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An introduction to measure and integration. 2nd ed
... Además, cualquier función Lipschitz es absolutamente continua. Las funciones absolutamente continuas poseen derivada finita en casi todas partes (ver por ejemplo [21,31]). Más aún, la funciónẋ(·) es una función integrable según Lebesgue y x(·) satisface el siguiente Teorema Fundamental del Cálculo, conocido como fórmula de Newton-Leibniz [31]: ...
... Las funciones absolutamente continuas poseen derivada finita en casi todas partes (ver por ejemplo [21,31]). Más aún, la funciónẋ(·) es una función integrable según Lebesgue y x(·) satisface el siguiente Teorema Fundamental del Cálculo, conocido como fórmula de Newton-Leibniz [31]: ...
... La celebrada función de Cantor-Vitali, o escalera del Diablo, es un contraejemplo para laúltima afirmación (ver [38,41]). Las observaciones anteriores hacen plausible la siguiente caracterización cuya prueba puede consultarse en [31,21]. Observe que (2.7) es consecuencia directa de la misma caracterización. ...
... To verify property (3), note that L 2 -convergence of {g km } implies that there exists a subsequence {g k l } of {g km } that converges tog a.e. [18,Thm. 8.5.14,Thm. ...
... −→ Ω. But this is now immediate from[18, Thm. 8.5.1], which states that if 1 ≤ p < ∞, f i → f a.e. and f i p → f p , Using this lemma, we can globalize Corollary 4.12.Proposition 4.16. ...
We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finite-dimensional, orientable manifold with respect to a natural metric called the metric. The primary motivation for studying this problem comes from Teichmueller theory, where similar considerations lead to a completion of the well-known Weil-Petersson metric. We give an application of the main theorem to the completions of Teichmueller space with respect to a class of metrics that generalize the Weil-Petersson metric.
... Proof in [16] for the first part. In [14] for the second part using Proposition 2.9. As mentioned above, in the proofs of both parts the countable additivity of m (Proposition 2.12) is used. ...
As shape analysis of the form presented in Srivastava and Klassen's textbook 'Functional and Shape Data Analysis' is intricately related to Lebesgue integration and absolute continuity, it is advantageous to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results about Lebesgue integration and absolute continuity. In particular, we review fundamental results connecting them to each other and to the kind of shape analysis, or more generally, functional data analysis presented in the aforementioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute continuity, are presented without proofs. However, a good number of results about absolute continuity and most results about functional data and shape analysis are presented with proofs. Actually, most missing proofs can be found in Royden's 'Real Analysis' and Rudin's 'Principles of Mathematical Analysis' as it is on these classic textbooks and Srivastava and Klassen's textbook that a good portion of these notes are based. However, if the proof of a result does not appear in the aforementioned textbooks, nor in some other known publication, or if all by itself it could be of value to the reader, an effort has been made to present it accordingly.
... This guarantees the existence and uniqueness of the polynomial for given points [10]. ...
Ensuring data integrity is a critical requirement in complex systems, especially in financial platforms where vast amounts of data must be consistently accurate and reliable. This paper presents a robust approach using polynomial interpolation methods to maintain data integrity across multiple indicators and dimensions.
... This guarantees the existence and uniqueness of the polynomial for given points [10]. ...
Ensuring data integrity is a critical requirement in complex systems, especially in financial platforms where vast amounts of data must be consistently accurate and reliable. This paper presents a robust approach using polynomial interpolation methods to maintain data integrity across multiple indicators and dimensions.
... Supposing we are able to define the notion of length for a class M of subsets of ℝ such that M includes intervals, we can consider functions of the type ∑ χAi Such functions are called simple measurable functions. For such a function we define the new integral by (2). Since mAi could be +∞, to make the sum in (2) meaningful, we assume that Ci ≥ 0 ∀i. ...
... The authors [1] introduced the concept of Edge complement graph and authors [2] proved some results related to these concepts. In [3], [4] and [5] similar concepts are explained with respect to graphs. In [6], it is proved that collection of subsets of a set is an abelian group, with respect to the symmetric difference(∆). ...
The collection of edge complement spanning subgraphs of a simple graph is an abelian group with respect to the symmetric difference operation.
