arXiv subjects

Thomas Schürmann

Publications and source records attributed to Thomas Schürmann.

At least 19 recordsLinked to original sources

arXiv2026

A First-Order Entropy Law for Canonical T-Complexity of Finite-Alphabet i.i.d. Sources

Let $W_N$ be an exact length-$N$ block from a strictly positive i.i.d. source $\mathbf p$ on a fixed finite alphabet. We prove that the canonical T-complexity $c_T$ satisfies \[ \frac{c_T(W_N)}{e^{-γ}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\mathbf p)$ is the source entropy in nats and $γ$ is the Euler-Mascheroni constant. The proof combines an exact length budget for canonical recovery, a critical-scale $E_1$ estimate for an ideal backward chain, and an exact finite-block boundary representation. An exact Doob-transform identity expresses the finite-boundary law relative to the ideal law conditioned at each step to avoid the current history-dependent successor codeword. A history-uniform renewal estimate then makes the telescoping endpoint density uniformly asymptotic to one, so no one-step approximation errors accumulate.

math.ST↗

arXiv2026

Static Einstein-Scalar Reconstruction: Compatibility and Descent

Prescribing both the areal-radius metric function $F(r)$ and the scalar profile $ϕ(r)$ generally overdetermines static Einstein-scalar reconstruction. The temporal and radial Einstein equations determine a redshift function and an algebraic radial source $V_{\mathrm{alg}}(r)$, but do not by themselves impose the angular equation or guarantee a single-valued scalar potential. On a connected static interval we formulate the corresponding membership criterion for independently prescribed pairs: a potential-independent residual $\mathcal{C}[F,ϕ]$ must vanish, and $V_{\mathrm{alg}}$ must descend across the full scalar image to one branch-independent $C^1$ potential, with stationarity at values attained on constant-profile intervals. The compatibility and descent requirements are independent. As a positive benchmark, the criterion recovers the logarithmic Matos-Guzmán-Núñez scalar halo with its exponential potential. At a radially regular center we compute the leading scalar backreaction on $F$. Applied to a rational kink profile with rational curvature interpolation, exact compatibility forces the scalar amplitude to vanish for every integer $n\geq2$, giving an ansatz-specific compatibility obstruction rather than a no-hair or classification theorem.

physics.gen-ph↗

arXiv2026

Non-Simple T-Prescriptions Yield T-Complexity Gains Infinitely Often

Titchener et al. (2005) asked whether non-simple T-prescriptions can achieve larger T-complexity than simple T-prescriptions for infinitely many finite values of the maximum codeword length. We answer this question affirmatively for every fixed alphabet of size at least two. In fact, we prove a stronger upper-envelope statement: for infinitely many lengths $N$, there exists a valid non-simple T-prescription of exact maximum codeword length $N$ whose T-complexity exceeds the best value attainable by any simple prescription with maximum codeword length at most $N$. Hence the unrestricted exact-length maximum over prescriptions is strictly larger than the corresponding simple exact-length maximum for infinitely many $N$. The proof is elementary. Once a copy pattern has been used, it cannot be selected again. Thus simple prescriptions with more and more steps must use ever more distinct words, which forces the minimal simple length thresholds to have infinitely many strict upward jumps. At each such jump, changing the last copy factor of a minimal simple prescription from $1$ to $2$ yields a valid non-simple prescription, thereby increasing T-complexity by $\log_2 3-1$ while staying below the next simple threshold.

cs.IT↗

arXiv2026

Intrinsic Heisenberg-type lower bounds on spacelike hypersurfaces in general relativity

In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann-Lüders projection onto a geodesic ball $B_Σ(r)$ of radius $r$ on a Cauchy slice $(Σ,h)$, with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation $σ_p$ and show that strict confinement to $B_Σ(r)$ enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue $λ_1$ of the Laplace-Beltrami operator on the ball, $σ_p \ge \hbar\sqrt{λ_1}$, and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus $|ψ|$ from phase-gradient fluctuations and clarifies how the spectral geometry of $(Σ,h)$ controls momentum uncertainty. Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, $σ_p r \ge π\hbar/2$, depending only on the proper radius $r$.

gr-qc↗

arXiv2025

Variance-Refined In-Diameter Lower Bound for the First Dirichlet Eigenvalue

Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold with nonempty boundary and $n\geq 2$. Assume that ${\mathrm{Ric}(M)\ge (n-1)K}$ for some ${K>0}$ and that $\partial M$ has nonnegative mean curvature with respect to the outward unit normal. Denote by $λ$ the first Dirichlet eigenvalue of the Laplacian. Ling's gradient-comparison method (Ling, 2006) provides an explicit lower bound for $λ$ in terms of $K$ and the in-diameter $\tilde d$ (twice the maximal distance from a point of $M$ to $\partial M$). We isolate the only step in Ling's argument that loses quantitative information: a Jensen-Hölder averaging that replaces a nonconstant one-dimensional comparison function by its mean. Using the uniform strong convexity of $x\to x^{-1/2}$ on $(0,1]$, we refine this averaging by a variance term and thereby retain part of the discarded oscillation. This yields an explicit closed-form in-diameter bound that is strictly stronger than Ling's estimate for every $K>0$.

