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Learning Machines 101 - A Gentle Introduction to Artificial Intelligence and Machine LearningEpisode Summary: In real life, there is no certainty. There are always exceptions. In this episode, two methods are discussed for making inferences in uncertain environments. In fuzzy set theory, a smart machine has certain beliefs about imprecisely defined concepts. In fuzzy measure theory, a smart machine has beliefs about precisely defined concepts but some beliefs are stronger. Read More »The post LM101-007: How to Reason About Uncertain Events using Fuzzy Set Theory and Fuzzy Measure Theory appeared first on Learning Machines 101.

In the theory of belief functions, the measure of uncertainty is an important concept, which is used for representing some types of uncertainty incorporated in bodies of evidence such as the discord and the non-specificity. For the non-specificity part, some traditional measures use for reference the Hartley measure in classical set theory; other traditional measures use the simple and heuristic function for joint use of mass assignments and the cardinality of focal elements. In this paper, a new non-specificity measure is proposed using lengths of belief intervals, which represent the degree of imprecision. Therefore, it has more intuitive physical meaning. It can be proved that our new measure can be rewritten in a general form for the non-specificity. Our new measure is also proved to be a strict non-specificity measure with some desired properties. Numerical examples, simulations, the related analyses and proofs are provided to show the characteristics and good properties of the new non-specificity definition. An example of an application of the new nonspecificity measure is also presented.

Source: Digital Library of India Scanning Centre: C-DAC, Noida Source Library: Ratan Tata Library, University Delhi Date Accessioned: 7/10/2015 20:55 The Digital Library of India was a project under the auspices of the Government of India.

An introduction to Measure Theory

An introduction to Measure Theory

In this article we obtain a fundamental measure functional for the model of aligned hard hexagons in the plane. Our aim is not just to provide a functional for a new, admittedly academic, model, but to investigate the structure of fundamental measure theory. A model of aligned hard hexagons has similarities with the hard disk model. Both share "lost cases", i.e. admit configurations of three particles in which there is pairwise overlap but not triple overlap. These configurations are known to be problematic for fundamental measure functionals, which are not able to capture their contribution correctly. This failure lies in the inability of these functionals to yield a correct low density limit of the third order direct correlation function. Here we derive the functional by projecting aligned hard cubes on the plane x+y+z=0. The correct dimensional crossover behavior of these functionals permits us to follow this strategy. The functional of aligned hard cubes, however, does not have lost cases, so neither had the resulting functional for aligned hard hexagons. The latter exhibits, in fact, a peculiar structure as compared to the one for hard disks. It depends on a uniparametric family of weighted densities through a new term not appearing in the functional for hard disks. Apart from studying the freezing of this system, we discuss the implications of the functional structure for new developments of fundamental measure theory.

In this article we obtain a fundamental measure functional for the model of aligned hard hexagons in the plane. Our aim is not just to provide a functional for a new, admittedly academic, model, but to investigate the structure of fundamental measure theory. A model of aligned hard hexagons has similarities with the hard disk model. Both share "lost cases", i.e. admit configurations of three particles in which there is pairwise overlap but not triple overlap. These configurations are known to be problematic for fundamental measure functionals, which are not able to capture their contribution correctly. This failure lies in the inability of these functionals to yield a correct low density limit of the third order direct correlation function. Here we derive the functional by projecting aligned hard cubes on the plane x+y+z=0. The correct dimensional crossover behavior of these functionals permits us to follow this strategy. The functional of aligned hard cubes, however, does not have lost cases, so neither had the resulting functional for aligned hard hexagons. The latter exhibits, in fact, a peculiar structure as compared to the one for hard disks. It depends on a uniparametric family of weighted densities through a new term not appearing in the functional for hard disks. Apart from studying the freezing of this system, we discuss the implications of the functional structure for new developments of fundamental measure theory.

