Phases in WLZZ matrix models

Preface.- Random Matrices and Number Theory. 1. Introduction. 2. 3. Characteristic polynomials of random unitary matrices 4. Other compact groups. 5. Families of L-functions and Symmetry. 6. Asymptotic expansions. References J.P. Keating.- 2D Quantum Gravity, Matrix Models and Graph Combinatorics. 1. Introduction. 2. Matrix models for 2D quantum gravity. The one-matrix model I: large N limit and the enumeration of planar graphs. 4. The trees behind the graphs. 5. The one-matrix model II: topological expansions and quantum gravity. 6. The combinatorics beyond matrix models: geodesic distance in planar graphs. 7. Planar graphs as spatial branching processes. 8. Conclusion P. Di Francesco.- Eigenvalue Dynamics, Follytons and Large N Limits of Matrices. References J. Arnlind, J. Hoppe.- Random Matrices and Supersymmetry in Disordered Systems. Supersymmetry method. 2. Wave functions fluctuations in a finite volume. Multifractality. Recent and possible future developments. Summary. Acknowledgements. References K.B. Efetov.- Hydrodynamics of Correlated Systems. 1. Introduction. 2. Instanton or rare fluctuation method. 3. Hydrodynamic approach. 4. Linearized hydrodynamics or bosonization. 5. EFP through an asymptotics of the solution. 6. Free fermions. 7. Calogero-Sutherland model. 8. Free fermions on the lattice. 9. Conclusion. Acknowledgements. Appendix: Hydrodynamic approach to non-Galilean invariant systems. Appendix: Exact results for EFP in some integrable models. References. A.G. Abanov.- QCD, Chiral Random Matrix Theory and Integrability. 1. Summary. 2. Introduction. 3. QCD. 4. The Dirac Spectrum in QCD. 5. Low Energy Limit of QCD. 6. Chiral RMT and the QCD Dirac Spectrum. 7. Integrability and the QCD Partition Function. 8. QCD at Finite Baryon Density. 9. Full QCD at Nonzero Chemical Potential. 10. Conclusions. Acknowledgements. References J.J. M. Verbaarschot.- Euclidan Random Matrices: Solved and Open Problems. 1. Introduction. 2. Basic definitions. 3. Physical motivations. 4. Field theory. 5. The simplest case. 6. Phonnos. References G. Parisi.- Matrix Models and Growth Processes. 1. Introduction. 2. Some ensembles of random matrices with complex eigenvalues. 3. Exact results at finite N. 4. Large N limit. 5. The matrix model as a growth problem. References A. Zabrodin.- Matrix Models and Topological Strings. 1. Introduction. 2. Matrix models. 3. Type B topological strings and matrix models. 4. Type A topological strings, Chern-Simons theory and matrix models M. Marino.- Matrix Models of Moduli Space. 1. Introduction. 2. Moduli Space of Riemann Surfaces and its Topology. 3. Quadratic Differentials and Fatgraphs. 4. The Penner model. 5. Penner Model and Matrix Gamma Function. 6. The Kontsevich Model. 7. Applications to String Theory. 8. Conclusions. References S. Mukhi.- Matrix Models and 2D String Theory. 1. Introduction. 2. An overview of string theory. 3. Strings in D-dimensional spacetime. 4. Discretized surfaces and 2D string theory. 5. An overview of observables. 6. Sample calculation: the disk one-point function. 7. Worldsheet description of matrix eigenvalues. 8. Further results. 9. Open problems. References E.J. Martinec.- Matrix Models as Conformal Field Theories. 1. Introduction and historical notes. 2. Hermitian matrix integral: saddle points and hyperellptic curves. 3. The hermitian matrix model as a chiral CFT. 4. Quasiclassical expansions: CFT on a hyperelliptic Riemann surface. 5. Generalization to chains of random matrices. References I.K. Kostov.-

A comparison of individual-based and matrix projection models for simulating yellow perch population dynamics in Oneida Lake, New York, USA

