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In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds. [ABSTRACT FROM AUTHOR]

Abstract We show that every planar graph G has a 2-fold 9-coloring. In particular, this implies that G has fractional chromatic number at most 9 2. This is the first proof (independent of the 4 Color Theorem) that there exists a constant k < 5 such that every planar G has fractional chromatic number at most k. [ABSTRACT FROM AUTHOR]

It is important to increase energy absorption of a tube under axial loading without using any reinforcement material. Also, such a process results directly in an increase of specific energy absorbing (SEA) capability of the tube. Based on this consideration, applicability of annular rolling process to thin-walled aluminium 6063-T5 tubes (D/t = 58/1.5) has been investigated experimentally and numerically. The study was conducted with two groups of tubular specimens. After determining the appropriate restricted fold length (RFL) value from short length tubes, the data were applied to long tubes. From the experimental studies of long-length annular-rolled tube, folding initiates within the RFL, and it enhances not only maximum folding, but also upper and lower force peaks in force-displacement curve. Thus, 23% improvement in SEA was obtained. Moreover, annular rolled tubes exhibited more outward folding with respect to that of the base tube, and this folding behaviour was used to explain results of the finite element (FE) analyses. It is concluded that annular rolling process is promising for thin-walled circular tubes for enhancing their energy absorption capabilities. [ABSTRACT FROM AUTHOR]

Flexural-slip is considered as an important mechanism during folding and a general conceptual and qualitative understanding has been provided by various field studies. However, quantitative evidence of the importance of the flexural-slip mechanism during fold evolution is sparse due to the lack of suitable strain markers. In this study, 2D finite element analysis is used to overcome these disadvantages and to simulate flexural-slip during visco-elastic buckle folding. Variations of single and multilayer layer fold configurations are investigated, showing that flexural-slip is most likely to occur in effective single layer buckle folds, where slip occurs between contacts of competent layers. Based on effective single layer buckle folds, the influence of the number of slip surfaces, the degree of mechanical coupling (based on the friction coefficient), and layer thickness, on the resulting slip distribution are investigated. The results are in agreement with the conceptual flexural-slip model and show that slip is initiated sequentially during the deformation history and is maximum along the central slip surface of the fold limb. The cumulative amount of slip increases as the number of slip surfaces is increased. For a lower degree of mechanical coupling increased slip results in different fold shapes, such as box folds, during buckling. In comparison with laboratory experiments, geometrical relationships and field observations, the numerical modeling results show similar slip magnitudes. It is concluded that flexural-slip should represent a significant contribution during buckle folding, affecting the resulting fold shape for increased amounts of slip. [ABSTRACT FROM AUTHOR]

Ancient Egyptian papyri are often folded, rolled up or kept as small packages, sometimes even sealed. Physically unrolling or unfolding these packages might severely damage them. We demonstrate a way to get access to the hidden script without physical unfolding by employing computed tomography and mathematical algorithms for virtual unrolling and unfolding. Our algorithmic approaches are combined with manual interaction. This provides the necessary flexibility to enable the unfolding of even complicated and partly damaged papyrus packages. In addition, it allows us to cope with challenges posed by the structure of ancient papyrus, which is rather irregular, compared to other writing substrates like metallic foils or parchment. Unfolding of packages is done in two stages. In the first stage, we virtually invert the physical folding process step by step until the partially unfolded package is topologically equivalent to a scroll or a papyrus sheet folded only along one fold line. To minimize distortions at this stage, we apply the method of moving least squares. In the second stage, the papyrus is flattened, which requires the definition of a medial surface. We have applied our software framework to several papyri. In this work, we present the results of applying our approaches to mockup papyri that were either rolled or folded along perpendicular fold lines. In the case of the folded papyrus, our approach represents the first attempt to address the unfolding of such complicated folds. [ABSTRACT FROM AUTHOR]

Given an arithmetic function ƒ : N → N, let the k-fold iterate of ƒ be defined by fo(n) = n and fk(n) = f(fk-i(n)) for each integer k ≥ 1. Let ω(1) = 0 and, for each integer n ≥ 2, let ω(n) stand for the number of distinct prime factors of n. Here, we examine the distribution of the functions ω(fk(n)) for various arithmetic functions ƒ. [ABSTRACT FROM AUTHOR]

