, we investigate the impact of lossy compression of CNNs by weight pruning and quantization, and lossless weight matrix representations based on source coding. We tested several combinations of these techniques on four benchmark datasets for classification and regression problems, achieving compression rates up to $165$ times, while preserving or improving the model performance.
We consider a set of solutions for a massless quartic scalar field, recently devised, that satisfy a massive dispersion relation. We show that such solutions have the property to give the correct behavior for the equation of state of the dark energy. It seen that conformal invariance is restored and the mass gap goes to zero on a time scale determined by the Hubble constant and the strength of the self-interaction of the scalar field. When conformal invariance is restored, the equation of state for the dark energy can apply.
Recently, we have found an exact solution to the full set of Dyson-Schwinger equations of the non-interacting part of the Higgs sector of the Standard Model obtained by solving the 1-point correlation function equation. In this work we extend this analysis considering also the other possible solution that is the one experimentally observed in the Standard Model. Indeed, the same set of Dyson-Schwinger equations can be exactly solved for the Standard Model with a constant as a solution for the 1-point correlation function. Differently from the Standard Model solution, the one we have found has a mass spectrum of a Kaluza-Klein particle. This could be a clue toward the identification of a further space dimension. Gap equations are obtained in both cases as also the running self-coupling equations.
We consider a simple model for the diffusion of heavy quarks in a hot bath, modeling the latter by an ensemble of oscillators distributed accorded to either a thermal distribution or to an out-of-equilibrium distribution with a saturation scale. Within this model it is easy to introduce memory effects by changing the distribution of oscillators: we model these by introducing a gaussian distribution, $dN/dω$, which can be deformed continuously from a $δ-$function giving a Markov dissipation to a broad kernel with memory. Deriving the equation of motion of the heavy quark in the bath we remark how dissipation comes out naturally as an effect of the back-reaction on the bath of oscillators. Moreover, the exact solution of this equation allows for the definition of a thermalization time as the time necessary to remove any memory of the initial condition. We find that the broadening the dissipative kernel while keeping the coupling fixed lowers the thermalization time. We also derive the fluctuation-dissipation theorem for the bath, and use this to estimate the kinematic regime in which momentum diffusion of the heavy quark dominates over drift: we find that diffusion is more important as long as $K_0/{\cal E}$ is small, where $K_0$ and ${\cal E}$ denote the initial energy of the heavy quark and the average energy of the bath respectively.
Multitask algorithms typically use task similarity information as a bias to speed up and improve the performance of learning processes. Tasks are learned jointly, sharing information across them, in order to construct models more accurate than those learned separately over single tasks. In this contribution, we present the first multitask model, to our knowledge, based on Hopfield Networks (HNs), named HoMTask. We show that by appropriately building a unique HN embedding all tasks, a more robust and effective classification model can be learned. HoMTask is a transductive semi-supervised parametric HN, that minimizes an energy function extended to all nodes and to all tasks under study. We provide theoretical evidence that the optimal parameters automatically estimated by HoMTask make coherent the model itself with the prior knowledge (connection weights and node labels). The convergence properties of HNs are preserved, and the fixed point reached by the network dynamics gives rise to the prediction of unlabeled nodes. The proposed model improves the classification abilities of singletask HNs on a preliminary benchmark comparison, and achieves competitive performance with state-of-the-art semi-supervised graph-based algorithms.
Motivated by applications in protein function prediction, we consider a challenging supervised classification setting in which positive labels are scarce and there are no explicit negative labels. The learning algorithm must thus select which unlabeled examples to use as negative training points, possibly ending up with an unbalanced learning problem. We address these issues by proposing an algorithm that combines active learning (for selecting negative examples) with imbalance-aware learning (for mitigating the label imbalance). In our experiments we observe that these two techniques operate synergistically, outperforming state-of-the-art methods on standard protein function prediction benchmarks.
The well-known Green's function method has been recently generalized to nonlinear second order differential equations. In this paper we study possibilities of exact Green's function solutions of nonlinear differential equations of higher order. We show that, if the nonlinear term satisfies a generalized homogeneity property, then the nonlinear Green's function can be represented in terms of the homogeneous solution. Specific examples and a numerical error analysis support the advantage of the method. We show how, for the Bousinesq and Kortweg-de Vries equations, we are forced to introduce higher order Green functions to obtain the solution to the inhomogeneous equation. The method proves to work also in this case supporting our generalization that yields a closed form solution to a large class of nonlinear differential equations, providing also a formula easily amenable to numerical evaluation.
We derive the critical temperature in a nonlocal Nambu-Jona-Lasinio model with the presence of a chiral chemical potential. The model we consider uses a form factor derived from recent studies of the gluon propagator in Yang-Mills theory and has the property to fit in excellent way the form factor arising from the instanton liquid picture for the vacuum of the theory. Nambu-Jona-Lasinio model is derived form quantum chromodynamics providing all the constants of the theory without any need for fits. We show that the critical temperature in this case always exists and increases as the square of the chiral chemical potential. The expression we obtain for the critical temperature depends on the mass gap that naturally arises from Yang-Mills theory at low-energy as also confirmed by lattice computations.
In this paper we study some classes of second order non-homogeneous nonlinear differential equations allowing a specific representation for nonlinear Green's function. In particular, we show that if the nonlinear term possesses a special multiplicativity property, then its Green's function is represented as the product of the Heaviside function and the general solution of the corresponding homogeneous equations subject to non-homogeneous Cauchy conditions. Hierarchies of specific non-linearities admitting this representation are derived. The nonlinear Green's function solution is numerically justified for the sinh-Gordon and Liouville equations. We also list two open problems leading to a more thorough characterizations of non-linearities admitting the obtained representation for the nonlinear Green's function.
