Arbitrary Dimension Research Articles

Finite element approximation of scalar curvature in arbitrary dimension

We analyze finite element discretizations of scalar curvature in dimension N ≥ 2 N \ge 2 . Our analysis focuses on piecewise polynomial interpolants of a smooth Riemannian metric g g on a simplicial triangulation of a polyhedral domain Ω ⊂ R N \Omega \subset \mathbb {R}^N having maximum element diameter h h . We show that if such an interpolant g h g_h has polynomial degree r ≥ 0 r \ge 0 and possesses single-valued tangential-tangential components on codimension-1 simplices, then it admits a natural notion of (densitized) scalar curvature that converges in the H − 2 ( Ω ) H^{-2}(\Omega ) -norm to the (densitized) scalar curvature of g g at a rate of O ( h r + 1 ) O(h^{r+1}) as h → 0 h \to 0 , provided that either N = 2 N = 2 or r ≥ 1 r \ge 1 . As a special case, our result implies the convergence in H − 2 ( Ω ) H^{-2}(\Omega ) of the widely used “angle defect” approximation of Gaussian curvature on two-dimensional triangulations, without stringent assumptions on the interpolated metric g h g_h . We present numerical experiments that indicate that our analytical estimates are sharp.

Regularized FJNW metric in arbitrary dimensions

Regularized FJNW metric in arbitrary dimensions

Enhancing some separability criteria via equiangular tight frames

Abstract Improving the detection ability of quantum entanglement is one of the es-
sential tasks in quantum computing and quantum information. Finite tight frames
play a fundamental role in a wide variety of areas, and generally, each application
requires a specific class of frames and is closely related to quantum measurement. It
is worth noting that a maximal set of complex equiangular vectors is closely related
to a symmetric informationally complete measurement. Hence, our goal in this work
is to propose a series of separability criteria assigned to a finite tight frame and some
well-known inequalities in different quantum systems, respectively. In addition, some
tighter criteria to detect entanglement for many-body quantum states are presented in
arbitrary dimensions. Finally, the effectiveness of the proposed entanglement detec-
tion criteria is illustrated through some detailed examples.

Exploring Nonlocal Correlations in Arbitrarily High-Dimensional Systems

Abstract The utilization of higher-dimensional systems holds promise for unveiling novel phenomena and enhancing the efficacy of practical tasks, such as entanglement-based quantum cryptography. However, detecting quantum correlations within qudit systems poses a formidable challenge. In this study, we delve into the intricacies of Bell’s nonlocality within higher-dimensional systems. We introduce a novel correlation function and leverage it to construct a family of Bell inequalities adapted for quantum systems of arbitrary dimensionality. By gauging the probability of local realism violation under random measurements as our primary metric, we explore the relation between pure state entanglement and nonlocality. Our findings align closely with predictions derived from the comprehensive polytope analysis via linear programming methods. While our focus predominantly centers on the two-qudit scenario, our methodology readily extends to the N -partite case, accommodating an arbitrary number of measurements per party.

The maxcut of the sunrise with different masses in the continuous Minkoskean dimensional regularisation

We evaluate the maxcut of the two loops sunrise amplitude with three different masses by direct use of the loop momenta in the Minkoskean (as opposed to the usual Euclidean) continuous dimension regularisation, obtaining in that way six related but different functions expressed in the form of one dimensional finite integrals. We then consider the 4th order homogeneous equation valid for the maxcut, and show that for arbitrary dimension d the six functions do satisfy the equation separately. We further discuss the d=2,3,4 cases, verifying that only four of them are linearly independent. The equal mass limit is also shortly considered.

A Dirichlet Process Model for Directional-linear Data with Application to Bloodstain Pattern Analysis

Directional data require specialized models because of the non-Euclidean nature of their domain. When a directional variable is observed jointly with linear variables, modeling their dependence adds an additional layer of complexity. A Bayesian nonparametric approach is introduced to analyze directional-linear data. Firstly, the projected normal distribution is extended to model the joint distribution of linear variables and a directional variable with arbitrary dimension projected from a higher-dimensional augmented multivariate normal distribution. The new distribution is called the semi-projected normal distribution (SPN) and can be used as the mixture distribution in a Dirichlet process model to obtain a more flexible class of models for directional-linear data. Then, a conditional inverse-Wishart distribution is proposed as part of the prior distribution to address an identifiability issue inherited from the projected normal and preserve conjugacy with the SPN. The SPN mixture model shows superior performance in clustering on synthetic data compared to the semi-wrapped Gaussian model. The experiments show the ability of the SPN mixture model to characterize bloodstain patterns. A hierarchical Dirichlet process model with the SPN distribution is built to estimate the likelihood of bloodstain patterns under a posited causal mechanism for use in a likelihood ratio approach to the analysis of forensic bloodstain pattern evidence.

