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We recall a natural framework to deal with local field theory in which bare amplitudes are completely finite. We first present the main general properties of this scheme, the so-called Taylor-Lagrange regularization scheme. We then investigate the consequences of this scheme on the calculation of perturbative radiative corrections to the Higgs mass within the Standard Model. Important consequences for the renormalization group equations are finally discussed.

We give a detailed general description of a recent geometrical discretisation scheme and illustrate, by explicit numerical calculation, the scheme's ability to capture topological features. The scheme is applied to the Abelian Chern-Simons theory and leads, after a necessary field doubling, to an expression for the discrete partition function in terms of untwisted Reidemeister torsion and of various triangulation dependent factors. The discrete partition function is evaluated computationally for various triangulations of $S^3$ and of lens spaces. The results confirm that the discretisation scheme is triangulation independent and coincides with the continuum partition function

In this highly speculative note we conjecture that it may be possible to understand features of coincident D-branes, such as the appearance of enhanced non-abelian gauge symmetry, in a purely geometric fashion, using a form of geometry known as scheme theory. We give a very brief introduction to some relevant ideas from scheme theory, and point out how these ideas work in special cases.

Gauge field theory of a horizontal symmetry of the group G = SU(2) X U(1) is developed so as to generalize the standard model of particle physics. All fermion and scalar fields are assumed to belong to doublets and singlets of the group in high energy regime. Mass matrices with four texture zeros of Dirac and Majorana types are systematically derived. In addition to seven scalar particles, the theory predicts existence of one peculiar vector particle which seems to play important roles in astrophysics and particle physics.

To resum large logarithms in multi-scale problems a generalization of $\MS$ is introduced allowing for as many renormalization scales as there are generic scales in the problem. In the new \lq\lq minimal multi-scale subtraction scheme'' standard perturbative boundary conditions become applicable. However, the multi-loop beta functions depend on the various renormalization scale ratios and a large logarithms resummation has to be performed on them. Using these improved beta functions the \lq\lq partial'' renormalization group equations corresponding to the renormalization point independence of physical quantities allows one to resum the logarithms. As an application the leading and next-to-leading order two-scale analysis of the effective potential in the $O(N)$-symmetric $\phi^4$-theory is performed. This calculation indicates that there is no stable vacuum in the broken phase of the theory for $1

We define an unstable equivariant motivic homotopy category for an algebraic group over a Noetherian base scheme. We show that equivariant algebraic $K$-theory is representable in the resulting homotopy category. Additionally, we establish homotopical purity and blow-up theorems for finite abelian groups.

We present an efficient perturbative method to obtain both static and dynamic polarizabilities and hyperpolarizabilities of complex electronic systems. This approach is based on the solution of a frequency dependent Sternheimer equation, within the formalism of time-dependent density functional theory, and allows the calculation of the response both in resonance and out of resonance. Furthermore, the excellent scaling with the number of atoms opens the way to the investigation of response properties of very large molecular systems. To demonstrate the capabilities of this method, we implemented it in a real-space (basis-set free) code, and applied it to benchmark molecules, namely CO, H2O, and paranitroaniline (PNA). Our results are in agreement with experimental and previous theoretical studies, and fully validate our approach.

Starting from the observation that one of the most successful methods for solving the Kohn-Sham equations for periodic systems -- the plane-wave method -- is a spectral method based on eigenfunction expansion, we formulate a spectral method designed towards solving the Kohn-Sham equations for clusters. This allows for efficient calculation of the electronic structure of clusters (and molecules) with high accuracy and systematic convergence properties without the need for any artificial periodicity. The basis functions in this method form a complete orthonormal set and are expressible in terms of spherical harmonics and spherical Bessel functions. Computation of the occupied eigenstates of the discretized Kohn-Sham Hamiltonian is carried out using a combination of preconditioned block eigensolvers and Chebyshev polynomial filter accelerated subspace iterations. Several algorithmic and computational aspects of the method, including computation of the electrostatics terms and parallelization are discussed. We have implemented these methods and algorithms into an efficient and reliable package called ClusterES (Cluster Electronic Structure). A variety of benchmark calculations employing local and non-local pseudopotentials are carried out using our package and the results are compared to the literature. Convergence properties of the basis set are discussed through numerical examples. Computations involving large systems that contain thousands of electrons are demonstrated to highlight the efficacy of our methodology. The use of our method to study clusters with arbitrary point group symmetries is briefly discussed.

