Geometric Arguments Research Articles

Asymptotic Behavior of Fronts and Pulses of the Bidomain Model

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we investigate the instability and asymptotic behavior of planar fronts and planar pulses of the bidomain Allen-Cahn equation and the bidomain FitzHugh-Nagumo equation in two spatial dimension. In previous work, it was shown that planar fronts of the bidomain Allen-Cahn equation can become unstable in contrast to the classical Allen-Cahn equation. We find that, after the planar front is destabilized, a rotating zigzag front develops whose shape can be explained by simple geometric arguments using a suitable Frank diagram. We also show that the Hopf bifurcation through which the front becomes unstable can be either supercritical or subcritical, by demonstrating a parameter regime in which a stable planar front and zigzag front can coexist. In our computational studies of the bidomain FitzHugh-Nagumo pulse solution, we show that the pulses can also become unstable much like the bidomain Allen-Cahn fronts. However, unlike the bidomain Allen-Cahn case, the destabilized pulse does not necessarily develop into a zigzag pulse. For certain choice of parameters, the destabilized pulse can disintegrate entirely. These studies are made possible by the development of a numerical scheme that allows for the accurate computation of the bidomain equation in a two dimensional strip domain of infinite extent.

The evolution of the heaviest supermassive black holes in jetted AGNs

ABSTRACT We present the space density evolution, from z = 1.5 up to z = 5.5, of the most massive ($M \ge 10^9\, \mathrm{M}_{\odot }$) black holes hosted in jetted active galactic nuclei (AGNs). The analysis is based on a sample of 380 luminosity-selected (λL1350 ≥ 1046 erg s−1 and $P_{5\, \text{GHz}}\ge 10^{27}$ W Hz−1) flat spectrum radio quasars (FSRQs) obtained from the Cosmic Lens All Sky Survey (CLASS). These sources are known to be face-on jetted AGNs (i.e. blazars) and can be exploited to infer the abundance of all the (misaligned) jetted AGNs, using a geometrical argument. We then compare the space density of the most massive supermassive black holes hosted in jetted AGNs with those present in the total population (mostly composed by non-jetted AGNs). We find that the space density has a peak at z ∼ 3, which is significantly larger than the value observed in the total AGN population with similar optical/UV luminosities (z ∼ 2.2), but not as extreme as the value previously inferred from X-ray-selected blazars (z ≳ 4). The jetted fraction (jetted AGNs/total AGNs) is overall consistent with the estimates in the local Universe (10–20 per cent) and at high redshift, assuming Lorentz bulk factors Γ ≈ 5. Finally, we find a marginal decrease in the jetted fraction at high redshifts (by a factor of ∼2). All these evidences point towards a different evolutionary path in the jetted AGNs compared to the total AGN population.

The Maximax Minimax Quotient Theorem

We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min programming on polytopes. A naïve solution would require solving four nested, possibly nonlinear, optimization problems. Instead, relying on numerous geometric arguments we determine an analytical solution to this problem. In the course of proving our main theorem, we also establish another optimization result stating that the minimum of a specific minimax optimization is located at a vertex of the constraint set.

Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle

We consider the Anderson model with Bernoulli potential on the three-dimensional (3D) lattice Z3, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework of Bourgain–Kenig and Ding–Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.

Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-element Galois field which is linear in both <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.

Finite-time thermodynamics of fluctuations in microscopic heat engines

Fluctuations of thermodynamic quantities become non-negligible and play an important role when the system size is small. We develop finite-time thermodynamics of fluctuations in microscopic heat engines whose environmental temperature and mechanical parameter are driven periodically in time. Within the slow-driving regime, this formalism universally holds in a coarse-grained time scale whose resolution is much longer than the correlation time of the fluctuations, and is shown to be consistent with the relation analogous to the fluctuation-dissipation relation. Employing a geometric argument, a scenario to simultaneously minimize both the average and fluctuation of the dissipation in the Carnot cycle is identified. For this simultaneous optimization, the existence of a zero eigenvalue of the singular metric for the scale invariant equilibrium state is found to be essential. Furthermore, we demonstrate that our optimized protocol can improve the dissipation and its fluctuation over the current experiment.

A note on the minimum number of red lines needed to pierce the intersections of blue lines

Let L be a set of n non-concurrent blue lines and let R be a set of m red lines in the real projective plane. In this note, using elementary geometric arguments, we show that if L∩R=∅ and there is a line from R through every intersection point of lines in L, then m≥411(n−1311). This lower bound improves the previous ones whenever n≥4. Also we give an application of this result.

Magneto-Mechanical Enhancement of Elastic Moduli in Magnetoactive Elastomers with Anisotropic Microstructures.

Magnetoactive elastomers (MAEs) have gained significant attention in recent years due to their wide range of engineering applications. This paper investigates the important interplay between the particle microstructure and the sample shape of MAEs. A simple analytical expression is derived based on geometrical arguments to describe the particle distribution inside MAEs. In particular, smeared microstructures are considered instead of a discrete particle distribution. As a consequence of considering structured particle arrangements, the elastic free energy is anisotropic. It is formulated with the help of the rule of mixtures. We show that the enhancement of elastic moduli arises not only from the induced dipole–dipole interactions in the presence of an external magnetic field but also considerably from the change in the particle microstructure.

