Sequential decomposition and policy iteration schemes for M-player games with partial weak coupling

In this paper, the open-loop and closed-loop local and remote stochastic nonzero-sum game (LRSNG) problem is investigated. Different from previous works, the stochastic nonzero-sum game problem under consideration is essentially a special class of two-person nonzero-sum game problem, in which the information sets accessed by the two players are inconsistent. More specifically, both the local player and the remote player are involved in the system dynamics, and the information sets obtained by the two players are different, and each player is designed to minimise its own cost function. For the considered LRSNG problem, both the open-loop and closed-loop Nash equilibrium are derived. The contributions of this paper are given as follows. Firstly, the open-loop optimal Nash equilibrium is derived, which is determined in terms of the solution to the forward and backward stochastic difference equations (FBSDEs). Furthermore, by using the orthogonal decomposition method and the completing square method, the feedback representation of the optimal Nash equilibrium is derived for the first time. Finally, the effectiveness of our results is verified by a numerical example.

This paper deals with a systematic way for modelling and visualising market power in liberalised electricity markets. The paper first introduces a new index termed the “Nodal Market Power” index, the NMP index. The NMP index is calculated based on the concept of “social welfare” in economics and the game theory in applied mathematics. The oligopoly electricity market is modelled through a non-cooperative game and the solution concept of the Nash equilibrium. The solution concept of Nash equilibrium is reformulated as an optimisation problem. To tackle the multiple Nash equilibria problem, the worst Nash equilibrium in terms of the social cost to the society is selected. Then after, the NMP index is calculated for each node of power system. A colour contour map is used for visualising the market power using the NMP index.

The problem of a dynamic Nash equilibrium traffic assignment with schedule delays on congested networks is formulated as an N-person nonzero-sum differential game in which each player represents an origin-destination pair. Optimality conditions are derived using a Nash equilibrium solution concept in the open-loop strategy space and given the economic interpretation as a dynamic game theoretic generalization of Wardrop's second principle. It is demonstrated that an open-loop Nash equilibrium solution converges to an instantaneous dynamic user equilibrium solution as the number of players for each origin-destination pair increases to infinity. An iterative algorithm is developed to solve a discrete-time version of the differential game and is used to numerically show the asymptotic behavior of open-loop Nash equilibrium solutions on a simple network. A Nash equilibrium solution is also analyzed on the 18-arc network. © 1998 John Wiley & Sons, Inc. Networks 32: 141–163, 1998