Speaker : Tatiana Toro

Speaker : Tatiana Toro

An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness and asymptotic analysis for fully nonlinear evolutionary game theoretic models. The model should be rich enough to include all classical nonlinearities, e.g., Beverton-Holt or Ricker type. For several such models formulated on the space of integrable functions, it is known that as the variance of the payoff kernel becomes small the solution converges in the long term to a Dirac measure centered at the fittest strategy; thus the limit of the solution is not in the state space of integrable functions. Starting with the replicator-mutator equation and a generalized logistic equation as bases, a general model is formulated as a dynamical system on the state space of finite signed measures. Well-posedness is established, and then it is shown that by choosing appropriate payoff kernels this model includes all classical density models, both selection and mutation, and discrete and continuous strategy (trait) spaces.

Capacitive mixing (CAPMIX) and capacitive deionization (CDI) are promising candidates for harvesting clean, renewable energy and for the energy efficient production of potable water, respectively. Both CAPMIX and CDI involve water-immersed porous carbon (supercapacitors) electrodes at voltages of the order of hundreds of millivolts, such that counter-ionic packing is important for the electric double layer (EDL) which forms near the surface of these porous materials. Thus, we propose a density functional theory (DFT) to model the EDL, where the White-Bear mark II fundamental measure theory functional is combined with a mean-field Coulombic and a mean spherical approximation-type correction to describe the interplay between dense packing and electrostatics, in good agreement with molecular dynamics simulations. We discuss the concentration-dependent potential rise due to changes in the chemical potential in capacitors in the context of an over-ideal theoretical description and its impact on energy harvesting and water desalination. Compared to less elaborate mean-field models our DFT calculations reveal a higher work output for blue-energy cycles and a higher energy demand for desalination cycles.

The NoMAD questionnaire is a measurement instrument to assess normalisation as an implementation outcome. The original English version of the questionnaire was developed by Finch and colleagues (2018).The NoMAD has been translated into several languages. This project brings NoMAD data from German health care innovation implementation projects together to derive a common German Normalization Process Theory Measure (G-NoMAD).

Conspiracy theories are everywhere, yet measuring them is tricky. Individuals endorsing these beliefs are not always willing to disclose their views in self-report questionnaires. In this study we design an instrument to distinguish CT-believers from non-believers, by measuring how quickly they react to conspiracy-related stimuli after the activation of a relevant topic.

Regularization functionals that lower level set boundary length when used with L^1 fidelity functionals on signal de-noising on images create artifacts. These are (i) rounding of corners, (ii) shrinking of radii, (iii) shrinking of cusps, and (iv) non-smoothing of staircasing. Regularity functionals based upon total curvature of level set boundaries do not create artifacts (i) and (ii). An adjusted fidelity term based on the flat norm on the current (a distributional graph) representing the density of curvature of level sets boundaries can minimize (iii) by weighting the position of a cusp. A regularity term to eliminate staircasing can be based upon the mass of the current representing the graph of an image function or its second derivatives. Densities on the Grassmann bundle of the Grassmann bundle of the ambient space of the graph can be used to identify patterns, textures, occlusion and lines.

Regularization functionals that lower level set boundary length when used with L^1 fidelity functionals on signal de-noising on images create artifacts. These are (i) rounding of corners, (ii) shrinking of radii, (iii) shrinking of cusps, and (iv) non-smoothing of staircasing. Regularity functionals based upon total curvature of level set boundaries do not create artifacts (i) and (ii). An adjusted fidelity term based on the flat norm on the current (a distributional graph) representing the density of curvature of level sets boundaries can minimize (iii) by weighting the position of a cusp. A regularity term to eliminate staircasing can be based upon the mass of the current representing the graph of an image function or its second derivatives. Densities on the Grassmann bundle of the Grassmann bundle of the ambient space of the graph can be used to identify patterns, textures, occlusion and lines.

Improving human systems integration through technologically advanced training and performance aids has become increasingly important to military transformation. Measures of improved cognitive and coordination processes arising from the employment of transformational tools are necessary to guide the refinement and future development of such technologies. In this paper we describe a Cognitive Load Theory approach to developing a combinatory measure of individual workload and team performance following an experimental intervention involving training and a Decision Support System. We discuss how indicators of what we term team decision efficiency can improve assessing the effectiveness of transformational processes and technologies.

