Geometric Arguments Research Articles

Weighted twisted inhomogeneous diophantine approximation

Abstract We prove a multidimensional weighted analogue of the well-known theorem of Kurzweil (1955) in the metric theory of inhomogeneous Diophantine approximation. Let ∑ i = 1 m α i = m and | ⋅ | α = max 1 ⩽ i ⩽ m | ⋅ | 1 / α i . Given an n-tuple of monotonically decreasing functions Ψ = ( ψ 1 , … , ψ n ) with ψ i : R + → R + such that each ψ i ( r ) → 0 as r → ∞ and fixed A ∈ R n × m define W A ( Ψ ) := { b ∈ [ 0 , 1 ] n : | A i ⋅ q − b i − p i | < ψ i ( | q | α ) ( 1 ⩽ i ⩽ n ) , for infinitely many ( p , q ) ∈ Z n × ( Z m ∖ { 0 } ) } . We prove that the set W A ( Ψ ) has zero-full Lebesgue measure under convergent–divergent sum conditions with some mild assumptions on A and the approximating functions Ψ. We also prove the Hausdorff dimension results for this set. Along with some geometric arguments, the main ingredients are the weighted ubiquity and weighted mass transference principle introduced recently by Kleinbock & Wang (2023 Adv. Math. 428 109154), and Wang & Wu (2021 Math. Ann. 381 243–317) respectively.

SPPs based nano refractive index sensor for sodium and potassium ion concentration detection in human blood

In this paper, a micro-nano sensing configuration constructed from MIM waveguide and a semicircular cavity with annular cavity (SCAC) based on surface plasmon polaritons(SPPs) is submitted. And the sensing property of the waveguide coupling construction is investigated using the finite element method. The optimization emulation results show that the broadband mode generated by the two rectangular cavities on the bus waveguide interacts with the narrowband mode generated by the main construction SCAC, and the radiation spectroscopy of the system appears to be significantly sharp and asymmetric, which is the Fano resonance phenomenon. The impact of external environmental changes on the capability of the sensing structure was investigated by altering the geometric argument of the designed structure afterwards. The final system can achieve a sensitivity of 2600 nm RIU−1 and a figure of merit (FOM) of 47.27. It was then applied to biosensing and its sensitivity as a human sodium and potassium ion concentration detection sensor was investigated and the best can be achieved at 0.415 mgdl−1 and 0.525 mgdl−1.

Lost, hidden, broken, cut-estimating and interpreting the shapes and masses of damaged assemblages of plesiosaur gastroliths.

Gastroliths are stones of uncertain purpose that are commonly found inside the rib cages of plesiosaur fossils worldwide. Gastroliths from four Alberta (Canada) plesiosaurs were studied to determine both their shapes and masses, and their mass fractions relative to body mass. One animal's set of gastroliths was 100% complete and fully visible, but the others showed varying degrees of loss, damage or obscuration, so estimations of their original states were needed. The studied animals were: Albertonectes vanderveldei, Fluvionectes sloanae, Nichollssaura borealis and Wapuskanectes betsynichollsae. The animals come from three different palaeoenvironments: open marine, near shore marine, and fluvial. Gastrolith shapes were classified as either xiphoid, cylindrical, discoidal or spherical based on observed and/or estimated dimensions. Although not all methods could be applied in all cases, gastrolith shapes and masses were estimated four different ways: (1) direct measurement and weighing of a subset and predicting the properties of the remaining obscured and hidden stones; (2) measuring triaxial ellipsoid dimensions of free stones to calculate volumes and multiplying by the mass density of chert; (3) measuring two visible triaxial dimensions of embedded stones, estimating the hidden third dimension three different ways, and then determining volumes and masses by calculation; and (4) predicting the density and mass of a densely packed cluster of small gastroliths using geometrical arguments. Total gastrolith mass never exceeded 0.2% of body mass in any plesiosaur, and is consistent with the idea that the amounts of gastroliths recovered with plesiosaurs would be ineffective as ballast. The largest plesiosaur in the sample had the largest single gastrolith and total gastrolith mass increases with body size. The shape characteristics of the gastroliths were different for different environments, but compositionally they are dominated by black cherts. A possible common source for the gastroliths was identified for the two geographically close and near-contemporanous Nichollssaura and Wapuskanectes.

