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We present a general formula that transforms any conic of the form Ax2+Bxy+Cy2+Dx+Ey+F=0, with B≠0, into A′(x′)2+C′(y′)2+D′x′+E′y′+F=0, without requiring the rotation angle θ. This directly eliminates the cross term xy, simplifying the rotated conics analysis. As consequences, we obtain new formulae that remove both rotations and translations, a novel proof of the discriminant criterion, improved expressions for eccentricity, and a detailed taxonomy of all loci described by the general conic equation.

In this paper, we bring forth several new general formulae in the classic study of conics in the Analytic Geometry: the coordinates of all vertices and focal points of arbitrary parabolas, ellipses, and hyperbolas; lengths for all latera recta from any non-degenerate conic section; equations describing straight lines whose limited-slope contents stand on exactly equal footing as focal axes, latera recta, and directrices from every non-degenerate conic section; and, respectively, these ones characterizing asymptotes for each non-degenerate hyperbola. All these general results work regardless of whether the conics in question are rotated or not on the Cartesian plane, because all of them depend only on the coefficients of the general conic equation, making the rotation angle irrelevant for the analysis of conic sections.

The construction of rational absolute nodal coordinate formulation (RANCF) elements is usually based on a linear transformation of non-uniform rational B-spline (NURBS) geometry. However, this linear transformation can lead to property transfer issues, which greatly reduce the modeling efficiency, especially for conic sections. To overcome this limitation, we first analyze the geometric constraints of conic sections and derive a unique defining equation in rational parametric form. A corresponding degree-elevation formula is also obtained. Using these results, we propose a direct definition method for RANCF elements that explicitly exploits the analytic properties of conic sections. The method provides fast and accurate expressions for the nodal coordinates and weights, and thus enables efficient modeling of RANCF elements for conic-section configurations. We also mitigate the arbitrariness in element definition by introducing, for the first time, the concept of a mapping factor K, which characterizes the mapping between the physical space and the parameter space. Based on this mapping factor, we establish a parameterization procedure for RANCF conic-section elements. An evaluation criterion for K is further proposed and used to define the optimal mapping factor Kopt, which yields an optimal parameterization and allows the construction of Kopt elements. Numerical examples demonstrate that, in large-deformation analyses of flexible systems, the proposed elements can achieve a given accuracy with fewer elements than conventional approaches.

Distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty. The distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker’s prior information. In this paper, we propose a unifying framework for modeling and solving distributionally robust optimization problems. We introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold. These ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases. They also allow us to characterize distributional families in terms of several classical and/or robust statistical indicators that have not yet been studied in the context of robust optimization. We determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable. We also provide tractable conservative approximations for problems that violate these conditions.

Freeze-fracture transmission electron microscopy study of the nanoscale structure of the so-called \"twist-bend\" nematic phase of the cyanobiphenyl (CB) dimer molecule CB(CH2)7CB reveals stripe-textured fracture planes that indicate fluid layers periodically arrayed in the bulk with a spacing of d ∼ 8.3 nm. Fluidity and a rigorously maintained spacing result in long-range-ordered 3D focal conic domains. Absence of a lamellar X-ray reflection at wavevector q ∼ 2π/d or its harmonics in synchrotron-based scattering experiments indicates that this periodic structure is achieved with no detectable associated modulation of the electron density, and thus has nematic rather than smectic molecular ordering. A search for periodic ordering with d ∼ in CB(CH2)7CB using atomistic molecular dynamic computer simulation yields an equilibrium heliconical ground state, exhibiting nematic twist and bend, of the sort first proposed by Meyer, and envisioned in systems of bent molecules by Dozov and Memmer. We measure the director cone angle to be θTB ∼ 25° and the full pitch of the director helix to be pTB ∼ 8.3 nm, a very small value indicating the strong coupling of molecular bend to director bend.

When uncontrollable resources fluctuate, optimal power flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas, and hydro plants) over control areas of transmission networks, can result in grid instability and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by our simulations of real grids, is considered undesirable in power engineering practice. Possibly, it can lead to a risky outcome that compromises grid stability—line tripping. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our chance-constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CC-OPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic redispatch. CC-OPF allows efficient implementation, e.g., solving a typical instance over the 2746-bus Polish network in 20 seconds on a standard laptop.

We propose a pivotal method for estimating high-dimensional sparse linear regression models, where the overall number of regressors p is large, possibly much larger than n, but only s regressors are significant. The method is a modification of the lasso, called the square-root lasso. The method is pivotal in that it neither relies on the knowledge of the standard deviation σ nor does it need to pre-estimate σ. Moreover, the method does not rely on normality or sub-Gaussianity of noise. It achieves near-oracle performance, attaining the convergence rate σ{(s/n) log p} 1/2 in the prediction norm, and thus matching the performance of the lasso with known σ. These performance results are valid for both Gaussian and non-Gaussian errors, under some mild moment restrictions. We formulate the square-root lasso as a solution to a convex conic programming problem, which allows us to implement the estimator using efficient algorithmic methods, such as interior-point and first-order methods.

In this paper we present the first example of a primitive, totally geodesic subvariety F ⊂ ℳg,n with dim(F) > 1. The variety we consider is a surface F ⊂ ℳ1,3 defined using the projective geometry of plane cubic curves. We also obtain a new series of Teichmüller curves in ℳ₄, and new SL₂(ℝ)-invariant varieties in the moduli spaces of quadratic differentials and holomorphic 1-forms.

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let

We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.