Matrix projection models and individual-based models (IBM) are commonly used for the analysis and management of fish populations. Matrix models break down the population into age or stage classes, while IBMs track individuals. I perform a series of quantitative comparisons between the predictions of the two modeling approaches using the IBM as the standard of comparison to demonstrate when individual variation, species interactions, and spatial heterogeneity adversely affect matrix model performance. I first evaluate the matrix approach for predicting yellow perch population responses when perch are involved in size-specific predator-prey interactions with walleye. I created density-dependent and stochastic age-structured and stage-within-age matrix models from an Oneida Lake walleye-yellow perch IBM, and then changed perch survival rates within the matrix models and IBM and compared their predicted responses. The matrix models simulated yellow perch responses reasonably well when density-dependent YOY survival was correctly defined. At least 20 years of data (IBM output) were needed to correctly estimate the density-dependent relationships in the matrix models. Second, I developed a 2-species matrix model by linking the elements between perch and walleye matrix models. The 2-species model simulated yellow perch prey responses reasonably well, but was unable to correctly predict walleye predator responses. Third, I developed a new IBM that simulated a 6-species tidal marsh community on a fine-scale spatial grid of habitat cells. The IBM was used to scale individual-level effects of lowered dissolved oxygen and habitat degradation to population-level responses, and used to estimate relatively simple stage-based matrix models for grass shrimp and gulf killifish populations. Equilibrium analysis of the simple matrix models was insufficient for predicting population responses. This study showed that stochastic, density-dependent matrix projection models were able to mimic density- dependent survival processes and species interactions relatively well, while equilibrium analysis of simple matrix models was inadequate. The matrix approach consistently had trouble estimating density-dependent and inter-specific growth relationships that were important for accurate model predictions. I recommend the use of IBMs and relatively complicated matrix models (stage-within-age, stochastic, density-dependent, multispecies) for simulation of fish population and community dynamics.

Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. We introduce the “integral projection model,” which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. We describe a method for fitting the model to data, and we apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection model avoids this problem and yields size-specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.

Relationships between fish population responses to changes in their vital rates and commonly available life history traits would be a powerful screening tool to guide management about species vulnerability, to focus future data collection on species and life stages of concern, and to aid in designing effective habitat enhancements. As an extension of previous analyses by others, I analyzed the responses to changes in fecundity and yearling survival of age-structured matrix and individual-based population models of 17 populations comprising 10 species. Simulations of the matrix models showed that the magnitude of population responses, but not the relative order of species sensitivity, depended on the state (sustainable or undergoing excessive removals) of the population. Matrix and individual-based models predicted population responses that appeared to be unrelated to their species-level life history traits when responses were plotted on a three-end-point life history surface. Density-dependent adult growth was added to the lake trout (Salvelinus namaycush) matrix model, and simulations demonstrated the potential importance to predicted responses of density-dependent processes outside the usual spawner–recruit relationship. Four reasons for the lack of relationship between population responses and life history traits related to inadequate population models, incorrect analysis, inappropriate life history model, and important site-specific factors are discussed.

The mechanistic approach has proven so far to be flexible and successful for simulation of the grinding process. The basic idea underlying mechanistic models, namely the matrix and population balance models, is based on the identification of natural events during grinding. Since each model has its own capabilities and limitations, their combined use may offer additional advantages on this aspect. In this study, the matrix model and the selection function, namely the probability of breakage of the population balance model, were combined through a MATLAB code to predict the size distribution of the grinding products of quartz, marble, quartzite and metasandstone. The modeling results were in very good agreement with the particle size distributions obtained after grinding the feeds in a ball mill.

Wilsonian approximated renormalization group for matrix and vector models in 2 < d < 4

Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. We introduce the “integral projection model,” which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. We describe a method for fitting the model to data, and we apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection model avoids this problem and yields size-specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.

Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wavefunctions can be reduced to the solution of a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For models with cubic and quartic interactions, the ground-state energy appears to show rapid convergence to the large-N limit. For the special case of a quartic model, we find explicitly an isospectral Hermitian matrix model. The Hermitian form for a vector model with O(N) symmetry can also be found, and shows many unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that PT-symmetric scalar field theories are asymptotically free.