We construct and study a generalized one-parameter class of scheme transformations, denoted $S_{R,m,k_1}$ with $m \ge 2$, with the property that an $S_{R,m,k_1}$ scheme transformation eliminates the $\ell$-loop terms in the beta function of a gauge theory from loop order $\ell=3$ to order $\ell=m+1$, inclusive. These scheme transformations are applied to the higher-loop calculation of the infrared zero of the beta function of an asymptotically free gauge theory with multiple fermions. We show that scheme transformations in this generalized class satisfy a set of criteria for physical acceptability over a larger range of numbers of fermions than previously studied scheme transformations. We also present an interesting modification of a different type of scheme transformation that removes the three-loop term in the beta function.

In this thesis, we propose a new software defined network monitoring scheme that provides the controller with a method to determine network states for the purpose of updating flow rules for network control and management. Network centrality and nodal influence metrics derived from the dual basis concept of the graph theory are used to monitor changes in a network. The proposed scheme uses a phantom node and the concept of a reference node to determine changes in these metrics in order to identify disconnected, congested, underutilized, and attacked nodes. The phantom node establishes a congestion threshold in the dual basis that is used to determine changes in node health and capacity due to network traffic. Multiple phantom nodes are used to produce multiple congestion thresholds for network monitoring. A congestion estimation method is proposed to estimate a node’s capacity used when it crosses the congestion threshold. Simulations are used to validate the concept of reference node, identification of node disconnections, congestion, and attacks, and the congestion estimation method.

We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. We aim at an effective action, where all field modes below the orbital scale are integrated out. We spell out the relevant degrees of freedom, in particular the rotational ones, and the associated symmetries. Building on these symmetries, we introduce the minimal coupling part of the point particle action in terms of gauge rotational variables, and construct the spin-induced nonminimal couplings, where we obtain the leading order couplings to all orders in spin. We specify the gauge for the rotational variables, where the unphysical degrees of freedom are eliminated already from the Feynman rules, and all the orbital field modes are integrated out. The equations of motion of the spin can be directly obtained via a proper variation of the action, and Hamiltonians may be straightforwardly derived. We implement this effective field theory for spin to derive all spin dependent potentials up to next-to-leading order to quadratic level in spin, namely up to the third post-Newtonian order for rapidly rotating compact objects. In particular, the proper next-to-leading order spin-squared potential and Hamiltonian for generic compact objects are also derived. For the implementations we use the nonrelativistic gravitational field decomposition, which is found here to eliminate higher-loop Feynman diagrams also in spin dependent sectors, and facilitates derivations. This formulation for spin is thus ideal for treatment of higher order spin dependent sectors.

We argue that the choice of an appropriate, massive, renormalization scheme can greatly improve the apparent convergence of perturbation theory at finite temperature. This is illustrated by the calculation of the pressure of a scalar field theory with quartic interactions, at 2-loop order. The result, almost identical to that obtained with more sophisticated resummation techniques, shows a remarkable stability as the coupling constant grows, in sharp contrast with standard perturbation theory.

Subdivergences constitute a major obstacle to the evaluation of Feynman integrals and an expression in terms of finite quantities can be a considerable advantage for both analytic and numeric calculations. We report on our implementation of the suggestion by F. Brown and D. Kreimer, who proposed to use a modified BPHZ scheme where all counterterms are single-scale integrals. Paired with parametric integration via hyperlogarithms, this method is particularly well suited for the computation of renormalization group functions and easily automated. As an application of this approach we compute the 6-loop beta function and anomalous dimensions of the $\phi^4$ model.

We propose a partial fraction decomposition scheme to the construction of hierarchical equations of motion theory for bosonic quantum dissipation systems. The expansion of Bose--Einstein function in this scheme shows similar properties as it applies for Fermi function. The performance of the resulting quantum dissipation theory is exemplified with spin--boson systems. In all cases we have tested the new theory performs much better, about an order of magnitude faster, than the best available conventional theory based on Matsubara spectral decomposition scheme.

We advocate the replacement of standard alphas(mu)-based QCD perturbation theory, in which the coupling and truncated perturbative predictions are dependent on the chosen renormalisation scheme, by a Lambda-based approach in which QCD observables are directly related to the dimensional transmutation parameter, Lambda, of the theory. The shortcomings of the standard approach are emphasised by formulating it in the Lambda-based language. We show how the Lambda-based approach can be extended to include massive quarks. We also consider the use of Lambda-based perturbation theory in the case when a resummation of large infrared logarithms is required, performing a phenomenological analysis for the e+e- event shape observables thrust and heavy jet mass. We show by fitting to data that smaller power corrections are required than in the standard approach.