Article Figures and data Abstract eLife digest Introduction Results Discussion Materials and methods Appendix 1: Model Details Appendix 2 Thermodynamics of assembly around a fluid cargo References Decision letter Author response Article and author information Metrics Abstract We computationally study how an icosahedral shell assembles around hundreds of molecules. Such a process occurs during the formation of the carboxysome, a bacterial microcompartment that assembles around many copies of the enzymes ribulose 1,5-bisphosphate carboxylase/ oxygenase and carbonic anhydrase to facilitate carbon fixation in cyanobacteria. Our simulations identify two classes of assembly pathways leading to encapsulation of many-molecule cargoes. In one, shell assembly proceeds concomitantly with cargo condensation. In the other, the cargo first forms a dense globule; then, shell proteins assemble around and bud from the condensed cargo complex. Although the model is simplified, the simulations predict intermediates and closure mechanisms not accessible in experiments, and show how assembly can be tuned between these two pathways by modulating protein interactions. In addition to elucidating assembly pathways and critical control parameters for microcompartment assembly, our results may guide the reengineering of viruses as nanoreactors that self-assemble around their reactants. https://doi.org/10.7554/eLife.14078.001 eLife digest Bacterial microcompartments are protein shells that are found inside bacteria and enclose enzymes and other chemicals required for certain biological reactions. For example, the carboxysome is a type of microcompartment that enables the bacteria to convert the products of photosynthesis into sugars. During the formation of a microcompartment, the outer protein shell assembles around hundreds of enzymes and chemicals. This formation process is tightly controlled and involves multiple interactions between the shell proteins and the cargo – the enzymes and other reaction ingredients – they will enclose. Understanding how to control which enzymes are encapsulated within microcompartments could help researchers to re-engineer the microcompartments so that they contain drugs or other useful products. Recent studies have used microscopy to visualize how microcompartments are assembled. However, most of the intermediate structures that form during assembly are too small and short-lived to be seen. It has therefore not been possible to explore in detail how shell proteins collect the necessary cargo and then assemble into an ordered shell with the cargo on the inside. Experiments alone are probably not enough to understand the process, especially since microcompartment assembly can currently only be studied within live cells or cellular extract. Within these complex environments it is difficult to determine the effect of any individual factor on the overall assembly process. Perlmutter, Mohajerani and Hagan have now taken a different approach by developing computational and theoretical models to explore how microcompartments assemble. Computer simulations showed that microcompartments could assemble by two pathways. In one pathway, the protein shell and cargo coalesce at the same time. In the other pathway, the cargo molecules first assemble into a large disordered complex, with the shell proteins attached on the outside. The shell proteins then assemble, carving out a piece of the cargo complex. The simulations showed that many factors affect how the shell assembles, such as the strengths of the interactions between the shell proteins and the cargo. They also identified a factor that controls how much cargo ends up inside the assembled shell. Perlmutter, Mohajerani and Hagan found that, in addition to revealing how microcompartments may assemble within their natural setting, the simulations provided guidance on how to re-engineer microcompartments to assemble around other components. This would enable researchers to create customizable compartments that self-assemble within bacteria or other host organisms, for example to carry out carbon fixation or make biofuels. A future challenge will be to investigate other aspects of microcompartment assembly, such as the factors that control the size of these compartments. https://doi.org/10.7554/eLife.14078.002 Introduction Encapsulation is a hallmark of biology. A cell must co-localize high concentrations of enzymes and reactants to perform the reactions that sustain life, and it must safely store genetic material to ensure long-term viability. While lipid-based organelles primarily fulfill these functions in eukaryotes, self-assembling protein shells take the lead in simpler organisms. For example, viruses surround their genomes with a protein capsid, while bacteria use large icosahedral shells known as bacterial microcompartments (BMCs) to sequester the enzymes and reactions responsible for particular metabolic pathways (Kerfeld et al., 2010; Axen et al., 2014; Shively et al., 1998; Bobik et al., 1999; Erbilgin et al., 2014; Petit et al., 2013; Price and Badger, 1991; Shively et al., 1973; Shively et al., 1973; Kerfeld and Erbilgin, 2015). Within diverse bacteria, BMC functions have been linked to bacterial growth, carbon fixation, symbiosis, or pathogenesis (Kerfeld and Erbilgin, 2015). Other protein-based compartments are found in bacteria and archea (e.g. encapsulins (Sutter et al., 2008) and gas vesicles (Pfeifer, 2012; Sutter et al., 2008)) and even eukaryotes (e.g. vault particles (Kickhoefer et al., 1998)), while some viruses may assemble around lipidic globules (Lindenbach and Rice, 2013; Faustino et al., 2014). Thus, understanding the factors that control microcompartment assembly and encapsulation is a central question in modern cell biology. From the perspectives of synthetic biology and nanoscience, there is great interest in reengineering BMCs or viruses as nanoreactors that spontaneously encapsulate enzymes and reagents in vitro (e.g. Luque et al., 2014; Douglas and Young, 1998; Rurup et al., 2014; Patterson et al., 2014; Patterson et al., 2012; Zhu et al., 2014; Rhee et al., 2011; Rurup et al., 2014; Wörsdörfer et al., 2012; Comas-Garcia et al., 2014), or as customizable organelles that assemble around a programmable set of core enzymes in vivo, introducing capabilities such as carbon fixation or biofuel production into bacteria or other organisms (e.g. Kerfeld and Erbilgin, 2015; Bonacci et al., 2012; Parsons et al., 2010; Choudhary et al., 2012; Lassila et al., 2014). However, the principles controlling such co-assembly processes have yet to be established, and it is not clear how to design systems to maximize encapsulation. In this article we take a step toward this goal, by developing theoretical and computational models that describe the dynamical encapsulation of hundreds of cargo molecules by self-assembling icosahedral shells. Although our models are general, we are motivated by recent experiments on a type of BMC known as the carboxysome (Kerfeld et al., 2010; Schmid et al., 2006; Iancu et al., 2007; Tanaka et al., 2008). Carboxysomes are large (40–400 nm), roughly icosahedral shells that encapsulate a dense complex of the enzyme ribulose-1,5-bisphosphate carboxylase/oxygenase (RuBisCO) and other proteins to facilitate the Calvin-Bensen-Bassham cycle in autotrophic bacteria (Price and Badger, 1991; Shively et al., 1973; Shively et al., 1973; Iancu et al., 2007; 2010; Kerfeld et al., 2010; Tanaka et al., 2008). Recently, striking microscopy experiments visualized β-carboxysome shells assembling on and budding from procarboxysomes (the condensed complex of RuBisCO and other proteins found in the interior of carboxysomes) (Cameron et al., 2013; Chen et al., 2013). Genomic analysis suggests that many BMCs with diverse functions assemble via similar pathways (Cameron et al., 2013; Kerfeld and Erbilgin, 2015). However, the mechanisms of budding and pinch-off to close the shell remain incompletely understood because of the small size and transient nature of assembly intermediates. Moreover, experiments suggest that α-carboxysomes (another form of carboxysome) assemble by a different mechanism, in which shell assembly encapsulates an initially diffuse pool of RuBisCO (Iancu et al., 2010; Cai et al., 2015). The factors determining which of these assembly pathways occurs are unknown. BMC assembly is driven by a complex interplay of interactions among the proteins forming the external shell and the interior cargo. It is difficult, with experiments alone, to parse these interactions for those mechanisms and factors