Twin-width I: Tractable FO Model Checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Strong Euler well-composedness

In this paper, we define a new flavour of well-composedness, called strong Euler well-composedness. In the general setting of regular cell complexes, a regular cell complex of dimension n is strongly Euler well-composed if the Euler characteristic of the link of each boundary cell is 1, which is the Euler characteristic of an $$(n-1)$$ ( n - 1 ) -dimensional ball. Working in the particular setting of cubical complexes canonically associated with $$n$$ n D pictures, we formally prove in this paper that strong Euler well-composedness implies digital well-composedness in any dimension $$n\ge 2$$ n ≥ 2 and that the converse is not true when $$n\ge 4$$ n ≥ 4 .

FOUR-DIMENSIONAL BALL IN A GRAPHIC REPRESENTATION

The paper presents a method for geometric modelling of a four-dimensional ball. For this, the regularities of the change in the shape of the projections of simple geometric images of two-dimensional and three-dimensional spaces during rotation are considered. Rotations of a segment and a circle around an axis are considered; it is shown that during rotation the shape of their projections changes from the maximum value to the degenerate projection. It was found that the set of points of the degenerate projection belongs to the axis of rotation, and each n-dimensional geometric image during rotation forms a body of a higher dimension, that is, one that belongs to (n + 1) -dimensional space. Identified regularities are extended to the four-dimensional space in which the ball is placed. It is shown that the axis of rotation of the ball will be a degenerate projection in the form of a circle, and the ball, when rotating, changes its size from a volumetric object to a flat circle, then increases again, but in the other direction (that is, it turns out), and then in reverse order to its original position. This rotation is more like a deformation, and such a ball of four-dimensional space is a hypersphere. For geometric modelling of the hypersphere and the possibility of its projection image, the article uses the vector model proposed by P.V. Filippov. The coordinate system 0xyzt is defined. The algebraic equation of the hypersphere is given by analogy with the three-dimensional space along certain coordinates of the center a, b, c, d. A variant of hypersection at t = 0 is considered, which confirms by equations obtaining a two-dimensional ball of three-dimensional space, a point (a ball of zero radius), which coincides with the center of the ball, or an imaginary ball. For the variant t = d, the equation of a two-dimensional ball is obtained, in which the radius is equal to R and the coordinates of all points along the 0t axis are equal to d. The variant of hypersection t = k turned out to be interesting, in which the equation of a two-dimensional sphere was obtained, in which the coordinates of all points along the 0t axis are equal to k, and the radius is . Horizontal vector projections of hypersection are constructed for different values of k. It is concluded that the set of horizontal vector projections of hypersections at t = k defines an ellipse.

The geometric invariants for the spectrum of the Stokes operator

For a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion for the integral of the trace of the Stokes semigroup $$e^{-t S}$$ e - t S as $$t\rightarrow 0^+$$ t → 0 + . These coefficients (i.e., spectral invariants) provide precise information for the volume of the domain $$\Omega $$ Ω and the surface area of the boundary $$\partial \Omega $$ ∂ Ω by the spectrum of the Stokes problem. As an application, we show that an n-dimensional ball is uniquely determined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.

A Noise Study of the PSW Signature Family: Patching DRS with Uniform Distribution †

At PKC 2008, Plantard et al. published a theoretical framework for a lattice-based signature scheme, namely Plantard–Susilo–Win (PSW). Recently, after ten years, a new signature scheme dubbed the Diagonal Reduction Signature (DRS) scheme was presented in the National Institute of Standards and Technology (NIST) PQC Standardization as a concrete instantiation of the initial work. Unfortunately, the initial submission was challenged by Yu and Ducas using the structure that is present on the secret key noise. In this paper, we are proposing a new method to generate random noise in the DRS scheme to eliminate the aforementioned attack, and all subsequent potential variants. This involves sampling vectors from the n-dimensional ball with uniform distribution. We also give insight on some underlying properties which affects both security and efficiency on the PSW type schemes and beyond, and hopefully increase the understanding on this family of lattices.

The Validaty and Practicality of Media for Teaching Atomic Bolls for Class X Students on the Material of Atomic Strubture

One effort to improve student learning outcomes is to choose teaching media that are appropriate and attract students' attention. The media developed is a three-dimensional ball teaching media.The purpose of this study is to develop three-dimensional atomic ball teaching media, to find out the validity level of three-dimensional atomic ball teaching media, to find out the practicalities of the three-dimensional atomic ball teaching media that have been developed. This research is a type of Research and Development (R & D) research using a 4D development model, namely define, design, develop, and dessiminatebut is limited to the develop stage. The teaching media developed were validated by media experts and material experts. The subject of this research trial was a chemistry teacher to find out the practicality of three-dimensional atomic balls teaching media, and 20 students of East Siantan 1 High School to find out the practicality of three-dimensional atomic balls. The method used to analyze data uses qualitative and quantitative descriptive techniques. The results showed that the validity of three-dimensional atomic ball teaching media for atomic structure material was obtained from the results of a media expert's assessment of 95.19% with very valid criteria, and the material expert's assessment was 81.23% with very valid criteria.The practicality test results were obtained from teacher responses of 92.5% with very practical criteria, and 81.41% student responses with very practical criteria. Based on the results of the study it can be concluded that atomic ball teaching media in three dimensions of atomic structure are worthy of being used as learning media by teachers and students.