... where the second equality follows from lim n→∞ −L(Y, X, b n ) and δ(Y, X) being nonnegative functions on YX (e.g., Proposition 5.2.6(ii) in Rana, 2002), and δ(Y, X) and lim n→∞ −L(Y, X, b n ) having finite expectation on YX c , since lim n→∞ b n t(X, Y ) > 0 on YX c and E[| − L(Y, X, b)|] < ∞ for all b ∈ Θ by Lemma 2. ...
Conditional distribution functions are important statistical objects for the analysis of a wide class of problems in econometrics and statistics. We propose flexible Gaussian representations for conditional distribution functions and give a concave likelihood formulation for their global estimation. We obtain solutions that satisfy the monotonicity property of conditional distribution functions, including under general misspecification and in finite samples. A Lasso-type penalized version of the corresponding maximum likelihood estimator is given that expands the scope of our estimation analysis to models with sparsity. Inference and estimation results for conditional distribution, quantile and density functions implied by our representations are provided and illustrated with an empirical example and simulations.
... Theorem B.1 (Leibniz Integral Rule, Theorem 5.4.12 in[46]). Let U be an open subset of R d and Ω be a measure space. ...
The blind deconvolution problem aims to recover a rank-one matrix from a set of rank-one linear measurements. Recently, Charisopulos et al. introduced a nonconvex nonsmooth formulation that can be used, in combination with an initialization procedure, to provably solve this problem under standard statistical assumptions. In practice, however, initialization is unnecessary. As we demonstrate numerically, a randomly initialized subgradient method consistently solves the problem. In pursuit of a better understanding of this phenomenon, we study the random landscape of this formulation. We characterize in closed form the landscape of the population objective and describe the approximate location of the stationary points of the sample objective. In particular, we show that the set of spurious critical points lies close to a codimension two subspace. In doing this, we develop tools for studying the landscape of a broader family of singular value functions, these results may be of independent interest.
... Moreover, in this work, almost every (where) and absolutely continuous are abbreviated as a.e. and a.c., respectively [28]. The following lemma is a version of the chain rule which follows from, e.g., [29,Th. 6.3.15]. ...
In this paper, we propose a novel method to construct hyper-rectangular over-approximations of reachable sets for a general class of linear uncertain systems whose nominal parameters and bounds on uncertainties are time-varying. The motivation of this paper is the need for relatively accurate yet computationally inexpensive over-approximation method that is practical in various control methods
especially abstraction-based control synthesis. We demonstrate the proposed method in an illustrative example and two control examples related to abstraction-based control synthesis. In these examples, the
proposed method shows excellent results in terms of representing reachable sets with reasonable accuracy and allowing the implementation of time-varying control input signals.
... Also the concept of hyperbolic valued signed measure is introduced and the hyperbolic version of Hahn and Jordan decomposition theorems are proved in this article. The proofs in this article are based on the book of I. K. Rana [8]. ...
... Now, using (2.8)-(2.11) and Generalized Lebesgue Theorem (see [22], Exercise 5.4.13) ...
... The first line is the definition of the Radon-Nikodym derivative on R d (see [38] ≤ 4 · W · Z · ||μ s−1 − µ s−1 || 2,s + 1 ...
We present bounds for the finite sample error of sequential Monte Carlo samplers on static spaces. Our approach explicitly relates the performance of the algorithm to properties of the chosen sequence of distributions and mixing properties of the associated Markov kernels. This allows us to give the first finite sample comparison to other Monte Carlo schemes. We obtain bounds for the complexity of sequential Monte Carlo approximations for a variety of target distributions including finite spaces, product measures, and log-concave distributions including Bayesian logistic regression. The bounds obtained are within a logarithmic factor of similar bounds obtainable for Markov chain Monte Carlo.
... The authors [6] introduced a new operation '⊻' as in definition 2.5 and proved some results on Cartesian product of vertex complement and vertex measurable graphs. In [7], the concept of Cartesian product of two sets was introduced in the field of measure theory and it has been proved that the Cartesian product × is a semi-algebra of the subsets of X × Y, where and are semi-algebras of X and Y respectively. In this paper, the authors have developed a graph analog of this concept. ...
... The concept of Cartesian product of two measurable spaces was introduced in the field of measure theory. In [2] it has been proved that 1 × 2 is the smallest σalgebra of subsets of such that the maps and defined by ( ) and ( ) respectively are measurable. In this paper we develop the graph analog of these concepts. ...
