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As supercomputers become larger with powerful Graphics Processing Unit (GPU), traditional direct eigensolvers struggle to keep up with the hardware evolution and scale efficiently due to communication and synchronization demands. Conversely, subspace eigensolvers, like the Chebyshev Accelerated Subspace Eigensolver (ChASE), have a simpler structure and can overcome communication and synchronization bottlenecks. ChASE is a modern subspace eigensolver that uses Chebyshev polynomials to accelerate the computation of extremal eigenpairs of dense Hermitian eigenproblems. In this work we show how we have modified ChASE by rethinking its memory layout, introducing a novel parallelization scheme, switching to a more performing communication-avoiding algorithm for one of its inner modules, and substituting the MPI library by the vendor-optimized NCCL library. The resulting library can tackle dense problems with size up to $N=\mathcal{O}(10^6)$, and scales effortlessly up to the full 900 nodes -- each one powered by 4$\times$A100 NVIDIA GPUs -- of the JUWELS Booster hosted at the J\"ulich Supercomputing Centre., Comment: 8 pages, accepted at the proceedings of ScalAH23 workshop within the SC23 conference

In the last decade, the use of Machine and Deep Learning (MDL) methods in Condensed Matter physics has seen a steep increase in the number of problems tackled and methods employed. A number of distinct MDL approaches have been employed in many different topics; from prediction of materials properties to computation of Density Functional Theory potentials and inter-atomic force fields. In many cases the result is a surrogate model which returns promising predictions but is opaque on the inner mechanisms of its success. On the other hand, the typical practitioner looks for answers that are explainable and provide a clear insight on the mechanisms governing a physical phenomena. In this work, we describe a proposal to use a sophisticated combination of traditional Machine Learning methods to obtain an explainable model that outputs an explicit functional formulation for the material property of interest. We demonstrate the effectiveness of our methodology in deriving a new highly accurate expression for the enthalpy of formation of solid solutions of lanthanides orthophosphates., Comment: 24 pages, 5 figures

As modern massively parallel clusters are getting larger with beefier compute nodes, traditional parallel eigensolvers, such as direct solvers, struggle keeping the pace with the hardware evolution and being able to scale efficiently due to additional layers of communication and synchronization. This difficulty is especially important when porting traditional libraries to heterogeneous computing architectures equipped with accelerators, such as Graphics Processing Unit (GPU). Recently, there have been significant scientific contributions to the development of filter-based subspace eigensolver to compute partial eigenspectrum. The simpler structure of these type of algorithms makes for them easier to avoid the communication and synchronization bottlenecks typical of direct solvers. The Chebyshev Accelerated Subspace Eigensolver (ChASE) is a modern subspace eigensolver to compute partial extremal eigenpairs of large-scale Hermitian eigenproblems with the acceleration of a filter based on Chebyshev polynomials. In this work, we extend our previous work on ChASE by adding support for distributed hybrid CPU-multi-GPU computing architectures. Our tests show that ChASE achieves very good scaling performance up to 144 nodes with 526 NVIDIA A100 GPUs in total on dense eigenproblems of size up to $360$k., Comment: Accepted for publication on the Proceedings of the PASC22 Conference

The last ten years have witnessed fast spreading of massively parallel computing clusters, from leading supercomputing facilities down to the average university computing center. Many companies in the private sector have undergone a similar evolution. In this scenario, the seamless integration of software and middleware libraries is a key ingredient to ensure portability of scientific codes and guarantees them an extended lifetime. In this work, we describe the integration of the ChASE library, a modern parallel eigensolver, into an existing legacy code for the first-principles computation of optical properties of materials via solution of the Bethe-Salpeter equation for the optical polarization function. Our numerical tests show that, as a result of integrating ChASE and parallelizing the reading routine, the code experiences a remarkable speedup and greatly improved scaling behavior on both multi- and many-core architectures. We demonstrate that such a modernized BSE code will, by fully exploiting parallel computing architectures and file systems, enable domain scientists to accurately study complex material systems that were not accessible before., Comment: 17 Pages plus 7 pages of supplemental information, 6 figures and 3 tables. To be submitted to Computer Physics Communications