BackgroundEducational experts commonly agree that tailor-made guidance is the most efficient way to foster the learning and developmental process of learners. Diagnostic assessments using cognitive diagnostic models (CDMs) have the potential to provide individual profiles of learners’ strengths and weaknesses on a fine-grained level that can enable educators to assess the current position of learners. However, to obtain this necessary information a strong connection has to be made between cognition (the intended competence), observation (the observed learners’ responses while solving the tasks), and interpretation (the inferences made based on the observed responses of learners’ underlying competencies). To secure this stringent evidence-based reasoning, a principled framework for designing a technology-based diagnostic assessment is required—such as the evidence-centred game design (ECgD).AimWith regard to a diagnostic assessment, three aspects are of particular importance according to the ECgD approach: (I) the selection of a measurable set of competence facets (so-called skills) and their grain-size, (II) the constructed pool of skill-based tasks, and (III) the clear and valid specified task to skill assignments expressed within the so-called Q matrix. The Q matrix represents the a priori assumption for running the statistical CDM-procedure for identifying learners’ individual competence/skill profiles. These three prerequisites are not simply set by researchers’ definition nor by experts’ common sense. Rather, they require their own separate empirical studies. Hence, the focus of this paper is to evaluate the appropriateness and coherence of these three aspects (I: skill, II: tasks, and III: Q matrix). This study is a spin-off project based on the results of the governmental ASCOT research initiative on visualizing apprentices’ work-related competencies for a large-scale assessment—in particular, the intrapreneurship competence of industrial clerks. With the development of a CDM I go beyond the IRT-scaling offering the prerequisites for identifying individuals’ skill profiles as a point of departure for an informative individual feedback and guidance to enhance students’ learning processes.MethodsTherefore, I shall use a triangulated approach to generate three empirically based Q matrix models from different sources (experts and target-group respondents), inquiry methods (expert ratings and think-aloud studies), and methods of analyses (frequency counts and a solver–non-solver comparison). Consequently, the four single Q matrix models (researchers’ Q matrix generated within the task construction process and the three empirically based Q matrix models) were additionally matched by different degrees of overlap for balancing the strengths and weaknesses of each source and method. By matching the patterns of the four single Q matrix models, the appropriateness of the set of intrapreneurship skills (I) and the pool of intrapreneurship tasks (II) were investigated. To identify and validate a reasonable proxy for the task to skill assignments for selecting the best fitting Q matrix model (III), the single as well as the matched Q matrix models where empirically contrasted against N = 919 apprentices’ responses won and scaled up within the ASCOT-project using psychometric procedures of cognitive diagnostic within the DINA (Haertel in J Educ Meas 26:301–323, 1989) model.ResultsThe pattern matching resulted in a set of seven skills and 24 tasks. The appropriateness of these results was emphasized by model fit values of the different Q matrix models. They show acceptable up to good sizes (SRMSR between .053 and .055). The best fitting model is a matched Q matrix of which the match is not that strict or smooth with regard to the degree of overlap.ConclusionsThe study provides a principled design for a technology-based diagnostic assessment. The systematic and extensive validation process offers empirical evidence for (I) the relevance and importance of the specified intrapreneurship skills, (II) tasks prompting the intended skills, and (III) the sophisticated proxy of real cognitive processes (in terms of the Q matrix), but also give hints for revision. This—within a diagnostic assessment—preliminary work aims at identifying the best-fitting Q matrix to enable the next step of depicting learners’ individual strengths and weaknesses on a sound basis.

A matrix model of the representation of spatial objects for the synthesis, reconstruction, and analysis of their shape is proposed. The model is built on the basis of discrete data about the object, such as, for example, raster images or readings of spatial scanners. Unlike similar voxel models, matrix models describe not the volume but the surfaces of objects and, while preserving the advantages of voxel models, such as simplicity and regularity of structure, eliminate their inherent redundancy. It is shown in the work that, while retaining information on the form sufficient for visualizing the object, the matrix model can occupy 1.5–3 times less memory (the comparison was carried out for models in the VOX format of the MagicaVoxel package). The conditions are established under which the matrix model remains more economical than the voxel model, and it is shown that these conditions are satisfied for practically significant cases. An algorithm for constructing a discrete matrix model based on a voxel is described. A general approach to solving the problem of the resampling of models of three-dimensional graphics objects is proposed, which does not depend on the dimension of the source data array. In the framework of this approach, the matrix model is resampled. The necessary transformations of the model matrices are described, including both resampling and requantization, which ensures their controlled accuracy of the representation of spatial objects. Procedures for monitoring and restoring integrity have also been developed for the proposed matrix model. The obtained conditions for monitoring the integrity of the model in practically significant cases (when the number of model elements is more than 153) can reduce the number of elements viewed, compared with the voxel model. The limitations of matrix models are established associated with the possible loss of information about a part of the surface hidden from an external observer

