Lattice Points Research Articles

Local Euler characteristics of $A_n$-singularities and their application to hyperbolicity

Wahl's local Euler characteristic measures the local contributions of a singularity to the usual Euler characteristic of a sheaf. Using tools from toric geometry, we study the local Euler characteristic of sheaves of symmetric differentials for isolated surface singularities of type $A_n$. We prove an explicit formula for the local Euler characteristic of the $m$th symmetric power of the cotangent bundle; this is a quasi-polynomial in $m$ of period $n+1$. We also express the components of the local Euler characteristic as a count of lattice points in a non-convex polyhedron, again showing it is a quasi-polynomial. We apply our computations to obtain new examples of algebraic quasi-hyperbolic surfaces in $\mathbb{P}^3$ of low degree. We show that an explicit family of surfaces with many singularities constructed by Labs has no genus $0$ curves for the members of degree at least $8$ and no curves of genus $0$ or $1$ for degree at least $10$.Comment: 28 pages, 7 figures

N-Dimensional Lattice Path Enumeration

Let V_n be a set of integer vectors. We enumerate lattice paths that only uses vectors in V_n. Unlike most lattice path enumeration problems, the number of dimensions isn't fixed and the vector set is dependent on the dimension. This requires us to follow a different approach in explicitly expressing the number of lattice paths from origin to any point in $n$-dimensional space. We notice that a special case of this problem corresponds to Fubini numbers, which count the number of weak orderings of a set consisting of $n$ elements. Then, we find the recursive relation of this sequence. Finally, we develop an algorithm that can be used to find the number of paths between any two points that do not touch the lattice points in R. The crucial part of our algorithm is that it doesn't rely on finding all paths and checking each path for usage of restricted points.

A central limit theorem for coefficients of L$L$‐functions in short intervals

AbstractAssuming the generalized Lindelöf hypothesis (GLH), a weak version of the generalized Ramanujan conjecture and a Rankin–Selberg type partial sum estimate, we establish the normality of the sum of coefficients of a general ‐function in short intervals of appropriate length. The novelty lies in the degree aspect under GLH. In particular, this generalizes the result of Hughes and Rudnick on lattice point counts in thin annuli.

Simulation of a liquid drop on a soft substrate.

A liquid drop resting on a soft substrate is numerically simulated as an energy minimization problem. The elastic substrate is modeled as a cubic lattice of mass-springs, to which an energy term controlling the change of volume is associated. The interfacial energy between three phases of solid, liquid, and vapor is also introduced. Under the constant volume constraint of the liquid drop, the total energy of the system is subjected to a numerical minimization process by which profiles of both substrate and drop are obtained. The numerical simulation enables the modeling of the wetting setup with various parameters associated with solid and liquid phases, including the Young's modulus and Poisson's ratio of the solid, the surface tension of the three phases, and geometrical parameters such as the contact radius or thickness of the solid. The direct outputs of the minimization process are the displacements of the solid lattice and boundary points of the liquid, by which the behavior of all relevant quantities, such as contact angles in the three phases, as well as the effective surface tension of the solid, can be quantitatively studied. The resulting displacements of the solid are compared with the exact solution of the elasticity equation under the assumption of no tangential traction, and a quite satisfactory agreement is observed. However, at larger Young's modulus or lower Poisson's ratios the agreement between the numerical results and the analytical solutions is lost in the vicinity of the contact points. Interestingly, a non-zero tangential traction in the vicinity of contact points is calculated by the numerical outputs, indicating that the assumption of zero tangential traction is not valid generally around the contact points. The effective surface tension at the contact points is calculated for an incompressible solid substrate, showing a linear increase with respect to the Young's modulus of the solid, as .

Doubling the rate: improved error bounds for orthogonal projection with application to interpolation

Convergence rates for L2 approximation in a Hilbert space H are a central theme in numerical analysis. The present work is inspired by Schaback (Math Comp, 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the best rate for functions in the “native space” H. Motivated by this, we obtain a general result for H-orthogonal projection onto a finite dimensional subspace of H: namely, that any known L2 convergence rate for all functions in H translates into a doubled L2 convergence rate for functions in a smoother normed space B, along with a similarly improved error bound in the H-norm, provided that L2, H and B are suitably related. As a special case we improve the known L2 and H-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the L2 convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space B.

Defect Chemistry Engineering to Regulate the Excellent ZTave Value of Nonstoichiometric Ca3-xCo4O9 (0 ≤ x ≤ 0.06) Ceramics.