that critically influence assembly pathways, especially due to the lack of an in vitro assembly system. Models which can correlate individual factors to their effect on assembly are therefore an important complement to experiments. Previous experimental and theoretical studies of encapsulation by icosahedral shells, e.g. the assembly of viral capsids around their nucleic acid genomes (e.g. Hu and Shklovskii, 2007; Kivenson and Hagan, 2010; Elrad and Hagan, 2010; Perlmutter et al., 2013; 2014; Mahalik and Muthukumar, 2012; Zhang et al., 2013; Zhang and Linse, 2013; Hagan, 2008; Devkota et al., 2009; Dixit et al., 2006; Borodavka et al., 2012; Dykeman et al., 2013; 2014; Zlotnick et al., 2013; Johnson et al., 2004; Patel et al., 2015; Cadena-Nava et al., 2012; Comas-Garcia et al., 2012; 2014; Garmann et al., 2014a; 2014b; Malyutin and Dragnea, 2013), have demonstrated that the structure of the cargo can strongly influence assembly pathways and products. However, BMCs assemble around a cargo which is topologically different from a nucleic acid — a fluid complex comprising many, noncovalently linked molecules. We demonstrate here that changing the cargo topology leads to new assembly pathways and different critical control parameters. We present phase diagrams and analysis of dynamical simulation trajectories showing how the thermodynamics, assembly pathways, and emergent structures depend on the interactions among shell proteins and cargo molecules. Within distinct parameter ranges, we observe two classes of assembly pathways, which resemble those suggested for respectively α- or β-carboxysomes. We find that tunability of cargo loading is a key functional difference between the two classes of pathways. Shells assembled around a diffuse cargo can be varied from empty (containing almost no cargo) to completely full, whereas assembly around a condensed, procarboxysome-like complex invariably produces full shells. While we find that the encapsulated cargo becomes ordered due to confinement, complete crystalline order in the globule before encapsulation inhibits budding. We discuss these results in the context of recent observations on carboxysome assembly, and their implications for engineering BMCs, viruses or drug delivery vehicles that assemble around a fluid cargo (e.g. Refs. [Kerfeld and Erbilgin, 2015; Parsons et al., 2010; Choudhary et al., 2012; Lassila et al., 2014; Luque et al., 2014; Douglas and Young, 1998; Rurup et al., 2014; Patterson et al., 2014; Patterson et al., 2012; Zhu et al., 2014; Rhee et al., 2011; Rurup et al., 2014; Wörsdörfer et al., 2012]). Results Our model system is motivated by icosahedral viral capsids and BMCs (Tanaka et al., 2008; Kerfeld et al., 2010). Since icosahedral symmetry can accommodate at most 60 identical subunits, formation of large icosahedral structures requires subunits to assemble into different local environments. The subunits can be grouped into pentamers and hexamers, with 12 pentamers at the icosahedron vertices and the remaining subunits in hexamers. Viruses typically assemble from small oligomers of the capsid protein, which we refer to as the basic assembly unit (Hagan, 2014). Recent AFM experiments demonstrated that hexamers are the basic assembly unit during the assembly of BMC shell facets (Sutter et al., 2016), and the carboxysome major shell proteins crystallize as pentamers and hexamers (Tanaka et al., 2008). Motivated by these observations, our model considers two basic assembly units, one a pentamer and the other a hexamer, with interactions designed so that the lowest energy structure corresponds to a truncated icosahedron with 12 pentamers and 20 hexamers (Figure 1). While BMCs generally have more hexamers, our model is intended to explore the general principles of assembly around a fluid cargo rather than model a specific system. Further details of the model and a thermodynamic analysis are given in section 3 and the appendices. Figure 1 Download asset Open asset Description of the model. (A) Each shell subunit contains 'Attractors' (green circles) on the perimeter, a 'Top' (tan circle, 'T') in the center above the plane, and a 'Bottom' (purple circle, 'B' below the plane). (B) Interactions between complementary Attractors drive subunit dimerization, with the Top-Top repulsions (tan arrow) tuned to favor the subunit-subunit angle in a complete shell. Complementary pairs of attractors are indicated by green arrows in (A) for the pentamer-hexamer interface and in (B) for the hexamer-hexamer interface. (C) Bottom psuedoatoms bind cargo molecules (terra cotta circles, 'C'), while excluder atoms (blue and brown pseudoatoms in (D)) placed in the plane of the pentagon experience excluded volume interactions with the cargo. (D) The positions of excluder atoms in the lowest energy shell geometry, a truncated icosahedron with 12 pentamers (blue) and 20 hexamers (brown). https://doi.org/10.7554/eLife.14078.003 To understand how assembly around multiple cargo molecules depends on the relative strengths of interactions between components, we performed dynamical simulations as a function of the parameters controlling shell subunit-subunit (εSS), shell subunit-cargo (εSC), and cargo-cargo (εCC) interaction strengths. All energy values are given in units of the thermal energy, kB⁢T. We focus on parameters for which shell subunit-subunit interactions are too weak to drive assembly in the absence of cargo (εSS≤4.5). Except where mentioned otherwise, the cargo diameter is set equal to the circumradius of a shell subunit. For the simulated density of cargo particles, the phase behavior (in the absence of shells) corresponds to a vapor at εCC=1.3, liquid-vapor phase coexistence for εCC∈[1.6,2.0] (the phase coexistence boundary is slightly below εCC=1.6), and a solid phase at εCC=3.0. We find that tuning εCC through phase coexistence dramatically alters the typical assembly process. Strong cargo interactions (εCC≥1.6) drive formation of a globule followed by assembly and budding of a shell, such as observed for β-carboxysomes (Figure 2A, Simulation Video 1), while under weak interactions (εCC<1.6) shell assembly usually proceeds in concert with cargo encapsulation (Figure 2B, Simulation Video 2), as suggested for assembly of α-carboxysomes. We now elaborate on these classes of assembly pathways, and how the resulting assembly products depend on parameter values. Figure 2 with 2 supplements see all Download asset Open asset Snapshots illustrating typical assembly trajectories. (A) Multi-step assembly involving an amorphous globule of cargo and shell subunits. (B) Single-step assembly, in which shell assembly drives local cargo condensation. and (C) when shell-cargo interactions are too weak to condense the cargo. The values of the cargo-cargo (εCC), shell subunit-cargo (εSC), and subunit-subunit (εSS) interaction strengths are listed above each panel (all energies are in units of the thermal energy kB⁢T), and the time (in units of 106 timesteps) is noted below each image. The color scheme here and throughout the manuscript is: Red=Cargo, Blue=Pentagon Excluder, Brown=Hexagon Excluder. Attractor and Bottom pseudoatoms are omitted to aid visibility. Videos of assembly trajectories are included below. https://doi.org/10.7554/eLife.14078.004 Video 1 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Animation of a typical simulation showing assembly around a cargo globule. Parameters are εCC=1.6, εSC=7, and εSS=2.5. https://doi.org/10.7554/eLife.14078.007 Video 2 Download asset This video cannot be played in place because your browser does support HTML5 video. You may still download the video for offline viewing. Download as MPEG-4 Download as WebM Download as Ogg Animation of a typical simulation showing simultaneous assembly and cargo condensation. Parameters are εCC=1.3, εSC=9, and εSS=3.5. https://doi.org/10.7554/eLife.14078.008 Assembly and budding from a cargo globule We begin by discussing assembly behavior when the cargo-cargo interactions are strong enough to drive equilibrium phase coexistence (εCC≥1.6). Near the phase boundary (εCC=1.6) a system of pure cargo particles is metastable on the timescales we simulate. However, for εSC>4, adding shell subunits drives nucleation of a cargo globule with shell subunits adsorbed on the surface. The subsequent fate of the globule depends on parameter values; typical simulation end-states are shown as a function of parameter values in Figure 3. For moderate interaction strengths (2.5≤εSS≤3.5) the globule grows to a large size, typically containing at least twice the cargo molecules that