Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a s...

https://arxiv.org/pdf/cond-mat/0505185.pdf

Characterization of complex networks: A survey of measurements

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements. Important related issues covered in this work comprise the representation of the evolution of complex networks in terms of trajectories in several measurement spaces, the analysis of the correlations between some of the most traditional measurements, perturbation analysis, as well as the use of multivariate statistics for feature selection and network classification. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the proper application and interpretation of measurements.

Thermophysical Properties Research Literature Retrieval Guide Edited by Y. S. Touloukian, J. K. Gerritsen and N. Y. Moore Second edition, revised and expanded. Book 1: Pp. xxi + 819. Book 2: Pp.621. Book 3: Pp. ix + 1315. (New York: Plenum Press, 1967.) n.p.

/pdf/heat-and-mass-transfer-1yaijfx8vp.pdf

Heat and Mass Transfer

Preface 1.General Considerations 2.Basic Processes in Atomization 3.Drop Size Distributions of Sprays 4.Atomizers 5.Flow in Atomizers 6.Atomizer Performance 7.External Spray Charcteristics 8.Drop Evaporation 9.Drop Sizing Methods Index

Atomization and Sprays

This paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry. The principal message is that such a theory is both relevant and viable. The first main issue is the problem of definition, which we next describe. Quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount. This requirement makes sense in every metric space. Given a homeomorphism f from a metric space X to a metric space Y , then for x∈X and r>0 set

/pdf/quasiconformal-maps-in-metric-spaces-with-controlled-24nmi0p2hm.pdf

Quasiconformal maps in metric spaces with controlled geometry

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A p -condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion). The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic Holder singularities demonstrate the sharpness of our machinery comparatively with known results.

