Characteristic Cone Research Articles

Movable Singularity of Some Hamiltonian Systems Associated with Blowup Phenomena

The motivation for this paper comes from a blowup problem for a semi-linear wave equation and a heat equation. Following an idea of J. Leray, we study a radially symmetric self-similar solution which has singularities on the characteristic cone. This naturally leads to the study of a Hamiltonian system called a profile equation. The novelty of this paper is that we focus on the movable singularity of the Hamiltonian system and we use Borel summability in constructing a singular solution. By “movable singularity” we mean that the singularity does not appear in the coefficients of the equation and depends on the respective solution. In the proof of our theorem we reduce the Hamiltonian system to a simpler form by a method similar to the so-called Birkhoff reduction. We obtain the parametrization of a singular solution by an elementary function. We also give applications to a semi-linear wave equation and a heat equation.

Simultaneous lesions of complex odontoma associated with dentigerous cyst: case report and critical review in CBCT

Objectives: To our knowledge, there is limited evidence reporting cystic lesions associated with odontoma. Therefore, this case report aims to describe the clinical, histological, and particularly the characteristic cone beam computed tomography (CBCT) diagnostic features . Case Report: A 20-year-old male patient presented with a complaint of a lump in the upper right jaw. There was no history of systemic disease. A panoramic radiograph showed a solid radio-opaque mass surrounded by an osteolytic lesion with a radio-opaque margin causing impaction of the maxillary canine. Given the limitations of panoramic radiography in assessing lesion extent, a CBCT examination was performed. Multiplanar reconstruction revealed a well-defined unilocular hyperdense lesion with a hypodense encapsulation and corticated margin, measuring 31.5 x 32 mm, expanding the buccal cortical plate and pushing the upper right canine tooth into the maxillary sinus cavity. The extensive expansion of the lesion required histopathological analysis to confirm the final diagnosis. The histopathological examination concluded a dentigerous cyst. Conclusion: Several reports in the literature discuss the development of cysts and tumors in the jaw, but some cases are very difficult to identify. The variability of lesions can complicate pattern identification, leading to misinterpretation of conventional radiographs, thus requiring additional CBCT evaluation.

Control of waves on Lorentzian manifolds with curvature bounds

We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds.

Electronic Band Engineering of Two-Dimensional Kagomé Polymers.

Two-dimensional conjugated polymers (2DCPs) are an emerging class of materials that exhibit properties similar to graphene yet do not have the limitation of zero bandgap. On-surface synthesis provides exceptional control on the polymerization reaction, allowing tailoring properties by choosing suitable monomers. Heteroatom-substituted triangulene 2DCPs constitute a playing ground for such a design and are predicted to exhibit graphene-like band structures with high charge mobility and characteristic Dirac cones in conduction or valence states. However, measuring these properties experimentally is challenging and requires long-range-ordered polymers, preferably with an epitaxial relationship with the substrate. Here, we investigate the electronic properties of a mesoscale-ordered carbonyl-bridged triphenylamine 2DCP (P2TANGO) and demonstrate the presence of a Dirac cone by combining angle-resolved photoemission spectroscopy (ARPES) with density functional theory (DFT) calculations. Moreover, we measure the absolute energy position of the Dirac cone with respect to the vacuum level. We show that the bridging functionality of the triangulene (ether vs carbonyl) does not significantly perturb the band structure but strongly affects the positioning of the bands with respect to the Au(111) states and allows control of the ionization energy of the polymer. Our results provide proof of the controllable electronic properties of 2DCPs and bring us closer to their use in practical applications.

Components of $$V(\rho ) \otimes V(\rho )$$ and Dominant Weight Polyhedra for Affine Kac–Moody Lie Algebras

AbstractKostant asked the following question: Let $$\mathfrak {g}$$ g be a simple Lie algebra over the complex numbers. Let $$\lambda $$ λ be a dominant integral weight. Then, $$V(\lambda )$$ V ( λ ) is a component of $$V(\rho )\otimes V(\rho )$$ V ( ρ ) ⊗ V ( ρ ) if and only if $$\lambda \le 2 \rho $$ λ ≤ 2 ρ under the usual Bruhat–Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei, the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac–Moody Lie algebra $$\mathfrak {g}$$ g associated with any simple Lie algebra $$\mathring{\mathfrak {g}}$$ g ˚ (up to a saturation factor). In fact, we prove the result for affine $$sl_n$$ s l n without any saturation factor. Our proof requires some additional techniques including the Goddard–Kent–Olive construction and study of the characteristic cone of non-compact polyhedra.