can be packaged within a complete shell. Adsorbed shell subunits then reversibly associate to form ordered clusters. Once a cluster acquires enough inter-subunit interactions to be a stable nucleus, it grows by coagulation of additional subunits or other adsorbed clusters. For the parameter set corresponding to Figure 2A, nucleation is fast in comparison to cluster growth, and thus two nuclei grow simultaneously. The last three images show the system immediately preceding and following detachment of the lower shell. Missing only one of its 32 subunits, the shell is connected to the remainder of the droplet only by a narrow neck of cargo. Insertion of the final subunit breaks the neck and completes shell detachment. The complete shell contains 120–130 cargo particles, which is slighty above random close packing (≈120 particles) but below fcc density (≈150 particles, see appendix 1.2). Figure 3 with 3 supplements see all Download asset Open asset Results of assembly around a cargo globule. (A) The most frequently observed assembly outcome is overlaid on a color map of the theoretical free energy density difference Δ⁢fassem (Equation (3)) between assembled shells and the unassembled globule. Results are plotted against the shell-cargo adsorption strength εSC and the shell-shell interaction strength εSS for indicated values of the cargo-cargo interaction strength εCC. (B) Representative snapshots of the predominant assembly outcomes shown in (A). https://doi.org/10.7554/eLife.14078.009 Figure 3—source data 1 List of all simulation outcomes for Figures 3A,5A. https://doi.org/10.7554/eLife.14078.010 Download elife-14078-fig3-data1-v3.zip Figure 3—source data 2 Criteria used to categorize assembly outcomes. The sizes of each cargo globule and shell assemblage, and associations between shell assemblages and cargo globules, were determined by clustering. The outcome was then categorized according to the criteria listed in this table. https://doi.org/10.7554/eLife.14078.011 Download elife-14078-fig3-data2-v3.zip Increasing the shell-shell interaction strength drives faster shell assembly and closure, thus limiting the size of the globule before budding. For the largest interaction strength we simulated (εSS=4.5) the globule typically does not exceed the size of a single shell, and multiple globules nucleate within the simulation box (Figure 2—figure supplement 1). This observation could place an upper bound on shell-shell interaction strengths, since multiple nucleation events were rare in the carboxysome assembly experiments (Cameron et al., 2013) (however, we discuss potential complicating factors within the cellular environment below). To quantify the relationship between assembly mechanism and parameter values, we calculate an assembly order parameter, defined as the maximum number of unassembled subunits adsorbed onto a globule during an assembly trajectory. The order parameter is shown as a function of the interaction strengths in Figure 4. For εCC≥1.6 and εSS≤3 we observe large values of the order parameter (e.g. >32, the red and yellow regions in Figure 4), which formation of a large amorphous globule with the to carboxysome shell assembly (Cameron et al., 2013). Figure with 1 supplement see all Download asset Open asset of assembly on shell-cargo and shell-shell interaction The assembly order parameter, defined as the maximum number of unassembled shell subunits adsorbed on a globule at any during a is shown as a function of εSC and εSS for indicated values of the cargo-cargo interaction εCC. of adsorbed unassembled subunits the two step assembly mechanism (Figure whereas values to simultaneous assembly and cargo (Figure Other assembly products of the parameter ranges, we observe classes of outcomes. weak shell-shell interactions to drive For and the cargo vapor phase is and the system no cargo on simulated timescales discuss below). cargo-cargo or shell-cargo interactions in unassembled where a cargo globule forms but the shell subunits on its to εSS we observe assembly on the leading to complete shells or two classes of In the first one or more shells almost but to from the droplet within simulated for when the subunit-cargo interaction does not a strong enough for the last to the cargo and close the shell. strong interactions drive the other of in which an of assembled shells the system of free subunits before any shells are In this it is also to observe in which within shells. the cargo-cargo interaction multiple narrow the parameter that leads to complete assembly and detachment. cargo globules nucleate at multiple within the simulation the of the the of εSC required for cargo resulting in shells a parameter We also observe a we refer to as in which shell assembly to the globule thus does not even to the attached The is especially for when the cargo even in the absence of shell encapsulation. For and of the number of nucleation we typically observe into a large globule. shell assembly and cargo For the cargo forms an equilibrium vapor phase in the absence of shell subunits. However, above values of εSS and the diffuse cargo molecules drive nucleation of shell The subsequent assembly depends on the shell-cargo interaction For εSC (Figure assembly only a cargo leading to but empty shells. For εSC (Figure 2B, and Simulation Video 2), the shell-cargo interactions drive local of cargo molecules. assembly and cargo then in the mechanism for assembly of α-carboxysomes (Iancu et al., 2010). Thus, tuning the shell-cargo interaction dramatically cargo with a from empty to shells around This the equilibrium (Figure by a complete shell to cargo molecules. This effect is to the of vapor below its inside of In assembly around a globule only full shells. Figure with 2 supplements see all Download asset Open asset Results of assembly around a cargo with weak interactions (A) The most frequently observed assembly outcome as a function of εSS and The of outcomes for is shown in Figure supplement and a data containing the outcome for each at each parameter set is included (Figure 3—source data 1). (B) Representative snapshots for the outcomes shown in (A). The complete shell outcomes are shown with the and to enable the encapsulated cargo. (C) The number of cargo molecules encapsulated by shells assembled in simulations is to the results of equilibrium simulations The results are all complete shells any assembled at each of the simulations were performed for simulations with and assembled shells at and were up to Assembly of full shells pathway, Figure or Figure is typically two of faster than assembly of empty shells (Figure This the key that the cargo in shell during all of molecules initially shell nucleation by interactions among clusters. the of a condensed globule a large for adsorption of additional subunits, the of subunits to the capsid, thus its The condensed cargo of the last subunits, which are by as noted for simulations of empty capsids et al., Figure how the products of assembly around cargo with weak interactions depends on parameters. While moderate parameter values lead to complete assembly, weak εSC and εSS of Figure shell leading to the In the of large εSC but weak εSS the shell-cargo interaction small disordered globules cargo particles, lower of Figure while under strong subunit and weak cargo interactions shells nucleate but cannot condense the leading to the complete but assembly for assembly around a strong interactions lead to and shells. However, the predominant of is now shell the cargo is below its the condensed globule leads to a on the shell subunits, which can the shell and thus closure of a shell. model The free energy model the parameter values required for shell assembly with no parameters map in Figure Since it is an equilibrium model and only considers the free energy difference between complete and unassembled it cannot between parameter values that lead to complete assembly or at the but simulation However, the thermodynamic does suggest that the simulations resulting in shells would on a We not show Δ⁢fassem in Figure because the globule is than assembled shells for εCC=1.3, but the of shells in our simulations roughly the of the equilibrium (Figure supplement 1). of other parameters or To investigate the results above depend on within our we performed of additional we performed simulations in which the between cargo diameter in shell subunit size was shown in Figure supplement assembly is most for our cargo diameter which the model was but assembly occurs for cargo varied a factor of we performed assembly simulations with cargo molecules with a

