Journal of Mathematical Analysis and Applications, Mar 1, 2016
Abstract Using interpolation properties of cones of general monotone functions, we prove the equi... more Abstract Using interpolation properties of cones of general monotone functions, we prove the equivalence of the L ( p , q ) norms of such functions and their Fourier transforms.
Given an interpolation couple (A0, A~), the approximation functional is dcfined E(t, a; Ao, A1) =... more Given an interpolation couple (A0, A~), the approximation functional is dcfined E(t, a; Ao, A1) = inf {[a-ao[a]Iaolao <= t}.
WIT Transactions on Ecology and the Environment, Jun 9, 2004
Fractional calculus provides novel mathematical tools for modeling physical and biological proces... more Fractional calculus provides novel mathematical tools for modeling physical and biological processes. The bioheat equation is often used as a first order model of heat transfer in biological systems. In this paper we describe formulation of bioheat transfer in one dimension in terms of fractional order differentiation with respect to time. The solution to the resulting fractional order partial differential equation reflects the interaction of the system with the dynamics of its response to surface or volume heating. An example taken from a study involving pulsating (on-off) cooling of a peripheral tissue region during laser surgery is used to illustrate the utility of the method. In the future we hope to interpret the physical basis of fractional derivatives using Constructal Theory, according to which, the geometry biological structures evolve as a result of the optimization process.
Publisher Summary This chapter discusses the Hilbert transform of the Gaussian. It highlights a c... more Publisher Summary This chapter discusses the Hilbert transform of the Gaussian. It highlights a connection between the Hilbert transform and the one-dimensional heat equation. The Hilbert transform is a linear operator that takes a function, u ( t ), and produces a function, H ( u )( t ), with the same domain. The Hilbert transform is a basic tool in Fourier analysis and provides a concrete means for realizing the harmonic conjugate of a given function or Fourier series. By taking A = 0 and B = 1/π gives Q ( u ), the conjugate Poisson kernel for the upper half plane.
Proceedings of the American Mathematical Society, 1991
Let D be a simply connected domain in the complex plane whose boundary T is a rectifiable simple ... more Let D be a simply connected domain in the complex plane whose boundary T is a rectifiable simple closed curve. Let {A(y)/y e T) and {B{y)l7 6 r} be interpolation families of Banach spaces. Let T be a linear operator mapping A(y) continuously into B(y). For z & D let Tz be the restriction of T to the interpolation space Az. Then {z € D/cod(Tz) = d < oo and dimKer(Tz) = 0} and {z € D/dimKer(Tz) = d < oo and Tz is onto B } are open sets.
We complete a characterization of homogeneous solutions of the beat equation begun by D. V. Widde... more We complete a characterization of homogeneous solutions of the beat equation begun by D. V. Widder. We determine regions of convergence for expansions of temperature functions in terms of the homogeneous solutions.
Journal of the Australian Mathematical Society, Feb 1, 1996
We prove that if u(x, t) is a solution of the one dimensional heat equation and if siu(x, i) is i... more We prove that if u(x, t) is a solution of the one dimensional heat equation and if siu(x, i) is its Appell transform, then u(x, t) has the semi-group (Huygens) property in a domain D if and only if s4u(x, t) has the semi-group property in a dual region. We apply this result to simplify and extend some results of Rosenbloom and Widder.
We develop Cauchy-Riemann equations for pairs of temperature functions with boundary values in L'... more We develop Cauchy-Riemann equations for pairs of temperature functions with boundary values in L'(R, dx/(1 + x2)).
This study reports the effects of an integrated instructional program (the Keystone Method) on th... more This study reports the effects of an integrated instructional program (the Keystone Method) on the students' performance in mathematics and reading, and tracks students' persistence and retention. The subject of the study was a large group of students in remedial mathematics classes at the college, willing to learn but lacking basic educational skills. The results show not only improvements in student outcomes in mathematics, but also gains in reading comprehension scores, as compared with the control group. These results were achieved at no cost to classroom retention. The persistence rates of the Keystone students were also higher for the subsequent terms.