Abstract—Let G1 and G2 be two simple graphs. Let (G1, 1) and (G2
, 2) be two vertex measure spaces. In this paper we introduce a σ algebra 1 × 2, which consists of all vertex induced sub graphs of G1 × G2, and it contains every vertex measurable rectangle graph of the form H1 × H2 , H1 ∈ 1 and H2 ∈ 2. Here, we prove 1 × 2 is the smallest
σ algebra of G1 × G2 such that the maps LG1: G1 × G2 → G1 and
LG2: G1 × G2 → G2 defined by LG1(H × K) = H and LG2(H × K) = K for all vertex measurable graphs H in G1 and K in G2 respectively are measurable.
... Therefore, ( ) ( ) ( ) ( ) 1 1 , ...
n this brief communication we present a new integral transform, so far un- known, which is applicable, for instance, to studying the kinetic theory of natural eigenmodes or transport excited in plasmas with bounded distribution functions such as in Q machines/plasma diodes or in the scrap-off layer of Tokamak fusion plasmas. The results are valid for functions of Lp {,σS ,μ}
function spaces—Lebesgue spaces, which are defined using a natural genera- lization of the p-norm for finite-dimensional vector spaces, where is the real set, σS is the σ -algebra of Lebesgue measurable sets, and μ the Le-
besguemeasure. AK:Lp[0,L]→Lp[0,L],sothat f →AK(f).Notethat, using a simpler notation, more natural/known to engineers, f could be consi- dered any piecewise continuous function, that is: f ∈ PC [0, L]. Here
PC [0, L] is a Euclidian space with the usual norm (inner product: f , f ) 2 ∫L 2
givenby: f = f,f 0 f=(x)dx [1]. Keywords
Integral Transform, Lebesgue Measures, Kinetic Theory of Bounded Plasmas, Natural Eigenmodes, Transport, Q-Machines, Plasma Diodes, Tokamak, Nuclear Fusion
... As promised in the above proof, let us state the following measure theoretic lemma, whose proof uses the classical Vitali convergence theorem [29,Theorem 8.5.14], and is exactly the same as the argument of [14,Lemma 5.3]: ...
Suppose (X,\o) is a compact K\"ahler manifold. We introduce and explore the metric geometry of the -Calabi Finsler structure on the space of K\"ahler metrics . After noticing that the -Calabi and -Mabuchi path length topologies on do not typically dominate each other, we focus on the finite entropy space , contained in the intersection of the -Calabi and -Mabuchi completions of and find that after a natural strengthening, the -Calabi and -Mabuchi topologies coincide on . As applications to our results, we give new convergence results for the K\"ahler--Ricci flow and the weak Calabi flow.
... Now by Lemma 3.2, both F + and F − are measures on F. Using Proposition 3.9.9 [3], the measures F + and F − can be extended to the σ-algebra Ω. Denote these extensions by F + * : Ω → [0, ∞) and F − * : Ω → [0, ∞), respectively. We claim that both F + * and F − * are absolutely continuous with respect to µ, as measures on the σ-algebra Ω. ...
We analyze the relationship between the absolute continuity of charges and the Henstock-Kurzweil integral on metric measure spaces. We also discuss a measure theoretic characterization of the Henstock-Kurzweil integral in terms of the Henstock variational measure, on such spaces.
... Then W is a measurable function of the two variables (s, t), increasing and continuous with respect to the second variable, and W(s, 1) = 1. The change of variable formula for the Lebesgue integral [Ran,Corollary 6.3.17] yields that ...
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the "power triangle function". The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
In this chapter, we shall give applications of set theory to other branches of mathematics. We shall assume that the reader is familiar with the subjects and give only preliminary definitions needed to understand the statements and proofs of the applications that we shall be presenting. Let I be an infinite set and for each let be a non-negative real number. We define Note that this extends the usual definition of the sum when each and J countable to uncountable sum of non-negative real numbers.
These lessons were written primarily for a third-year college math student. It will also
be useful for students of engineering and physical sciences as the Laplace transform and Fourier transform are a very useful tool for them. These lessons require nothing more than a basic knowledge of calculus.
A prey-predator model with square root functional response and a logistic growth delay for the prey population are taken into account
If V 1 , … , V n {V_{1},\dots,V_{n}} are translations of a Vitali set by rational numbers, then we prove that ⋃ i = 1 n V i {\bigcup_{i=1}^{n}V_{i}} contains no measurable subset of positive measure. This provides a decomposition of ℝ {{\mathbb{R}}} as a countable union of disjoint sets, any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function δ : ℝ → ( 0 , + ∞ ) {\delta:{\mathbb{R}}\rightarrow(0,+\infty)} for which there is no measurable function f : ℝ → ℝ {f:{\mathbb{R}}\rightarrow{\mathbb{R}}} satisfying 0 < f ≤ δ {0<f\leq\delta} , on any measurable set of positive Lebesgue measure.