weak interaction is one of the least understood topics in particle physics. is one of the four kinds of force that physicists recognize in nature, and, as its name indicates, it is much weaker than two of the others, the strong (nuclear) interaction and electromagnetism. has therefore been difficult to study because its effects are usually masked by the two stronger forces. In a few years the world will have at least two particle accelerators with capacities of hundreds of billions of electron-volts energy, and physicists hope that these machines may allow the beginning of a new era of systematic and vigorous investigation of the weak force. There is a finite chance that in our lifetime we'll see something interesting in the weak David B. Cline of the University of Wisconsin told the recent Coral Gables Conference on Fundamental Interactions at High Energy at the University of Miami. first manifestation of the weak interaction that physicists noticed was nuclear beta decay. In beta decay a neutron inside an atomic nucleus turns into a proton, emitting by the way an electron (called in the old days a beta ray) and an antineutrino. In recent years many other particles have been found that decay radioactively under the influence of the weak interaction. From these decay activities alone it is hard to arrive at a good mathematical description of the weak force or to test rival theories. That can be better done by investigating how the force acts in particle collisions in accelerator experiments. But the two stronger forces blot out the effect of the weak force unless one party to the collision is a neutrino. (Neutrinos are the only known kind of particle that responds to the weak force but to neither of the two stronger ones.) To do experiments with neutrinos requires a copious and energetic beam of them. world's two new accelerators (the National Accelerator Laboratory now in operation at Batavia, Ill., and CERN II beginning construction near Geneva) will provide such beams. They may be used to study the weak force itself and to use it as a means of probing the structure of other particles. The new accelerators may very well provide a new era in neutrino spectroscopy, A. K. Mann of NAL told the Coral Gables Conference. It may be the new optics of the remainder of this century. One of the most important questions to be decided by forthcoming experiments is whether the weak force is a local interaction or a nonlocal one. A local interaction is one in which the two interacting particles affect each other directly, rather like two billiard balls colliding. In a nonlocal interaction, the interacting particles exchange an intermediarly particle, and this particle mediates the effect of one to the other. In the strong interaction, the intermediary particle is a virtual meson; in electromagnetism it is a virtual photon. Formulations of both kinds have been suggested for the weak interaction. In the traditional nonlocal formulation, the intermediary is called the intermediate vector boson. Lately some theorists, most prominently T. D. Lee of Columbia University (SN: 10/9/71, p. 252) and Steven Weinberg of Massachusetts Institute of Technology, have been trying to work out a combined theory of the weak interaction plus electromagnetism. In this case there is a family of intermediaries among which both the vector boson and the photon find a place. local formulations arouse dismay because in them the particles most concerned with the weak interaction, the class called leptons (the electron, the muon, the two kinds of neutrino and their respective antiparticles), are point particles; like a formal geometric point, they have zero dimensions. is difficult to visualize a body with mass that does not occupy space, and, when one tries, all sorts of disturbing paradoxes come forth: particles have infinite mass densities, infinite self-energies, infinite electric charge densities and other infinities, some of which appear to be at variance with known properties of the particles. A nonlocal weak interaction allows the particles to occupy space. also does away with the troublesome infinities. Besides its esthetic qualities, what evidence is now in hand tends to incline some people toward the nonlocal theory. The point theory of the weak interaction cannot be exact, says Mann. He and others expect that the inexactness will become clear in the new range of energies about to be opened up. At energies near 300 billion electron-volts (GeV) something new may happen, he suggests. Some structure may appear in the interaction: maybe a vector boson propagator, maybe new kinds of leptons, maybe new neutrinos, maybe higher~~~~~~~~~~~~~~~~~~~~~~~~~~~~~lr i,.,...,...,.~~~~~~~~~~~~~~~~-,-----------------'>.' s