Networks that have power-law connectivity, commonly referred to as the scale-free networks, are an important class of complex networks. A heterogeneous mean-field approximation has been previously proposed for the Ising model of the Barab\'{a}si-Albert model of scale-free networks with classical spins on the nodes wherein it was shown that the critical temperature for such a system scales logarithmically with network size. For finite sizes, there is no criticality for such a system and hence no true phase transition in terms of singular behavior. Further, in the thermodynamic limit, the mean-field prediction of an infinite critical temperature for the system may exclude any true phase transition even then. Nevertheless, with an eye on potential applications of the model on biological systems that are generally finite, one may still try to find approximations that describe the relevant observables quantitatively. Here we present an alternative, approximate formulation for the description of the Ising model of a Barab\'{a}si-Albert Network. Using the classical definition of magnetization, we show that Ising models on a network can be well-approximated by a long-range interacting homogeneous Ising model wherein each node of the network couples to all other spins with a strength determined by the mean degree of the Barab\'{a}si-Albert Network. In such an effective long-range Ising model of a Barab\'{a}si-Albert Network, the critical temperature is directly proportional to the number of preferentially attached links added to grow the network. The proposed model describes the magnetization of the majority of the sites with average or smaller than average degree better compared to the heterogeneous mean-field approximation. The long-range Ising model is the only homogeneous description of Barab\'{a}si-Albert networks that we know of., Comment: 30 pages, 6 figures, 2 tables, preliminary results of this work has been presented in the form of a talk at the Conference on Complex Systems 2019

We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert's pseudo-charge method [M. Weinert, J. Math. Phys. 22, 2433 (1981)] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert's convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation., Comment: submitted to the Journal of Mathematical Physics

In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of so-called rational filters, we introduce a systematic method to design these filters by minimizing the worst-case convergence ratio and eliminate the parametric dependence on weight functions. Further, we provide an efficient way to deal with the box-constraints which play a central role for the use of iterative linear solvers in contour-based eigensolvers. Indeed, these parameter-free filters consistently minimize the number of iterations and the number of FLOPs to reach convergence in the eigensolver. As a byproduct, our rational filters allow for a simple solution to load balancing when the solution of an interior eigenproblem is approached by the slicing of the sought after spectral interval., Comment: 23 pages, 7 figures

The central question of systems biology is to understand how individual components of a biological system such as genes or proteins cooperate in emerging phenotypes resulting in the evolution of diseases. As living cells are open systems in quasi-steady state type equilibrium in continuous exchange with their environment, computational techniques that have been successfully applied in statistical thermodynamics to describe phase transitions may provide new insights to emerging behavior of biological systems. Here we will systematically evaluate the translation of computational techniques from solid-state physics to network models that closely resemble biological networks and develop specific translational rules to tackle problems unique to living systems. Hence we will focus on logic models exhibiting only two states in each network node. Motivated by the apparent asymmetry between biological states where an entity exhibits boolean states i.e. is active or inactive, we present an adaptation of symmetric Ising model towards an asymmetric one fitting to living systems here referred to as the modified Ising model with gene-type spins. We analyze phase transitions by Monte Carlo simulations and propose mean-field solution of modified Ising model of a network type that closely resembles real-world network, the Barab\'{a}si-Albert model of scale-free networks. We show that asymmetric Ising models show similarities to symmetric Ising models with external field and undergoes a discontinuous phase transition of the first-order and exhibits hysteresis. The simulation setup presented here can be directly used for any biological network connectivity dataset and is also applicable for other networks that exhibit similar states of activity. This is a general statistical method to deal with non-linear large scale models arising in the context of biological systems and is scalable to any network size., Comment: 30 pages, 8 figures, presented in poster form in SBHD'18, LA, USA; as a talk in SIAM CSE'19