College (Illinois) conducted an experiment in remedial mathematics classes at a community college... more College (Illinois) conducted an experiment in remedial mathematics classes at a community college. The researchers wanted to investigate whether, while teaching mathematics in a rigorous manner, they could also use their instruction to enhance the students' academic and workplace skills. Remedial mathematics constitutes a large part of the curriculum in postsecondary institutions across the U.S. In a national study of college-level remediation in 1990, it was found that 21 percent of all entering college freshmen were enrolled in a remedial mathematics course. The project used mathematics classes to enhance the student' work and study habits and improve their concentration skills. This, in turn, produced good results in mathematics, and greatly improved the students' reading comprehension scores. Their general experience as educators had convinced them that unsuccessful students are held back by behavior patterns which inhibit learning. In their program, they addressed the behavior patterns of students in various ways. For example, to increase the students' attention spans, they administered time-pressured quizzes, which required work done with full concentration. (Contains 46 references.) (JA)
Marcel Riesz in Lund.- Interpolation of tent spaces and applications.- Toeplitz liftings of hanke... more Marcel Riesz in Lund.- Interpolation of tent spaces and applications.- Toeplitz liftings of hankel forms.- Some recent developments in fourier analysis and HP theory on product domains - II.- A unified approach to atomic decompositions via integrable group representations.- The work of coifman and semmes on complex interpolation, several complex variables, and PDE's.- Interpolation of quasinormed spaces by the complex method.- New and old function spaces.- A note on choquet integrals with respect to Hausdorff capacity.- Besov norms of rational functions.- Functions of bounded mean oscillation and Hausdorff-Young type theorems.- Remarks on local function spaces.- The classes Va are monotone.- Hardy-Sobolev spaces and Besov spaces with a function parameter.- An extension of fourier type to quasi-Banach spaces.- ?-Moduli and interpolation.- Interpolation spaces and non-linear approximation.- Atomic decompositions in Hardy spaces on bounded lipschitz domains.- The ?-transform and applications to distribution spaces.- Approximation by solutions of elliptic boundary value problems in various function spaces.- Interpolation of subspaces and quotient spaces by the complex method.- On interpolation of multi-linear operators.- Markov's inequality and local polynomial approximation.- Two weights weak type inequality for the maximal function in the zygmund class.- Banach envelopes of some interpolation quasi-banach spaces.- A construction of eigenvectors for the canonical isometry.- Smoothness of Schmidt functions of smooth Hankel operators.- Real interpolation between some operator ideals.- Direct and converse theorems for spline and rational approximation and besov spaces.- Interpolation of some analytic families of operators.- Function spaces on lie groups and on analytic manifolds.- Equivalent normalizations of Sobolev and Nikol'ski? spaces in domains. boundary values and extension.- Remarks on interpolation of subspaces.- Spectral analysis in spaces of continuous functions.- Lipschitz spaces and interpolating polynomials on subsets of euclidean space.- Problem section.- Errata.
Journal of Mathematical Analysis and Applications, Mar 1, 2016
Abstract Using interpolation properties of cones of general monotone functions, we prove the equi... more Abstract Using interpolation properties of cones of general monotone functions, we prove the equivalence of the L ( p , q ) norms of such functions and their Fourier transforms.
Given an interpolation couple (A0, A~), the approximation functional is dcfined E(t, a; Ao, A1) =... more Given an interpolation couple (A0, A~), the approximation functional is dcfined E(t, a; Ao, A1) = inf {[a-ao[a]Iaolao <= t}.
WIT Transactions on Ecology and the Environment, Jun 9, 2004
Fractional calculus provides novel mathematical tools for modeling physical and biological proces... more Fractional calculus provides novel mathematical tools for modeling physical and biological processes. The bioheat equation is often used as a first order model of heat transfer in biological systems. In this paper we describe formulation of bioheat transfer in one dimension in terms of fractional order differentiation with respect to time. The solution to the resulting fractional order partial differential equation reflects the interaction of the system with the dynamics of its response to surface or volume heating. An example taken from a study involving pulsating (on-off) cooling of a peripheral tissue region during laser surgery is used to illustrate the utility of the method. In the future we hope to interpret the physical basis of fractional derivatives using Constructal Theory, according to which, the geometry biological structures evolve as a result of the optimization process.
Publisher Summary This chapter discusses the Hilbert transform of the Gaussian. It highlights a c... more Publisher Summary This chapter discusses the Hilbert transform of the Gaussian. It highlights a connection between the Hilbert transform and the one-dimensional heat equation. The Hilbert transform is a linear operator that takes a function, u ( t ), and produces a function, H ( u )( t ), with the same domain. The Hilbert transform is a basic tool in Fourier analysis and provides a concrete means for realizing the harmonic conjugate of a given function or Fourier series. By taking A = 0 and B = 1/π gives Q ( u ), the conjugate Poisson kernel for the upper half plane.
Proceedings of the American Mathematical Society, 1991
Let D be a simply connected domain in the complex plane whose boundary T is a rectifiable simple ... more Let D be a simply connected domain in the complex plane whose boundary T is a rectifiable simple closed curve. Let {A(y)/y e T) and {B{y)l7 6 r} be interpolation families of Banach spaces. Let T be a linear operator mapping A(y) continuously into B(y). For z & D let Tz be the restriction of T to the interpolation space Az. Then {z € D/cod(Tz) = d < oo and dimKer(Tz) = 0} and {z € D/dimKer(Tz) = d < oo and Tz is onto B } are open sets.