Nature of radio-wave radiation from particle cascades

The nature of the radio-wave radiation generated by particle cascades in both the Earth's atmosphere and dense media such as ice has, historically, been much debated. This situation changed in the early 2010's, with the community converging on the common terminology of "geomagnetic" and "Askaryan" radiation to describe the two emission mechanisms. However, this convergence arose from discussions at various conferences and workshops, and was ultimately reached through agreement between simulation codes and experimental measurements. In this article therefore, I use relatively simple geometrical arguments, and a minimum of calculations based on single particle tracks, to explain the nature of radiation from extensive air showers (EAS) and cascades in dense media such as ice. I identify well-determined frequency regimes where the radiation from the Askaryan effect will be bremsstrahlung-like and Cherenkov-like, being respectively below/above 1 GHz in EAS and 100 MHz in dense media; and where geomagnetic emission will be transverse-current-like and where it will resemble synchrotron radiation, respectively below/above a few GHz in EAS, depending on the height of cascade development. I suggest how these transitions in the nature of the emission may be experimentally observed.

Thermodynamic Stability Implies Causality.

The stability conditions of a relativistic hydrodynamic theory can be derived directly from the requirement that the entropy should be maximized in equilibrium. Here, we use a simple geometrical argument to prove that, if the hydrodynamic theory is stable according to this entropic criterion, then localized perturbations to the equilibrium state cannot propagate outside their future light cone. In other words, within relativistic hydrodynamics, acausal theories must be thermodynamically unstable, at least close to equilibrium. We show that the physical origin of this deep connection between stability and causality lies in the relationship between entropy and information. Our result may be interpreted as an "equilibrium conservation theorem," which generalizes the Hawking-Ellis vacuum conservation theorem to finite temperature and chemical potential.

Positive output controllability of linear discrete–time invariant systems

Abstract This paper studies the output controllability of discrete linear time invariant systems (LTI) with non-negative input constraints. Some geometrical arguments and positive invariance concepts are used to derive the necessary and/or sufficient conditions for the positive output controllability of discrete LTI systems. The paper also provides several academic examples, which support the theoretical results.

Investigating the exceptionality of scattered polynomials

Scattered polynomials over a finite field Fqn have been introduced by Sheekey in 2016, and a central open problem regards the classification of those that are exceptional. So far, only two families of exceptional scattered polynomials are known. Very recently, Longobardi and Zanella weakened the property of being scattered by introducing the notion of L-qt-partially scattered and R-qt-partially scattered polynomials, for t a divisor of n. Indeed, a polynomial is scattered if and only if it is both L-qt-partially scattered and R-qt-partially scattered. In this paper, by using techniques from algebraic geometry over finite fields and function fields theory, we show that the property which is the hardest to be preserved is the L-qt-partially scattered one. We investigate a large family F of R-qt-partially scattered polynomials, containing examples of exceptional R-qt-partially scattered polynomials, which turn out to be connected with linear sets of so-called pseudoregulus type. We introduce two different notions of equivalence preserving the property of being R-qt-partially scattered. Many polynomials in F are inequivalent and geometric arguments are used to determine their equivalence classes under the action of ΓL(2n/t,qt).

Almost color-balanced perfect matchings in color-balanced complete graphs

For a graph G and a not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges of G of color i, and call G color-balanced if mi(G)=mj(G) for every two colors i and j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, for instance, is equivalent to the statement that every properly n-edge colored complete bipartite graph Kn,n has a color-balanced perfect matching. We contribute some results on the question posed by Kittipassorn and Sinsap (arXiv:2011.00862v1) whether every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M. For a perfect matching M of K2kn, a natural measure for the total deviation of M from being color-balanced is f(M)=∑i=1k|mi(M)−n|. While not every k-edge colored color-balanced complete graph K2kn has a color-balanced perfect matching M, that is, a perfect matching with f(M)=0, we prove the existence of a perfect matching M with f(M)=O(kknln⁡(k)) for general k and f(M)≤2 for k=3; the case k=2 has already been studied earlier. An attractive feature of the problem is that it naturally invites the combination of a combinatorial approach based on counting and local exchange arguments with probabilistic and geometric arguments.

Key Parameters That Determine the Magnitude of the Decrease in Current in Nanopore Blockade Sensors.

Nanopore blockade sensors were developed to address the challenges of sensitivity and selectivity for conventional nanopore sensors. To date, the parameters affecting the current of the sensor have not been elucidated. Herein, the impacts of nanopore size and charge and the shape, size, surface charge, and aggregation state of magnetic nanoparticles were assessed. The sensor was tolerant to all parameters contrary to predictions from electronic or geometric arguments on the current change. Theoretical models showed the greater importance of the polymers around nanoparticles and the access resistance of nanopores to the current, when compared with translocation-based nanopore sensors. The signal magnitude was dominated by the change in access resistance of ∼4.25 MΩ for all parameters, resulting in a robust system. The findings provide understandings of changes in current when nanopores are blocked, like in RNA trapping or nanopore blockade sensors, and are important for designing sensors based on nanopore blockades.