The Green's function method which has been originally proposed for linear systems has several extensions to the case of nonlinear equations. A recent extension has been proposed to deal with certain applications in quantum field theory. The general solution of second order nonlinear differential equations is represented in terms of a so-called short time expansion. The first term of the expansion has been shown to be an efficient approximation of the solution for small values of the state variable. The proceeding terms contribute to the error correction. This paper is devoted to extension of the short time expansion solution to non-linearities depending on the first derivative of the unknown function. Under a proper assumption on the nonlinear term, a general representation for Green's function is derived. It is also shown how the knowledge of nonlinear Green's function can be used to study the spectrum of the nonlinear operator. Particular cases and their numerical analysis support the advantage of the method. The technique we discuss grants to obtain a closed form analytic solution for non-homogeneous non-linear PDEs so far amenable just to numerical solutions. This opens up the possibility of several applications in physics and engineering.
This paper is about the surprising connection between the Fourier heat equation and the Schrödinger wave equation. In fact, if the independent "time" variable in the heat equation is replaced by the time variable multiplied by $i=\sqrt{-1}$, the heat equation becomes the Schrödinger equation. Two quite different physical phenomena are put in close connection: the heat diffusion in a material and the probability amplitude of particles in an atom. It is a fact of life that the movements of a small particle floating randomly in a fluid, the well-known Brownian motion, is regulated by the Fourier equation while the probabilistic behavior of the matter around us, the quantum world, is driven by the Schrödinger equation but no known stochastic process seems at work here. The apparent simplicity of the formal connection by a "time-rotation", a Wick rotation as it is commonly known, seems to point otherwise. Why this connection? Is there any physical intuitive explanation? Is there any practical value? In this paper, the authors attempt to shed some light on the above questions. The recent concept of volume quantization in noncommutative geometry, due to Connes, Chamseddine and Mukhanov, points again to stochastic processes also underlying the quantum world making Fourier and Schröodinger strict relatives.
Public repositories for genome and proteome annotations, such as the Gene Ontology (GO), rarely stores negative annotations, i.e. proteins not possessing a given function. This leaves undefined or ill defined the set of negative examples, which is crucial for training the majority of machine learning methods inferring proteins functions. Automated techniques to choose reliable negative proteins are thereby required to train accurate function prediction models. This study proposes the first extensive analysis of the temporal evolution of protein annotations in the GO repository. Novel annotations registered through the years have been analyzed to verify the presence of annotation patterns in the GO hierarchy. Our research supplied fundamental clues about proteins likely to be unreliable as negative examples, that we verified into a novel algorithm of our own construction, validated on two organisms in a genome wide fashion against approaches proposed to choose negative examples in the context of functional prediction.
We show that a criterion for confinement, based on the BRST invariance, holds in four dimensions, by solving a non-Abelian gauge theory with a set of exact solutions. The confinement condition we consider was obtained by Kugo and Ojima some decades ago. The current understanding of gauge theories permits us to apply the techniques straightforwardly for checking the validity of this criterion. In this way, we are able to show that the non-Abelian gauge theory is confining and that confinement is rooted in the BRST invariance and asymptotic freedom.
We solve exactly the Dyson-Schwinger equations for Yang-Mills theory in 3 and 4 dimensions. This permits us to obtain the exact correlation functions till order 2. In this way, the spectrum of the theory is straightforwardly obtained and comparison with lattice data can be accomplished. The results are in exceedingly good agreement with an error well below 1\%. This extends both to 3 and 4 dimensions and varying the degree of the gauge group. These results provide a strong support to the value of the lattice computations and show once again how precise can be theoretical computations in quantum field theory.
Hawking radiation is generally derived using a non-interacting field theory. Some time ago, Leahy and Unruh showed that, in two dimensions with a Schwarzschild geometry, a scalar field theory with a quartic interaction gets the coupling switched off near the horizon of the black hole. This would imply that interaction has no effect on Hawking radiation and free theory for particles can be used. Recently, a set of exact classical solutions for the quartic scalar field theory has been obtained. These solutions display a massive dispersion relation even if the starting theory is massless. When one considers the corresponding quantum field theory, this mass gap becomes a tower of massive excitations and, at the leading order, the theory is trivial. We apply these results to Hawking radiation for a Kerr geometry and prove that the Leahy-Unruh effect is at work. Approaching the horizon the scalar field theory has the mass gap going to zero. We devise a technique to study the interacting scalar theory very near the horizon increasing the coupling. As these solutions are represented by a Fourier series of plane waves, Hawking radiation can be immediately obtained with well-known techniques. These results open a question about the behavior of the Standard Model of particles very near the horizon of a black hole where the interactions turn out to be switched off and the electroweak symmetry could be restored.
The recent analysis on noncommutative geometry, showing quantization of the volume for the Riemannian manifold entering the geometry, can support a view of quantum mechanics as arising by a stochastic process on it. A class of stochastic processes can be devised, arising as fractional powers of an ordinary Wiener process, that reproduce in a proper way a stochastic process on a noncommutative geometry. These processes are characterized by producing complex values and so, the corresponding Fokker-Planck equation resembles the Schroedinger equation. Indeed, by a direct numerical check, one can recover the kernel of the Schroedinger equation starting by an ordinary Brownian motion. This class of stochastic processes needs a Clifford algebra to exist. In four dimensions, the full set of Dirac matrices is needed and the corresponding stochastic process in a noncommutative geometry is easily recovered as is the Dirac equation in the Klein-Gordon form being it the Fokker--Planck equation of the process.