math.DG↗

arXiv2025

Intrinsic Heisenberg Lower Bounds on Schwarzschild and Weyl-Class Spacelike Slices

We establish a coordinate-invariant Heisenberg-type lower bound for quantum states strictly localized in geodesic balls of radius $r_g$ on horizon-regular spacelike slices of static, spherically symmetric, asymptotically flat (AF) black-holes. Via a variance-eigenvalue equivalence the momentum uncertainty reduces to the first Dirichlet eigenvalue of the Laplace-Beltrami operator, yielding a slice-uniform Hardy baseline $σ_p r_g \ge \hbar/2$ under mild convexity assumptions on the balls; the bound is never attained and admits a positive gap both on compact interior regions and uniformly far out. For the Schwarzschild Painlevé-Gullstrand (PG) slice, whose induced 3-geometry is Euclidean, one recovers the exact Euclidean scale $σ_p r_g \ge π\hbar$, which is optimal among all admissible slices. The entire construction extends across the black-hole horizon, and it transfers to the static axisymmetric Weyl class, where the Hardy floor, strict gap, and AF $π$-scale persist (a global PG-like optimum need not exist).

gr-qc↗

arXiv2025

Robust Areal Thermodynamics of the Schwarzschild Black Hole with Robin Boundary Conditions and Weyl Asymptotics

We formulate an areal thermodynamics for the Schwarzschild black hole that takes the horizon area as the sole macroscopic variable. Quantizing ultrarelativistic interior modes on a regular spacelike slice with a Robin boundary at a stretched horizon leads to a self-adjoint Laplace-Beltrami problem with Heun-type quantization. A maximum-entropy area ensemble introduces an intensive areal temperature $T_A$, and Weyl/heat-kernel asymptotics control the resulting statistical mechanics. The leading equations of state follow universally from the spatial Weyl volume coefficient: in a canonical ensemble of $N$ ultrarelativistic bosons one finds $A = 3 N k_B T_A$ up to a boundary-dependent constant, while in the massless grand-canonical sector $A \propto T_A^{4}$ with a generalized Planck spectrum and a Wien displacement relation. These scaling exponents are insensitive to Dirichlet/Neumann/Robin data and to the foliation; only numerical prefactors vary. Embedding the construction into a static four-dimensional background via Matsubara factorization reproduces the 4D Weyl law and yields a finite matter entropy $S_{\mathrm{rad}} \propto A^{3/4}$, parametrically subleading to the Bekenstein-Hawking term after standard renormalization. The framework provides a concise, mathematically controlled bridge between interior spectral data and macroscopic area relations, clarifying the scope and limitations of areal thermodynamics.

gr-qc↗

arXiv2025

The Extended Uncertainty Principle from a Projector-Valued Measurement Perspective

We revisit the Extended Uncertainty Principle (EUP) from an operational viewpoint, replacing wavefunction-based widths with apparatus-defined position constraints such as a finite slit of width $Δx$ or a geodesic ball of radius $R$. Using Hermitian momentum operators consistent with the EUP algebra, we prove a sharp lower bound on the product of momentum spread and preparation size in one dimension and show that it reduces smoothly to the standard quantum limit as the deformation vanishes. We then extend the construction to the dimensions two and three on spaces of constant curvature and obtain the corresponding bound for spherical confinement, clarifying its geometric meaning via an isometry to $S^2$ and $S^3$. The framework links curvature-scale effects to operational momentum floors and suggests concrete tests in diffraction, cold-atom, and optomechanical settings.

quant-ph↗

arXiv2023

On the experimental verification of the uncertainty principle of position and momentum

Historically, Kennard was the first to choose the standard deviation as a quantitative measure of uncertainty, and neither he nor Heisenberg explicitly explained why this choice should be appropriate from the experimental physical point of view. If a particle is prepared by a single slit of spatial width $Δx$, it has been shown that a finite standard deviation $σ_p<\infty$ can only be ensured if the wave-function is zero at the edge of $Δx$, otherwise it does not exist [8]. Under this circumstances the corresponding sharp inequality is $σ_p Δx\geq π\hbar$. This bound will be reconsidered from the mathematical point of view in terms of a variational problem in Hilbert space and will furthermore be tested in a 4f-single slit diffraction experiment of a laser beam. Our results will be compared with a laser-experiment recently given by Fernández-Guasti (2022) [9].

quant-ph↗

arXiv2022

On momentum operators given by Killing vectors whose integral curves are geodesics

We consider momentum operators on intrinsically curved manifolds. Given that the momentum operators are Killing vector fields whose integral curves are geodesics, it is shown that the corresponding manifold is either flat, or otherwise of compact type with positive constant sectional curvature and dimension equal to 1, 3 or 7. Explicit representations of momentum operators and the associated Casimir element will be discussed for the 3-sphere $S^3$. It will be verified that the structure constants of the underlying Lie algebra are proportional to $2\hbar/R$, where $R$ is the curvature radius of $S^3$. This results in a countable energy and momentum spectrum of freely moving particles in $S^3$. It is shown that the maximum resolution of the possible momenta is given by the de-Broglie wave length $λ_R=πR$, which is identical to the diameter of the manifold. The corresponding covariant position operators are defined in terms of geodesic normal coordinates and the associated commutator relations of position and momentum are established.