In this paper, a model for assessment and intervention is presented. This model explains how to perform theory- and evidence- based as well as practice-based assessment and intervention. The assessment model applies a holistic approach to treatment planning, which includes recognition of the influence of community, school, peers, family and the functional and structural domains of personality at the behavioural, phenomenological, intra-psychic and biophysical level in a dialectical developmental process. One important aspect of the theoretical basis for preparation of this model is that the child's personality must be understood in terms of the historical development. This assessment and intervention model focuses equally on strength and virtues as well as obstacles against development within the individual and its relations. Finally, guidelines for how to measure outcome are provided. (Contains 6 figures and 35 footnotes.)

This paper considered the fairness of mutual exclusion elements, the most important building block for any arbiter, A probabilistic choice set model was introduced to capture the choice behavior of such elements. Using this model on infinite sequences we defined a probabilistic notion of fairness, and shown that mutual exclusion elements are fair in general, provided that a simple assumption about their probabilistic behavior is satisfied. (Any well-designed mutual exclusion element does satisfy the assumption.) We extended this result to establish the fairness of a wide class of arbiters including virtually all known non-prioritized multi-input designs. This essentially settles the weak fairness question for non-prioritized arbiters; in general such arbiters are fair in a sense that is very close to the standard notion of weak fairness.

In this paper we develop a notion of measure theory over boolean toposes which is analogous to noncommutative measure theory, i.e. to the theory of von Neumann algebras. This is part of a larger project to study relations between topos theory and noncommutative geometry. The main result is a topos theoretic version of the modular time evolution of von Neumann algebra which take the form of a canonical R+*-principal bundle over any integrable locally separated boolean topos.

Summary of abstract Field theory models including gauge theories with SSB are presented where the energy density of the true vacuum state (TVS) is zero without fine tuning. The above models are constructed in the gravitational theory where a measure of integration \Phi in the action is not necessarily \sqrt{-g} but it is determined dynamically through additional degrees of freedom. The ratio \Phi/\sqrt{-g} is a scalar field which can be solved in terms of the matter degrees of freedom due to the existence of a constraint. We study a few explicit field theory models where it is possible to combine the solution of the cosmological constant problem with: 1) possibility for inflationary scenario for the early universe; 2) spontaneously broken gauge unified theories (including fermions). The models are free from the well known problem of the usual scalar-tensor theories in what is concerned with the classical GR tests. The only difference of the field equations in the Einstein frame from the canonical equations of the selfconsistent system of Einstein's gravity and matter fields, is the appearance of the effective scalar field potential which vanishes in TVS without fine tuning.

Summary of abstract Field theory models including gauge theories with SSB are presented where the energy density of the true vacuum state (TVS) is zero without fine tuning. The above models are constructed in the gravitational theory where a measure of integration \Phi in the action is not necessarily \sqrt{-g} but it is determined dynamically through additional degrees of freedom. The ratio \Phi/\sqrt{-g} is a scalar field which can be solved in terms of the matter degrees of freedom due to the existence of a constraint. We study a few explicit field theory models where it is possible to combine the solution of the cosmological constant problem with: 1) possibility for inflationary scenario for the early universe; 2) spontaneously broken gauge unified theories (including fermions). The models are free from the well known problem of the usual scalar-tensor theories in what is concerned with the classical GR tests. The only difference of the field equations in the Einstein frame from the canonical equations of the selfconsistent system of Einstein's gravity and matter fields, is the appearance of the effective scalar field potential which vanishes in TVS without fine tuning.