The non - accelerator observation of the long - range strong nuclear interaction is presented. We have studied the low - temperature spectra (reflection and luminescence) of the LiH (without strong interaction in hydrogen nucleus) and LiD (with strong interaction in deuterium nucleus) crystals which are different by term of one neutron from each other. The experimental observation of isotopic shift (103 meV) of the phononless free excitons emission line in LiD crystals is a direct manifestation of the long - range strong nuclear interaction. Such conclusion is made to the fact that the gravitation, electromagnetic and weak interactions are the same in both kind crystals, it only emerges the strong interaction in deuterium nucleus. As far as Born - Oppenheimer approximation does not work in isotope effect, we tentative connect our experimental observation with long - range hadron - lepton interaction. Most important study of the LiHx D1-x mixed crystals is the first measurement of the long - range force dependence of strong nuclear interaction on the distance between nucleons in deuterium nucleus

A possible goal of fundamental physics is to reduce all natural phenomena to a set of basic laws and theories which, at least in principle, can quantitatively reproduce and predict experimental observations. At the microscopic level all the phenomenology of matter and radiation, including molecular, atomic, nuclear, and subnuclear physics, can be understood in terms of three classes of fundamental interactions: strong, electromagnetic, and weak interactions. For all material bodies on the Earth and in all geological, astrophysical, and cosmological phenomena, a fourth interaction, the gravitational force, plays a dominant role, but this remains negligible in atomic and nuclear physics. In atoms, the electrons are bound to nuclei by electromagnetic forces, and the properties of electron clouds explain the complex phenomenology of atoms and molecules. Light is a particular vibration of electric and magnetic fields (an electromagnetic wave). Strong interactions bind the protons and neutrons together in nuclei, being so strongly attractive at short distances that they prevail over the electric repulsion due to the like charges of protons. Protons and neutrons, in turn, are composites of three quarks held together by strong interactions occur between quarks and gluons (hence these particles are called “hadrons” from the Greek word for “strong”). The weak interactions are responsible for the beta radioactivity that makes some nuclei unstable, as well as the nuclear reactions that produce the enormous energy radiated by the stars, and in particular by our Sun. The weak interactions also cause the disintegration of the neutron, the charged pions, and the lightest hadronic particles with strangeness, charm, and beauty (which are “flavour” quantum numbers), as well as the decay of the top quark and the heavy charged leptons (the muon μ− and the tau τ−). In addition, all observed neutrino interactions are due to these weak forces.