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We implement the effective field theory for gravitating spinning objects in the post-Newtonian scheme at the next-to-next-to-leading order level to derive the gravitational spin-orbit interaction potential at the third and a half post-Newtonian order for rapidly rotating compact objects. From the next-to-next-to-leading order interaction potential, which we obtain here in a Lagrangian form for the first time, we derive straightforwardly the corresponding Hamiltonian. The spin-orbit sector constitutes the most elaborate spin dependent sector at each order, and accordingly we encounter a proliferation of the relevant Feynman diagrams, and a significant increase of the computational complexity. We present the evaluation of the interaction potential, going over contributing Feynman diagrams. The computation is carried out in terms of the nonrelativistic gravitational fields, together with the various gauge choices included in the effective field theory for gravitating spinning objects, which optimize the calculation. In addition, we automatize the effective field theory computations, and carry out the automated computations in parallel. Such automated effective field theory computations would be most useful to obtain higher order post-Newtonian corrections. We compare our Hamiltonian to the ADM Hamiltonian, and arrive at a complete agreement between the ADM and effective field theory results. We provide complete gauge invariant relations among the binding energy, angular momentum, and orbital frequency of an inspiralling binary with generic compact spinning components to third and a half post-Newtonian order. The derivation presented here is essential to obtain further higher order post-Newtonian corrections, and to reach the accuracy level required for the successful detection of gravitational radiation.

We discuss an efficient scheme for obtaining spin-polarized quasi-particle excitation energies within the general framework of the density functional theory (DFT). Our approach is to correct the DFT eigenvalues via the electrostatic energy of a majority or minority spin electron resulting from its interaction with the associated exchange and correlation holes by using appropriate spin-resolved pair correlation functions. A version of the method for treating systems with localized orbitals, including the case of partially filled metallic bands, is considered. Illustrative results on Cu are presented.

In the mid Sixties Edward Nelson proved the existence of a consistent quantum field theory that describes the Yukawa-like interaction of a non-relativistic nucleon field with a relativistic meson field. Since then it is thought, despite the renormalization procedure involved in the construction, that the quantum dynamics should be governed in the classical limit by a Schr\"odinger-Klein-Gordon system with Yukawa coupling. In the present paper we prove this fact in the form of a Bohr correspondence principle. Besides, our result enlighten the nature of the renormalization method employed in this model which we interpret as a strategy that allows to put the related classical Hamiltonian PDE in a normal form suitable for a canonical quantization.

Using quantum electrodynamics as an example, a dependence of physical predictions of quantum field theory in a finite perturbation theory order on the choice of renormalization scheme is studied. It is shown that On-Mass-Shell renormalization scheme is distinguished in quantum electrodynamics not only due to the agreement of the theory predictions with experimental results but also due to a set of specific theoretical properties. Thus performing renormalization procedure in other theories, it seems reasonable to use some of On-Mass-Shell renormalization scheme prescriptions.

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relations. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions.

This is the first of three reports investigating the muzzle blast fields from the firing of Army weapons. This report presents theory and validation obtained by comparing analysis and muzzle blast field data. This first report limits investigation to the forward hemisphere of the blast field. The theory ganerally follows the data within the experimental error. The advantages of the theory are: its independence of empirical constants: and, the ease with which the computations are performed. Versions of the theory exist on minicomputers and on programmable pocket calculators.

We test third-order standard perturbation theory (SPT) as an approximation to non-linear cosmological structure formation. A novel approach is used to numerically calculate the three-dimensional dark matter density field using SPT from the initial conditions of two high-resolution cosmological simulations. The calculated density field is compared to the non-linear dark matter field of the simulations both point-by-point and statistically. For smoothing scales above 8 Mpc/h it shows a good agreement up to redshift 0. We present a simple fitting formula to relate the linear and non-linear density contrast that accurately recovers the non-linear time evolution for 0

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In a nonrelativistic contact four-fermion model we show that simple regularisation prescriptions together with a definite fine-tuning of the cut-off-parameter dependence of ``bare'' quantities give the exact solutions for the two-particle sector and Goldstone modes. Their correspondence with the self-adjoint extension into Pontryagin space is established leading to self-adjoint semi-bounded Hamiltonians in three-particle sectors as well. Renormalized Faddeev equations for the bound states with Fredholm properties are obtained and analysed.