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Warped time-frequency systems have recently been introduced as a class of structured continuous frames for functions on the real line. Herein, we generalize this framework to the setting of functions of arbitrary dimensionality. After showing that the basic properties of warped time-frequency representations carry over to higher dimensions, we determine conditions on the warping function which guarantee that the associated Gramian is well-localized, so that associated families of coorbit spaces can be constructed. We then show that discrete Banach frame decompositions for these coorbit spaces can be obtained by sampling the continuous warped time-frequency systems. In particular, this implies that sparsity of a given function f in the discrete warped time-frequency dictionary is equivalent to membership of f in the coorbit space. We put special emphasis on the case of radial warping functions, for which the relevant assumptions simplify considerably.

Unified approach to reciprocal matrices with Kippenhahn curves containing elliptical components

We consider tridiagonal matrices ( a ij ) i , j = 1 n with constant main diagonal and such that a i , i + 1 a i + 1 , i = 1 for i = 1 , … , n − 1 . For these matrices, criteria are established under which their Kippenhahn curves contain elliptical components or even consist completely of such. These criteria are in terms of a system of homogeneous polynomial equations in variables ( | a j , j + 1 | − | a j + 1 , j | ) 2 , and established via a unified approach across arbitrary dimensions. The results are illustrated, and specific numerical examples are provided for n = 7, thus generalizing earlier work in the lower-dimensional setting.

Winning of inhomogeneous bad for curves

AbstractWe prove the absolute winning property of weighted inhomogeneous badly approximable vectors on nondegenerate analytic curves. This answers a question by Beresnevich, Nesharim, and Yang. In particular, our result is an inhomogeneous version of the main result in (Duke Math J 171(14), 2022) by Beresnevich, Nesharim, and Yang. Also, the generality of the inhomogeneous part that we considered extends the previous result in An et al. (Adv Math 324:148–202, 2018). Moreover, our results even contribute to classical results, namely establishing the inhomogeneous Schmidt’s conjecture in arbitrary dimensions.

Efficient Sampling From the Watson Distribution in Arbitrary Dimensions

In this paper, we present two efficient methods for sampling from the Watson distribution in arbitrary dimensions. The first method adapts the rejection sampling algorithm from Kent et al. (2018), originally designed for Bingham distributions, using angular central Gaussian envelopes. For the Watson distribution, we derive a closed-form expression for the parameters that maximize sampling efficiency, which is further investigated and bounded by asymptotic results. This approach avoids the curse of dimensionality through a smart matrix inversion, enabling fast runtimes even in high dimensions. The second method, based on Saw (1978), employs adaptive rejection sampling from a projected distribution. This algorithm is also effective in all dimensions and offers rapid sampling capabilities. Finally, our simulation study compares the two main methods, revealing that each excels under different conditions: the first method is more efficient for small samples or large dimensions, while the second performs better with larger samples and more concentrated distributions. Both algorithms are available in the R package watson on CRAN. Supplementary materials for this article are available online.

Anomalies in covariant fracton theories

Covariant (Lorentz invariant) fracton matter, minimally coupled and charged under a symmetric rank two gauge tensor, is considered. The gauge transformations correspond to linearized longitudinal diffeomorphisms. Consistent possible anomalies are computed using the cohomology method. They depend only on the gauge field, treated as a background field, and on the gauge parameter, promoted to an anticommuting scalar ghost field. The problem is phrased in terms of polyforms, whose total degree is the sum of the form degree and of the ghost number. The most general anomaly in two dimensions and in four dimensions is computed, and an anomaly in arbitrary dimensions is individuated. In conclusion, it is shown that a simple higher-derivative scalar field theory is an example of covariant fracton matter. Published by the American Physical Society 2024

Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries

AbstractWhile there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves and combustion theory. For such critical points —which can be obtained as limits of classical solutions or limits of a singular perturbation problem—it has been open since (Weiss, 2003) whether the singular set can be large and what equation the measure satisfies, except for the case of two dimensions. In the present result we use recent techniques such as a frequency formula for the Bernoulli problem as well as the celebrated Naber–Valtorta procedure to answer this more than 20 year old question in an affirmative way: For a closed class we call variational solutions of the Bernoulli problem, we show that the topological free boundary (including degenerate singular points , at which as ) is countably ‐rectifiable and has locally finite ‐measure, and we identify the measure completely. This gives a more precise characterization of the free boundary of in arbitrary dimension than was previously available even in dimension two. We also show that limits of (not necessarily minimizing) classical solutions as well as limits of critical points of a singularly perturbed energy are variational solutions, so that the result above applies directly to all of them.

General features and contact modeling of $N$-body isolated resonances near threshold

NN-body non efimovian bound or quasi-bound states for particles with short range interactions are considered in arbitrary dimensions. The different resonance regimes near the threshold are depicted by using a generalization of the effective range approximation. This two-parameter description can be used in various contexts from ultracold to hadronic physics. The universal character of these states makes it possible a formulation in terms of a contact model. The singularity at the contact imposes the introduction of a modified scalar product to solve the normalization catastrophe and to restore the self-adjoint character of the model. An equivalence with the standard scalar product used for realistic finite range models is derived.

Can a chemotaxis-consumption system recover from a measure-type aggregation state in arbitrary dimension?