Predicting Heteropolymer Interactions: Demixing and Hypermixing of Disordered Protein Sequences

Cells contain multiple condensates which spontaneously form due to the heterotypic interactions between their components. Although the proteins and disordered region sequences that are responsible for condensate formation have been extensively studied, the rule of interactions between the components that allow demixing, i.e., the coexistence of multiple condensates, is yet to be elucidated. Here, we construct an effective theory of the interaction between heteropolymers by fitting it to the molecular dynamics simulation results obtained for more than 200 sequences sampled from the disordered regions of human proteins. We find that the sum of amino acid pair interactions across two heteropolymers predicts the Boyle temperature qualitatively well, which can be quantitatively improved by the dimer pair approximation, where we incorporate the effect of neighboring amino acids in the sequences. The improved theory, combined with the finding of a metric that captures the effective interaction strength between distinct sequences, allowed the selection of up to three disordered region sequences that demix with each other in multicomponent simulations, as well as the generation of artificial sequences that demix with a given sequence. The theory points to a generic sequence design strategy to demix or hypermix thanks to the low-dimensional nature of the space of the interactions that we identify. As a consequence of the geometric arguments in the space of interactions, we find that the number of distinct sequences that can demix with each other is strongly constrained, irrespective of the choice of the coarse-grained model. Altogether, we construct a theoretical basis for methods to estimate the effective interaction between heteropolymers, which can be utilized in predicting phase separation properties as well as rules of assignment in the localization and functions of disordered proteins. Published by the American Physical Society 2024

Almost $\iota$-complexes as immersed curves

This paper uses immersed curves to study homology 3-spheres and their corresponding almost \iota -complexes. A new proof of the classification of almost \iota -complexes is given using a geometric argument. This is then used to reprove the existence of an infinite rank summand in the homology cobordism group, \Theta_{\mathbb{Z}}^{3} . This paper further demonstrates a new integer homomorphism from the homology cobordism group, P_{\omega}: \Theta_{\mathbb{Z}}^{3} \rightarrow \mathbb{Z} , conjectured by Dai, Hom, Stoffregen, and Truong. The application of immersed curves to the study of almost \iota -complexes admits several streamlined proofs and could inspire applications to similar problems.

Investigation of thermal damage in explosive bridgewire detonators via discrete element method simulations

AbstractExploding bridgewire (EBW) detonators are used to rapidly and reliably initiate energetic reactions by exploding a bridgewire via Joule heating. While the mechanisms of EBW detonators have been studied extensively in nominal conditions, comparatively few studies have addressed thermally damaged detonator operability. We present a mesoscale simulation study of thermal damage in a representative EBW detonator, using discrete element method (DEM) simulations that explicitly account for individual particles in the pressed explosive powder. We use a simplified model of melting, where solid spherical particles undergo uniform shrinking, and fluid dynamics are ignored. The subsequent settling of particles results in the formation of a gap between the solid powder and the bridgewire, which we study under different conditions. In particular, particle cohesion has a significant effect on gap formation and settling behavior, where sufficiently high cohesion leads to coalescence of particles into a free‐standing pellet. This behavior is qualitatively compared to experimental visualization data, and simulations are shown to capture several key changes in pellet shape. We derive a minimum and maximum limit on gap formation during melting using simple geometric arguments. In the absence of cohesion, results agree with the maximum gap size. With increasing cohesion, the gap size decreases, eventually saturating at the minimum limit. We present results for different combinations of interparticle cohesion and detonator orientations with respect to gravity, demonstrating the complex behavior of these systems and the potential for DEM simulations to capture a range of scenarios.