In this paper we propose an accurate, highly parallel algorithm for the generalized eigendecomposition of a matrix pair $(H, S)$, given in a factored form $(F^{\ast} J F, G^{\ast} G)$. Matrices $H$ and $S$ are generally complex and Hermitian, and $S$ is positive definite. This type of matrices emerges from the representation of the Hamiltonian of a quantum mechanical system in terms of an overcomplete set of basis functions. This expansion is part of a class of models within the broad field of Density Functional Theory, which is considered the golden standard in condensed matter physics. The overall algorithm consists of four phases, the second and the fourth being optional, where the two last phases are computation of the generalized hyperbolic SVD of a complex matrix pair $(F,G)$, according to a given matrix $J$ defining the hyperbolic scalar product. If $J = I$, then these two phases compute the GSVD in parallel very accurately and efficiently., Comment: The supplementary material is available at https://web.math.pmf.unizg.hr/mfbda/papers/sm-SISC.pdf due to its size. This revised manuscript is currently being considered for publication

Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties which are pertinent to the entire sequence, and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present ChASE, a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree which further reduces the necessary FLOPs. ChASE is written in C++ using the modern software engineering concepts which favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence's spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and it is easily extensible to other parallel computing architectures., Comment: 33 pages. Submitted to ACM TOMS

Legacy codes in computational science and engineering have been very successful in providing essential functionality to researchers. However, they are not capable of exploiting the massive parallelism provided by emerging heterogeneous architectures. The lack of portable performance and scalability puts them at high risk: either they evolve or they are doomed to disappear. One example of legacy code which would heavily benefit from a modern design is FLEUR, a software for electronic structure calculations. In previous work, the computational bottleneck of FLEUR was partially re-engineered to have a modular design that relies on standard building blocks, namely BLAS and LAPACK. In this paper, we demonstrate how the initial redesign enables the portability to heterogeneous architectures. More specifically, we study different approaches to port the code to architectures consisting of multi-core CPUs equipped with one or more coprocessors such as Nvidia GPUs and Intel Xeon Phis. Our final code attains over 70\% of the architectures' peak performance, and outperforms Nvidia's and Intel's libraries. Finally, on JURECA, the supercomputer where FLEUR is often executed, the code takes advantage of the full power of the computing nodes, attaining $5\times$ speedup over the sole use of the CPUs., Comment: 22 pages, submitted to special issue of CCPE

Spiking neuronal networks are usually simulated with three main simulation schemes: the classical time-driven and event-driven schemes, and the more recent hybrid scheme. All three schemes evolve the state of a neuron through a series of checkpoints: equally spaced in the first scheme and determined neuron-wise by spike events in the latter two. The time-driven and the hybrid scheme determine whether the membrane potential of a neuron crosses a threshold at the end of of the time interval between consecutive checkpoints. Threshold crossing can, however, occur within the interval even if this test is negative. Spikes can therefore be missed. The present work derives, implements, and benchmarks a method for perfect retrospective spike detection. This method can be applied to neuron models with affine or linear subthreshold dynamics. The idea behind the method is to propagate the threshold with a time-inverted dynamics, testing whether the threshold crosses the neuron state to be evolved, rather than vice versa. Algebraically this translates into a set of inequalities necessary and sufficient for threshold crossing. This test is slower than the imperfect one, but faster than an alternative perfect tests based on bisection or root-finding methods. Comparison confirms earlier results that the imperfect test rarely misses spikes (less than a fraction $1/10^8$ of missed spikes) in biologically relevant settings. This study offers an alternative geometric point of view on neuronal dynamics., Comment: 9 figures, Preliminary results in proceedings of the Bernstein Conference 2016

Rational filter functions can be used to improve convergence of contour-based eigensolvers, a popular family of algorithms for the solution of the interior eigenvalue problem. We present a framework for the optimization of rational filters based on a non-convex weighted Least-Squares scheme. When used in combination with the FEAST library, our filters out-perform existing ones on a large and representative set of benchmark problems. This work provides a detailed description of: (1) a set up of the optimization process that exploits symmetries of the filter function for Hermitian eigenproblems, (2) a formulation of the gradient descent and Levenberg-Marquardt algorithms that exploits the symmetries, (3) a method to select the starting position for the optimization algorithms that reliably produces effective filters, (4) a constrained optimization scheme that produces filter functions with specific properties that may be beneficial to the performance of the eigensolver that employs them.

