Sequence Of Flips Research Articles

A note on a flip-connected class of generalized domino tilings of the box [0,2]n

Let n,d∈N and n>d. An (n−d)-domino is a box I1×⋯×In such that Ij∈{[0,1],[1,2]} for all j∈N⊂[n] with |N|=d and Ii=[0,2] for every i∈[n]∖N. If A and B are two (n−d)-dominoes such that A∪B is an (n−(d−1))-domino, then A,B is called a twin pair. If C,D are two (n−d)-dominoes which form a twin pair such that A∪B=C∪D and {C,D}≠{A,B}, then the pair C,D is called a flip of A,B. A family D of (n−d)-dominoes is a tiling of the box [0,2]n if interiors of every two members of D are disjoint and ⋃B∈DB=[0,2]n. An (n−d)-domino tiling D′ is obtained from an (n−d)-domino tiling D by a flip, if there is a twin pair A,B∈D such that D′=(D∖{A,B})∪{C,D}, where C,D is a flip of A,B. A family of (n−d)-domino tilings of the box [0,2]n is flip-connected, if for every two members D,E of this family the tiling E can be obtained from D by a sequence of flips. In the paper a flip-connected class of (n−d)-domino tilings of the box [0,2]n is described.

Flips on homologous orientations of surface graphs with prescribed forbidden facial cycles

Let G be a graph embedded on an orientable surface. Given a class C of facial cycles of G as a forbidden class, we give a sufficient and necessary condition such that an α-orientation (orientation with prescribed out-degrees) of G can be transformed into another by a sequence of flips on non-forbidden cycles and further give an explicit formula for the minimum number of such flips.

A state-flipped approach to complete synchronization of Boolean networks

State flipping is an effective control method that has just recently been proposed. In this paper, the state-flipped mechanism is introduced to implement the complete synchronization problem of BNs by use of the semi-tensor product (STP) of matrices. Firstly, an algebraic expression of BNs with state-flipped mechanism is constructed, and under such representation, the validation conditions for the existence of the complete synchronization of drive-response BNs are derived. Subsequently, the synchronization results with state-flipped control of the above system is further extended to more general BNs by analyzing the synchronous flip sequence. Meanwhile, some algorithms are developed to find and determine the corresponding sequence of synchronous flips and the set of minimum synchronous flips. Finally, several typical numerical examples are offered in detail to demonstrate the validity of obtained results. To sum up, the current research may further enrich and develop the study of network synchronization problem.

Flip graphs for infinite type surfaces

We associate to triangulations of infinite type surfaces a type of flip graph where simultaneous flips are allowed. Our main focus is on understanding exactly when two triangulations can be related by a sequence of flips. A consequence of our results is that flip graphs for infinite type surfaces have uncountably many connected components.

Flip Graphs of Stacked and Flag Triangulations of the 2-Sphere

It is well-known that the flip graph of $n$-vertex triangulated $2$-spheres is connected, i.e., each pair of $n$-vertex triangulated $2$-spheres can be turned into each other by a sequence of edge flips for each $n\ge 4$. In this article, we study various induced subgraphs of this graph. In particular, we prove that the subgraph of $n$-vertex flag $2$-spheres distinct from the double cone is still connected. In contrast, we show that the subgraph of $n$-vertex stacked $2$-spheres has at least as many connected components as there are trees on $\lfloor\frac{n-5}{3}\rfloor$ nodes with maximum node-degree at most four.

Domino tilings of cylinders: the domino group and connected components under flips

We consider domino tilings of three-dimensional cubiculated regions. A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is an integer associated to each tiling, which is invariant under flips. A balanced quadriculated disk $D$ is regular if whenever two tilings $t_0$ and $t_1$ of $D \times [0,N]$ have the same twist then $t_0$ and $t_1$ can be joined by a sequence of flips provided some extra vertical space is allowed. We define the domino group of a quadriculated disk and prove that $D$ is regular if and only if its domino group is isomorphic to $Z \oplus Z/(2)$. We prove that a rectangle $D = [0,L] \times [0,M]$ with $LM$ even is regular if and only if $\min\{L,M\} \ge 3$ and conjecture that in general large disks are regular. In the cases where $D$ is not regular we prove partial results concerning the structure of the domino group: the group is not abelian and has exponential growth. We also prove that if $D$ is regular then the extra vertical space necessary to join by flips two tilings of $D \times [0,N]$ with the same twist depends only on $D$, not on the height $N$.

On the canonical bundle formula and log abundance in positive characteristic

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing conjecture for three-dimensional klt pairs over any algebraically closed field k of characteristic $$p>5$$ , which in turn implies the termination of any sequence of three-dimensional flips in the pseudo-effective case. We also show the log abundance conjecture for threefolds over k when the nef dimension is not maximal, and the base point free theorem for threefolds over $$\mathbb {{\overline{F}}}_p$$ when $$p>2$$ .

Biases and Variability from Costly Bayesian Inference.