Geometry of spectral bounds of curves of unitary operators

This article presents a new proof of a theorem concerning bounds of the spectrum of the product of unitary operators and a generalization for differentiable curves of this theorem. The proofs involve metric geometric arguments in the group of unitary operators and the sphere where these operators act.

Multi-entropy at low Renyi index in 2d CFTs

For a static time slice of AdS_33 we describe a particular class of minimal surfaces which form trivalent networks of geodesics. Through geometric arguments we provide evidence that these surfaces describe a measure of multipartite entanglement. By relating these surfaces to Ryu-Takayanagi surfaces it can be shown that this multipartite contribution is related to the angles of intersection of the bulk geodesics. A proposed boundary dual [Phys. Rev. D 106, 126001 (2022), J. High Energy Phys. 08, 202 (2023), J. High Energy Phys. 05, 008 (2023)], the multi-entropy, generalizes replica trick calculations involving twist operators by considering monodromies with finite group symmetry beyond the cyclic group used for the computation of entanglement entropy. We make progress by providing explicit calculations of Renyi multi-entropy in two dimensional CFTs and geometric descriptions of the replica surfaces for several cases with low genus. We also explore aspects of the free fermion and free scalar CFTs. For the free fermion CFT we examine subtleties in the definition of the twist operators used for the calculation of Renyi multi-entropy. In particular the standard bosonization procedure used for the calculation of the usual entanglement entropy fails and a different treatment is required.

A Unified Geometric Proof of Aristarchus’ Inequality

Summary We present a refined, simplified, and unified proof of Aristarchus’ inequality, without resorting to trigonometry, calculus or extended geometric arguments.

Convex bodies associated to linear series of adelic divisors on quasi-projective varieties

AbstractIn this article we define and study convex bodies associated to linear series of adelic divisors over quasi-projective varieties that have been introduced recently by Yuan and Zhang. Restricting our attention to big adelic divisors, we deduce properties of volumes obtained by Yuan and Zhang using different convex geometric arguments. We go on to define augmented base loci and restricted volumes of adelic divisors following the works of Michael Nakamaye and develop a similar study using convex bodies to obtain analogous properties for restricted volumes. We closely follow methods developed originally by Lazarsfeld and Mustaţă

Resolution analysis of a volumetric coded aperture X-ray diffraction imaging system.

A coded aperture X-ray diffraction (XRD) imaging system can measure the X-ray diffraction form factor from an object in three dimensions -X, Y and Z (depth), broadening the potential application of this technology. However, to optimize XRD systems for specific applications, it is critical to understand how to predict and quantify system performance for each use case. The purpose of this work is to present and validate 3D spatial resolution models for XRD imaging systems with a detector-side coded aperture. A fan beam coded aperture XRD system was used to scan 3D printed resolution phantoms placed at various locations throughout the system's field of view. The multiplexed scatter data were reconstructed using a model-based iterative reconstruction algorithm, and the resulting volumetric images were evaluated using multiple resolution criteria to compare against the known phantom resolution. We considered the full width at half max and Sparrow criterion as measures of the resolution and compared our results against analytical resolution models from the literature as well as a new theory for predicting the system resolution based on geometric arguments. We show that our experimental measurements are bounded by the multitude of theoretical resolution predictions, which accurately predict the observed trends and order of magnitude of the spatial and form factor resolutions. However, we find that the expected and observed resolution can vary by approximately a factor of two depending on the choice of metric and model considered. We observe depth resolutions of 7-16 mm and transverse resolutions of 0.6-2 mm for objects throughout the field of view. Furthermore, we observe tradeoffs between the spatial resolution and XRD form factor resolution as a function of sample location. The theories evaluated in this study provide a useful framework for estimating the 3D spatial resolution of a detector side coded aperture XRD imaging system. The assumptions and simplifications required by these theories can impact the overall accuracy of describing a particular system, but they also can add to the generalizability of their predictions. Furthermore, understanding the implications of the assumptions behind each theory can help predict performance, as shown by our data's placement between the conservative and idealized theories, and better guide future systems for optimized designs.