A new measure of intrinsic time is introduced which broadens the endochronic theory and lends it a wider predictive scope. Idealized plastic models are shown to be constitutive subsets of the general theory and the phenomenon of yield is proved to be a consequence of a particular definition of the intrinsic time measure in terms of the plastic strain tensor. Various versions of the classical plasticity theory are shown to be asymptotic cases of the endochronic theory. In particular the kinematic hardening model, the isotropic hardening model as well as their combinations, are derivable directly from the general theory. In addition, the translation vector of the yield surface in stress space is found to be a constitutive property given by a linear functional of the history of the plastic strain. The Prager and Ziegler rules are immediately obtainable as special cases. It is also shown that the essential features of plastic response under conditions of 'stress reversals', are contained in the constitutive equation whose form remains invariant of the deformation history. It is believed that this is the first time that one single constitutive equation has been shown to predict correctly the essential features of plastic response under conditions of loading, unloading and reloading. However, when the intrinsic time measure is precisely equal to the norm of the increment of the plastic strain tensor, the constitutive unity, spoken of above, no longer exists.

A new measure of intrinsic time is introduced which broadens the endochronic theory and lends it a wider predictive scope. Idealized plastic models are shown to be constitutive subsets of the general theory and the phenomenon of yield is proved to be a consequence of a particular definition of the intrinsic time measure in terms of the plastic strain tensor. Various versions of the classical plasticity theory are shown to be asymptotic cases of the endochronic theory. In particular the kinematic hardening model, the isotropic hardening model as well as their combinations, are derivable directly from the general theory. In addition, the translation vector of the yield surface in stress space is found to be a constitutive property given by a linear functional of the history of the plastic strain. The Prager and Ziegler rules are immediately obtainable as special cases. It is also shown that the essential features of plastic response under conditions of 'stress reversals', are contained in the constitutive equation whose form remains invariant of the deformation history. It is believed that this is the first time that one single constitutive equation has been shown to predict correctly the essential features of plastic response under conditions of loading, unloading and reloading. However, when the intrinsic time measure is precisely equal to the norm of the increment of the plastic strain tensor, the constitutive unity, spoken of above, no longer exists.

It is known that hyperbolic dynamical systems admit a unique invariant probability measure with maximal entropy. We prove an effective version of this statement and use it to estimate an upper bound for Hausdorff dimension of exceptional sets arising from dynamics.

In the literature concerning the monopole matter, three gauges: Dirac, Schwinger, and Wu-Yang's, have been contrasted to each other, and the Wu-Yang's often appears as the most preferable one. The article aims to analyse this view by interpreting the monopole situation in terms of the conventioal Fourier series theory; in particular, having relied on the Dirichlet theorem. It is shown that the monopole case can be labelled as a very spesific and even rather simple class of problems in the frame of that theory: all the three monopole gauges amount to practicaly the same one-dimentional problem for functions given on the interval [0,pi] having a single point of discontinuity; these three vary only in its location.In addition, some general aspects of the Aharonov-Bohm effect are discussed; the way of how any singular potentials such as monopole's, being allowed in physics, touche the essence of the physical gauge principle itself is considered.

This tutorial is an informal introduction to measure theory for people who are interested in reading papers that use measure theory. The tutorial assumes one has had at least a year of college-level calculus, some graduate level exposure to random processes, and familiarity with terms like �closed� and �open.� The focus is on the terms and ideas relevant to applied probability and information theory. There are no proofs and no exercises. Lecture Notes Collection FreeScience.info ID2240 Obtained from https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2006-0008.pdf http://www.freescience.info/go.php?pagename=books&id=2240

Speaker : Tatiana Toro

Introduction to Real Analysis and Measure theory

To the Mathematics Editors of Internet Archive: I am contacting you as the only child of the mathematician Aurel Wintner (1903-1958), whose monograph “Spektraltheorie der Unendlichen Matrizen” you most kindly have added to your collection. My father published well over 400 papers and monographs in addition to his books. Nearly all of this material appeared in well-known journals and is easily available. However, during and shortly after WWII he independently published 4 monographs that only were distributed by mail to those who requested copies. It occurred to me, therefore, that these 4 items would be worth contributing to Internet Archive as well, along with the Spektraltheorie, which had gone out of print. As in that case, so with these — as a chemist I cannot actually judge this material, but from the comments of mathematician friends I believe that all of it is highly original. My father held the copyright to each; there was no publisher or journal involved; and I inherited all of his copyrights. Thus, there is no problem along those lines. Claude Wintner

