Narrow Results

We say a family of strings over an alphabet is an UMFF if every string has a unique maximal factorization over . Foundational work by Chen, Fox and Lyndon established properties of the Lyndon circ-UMFF, which is based on lexicographic ordering. Commencing with the circ-UMFF related to V-order, we then proved analogous factorization families for a further 32 Block-like binary orders. Here we distinguish between UMFFs and circ-UMFFs, and then study the structural properties of circ-UMFFs. These properties give rise to the complete construction of any circ-UMFF. We prove that any circ-UMFF is a totally ordered set and a factorization over it must be monotonic. We define atom words and initiate a study of u, v-atoms. Applications of circ-UMFFs arise in string algorithmics.

Sequences (L_n | n ≥ k), called streams, of regular languages L_n are considered, where k is some small positive integer, n is the state complexity of L_n, and the languages in a stream differ only in the parameter n, but otherwise, have the same properties. The following measures of complexity are proposed for any stream: (1) the state complexity n of L_n, that is, the number of left quotients of L_n (used as a reference); (2) the state complexities of the left quotients of L_n; (3) the number of atoms of L_n; (4) the state complexities of the atoms of L_n; (5) the size of the syntactic semigroup of L_n; and the state complexities of the following operations: (6) the reverse of L_n; (7) the star of L_n; (8) union, intersection, difference and symmetric difference of L_m and L_n; and (9) the concatenation of L_m and L_n. A stream that has the highest possible complexity with respect to these measures is then viewed as a most complex stream. The language stream (U_n(a, b, c) | n ≥ 3) is defined by the deterministic finite automaton with state set {0, 1, … , n−1}, initial state 0, set {n−1} of final states, and input alphabet {a, b, c}, where a performs a cyclic permutation of the n states, b transposes states 0 and 1, and c maps state n − 1 to state 0. This stream achieves the highest possible complexities with the exception of boolean operations where m = n. In the latter case, one can use U_n(a, b, c) and U_n(b, a, c), where the roles of a and b are interchanged in the second language. In this sense, U_n(a, b, c) is a universal witness. This witness and its extensions also apply to a large number of combined regular operations.

A regular language $L$ is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each $n\ge 4$, there exists a ternary witness of state complexity $n$ that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has ${(n-1)}^n$ elements and requires at least $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.

In view of difficulties and questions of quantum mechanics in description of motion of electrons in hydrogen atom, we here established their nonlinear theory of motion based on the true motions of electron and nucleon and the real interactions between them, in which the motion of electron is depicted by a nonlinear Schrödinger equation with a Coulomb potential, the nonlinear interaction b∣φ∣²φ is produced by the change of Coulomb interaction between nucleon and electron due to the motion of nucleon. Thus the natures of the electron are thoroughly changed relative to those in quantum mechanics due to the nonlinear interactions, it not only is stable and localized, but also possesses a wave–corpuscle duality. Meanwhile, if its eigenenergy is still quantized and distributed in accordance with the energy levels, then we can use the new theory to explain perfectly the spectrum features of hydrogen atom, which resembles quantum mechanics, but its sizes of eigenenergy are depressed relative to that in quantum mechanics. This means that the nonlinear interaction enhances the localized and stable nature of the electron. Therefore, the new nonlinear theory is successful and correct to the hydrogen atom.

A θ-curve is called almost trivial if it does not contain any non-trivial knot. A θ-curve is called strongly almost trivial if it has a planar projection which does not contain a projection of any non-trivial knot. In this paper, we introduce a method to present strongly almost trivial θ-curves. We also give an almost trivial θ-curve which may not be strongly almost trivial.

In the present paper, we construct Khovanov homology theory with arbitrary coefficients for arbitrary virtual knots. We give a definition of the complex, which is homotopy equivalent to the initial Khovanov complex in the classical case; our definition works in the virtual case as well. The method used in this work allows us to construct a Khovanov homology theory for "twisted virtual knots" in the sense of Bourgoin and Viro [4, 27] (in particular, for knots in RP³). We also generalize some results of the Khovanov homology for virtual knots with arbitrary atoms (Wehrli and Kofman–Champanerkar spanning tree, minimality problems, Frobenius extensions) and orientable ones (Rasmussen's invariant).

In this paper, we give a new proof of Vassilievs planarity criterion of a graph with vertices of degree 4 and an additional cross structure. It is also established that all height atoms are orientable. The criterion for an atom to be an orientable one is established in terms of its $f$-graph.

Some problems connected with the theory of qualitative investigation of integrable Hamiltonian systems are discussed in this paper. The main topological object under consideration is the Liouville foliation either on the whole phase space of the system or on some part of this space.

This paper contains main results on the topological classification of integrable Hamiltonian systems (in other words, on the classification of corresponding Liouville foliations). All results are cited without proof.

The main aim of this paper is to formulate a number of problems which can be solved with the help of a computer, for example, algorithmic classification of topological objects coding singularities of the system, visualization of various effects which is not completely described yet, calculation of invariants for the specific integrable systems arising in mechanics and mathematical physics.