A Theory of Best Choice Selection through Objective Arguments Grounded in Linear Response Theory Concepts

In this study, we propose how to use objective arguments grounded in statistical mechanics concepts in order to obtain a single number, obtained after aggregation, which would allow for the ranking of “agents”, “opinions”, etc., all defined in a very broad sense. We aim toward any process which should a priori demand or lead to some consensus in order to attain the presumably best choice among many possibilities. In order to specify the framework, we discuss previous attempts, recalling trivial means of scores—weighted or not—Condorcet paradox, TOPSIS (Technique for Order Preference by Similarity to Ideal Solution), etc. We demonstrate, through geometrical arguments on a toy example and with four criteria, that the pre-selected order of criteria in previous attempts makes a difference in the final result. However, it might be unjustified. Thus, we base our “best choice theory” on the linear response theory in statistical physics: we indicate that one should be calculating correlations functions between all possible choice evaluations, thereby avoiding an arbitrarily ordered set of criteria. We justify the point through an example with six possible criteria. Applications in many fields are suggested. Furthermore, two toy models, serving as practical examples and illustrative arguments are discussed.

Observations of specimen morphology effects on near-zone-axis convergent-beam electron diffraction patterns.

This work presents observations of symmetry breakages in the intensity distributions of near-zone-axis convergent-beam electron diffraction (CBED) patterns that can only be explained by the symmetry of the specimen and not the symmetry of the unit cell describing the atomic structure of the material. The specimen is an aluminium-copper-tin alloy containing voids many tens of nanometres in size within continuous single crystals of the aluminium host matrix. Several CBED patterns where the incident beam enters and exits parallel void facets without the incident beam being perpendicular to these facets are examined. The symmetries in their intensity distributions are explained by the specimen morphology alone using a geometric argument based on the multislice theory. This work shows that it is possible to deduce nanoscale morphological information about the specimen in the direction of the electron beam - the elusive third dimension in transmission electron microscopy - from the inspection of CBED patterns.

2D lattice animal formation in the irreversible aggregation of patchy particles

With the self-assembly of patchy colloids, we can engineer many complex structures. In the present work, we are modeling spherical particles with 2 diagonally opposite patches using the Kern-Frenkel potential coupled with isotropic square well potential having different interaction ranges. The isotropic square well potential formed reversible bonds while the patches were irreversibly bonded. A state diagram is calculated where we observed a range of structures from a 1D chain, 2D planar cluster, and 3D random clusters depending on the width of the patch. These structures are controlled by the number of bonds each patch can form for a given patch angle. The range of the patch width can be analytically calculated using geometric arguments to identify different types of clusters, which agree with our simulations. The aggregation process was diffusion-limited, forming clusters with fractal dimension 2 in the low volume fraction limit, otherwise known as flocculation region, and 2.5 in the high volume fraction limit or percolation region, irrespective of the type of clusters formed. In the flocculation limit of the DLCA aggregation model, we expect a fractal dimension of 1.8, contrary to that, we observed the formation of lattice animals having a fractal dimension of 2. The presence of lattice animals is confirmed using the scaling laws of size distribution and chemical length of the clusters formed in the flocculation limit.

A Metric Fixed Point Theorem and Some of Its Applications

A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.

A lower bound for the complex flow number of a graph: A geometric approach

AbstractLet be a real number. A complex nowhere‐zero ‐flow on a graph is an orientation of together with an assignment such that, for all , the Euclidean norm of the complex number lies in the interval and, for every vertex, the incoming flow is equal to the outgoing flow. The complex flow number of a bridgeless graph , denoted by , is the minimum of the real numbers such that admits a complex nowhere‐zero ‐flow. The exact computation of seems to be a hard task even for very small and symmetric graphs. In particular, the exact value of is known only for families of graphs where a lower bound can be trivially proved. Here, we use geometric and combinatorial arguments to give a nontrivial lower bound for in terms of the odd‐girth of a cubic graph (i.e., the length of a shortest odd cycle) and we show that this lower bound is tight. This result relies on the exact computation of the complex flow number of the wheel graph . In particular, we show that for every odd , the value of arises from one of three suitable configurations of points in the complex plane according to the congruence of modulo 6.

