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].

The reverse Holder inequality for an elementary function

For a positive function $f$ on the interval $[0,1]$, the power mean of order $p\in\mathbb R$ is defined by \smallskip\centerline{$\displaystyle\|\, f\,\|_p=\left(\int_0^1 f^p(x)\,dx\right)^{1/p}\quad(p\ne0),\qquad\|\, f\,\|_0=\exp\left(\int_0^1\ln f(x)\,dx\right).$} Assume that $0<A<B$, $0<\theta<1$ and consider the step function$g_{A<B,\theta}=B\cdot\chi_{[0,\theta)}+A\cdot\chi_{[\theta,1]}$, where $\chi_E$ is the characteristic function of the set $E$. Let $-\infty<p<q<+\infty$. The main result of this work consists in finding the term \smallskip\centerline{$\displaystyleC_{p<q,A<B}=\max\limits_{0\le\theta\le1}\frac{\|\,g_{A<B,\theta}\,\|_q}{\|\,g_{A<B,\theta}\,\|_p}.$} \smallskip For fixed $p<q$, we study the behaviour of $C_{p<q,A<B}$ and $\theta_{p<q,A<B}$ with respect to $\beta=B/A\in(1,+\infty)$.The cases $p=0$ or $q=0$ are considered separately. The results of this work can be used in the study of the extremal properties of classes of functions, which satisfy the inverse H\"older inequality, e.g. the Muckenhoupt and Gehring ones. For functions from the Gurov-Reshetnyak classes, a similar problem has been investigated in~[4].

Optimal Random Packing of Spheres and Extremal Effective Conductivity

A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

W-representation of Rainbow tensor model

We analyze the rainbow tensor model and present the Virasoro constraints, where the constraint operators obey the Witt algebra and null 3-algebra. We generalize the method of W-representation in matrix model to the rainbow tensor model, where the operators preserving and increasing the grading play a crucial role. It is shown that the rainbow tensor model can be realized by acting on elementary function with exponent of the operator increasing the grading. We derive the compact expression of correlators and apply it to several models, i.e., the red tensor model, Aristotelian tensor model and r = 4 rainbow tensor model. Furthermore, we discuss the case of the non-Gaussian red tensor model and present a dual expression for partition function through differentiation.

Deriving and Using Descriptors of Elementary Functions in Rational Protein Design

The rational design of proteins with desired functions requires a comprehensive description of the functional building blocks. The evolutionary conserved functional units constitute nature's toolbox; however, they are not readily available to protein designers. This study focuses on protein units of subdomain size that possess structural properties and amino acid residues sufficient to carry out elementary reactions in the catalytic mechanisms. The interactions within such elementary functional loops (ELFs) and the interactions with the surrounding protein scaffolds constitute the descriptor of elementary function. The computational approach to deriving descriptors directly from protein sequences and structures and applying them in rational design was implemented in a proof-of-concept DEFINED-PROTEINS software package. Once the descriptor is obtained, the ELF can be fitted into existing or novel scaffolds to obtain the desired function. For instance, the descriptor may be used to determine the necessary spatial restraints in a fragment-based grafting protocol. We illustrated the approach by applying it to well-known cases of ELFs, including phosphate-binding P-loop, diphosphate-binding glycine-rich motif, and calcium-binding EF-hand motif, which could be used to jumpstart templates for user applications. The DEFINED-PROTEINS package is available for free at https://github.com/MelvinYin/Defined_Proteins.

Studies on population balance equation involving aggregation and growth terms via symmetries

The population balance equation (PBE) is one of the most popular integro-differential equations modeled for several industrial processes. The solution to this equation is usually solved using a numerical approach as the analytical solutions of such equations are not obtained easily. Typically, the available analytical solutions are limited and are based on momentous Laplace transform. In this study, the reduced equations of the PBE are obtained via the group analysis method. Two particulate cases involving aggregation, growth and nucleation are selected, the determining equations are solved and the reduced equations are solved via approximate methods. The approximate method involves the target solution of the nonlinear evolution equation, here the PBE, to be expressed as a polynomial in an elementary function which satisfies a particular ordinary differential equation termed as an auxiliary equation.

Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions

We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was regarded as the base separate from the basis of the complex number. However, considering the properties of the quaternion, we propose two types of step derivatives in this study. The step derivative is first defined in the j direction, which includes a quaternion. Furthermore, the step derivative in the j+k2 direction is determined using the rule between bases i, j, and k defined in the quaternion. We present examples in which the definition of the j-step derivative and (j,k)-step derivative are applied to elementary functions ez, sinz, and cosz.

Numerical Differentiation and Integration

There are several reasons why numerical differentiation and integration are used. The function that integrates f (x) can be known only in certain places, which is done by taking a sample. Some supercomputers and other computer applications sometimes need numerical integration for this very reason. The formula for the function to be integrated may be known, but it may be difficult or impossible to find the antiderivation that is an elementary function. One example is the function f (x) = exp (−x2), an antiderivation that cannot be written in elementary form. It is possible to find antiderivation symbolically, but it is much easier to find a numerical approximation than to calculate antiderivation (anti-derivative). This can be used if antiderivation is given as an unlimited array of products, or if the budget would require special features that are not available to computers.