Topology and Analysis Forum

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Participate in expert discussion on topology and analysis topics. This includes point-set topology, Real, complex, harmonic and functional analysis. Also measure and integration theory.
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Insights A Path to Fractional Integral Representations of Some Special Functions
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I Exercise 2.23 in Folland's real analysis text
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I 3-sphere with Ricci flow
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I Regarding dominated convergence theorem in Folland
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I On Borel sets of the extended reals
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I Basic question on 'bounded implies totally bounded'
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I Open balls dense in closed balls in Euclidean space
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I Show that a "cross" is not a topological manifold
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I Tough lemma on locally finite refinement
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A Cubical cup poduct with a unit 1-cochain and the coboundary operator
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I Compact support functions and law of a random variable
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I Uniform closure of algebra of bounded functions is uniformly closed
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I Parallel rectangles contained in oblique rectangle
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A Question about different statements of Picard Theorem
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I Question about convex property in Jensen's inequality
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I A non-empty intersection of closures of level sets implies discontinuity
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A Proving that this integral is divergent
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A Question about covering map of punctured unit disk
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A Question about lifting of a function
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A Question about existence of path-lifting property
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I Riemann integrability and uniform convergence
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MHB Integration in Polar Coordinates (Fubini/Tonelli)
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I On theorem 1.19 in Folland's and completion of measure
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I Are slices of measurable functions measurable?
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I On integral of simple function and representation
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I Collection of finite unions of half-open intervals form an algebra
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I Foliation of the 3-torus
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I Help with basic transfinite induction proof
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I On two definitions of locally compact
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I Showing equivalence of two definitions of essential supremum
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I Two basic results from measure theory -- on volumes of rectangles
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I In which sense(s) do square integrable functions go to zero at infinity?
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A Functional Analysis for Physics in 2024
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I Question about branch of logarithms
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I Verify the completion of continuous compactly supported functions
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I Fiber bundle homeomorphism with the fiber
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I On deriving the (inverse) Fourier transform from Fourier series
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I Prove projection of a measurable set from product space is measurable
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B Questions on a numberphile video
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I "Magic" regulating functions for divergent series
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I ##SU(2, \mathbb C)## parametrization using Euler angles
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I Regarding Wirtinger's inequality
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I ##SL(2,\mathbb R)## Lie group as manifold
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I Is the Projection Restriction to a Linear Subspace a Homeomorphism?
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I New Video Course on General Topology on YouTube
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I ##SU(2)## homeomorphic with ##\mathbb S^3##
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I Bloch Analysis proof of Theorem 2.5.5 (Definition by recursion)
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I How do I use the four axioms of a neighborhood to define an open set?
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I How to define an open set using the four axioms of a neighborhood
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I Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?
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I On differentiability and Fourier coefficients (Vretblad's text)
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A Complex analysis -- Essential singularity
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I On limit of convolution of function with a summability kernel
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I Fourier coefficients of convolution
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I Outer measure .... Axler, Result 2.14 .... Another Question ....
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I Bounding modulus of complex logarithm times complex power function
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B Confusion with the basics of Topology (Poincare conjecture)
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I Is a Locally One-to-One Proper Map Globally Bijective?
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I On a remark regarding the Cauchy integral formula
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I Infinity Numbers (Expanded P-Adic Numbers)
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I Why is this function not ##L^1(\mathbb{R} \times \mathbb{R})##?
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B Why is the set {(x,y)∈Ω×R|y=f(x)} a manifold?
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A Topology - Boundary of a ball without a point
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I Feedback for my YouTube Videos on Real Analysis
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B Understanding Dedekind Cuts: Exploring Addition and Negative Numbers
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I Using Residues (Complex Analysis) to compute partial fractions
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I Is the Set of Integer Outputs of sin(x) Sequentially Compact in ℝ?
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I Circuits or edge-disjoint unions of circuits in a connected graph
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I Can you somehow make left handed helix into a right handed?
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MHB Outer measure .... Axler, Result 2.8 ....
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I Calculate the residue of 1/Cosh(pi.z) at z = i/2 (complex analysis)
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B Mapping wave forms to sphere, does wave form y=0 have a reflection?
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B Aperiodic Tiling with a single tile shape
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A Regarding fibrations between smooth manifolds
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I Geometry of series terms of the Riemann Zeta Function
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I I need a mapping from the unit Hypercube [0,1]^n to a given simplex
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I Discrete Subrings of Complex Numbers: Topology of Rings
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MHB Checking Proof of Theorem 6.2.8 Part (ii)
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I I would like opinions of the latest draft of my note - Integration
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I Upper and Lower Darboux Sum Inequality
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I Original definition of Riemann Integral and Darboux Sums
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MHB How to Prove \(\text{vol}_n (B) = \text{vol}_n (B') + \text{vol}_n (B'')\)?
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A Infinite series of this type converges?
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I In a 3-dimension isoperimetric problem, a ball maximizes the volume
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I A curve that does not meet rational points
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I Create a surjective function from [0,1]^n→S^n
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I Lars Olsen proof of Darboux's Intermediate Value Theorem for Derivatives
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A Norm 2, f Integrable function, show: ##||f-g||_2<\epsilon##
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I Does topology distinguish between real and imaginary dimensions?
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I Understanding the Role of the Identity Map in Fundamental Group Theory
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MHB How to Prove a Complex Number Equation and Its Trajectory Forms a Circle?
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I Understanding an argument in Intermediate Value Theorem
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I Are there any elementary functions of norms that are still norms?
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I Applications of measurement of commutativity
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MHB Can This Forgotten Formula Simplify the Laplace Transform of \(y(t)^3\)?
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Is the Integral Inequality Possible to Prove for Certain Parameters?
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A Ref: "Standard" Action of S^1 on S^n ?
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B Is the Texan Conjecture a mathematical truth or a myth?
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MHB The Supremum of a Set is not an Interior Point
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I If a sequence converges, then all subsequences of it have same limit
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MHB How Can I Tackle Difficult Questions Effectively?
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Topology and Analysis

Welcome to the Topology forum!

Topology and analysis are two branches of mathematics that investigate the properties of spaces and functions, respectively. Topology is concerned with the study of the properties that remain unchanged under continuous deformations, such as stretching or bending, but not tearing. It explores concepts like continuity, connectedness, and compactness.

Analysis, on the other hand, is focused on understanding the behavior of mathematical functions and sequences. It deals with limits, continuity, differentiation, and integration, providing tools for a detailed examination of functions. Both topology and analysis play integral roles in diverse areas of mathematics and find applications in various scientific and engineering disciplines.

Together, they contribute to a deeper understanding of the structure and properties of mathematical objects.
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