Narrow Results

The tensor formulation for the effective theory of QCD vector resonances, whose model we denote by TEVR, is given by an antisymmetric tensor field and describes spin 1 particles. Our goal is to show, by different approaches, that the Abelian version of this model presents the so-called “spin jumping” when we consider its massless limit. Classically we find, by the use of the equations of motion and the Hamiltonian constraint analysis, that the massive phase of the model describes spin 1 particles while its massless phase describes spin 0 particles. From the quantum point of view, we derive these conclusions via tree-level unitarity analysis and the master action approach.

We introduce a new class of two-dimensional gravity models using ideas motivated by the Teleparallel Equivalent of General Relativity. This leads to a rather natural formulation of a theory that has close links with Jackiw–Teitelboim gravity. After introducing the theory and discussing its vacuum solutions, we present the Hamiltonian analysis. This implies the presence of a single dynamical degree of freedom, which is in sharp contrast to General Relativity, where there are no degrees of freedom in two spacetime dimensions. Our approach can be extended to various other lower-dimensional gravity theories and thus could be of wider interest.

In this paper, we analyzed charged particle transverse momentum spectra in high multiplicity events in proton–proton and nucleus–nucleus collisions at LHC energies from the ALICE experiment using the color string percolation model (CSPM). The color reduction factor and associated string density parameters are extracted for various multiplicity classes in $pp$ collisions and centrality classes for heavy-ion collisions at various LHC energies to study the effect of collision geometry and collision energy. These parameters are used to extract the thermodynamical quantities temperature and the energy density of the hot nuclear matter. A universal scaling is observed in initial temperature when studied as a function of charged particle multiplicity scaled by transverse overlap area. From the measured initial energy density $𝜀$ and the initial temperature T, a dimensionless quantity $𝜀/T^4$ is constructed which is used to obtain the degrees of freedom (DOF) of the deconfined phase. A two-step behavior and a sudden increase in DOF of $\sim$47 for the ideal gas, above the hadronization temperature (T$\approx$ 210MeV), are observed in case of heavy-ion collisions at LHC energies.

The main goal of this work is to develop a robust framework for an exhaustive description of essential properties of a fractal object. For this purpose, the inherent features of fractal sets are scrutinized. The topological, metrological, morphological, and topographical attributes of fractal systems are delineated. The criteria of the fractal connectedness are established. The characteristics of the fractal connectivity and ramification are ascertained. The index of the fractal loopiness is introduced. The quantifications of the fractal heterogeneity, lacunarity, and anisotropy are briefly sketched out. A set of key attributes which enable a proper characterization of fractal system are suggested.

Multipath propagation effects encountered in mobile wireless channels provide additional degrees of freedom [Ganesan et al., 2000] that can be exploited via appropriate signaling and reception. In this work, based on a set of measurements conducted at Eurecom, we consider a novel approach of analyzing an Ultra Wide Band (UWB) indoor radio propagation channel by performing an eigen-decomposition and observing the scaling of the number of significant eigenvalues with the channel bandwidth by using the Akaike information criterion (AIC) and Minimum Description Length (MDL). These criterion are applied to estimate the number of degrees of freedom (DoF) of an UWB in an indoor environment. We evaluate our approach under both scenarios, line-of-sight (LOS) and non-line-of-sight (NLOS). Based on AIC and the MDL criterion, we find that this number is large; this has important consequences on the receiver design. We show also, that in opposition to the accepted idea in the literature, the number of Degrees of Freedom (DoF) does not increase linearly with the channel bandwidth. Hence, the results of the estimation of UWB channel entropy confirm that the number of DoF increases sub-linearly with the channel bandwidth.

We give a short review on the status of research on the theoretical foundations of $f(𝕋)$ gravity theories. We discuss recent results on perturbative and non-perturbative approaches, causality and degrees of freedom, and discuss future directions to follow.

This paper is concerned with the approximation of data using linear models for which there is additional information to be incorporated into the analysis. This situation arises, for example, in Tikhonov regularisation, model averaging, Bayesian inference and approximation with Gaussian process models. In a least-squares approximation process, the m-vector of data y is approximated by a linear function ŷ = Hy of the data, where H is an m × m matrix related to the observation matrix. The effective number of degrees of freedom associated with the model is given by the sum of the eigenvalues of H. For standard linear least-squares regression, the matrix H is a projection and has n eigenvalues equal to 1 and all others zero, where n is the number of parameters in the model. Incorporating prior information about the parameters reduces the effective number of degrees of freedom since the ability of the model to approximate the data vector y is partially constrained by the prior information. We give a general approach for providing bounds on the effective number of degrees of freedom for models with prior information on the parameters and illustrate the approach on common problems. In particular, we show how the effective number of degrees of freedom depends on spatial or temporal correlation lengths associated with Gaussian processes. The correlation lengths are seen to be tuning parameters used to match the model degrees of freedom to the (generally unknown) number of degrees of freedom associated with the system giving rise to the data. We also show how Gaussian process models can be related to Tikhonov regularisation for an appropriate set of basis functions.