quant-ph↗

arXiv2020

On the uncertainty principle in Rindler and Friedmann spacetimes

We revise the extended uncertainty relations for the Rindler and Friedmann spacetimes recently discussed by Dabrowski and Wagner in [9]. We reveal these results to be coordinate dependent expressions of the invariant uncertainty relations recently derived for general 3-dimensional spaces of constant curvature in [10]. Moreover, we show that the non-zero minimum standard deviations of the momentum in [9] are just artifacts caused by an unfavorable choice of coordinate systems which can be removed by standard arguments of geodesic completion.

gr-qc↗

arXiv2018

Uncertainty principle on 3-dimensional manifolds of constant curvature

We consider the Heisenberg uncertainty principle of position and momentum in 3-dimensional spaces of constant curvature $K$. The uncertainty of position is defined coordinate independent by the geodesic radius of spherical domains in which the particle is localized after a von Neumann-Lüders projection. By applying mathematical standard results from spectral analysis on manifolds, we obtain the largest lower bound of the momentum deviation in terms of the geodesic radius and $K$. For hyperbolic spaces, we also obtain a global lower bound $σ_p\geq |K|^\frac{1}{2}\hbar$, which is non-zero and independent of the uncertainty in position. Finally, the lower bound for the Schwarzschild radius of a static black hole is derived and given by $r_s\geq 2\,l_P$, where $l_P$ is the Planck length.

quant-ph↗

arXiv2017

A note on the best invariant estimation of continuous probability distributions under mean square loss

We consider the nonparametric estimation problem of continuous probability distribution functions. For the integrated mean square error we provide the statistic corresponding to the best invariant estimator proposed by Aggarwal (1955) and Ferguson (1967). The table of critical values is computed and a numerical power comparison of the statistic with the traditional Cramér-von Mises statistic is done for several representative distributions.

physics.data-an↗

arXiv2017

On Biased Correlation Estimation

In general, underestimation of risk is something which should be avoided as far as possible. Especially in financial asset management, equity risk is typically characterized by the measure of portfolio variance, or indirectly by quantities which are derived from it. Since there is a linear dependency of the variance and the empirical correlation between asset classes, one is compelled to control or to avoid the possibility of underestimating correlation coefficients. In the present approach, we formalize common practice and classify these approaches by computing their probability of underestimation. In addition, we introduce a new estimator which is characterized by having the advantage of a constant and controllable probability of underestimation. We prove that the new estimator is statistically consistent.

q-fin.ST↗

arXiv2017

Scaling behaviour of entropy estimates

Entropy estimation of information sources is highly non trivial for symbol sequences with strong long-range correlations. The rabbit sequence, related to the symbolic dynamics of the nonlinear circle map at the critical point as well as the logistic map at the Feigenbaum point have been argued to exhibit long memory tails. For both dynamical systems the scaling behavior of the block entropy of order n has been shown to increase like as log(n). In contrast to probabilistic concepts, we investigate the scaling behavior of certain non-probabilistic entropy estimation schemes suggested by Lempel and Ziv in the context of algorithmic complexity and data compression. These are applied in a sequential manner with the scaling variable being the length N of the sequence. We determine the scaling law for the Lempel-Ziv entropy estimate applied to the case of the critical circle map and the logistic map at the Feigenbaum point in a binary partition.

cond-mat.stat-mech↗

arXiv2017

A note on the violation of Bell's inequality

With Bell's inequalities one has a formal expression to show how essentially all local theories of natural phenomena that are formulated within the framework of realism may be tested using a simple experimental arrangement. For the case of entangled pairs of spin-1/2 particles we propose an alternative measurement setup which is consistent to the necessary assumptions corresponding to the derivation of the Bell inequalities. We find that the Bell inequalities are never violated with respect to our suggested measurement process.

physics.gen-ph↗

arXiv2016

The uncertainty principle in terms of isoperimetric inequalities

Simultaneous measurements of position and momentum are considered in $n$ dimensions. We find, that for a particle whose position is strictly localized in a compact domain $D\subset \mathbb{R}^n$ (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by $σ_p \geq λ_1^{1/2}\hbar$, while $λ_1$ is the first Dirichlet eigenvalue of the Laplacian on $D$.

quant-ph↗

arXiv2015

A note on entropy estimation

We compare an entropy estimator $H_z$ recently discussed in [10] with two estimators $H_1$ and $H_2$ introduced in [6][7]. We prove the identity $H_z \equiv H_1$, which has not been taken into account in [10]. Then, we prove that the statistical bias of $H_1$ is less than the bias of the ordinary likelihood estimator of entropy. Finally, by numerical simulation we verify that for the most interesting regime of small sample estimation and large event spaces, the estimator $H_2$ has a significant smaller statistical error than $H_z$.

physics.data-an↗