When humans infer underlying probabilities from stochastic observations, they exhibit biases and variability that cannot be explained on the basis of sound, Bayesian manipulations of probability. This is especially salient when beliefs are updated as a function of sequential observations. We introduce a theoretical framework in which biases and variability emerge from a trade-off between Bayesian inference and the cognitive cost of carrying out probabilistic computations. We consider two forms of the cost: a precision cost and an unpredictability cost; these penalize beliefs that are less entropic and less deterministic, respectively. We apply our framework to the case of a Bernoulli variable: the bias of a coin is inferred from a sequence of coin flips. Theoretical predictions are qualitatively different depending on the form of the cost. A precision cost induces overestimation of small probabilities, on average, and a limited memory of past observations, and, consequently, a fluctuating bias. An unpredictability cost induces underestimation of small probabilities and a fixed bias that remains appreciable even for nearly unbiased observations. The case of a fair (equiprobable) coin, however, is singular, with non-trivial and slow fluctuations in the inferred bias. The proposed framework of costly Bayesian inference illustrates the richness of a ‘resource-rational’ (or ‘bounded-rational’) picture of seemingly irrational human cognition.

Domino Tilings of Cylinders: Connected Components under Flips and Normal Distribution of the Twist

We consider domino tilings of $3$-dimensional cubiculated regions. A three-dimensional domino is a $2\times 1\times 1$ rectangular cuboid. We are particularly interested in regions of the form $\mathcal{R}_N = \mathcal{D} \times [0,N]$ where $\mathcal{D} \subset \mathbb{R}^2$ is a fixed quadriculated disk. In dimension $3$, the twist associates to each tiling $\mathbf{t}$ an integer $\operatorname{Tw}(\mathbf{t})$. We prove that, when $N$ goes to infinity, the twist follows a normal distribution.
 A flip is a local move: two neighboring parallel dominoes are removed and placed back in a different position. The twist is invariant under flips. A quadriculated disk $\mathcal{D}$ is regular if, whenever two tilings $\mathbf{t}_0$ and $\mathbf{t}_1$ of $\mathcal{R}_N$ satisfy $\operatorname{Tw}(\mathbf{t}_0) = \operatorname{Tw}(\mathbf{t}_1)$, $\mathbf{t}_0$ and $\mathbf{t}_1$ can be joined by a sequence of flips provided some extra vertical space is allowed.
 Many large disks are regular, including rectangles $\mathcal{D} = [0,L] \times [0,M]$ with $LM$ even and $\min\{L,M\} \ge 3$. For regular disks, we describe the larger connected components under flips of the set of tilings of the region $\mathcal{R}_N = \mathcal{D} \times [0,N]$. As a corollary, let $p_N$ be the probability that two random tilings $\mathbf{T}_0$ and $\mathbf{T}_1$ of $\mathcal{D} \times [0,N]$ can be joined by a sequence of flips conditional to their twists being equal. Then $p_N$ tends to $1$ if and only if $\mathcal{D}$ is regular.
 Under a suitable equivalence relation, the set of tilings has a group structure, the {\em domino group} $G_{\mathcal{D}}$. These results illustrate the fact that the domino group dictates many properties of the space of tilings of the cylinder $\mathcal{R}_N = \mathcal{D} \times [0,N]$, particularly for large $N$.

Why Humans Are Vulnerable to Conspiracy Theories.

Humans seem drawn to dark conspiracy theories, often in favor of the simple truth In response to the COVID-19 pandemic, misinformation and conspiracy theorizing have surged President Trump, for example, praised the antimalarial drug hydroxychloroquine as a "game changer" despite scant empirical evidence of its efficacy and safety for the treatment of SARS-CoV-2 Others in the administration have promoted the unsubstantiated theory that SARS-CoV-2 originated in a Chinese lab, despite scientific consensus that the virus likely originated in an animal source before zoonotic transfer and that no evidence indicates that the virus emerged through deliberate lab manipulation of a related virus Here, Friedman discusses why humans are vulnerable to conspiracy theories

Fano 4-folds with rational fibrations

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X-->Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a non-zero, non-big movable divisor on X. Our main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X-->Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or rho(X) is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

Fano 4-folds with rational fibrations

We study (smooth, complex) Fano 4-folds X having a rational contraction of fiber type, that is, a rational map X −−→Y that factors as a sequence of flips followed by a contraction of fiber type. The existence of such a map is equivalent to the existence of a nonzero, nonbig movable divisor on X. Our main result is that if Y is not ℙ1 or ℙ2, then the Picard number ρX of X is at most 18, with equality only if X is a product of surfaces. We also show that if a Fano 4-fold X has a dominant rational map X −−→Z, regular and proper on an open subset of X, with dim(Z)=3, then either X is a product of surfaces, or ρX is at most 12. These results are part of a program to study Fano 4-folds with large Picard number via birational geometry.

A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm (with O(n^8) being a crude bound on the run-time) to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.

Geometric quadrangulations of a polygon

We deal with a geometric quadrangulation of a polygon P. We define a new notion called “the spirality” of P, which measures how close P is from being a convex polygon. Using the spirality, we describe (1) a condition of P to admit a geometric quadrangulation, and (2) a condition of P guaranteeing that any two geometric quadrangulations on P can be related by a sequence of edge flips.