Starting from fixed-order perturbation theory (FOPT) we derive expressions for the heavy-flavour components of the deep-inelastic structure functions FL and F2 in the variable-flavour number scheme (VFNS). These expressions are valid in all orders of perturbation theory. This derivation establishes a relation between the parton densities parametrized at N and N light flavours. The consequences for the existing parametrizations of the parton densities are discussed. Further we show that in charm electroproduction the exact and asymptotic expressions for the heavy-quark coefficient functions yield identical results for F2 when Q^2>20 (GeV/c)^2. We also study the differences between the FOPT and the VFNS descriptions for F2. It turns out that the charm structure function in the VFNS is larger than the one obtained in FOPT over the whole Q^2-range. Furthermore inspection of the perturbation series reveals that the higher order corrections in the VFNS are smaller than those present in FOPT for Q^2>10 (GeV/c)^2. Therefore the VFNS gives a better prediction for the charm structure function at large Q^2-values than FOPT.

We demonstrate that Optimal Control Theory (OCT) with a state-dependent constraint which depends on the state of the system at each instant can reproduce the famous counterintuitive mechanism of Stimulated Raman adiabatic passage (STIRAP). We examine this behavior in a {\Lambda}-type three-level system and we show that could be applied for sequentially coupled many-level systems. We study the robustness of the two methods with respect to pulse fluctuations and the decays. We show that new OCT formulation appears to be more robust than STIRAP when a perturbation is introduced in the pulses. Such method is of great use for systems involving coherence loss such as molecular systems with dissociation or ionization limits. It also may find potential applications in the control of chemical reactions, quantum optics, and quantum information processing.

The next-to-next-to-leading order spin-squared interaction potential for generic compact binaries is derived for the first time via the effective field theory for gravitating spinning objects in the post-Newtonian scheme. The spin-squared sector is an intricate one, as it requires the consideration of the point particle action beyond minimal coupling, and mainly involves the spin-squared worldline couplings, which are quite complex, compared to the worldline couplings from the minimal coupling part of the action. This sector also involves the linear in spin couplings, as we go up in the nonlinearity of the interaction, and in the loop order. Hence, there is an excessive increase in the number of Feynman diagrams, of which more are higher loop ones. We provide all the Feynman diagrams and their values. The beneficial nonrelativistic gravitational fields are employed in the computation. This spin-squared correction, which enters at the fourth post-Newtonian order for rapidly rotating compact objects, completes the conservative sector up to the fourth post-Newtonian accuracy. The robustness of the effective field theory for gravitating spinning objects is shown here once again, as demonstrated in a recent series of papers by the authors, which obtained all spin dependent sectors, required up to the fourth post-Newtonian accuracy. The effective field theory of spinning objects allows to directly obtain the equations of motion, and the Hamiltonians, and these will be derived for the potential obtained here in a forthcoming paper.

We present a computationally tractable scheme of time-dependent transport phenomena within open-boundary time-dependent density-functional-theory. Within this approach all the response properties of a system are determined from the time-propagation of the set of ``occupied'' Kohn-Sham orbitals under the influence of the external bias. This central idea is combined with an open-boundary description of the geometry of the system that is divided into three regions: left/right leads and the device region (``real simulation region''). We have derived a general scheme to extract the set of initial states in the device region that will be propagated in time with proper transparent boundary-condition at the device/lead interface. This is possible due to a new modified Crank-Nicholson algorithm that allows an efficient time-propagation of open quantum systems. We illustrate the method in one-dimensional model systems as a first step towards a full first-principles implementation. In particular we show how a stationary current develops in the system independent of the transient-current history upon application of the bias. The present work is ideally suited to study ac transport and photon-induced charge-injection. Although the implementation has been done assuming clamped ions, we discuss how it can be extended to include dissipation due to electron-phonon coupling through the combined simulation of the electron-ion dynamics as well as electron-electron correlations.