A Novel Experiment Approach for Measurement Breakup Length, Cone Angle, Sheet Velocity, and Film Thickness in Swirl Air-Blast Atomizers

<div>Measuring the dynamic parameters of liquid fragments generated in the near-field of atomizing sprays poses a significant challenge due to the random nature of the fragments, the instability of the spray, and the limitations of current measuring technology. Precise determination of these parameters can aid in improving the control of the atomization process, which is necessary for providing suitable spray structures with appropriate flow rates and droplet size distributions for various applications such as those used in heat engines. In piston and gas turbine engines, controlling spray characteristics such as penetration, cone angle, particle size, and droplet size distribution is crucial to improve combustion efficiency and decrease exhaust emissions. This can be accomplished by adjusting the structural and/or operating parameters of the fuel supply system. This article aims to measure the breakup length, spray cone angle, axial velocity, breakup time, and liquid sheet film thickness for a swirl air-blast atomizer used in a gas-steam engine. The measurement was conducted using a shadowgraph imaging system developed specifically for this study, consisting of a high-speed camera, a lens, and a light source. While lasers are commonly used as light sources in the literature, this study utilized a special LED high-speed pulse light generator, which is cheaper, easier to handle, and provides a more uniform background. Images were processed using a MATLAB code developed for this study. Although the breakup zone is naturally random and the breakup location significantly varies with time, the novel method developed in this study helps quantify critical parameters under different operating conditions.</div>

A sufficient and necessary condition for $${\mathcal {A}}$$-quasiaffinity

AbstractWe consider a homogeneous differential operator $${\mathcal {A}}$$ A and show an improved version of Murat’s condition for $${\mathcal {A}}$$ A -quasiaffine functions, provided that the operator $${\mathcal {A}}$$ A satisfies the constant rank condition. As a consequence, we obtain that affinity along the characteristic cone of $${\mathcal {A}}$$ A implies $${\mathcal {A}}$$ A -quasiaffinity if $${\mathcal {A}}$$ A admits a first order potential.

Hall effect of ferro/antiferromagnetic wallpaper fermions

Nonsymmorphic crystals can host characteristic double surface Dirac cones with fourfold degeneracy on the Dirac points, called wallpaper fermion, protected by wallpaper group symmetry. We clarify the charge and spin Hall effect of wallpaper fermions in the presence of the (anti)ferromagnetism. Based on a four-sublattice model, we construct the effective Hamiltonian of wallpaper fermions coupled with the ferromagnetic or antiferromagnetic moment. Both ferromagnetic and antiferromagnetic moments induce an energy gap for the wallpaper fermions, leading to quantized (spin) Hall conductivity. The ferromagnetic wallpaper fermion induces the Hall conductivity quantized into ${e}^{2}/h$, which is twice that for a single Dirac fermion on the surface of topological insulators. On the other hand, the spin Hall conductivity decays and reaches to be a finite value as the antiferromagnetic coupling increases. We also show that the results above are valid for a general model of wallpaper fermions from symmetry consideration.

Solutions with exponential singularity for (3 + 1)-D Protter problems

In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we study Protter problems in the hyperbolic part of the domain. Unlike the planar analogues, the four-dimensional variant is not well-posed for classical solutions. The problem is not Fredholm — there is an infinite number of necessary conditions for classical solvability. Alternatively, the notion of a generalized solution that may have singularities was introduced. It is known that for smooth right-hand sides, the uniquely determined generalized solution may have a power-type growth at one boundary point. The singularity is isolated at the vertex of the boundary characteristic light cone and does not propagate along the cone. Here, we construct a new singular solution with an exponential growth at the point where the singularity appears.

Orphan gene expressed in flame cone cells uniquely found in seahorse epithelium.

The seahorse is one of the most unique teleost fishes in its morphology. The body is surrounded by bony plates and spines, and the male fish possess a brooding organ, called the brood pouch, on their tail. The surfaces of the brood pouch and the spines are surrounded by characteristic so-called flame cone cells. Based on our histological observations, flame cone cells are present in the seahorse Hippocampus abdominalis, but not in the barbed pipefish Urocampus nanus or the seaweed pipefish Syngnathus schlegeli, both of which belong to the same family as the seahorse. In the flame cone cells, we observed expression of an "orphan gene" lacking homologs in other lineages. This gene, which we named the proline-glycine rich (pgrich) gene, codes for an amino acid sequence composed of repetitive units. In situ hybridization and immunohistochemical analyses detected pgrich-positive signals from the flame cone cells. Based on a survey of the genome sequences of 15 teleost species, the pgrich gene is only found from some species of Syngnathiformes (namely, the genera Syngnathus and Hippocampus). The amino acid sequence of the seahorse PGrich is somewhat similar to the sequence deduced from the antisense strand of elastin. Furthermore, there are many transposable elements around the pgrich gene. These results suggest that the pgrich gene may have originated from the elastin gene with the involvement of transposable elements and obtained its novel function in the flame cone cells during the evolution of the seahorse.