An innovative method of shaping free form buildings composed of many original complete parts roofed with profiled steel sheets transformed effectively into hyperbolic paraboloid strips is proposed in the paper. The method uses tetrahedral compositions of various kinds connected one to another with their relevant faces contained in planes of a spatial network called control structure. The control structures enable: a) controlling structural systems and architectural forms of the complex buildings as well as their morphological systems, b) arranging regularly the roof shells in three-dimensional space, c) integrating the roof and elevation forms as well as their consistent composition into the natural and built environments. The effectiveness of the transformations consists in the fact that displacements of adjacent folds in the sheeting are big, whereas strain is small. The effective transformations affect the fold's shape in such a way that the folds are contracted in the middle on their length and stretched at their crosswise ends depending on supporting conditions produced by curvature and a mutual location of skew directrices. The supporting conditions are diversified along each directrix, so the diversified arrangement of the supporting points of the shell folds to the skew directrices has to be calculated. For this purpose, an innovative computer program was created. Because of the diversified fold shape changes, two individual shell sheeting's cannot be connected one to another with their transversal borders to obtain a smooth shell. However, many shells can be set together with their transversal edges to achieve a compound ribbed roof of medium or large span. Thus, the resultant shell roof has to be characterized by edges disturbing its smoothness. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of -fold obstinate ideals of an -algebra and investigate the relations between the obstinate ideals and the other ideals of an -algebras and also we consider the quotient algebras induced by -fold obstinate ideals and prove some related theorems. In addition, the extension theorem of -fold obstinate ideals are obtained. Also, several characterizations of these -fold ideals are given. Finally, we construct algorithm for studing the structure of the -fold obstinate ideals in finite -algebras. [ABSTRACT FROM AUTHOR]

The article discusses the spatial occurrence of folding instability of moduli-patterned and bilayered membrane under compressive stresses. Topics discussed include ordered micropatterns of thin metal film on soft substrate PDMS (polydimethylsiloxane), pattern formation with mechanical instability, and controlling wrinkles or folds with help of heterogeneous induction of pre-strain.

Over the past few decades, folding paper has extended beyond the origami deployable applications to reach the engineering field. Nevertheless, mechanical information about paper behavior is still lacking, especially during folding/unfolding. This article proposes an approach to characterize the paper fold behavior in order to extract the material data that will be needed for the simulation of folding and to go a step further the single kinematics of origami mechanisms. The model developed herein from simple experiments for the fold behavior relies on a macroscopic local hinge with a nonlinear torsional spring. Though validated with only straight folds, the model is still applicable in the case of curved folds thanks to the locality principle of the mechanical behavior. The influence of both the folding angle and the fold length is extracted automatically from a set of experimental values exhibiting a deterministic behavior and a variability due to the folding process. The goal is also to propose a methodology that may extend the simple case of the paper crease, or even the case of thin material sheets, and may be adapted to other identification problems. [ABSTRACT FROM AUTHOR]

The origami waterbomb base (WB) is a single-vertex bistable mechanism that can be generalized to accommodate various geometric, kinematic, and kinetic needs. The traditional WB consists of a square sheet that has four mountain folds alternating with five valley folds (eight folds total) around the vertex in the center of the sheet. This special case mechanism can be generalized to create two classes of waterbomb-base-type mechanisms that allow greater flexibility for potential application. The generalized WB maintains the pattern of alternating mountain and valley folds around a central vertex but it is not restricted to eight total folds. The split-fold waterbomb base (SFWB) is made by splitting each fold of a general WB into two "half folds" of the same variety as the parent fold. This study develops kinematic, potential energy, and force-deflection models for the rigid-foldable, developable, symmetric cases of the generalized WB and the SFWB, and investigates the relative effects of numbers of folds and split-fold panel size, on device behavior. The effect of selected key parameters is evaluated, and equations are provided to enable the exploration of other important parameters that may be of interest in the design and analysis of specific mechanisms. The similarities and differences between the two general forms are discussed, including tunability of the bistable and force-deflection behavior of each. [ABSTRACT FROM AUTHOR]

