Basic Notions of Algebraic Topology

AbstractDementia is related to the cellular accumulation of β-amyloid plaques, tau aggregates, or α-synuclein aggregates, or to neurotransmitter deficiencies in the dopaminergic and cholinergic pathways. Cellular and neurochemical changes are both involved in dementia pathology. However, the role of dopaminergic and cholinergic networks in metabolic connectivity at different stages of dementia remains unclear. The altered network organisation of the human brain characteristic of many neuropsychiatric and neurodegenerative disorders can be detected using persistent homology network (PHN) analysis and algebraic topology. We used 18F-fluorodeoxyglucose positron emission tomography (18F-FDG PET) imaging data to construct dopaminergic and cholinergic metabolism networks, and used PHN analysis to track the evolution of these networks in patients with different stages of dementia. The sums of the network distances revealed significant differences between the network connectivity evident in the Alzheimer’s disease and mild cognitive impairment cohorts. A larger distance between brain regions can indicate poorer efficiency in the integration of information. PHN analysis revealed the structural properties of and changes in the dopaminergic and cholinergic metabolism networks in patients with different stages of dementia at a range of thresholds. This method was thus able to identify dysregulation of dopaminergic and cholinergic networks in the pathology of dementia.

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Dynamics of Ion Channels via Non-Hermitian Quantum Mechanics

Entropy ◽

10.3390/e23010125 ◽

2021 ◽

Vol 23 (1) ◽

pp. 125

Author(s):

Tobias Gulden ◽

Alex Kamenev

Keyword(s):

Quantum Mechanics ◽

Algebraic Topology ◽

Riemann Surfaces ◽

Surrounding Medium ◽

Coulomb Systems ◽

Dielectric Constants ◽

Ion Interactions ◽

Multivalent Ions ◽

Electric Filed ◽

Entropic Effects

We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels.

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Algebraic Topology.

American Mathematical Monthly ◽

10.2307/2319113 ◽

1973 ◽

Vol 80 (4) ◽

pp. 449

Author(s):

M. J. Powers ◽

C. R. F. Maunder

Keyword(s):

Algebraic Topology

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THE MAIN TRENDS OF ALGEBRAIC TOPOLOGY AND ALGEBRAIC GEOMETRY

Russian Mathematical Surveys ◽

10.1070/rm1964v019n06abeh001164 ◽

1964 ◽

Vol 19 (6) ◽

pp. 67-73

Author(s):

S P Novikov ◽

I I Pyatetskii-Shapiro ◽

I R Shafarevich

Keyword(s):

Algebraic Geometry ◽

Algebraic Topology

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Algebraic Topology: A First Course.

American Mathematical Monthly ◽

10.2307/2974465 ◽

1996 ◽

Vol 103 (9) ◽

pp. 819

Author(s):

William S. Massey ◽

William Fulton

Keyword(s):

Algebraic Topology

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Formal groups and Z -entropies

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2016.0143 ◽

2016 ◽

Vol 472 (2195) ◽

pp. 20160143 ◽

Cited By ~ 13

Author(s):

Piergiulio Tempesta

Keyword(s):

Algebraic Topology ◽

Formal Group ◽

Point Of View ◽

Trace Form ◽

Formal Groups ◽

Form Class ◽

New Family ◽

Mathematical Properties ◽

Composition Process ◽

Group Law

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z -entropy is composable (Tempesta 2016 Ann. Phys. 365 , 180–197. ( doi:10.1016/j.aop.2015.08.013 )). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon–Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

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RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500174 ◽

2012 ◽

Vol 14 (03) ◽

pp. 1250017 ◽

Cited By ~ 10

Author(s):

LEONARDO CABRER ◽

DANIELE MUNDICI

Keyword(s):

Abelian Group ◽

Algebraic Topology ◽

Lattice Structure ◽

Order Unit ◽

Finitely Generated ◽

Lattice Order ◽

Translation Invariant ◽

Projective Lattice ◽

Rational Polyhedra ◽

Finitely Presented

An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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Algebraic Topology

10.1142/12586 ◽

2022 ◽

Author(s):

Marco Grandis

Keyword(s):

Algebraic Topology

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