Characteristic, C-Characteristic and Positive Cones in Hyperfields

We study the notions of the positive cone, characteristic and C-characteristic in (Krasner) hyperfields. We demonstrate how these interact in order to produce interesting results in the theory of hyperfields. For instance, we provide a criterion for deciding whether certain hyperfields cannot be obtained via Krasner’s quotient construction. We prove that any positive integer (larger than 1) can be realized as the characteristic of some infinite hyperfield and an analogous result for the C-characteristic. Finally, we study the (directed) graph associated with the strict partial order induced by a positive cone in a hyperfield in various examples.

On second-order conic programming duals for robust convex quadratic optimization problems

In this paper, we deal with second-order conic programming (SOCP) duals for a robust convex quadratic optimization problem with uncertain data in the constraints. We first introduce a SOCP dual problem for this robust convex quadratic optimization problem with polytopic uncertain sets. Then, we obtain a zero duality gap result between this robust convex quadratic optimization problem and its dual problem in terms of a new robust type characteristic cone constraint qualification. We also construct a SOCP dual problem for this robust convex quadratic optimization problem with norm-constrained uncertain sets and obtain the corresponding zero duality gap result between them. Moreover, some numerical examples are given to explain the obtained results.

Robust dual topological insulator phase in NaZnBi

Topological insulators are characterized by {Bbb Z}_2 indices, and their metallic surface states are protected by time-reversal symmetry. In topological crystalline insulators, on the other hand, crystal symmetry plays a key role in protecting surface states. Therefore, perturbation breaking time-reversal or crystal symmetry induces a phase transition to a conventional band insulator. In dual topological insulators (DTIs), in which time-reversal and crystal symmetries coexist, the topological character is more robust against perturbation. Here, we propose that NaZnBi is a new DTI with {Bbb Z}_2 invariants (nu _0;nu _1nu _2nu _3) = (1;000) and odd mirror Chern numbers ±1. We find that the characteristic Dirac cone is preserved even if either time-reversal or mirror symmetry is broken, verifying the robustness of the DTI phase. Given that gapless surface states can be experimentally observable even under perturbations that break any lattice symmetry or time-reversal, NaZnBi can be a good candidate material for future device applications.

Realizing Kagome Band Structure in Two-Dimensional Kagome Surface States of RV_{6}Sn_{6} (R=Gd, Ho).

We report angle resolved photoemission experiments on a newly discovered family of kagome metals RV_{6}Sn_{6} (R=Gd, Ho). Intrinsic bulk states and surface states of the vanadium kagome layer are differentiated from those of other atomic sublattices by the real-space resolution of the measurements with a small beam spot. Characteristic Dirac cone, saddle point, and flat bands of the kagome lattice are observed. Our results establish the two-dimensional (2D) kagome surface states as a new platform to investigate the intrinsic kagome physics.

Abstracts and Case Studies for the College of American Pathologists 2021 (CAP21) Annual Meeting

Abstracts and Case Studies for the College of American Pathologists 2021 (CAP21) Annual Meeting

On solving a class of fractional semi-infinite polynomial programming problems

In this paper, we study a class of fractional semi-infinite polynomial programming (FSIPP) problems, in which the objective is a fraction of a convex polynomial and a concave polynomial, and the constraints consist of infinitely many convex polynomial inequalities. To solve such a problem, we first reformulate it to a pair of primal and dual conic optimization problems, which reduce to semidefinite programming (SDP) problems if we can bring sum-of-squares structures into the conic constraints. To this end, we provide a characteristic cone constraint qualification for convex semi-infinite programming problems to guarantee strong duality and also the attainment of the solution in the dual problem, which is of its own interest. In this framework, we first present a hierarchy of SDP relaxations with asymptotic convergence for the FSIPP problem whose index set is defined by finitely many polynomial inequalities. Next, we study four cases of the FSIPP problems which can be reduced to either a single SDP problem or a finite sequence of SDP problems, where at least one minimizer can be extracted. Then, we apply this approach to the four corresponding multi-objective cases to find efficient solutions.