The following sections are included:

• INTRODUCTION

• POPULATION PARAMETERS AND THE REGRESSION MODELS
  • Data Description
  • Building the Population Regression Model
  • Sample Versus Population Regression Model

• THE LEAST-SQUARES ESTIMATION OF α AND β
  • Scatter Diagram
  • The Method of Least Squares
  • Estimation of Intercept and Slope

• STANDARD ASSUMPTIONS FOR LINEAR REGRESSION

• THE STANDARD ERROR OF ESTIMATE AND THE COEFFICIENT OF DETERMINATION
  • Variance Decomposition
  • Standard Error of Residuals (Estimate)
  • The Coefficient of Determination

• THE BIVARIATE NORMAL DISTRIBUTION AND CORRELATION ANALYSIS
  • The Sample Correlation Coefficient
  • The Relationship Between r and b
  • The Relationship Between r and R²

• Summary

• Appendix 13A Derivation of Normal Equations and Optimal Portfolio Weights

• Appendix 13B The Derivation of Equation 13.16

• Appendix 13C The Bivariate Normal Density Function
  • Using a Mathematics Aptitude Test to Predict Grade in Statistics

• Appendix 13D American Call Option and the Bivariate Normal CDF
  • Valuating American Option

• Questions and Problems

This chapter is an introduction to complexity theory (encompassing chaos — a subset of complexity), a nascent domain, although, it possesses a historical root. Some fundamental properties of chaos/complexity (including complexity mindset, nonlinearity, interconnectedness, interdependency, far-from-equilibrium, butterfly effect, determinism/in-determinism, unpredictability, bifurcation, deterministic chaotic dynamic, complex dynamic, complex adaptive dynamic, dissipation, basin of attraction, attractor, chaotic attractor, strange attractor, phase space, rugged landscape, red queen race, holism, self-organization, self-transcending constructions, scale invariance, historical dependency, constructionist hypothesis and emergence), and its development are briefly examined. In particular, the similarities (sensitive dependence on initial conditions, unpredictability) and differences between deterministic chaotic systems (DCS) and complex adaptive systems (CAS) are analyzed. The edge of emergence (2^nd critical value, a new concept) is also conceived to provide a more comprehensive explanation of the complex adaptive dynamic (CAD) and emergence. Subsequently, a simplified system spectrum is introduced to illustrate the attributes, and summarize the relationships of the various categories of common systems.

Next, the recognition that human organizations are nonlinear living systems (high finite dimensionality CAS) with adaptive and thinking agents is examined. This new comprehension indicates that a re-calibration in thinking is essential. In the human world, high levels of human intelligence/consciousness (the latent impetus that is fundamentally stability-centric) drives a redefined human adaptive and evolution dynamic encompassing better potentials of self-organization or self-transcending constructions, autocatalysis, circular causation, localized spaces/networks, hysteresis, futuristic, and emergent of new order (involving a multi-layer structure and dynamic) — vividly indicating that intelligence/consciousness-centric is extremely vital. Simultaneously, complexity associated properties/characteristics in human organizations must be better scrutinized and exploited — that is, establishing appropriate complexity-intelligence linkages is a significant necessity. In this respect, nurturing of the intelligence mindset and developing the associated paradigmatic shift is inevitable.

A distinct attempt (the basic strategic approach) of the new intelligence mindset is to organize around human intrinsic intelligence — intense intelligence-intelligence linkages that exploits human intelligence/consciousness sources individually and collectively by focusing on intelligence/consciousness-centricity, complexity-centricity, network-centricity, complexity-intelligence linkages, collective intelligence, org-consciousness, complex networks, spaces of complexity (better risk management <=> new opportunities <=> higher sustainability) and prepares for punctuation points (better crisis management <=> collectively more intelligent <=> higher resilience/sustainability) concurrently — illustrating the significance of self-organizing capability and emergence-intelligence capacity. The conceptual development introduced will serve as the basic foundation of the intelligent organization (IO) theory.