From the perspective of perturbation theory, we propose a systematic procedure for the evaluation of the derivative discontinuity (DD) of the exchange-correlation energy functional in Kohn-Sham density functional theory (KS-DFT), wherein the exact DD can in principle be obtained by summing up all the perturbation corrections to infinite order. Truncation of the perturbation series at low order yields an efficient scheme for obtaining the approximate DD. While the zeroth-order theory yields a vanishing DD, the first-order correction to the DD can be expressed as an explicit universal functional of the ground-state density and the KS lowest unoccupied molecular orbital density, allowing the direct evaluation of the DD in the standard KS method without extra computational cost. The fundamental gap can be predicted by adding the estimated DD to the KS gap. This scheme is shown to be accurate in the prediction of the fundamental gaps for a wide variety of atoms and molecules.

In the standard theory of neutrino oscillations it is supposed that physical observed neutrino states $\nu_{e}, \nu_{\mu}, \nu_{\tau}$ have no definite masses, that they are initially produced as a mixture of the $\nu_{1}, \nu_{2}, \nu_{3}$ neutrino states (are produced as a wave packet), and that neutrino oscillations are the real ones. Then this wave packet must decompose at a definite distance into constituent parts and neutrino oscillations must disappear. It was shown that these suppositions lead to violation of the law of energy and momentum conservation. An alternative scheme of neutrino oscillations obtained within the framework of particle physics has been considered where the above mentioned shortcomings are absent, the oscillations of neutrinos with equal masses are the real ones, and the oscillations of neutrinos with different masses are virtual ones. Expressions for probabilities of neutrino transitions (oscillations) in the alternative (corrected) scheme, are given.

We summarize the recent progress in a new approach to precision LHC physics based on the IR-improved DGLAP-CS theory as it relates to a new MC friendly exponentiated scheme for precision calculation of higher order corrections to LHC physics in which IR singularities from both QED and QCD are canceled to all orders in \alpha and in \alpha_s simultaneously in the presence of rigorous shower/ME matching. We present the first MC data comparing the implied new showers themselves with the standard ones using the HERWIG6.5 MC event generator as a test case at LHC energies.

We demonstrate how the separation of the total energy of a self-bound system into a functional of the internal one-body Fermionic density and a function of an arbitrary wave vector describing the center-of-mass kinetic energy can be used to set-up an "internal" Kohn-Sham scheme.

We study the standard one-component $\varphi^4$-theory in four dimensions. A renormalized coupling is defined in a finite size renormalization scheme which becomes the standard scheme of the broken phase for large volumes. Numerical simulations are reported using the worm algorithm in the limit of infinite bare coupling. The cutoff dependence of the renormalized coupling closely follows the perturbative Callan Symanzik equation and the triviality scenario is hence further supported.

The study provided research and experimental verification of the hypothetic-deductive scheme for teaching the theory of quantum physics to secondary school students. The study highlighted the technologies and didactic tools, which help to boost the effectiveness of teaching physics to students and develop their critical thinking. It was found that the hypothetic-deductive scheme is based on the concept of critical rationalism and it has such stages: problem-hypotheses-rational criticism-choice of hypothesis-rational criticism of the new theory-new problem. The study described the stages of its implementation for teaching students the basics of quantum physics. To prove the effectiveness of the suggested scheme, the study carried out a comparative experiment in which 479 students took part (there were 241 students in experimental classes and 238 students in control classes). The statistical procession and interpretation of the experiment results were carried out with Pearson’s Chi-squared test. After the experiment, the quality of learning achievements that corresponds to the sufficient and high levels is 62.87% in the experimental classes, whereas in the control classes this figure is 52.48%. In further prospect, researchers plan to check the effectiveness of the suggested scheme to study other theories, which are studied in secondary school.

The study provided research and experimental verification of the hypothetic-deductive scheme for teaching the theory of quantum physics to secondary school students. The study highlighted the technologies and didactic tools, which help to boost the effectiveness of teaching physics to students and develop their critical thinking. It was found that the hypothetic-deductive scheme is based on the concept of critical rationalism and it has such stages: problem-hypotheses-rational criticism-choice of hypothesis-rational criticism of the new theory-new problem. The study described the stages of its implementation for teaching students the basics of quantum physics. To prove the effectiveness of the suggested scheme, the study carried out a comparative experiment in which 479 students took part (there were 241 students in experimental classes and 238 students in control classes). The statistical procession and interpretation of the experiment results were carried out with Pearson’s Chi-squared test. After the experiment, the quality of learning achievements that corresponds to the sufficient and high levels is 62.87% in the experimental classes, whereas in the control classes this figure is 52.48%. In further prospect, researchers plan to check the effectiveness of the suggested scheme to study other theories, which are studied in secondary school.