/pdf/weighted-sobolev-spaces-and-embedding-theorems-4ud9ud011b.pdf

Weighted Sobolev spaces and embedding theorems

1. Preliminary Information about Integration Theory.- 1. Notation and Terminology.- 1.1. Sets in Rn.- 1.2. Classes of Functions in Rn.- 2. Some Auxiliary Information about Sets and Functions in Rn.- 2.1. Averaging of Functions.- 2.2. The Whitney Partition Theorem.- 2.3. Partition of Unitiy.- 3. General Information about Measures and Integrals.- 3.1. Notion of a Measure.- 3.2. Decompositions in the Sense of Hahn and Jordan.- 3.3. The Radon-Nikodym Theorem and the Lebesgue Decomposition of Measure.- 4. Differentiation Theorems for Measures in Rn.- 4.1. Definitions.- 4.2. The Vitali Covering Lemma.- 4.3. The Lp-Continuity Theorem for Functions of the Class Lp,loc.- 4.4. The Differentiability Theorem for the Measure in Rn.- 5. Generalized Functions.- 5.1. Definition and Examples of Generalized Functions.- 5.2. Operations with Generalized Functions.- 5.3. Support of a Generalized Function. The Order of Singularity of a Generalized Function.- 5.4. The Generalized Function as a Derivative of the Usual Function. Averaging Operation.- 2. Functions with Generalized Derivatives.- 1. Sobolev-Type Integral Representations.- 1.1. Preliminary Remarks.- 1.2. Integral Representations in a Curvilinear Cone.- 1.3. Domains of the Class J.- 1.4. Integral Representations of Smooth Functions in Domains of the Class J.- 2. Other Integral Representations.- 2.1. Sobolev-Type Integral Representations for Simple Domains.- 2.2. Differential Operators with the Complete Integrability Condition.- 2.3. Integral Representations of a Function in Terms of a System of Differential Operators with the Complete Integrability Condition.- 2.4. Integral Representations for the Deformation Tensor and for the Tensor of Conformal Deformation.- 3. Estimates for Potential-Type Integrals.- 3.1. Preliminary Information.- 3.2. Lemma on the Compactness of Integral Operators.- 3.3. Basic Inequalities.- 4. Classes of Functions with Generalized Derivatives.- 4.1. Definition and the Simplest Properties.- 4.2. Integral Representations for Elements of the Space W?1,locl.- 4.3. The Imbedding Theorem.- 4.4. Corollaries of Theorem 4.2. Normalization of the Spaces Wpl(U).- 4.5. Approximation of Functions from Wpl by Smooth Functions.- 4.6. Change of Variables for Functions with Generalized Derivatives.- 4.7. Compactness of the Imbedding Operators.- 4.8. Estimates with a Small Coefficient for the Norm in Lpl.- 4.9. Functions of One Variable.- 4.10. Differential Description of Convex Functions.- 4.11. Functions Satisfying the Lipschitz Condition.- 5. Theorem on the Differentiability Almost Everywhere.- 5.1. Definitions.- 5.2. Auxiliary Propositions.- 5.3 The Main Result.- 5.4. Corollaries of the General Theorem on the Differentiability Almost Everywhere.- 5.5. The Behaviour of Functions of the Class Wpl on Almost All Planes of Smaller Dimensionality.- 5.6. The ACL-Classes.- 3. Nonlinear Capacity.- 1. Capacity Induced by a Linear Positive Operator.- 1.1. Definition and the Simplest Properties.- 1.2. Capacity as the Outer Measure.- 1.3. Sets of Zero Capacity.- 1.4. Extension of the Set of Admissible Functions.- 1.5. Extremal Function for Capacity.- 1.6. Comparison of Various Capacities.- 2. The Classes W(T, p, V).- 2.1. Definition of Classes.- 2.2. Theorems of Egorov and Luzin for Capacity.- 2.3. Dual (T, p)-Capacity, p > 1. Definition and Basic Properties.- 2.4. Calculation of Dual (T, p)-Capacity.- 3. Sets Measurable with Respect to Capacity.- 3.1. Definition and the Simplest Properties of Generalized Capacity.- 3.2. (T, p)-Capacity as Generalized Capacity.- 4. Variational Capacity.- 4.1. Definition of Variational Capacity.- 4.2. Comparison of Variational Capacity and (T, p)-Capacity.- 4.3. Sets of Zero Variational Capacity.- 4.4. Examples of Variational Capacity.- 4.5. Refined Functions.- 4.6. Theorems of Imbedding into the Space of Continuous Functions.- 5. Capacity in Sobolev Spaces.- 5.1. Three Types of Capacity.- 5.2. Extremal Functions for Capacity.- 5.3. Capacity and the Hausdorff h-Measure.- 5.4. Sufficient Conditions for the Vanishing of (l, p)-Capacity.- 6. Estimates of [l, p]-Capacity for Some Pairs of Sets.- 6.1. Estimates of Capacity for Spherical Domains.- 6.2. Estimates of Capacity for Pairs of Continuums Connecting Concentric Spheres.- 7. Capacity in Besov-Nickolsky Spaces.- 7.1. Preliminary Information.- 7.2. Capacities in bl,p,?,G,hl. Simplest Properties.- 7.3. Comparison of Capacity of a Pair of Points to Capacity of a Point Relative to a Complement of a Ball.- 7.4. Capacity of the Spherical Layer.- 4. Density of Extremal Functions in Sobolev Spaces with First Generalized Derivatives.- 1. Extremal Functions for (l, p)-Capacity.- 1.1. Simplest Properties of Extremal Functions.- 1.2. The Dirichlet Problem and Extremal Functions.- 1.3. Extremal Functions for Pairs of Smooth Compacts.- 2. Theorem on the Approximation of Functions from Lpl by Extremal Functions.- 2.1. Auxiliary Statements.- 2.2. The Class Extp(G).- 2.3. Proof of the Theorem on Approximation.- 2.4. Representation in Form of a Series.- 3. Removable Singularities for the Spaces Lpl (G).- 3.1. Two Ways of Describing Removable Singularities.- 3.2. Properties of NCp-Sets. Localization Principle.- 5. Change of Variables.- 1. Multiplicity of Mapping, Degree of Mapping, and Their Analogies.- 1.1. The Multiplicity Function of Mapping.- 1.2. The Approximate Differential.- 1.3. The K-Differential.- 1.4. The Change of Variable Theorem for the Multiplicity Function.- 1.5. The Degree of Mapping.- 2. The Change of Variable in the Integral for Mappings of Sobolev Spaces.- 2.1. The Change of Variable Theorem for Continuous Mappings of the Class Lnl.- 2.2. The Linking Index.- 2.3. The Change of Variable Theorem for Discontinuous Mappings of the Class Lnl.- 3. Sufficient Conditions of Monotonicity and Continuity for the Approximation Functions of the Class Lnl.- 4. Invariance of the Spaces Lpl(G)(Lnl(G)) for Quasiisometric (Quasiconformal) Homeomorphisms.- 4.1. Preliminary Information on the Mappings.- 4.2. Differentiation of Composition.- 4.3. Representation of Operators Preserving the Order.- 6. Extension of Differentiate Functions.- 1. Arc Diameter Condition.- 1.1. Analysis of the Ahlfors Condition.- 1.2. The Arc Diameter Condition.- 1.3. Properties of Domains Satisfying the Arc Diameter Condition.- 2. Necessary Extension Conditions for Seminormed Spaces.- 2.1. The Extension Operator. Capacitary Extension Condition.- 2.2. Additional Properties of Capacity.- 2.3. The Invisibility Condition.- 2.4. The Extension Theorem.- 2.5. Verification of the Conditions of the Theorem for the Spaces Lpl (G), Wpl(G).- 3. Necessary Extension Conditions for Sobolev Spaces.- 3.1. Necessary Extension Conditions for Lpl, Wpl at lp=n.- 3.2. Necessary Conditions for Lpl, Wpl at 1 ? lp ? 2 in Plane Domains.- 3.3 Necessary Conditions Different from the Arc Diameter Condition.- 3.4. Refinement for the Space Wpl.- 4. Necessary Extension Conditions for Besov and Nickolsky Spaces.- 4.1. Extension Theorem for lp > n.- 4.2. Extension Conditions for lp = n.- 5. Sufficient Extension Conditions.- 5.1. Quasiconformal Extension.- 5.2. Extension Conditions for Sobolev Classes.- 5.3. Example of Estimating the Norm of an Extension Operator.- 5.4. The Extension Condition for Nickolsky-Besov Spaces.- Comments.- References.

Quasiconformal Mappings and Sobolev Spaces

Quasiconformal mappings and spaces of functions with generalized first derivatives

In the present paper we study embedding operators for weighted Sobolev spaces whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient conditions for boundedness and compactness of the embedding operators are obtained for smooth domains and domains with boundary singularities. The proposed method is based on the concept of 'generalized' quasiconformal homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the homeomorphism type depends on the choice of the corresponding weighted Sobolev space. Such classes of homeomorphisms induce bounded composition operators for weighted Sobolev spaces. With the help of these homeomorphism classes the embedding problem for non-smooth domains is reduced to the corresponding classical embedding problem for smooth domains. Examples of domains with anisotropic H\"older singularities demonstrate sharpness of our machinery comparatively with known results.

https://www.cs.bgu.ac.il/~goldfarb/Papers_html/SGGKFM_CF.pdf

Vladimir Gol'dshtein

Papers

Weighted Sobolev spaces and embedding theorems

Quasiconformal Mappings and Sobolev Spaces

Quasiconformal mappings and spaces of functions with generalized first derivatives

Weighted Sobolev spaces and embedding theorems

Thermal Ignition Analysis of a Monodisperse Spray with Radiation