Depth lower bounds in Stabbing Planes for combinatorial principles

Stabbing Planes (also known as Branch and Cut) is a proof system introduced very recently which, informally speaking, extends the DPLL method by branching on integer linear inequalities instead of single variables. The techniques known so far to prove size and depth lower bounds for Stabbing Planes are generalizations of those used for the Cutting Planes proof system. For size lower bounds these are established by monotone circuit arguments, while for depth these are found via communication complexity and protection. As such these bounds apply for lifted versions of combinatorial statements. Rank lower bounds for Cutting Planes are also obtained by geometric arguments called protection lemmas. In this work we introduce two new geometric approaches to prove size/depth lower bounds in Stabbing Planes working for any formula: (1) the antichain method, relying on Sperner's Theorem and (2) the covering method which uses results on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz. We demonstrate their use on classes of combinatorial principles such as the Pigeonhole principle, the Tseitin contradictions and the Linear Ordering Principle. By the first method we prove almost linear size lower bounds and optimal logarithmic depth lower bounds for the Pigeonhole principle and analogous lower bounds for the Tseitin contradictions over the complete graph and for the Linear Ordering Principle. By the covering method we obtain a superlinear size lower bound and a logarithmic depth lower bound for Stabbing Planes proof of Tseitin contradictions over a grid graph.

Tunneling splittings in the vibrationally excited states of water trimer.

Tunneling splitting (TS) patterns in vibrationally excited states of the water trimer are calculated, taking into account six tunneling pathways that describe the flips of free OH bonds and five bifurcation mechanisms that break and reform hydrogen bonds in the trimer ring. A tunneling matrix (TM) model is used to derive the energy shifts due to tunneling in terms of the six distinct TM elements in symbolic form. TM elements are calculated using the recently-developed modified WKB (Wentzel-Kramers-Brillouin) method in full dimensionality. Convergence was achieved for the lowest six excited vibrational modes. Bifurcation widths of the pseudorotational quartets are shown to be of comparable size to the ground-state widths, obtained using instanton theory, or increased for some particular modes of vibration. The largest increase is obtained for the excited out-of-phase flip of two adjacent water monomers with free OH bonds pointing in opposite directions relative to the ring plane. Bifurcation widths in (D2O)3 are found to be two orders of magnitude smaller than in (H2O)3. Geometrical arguments were used to explain the order of states in some TS multiplets in vibrationally excited water trimers.

Unidirectional Freezing of Polymer Solution Droplets.

Ice templating provides a means of generating textures with a well-defined topography. Recent applications involve the freezing of water droplets, with or without colloids, on flat or textured surfaces. An interesting feature of water droplets freezing on a substrate is the formation of a pointy tip at a constant angle, regardless of the substrate temperature, surface energy, or droplet volume. Here, by adding the polymer to water, we demonstrate how to manipulate and even prevent the formation of such an icy tip. We find that the sharpness of the tip decreases with increasing polymer concentration until completely disappearing above the overlap concentration, while the total freezing time increases concomitantly. Building on these observations, we combined simple geometrical arguments with heat flux measurements to model and connect the spatial and temporal evolution of polymer droplets under unidirectional freezing. Together our results provide new ways to control the shape of frozen droplets for ice templating or microstructure fabrication, with applications in tissue engineering, separation membranes, and soft robotics.

A SCALING LAW RELATING THE RATE OF DESTRUCTION OF A SOLID TUMOR AND THE FRACTAL DIMENSION OF ITS BOUNDARY

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.