The paper provides sufficient conditions for monotonicity, subhomogeneity and concavity of vector-valued Hammerstein integral operators over compact domains, as well as for the persistence of these properties under numerical discretizations of degenerate kernel type. This has immediate consequences on the dynamics of Hammerstein integrodifference equations and allows to deduce a local-global stability principle.
I have included freely the exercises and their solutions of Chapter 5 of my book entitled "An operator theory problem book".
Bibliography; Author Index; Subject Index
A traditional random variable X is a function that maps from a stochastic process to the real line. Here, "real line" refers to the structure (R,<=,|x-y|), where R is the set of real numbers, <= is the standard linear order relation on R, and d(x,y)=|x-y| is the usual metric on R. The traditional expectation value E(X) of X is then often a poor choice of a statistic when the stochastic process that X maps from is a structure other than the real line or some substructure of the real line. If the stochastic process is a structure that is not linearly ordered (including structures totally unordered) and/or has a metric space geometry very different from that induced by the usual metric, then statistics such as E(X) are often of poor quality with regards to qualitative intuition and quantitative variance (expected error) measurements. For example, the traditional expected value of a fair die is E(X)=(1/6)(1+2+...+6)=3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of [ooo] (die face value "3") or [oooo] (dice face value "4") more than we expect the outcome of say [o] or [oo]. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. On a fair die, [oo] is not greater or less than [o]; rather [oo] and [o] are simply symbols without order. Moreover, [o] is not "closer" to [oo] than it is to [ooo]; rather, [o], [oo], and [ooo] are simply symbols without any inherit order or metric geometry. This paper proposes an alternative statistical system, based somewhat on graph theory, that takes into account the order structure and metric geometry of the underlying stochastic process.
A traditional random variable X is a function that maps from a stochastic process to the real line. Here, "real line" refers to the structure (R,<=,|x-y|), where R is the set of real numbers, <= is the standard linear order relation on R, and d(x,y)=|x-y| is the usual metric on R. The traditional expectation value E(X) of X is then often a poor choice of a statistic when the stochastic process that X maps from is a structure other than the real line or some substructure of the real line. If the stochastic process is a structure that is not linearly ordered (including structures totally unordered) and/or has a metric space geometry very different from that induced by the usual metric, then statistics such as E(X) are often of poor quality with regards to qualitative intuition and quantitative variance (expected error) measurements. For example, the traditional expected value of a fair die is E(X)=(1/6)(1+2+...+6)=3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of [ooo] (die face value "3") or [oooo] (dice face value "4") more than we expect the outcome of say [o] or [oo]. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. On a fair die, [oo] is not greater or less than [o]; rather [oo] and [o] are simply symbols without order. Moreover, [o] is not "closer" to [oo] than it is to [ooo]; rather, [o], [oo], and [ooo] are simply symbols without any inherit order or metric geometry. This paper proposes an alternative statistical system, based somewhat on graph theory, that takes into account the order structure and metric geometry of the underlying stochastic process.
This paper investigates and illustrates the bifurcation behavior of various time-periodic integrodifference equations, a class of popular models in spatial ecology. Our approach requires a combination of analytical methods developed in a previous paper, and numerical tools based on Nyström discretization of the integral operators involved.
This book is for third and fourth year university mathematics students (and Master students) as well as lecturers and tutors in mathematics and anyone who needs the basic facts on Operator Theory (e.g. Quantum Mechanists). The main setting for bounded linear operators here is a Hilbert space. There is, however, a generous part on General Functional Analysis (not too advanced though). There is also a chapter on Unbounded Closed Operators.
The book is divided into two parts. The first part contains essential background on all of the covered topics with the sections: True or False Questions, Exercises, Tests and More Exercises. In the second part, readers may find answers and detailed solutions to the True or False Questions, Exercises and Tests.
Another virtue of the book is the variety of the topics and the exercises and the way they are tackled. In many cases, the approaches are different from what is known in the literature. Also, some very recent results from research papers are included.
