#functionalanalysis — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #functionalanalysis, aggregated by home.social.
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The hierarchy of mathematical spaces
These diagrams illustrate the hierarchical relationships and nesting of various mathematical spaces used in functional analysis and geometry. They demonstrate how specific structures, such as Hilbert and Banach spaces, are specialized subsets of broader categories like normed linear spaces and metric spaces. The visuals highlight the essential properties required for each classification, ranging from basic topological sets to complex systems involving inner products and completeness. By organizing these concepts into flowcharts and Venn diagrams, the sources clarify how adding constraints like distance, magnitude, or orthogonality transforms one type of space into another. Ultimately, the collection provides a comprehensive map of how abstract vector spaces relate to concrete examples like Euclidean space.
#educationalcontent #mathematics #functionalanalysis #mathematicalspaces
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`A substantial part of the research in functions of one complex variable in the 20th century was focused on Nevanlinna theory. One direction of this research was to find out whether the main conclusions of Nevanlinna theory are best possible. For example, the Inverse Problem of Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points. This was solved by David Drasin in 1976.`
https://en.wikipedia.org/wiki/Nevanlinna_theory#Further_development
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Tenure-track opening @ U. Colorado Boulder Dept. of Math!
Esp. (but not only) looking for:
algebraic geometry
homotopy theory
foundations
functional analysis
number theory
interdisciplinary collab. b/w math & computer science or the math of quantum physicshttps://www.mathjobs.org/jobs/list/27231
Please help spread the word!
#AlgebraicGeometry #HomotopyTheory #AlgebraicTopology #FoundationsOfMath #FunctionalAnalysis #NumberTheory #ComputerScience #Quantum #Math
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Dr. Miller has tipped me off that the suggested text for his upcoming will be:Berberian, Sterling Khazag. Introduction To Hilbert Space. Oxford University Press, 1961. Reprint Literary Licensing, 2012.
He’s not happy that it ignores measure theory as a means to introduce the Lebesque integral, so he’ll be supplementing that with additional notes. I’ve ordered a used copy of the 1st edition, but there are also versions from AMS as well as a more recent reprint from 2012.
He also suggested that Debnath & Mikusinski was pretty good, albeit more expensive than he would like in addition to not being a fan some of their approaches to topics.
Debnath, Lokenath, and Piotr Mikusinski. Introduction to Hilbert Spaces with Applications. 3rd ed., Academic Press, 2005.
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Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions
Looking for some serious entertainment with an intellectual bent on Tuesday nights this fall? Professor Michael Miller has got you covered in multiple dimensions. Dr. Miller has now listed his mathematics offering for Fall 2025 at UCLA Extension. It's Introduction to Hilbert Spaces: An Adventure In Infinite Dimensions (MATH 900). As always, it will be presented in lectures on Tuesday nights from 7:00 PM to 10:00 PM with a short break in the middle. The class runs from September 23 - […] -
In the past few weeks I have been trying to understand the eigenvalue problem (time-independent Schrödinger equation)
–𝑢'' + λ (cos 𝑥 + cos τ𝑥) 𝑢 = 𝐸𝑢
where λ is a parameter, 𝐸 is the eigenvalue (blame the physicists for the notation), τ is the golden ratio and the problem is posed on the infinite line. The motivation comes from quasicrystals.
Some solutions are localized around a minimum of the potential, but the none of the corresponding eigenvalues are isolated.
At higher energies, solutions spread out over the whole line, giving rise to the absolutely continuous spectrum which is a Cantor set.
This is wild, at least for me, but partially supported by my own computations and functional analysis results. But I am not fully confident of the former and struggling to understand the latter, so I am not sure whether this picture is complete or even correct.
The more I look into it, the less I understand ... any pointers are appreciated.
