A Primer of Real Analytic Functions

Krantz, Steven G.; Parks, Harold R.

doi:10.1007/978-0-8176-8134-0

Cited by 454 publications

(101 citation statements)

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“…The right side is an integral with respect to x of an analytic function of (s, β, x), and hence G is analytic on (0, t) × R K by Proposition 2.2.3 of Krantz and Parks [62]. Since this is true for every t < T , G is analytic on (0, T ) × R K .…”

Section: Theorem A2 (The Analytic Implicit Function Theoremmentioning

confidence: 84%

“…Proof: Lojasiewicz's Structure Theorem for Varieties (Theorem 6.3.3 of Krantz and Parks [62]) states that if U is an open set in R n and F : U → R is analytic, then for every x 0 ∈ U, there exists a neighborhood V x 0 of x 0 such that either F (x) = 0 for all x ∈ V x 0 or {x ∈ V x 0 : F (x) = 0} is a finite union of real algebraic varieties of dimension < n. If F (x) = 0 for all x ∈ V x 0 and y ∈ U, there is a ray that passes through V x 0 and through y; the restriction of F to the ray is an analytic function of a single variable, and it vanishes on an interval (the intersection of the ray with the set V 0 ); since it is well known that an analytic function of one variable that vanishes on an interval is identically zero, we must have F (y) = 0 for all y ∈ U and we are done. On the other hand, if {x ∈ V : F (x) = 0} is a finite union of algebraic varieties of dimension < n, {x ∈ V x 0 : F (x) = 0} has Lebesgue measure zero.…”

Section: A Real Analytic Functions Of Several Variablesmentioning

confidence: 99%

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Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

2008

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We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and of a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information. We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy.

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Section: Theorem A2 (The Analytic Implicit Function Theoremmentioning

confidence: 84%

Section: A Real Analytic Functions Of Several Variablesmentioning

confidence: 99%

Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

2008

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“…By Lojasiewicz's structure theorem for real analytic varieties (see e.g. [24,Theorem 6.3.3,p. 168]), if Q is a small enough neighborhood of a point x 0 ∈ ∂Ω + ∞ , we have that…”

Section: Next We Prove the Statements (A)-(g)mentioning

confidence: 99%

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Azzam

Mourgoglou

Tolsa

2017

Comm Pure Appl Math

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“…In particular, g(0) = 0. Analyticity of g(·) on the positive semi-axis follows directly by the Inverse Function Theorem (for analytic functions: see for example [22,Sec. 2.5]).…”

Section: 4mentioning

confidence: 99%

Generalized Poland–Scheraga denaturation model and two-dimensional renewal processes

Giacomin

Khatib

2017

Stochastic Processes and their Applications

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Abstract. The Poland-Scheraga model describes the denaturation transition of two complementary -in particular, equally long -strands of DNA, and it has enjoyed a remarkable success both for quantitative modeling purposes and at a more theoretical level. The solvable character of the homogeneous version of the model is one of features to which its success is due. In the bio-physical literature a generalization of the model, allowing different length and non complementarity of the strands, has been considered and the solvable character extends to this substantial generalization. We present a mathematical analysis of the homogeneous generalized Poland-Scheraga model. Our approach is based on the fact that such a model is a homogeneous pinning model based on a bivariate renewal process, much like the basic Poland-Scheraga model is a pinning model based on a univariate, i.e. standard, renewal. We present a complete analysis of the free energy singularities, which include the localization-delocalization critical point and (in general) other critical points that have been only partially captured in the physical literature. We obtain also precise estimates on the path properties of the model. 2010 Mathematics Subject Classification: 60K35, 82D60, 92C05, 60K05, 60F10

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A Primer of Real Analytic Functions

Cited by 454 publications

References 0 publications

Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Generalized Poland–Scheraga denaturation model and two-dimensional renewal processes

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