A Note on One-dimensional Varieties Over the Complex p-adic Field

AbstractFor the Markov property of a multivariate process, a necessary and suficient condition on the multidimensional copula of the finite-dimensional distributions is given. This establishes that the Markov property is solely a property of the copula, i.e., of the dependence structure. This extends results by Darsow et al. [11] from dimension one to the multivariate case. In addition to the one-dimensional case also the spatial copula between the different dimensions has to be taken into account. Examples are also given.

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FRACTAL DIMENSIONS OF WEYL–MARCHAUD FRACTIONAL DERIVATIVE OF CERTAIN ONE-DIMENSIONAL FUNCTIONS

Fractals ◽

10.1142/s0218348x19501147 ◽

2019 ◽

Vol 27 (07) ◽

pp. 1950114

Author(s):

Y. S. LIANG ◽

N. LIU

Keyword(s):

Fractional Derivative ◽

Fractal Dimensions ◽

Continuous Functions ◽

Box Dimension ◽

One Dimensional ◽

Dimension One

Fractal dimensions of Weyl–Marchaud fractional derivative of certain continuous functions are investigated in this paper. Upper Box dimension of Weyl–Marchaud fractional derivative of certain continuous functions with Box dimension one has been proved to be no more than the sum of one and its order.

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Projector augmented-wave method: an analysis in a one-dimensional setting

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019017 ◽

2020 ◽

Vol 54 (1) ◽

pp. 25-58

Author(s):

Mi-Song Dupuy

Keyword(s):

Numerical Analysis ◽

Eigenvalue Problem ◽

Ab Initio Calculations ◽

Wave Method ◽

One Dimensional ◽

The One ◽

Dimension One ◽

Projector Augmented Wave ◽

Periodic Setting ◽

Periodic Schrödinger Operator

In this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian H by a pseudo-Hamiltonian HPAW via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as H and smoother eigenfunctions. In practice, the pseudo-Hamiltonian HPAW has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schrödinger operator with double Dirac potentials.

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Cohomology bounds for sheaves of dimension one

International Journal of Mathematics ◽

10.1142/s0129167x14501031 ◽

2014 ◽

Vol 25 (11) ◽

pp. 1450103 ◽

Cited By ~ 2

Author(s):

Jinwon Choi ◽

Kiryong Chung

Keyword(s):

Projective Plane ◽

Moduli Space ◽

Hilbert Scheme ◽

Closed Subset ◽

Projective Variety ◽

Global Section ◽

Sharp Bounds ◽

One Dimensional ◽

Dimension One

We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

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A classification of explosions in dimension one

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708080486 ◽

2009 ◽

Vol 29 (2) ◽

pp. 715-731 ◽

Cited By ~ 3

Author(s):

E. SANDER ◽

J. A. YORKE

Keyword(s):

Sufficient Conditions ◽

Global Bifurcation ◽

Necessary And Sufficient Conditions ◽

Homoclinic Tangency ◽

Discontinuous Change ◽

One Dimensional ◽

One Dimensional Maps ◽

Dimension One ◽

Necessary And Sufficient

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

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A CRITERION FOR ERDŐS SPACES

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091504000823 ◽

2005 ◽

Vol 48 (3) ◽

pp. 595-601 ◽

Cited By ~ 10

Author(s):

Jan J. Dijkstra

Keyword(s):

Hilbert Space ◽

Banach Spaces ◽

Real Hilbert Space ◽

Topological Dimension ◽

One Dimensional ◽

The Real ◽

General Space ◽

Dimension One ◽

Totally Disconnected

AbstractIn 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.

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Entropy in dimension one

Frontiers in Complex Dynamics ◽

10.23943/princeton/9780691159294.003.0015 ◽

2014 ◽

Cited By ~ 2

Author(s):

William P. Thurston

Keyword(s):

Topological Entropy ◽

Finite Interval ◽

Algebraic Integer ◽

Small Letter ◽

One Dimensional ◽

Original Manuscript ◽

Postcritically Finite ◽

Perron Number ◽

One Dimensional Maps ◽

Dimension One

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.

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One-Dimensional Case

Optimal Transport Methods in Economics ◽

10.23943/princeton/9780691172767.003.0004 ◽

2016 ◽

Author(s):

Alfred Galichon

Keyword(s):

Explicit Solution ◽

Dimensional Case ◽

Assortative Matching ◽

Optimal Assignment ◽

Primal Problem ◽

One Dimensional ◽

Quantile Transform ◽

The One ◽

Dimension One ◽

Kantorovich Problem

This chapter considers the Monge–Kantorovich problem in the one-dimensional case, when both the worker and the job are characterized by a scalar attribute. The important assumption of positive assortative matching, or supermodularity of the matching surplus, is introduced and discussed. As a consequence, the primal problem has an explicit solution (an optimal assignment) which is related to the probabilistic notion of a quantile transform, and the dual problem also has an explicit solution (a set of equilibrium prices), which are obtained from the solution to the primal problem. As a consequence, the Monge–Kantorovich problem is explicitly solved in dimension one under the assumption of positive assortative matching.

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Automatic continuity and second order cohomology

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700001968 ◽

2000 ◽

Vol 68 (2) ◽

pp. 231-243

Author(s):

Volker Runde

Keyword(s):

Hochschild Cohomology ◽

Banach Algebras ◽

Locally Compact Abelian Group ◽

Dense Subspace ◽

Automatic Continuity ◽

One Dimensional ◽

First Case ◽

Hochschild Cohomology Group ◽

The One ◽

Dimension One

AbstractMany Banach algebras A have the property that, although there are discontinuous homomorphisms from A into other Banach algebras, every homomorphism from A into another Banach algebra is automatically continuous on a dense subspace—preferably, a subalgebra—of A. Examples of such algebras are C*-algebras and the group algebras L1(G), where G is a locally compact, abelian group. In this paper, we prove analogous results for , where E is a Banach space, and . An important rôle is played by the second Hochschild cohomology group of and , respectively, with coefficients in the one-dimensional annihilator module. It vanishes in the first case and has linear dimension one in the second one.

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The topological line of ABJ(M) theory

Journal of High Energy Physics ◽

10.1007/jhep06(2021)091 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Nicola Gorini ◽

Luca Griguolo ◽

Luigi Guerrini ◽

Silvia Penati ◽

Domenico Seminara ◽

...

Keyword(s):

Partition Function ◽

Central Charge ◽

Exact Representation ◽

One Dimensional ◽

Topological Sector ◽

The Matrix ◽

The One ◽

The Masses ◽

Dimension One ◽

M Theory

Abstract We construct the one-dimensional topological sector of $$ \mathcal{N} $$ N = 6 ABJ(M) theory and study its relation with the mass-deformed partition function on S3. Supersymmetric localization provides an exact representation of this partition function as a matrix integral, which interpolates between weak and strong coupling regimes. It has been proposed that correlation functions of dimension-one topological operators should be computed through suitable derivatives with respect to the masses, but a precise proof is still lacking. We present non-trivial evidence for this relation by computing the two-point function at two-loop, successfully matching the matrix model expansion at weak coupling and finite ranks. As a by-product we obtain the two-loop explicit expression for the central charge cT of ABJ(M) theory. Three- and four-point functions up to one-loop confirm the relation as well. Our result points towards the possibility to localize the one-dimensional topological sector of ABJ(M) and may also be useful in the bootstrap program for 3d SCFTs.

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