Generic Point Research Articles

Nonlinear Thermoelastic Numerical Frequency Analysis and Experimental Verification of Cutout Abided Laminated Shallow Shell Structure

Abstract The cutout and temperature loading influences on the nonlinear frequencies of the laminated shell structures are predicted numerically using two different types of geometrical nonlinear strain-displacement relationships to count the large deformation. The displacement of any generic point on the structural panel is derived using the third-order shear deformation theory (TSDT). Moreover, the direct iterative method has been adopted to obtain the nonlinear eigenvalues in conjunction with the isoparametric finite element (FE) steps. The present analysis includes the effect of temperature and the temperature-dependent composite elastic properties on the thermoelastic frequencies. This study intends to establish the Green-Lagrange type of nonlinear strain's efficacy in computing the nonlinear frequency of layered structure with and without cutout instead of von-Karman strain kinematics. The numerical model's validity has been established by comparing the results to previously published results. In addition, experimentally obtained fundamental frequency values of a few modes are compared to numerical proposed numerical results under the thermal loading. Finally, the effects of cutout (shape and size) and the associated structural geometrical parameters on the nonlinear thermal frequency responses of the laminated structure are expressed in the final output form.

On Bowen's entropy inequality and almost specification for flows

We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is ergodic. We propose a definition of almost specification property for flows and prove that a continuous flow has the almost specification property if the time-1 map satisfies this property. Using Bowen's inequality for flows, we show that every continuous flow with the almost specification property is saturated, extending a result of Meson and Vericat in [22]. Under the same hypotheses, we extend a result of Thompson on the entropy of irregular points in [29].

On log-growth of solutions of $p$-adic differential equations with $p$-adic exponents

We consider a differential system x\frac{d}{dx} Y=GY , where G is a m\times m matrix whose coefficients are power series which converge and are bounded on the open unit disc D(0,1^-) . Assume that G(0) is a diagonal matrix with p -adic integer coefficients. Then there exists a solution matrix of the form Y=F \exp(G(0)\log x) at x=0 if all differences of exponents of the system are p -adically non-Liouville numbers. We give an example where F is analytic on the p -adic open unit disc and has log-growth greater than m . Under some conditions, we prove that if a solution matrix at a generic point has log-growth \delta , then F has log-growth \delta .

Holomorphic isometric maps from the complex unit ball to reducible bounded symmetric domains

Abstract The first part of the paper studies the boundary behavior of holomorphic isometric mappings F = ( F 1 , … , F m ) {F=(F_{1},\dots,F_{m})} from the complex unit ball 𝔹 n {\mathbb{B}^{n}} , n ≥ 2 {n\geq 2} , to a bounded symmetric domain Ω = Ω 1 × ⋯ × Ω m {\Omega=\Omega_{1}\times\cdots\times\Omega_{m}} up to constant conformal factors, where Ω i ′ {\Omega_{i}^{\prime}} s are irreducible factors of Ω. We prove every non-constant component F i {F_{i}} must map generic boundary points of 𝔹 n {\mathbb{B}^{n}} to the boundary of Ω i {\Omega_{i}} . In the second part of the paper, we establish a rigidity result for local holomorphic isometric maps from the unit ball to a product of unit balls and Lie balls.

Solving Natural Conic Formulations with Hypatia.jl

Many convex optimization problems can be represented through conic extended formulations (EFs) using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. However, EFs are often significantly larger and more complex than equivalent conic natural formulations (NFs) represented using the much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement easily computable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. Our goal is to establish whether a generic conic interior point solver supporting NFs can outperform an advanced conic solver specialized for EFs across a variety of applied problems. We introduce Hypatia, a highly configurable open-source conic primal-dual interior point solver written in Julia and accessible through JuMP. Hypatia has a generic interface for exotic cones, some of which we define here. For seven applied problems, we introduce NFs using these cones and construct EFs that are necessarily larger and more complex. Our computational experiments demonstrate the advantages, especially in terms of solve time and memory usage, of solving the NFs with Hypatia compared with solving the EFs with either Hypatia or MOSEK 9.

Dynamical consequences of 1-form symmetries and the exceptional Argyres-Douglas theories

Higher-form symmetries have proved useful in constraining the dynamics of a number of quantum field theories. In the context of the Argyres-Douglas (AD) theories of the (G, G′) type, we find that the 1-form symmetries are invariant under the Higgs branch flow, and that they are captured by the non-Higgsable sector at a generic point on the Higgs branch of the AD theory in question. As a consequence, dimensional reduction of an AD theory with a non-trivial 1-form symmetry to 3d leads to a free sector. We utilize these observations, along with other results, to propose systematically the mirror theories for the AD theories of the (An, Em) type. As a by-product of these findings, we discover many important results: the Flip-Flip duality for all T[G] theories with simply-laced group G, including the exceptional ones; the class mathcal{S} descriptions of exceptional affine Dynkin diagram such that all gauge groups are special unitary; the universality of the mirror theories for {D}_{h_G^{vee }} (G) with {h}_G^{vee } the dual Coxeter number of G; and the triviality of the 2-group structure in the (An, Em) theories.