We present a scheme-independent calculation of the infrared value of the anomalous dimension of the fermion bilinear, $\gamma_{\bar\psi\psi,IR}$ in an SU(3) gauge theory as a function of the number of fermions, $N_f$, via a series expansion in powers of $\Delta_f$, where $\Delta_f=(16.5-N_f)$, to order $\Delta_f^4$. We perform an extrapolation to obtain the first determination of the exact $\gamma_{\bar\psi\psi,IR}$ from continuum field theory. The results are compared with calculations of the $n$-loop values of this anomalous dimension from series in powers of the coupling and from lattice measurements.

We give a reinterpretation of the matrix theory discussed by Moore, Nekrasov and Shatashivili (MNS) in terms of the second quantized operators which describes the homology class of the Hilbert scheme of points on surfaces. It naturally relate the contribution from each pole to the inner product of orthogonal basis of free boson Fock space. These basis can be related to the eigenfunctions of Calogero-Sutherland (CS) equation and the deformation parameter of MNS is identified with coupling of CS system. We discuss the structure of Virasoro symmetry in this model.

We apply analytic perturbation theory in next-to-next-to-leading order to inclusive semileptonic $\tau$-decay and study the renormalization scheme dependence. We argue that the renormalization scheme ambiguity is considerably reduced in the analytic perturbation theory framework and we obtain a rather stable theoretical prediction.

We introduce a momentum subtraction scheme which obeys the power counting of Kaplan, Savage, and Wise (KSW), developed for systems with large scattering lengths, $a$. Unlike the power divergence subtraction scheme, coupling constants in this scheme obey the KSW scaling for all $\mu_R > 1/a$. We comment on the low-energy theorems derived by Cohen and Hansen. We conclude that there is no obstruction to using perturbative pions for momenta $p>m_\pi$.

The IR/UV mixing in the non-commutative (NC) field theory is investigated in Carlson-Carone-Zobin (CCZ) formalism of Lorentz-invariant NC field theory provided that the fields are `independent' of the `internal' coordinates $\theta^{\mu\nu}$. A new regularization scheme called NC regularizatioon is then proposed, which removes the Lorentz-invariant IR singularity from the theory. It requires the usual UV limit $\Lambda\to \infty$ to be accompanied with the commutative limit $a\to 0$ with $\Lambda^2a^2$ fixed, where $a$ is the length parameter in the theory. The new UV limit gives the usual renormalized amplitude of the one-loop self-energy diagram of $\phi^3$ model. It is shown that the new regularization is gauge-invariant, that is, the non-transverse part of the vacuum polarization in QED is automatically transverse in Lorentz-invariant NCQED but the two transverse pieces, one of which is already transverse in QED, possesses Lorentz-invariant IR singularity which should be `subtracted off' at zero external momentum squared. The subtraction leads to the same result as the renormalized one by Pauli-Villars or dimensional regularizations. Other diagrams with three-point vertices which contribute to the photon self-energy in Lorentz-non-invariant NCQED all vanish due to Lorentz invariance under the assumption adopted, while the tadpole diagram gives a finite contribution to the charge renormalization which vanishes if $ Lambda^2a^2\to 0$. Lorentz-invariant NC $\phi^4$ and scalar Yukawa models are also discussed in the one-loop approximation. A comment is made that Lorentz-invariance might lead to a decoupling of U(1) part from SU(N) in NC U(N) gauge theory.

The rapid advances in the field of optimization methods in many pure and applied science pose the difficulty of keeping track of the developments as well as selecting an appropriate technique that best suits the problem in-hand. From a practitioner point of view is rightful to wander "which optimization method is the best for my problem?". Looking at the optimization process as a "system" of intercon- nected parts, in this paper are collected some ideas about how to tackle an optimization problem using a class of tools from evolutionary computations called Genetic Algorithms. Despite the number of optimization techniques available nowadays the author of this paper thinks that Genetic Algorithms still play a central role for their versatility, robustness, theoretical framework and simplicity of use. The paper can be considered a "collection of tips" (from literature and personal experience) for the non-computer-scientist that has to deal with optimization problems both in the science and engineering practice. No original methods or algorithms are proposed.

Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.