Cobalt-based oxides have attracted significant attention as p-type thermoelectric materials due to their wide operational temperature range. However, their low average figure of merit (ZTave) value has hindered service performance. A series of cation vacancies as Ca-active sites were introduced into Ca3-xCo4O9 (0 ≤ x ≤ 0.06) by defect chemistry engineering to regulate the excellent ZTave value. The Ca-active sites of Ca3-xCo4O9 ceramics induced lattice distortion and point defects, which were unequivocally confirmed by the iDPC-STEM results. The power factor from 0.18 mW/(m·K2) to 0.38 mW/(m·K2) and the electrical conductivity from 54.8 to 108.3 S/cm were achieved for the Ca2.96Co4O9 sample. A notable ZT value of 0.32 was obtained at 1073 K. Furthermore, the high ZTave of approximately 0.166 reached within the temperature range of 323-1073 K, representing a 3.25 times improvement compared to pure Ca3Co4O9 ceramics. This study highlights defect chemistry engineering as an effective strategy for optimizing the thermoelectric performance of nonstoichiometric Ca3Co4O9 ceramics, offering promising prospects for the development of Ca3Co4O9-based thermoelectric materials.

Constraining primordial non-Gaussianity with Density-Split Clustering

Obtaining tight constraints on primordial non-Gaussianity (PNG) is a key step in discriminating between different models for cosmic inflation. The constraining power from large-scale structure (LSS) measurements is expected to overtake that from cosmic microwave background (CMB) anisotropies with the next generation of galaxy surveys including the Dark Energy Spectroscopic Instrument (DESI) and Euclid. We consider whether Density-Split Clustering (DSC) can help improve PNG constraints from these surveys for local, equilateral and orthogonal types. DSC separates a surveyed volume into regions based on local density and measures the clustering statistics within each environment. Using the Quijote simulations and the Fisher information formalism, we compare PNG constraints from the standard halo power spectrum, DSC power spectra and joint halo/DSC power spectra. We find that the joint halo/DSC power spectra outperform the halo power spectrum by factors of ∼ 1.4, 8.8, and 3.6 for local, equilateral and orthogonal PNG, respectively. This is driven by the higher-order information that DSC captures on small scales. We find that applying DSC to a halo field does not allow sample variance cancellation on large scales by providing multiple tracers of the same volume with different local PNG responses. Additionally, we introduce a Fourier space analysis for DSC and study the impact of several modifications to the pipeline, such as varying the smoothing radius and the number of density environments and replacing random query positions with lattice points.

Substitutional Platinum‐Phosphorus Solid Solution by Phosphidation of Worm‐Like Pt Nanoparticles Using Tri‐n‐octylphosphine

AbstractHerein, the investigations into the phosphidation of platinum (Pt) nanoparticles are reported using tri‐n‐octylphosphine (TOP) at elevated temperature in an organic solvent, and identify a unique phenomenon not addressed before: The phosphorus (P) atoms can replace partial Pt atoms from their lattice points to form a substitutional Pt‐P solid solution. The rationality is authenticated for forming substitutional Pt–P solid solution via P doping by various characterizations and density functional theory (DFT) calculations, both of which suggests that the maximum P content in the Pt–P solid solution is approximately ≈10% for maintaining the stability of the face‐centered cubic crystal structure.

On-Chip Photonic Simulating Band Structures toward Arbitrary-Range Coupled Frequency Lattices.

Photonic simulators are increasingly used to study physical systems for their affluent manipulable degrees of freedom. The advent of photonic chips offers a promising path towards compact and configurable simulators. Thin-film lithium niobate chips are particularly well suited for this purpose due to the high electro-optic coefficient, which allows for the creation of lattices in the frequency domain. Here, we fabricate and periodically modulate an on-chip resonator to observe band structures. The employed modulation rates are lower than the resonator linewidth, resulting in the inclusion of multiple lattice points within one resonant peak. This alleviates the difficulty of applying and detecting multiharmonic signals which are conventionally of ultrahigh frequency on chips and enables us to simulate structures with arbitrary-range coupling. As examples, we showcase the simulation of nanotubes along several directions where the required frequencies are reduced by more than 3 orders of magnitude (up to reduce near 100GHz to around 10MHz in our examples). Encompassing various models equipped with a gauge potential, our experiments demonstrate an effective and technically feasible scenario which may bolster the development of on-chip photonic simulators complementing existing techniques.