To the Mathematics Editors of Internet Archive: I am contacting you as the only child of the mathematician Aurel Wintner (1903-1958), whose monograph “Spektraltheorie der Unendlichen Matrizen” you most kindly have added to your collection. My father published well over 400 papers and monographs in addition to his books. Nearly all of this material appeared in well-known journals and is easily available. However, during and shortly after WWII he independently published 4 monographs that only were distributed by mail to those who requested copies. It occurred to me, therefore, that these 4 items would be worth contributing to Internet Archive as well, along with the Spektraltheorie, which had gone out of print. As in that case, so with these — as a chemist I cannot actually judge this material, but from the comments of mathematician friends I believe that all of it is highly original. My father held the copyright to each; there was no publisher or journal involved; and I inherited all of his copyrights. Thus, there is no problem along those lines. Claude Wintner

We investigate the no-boundary measure in the context of moduli stabilization. To this end, we first show that for exponential potentials, there are no classical histories once the slope exceeds a critical value. We also investigate the probability distributions given by the no-boundary wave function near maxima of the potential. These results are then applied to a simple model that compactifies 6D to 4D (HBSV model) with fluxes. We find that the no-boundary wave function effectively stabilizes the moduli of the model. Moreover, we find the a priori probability for the cosmological constant in this model. We find that a negative value is preferred, and a vanishing cosmological constant is not distinguished by the probability measure. We also discuss the application to the cosmic landscape. Our preliminary arguments indicate that the probability of obtaining anti de Sitter space is vastly greater than for de Sitter.

We investigate the no-boundary measure in the context of moduli stabilization. To this end, we first show that for exponential potentials, there are no classical histories once the slope exceeds a critical value. We also investigate the probability distributions given by the no-boundary wave function near maxima of the potential. These results are then applied to a simple model that compactifies 6D to 4D (HBSV model) with fluxes. We find that the no-boundary wave function effectively stabilizes the moduli of the model. Moreover, we find the a priori probability for the cosmological constant in this model. We find that a negative value is preferred, and a vanishing cosmological constant is not distinguished by the probability measure. We also discuss the application to the cosmic landscape. Our preliminary arguments indicate that the probability of obtaining anti de Sitter space is vastly greater than for de Sitter.

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Digital Repository of Scientific Institutes (RCIN) was established and is being developed mainly as an outcome of two projects acquired and jointly accomplished by a group of Polish scientific institutes.

This monograph is focused on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. Although it is a survey, it also includes various new recent results derived by the authors. The first part of this article introduces some classical concentration inequalities for martingales, and it also derives some recent refinements of these inequalities. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, and some other aspects that are related to wireless communications and coding. The second part of this article introduces the entropy method for deriving concentration inequalities for functions of many independent random variables, and it also exhibits its multiple connections to information theory. The basic ingredients of the entropy method are discussed first in conjunction with the closely related topic of logarithmic Sobolev inequalities. This discussion is complemented by a related viewpoint based on probability in metric spaces. This viewpoint centers around the so-called transportation-cost inequalities, whose roots are in information theory. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, the tutorial addresses several applications of the entropy method and related information-theoretic tools to problems in communications and coding. These include strong converses for several source and channel coding problems, empirical distributions of good channel codes with non-vanishing error probability, and an information-theoretic converse for concentration of measure.

After shortly reviewing the fundamentals of approach theory as introduced by R. Lowen in 1989, we show that this theory is intimately related with the well-known Wasserstein metric on the space of probability measures with a finite first moment on a complete and separable metric space. More precisely, we introduce a canonical approach structure, called the contractive approach structure, and prove that it is metrized by the Wasserstein metric. The key ingredients of the proof of this result are Dini's Theorem, Ascoli's Theorem, and the fact that the class of real-valued contractions on a metric space has some nice stability properties. We then combine the obtained result with Prokhorov's Theorem to establish inequalities between the relative Hausdorff measure of non-compactness for the Wasserstein metric and a canonical measure of non-uniform integrability.

