Search results for "Mathematical physics"
showing 10 items of 2687 documents
F-signature of pairs: Continuity, p-fractals and minimal log discrepancies
2011
This paper contains a number of observations on the {$F$-signature} of triples $(R,\Delta,\ba^t)$ introduced in our previous joint work. We first show that the $F$-signature $s(R,\Delta,\ba^t)$ is continuous as a function of $t$, and for principal ideals $\ba$ even convex. We then further deduce, for fixed $t$, that the $F$-signature is lower semi-continuous as a function on $\Spec R$ when $R$ is regular and $\ba$ is principal. We also point out the close relationship of the signature function in this setting to the works of Monsky and Teixeira on Hilbert-Kunz multiplicity and $p$-fractals. Finally, we conclude by showing that the minimal log discrepancy of an arbitrary triple $(R,\Delta,\b…
PT-symmetry and Schrödinger operators. The double well case
2015
We study a class of $PT$-symmetric semiclassical Schrodinger operators, which are perturbations of a selfadjoint one. Here, we treat the case where the unperturbed operator has a double-well potential. In the simple well case, two of the authors have proved in [6] that, when the potential is analytic, the eigenvalues stay real for a perturbation of size $O(1)$. We show here, in the double-well case, that the eigenvalues stay real only for exponentially small perturbations, then bifurcate into the complex domain when the perturbation increases and we get precise asymptotic expansions. The proof uses complex WKB-analysis, leading to a fairly explicit quantization condition.
F-singularities via alterations
2011
For a normal F-finite variety $X$ and a boundary divisor $\Delta$ we give a uniform description of an ideal which in characteristic zero yields the multiplier ideal, and in positive characteristic the test ideal of the pair $(X,\Delta)$. Our description is in terms of regular alterations over $X$, and one consequence of it is a common characterization of rational singularities (in characteristic zero) and F-rational singularities (in characteristic $p$) by the surjectivity of the trace map $\pi_* \omega_Y \to \omega_X$ for every such alteration $\pi \: Y \to X$. Furthermore, building on work of B. Bhatt, we establish up-to-finite-map versions of Grauert-Riemenscheneider and Nadel/Kawamata-V…
Plenty of big projections imply big pieces of Lipschitz graphs
2020
I prove that a closed $n$-regular set $E \subset \mathbb{R}^{d}$ with plenty of big projections has big pieces of Lipschitz graphs. This answers a question of David and Semmes.
Reciprocal lower bound on modulus of curve families in metric surfaces
2019
We prove that any metric space $X$ homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure satisfies a reciprocal lower bound on modulus of curve families associated to a quadrilateral. More precisely, let $Q \subset X$ be a topological quadrilateral with boundary edges (in cyclic order) denoted by $\zeta_1, \zeta_2, \zeta_3, \zeta_4$ and let $\Gamma(\zeta_i, \zeta_j; Q)$ denote the family of curves in $Q$ connecting $\zeta_i$ and $\zeta_j$; then $\text{mod} \Gamma(\zeta_1, \zeta_3; Q) \text{mod} \Gamma(\zeta_2, \zeta_4; Q) \geq 1/\kappa$ for $\kappa = 2000^2\cdot (4/\pi)^2$. This answers a question concerning minimal hypotheses under which a metric space admits a quasiconfor…
Quantum systems with fractal spectra
2002
Abstract We study Hamiltonians with singular spectra of Cantor type with a constant ratio of dissection and show strict connections between the decay properties of the states in the singular subspace and the algebraic number theory. More specifically, we study the decay properties of free n-particle systems and the computability of decaying and non-decaying states in the singular continuous subspace.
The $p\lambda n$ fractal decomposition: Nontrivial partitions of conserved physical quantities
2015
A mathematical method for constructing fractal curves and surfaces, termed the $p\lambda n$ fractal decomposition, is presented. It allows any function to be split into a finite set of fractal discontinuous functions whose sum is equal everywhere to the original function. Thus, the method is specially suited for constructing families of fractal objects arising from a conserved physical quantity, the decomposition yielding an exact partition of the quantity in question. Most prominent classes of examples are provided by Hamiltonians and partition functions of statistical ensembles: By using this method, any such function can be decomposed in the ordinary sum of a specified number of terms (g…
Accessible parts of boundary for simply connected domains
2018
For a bounded simply connected domain $\Omega\subset\mathbb{R}^2$, any point $z\in\Omega$ and any $0<\alpha<1$, we give a lower bound for the $\alpha$-dimensional Hausdorff content of the set of points in the boundary of $\Omega$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $\alpha$, in terms of the distance of $z$ to $\partial\Omega$. In fact this set in the boundary contains the intersection $\partial\Omega_z\cap\partial\Omega$ of the boundary of a John sub-domain $\Omega_z$ of $\Omega$, centered at $z$, with the boundary of $\Omega$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obta…
Fundamental isomorphism theorems for quantum groups
2017
The lattice of subgroups of a group is the subject of numerous results revolving around the central theme of decomposing the group into "chunks" (subquotients) that can then be compared to one another in various ways. Examples of results in this class would be the Noether isomorphism theorems, Zassenhaus' butterfly lemma, the Schreier refinement theorem for subnormal series of subgroups, the Dedekind modularity law, and last but not least the Jordan-H\"older theorem. We discuss analogues of the above-mentioned results in the context of locally compact quantum groups and linearly reductive quantum groups. The nature of the two cases is different: the former is operator algebraic and the latt…
Tailoring a pair of pants
2021
Abstract We show how to deform the map Log : ( C ⁎ ) n → R n such that the image of the complex pair of pants P ∘ ⊂ ( C ⁎ ) n is the tropical hyperplane by showing an (ambient) isotopy between P ∘ ⊂ ( C ⁎ ) n and a natural polyhedral subcomplex of the product of the two skeleta S × Σ ⊂ A × C of the amoeba A and the coamoeba C of P ∘ . This lays the groundwork for having the discriminant to be of codimension 2 in topological Strominger-Yau-Zaslow torus fibrations.