Decisive role of organic fluorophore and surface defect state in the photoluminescence of carbon quantum dots

Carbon quantum dots (CQDs) hold significant promise for applications in biological imaging, sensing, and optoelectronic devices owing to their superior photostability and low toxicity. Nevertheless, the elucidation of their photoluminescence mechanism remains an open question, necessitating further comprehensive investigation. In this Letter, CQDs exhibiting ultraviolet (UV) and white fluorescence were isolated through silica gel column chromatography separation of the crude product obtained from a one-step solvothermal synthesis. CQDs with different luminescent properties exhibit the same crystal structure and similar particle size distributions. Both CQDs exhibit orthorhombic structure where C60/C70 molecules are located at lattice points, having average particle sizes of 2.71 and 2.98 nm, respectively. Consequently, the luminescent properties of the synthesized CQDs are predominantly governed by their surface structure. The results of microstructure characterization and spectroscopic analysis demonstrate that the UV emission originates from the C(=O)OH and C–O–C related luminescent moieties within organic fluorophores, and the blue emission band is attributed to defect states related to surface group C–O–C, while the green/yellow emission arises from C(=O)O related surface defect levels. These observations have gained a profound understanding of the luminescent genesis of CQDs, broadened the luminescence coverage wavelength range of CQDs, and enriched the family of CQDs materials.

A novel deterministic sampling approach for the reliability analysis of high-dimensional structures

Overcoming the “curse of dimensionality” in high-dimensional reliability analysis is still an enduring challenge. This paper proposes an innovative deterministic sampling method designed to overcome this challenge. The approach starts with a two-dimensional uniform point set, generated using the good lattice point method. This set is then refined through the cutting method to produce a specific number of points. A novel generating vector is computed based on this method, enabling the generation of the targeted high-dimensional point set through a strategic dimension-by-dimension mapping. Notably, this method eliminates the need for complex congruence computation and primitive root optimization, enhancing its efficiency for high-dimensional sampling. The resulting point set is deterministic and uniform, greatly reducing variability in reliability analysis. Then, the proposed approach is integrated into the fractional exponential moment-based maximum entropy method with the Box–Cox transform. This integration efficiently recovers the probability distribution for the limit state function (LSF) with high-dimensional inputs, enabling precise assessment of the failure probability. The efficacy of the proposed method is demonstrated through three high-dimensional numerical examples, involving both explicit and implicit LSFs, highlighting its applicability for high-dimensional reliability analysis of structures.

Observation of higher-order gyrotropic modes and energy transfer in cylindrical ferromagnetic nanodots-based square lattices

We investigated and reported on a wide variety of gyrotropic modes in both isolated thick cylindrical magnetic vortices and lattices of such magnetic vortices arranged in square-lattice structures. In significantly thicker magnetic vortices, we see a higher-order flexure modes in addition to the uniform gyrotropic modes. We thoroughly examined the variation of the frequency and intensity of these modes with the thickness of the magnetic nanodots forming a vortex. We specifically looked at how the thickness, diameter, and aspect ratio of a single magnetic vortex and magnetic vortex lattice in various configurations affect the mode frequencies. When the magnetic vortex at the center is excited by an a.c. spin-polarized current with an excitation frequency equal to the collective gyrotropic frequencies of the lattice, we discuss the dynamics of the transfer of magnetic energy at three high symmetry points of the three-dimensional lattice of Permalloy nanodots with a vortex structure. The energy transmission in these lattice systems is estimated using the power and phase distributions for such configurations. Lattice configurations with substantially thicker nanodots consisting of uniform modes and higher-order flexure modes in the Eigen spectrum have transmission with uniform modes in specific directions and suppression in others, acting as a magnonic filter, and no transmission at all with higher-order modes. Such magnonic vortex lattice structures seek to improve the future prospects of technology and provide emerging applications in information processing devices, magnonic filters, three-dimensional waveguides, and magnonic crystals.

On accumulated spectrograms for Gabor frames

Analogs of classical results on accumulated spectrograms, the sum of spectrograms of eigenfunctions of localization operators, are established for Gabor multipliers. We show that the lattice ℓ1 distance between the accumulated spectrogram and the indicator function of the Gabor multiplier mask is bounded by the number of lattice points near the boundary of the mask and that this bound is sharp in general. The methods developed for the proofs are also used to show that the Weyl-Heisenberg ensemble restricted to a lattice is hyperuniform when the Gabor frame is tight.