The over-all reactive process is assumed to follow a two-step mechanism each characterized by a different kind of interaction: (i) the weak interaction that takes care of the product rotational distribution; and (ii) the strong interaction that governs the product vibrational distribution. If we make this assumption it is possible to couple three-dimensional classical rotational distributions with collinear quantum mechanical vibrational distributions. This procedure leads to a modified vibrational distribution which is found to be in agreement with experiment.

Correlated relaxed equilibria in nonzero-sum linear differential games

This paper investigates the problem of joint base station selection and resource allocation in an orthogonal frequency division multiple access (OFDMA) heterogeneous cellular network. The original throughput maximization problem is NP-hard and we propose solving it by using game theoretic stochastic learning approaches. To this end, we first transform the original problem into a tractable form, which has a weighted utility function. Then we prove that an exact potential game applies and it exists the best Nash equilibria which is a near optimal solution of the original problem when an efficient solution method of the weights is employed. To obtain the optimal solution, we redesign the utility function by leveraging a state space to formulate the original problem into an ordinal state based potential game, which is proved that it exists a recurrent state equilibrium point that maximizes system throughput. Furthermore, we propose two different variants of Max-logit learning algorithm based on these two games respectively: one is a simultaneous learning algorithm with less information exchange, which achieves the best Nash equilibria point of the exact potential game and the other is an efficient learning algorithm for the ordinal state based potential game, which can converge to the global optimization solution. Finally, numerical results are given to validate that theoretical findings.

This study aims to characterize and clarify a pentagonal fuzzy continuous static game (PF-CSG) that constraints and cost functions are fuzzy numbers. Pentagonal fuzzy numbers characterize their fuzzy parameters. The α − Pareto optimal solution concept has specified. The decomposition approach has applied to decompose the problem into subproblems each of them having smaller and independent subproblems. In addition, the Nash equilibrium solution concept was used to obtain the solutions of these subproblems. The advantages of this study are the players independently without collaboration with any of the others and that each player seeks to minimize the cost function. Also, the information available to each player consists of the cost function and constraints. An illustrated numerical example has discussed for proper understanding and interpretation of the proposed concept.

The branching ratios of the decay of the hyperons, the ratios of the mean life times of the decay of various hyperons and the ratios of the life times of the hyperons to that of the K+-meson are calculated by using the universal Ferm weak interaction proposed by Feynman and Gell-Mann and the strong interaction proposed by Salam. The perturbation theory is used with a cut off in the region of large momentum. By using the coupling constants of the strong interaction given by Warner and choosing suitable values for the other two coupling constants remaing to be fixed most of the results of our calculation agree fairly well with the experiments. The worst result differs from the experimental Values at most bya facto of 2. The theoretical values of the branching ratios of K+-decay is derivedfrom this set of new coupling constants shown better agreement with the experiment than the results obtained in the previous paper.Our, calculation shows that the universal Fermi weak interaction of V-A type gives quite satisfactory explanation of the decay phenomena of K-mesons and the hyperons. It is also evident from our calculation, that these decays must be regarded as phenomena brought about by the joint action of the strong and weak interactions and that there exist two classes of strong interactions: the "very strong" one and the "medium strong" one. Our results favour the assumption that the ∧ and ∑ have the same parity. It also shows, that of the 4 terms of the strong interaction Hamiltonian given by Salam involving the K-meson at least two terma involve γ5 matrix.

To synthesize fixed-final-time control-constrained optimal controllers for discrete-time nonlinear control-affine systems, a single neural network (NN)-based controller called the Finite-horizon Single Network Adaptive Critic is developed in this paper. Inputs to the NN are the current system states and the time-to-go, and the network outputs are the costates that are used to compute optimal feedback control. Control constraints are handled through a nonquadratic cost function. Convergence proofs of: 1) the reinforcement learning-based training method to the optimal solution; 2) the training error; and 3) the network weights are provided. The resulting controller is shown to solve the associated time-varying Hamilton-Jacobi-Bellman equation and provide the fixed-final-time optimal solution. Performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite-time attitude maneuver subject to control bounds. The new formulation has great potential for implementation since it consists of only one NN with single set of weights and it provides comprehensive feedback solutions online, though it is trained offline.

It has been pointed out that the group method of data handling algorithm has a disadvantage that predicts an abnormal result in the event of shortage of modelling data. This paper presents a method of modelling that avoids the emergence of abnormal predicted values. For estimation of partial polynomial parameters, a bad data suppression estimator is employed. It is based on a non-quadratic cost function that reduces to the weighted least squares estimator in the absence of bad data. Real data are used to illustrate that this method prevents abnormal results.

Dual-energy computed tomography (DECT) makes it possible to get two fractions of basis materials without segmentation. One is the soft-tissue equivalent water fraction and the other is the hard-matter equivalent bone fraction. Practical DECT measurements are usually obtained with polychromatic x-ray beams. Existing reconstruction approaches based on linear forward models without counting the beam polychromaticity fail to estimate the correct decomposition fractions and result in beam-hardening artifacts (BHA). The existing BHA correction approaches either need to refer to calibration measurements or suffer from the noise amplification caused by the negative-log preprocessing and the ill-conditioned water and bone separation problem. To overcome these problems, statistical DECT reconstruction approaches based on nonlinear forward models counting the beam polychromaticity show great potential for giving accurate fraction images. This work proposes a full-spectral Bayesian reconstruction approach which allows the reconstruction of high quality fraction images from ordinary polychromatic measurements. This approach is based on a Gaussian noise model with unknown variance assigned directly to the projections without taking negative-log. Referring to Bayesian inferences, the decomposition fractions and observation variance are estimated by using the joint maximum a posteriori (MAP) estimation method. Subject to an adaptive prior model assigned to the variance, the joint estimation problem is then simplified into a single estimation problem. It transforms the joint MAP estimation problem into a minimization problem with a nonquadratic cost function. To solve it, the use of a monotone conjugate gradient algorithm with suboptimal descent steps is proposed. The performance of the proposed approach is analyzed with both simulated and experimental data. The results show that the proposed Bayesian approach is robust to noise and materials. It is also necessary to have the accurate spectrum information about the source-detector system. When dealing with experimental data, the spectrum can be predicted by a Monte Carlo simulator. For the materials between water and bone, less than 5% separation errors are observed on the estimated decomposition fractions. The proposed approach is a statistical reconstruction approach based on a nonlinear forward model counting the full beam polychromaticity and applied directly to the projections without taking negative-log. Compared to the approaches based on linear forward models and the BHA correction approaches, it has advantages in noise robustness and reconstruction accuracy.