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in many ways, #mathematical #physics is based on the #smoothing properties of the #integral
#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein
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in many ways, #mathematical #physics is based on the #smoothing properties of the #integral
#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein
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in many ways, #mathematical #physics is based on the #smoothing properties of the #integral
#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein
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in many ways, #mathematical #physics is based on the #smoothing properties of the #integral
#integrableSystems #mathematicalPhysics #complexAnalysis #spectralTheory #harmonicAnalysis #functionalAnalysis #mathematicalAnalysis #differentialEquations #ODEs #PDEs #SDEs #DEs #equations #BrownianMotion #LangevinDynamics #dynamics #Langevin #StochasticDifferentialEquations #StochasticProcesses #WienerProcess #OrnsteinUhlenbeck #HarmonicOscillator #WaveEquation #Newton #Newtonian #Maxwell #Einstein
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`To prove #boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous #operator from L1 to the space of #functions of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the #singular #integral operator is bounded on all Lp for 1 < p < ∞.`
https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory
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`To prove #boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous #operator from L1 to the space of #functions of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the #singular #integral operator is bounded on all Lp for 1 < p < ∞.`
https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory
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`To prove #boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous #operator from L1 to the space of #functions of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the #singular #integral operator is bounded on all Lp for 1 < p < ∞.`
https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory
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`To prove #boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz. This method showed that the operator defined a continuous #operator from L1 to the space of #functions of weak L1. The Marcinkiewicz interpolation theorem and duality then implies that the #singular #integral operator is bounded on all Lp for 1 < p < ∞.`
https://en.wikipedia.org/wiki/Singular_integral_operators_of_convolution_type#General_theory
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youtube.com/@xenaproject
for #mathematics #mathsbasics#mathematicallogic #sets #logic #topology #numbertheory #linearalgebra
#probability #measuretheory
#functionalanalysis -
youtube.com/@xenaproject
for #mathematics #mathsbasics#mathematicallogic #sets #logic #topology #numbertheory #linearalgebra
#probability #measuretheory
#functionalanalysis -
youtube.com/@xenaproject
for #mathematics #mathsbasics#mathematicallogic #sets #logic #topology #numbertheory #linearalgebra
#probability #measuretheory
#functionalanalysis -
youtube.com/@xenaproject
for #mathematics #mathsbasics#mathematicallogic #sets #logic #topology #numbertheory #linearalgebra
#probability #measuretheory
#functionalanalysis -
Are you keen to read about results in #MathematicalPhysics and Analysis of #PDEs? Take a look at the paper of our previous workshop participant! 🧐
#FunctionalAnalysis #SpectralTheory
@univienna
https://arxiv.org/pdf/2303.04527.pdf -
We thank Prof Elliott Lieb for taking a long journey from @princeton Princeton
University and giving us an insight into his research work. 🔎 He is a living example of that the age is only a number and you can still accomplish a lot at the age of 91. 🎉
Watch his lecture on YouTube 🔜 www.youtube.com/@ESIVienna#MathematicalPhysics #Bosegas #condensedmatter #statisticalmechanics #statmech #functionalanalysis
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We thank Prof Elliott Lieb for taking a long journey from @princeton Princeton
University and giving us an insight into his research work. 🔎 He is a living example of that the age is only a number and you can still accomplish a lot at the age of 91. 🎉
Watch his lecture on YouTube 🔜 www.youtube.com/@ESIVienna#MathematicalPhysics #Bosegas #condensedmatter #statisticalmechanics #statmech #functionalanalysis
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Today in diagrams I liked: This "unit circle" imagination of Lp space for different values of p. Twas a meme that brought me down this Wikipedia rabbit hole but I have been staring at this picture for a very long while now 😂
https://en.m.wikipedia.org/wiki/Lp_space
An example is given of where Euclidean distance falls short: taxi drivers need to use rectilinear distance in gridded cities!
#illustration #math #computerScience #topology #distance #functionalAnalysis #learning #mathematics #diagram #design
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I can read C.
I have never done any C#.
I can write passable C++.
But my expertise is in C*. -
CW: Dirichlet series, link to twitter thread
I have some #math thoughts to add to this #twitter thread about dense flows on the torus
https://twitter.com/math_vet/status/1589680817101365248
You can do this in more dimensions (and in infinitely many, you can connect things to Dirichlet series)!
