Decomposition Theorem Research Articles

A prime decomposition theorem for string links in a thickened surface

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link [Formula: see text], where [Formula: see text] is a compact orientable (not necessarily closed) surface other than [Formula: see text], can be written in the form [Formula: see text], where [Formula: see text] is prime string link defined up to braid equivalence, and the decomposition is unique up to possibly permuting the order of factors in its right-hand side.

The impact of pinning points on memorylessness in Lévy random bridges

Abstract Random bridges have gained significant attention in recent years due to their potential applications in various areas, particularly in information-based asset pricing models. This paper aims to explore the potential influence of the pinning point’s distribution on the memorylessness and stochastic dynamics of the bridge process. We introduce Lévy bridges with random length and random pinning points, and analyze their Markov property. Our study demonstrates that the Markov property of Lévy bridges depends on the nature of the distribution of their pinning points. The law of any random variables can be decomposed into singular continuous, discrete, and absolutely continuous parts with respect to the Lebesgue measure (Lebesgue’s decomposition theorem). We show that the Markov property holds when the pinning points’ law does not have an absolutely continuous part. