Minkowski Sum Based Lattice Construction for Multivariate Simultaneous Coppersmith’s Technique and Applications to RSA

Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

Journal of High Energy Physics ◽

10.1007/jhep10(2020)054 ◽

2020 ◽

Vol 2020 (10) ◽

Cited By ~ 1

Author(s):

Song He ◽

Zhenjie Li ◽

Prashanth Raman ◽

Chi Zhang

Keyword(s):

Large Class ◽

Configuration Space ◽

Moduli Space ◽

Cluster Algebras ◽

Configuration Spaces ◽

Minkowski Sum ◽

Canonical Forms ◽

Infinite Class ◽

Special Cases ◽

Generalized Associahedra

Abstract Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

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G. Czédli’s tolerance factor lattice construction and weak ordered relations

Algebra Universalis ◽

10.1007/s00012-021-00712-x ◽

2021 ◽

Vol 82 (2) ◽

Author(s):

Sándor Radeleczki

Keyword(s):

Concept Lattice ◽

Tolerance Factor ◽

Macneille Completion ◽

Lattice Construction

AbstractG. Czédli proved that the blocks of any compatible tolerance T of a lattice L can be ordered in such a way that they form a lattice L/T called the factor lattice of L modulo T. Here we show that the Dedekind–MacNeille completion of the lattice L/T is isomorphic to the concept lattice of the context (L, L, R), where R stands for the reflexive weak ordered relation $$ \mathord {\le } \circ T$$ ≤ ∘ T . Weak ordered relations constitute the generalization of the ordered relations introduced by S. Valentini. Reflexive weak ordered relations can be characterized as compatible reflexive relations $$R\subseteq L^{2}$$ R ⊆ L 2 satisfying $$R=\ \mathord {\le } \circ R\circ \mathord {\le } $$ R = ≤ ∘ R ∘ ≤ .

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Analysis of Lattice Single Layer Cylindrical Structures

International Journal of Space Structures ◽

10.1177/026635118700200203 ◽

1987 ◽

Vol 2 (2) ◽

pp. 87-91 ◽

Cited By ~ 5

Author(s):

Istvan HegedüS

Keyword(s):

Stress Distribution ◽

Matrix Method ◽

Regular Ring ◽

Lattice Structure ◽

Single Layer ◽

Cylindrical Structures ◽

Lattice Construction

A matrix method is presented for the calculation of bar forces in a single layer lattice cylinder composed of regular ring polygons and symmetrically arranged bracing bars. Substantial differences occur between the stress distribution in the lattice structure and that of the membrane cylinder under the same load. Therefore, a membrane cylinder cannot be considered as a replacement continuum for the lattice construction. The purpose of the paper is to draw attention to the danger in the utilisation, without due caution, of this analogy.

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(Symplectic) leaves and (5d Higgs) branches in the Poly(go)nesian Tropical Rain Forest

Journal of High Energy Physics ◽

10.1007/jhep11(2020)124 ◽

2020 ◽

Vol 2020 (11) ◽

Author(s):

Marieke van Beest ◽

Antoine Bourget ◽

Julius Eckhard ◽

Sakura Schäfer-Nameki

Keyword(s):

Rain Forest ◽

Tropical Rain Forest ◽

Gauge Theories ◽

Edge Coloring ◽

Field Theories ◽

Minkowski Sum ◽

Coulomb Branch ◽

Superconformal Field Theories ◽

Minkowski Sums ◽

Symplectic Leaves

Abstract We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the (p, q) 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we identify the symplectic leaves of the Higgs branch and thereby the entire foliation structure. In the case of strictly toric polygons, this approach reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums.

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Building symmetry-enriched topological phases from a bipartite lattice construction and anyon condensation

Physical Review B ◽

10.1103/physrevb.98.214416 ◽

2018 ◽

Vol 98 (21) ◽

Cited By ~ 1

Author(s):

Jong Yeon Lee ◽

Ari M. Turner ◽

Ashvin Vishwanath

Keyword(s):

Topological Phases ◽

Lattice Construction

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Statistical Tolerance Analysis With Sensitivities Established With Tolerance-Maps

Volume 2B: 44th Design Automation Conference ◽

10.1115/detc2018-85108 ◽

2018 ◽

Author(s):

Aniket N. Chitale ◽

Joseph K. Davidson ◽

Jami J. Shah

Keyword(s):

Tolerance Analysis ◽

Minkowski Sum ◽

Coordinate Frame ◽

Target Feature ◽

Functional Feature ◽

Design Function ◽

Statistical Tolerance ◽

The One ◽

The Relationship ◽

Statistical Tolerance Analysis

The purpose of math models for tolerances is to aid a designer in assessing relationships between tolerances that contribute to variations of a dependent dimension that must be controlled to achieve some design function and which identifies a target (functional) feature. The T-Maps model for representing limits to allowable manufacturing variations is applied to identify the sensitivity of a dependent dimension to each of the contributing tolerances to the relationship. The method is to choose from a library of T-Maps the one that represents, in its own local (canonical) reference frame, each contributing feature and the tolerances specified on it; transform this T-Map to a coordinate frame centered at the target feature; obtain the accumulation T-Map for the assembly with the Minkowski sum; and fit a circumscribing functional T-Map to it. The fitting is accomplished numerically to determine the associated functional tolerance value. The sensitivity for each contributing tolerance-and-feature combination is determined by perturbing the tolerance, refitting the functional map to the accumulation map, and forming a ratio of incremental tolerance values from the two functional T-Maps. Perturbing the tolerance-feature combinations one at a time, the sensitivities for an entire stack of contributing tolerances can be built. For certain classes of loop equations, the same sensitivities result by fitting the functional T-Map to the T-Map for each feature, one-by-one, and forming the overall result as a scalar sum. Sensitivities help a designer to optimize tolerance assignments by identifying those tolerances that most strongly influence the dependent dimension at the target feature. Since the fitting of the functional T-Map is accomplished by intersection of geometric shapes, all the T-Maps are constructed with linear half-spaces.

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Calculus rules for combinations of ellipsoids and applications

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700012211 ◽

1993 ◽

Vol 47 (1) ◽

pp. 1-12 ◽

Cited By ~ 3

Author(s):

Alberto Seeger

Keyword(s):

Convex Hull ◽

Convex Polytope ◽

Finite Family ◽

Minkowski Sum ◽

Calculus Rules ◽

Ellipsoidal Approximations

We derive formulas for the Minkowski sum, the convex hull, the intersection, and the inverse sum of a finite family of ellipsoids. We show how these formulas can be used to obtain inner and outer ellipsoidal approximations of a convex polytope.

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