We apply a theorem of Wick to rewrite certain classes of exponential measures on random graphs as integrals of Feynman-Gibbs type, on the real line. The analytic properties of these measures can then be studied in terms of phase transitions; spaces of scale-free trees are a particularly interesting example.

We apply a theorem of Wick to rewrite certain classes of exponential measures on random graphs as integrals of Feynman-Gibbs type, on the real line. The analytic properties of these measures can then be studied in terms of phase transitions; spaces of scale-free trees are a particularly interesting example.

Speaker : Tatiana Toro

Howard Gardner's multiple intelligence (MI) theory is an alternative to the unitary concept of general intelligence, but it lacks a practical, reliable, and valid method of assessment. The Hillside Assessment of Perceived Intelligences (HAPI) is an attempt to measure the seven constructs of MI theory with an objective, psychometrically sound instrument. Previous investigations have shown that the HAPI provides a reliable profile, but it has not been clear how much validity this profile holds. This investigation into the concurrent validity of the HAPI with other instruments found the patterns of correlation coefficients to be generally supportive of the HAPI scales. Subjects were 338 undergraduate and graduate students and 45 adults from adult education or community volunteers. However, an examination of the construct validity of the scales using contrasted groups produced mixed results. Appendixes list the core components of MI theory and HAPI scale definitions. Four tables are included. (Contains 20 references.) (Author/SLD)

Howard Gardner's multiple intelligence (MI) theory is an alternative to the unitary concept of general intelligence, but it lacks a practical, reliable, and valid method of assessment. The Hillside Assessment of Perceived Intelligences (HAPI) is an attempt to measure the seven constructs of MI theory with an objective, psychometrically sound instrument. Previous investigations have shown that the HAPI provides a reliable profile, but it has not been clear how much validity this profile holds. This investigation into the concurrent validity of the HAPI with other instruments found the patterns of correlation coefficients to be generally supportive of the HAPI scales. Subjects were 338 undergraduate and graduate students and 45 adults from adult education or community volunteers. However, an examination of the construct validity of the scales using contrasted groups produced mixed results. Appendixes list the core components of MI theory and HAPI scale definitions. Four tables are included. (Contains 20 references.) (Author/SLD)

Howard Gardner's multiple intelligence (MI) theory is an alternative to the unitary concept of general intelligence, but it lacks a practical, reliable, and valid method of assessment. The Hillside Assessment of Perceived Intelligences (HAPI) is an attempt to measure the seven constructs of MI theory with an objective, psychometrically sound instrument. Previous investigations have shown that the HAPI provides a reliable profile, but it has not been clear how much validity this profile holds. This investigation into the concurrent validity of the HAPI with other instruments found the patterns of correlation coefficients to be generally supportive of the HAPI scales. Subjects were 338 undergraduate and graduate students and 45 adults from adult education or community volunteers. However, an examination of the construct validity of the scales using contrasted groups produced mixed results. Appendixes list the core components of MI theory and HAPI scale definitions. Four tables are included. (Contains 20 references.) (Author/SLD)

We consider non-linear elliptic equations having a measure in the right hand side, of the type $ \divo a(x,Du)=\mu, $ and prove differentiability and integrability results for solutions. New estimates in Marcinkiewicz spaces are also given, and the impact of the measure datum density properties on the regularity of solutions is analyzed in order to build a suitable Calder\'on-Zygmund theory for the problem. All the regularity results presented in this paper are provided together with explicit local a priori estimates.

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates $L^n$ of the transfer operator. This was previously an intractable problem. Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points. In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of $\sum_{j=1}^nL^j$) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for $L^n$ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.

One obtains Bell's inequalities if one posits a hypothetical joint probability distribution, or {\it measure}, whose marginals yield the probabilities produced by the spin measurements in question. The existence of a joint measure is in turn equivalent to a certain causality condition known as ``screening off''. We show that if one assumes, more generally, a joint {\it quantal measure}, or ``decoherence functional'', one obtains instead an analogous inequality weaker by a factor of $\sqrt{2}$. The proof of this ``Tsirel'son inequality'' is geometrical and rests on the possibility of associating a Hilbert space to any strongly positive quantal measure. These results lead both to a {\it question}: ``Does a joint measure follow from some quantal analog of `screening off'?'', and to the {\it observation} that non-contextual hidden variables are viable in histories-based quantum mechanics, even if they are excluded classically.