We consider the chemotaxis-consumption system \[ ( ⋆ ) { u t a m p ; = Δ u − χ ∇ ⋅ ( u ∇ v ) v t a m p ; = Δ v − u v \begin {cases} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) \\ v_t &= \Delta v - uv \end {cases} \tag {$\star $} \] in a smooth bounded domain Ω ⊆ R n \Omega \subseteq \mathbb {R}^n , n ≥ 2 n \geq 2 , with parameter χ > 0 \chi > 0 and Neumann boundary conditions. It is well known that, for sufficiently smooth nonnegative initial data and under a smallness condition for the initial state of v v , solutions of the above system never blow up and are even globally bounded. Going in a sense a step further in this paper, we ask the question whether the system can even recover from an initial state that already resembles measure-type blowup. To answer this, we show that, given an arbitrarily large positive Radon measure u 0 u_0 with u 0 ( Ω ¯ ) > 0 u_0(\overline {\Omega }) > 0 as the initial data for the first equation and a nonnegative L ∞ ( Ω ) L^\infty (\Omega ) function v 0 v_0 with \[ 0 > ‖ v 0 ‖ L ∞ ( Ω ) > π χ 2 n 0 > \|v_0\|_{L^{\infty }(\Omega )} > \frac {\pi }{\chi } \sqrt {\frac {2}{n}} \] as initial data for the second equation, it is still possible to construct a global classical solution to the above system. Notably, the above condition on the initial data appears to be weaker than those required in all previous works on ( ⋆ \star ) even in frameworks of smooth initial data.

An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension

An induced limitation in the reconstruction step for Euler equations of compressible gas dynamics in arbitrary dimension

A new algorithm for Gröbner bases conversion

A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.The new algorithm has been implemented in Maple and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in Maple. In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.

Controlling and optimizing the transport (search) efficiency with local information on a class of scale-free trees.

The scale-free trees are fundamental dynamics networks with extensive applications in material and engineering fields owing to their high reliability and low power consumption characteristics. Controlling and optimizing transport (search) efficiency on scale-free trees has attracted much attention. In this paper, we first introduce degree-dependent weighted tree by assigning each edge (x,y) a weight wxy=(dxdy)θ, with dx and dy being the degree of nodes x and y, and θ being a controllable parameter. Then, we design a parameterized biased random walk strategy with the transition probability depending on the local information (the degree of neighboring nodes) and a parameter θ. Finally, we evaluate analytically the global mean first-passage time, which is an important indicator for measuring the transport (search) efficiency on the underlying networks, and find the interval for parameter θ where transport (search) efficiency can be improved on a class of scale-free trees. We also analyze the (transfinite) walk dimension for our biased random walk on the scale-free trees and find one can obtain arbitrary transfinite walk dimension in an interval by properly tuning the biased parameter θ. The results obtained here would shed light on controlling and optimizing transport (search) efficiency on objects with scale-free tree structures.

Tempered fractional differential equations on hyperbolic space

Abstract In this paper we study linear fractional differential equations involving tempered Caputo-type derivatives in the hyperbolic space. We consider in detail the three-dimensional case for its simple and useful structure. We also discuss the probabilistic meaning of our results in relation to the distribution of an hyperbolic Brownian motion time-changed with the inverse of a tempered stable subordinator. The generalization to an arbitrary dimension n can be easily obtained. We also show that it is possible to construct a particular solution for the non-linear porous-medium type tempered equation by using elementary functions.

Magnetic Curvature and Existence of a Closed Magnetic Geodesic on Low Energy Levels

Abstract To a Riemannian manifold $(M,g)$ endowed with a magnetic form $\sigma $ and its Lorentz operator $\Omega $ we associate an operator $M^{\Omega }$, called the magnetic curvature operator. Such an operator encloses the classical Riemannian curvature of the metric $g$ together with terms of perturbation due to the magnetic interaction of $\sigma $. From $M^{\Omega }$ we derive the magnetic sectional curvature $\textrm{Sec}^{\Omega }$ and the magnetic Ricci curvature $\textrm{Ric}^{\Omega }$ that generalize in arbitrary dimension the already known notion of magnetic curvature previously considered by several authors on surfaces. On closed manifolds, under the assumption of $\textrm{Ric}^{\Omega }$ being positive on an energy level below the Mañé critical value, with a Bonnet–Myers argument, we establish the existence of a contractible periodic orbit. In particular, when $\sigma $ is nowhere vanishing, this implies the existence of a contractible periodic orbit on every energy level close to zero. Finally, on closed oriented even dimensional manifolds, we discuss about the topological restrictions that appear when one requires $\textrm{Sec}^{\Omega }$ to be positive.

Exactly solvable Stuart-Landau models in arbitrary dimensions.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equationsin the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same. Oscillatory dynamics is asymptotically confined to a hypersphere S^{D-1} and is characterised by extreme multistability, namely the coexistence of an infinite number of limiting orbits, each of which has the geometry of a torus T^{N} on which the motion is either periodic or quasiperiodic. We also comment on similar Clifford extensions of other limit cycle oscillator systems and their generalizations.