A convergent evolving finite element algorithm for Willmore flow of closed surfaces

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional surfaces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of second-order equations, and uses these evolving geometric quantities in the velocity law interpolated to the finite element space. This numerical method admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The error analysis combines stability estimates and consistency estimates to yield optimal-order \(H^1\)-norm error bounds for the computed surface position, velocity, normal vector and mean curvature. The stability analysis is based on the matrix–vector formulation of the finite element method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments illustrate and complement the theoretical results.

Precisely Encoding Geometric Features into Discrete Linear Polymer Chains for Robust Structural Engineering.

Molecular shape is an essential parameter that regulates the self-organization and recognition process, which has not yet been well appreciated and exploited in block polymers due to the lack of precise and efficient modulation methods. This work (i) develops a robust approach to break the intrinsic symmetry of linear polymers by introducing geometric features into otherwise homogeneous chains and (ii) quantitatively highlights the critical contribution of molecular geometry/architecture to the self-assembly behaviors. Iteratively connecting homologous monomers of different side chains according to pre-designed sequences generates discrete polymers with exact chemical structure, uniform chain length, and programmable side-chain gradient along the backbone, which transcribes into diverse shapes. The precise chemistry eliminates all the defects and heterogeneities, providing a delicate platform for fundamental inquiries into the role of molecular geometry. A rich collection of unconventional complex phases, including Frank-Kasper A15 and σ phases, as well as a dodecagonal quasicrystal phase, were captured in these rigorous single-component systems. The self-assembly behaviors are strikingly sensitive to subtle variations of geometry, such that simply migrating a few methylene units among the side chains would generate substantial differences in lattice size or phase stability, or even trigger a phase transition toward distinct structures. The phenomena can be rationalized with a geometric argument that nonuniform side chain distribution leads to conformational mismatch between two immiscible blocks, resulting in varied interfacial curvatures and distinct lattice symmetries. The profound contribution demonstrates that molecular geometry is an effective and robust parameter for structural engineering.

Remarks on the 3D Stokes eigenvalue problem under Navier boundary conditions

We study the Stokes eigenvalue problem under Navier boundary conditions in C^{1,1}-domains Omega subset mathbb {R}^3. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple, it has multiplicity three. We apply these results to show the validity/failure of a suitable Poincaré-type inequality. The proofs are obtained by combining analytic and geometric arguments.

A Five Distance Theorem for Kronecker Sequences

Abstract The three-distance theorem (also known as the three-gap theorem or Steinhaus problem) states that, for any given real number $\alpha $ and integer $N$, there are at most three values for the distances between consecutive elements of the Kronecker sequence $\alpha , 2\alpha ,\ldots , N\alpha $ mod 1. In this paper, we consider a natural generalization of the three-distance theorem to the higher-dimensional Kronecker sequence $\vec \alpha , 2\vec \alpha ,\ldots , N\vec \alpha $ modulo an integer lattice. We prove that in 2D, there are at most five values that can arise as a distance between nearest neighbors, for all choices of $\vec \alpha $ and $N$. Furthermore, for almost every $\vec \alpha $, five distinct distances indeed appear for infinitely many $N$ and hence five is the best possible general upper bound. In higher dimensions, we have similar explicit, but less precise, upper bounds. For instance, in 3D, our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalization of the gap length in 1D. For large cone angles, we use geometric arguments to produce explicit bounds directly analogous to the three-distance theorem. For small cone angles, we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (1) unbounded for almost all $\vec \alpha $ and (2) bounded for $\vec \alpha $ that satisfy certain Diophantine conditions.

Influence of drug/lipid interaction on the entrapment efficiency of isoniazid in liposomes for antitubercular therapy: a multi-faced investigation

Isoniazid (INH) is one of the primary drugs used in tuberculosis treatment and its encapsulation in liposomal vesicles can both improve its therapeutic index and minimize toxicity. Here we consider mixtures of hydrogenated soy phosphatidylcholine-phosphatidylglycerol (HSPC–DPPG) to get novel biocompatible liposomes for INH delivery. We determined INH encapsulation efficiency by coupling for the first time UV and Laser Transmission Spectroscopy and we showed that HSPC–DPPG liposomes can load more INH than expected from simple geometrical arguments, thus suggesting the presence of drug-lipid association. To focus on this aspect, which has never been explored in liposomal formulations, we employed several complementary techniques, such as dynamic and static light scattering, calorimetry and surface pressure measurements on lipid monolayers. We find that INH-lipid interaction increases the entrapment capability of liposomes due to INH adsorption. Moreover, the preferential INH-HSPC dipole–dipole interaction promotes the modification of lipid ordering, favoring the formation of HSPC-richer domains in excess of DPPG. Our findings highlight how investigating the fundamental aspects of drug-lipid interactions is of paramount importance for the optimal design of liposomal nanocarriers.

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.