Origami patterns, including the rigid origami patterns in which flat inflexible sheets are joined by creases, are primarily created for zero-thickness sheets. In order to apply them to fold structures such as roofs, solar panels, and space mirrors, for which thickness cannot be disregarded, various methods have been suggested. However, they generally involve adding materials to or offsetting panels away from the idealized sheet without altering the kinematic model used to simulate folding. We develop a comprehensive kinematic synthesis for rigid origami of thick panels that differs from the existing kinematic model but is capable of reproducing motions identical to that of zero-thickness origami. The approach, proven to be effective for typical origami, can be readily applied to fold real engineering structures. [ABSTRACT FROM AUTHOR]

For an arbitrary statistical mixture of τ-quantizations, we obtain an explicit expression for the leading parts of the equilibrium density matrix and for the corresponding Wigner function of a harmonic oscillator in the approach of Feynman approximations using the Chernoff theorem. Taking the oscillator Hamiltonian as an example, we determine the convergence rate for approximations of means of operators of observables depending on the approximation order and depending on the quantization rule. We demonstrate that the convergence rate of approximations of the mean of the energy operator is not uniform with respect to the Gibbs parameter. [ABSTRACT FROM AUTHOR]

The paper presents the usage of folded surfaces as parts of a building system. This type of surfaces is not often used in constructions, even though the structures get to have a very special and spectacular design. The authors present some of the most known structures using the folded surfaces as a building component. [ABSTRACT FROM AUTHOR]

We study some symplectic geometric aspects of rationally connected 4-folds. As a corollary, we prove that any smooth projective 4-fold symplectic deformation equivalent to a Fano 4-fold of pseudo-index at least 2 or a rationally connected 4-fold whose second Betti number is 2 is rationally connected. [ABSTRACT FROM AUTHOR]

Piecewise Straight Fold is proposed for large solar sail membranes to simultaneously realize the high packaging efficiency and the simple folding. The fold pattern is based on Spiral Fold to reduce the deviation of the fold line, which improve the packaging efficiency. In addition, the fold pattern consists of piecewise straight fold lines, which approximate the Spiral Fold line, in order to simplify the folding. A simple manufacturing process of Piecewise Straight Fold is developed based on the Z-fold membrane. The preliminary experimental results show the feasibility of the Piecewise Straight Fold and its simple manufacturing process, where the high packaging efficiency is also verified. The wrapping fold experiments for the solar power sail membrane is demonstrated by using the Piecewise Straight Fold. In the wrapping fold experiments, the applicability of the Piecewise Straight Fold to the solar power sail membrane is verified. [ABSTRACT FROM AUTHOR]

We present a numerical study of the shape taken by a spherical elastic surface when the volume it encloses is decreased. For the range of 2D parameters where such a surface may model a thin shell of an isotropic elastic material, the mode of deformation that develops a single depression is investigated in detail. It occurs via buckling from sphere toward an axisymmetric dimple, followed by a second buckling where the depression loses its axisymmetry through folding along portions of meridians. For the thinnest shells, a direct transition from the spherical conformation to the folded one can be observed. We could exhibit unifying master curves for the relative volume variation at which first and second buckling occur, and clarify the role of Poisson's ratio. In the folded conformation, the number of folds and inner pressure are investigated, allowing us to infer shell features from mere observation and/or knowledge of external constraints. [ABSTRACT FROM AUTHOR]

The article offers information on hexaflexagons, edge flexagons and point flexagons. It states that hexaflexagon is an example of edge flexagon, a folded band of hinged convex polygon. It says that certain edge flexagons can be classified using their characteristic polygon, which determines the association of hinge edges and vertex figures. Moreover, it considers point flexagons as the two-folds of edge flexagons.

Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewise-smooth flows. These are global bifurcations which occur when distinguished orbits become tangent to surfaces of discontinuity, called switching manifolds. The key idea of the paper is to attribute sliding bifurcations to singularities in the manifold's projection along the flow, namely, to points where the projection contains folds, cusps, and two-folds (saddles and bowls). From the possible local configurations of orbits we obtain sliding bifurcations. In this way we derive a complete classification of generic one-parameter sliding bifurcations at a smooth codimension one switching manifold in n dimensions for n 3. We uncover previously unknown sliding bifurcations, all of which are catastrophic in nature. We also describe how the method can be extended to sliding bifurcations of codimension two or higher. [ABSTRACT FROM AUTHOR]

The article discusses the use of origami techniques in science as profiled in the same issue, focusing on research by Felton et al. into the construction of self-folding robots and by Silverberg et al. that developed a mechanical metamaterial from tessellated units.