How Do You Treat Trigeminal Neuralgia in Your Practice?

How Do You Treat Trigeminal Neuralgia in Your Practice?

Nature of the Poynting Vector Field Singularities in Resonant Light Scattering by Nanoparticles.

Singularities of the Poynting vector field subwavelength patterns in resonant light scattering by nanoparticles are discussed and classified. There are two generic types of the singularities, namely, (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in proximity to the singular points. The connection of these types of singularities to the topology of the singular points, space dimension (3D vs. 2D), and energy conservation law are revealed. In particular, it is shown that in 2D cases in non-dissipative media, the energy conservation reduces the possible types of generic singular points to saddles and centers only. In 3D cases, a universal expression connecting different components of the Poynting vector and valid for any generic singularities is derived and numerically checked for various types of singular points.

Phase transitions of correlations in black hole geometries

We study the holographic realization of optimized correlation measures---measures of quantum correlation that generalize elementary entropic formulas---in two-dimensional thermal states dual to spacetimes with a black hole horizon. We consider the symmetric bipartite optimized correlation measures: the entanglement of purification, Q-correlation, R-correlation, and squashed entanglement, as well as the mutual information, a nonoptimized correlation measure, and identify the bulk surface configurations realizing their geometric duals over the parameter space of boundary region sizes and the black hole radius. This parameter space is divided into phases associated with given topologies for these bulk surface configurations, and first-order phase transitions occur as a new topology of bulk surfaces becomes preferred. The distinct phases can be associated with different degrees of correlation between the boundary regions and the thermal environment. The Q-correlation has the richest behavior, with a structure of nested optimizations leading to two topologically distinct bulk surface configurations being equally valid as geometric duals at generic points in the phase diagram.

Generic Property of the Partial Calmness Condition for Bilevel Programming Problems

The partial calmness for the bilevel programming problem (BLPP) is an important condition which ensures that a local optimal solution of BLPP is a local optimal solution of a partially penalized problem where the lower level optimality constraint is moved to the objective function and hence a weaker constraint qualification can be applied. In this paper we propose a sufficient condition in the form of a partial error bound condition which guarantees the partial calmness condition. We analyse the partial calmness for the combined program based on the Bouligand (B-) and the Fritz John (FJ) stationary conditions from a generic point of view. Our main result states that the partial error bound condition for the combined programs based on B and FJ conditions are generic for an important setting with applications in economics and hence the partial calmness for the combined program is not a particularly stringent assumption. Moreover we derive optimality conditions for the combined program for the generic case without any extra constraint qualifications and show the exact equivalence between our optimality condition and the one by Jongen and Shikhman given in implicit form. Our arguments are based on Jongen, Jonker and Twilt's generic (five type) classification of the so-called generalized critical points for one-dimensional parametric optimization problems and Jongen and Shikhman's generic local reductions of BLPPs.

On the Grothendieck-Serre conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is trivial if it is trivial over the generic point of $U$. We generalize some other related results from the simple simply-connected case to the case of arbitrary reductive group schemes.

Discriminative Shape Feature Pooling in Deep Neural Networks.

Although deep learning approaches are able to generate generic image features from massive labeled data, discriminative handcrafted features still have advantages in providing explicit domain knowledge and reflecting intuitive visual understanding. Much of the existing research focuses on integrating both handcrafted features and deep networks to leverage the benefits. However, the issues of parameter quality have not been effectively solved in existing applications of handcrafted features in deep networks. In this research, we propose a method that enriches deep network features by utilizing the injected discriminative shape features (generic edge tokens and curve partitioning points) to adjust the network’s internal parameter update process. Thus, the modified neural networks are trained under the guidance of specific domain knowledge, and they are able to generate image representations that incorporate the benefits from both handcrafted and deep learned features. The comparative experiments were performed on several benchmark datasets. The experimental results confirmed our method works well on both large and small training datasets. Additionally, compared with existing models using either handcrafted features or deep network representations, our method not only improves the corresponding performance, but also reduces the computational costs.

Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space

We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes (c1,c2,c3)=(−1,2n,0) and (c1,c2,c3)=(0,n,0) always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes (c1,c2,c3)=(−1,2,m) for non negative values for m; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.

Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

Determinantal polynomial systems are those involving maximal minors of some given matrix. An important situation where these arise is the computation of the critical values of a polynomial map restricted to an algebraic set. This leads directly to a strategy for, among other problems, polynomial optimisation.Computing Gröbner bases is a classical method for solving polynomial systems in general. For practical computations, this consists of two main stages. First, a Gröbner basis is computed with respect to a DRL (degree reverse lexicographic) ordering. Then, a change of ordering algorithm, such as Sparse-FGLM, designed by Faugère and Mou, is used to find a Gröbner basis of the same system but with respect to a lexicographic ordering. The complexity of this latter step, in terms of the number of arithmetic operations in the ground field, is O(mD2), where D is the degree of the ideal generated by the input and m is the number of non-trivial columns of a certain D×D matrix.While asymptotic estimates are known for m in the case of generic polynomial systems, thus far, the complexity of Sparse-FGLM was unknown for the class of determinantal systems.By assuming Fröberg's conjecture, thus ensuring that the Hilbert series of generic determinantal ideals have the necessary structure, we expand the work of Moreno-Socías by detailing the structure of the DRL staircase in the determinantal setting. Then we study the asymptotics of the quantity m by relating it to the coefficients of these Hilbert series. Consequently, we arrive at a new bound on the complexity of the Sparse-FGLM algorithm for generic determinantal systems and, in particular, for generic critical point systems.We consider the ideal inside the polynomial ring K[x1,…,xn], where K is some infinite field, generated by p generic polynomials of degree d and the maximal minors of a p×(n−1) polynomial matrix with generic entries of degree d−1. Then, in this setting, for the case d=2 and for n≫p we establish an exact formula for m in terms of n and p. Moreover, for d≥3, we give a tight asymptotic formula, as n→∞, for m in terms of n,p and d.

CR-manifolds of infinite type in the sense of Bloom and Graham

An analogue of the Bloom–Graham theorem for germs of real analytic CR-manifolds of infinite type is devised, and a certain standard form to which they can be transformed (a reduced form) is described. The concept of Bloom–Graham type is refined (as a stratified type). The refined type is also holomorphically invariant. The concept of a quasimodel surface is introduced and it is shown that for biholomorphically equivalent manifolds such surfaces are quasilinearly equivalent. A criterion for the Lie algebra of infinitesimal holomorphic automorphisms to be finite-dimensional is obtained in the case when the type is uniformly infinite (that is, infinite at all points). In combination with the criterion of a finite-dimensional automorphism algebra for manifolds of finite type almost everywhere, this yields a complete criterion for this algebra to be finite-dimensional. The sets of fixed Blooom–Graham type are shown to be semi-analytic and the type of a generic point (lying outside a proper analytic subset) is minimal in a certain sense.

Existence of Two View Chiral Reconstructions

A fundamental question in computer vision is whether a set of point pairs is the image of a scene that lies in front of two cameras. Such a scene and the cameras together are known as a chiral reconstruction of the point pairs. In this paper, we provide a complete classification of $k$ point pairs for which a chiral reconstruction exists. The existence of chiral reconstructions is equivalent to the nonemptiness of certain semialgebraic sets. We describe these sets and develop tools to certify their nonemptiness. For up to three point pairs, we prove that a chiral reconstruction always exists while the set of five or more point pairs that do not have a chiral reconstruction is Zariski-dense. We show that for five generic point pairs, the chiral region is bounded by line segments in a Schläfli double six on a cubic surface with 27 real lines. Four point pairs have a chiral reconstruction unless they belong to two nongeneric combinatorial types, in which case they may or may not.

A Dimensional Consistency Aware Time Domain Analysis of the Generic Fractional Order Biquadratic System

In this research, the time domain analysis of the fractional order biquadratic system with nonzero input and nonzero damping ratio has been performed. Unlike the previous works, the analysis has been generically done with dimensional consistency awareness without referring to any specific physical system where nonzero input and nonzero damping ratio have been allowed. The fractional differential equation of the system has been derived and analytically solved. The physical measurability of the dimensions of the fractional derivative terms which have been defined in Caputo sense, and response with significantly different dynamic from its dimensional consistency ignored counterpart have been obtained due to our dimensional consistency awareness. The resulting solution is applicable to the fractional biquadratic systems of any kind with any physical nature. Based on such solution and numerical simulations, the influence of the fractional order parameter to all major time domain parameters have been studied in detailed. The obtain results provide insight to the fractional order biquadratic system with dimensional consistency awareness in a generic point of view.

Specializing Brauer classes in Picard schemes

Let X be a smooth projective curve defined over a field k . The existence of tautological line bundles on X × Pic X / k is obstructed by a Brauer class α ∈ Br ( Pic X / k ) . We show that α splits at the generic points of various naturally defined loci in Pic X / k —the theta divisor and the generalized theta divisors associated with degree 2 g − 2 , semi-stable rank 2 vector bundles on X . We study a rank 2 analogue of Franchetta's theorem, and give explicit constructions of degree 2 g − 2 , semi-stable rank 2 vector bundles on the generic genus g curve.

The six functors for Zariski-constructible sheaves in rigid geometry

We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves.

Cutting and gluing with running couplings in N=2 QCD

We consider the order parameter $u=\left<{\rm Tr}\phi^2\right>$ as function of the running coupling constant $\tau \in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $\tau$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $\Gamma$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $\Gamma\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.