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We consider an asymptotically free vectorial SU($N_c$) gauge theory with $N_f$ massless fermions in a representation $R$, having an infrared fixed point (IRFP) of the renormalization group at $\alpha_{IR}$ in the conformal non-Abelian Coulomb phase. The cases with $R$ equal to the fundamental, adjoint, and symmetric rank-2 tensor representation are considered. We present scheme-independent calculations of the anomalous dimension $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^4)$ and $\beta'_{IR}$ to $O(\Delta_f^5)$ at this IRFP, where $\Delta_f$ is an $N_f$-dependent expansion parameter. Comparisons are made with conventional $n$-loop calculations and lattice measurements. As a test of the accuracy of the $\Delta_f$ expansion, we calculate $\gamma_{\bar\psi\psi,IR}$ to $O(\Delta_f^3)$ in ${\cal N}=1$ SU($N_c$) supersymmetric quantum chromodynamics and find complete agreement, to this order, with the exactly known expression. The $\Delta_f$ expansion also avoids a problem in which an IRFP may not be manifest as an IR zero of a higher $n$-loop beta function.

We develop a nonperturbative dynamical theory (NDT) that is useful for treating nonequilibrium transport in a system with strong correlation. We apply our NDT to the single-impurity Anderson model in equilibrium to check its reliability by comparing with the results of numerical renormalization group method (NRG). We finally suggest a self-consistent loop to calculate the current in a lead-dot-lead system with Kondo coupling.

Despite of its huge successes in vast amount of applications, the Kohn-Sham scheme of density functional theory (DFT-Kohn-Sham) has not been able to get reliable ionization potentials (IP) for semiconductors, due to self-interaction error in the local density approximation (LDA) and generalized gradient approximations (GGA), and the difficulty of using asymptotically long-ranged potentials for surface calculations. An approximate optimize effective potential (OEP), the Becke-Johnson'06 exchange, is used to explore the capability of OEP to calculate semiconductor IP with a surface technique suitable for both short- and long-ranged potentials. Combined with the LDA correlation, the approximate OEP has achieved an IP accuracy for 17 semiconductors which is similar to the much more sophisticated $GW$ approximation (GWA), with the computational cost of only LDA/GGA. For the first time, this opens the way for systematic calculation of semiconductor IP's by the DFT-Kohn-Sham scheme. The long-range part of the $v_{xc}$ is not as important for solid surface as for atoms and molecules since the wave functions are mainly bound to the slab while less exposed to the asymptotic region. Nevertheless, it contributes significantly to the total IP and therefore shall not be simply ignored. Without including its effects, it is unlikely that the same GWA procedure could achieve universally accurate band gaps and accurate IP's at the same time, as is indeed seen in practical calculations.

In this paper we compute the convolution algebra in the equivariant K-theory of the Hilbert scheme of A^2. We show that it is isomorphic to the elliptic Hall algebra, and hence to the spherical DAHA of GL_\infty. We explain this coincidence via the geometric Langlands correspondence for elliptic curves, by relating it also to the convolution algebra in the equivariant K-theory of the commuting variety. We also obtain a few other related results (action of the elliptic Hall algebra on the K-theory of the moduli space of framed torsion free sheaves over P^2, virtual fundamental classes, shuffle algebras,...).

In this work a practical scheme is developed for the first-principles study of time-dependent quantum transport. The basic idea is to combine the transport master-equation with the well-known time-dependent density functional theory. The key ingredients of this paper include: (i) the partitioning-free initial condition and the consideration of the time-dependent bias voltages which base our treatment on the Runge-Gross existence theorem; (ii) the non-Markovian master equation for the reduced (many-body) central system (i.e. the device); and (iii) the construction of Kohn-Sham master equation for the reduced single-particle density matrix, where a number of auxiliary functions are introduced and their equations of motion (EOM) are established based on the technique of spectral decomposition. As a result, starting with a well-defined initial state, the time-dependent transport current can be calculated simultaneously along the propagation of the Kohn-Sham master equation and the EOM of the auxiliary functions.

We generalize the recently developped "internal" Density Functional Theory (DFT) and Kohn-Sham scheme to multicomponent systems. We obtain a general formalism, applicable for the description of multicomponent self-bound systems (as molecules where the nuclei are treated explicitely, atomic nuclei and mix of 3He and 4He droplets), where the fundamental translational symmetry has been treated correctly. The main difference with traditional DFT is the explicit inclusion of center-of-mass correlations in the functional. A large part of the paper is dedicated to the application to molecules, which permits among other to clarify the approximations that underly traditional DFT.