In this paper we focus on the integration of high-performance numerical libraries in ab initio codes and the portability of performance and scalability. The target of our work is FLEUR, a software for electronic structure calculations developed in the Forschungszentrum J\"ulich over the course of two decades. The presented work follows up on a previous effort to modernize legacy code by re-engineering and rewriting it in terms of highly optimized libraries. We illustrate how this initial effort to get efficient and portable shared-memory code enables fast porting of the code to emerging heterogeneous architectures. More specifically, we port the code to nodes equipped with multiple GPUs. We divide our study in two parts. First, we show considerable speedups attained by minor and relatively straightforward code changes to off-load parts of the computation to the GPUs. Then, we identify further possible improvements to achieve even higher performance and scalability. On a system consisting of 16-cores and 2 GPUs, we observe speedups of up to 5x with respect to our optimized shared-memory code, which in turn means between 7.5x and 12.5x speedup with respect to the original FLEUR code.

The conceptual framework provided by the functional Renormalization Group (fRG) has become a formidable tool to study correlated electron systems on lattices which, in turn, provided great insights to our understanding of complex many-body phenomena, such as high- temperature superconductivity or topological states of matter. In this work we present one of the latest realizations of fRG which makes use of an adaptive numerical quadrature scheme specifically tailored to the described fRG scheme. The final result is an increase in performance thanks to improved parallelism and scalability., Comment: 14 pages, 4 figures, accepted for publication in the proceedings of the JHPCS'16

Many physical properties of metals can be understood in terms of the free electron model, as proven by the Wiedemann-Franz law. According to this model, electronic thermal conductivity ($\kappa_{el}$) can be inferred from the Boltzmann transport equation (BTE). However, the BTE does not perform well for some complex metals, such as Cu. Moreover, the BTE cannot clearly describe the origin of the thermal energy carried by electrons or how this energy is transported in metals. The charge distribution of conduction electrons in metals is known to reflect the electrostatic potential (EP) of the ion cores. Based on this premise, we develop a new methodology for evaluating $\kappa_{el}$ by combining the free electron model and non-equilibrium ab initio molecular dynamics (NEAIMD) simulations. We demonstrate that the kinetic energy of thermally excited electrons originates from the energy of the spatial electrostatic potential oscillation (EPO), which is induced by the thermal motion of ion cores. This method directly predicts the $\kappa_{el}$ of pure metals with a high degree of accuracy., Comment: 7 pages, 3 figures, with Supplementary Information of 19 pages, 7 figures and 7 tables

One of the greatest efforts of computational scientists is to translate the mathematical model describing a class of physical phenomena into large and complex codes. Many of these codes face the difficulty of implementing the mathematical operations in the model in terms of low level optimized kernels offering both performance and portability. Legacy codes suffer from the additional curse of rigid design choices based on outdated performance metrics (e.g. minimization of memory footprint). Using a representative code from the Materials Science community, we propose a methodology to restructure the most expensive operations in terms of an optimized combination of dense linear algebra kernels. The resulting algorithm guarantees an increased performance and an extended life span of this code enabling larger scale simulations., Comment: Second revised version. Corrected notation. Added acknowledgment. 30 pages, 2 figures and two tables. Submitted to a Special Issue of Computer Physics Communication

In many scientific applications the solution of non-linear differential equations are obtained through the set-up and solution of a number of successive eigenproblems. These eigenproblems can be regarded as a sequence whenever the solution of one problem fosters the initialization of the next. In addition, in some eigenproblem sequences there is a connection between the solutions of adjacent eigenproblems. Whenever it is possible to unravel the existence of such a connection, the eigenproblem sequence is said to be correlated. When facing with a sequence of correlated eigenproblems the current strategy amounts to solving each eigenproblem in isolation. We propose a alternative approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials (ChFSI). The resulting eigensolver is optimized by minimizing the number of matrix-vector multiplications and parallelized using the Elemental library framework. Numerical results show that ChFSI achieves excellent scalability and is competitive with current dense linear algebra parallel eigensolvers., Comment: 23 Pages, 6 figures. First revision of an invited submission to special issue of Concurrency and Computation: Practice and Experience

Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often an exact count is not necessary and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure., Comment: 24 pages and 8 figures. Submitted to Numerical Linear Algebra with Applications

Mathematical operators whose transformation rules constitute the building blocks of a multi-linear algebra are widely used in physics and engineering applications where they are very often represented as tensors. In the last century, thanks to the advances in tensor calculus, it was possible to uncover new research fields and make remarkable progress in the existing ones, from electromagnetism to the dynamics of fluids and from the mechanics of rigid bodies to quantum mechanics of many atoms. By now, the formal mathematical and geometrical properties of tensors are well defined and understood; conversely, in the context of scientific and high-performance computing, many tensor- related problems are still open. In this paper, we address the problem of efficiently computing contractions among two tensors of arbitrary dimension by using kernels from the highly optimized BLAS library. In particular, we establish precise conditions to determine if and when GEMM, the kernel for matrix products, can be used. Such conditions take into consideration both the nature of the operation and the storage scheme of the tensors, and induce a classification of the contractions into three groups. For each group, we provide a recipe to guide the users towards the most effective use of BLAS., Comment: 27 Pages, 7 figures and additional tikz generated diagrams. Submitted to Applied Mathematics and Computation

In one of the most important methods in Density Functional Theory - the Full-Potential Linearized Augmented Plane Wave (FLAPW) method - dense generalized eigenproblems are organized in long sequences. Moreover each eigenproblem is strongly correlated to the next one in the sequence. We propose a novel approach which exploits such correlation through the use of an eigensolver based on subspace iteration and accelerated with Chebyshev polynomials. The resulting solver, parallelized using the Elemental library framework, achieves excellent scalability and is competitive with current dense parallel eigensolvers., Comment: Submitted to 10th International Conference on Parallel Processing and Applied Mathematics(PPAM 2013)

In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct cycles have either been believed to be independent or at most very loosely connected. In a recent study [7], it was demonstrated that, contrary to belief, successive eigenproblems in a sequence are strongly correlated with one another. In particular, by monitoring the subspace angles between eigenvectors of successive eigenproblems, it was shown that these angles decrease noticeably after the first few iterations and become close to collinear. This last result suggests that we can manipulate the eigenvectors, solving for a specific eigenproblem in a sequence, as an approximate solution for the following eigenproblem. In this work we present results that are in line with this intuition. We provide numerical examples where opportunely selected block iterative eigensolvers benefit from the reuse of eigenvectors by achieving a substantial speed-up. The results presented will eventually open the way to a widespread use of block iterative eigensolvers in ab initio electronic structure codes based on the LAPW approach., Comment: 12 Pages, 5 figures. Accepted for publication on Computer Physics Communications

We analyze the FEAST method for computing selected eigenvalues and eigenvectors of large sparse matrix pencils. After establishing the close connection between FEAST and the well-known Rayleigh-Ritz method, we identify several critical issues that influence convergence and accuracy of the solver: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals. We complement the study with numerical examples, and hint at possible improvements to overcome the existing problems., Comment: 11 Pages, 5 Figures. Submitted to Journal of Computational and Applied Mathematics

We compare two approaches to compute a portion of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism of the method to the performance of the solvers. The experimental results on a state-of-the-art 8-core platform, equipped with a graphics processing unit (GPU), reveal that in real applications, iterative Krylov-subspace methods can be a competitive approach also for the solution of dense problems., Comment: 5 tables and 4 figures. In press by Applied Mathematics and Computation. Accepted version