Fakin’ Flips

SummaryA random sequence of many coin flips should have some long runs where the flips have the same value. How long? We develop a recurrence relation to compute the probability distribution of the length of the longest run.

On tetrahedralisations of generalised Chazelle polyhedra with interior Steiner points

In 1984, Chazelle constructed a family of non-convex polyhedra which can be parametrised by the number N of the defining line segments and the vertical shift ε between the two saddle surfaces. Such a polyhedron is known as the Chazelle polyhedron (Chazelle, 1984). It establishes a quadratic lower bound on the minimum number of convex pieces for the 3d convex decomposition problem. Chazelle polyhedra are not decomposable without using addition points, so-called Steiner points. In this paper, we study the problem of triangulating a Chazelle polyhedron with only interior Steiner points. It is motivated by a crucial step in tetrahedral mesh generation in which a set of arbitrary constraints (edges or faces) need to be entirely preserved. We first “cut off” the volume of the Chazelle polyhedron by removing the regions that can be tetrahedralised without adding Steiner points. This leads to a 3d non-convex polyhedron whose vertices belong to the two shifted saddle surfaces. We call it the reduced Chazelle polyhedron. We then give a set of (N+1)2 interior Steiner points that ensures the existence of a tetrahedralisation of the reduced Chazelle polyhedron with 4(N+1) input line segments. The proof uses an intuitive relation between a 3d tetrahedralisation and a sequence of 2d flips. In particular, we design an edge splitting and flipping algorithm and prove that it will result a tetrahedralisation of the reduced Chazelle polyhedron. Finally, we construct a family of polyhedra, so-called generalised Chazelle polyhedra in which the set of reduced Chazelle polyhedra is a subset of it. We prove that our placement of interior Steiner points also applies to tetrahedralise generalised Chazelle polyhedra.

Perceptions of randomness in binary sequences: Normative, heuristic, or both?

When people consider a series of random binary events, such as tossing an unbiased coin and recording the sequence of heads (H) and tails (T), they tend to erroneously rate sequences with less internal structure or order (such as HTTHT) as more probable than sequences containing more structure or order (such as HHHHH). This is traditionally explained as a local representativeness effect: Participants assume that the properties of long sequences of random outcomes—such as an equal proportion of heads and tails, and little internal structure—should also apply to short sequences. However, recent theoretical work has noted that the probability of a particular sequence of say, heads and tails of length n, occurring within a larger (>n) sequence of coin flips actually differs by sequence, so P(HHHHH) <P(HTTHT). In this alternative account, people apply rational norms based on limited experience. We test these accounts. Participants in Experiment 1 rated the likelihood of occurrence for all possible strings of 4, 5, and 6 observations in a sequence of coin flips. Judgments were better explained by representativeness in alternation rate, relative proportion of heads and tails, and sequence complexity, than by objective probabilities. Experiments 2 and 3 gave similar results using incentivized binary choice procedures. Overall the evidence suggests that participants are not sensitive to variation in objective probabilities of a sub-sequence occurring; they appear to use heuristics based on several distinct forms of representativeness.

Simplicial moves on balanced complexes

We introduce a notion of cross-flips: local moves that transform a balanced (i.e., properly (d+1)-colored) triangulation of a combinatorial d-manifold into another balanced triangulation. These moves form a natural analog of bistellar flips (also known as Pachner moves). Specifically, we establish the following theorem: any two balanced triangulations of a closed combinatorial d-manifold can be connected by a sequence of cross-flips. Along the way we prove that for every m≥d+2 and any closed combinatorial d-manifold M, two m-colored triangulations of M can be connected by a sequence of bistellar flips that preserve the vertex colorings.

Duality, Bridgeland wall-crossing and flips of secant varieties

Let [Formula: see text] denote the [Formula: see text]-uple Veronese surface. After studying some general aspects of the wall-crossing phenomena for stability conditions on surfaces, we are able to describe a sequence of flips of the secant varieties of [Formula: see text] by embedding the blow-up [Formula: see text] into a suitable moduli space of Bridgeland semistable objects on [Formula: see text].

Influence of thermal vibrations on polarization switching in the model of local fields

A microscopic model to simulate the polarization dynamics, the model of local fields, is improved by considering thermal vibrations. The model is based on a sequence of single dipole flips which are thermally activated. The time to flip a single dipole depends on its deterministic transition rate which depends on the local electric field and on a probabilistic factor. In each step, the dipole with the shortest flip time is switched. Thermal vibrations of the dipoles cause changes of the distances between the dipoles. The variation of distances effects variations of the local field at the dipoles. In the framework of the extended model, these variations are considered by multiplying the local fields in each step with a Gaussian distributed random number. The model is applied to simulate polarization switching and polarization hysteresis loops of two and of three dimensional systems based on the barium titanate structure. The simulations yield intrinsic dead layers close to the electrodes and around defects which cannot be switched even in very strong fields. These nonswitchable layers are nuclei for domains and thus nuclei for polarization switching. The switching time of the system vastly decreases with the amplitude of the thermal vibrations. Moreover, the thermal vibrations enable the polarization switching in low external fields and decrease the coercive fields.