Let be a metric space with doubling measure, L a nonnegative self-adjoint operator in satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type p_ω∈(0, 1] and ρ(t) = t^-1/ω^-1(t^-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space via the Lusin area function associated to the heat semigroup, and the BMO-type space . The authors then establish the duality between and ; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space . Characterizations of , including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let and L = -Δ+V be a Schrödinger operator, where is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L^-1/2 is bounded from H_{ω, L}(ℝⁿ) to L(ω). Moreover, if there exist q₁, q₂∈(0, ∞) such that q₁<1<q₂ and [ω(t^q₂)]^q₁ is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space H_{ω, L}(ℝⁿ), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = t^p for all t ∈ (0, ∞) and p ∈ (0, 1).

Let φ : ℝⁿ × [0,∞) → [0,∞) be a growth function such that φ(x, ⋅) is nondecreasing, φ(x, 0) = 0, φ(x, t) > 0 when t > 0, lim_t→∞φ(x, t) = ∞, and φ(⋅, t) is a Muckenhoupt A_∞(ℝⁿ) weight uniformly in t. In this paper, the authors establish the Lusin area function and the molecular characterizations of the Musielak–Orlicz Hardy space H_φ(ℝⁿ) introduced by Luong Dang Ky via the grand maximal function. As an application, the authors obtain the φ-Carleson measure characterization of the Musielak–Orlicz BMO-type space BMO_φ(ℝⁿ), which was proved to be the dual space of H_φ(ℝⁿ) by Luong Dang Ky.

Recently, both the bilinear decompositions $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$ and $h^1({ℝ}^n)\times \mathrm{bmo}({ℝ}^n)\subset L^1({ℝ}^n)+h^{log}({ℝ}^n)$ were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$, a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)$. The relationship $h_{\ast }^{\mathrm{\Phi }}({ℝ}^n)⊊h^{log}({ℝ}^n)$ is also clarified.

Let $(R,M)$ be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of $R$. Special attention is given to the set of atoms in $M\setminus M^2$ and to the existence of atoms in $M^2$. While our main interest is in local Cohen–Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight non-associate atoms having an atom in $M^2$.

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of $\delta$-type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of $\delta$-type Calderón–Zygmund operators on these Hardy-type spaces are also new.

Snider initiated the study of lattices of the class of radicals, in the sense of Kurosh and Amitsur, of associative rings. Various authors continued the investigation in more general universal classes. Recently, Fernández-Alonso et al. studied the lattice of all preradicals in R-Mod. Our definition of a preradical is weaker than theirs. In this paper, we consider the lattices of ideal maps 𝕀, preradical maps ℙ, Hoehnke radical maps ℍ and Plotkin radical maps 𝔹 in any universal class of Ω-groups (of the same type). We show that 𝕀 is a complete and modular lattice which contains atoms. In general, 𝕀 is not atomic. 𝕀 contains ℙ as a complete and atomic sublattice, whereas ℍ and 𝔹 are not sublattices of 𝕀. In its own right, ℍ is a complete and atomic lattice and 𝔹 is a complete lattice. We identify subclasses of 𝕀, ℙ and ℍ that are sublattices or preserve the meet (or join) of these respective lattices.

In this paper, we study different properties of Stone almost distributive lattice (Stone ADL), prove basic facts on a Stone ADL and derive a necessary and sufficient conditions for an ADL L to be a Stone ADL.

Some people think that carbon and sustainable development are not compatible. This textbook shows that carbon dioxide (CO₂) from the air and bio-carbon from biomass are our best allies in the energy transition, towards greater sustainability. We pose the problem of the decarbonation (or decarbonization) of our economy by looking at ways to reduce our dependence on fossil carbon (coal, petroleum, natural gas, bitumen, carbonaceous shales, lignite, peat). The urgent goal is to curb the exponential increase in the concentration of carbon dioxide in the atmosphere and hydrosphere (Figures 1.1 and 1.2) that is directly related to our consumption of fossil carbon for our energy and materials The goal of the Paris agreement (United Nations COP 21, Dec. 12, 2015) of limiting the temperature increase to 1.5 degrees (compared to the pre-industrial era, before 1800) is becoming increasingly unattainable (Intergovermental Panel on Climate Change (IPCC), report of Aug. 6, 2021). On Aug. 9, 2021 Boris Johnson, prime minister of the United Kingdom, declared that coal needs to be consigned to history to limit global warming. CO₂ has an important social cost…

Over the last decade, scanning probe microscopy (SPM), including scanning tunneling microscopy (STM) and atomic force microscopy (AFM), has become a powerful manipulation technique by virtue of its ability to interact with individual adsorbed nanoparticles with nanoscale precision on the surface. In this article, the principles, procedures and applications of both STM and AFM-based technologies for manipulation of atoms, molecules, and nanoclusters are reviewed with an emphasis on their ability to create a wide variety of nanostructures. In the manipulation of single atoms and molecules, the interaction among the atoms/molecules, surface, and tip are specifically discussed first. The approach for positioning the atom/molecule from and to the desired locations and precisely controlling its movement is also elaborated for each specific manipulation technique. The applications of these techniques for fabricating different nanostructures and nanosystems are then presented. In the manipulation of nanoclusters, different nanocluster-substrate pairs in different environments with their potential applications in electronics, biology, and medicine are specifically evaluated. Finally, concluding remarks are provided, where the scopes for technological improvement and future research are recommended.