Density Functional Theory (DFT) is one of the most used ab initio theoretical frameworks in materials science. It derives the ground state properties of a multi-atomic ensemble directly from the computation of its one-particle density \nr .In DFT-based simulations the solution is calculated through a chain of successive self-consistent cycles; in each cycle a series of coupled equations (Kohn-Sham) translates to a large number of generalized eigenvalue problems whose eigenpairs are the principal means for expressing \nr. A simulation ends when \nr\ has converged to the solution within the required numerical accuracy. This usually happens after several cycles, resulting in a process calling for the solution of many sequences of eigenproblems. In this paper, the authors report evidence showing unexpected correlations between adjacent eigenproblems within each sequence. By investigating the numerical properties of the sequences of generalized eigenproblems it is shown that the eigenvectors undergo an "evolution" process. At the same time it is shown that the Hamiltonian matrices exhibit a similar evolution and manifest a specific pattern in the information they carry. Correlation between eigenproblems within a sequence is of capital importance: information extracted from the simulation at one step of the sequence could be used to compute the solution at the next step. Although they are not explored in this work, the implications could be manifold: from increasing the performance of material simulations, to the development of an improved iterative solver, to modifying the mathematical foundations of the DFT computational paradigm in use, thus opening the way to the investigation of new materials., Comment: 11 Pages and 6 figures. Added references, appendix and figures. Accepted for publication by Computer Physics Communications

In this short paper, the authors report a new computational approach in the context of Density Functional Theory (DFT). It is shown how it is possible to speed up the self-consistent cycle (iteration) characterizing one of the most well-known DFT implementations: FLAPW. Generating the Hamiltonian and overlap matrices and solving the associated generalized eigenproblems $Ax = \lambda Bx$ constitute the two most time-consuming fractions of each iteration. Two promising directions, implementing the new methodology, are presented that will ultimately improve the performance of the generalized eigensolver and save computational time., Comment: To appear in the proceedings of 8th International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 2010)

One of the most used approaches in simulating materials is the tight-binding approximation. When using this method in a material simulation, it is necessary to compute the eigenvalues and eigenvectors of the Hamiltonian describing the system. In general, the system possesses few explicit symmetries. Due to them, the problem has many degenerate eigenvalues. The ambiguity in choosing a orthonormal basis of the invariant subspaces, associated with degenerate eigenvalues, will result in eigenvectors which are not invariant under the action of the symmetry operators in matrix form. A meaningful computation of the eigenvectors needs to take those symmetries into account. A natural choice is a set of eigenvectors, which simultaneously diagonalizes the Hamiltonian and the symmetry matrices. This is possible because all the matrices commute with each other. The simultaneous eigenvectors and the corresponding eigenvalues will be in a parametrized form in terms of the lattice momentum components. This functional dependence of the eigenvalues is the dispersion relation and describes the band structure of a material. Therefore it is important to find this functional dependence in any numerical computation related to material properties., Comment: To appear in the proceedings of the 7th International Conference on Computational Methods in Science and Engineering (ICCMSE '09)

We deconstruct the fifth dimension of 5D SCQD with general numbers of colors and flavors and general 5D Chern-Simons level; the latter is adjusted by adding extra quarks to the 4D quiver. We use deconstruction as a non-stringy UV completion of the quantum 5D theory; to prove its usefulness, we compute quantum corrections to the SQCD_5 prepotential. We also explore the moduli/parameter space of the deconstructed SQCD_5 and show that for |K_CS| < N_F/2 it continues to negative values of 1/(g_5)^2. In many cases there are flop transitions connecting SQCD_5 to exotic 5D theories such as E0, and we present several examples of such transitions. We compare deconstruction to brane-web engineering of the same SQCD_5 and show that the phase diagram is the same in both cases; indeed, the two UV completions are in the same universality class, although they are not dual to each other. Hence, the phase structure of an SQCD_5 (and presumably any other 5D gauge theory) is inherently five-dimensional and does not depends on a UV completion., Comment: LaTeX+PStricks, 108 pages, 41 colored figures. Please print in color