As shape analysis of the form presented in Srivastava and Klassen’s
textbook “Functional and Shape Data Analysis” is intricately related
to Lebesgue integration and absolute continuity, it is advantageous
to have a good grasp of the latter two notions. Accordingly, in these
notes we review basic concepts and results about Lebesgue integration
and absolute continuity. In particular, we review fundamental results
connecting them to each other and to the kind of shape analysis,
or more generally, functional data analysis presented in the aforementioned textbook, in the process shedding light on important aspects of all three notions. Many well-known results, especially most results about Lebesgue integration and some results about absolute continuity, are presented without proofs. However, a good number of results about absolute continuity and most results about functional data and shape analysis are presented with proofs. Actually, most missing proofs can be found in Royden’s “Real Analysis” and Rudin’s “Principles of Mathematical Analysis” as it is on these classic textbooks
and Srivastava and Klassen’s textbook that a good portion of these
notes are based. However, if the proof of a result does not appear in
the aforementioned textbooks, nor in some other known publication,
or if all by itself it could be of value to the reader, an effort has been
made to present it accordingly.
This thesis broadly concerns colloidal particle simulation which plays an important role in understanding two-phase flows. More specifically, we track the particles inside a turbulent flow and model their dynamics as a stochastic process, their interactions as perfectly elastic collisions where the influence of the flow is modelled by a drift on the velocity term. By coupling each particle and considering their relative position and velocity, the perfectly elastic collision becomes a specular reflection condition. We put forward a time discretisation scheme for the resulting Lagrange system with specular boundary conditions and prove that the convergence rate of the weak error decreases at most linearly in the time discretisation step. The evidence is based on regularity results of the Feynman-Kac PDE and requires some regularity on the drift. We numerically experiment a series of conjectures, amongst which the weak error linearly decreasing for drifts that do not comply with the theorem conditions. We test the weak error convergence rate for a Richardson Romberg extrapolation. We finally deal with Lagrangian/Brownian approximations by considering a Lagrangian system where the velocity component behaves as a fast process. We control the weak error between the position of the Lagrangian system and an appropriately chosen uniformly elliptic diffusion process and subsequently prove a similar control by introducing a specular reflecting boundary on the Lagrangian and an appropriate reflection on the elliptic diffusion.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the " power triangle inequality ". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
Let Ω ⊂ ℝn, n ≥ 2, be a bounded domain containing the origin. Applying the Mountain Pass Theorem and a singular version of the generalized Moser-Trudinger inequality we prove the existence of a non-trivial weak solution to the problem u ∈ W01LΦ(Ω) and -div ( Φ(|▽u|) ▽u |▽u| ) = f(x,u)/|x|a in Ω, where a ∈ [0,n) , Φ is a Young function such that the space W01 LΦ(Ω) is embedded into exponential or multiple exponential Orlicz space and f(x,t) has the corresponding critical growth.
An account is given of the Riemann integral for real-valued functions defined on intervals of the real line, a rapid development of the topic made possible by use of the Darboux approach in place of that originally adopted by Riemann. The sense in which integration is the inverse of differentiation is investigated. To cope with the demands of the later chapters the improper Riemann integral is introduced. Uniform convergence of sequences and series is defined and its usefulness in interchanging integration and limits established; to help with circumstances in which uniform convergence is not present, Arzelà’s theorem is proved.
Fractals, Point Processes, Fractal-Based Point Processes, Problems
This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change-of-variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young’s inequality and Hardy-Littlewood-Sobolev inequality, are proven. Further topics include the Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems including Marcinkiewicz’s theorem, and the definition of Lebesgue points and the Lebesgue differentiation theorem. Each chapter ends with a large number of exercises and detailed solutions. A comprehensive appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. It also provides more advanced material such as some basic properties of cardinals and ordinals which are useful for the study of measurability.
Let Ω ⊂ ℜn , n 2, be a bounded connected domain of the class C 1, for some ∈ (0, 1]. Applying the generalized Moser-Trudinger inequality without boundary condition, the Mountain Pass Theorem and the Ekeland Variational Principle, we prove the existence and multiplicity of nontrivial weak solutions to the problem where Φ is a Young function such that the space W 1L Φ(Ω) is embedded into exponential or multiple exponential Orlicz space, the nonlinearity f(x, t) has the corresponding critical growth, V (x) is a continuous potential, h ∈ (L Φ(Ω))* is a nontrivial continuous function, µ 0 is a small parameter and n denotes the outward unit normal to ∂Ω.
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