Steady flow in a rapidly rotating spheroid with weak precession: Part 2

This is a continuation of our companion paper (Kida 2020b Fluid Dyn. Res.) in which the steady flow in a precessing spheroid is studied in the strong spin and weak precession limit, and the elliptic flow with uniform vorticity, the direction of which is perpendicular to the spin axis on the plane spanned by the spin and the precession axes, is shown to be excited in the interior inviscid region. In this paper we examine the boundary-layer flow connected to this elliptic flow and also the higher-order inviscid flow modified by this boundary-layer flow. The explicit expression of the velocity and pressure fields is obtained for arbitrary aspect ratio c of the polar and equatorial radii of the spheroid. The boundary-layer approximation breaks down at two critical circles on which the boundary-layer thickness as well as the normal component of velocity diverge to infinity. The flow field in the neighborhood (called the critical regions) of the critical circles is analyzed to find the same structure, up to scales, as that for a sphere. The radial component of velocity in the critical region induces the conical shear layers along the characteristic cones in the inviscid region. They reflect on the spheroid surface, and the reflection takes place endlessly in general. On the other hand, it is proved that for a spheroid of which the aspect ratio is expressed as (m/n being an irreducible fraction) the reflection terminates at (n − 2) times and the conical shear layers form a regular pattern. Since such special values of c are distributed densely over all the positive numbers, we may observe practically always a regular pattern of the conical shear layers. Such closed conical shear layers are clearly visualized by the pressure field numerically calculated in the asymptotic formulation. Furthermore, the total angular momentum of the steady flow is expressed explicitly up to the non-trivial leading order for all the three components over the whole range of the aspect ratio.

Singularities generated by the triple interaction of semilinear conormal waves

We study the local propagation of conormal singularities for solutions of semilinear wave equations $\square u = P(y, u)$, where $P(y, u)$ is a polynomial of degree $N \geq 3$ in $u$ with $C^\infty(\mathbb{R}^3_y)$ coefficients. We know from the work of Melrose & Ritter and Bony that if u is conormal to three waves which intersect transversally at point $q$, then after the triple interaction $u(y)$ is a conormal distribution with respect to the three waves and the characteristic cone $Q$ with vertex at $q$. We compute the principal symbol of $u$ at the cone and away from the hypersurfaces. We show that if $\partial_u^3 P (q, u(q)) \neq 0$, $u$ is an ellipitic conormal distribution.

On the Active Flux Scheme for Hyperbolic PDEs with Source Terms

The Active Flux scheme is a Finite Volume scheme with additional point values distributed along the cell boundary. It is third order accurate and does not require a Riemann solver: the continuous reconstruction serves as initial data for the evolution of the points values. The intercell flux is then obtained from the evolved values along the cell boundary by quadrature. This paper focuses on the conceptual extension of Active Flux to include source terms, and thus for simplicity assumes the homogeneous part of the equations to be linear. To a large part, the treatment of the source terms is independent of the choice of the homogeneous part of the system. Additionally, only systems are considered which admit characteristics (instead of characteristic cones). This is the case for scalar equations in any number of spatial dimensions and systems in one spatial dimension. Here, we succeed to extend the Active Flux method to include (possibly nonlinear) source terms while maintaining third order accuracy of the method. This requires a novel (approximate) operator for the evolution of point values and a modified update procedure of the cell average. For linear acoustics with gravity, it is shown how to achieve a well-balanced / stationarity preserving numerical method.

Geographical cline and inter‐seaside difference in cone characteristics related to climatic conditions of old planted Pinus thunbergii populations throughout Japan

AbstractAlthough cone morphology has been mainly utilized for taxonomy or forestry purposes, characterization of the environmental factors that influence its variation is still insufficient. Major conifers distributed over diverse climatic conditions like the Japanese archipelago could show cone morphology variation among populations related to large differences in key influential factors such as temperature, snow, irradiance and sunshine. Geographical variation in cone characteristics (cone size, seed production, seed productivity per cone and seed size) was examined across the species distribution in 24 old planted Pinus thunbergii populations (479 trees) along two major seasides in Japan (Pacific Ocean side and Japan Sea side). Variance components of cone characteristics explained by seaside were at similar levels to those for populations nested within seasides. Populations on the Japan Sea side produced cones that had larger size (length and width), higher seed production (number of filled seeds per cone and seed mass per cone) and seed productivity per cone (seed‐to‐ovule ratio). Analysis of covariance showed that significant latitudinal clines observed in cone size and seed productivity per cone were mostly a result of inter‐seaside differences. Linear mixed model analysis detected that geographical variation in cone size, seed production and seed productivity per cone in the populations were affected significantly by low temperature, more snow, less solar irradiance and less sunshine time. Large cone sizes and high seed productivity per cone found in the populations on the Japan Sea side could be explained by a genetic or plastic response to maintain reproductive success.