We study an example of Grand Unified Theory (GUT), known as trinification, which was first introduced in 1984 by S.Glashow. This model has the GUT gauge group as $[SU(3)]^3$ with a discrete $\mathbb{Z}_3$ to ensure the couplings are unified at the GUT scale. In this letter we consider this trinification model in its minimal formulation and investigate its robustness in the context of cosmology. In particular we show that for a large set of the parameter space the model doesn't seem to provide a Dark Matter candidate compatible with cosmological data., Comment: To appear in the LXXXVI session of the "Les Houches" summer school. 9 pages, 2 graphs

Abelian quiver gauge theories provide nonsupersymmetric candidates for the conformality approach to physics beyond the standard model. Written as ${\cal N}=0$, $U(N)^n$ gauge theories, however, they have mixed $U(1)_p U(1)_q^2$ and $U(1)_p SU(N)_q^2$ triangle anomalies. It is shown how to construct explicitly a compensatory term $\Delta{\cal L}_{comp}$ which restores gauge invariance of ${\cal L}_{eff} = {\cal L} + \Delta {\cal L}_{comp}$ under $U(N)^n$. It can lead to a negative contribution to the U(1) $\beta$-function and hence to one-loop conformality at high energy for all dimensionless couplings., Comment: 15 pages. minor clarifications

If one tries to view de Sitter as a true (as opposed to a meta-stable) vacuum, there is a tension between the finiteness of its entropy and the infinite-dimensionality of its Hilbert space. We invetsigate the viability of one proposal to reconcile this tension using $q$-deformation. After defining a differential geometry on the quantum de Sitter space, we try to constrain the value of the deformation parameter by imposing the condition that in the undeformed limit, we want the real form of the (inherently complex) quantum group to reduce to the usual SO(4,1) of de Sitter. We find that this forces $q$ to be a real number. Since it is known that quantum groups have finite-dimensional representations only for $q=$ root of unity, this suggests that standard $q$-deformations cannot give rise to finite dimensional Hilbert spaces, ruling out finite entropy for q-deformed de Sitter., Comment: 10 pages, v2: references added, v3: minor corrections, abstract and title made more in-line with the result, v4: published version

With a view towards applications for de Sitter, we construct the multi-parametric $q$-deformation of the $so(5,\IC)$ algebra using the Faddeev-Reshetikhin-Takhtadzhyan (FRT) formalism., Comment: v4: cosmetic changes from the published version

We build a matrix model of a chiral [SU(N)]^K gauge theory (5D SQCD deconstructed down to 4D) using random unitary matrices to model chiral bifundamental fields (N,N-bar) (without (N-bar,N)). We verify the duality by matching the loop equation of the matrix model to the anomaly equations of the gauge theory. Then we evaluate the matrix model's free energy and use it to derive the effective superpotential for the gaugino condensates., Comment: 41 pages, LaTeX, PStricks figures

Dimensional deconstruction of 5D SQCD with general n_c, n_f and k_CS gives rise to 4D N=1 gauge theories with large quivers of SU(n_c) gauge factors. We construct the chiral rings of such [SU(n_c)]^N theories, off-shell and on-shell. Our results are broadly similar to the chiral rings of single U(n_c) theories with both adjoint and fundamental matter, but there are also some noteworthy differences such as nonlocal meson-like operators where the quark and antiquark fields belong to different nodes of the quiver. And because our gauge groups are SU(n_c) rather than U(n_c), our chiral rings also contain a whole zoo of baryonic and antibaryonic operators., Comment: 93 pages, LaTeX, PSTricks macros; 1 reference added in v2

This is a short report of the entire work developed during the study of the possible realizations of the "Top Mode" Standard Model. Here it is examined the breaking of internal symmetries from another point of view showing that is possible to reproduce the gauged electroweak panorama of the traditional Standard Model in a exhaustive and selfconsistent way. The result is reached applying the main futures of the Nambu-Jona Lasinio (NJL) mechanism to an electroweak invariant Lagrangian. In this context the use of functional formalism for composite operators naturally leads to a different dynamical approach. Meanwhile the Higgs mechanism acts on the Lagrangian form, a NJL like model looks directly at the physics of the system showing the real dynamical content hidden in the Green functions of the theory., Comment: 18 Pages, no figures, LaTex, corrected typos, references removed