Topological symmetry, background independence and matrix models

Matrix-model description of [formula omitted] gauge theories with non-hyperelliptic Seiberg–Witten curves

The ability to harmonize data sources with varying temporal, spatial, and ecosystem measurements (e.g. forest structure to soil organic carbon) for creation of terrestrial carbon baselines is paramount to refining the monitoring of terrestrial carbon stocks and stock changes. In this study, we developed and examined the short- (5 years) and long-term (30 years) performance of matrix models for incorporating light detection and ranging (LiDAR) strip samples and time-series Landsat surface reflectance high-level data products, with field inventory measurements to predict aboveground biomass (AGB) dynamics for study sites across the eastern USA—Minnesota (MN), Maine (ME), Pennsylvania-New Jersey (PANJ) and South Carolina (SC). The rows and columns of the matrix were stand density (i.e. number of trees per unit area) sorted by inventory plot and by species group and diameter class. Through model comparisons in the short-term, we found that average stand basal area (B) predicted by three matrix models all fell within the 95% confidence interval of observed values. The three matrix models were based on (i) only field inventory variables (inventory), (ii) LiDAR and Landsat-derived metrics combined with field inventory variables (LiDAR + Landsat + inventory), and (iii) only Landsat-derived metrics combined with field inventory variables (Landsat + inventory), respectively. In the long term, predicted AGB using LiDAR + Landsat + inventory and Landsat + inventory variables had similar AGB patterns (differences within 7.2 Mg ha−1) to those predicted by matrix models with only inventory variables from 2015–2045. When considering uncertainty derived from fuzzy sets all three matrix models had similar AGBs (differences within 7.6 Mg ha−1) by the year 2045. Therefore, the use of matrix models enabled various combinations of LiDAR, Landsat, and field data, especially Landsat data, to estimate large-scale AGB dynamics (i.e. central component of carbon stock monitoring) without loss of accuracy from only using variables from forest inventories. These findings suggest that the use of Landsat data alone incorporating elevation (E), plot slope (S) and aspect (A), and site productivity (C) could produce suitable estimation of AGB dynamics (ranging from 67.1–105.5 Mg ha−1 in 2045) to actual AGB dynamics using matrix models. Such a framework may afford refined monitoring and estimation of terrestrial carbon stocks and stock changes from spatially explicit to spatially explicit and spatially continuous estimates and also provide temporal flexibility and continuity with the Landsat time series.

The article defines the role of evaluation of investment attractiveness of the enterprise. The main directions of methodological approaches to the assessment of investment attractiveness of the enterprise are given, their peculiarities and purpose are defined. The main disadvantages of methodological approaches to the assessment of investment attractiveness are revealed. It is proposed to use a matrix model to evaluate the investment attractiveness of an industrial enterprise. The necessity of using precisely a matrix model of complex nature to assess the investment attractiveness of industrial enterprises based on the use of quantitative and qualitative factors is substantiated. The main advantages of the matrix model of estimation of investment attractiveness of the enterprise are determined. A methodological toolkit for estimating the investment attractiveness of an industrial enterprise on the basis of a matrix model has been improved, taking into account the current state of domestic industrial enterprises and identifying problematic areas of activity. The matrix model has been supplemented with a matrix of quantitative indicators by a group of property condition indicators, which will allow the investor to take a more objective approach to assessing the investment attractiveness of an industrial enterprise. Quantitative indicators of investment attractiveness are grouped and their marginal evaluation criteria are given. The general index of quantitative characteristic of investment attractiveness of industrial enterprise and possible variants of estimation of its financial condition are determined. The qualitative indicators of estimation of investment attractiveness on the basis of the matrix model are given, they are given weighting coefficients and their boundary intervals are defined. On the basis of qualitative and quantitative indicators, the levels of the generalized index of investment attractiveness of an industrial enterprise according to the matrix model have been formed, which will allow to prove to the investor the expediency of investing in an industrial enterprise. Keywords investment; investment attractiveness; investment potential; enterprise; matrix model.