Measure Theory - Paul Halmos I have no monetary benefit from this work and it is intended for non-commercial uses only

We analyze the structure of the Fundamental Measure Theory for the free energy density functional of hard sphere mixtures. A comparative study of the different versions of the theory, and other density functional approaches, is done in terms of their generic form for the three-points direct correlation function, which shows clearly the main advantages and problems of the different approximations. A recently developed version for the monocomponent case is extended to mixtures of hard spheres with different radii, and a new prescription is presented to obtain the exact dimensional crossover of those mixtures in the one-dimensional (1D) limit. Numerical results for planar wall-fluid interfaces and for the 1D fluid are presented.

In a certain (non-commutative) version of large-N SU(N) Yang-Mills theory there are special Wilson loops, called twistor Wilson loops for geometrical reasons, whose v.e.v. is independent on the parameter that occurs in their operator definition. There is a semigroup that acts on the parameter by rescaling and on the functional measure, resolved into anti-selfdual orbits by a non-supersymmetric version of the Nicolai map, by contracting the support of the measure. As a consequence the twistor Wilson loops are localized on the fixed points of the semigroup of contractions. This localization is a non-supersymmetric analogue of the localization that occurs in the Nekrasov partition function of the n=2 SUSY YM theory on the fixed points of a certain torus action on the moduli space of (non-commutative) instantons. One main consequence of the localization in the large-N YM case, as in the n=2 SUSY YM case, is that the beta function of the Wilsonian coupling constant in the anti-selfdual variables is one-loop exact. Consequently the large-N Yang-Mills canonical beta function has a NSVZ form that reproduces the first two universal perturbative coefficients.

Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and decomposition in generalized surface space) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently for data analysis (e.g., the integral area invariant). The flat norm measures distance between currents (generalized surfaces) by decomposing them in a way that is robust to noise. The simplicial deformation theorem shows currents can be approximated on a simplicial complex, generalizing the classical cubical deformation theorem and proving sharper bounds than Sullivan's convex cellular deformation theorem. Computationally, the discretized flat norm can be expressed as a linear programming problem and solved in polynomial time. Furthermore, the solution is guaranteed to be integral for integral input if the complex satisfies a simple topological condition (absence of relative torsion). This discretized integrality result yields a similar statement for the continuous case: the flat norm decomposition of an integral 1-current in the plane can be taken to be integral, something previously unknown for 1-currents which are not boundaries of 2-currents. Nonasymptotic densities (integral area invariants) taken along the boundary of a shape are often enough to reconstruct the shape. This result is easy when the densities are known for arbitrarily small radii but that is not generally possible in practice. When only a single radius is used, variations on reconstruction results (modulo translation and rotation) of polygons and (a dense set of) smooth curves are presented.

Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and decomposition in generalized surface space) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently for data analysis (e.g., the integral area invariant). The flat norm measures distance between currents (generalized surfaces) by decomposing them in a way that is robust to noise. The simplicial deformation theorem shows currents can be approximated on a simplicial complex, generalizing the classical cubical deformation theorem and proving sharper bounds than Sullivan's convex cellular deformation theorem. Computationally, the discretized flat norm can be expressed as a linear programming problem and solved in polynomial time. Furthermore, the solution is guaranteed to be integral for integral input if the complex satisfies a simple topological condition (absence of relative torsion). This discretized integrality result yields a similar statement for the continuous case: the flat norm decomposition of an integral 1-current in the plane can be taken to be integral, something previously unknown for 1-currents which are not boundaries of 2-currents. Nonasymptotic densities (integral area invariants) taken along the boundary of a shape are often enough to reconstruct the shape. This result is easy when the densities are known for arbitrarily small radii but that is not generally possible in practice. When only a single radius is used, variations on reconstruction results (modulo translation and rotation) of polygons and (a dense set of) smooth curves are presented.