We complete a characterization of homogeneous solutions of the beat equation begun by D. V. Widde... more We complete a characterization of homogeneous solutions of the beat equation begun by D. V. Widder. We determine regions of convergence for expansions of temperature functions in terms of the homogeneous solutions.
Journal of the Australian Mathematical Society, Feb 1, 1996
We prove that if u(x, t) is a solution of the one dimensional heat equation and if siu(x, i) is i... more We prove that if u(x, t) is a solution of the one dimensional heat equation and if siu(x, i) is its Appell transform, then u(x, t) has the semi-group (Huygens) property in a domain D if and only if s4u(x, t) has the semi-group property in a dual region. We apply this result to simplify and extend some results of Rosenbloom and Widder.
We develop Cauchy-Riemann equations for pairs of temperature functions with boundary values in L'... more We develop Cauchy-Riemann equations for pairs of temperature functions with boundary values in L'(R, dx/(1 + x2)).
This study reports the effects of an integrated instructional program (the Keystone Method) on th... more This study reports the effects of an integrated instructional program (the Keystone Method) on the students' performance in mathematics and reading, and tracks students' persistence and retention. The subject of the study was a large group of students in remedial mathematics classes at the college, willing to learn but lacking basic educational skills. The results show not only improvements in student outcomes in mathematics, but also gains in reading comprehension scores, as compared with the control group. These results were achieved at no cost to classroom retention. The persistence rates of the Keystone students were also higher for the subsequent terms.
College (Illinois) conducted an experiment in remedial mathematics classes at a community college... more College (Illinois) conducted an experiment in remedial mathematics classes at a community college. The researchers wanted to investigate whether, while teaching mathematics in a rigorous manner, they could also use their instruction to enhance the students' academic and workplace skills. Remedial mathematics constitutes a large part of the curriculum in postsecondary institutions across the U.S. In a national study of college-level remediation in 1990, it was found that 21 percent of all entering college freshmen were enrolled in a remedial mathematics course. The project used mathematics classes to enhance the student' work and study habits and improve their concentration skills. This, in turn, produced good results in mathematics, and greatly improved the students' reading comprehension scores. Their general experience as educators had convinced them that unsuccessful students are held back by behavior patterns which inhibit learning. In their program, they addressed the behavior patterns of students in various ways. For example, to increase the students' attention spans, they administered time-pressured quizzes, which required work done with full concentration. (Contains 46 references.) (JA)
Marcel Riesz in Lund.- Interpolation of tent spaces and applications.- Toeplitz liftings of hanke... more Marcel Riesz in Lund.- Interpolation of tent spaces and applications.- Toeplitz liftings of hankel forms.- Some recent developments in fourier analysis and HP theory on product domains - II.- A unified approach to atomic decompositions via integrable group representations.- The work of coifman and semmes on complex interpolation, several complex variables, and PDE's.- Interpolation of quasinormed spaces by the complex method.- New and old function spaces.- A note on choquet integrals with respect to Hausdorff capacity.- Besov norms of rational functions.- Functions of bounded mean oscillation and Hausdorff-Young type theorems.- Remarks on local function spaces.- The classes Va are monotone.- Hardy-Sobolev spaces and Besov spaces with a function parameter.- An extension of fourier type to quasi-Banach spaces.- ?-Moduli and interpolation.- Interpolation spaces and non-linear approximation.- Atomic decompositions in Hardy spaces on bounded lipschitz domains.- The ?-transform and applications to distribution spaces.- Approximation by solutions of elliptic boundary value problems in various function spaces.- Interpolation of subspaces and quotient spaces by the complex method.- On interpolation of multi-linear operators.- Markov's inequality and local polynomial approximation.- Two weights weak type inequality for the maximal function in the zygmund class.- Banach envelopes of some interpolation quasi-banach spaces.- A construction of eigenvectors for the canonical isometry.- Smoothness of Schmidt functions of smooth Hankel operators.- Real interpolation between some operator ideals.- Direct and converse theorems for spline and rational approximation and besov spaces.- Interpolation of some analytic families of operators.- Function spaces on lie groups and on analytic manifolds.- Equivalent normalizations of Sobolev and Nikol'ski? spaces in domains. boundary values and extension.- Remarks on interpolation of subspaces.- Spectral analysis in spaces of continuous functions.- Lipschitz spaces and interpolating polynomials on subsets of euclidean space.- Problem section.- Errata.
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