Local structural disorder introduced by Cr in fcc high-/medium-entropy alloys consisting of 3d transition metal elements

High-/medium-entropy alloys (H/MEAs) have attracted attention for their excellent mechanical properties. According to first-principles calculations, the atomic displacements from their lattice points are element-dependent and correlated with the yield strength of H/MEAs. To investigate experimentally the element dependence of the local structure, we measured the extended x-ray absorption fine structure in seven kinds of face-centered cubic H/MEAs consisting of 3d transition metal elements. Our experimental results indicate that the local disorder around chromium (Cr) is larger than that around other elements, which is a common feature in all the measured fcc H/MEAs, aligning with prior theoretical studies indicating larger displacements of Cr in comparison to other elements. We discuss the mechanism underlying the local structural disorder induced by the constituent element Cr.

Research on match fluctuation prediction and strategy based on OFS algorithm

The aim of this paper is to construct a model for predicting on-field fluctuations in matches by using OFS algorithm, combined with random forest and lattice point search algorithms, and to analyse the key factors in order to improve the coach's decision-making ability in matches. Cross-validation and hyper-parameter tuning techniques were used during the model building process to find the best hyper-parameter combinations through lattice point search to optimise the model performance and generalisation ability. The study provides a detailed description of the model implementation process, including data processing, feature selection, parameter setting, model building, training and evaluation. Meanwhile, a series of targeted recommendations are proposed to address the variability of match fluctuations, including the analysis of past matches, the focus on recent performance, the assessment of psychological quality, the study of opponent's weaknesses, the flexible response to changes in matches and the maintenance of focus, with a view to helping athletes and coaches to formulate a more effective match strategy and to achieve better match results.

Some Improvements on Good Lattice Point Sets.

Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. The generalized good lattice point (GGLP) sets obtained from linear-level permutations of GLP sets have demonstrated that the permutation does not reduce the criterion maximin distance. This paper confirms that linear-level permutation may lead to greater mixture discrepancy. Nevertheless, GGLP sets can still enhance the space-filling property of GLP sets under various criteria. For small-sized cases, the KL divergence from the uniform distribution of GGLP sets is lower than that of the initial GLP sets, and there is nearly no difference for large-sized points, indicating the similarity of their distributions. This paper incorporates a threshold-accepting algorithm in the construction of GGLP sets and adopts Frobenius distance as the space-filling criterion for large-sized cases. The initial GLP sets have been included in many monographs and are widely utilized. The corresponding GGLP sets are partially included in this paper and will be further calculated and posted online in the future. The performance of GGLP sets is evaluated in two applications: computer experiments and representative points, compared to the initial GLP sets. It shows that GGLP sets perform better in many cases.

Photonic crystal fibers based on Dirac point-guiding

—In this work photonic crystal fibers of different cross-sectional patterns are studied. Dirac points of these various lattices are explored, propagation diagrams showing positions of the Dirac spectrums are obtained, and their application in fiber guiding is discussed. A quasi-3D FDTD simulation method, which is simpler and more efficient than a commercial 3D FDTD simulation software, is developed for photonic crystal fiber. This method is then applied to the new photonic crystal fiber proposed in [Fiber guiding at the Dirac frequency beyond photonic bandgaps, Light Sci. & App. 4 (2015) e304.]. Dirac-point guidance, which is based on Dirac localization of photons, is verified for these fibers.

Analysis of differential scanning calorimetry data for aged plutonium.

Differential scanning calorimetry data for samples of a 52 year old plutonium alloy with 3.3 at. % Ga that were heated beyond the melting point is analyzed using transition state theory to find activation energies for the δ to ɛ and ɛ to liquid phase transitions. A Bayesian statistical method involving a Gaussian process model is used to find mean values and confidence intervals for the activation energies. The activation energy for the δ to ɛ phase transition increases by 3.3 ± 3.8% per decade, relative to the case when all age related plutonium lattice point defects have been removed through annealing. The corresponding increase in activation energy for the ɛ to liquid transition is shown to be 7.1 ± 1.8% per decade. It is postulated that the change in activation energy with age for both phase transitions is caused, in part, by the accumulation of the same type of lattice point defects associated with the observed increase in elastic bulk modulus over time.

Semistable torsion classes and canonical decompositions in Grothendieck groups

AbstractWe study two classes of torsion classes that generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen–Fei. More strongly, for ‐tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen–Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type .

Lattice points on polyominoes of inversion sequences

Inversion sequences of length n are positive integer sequences e 1 e 2 …e n such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations.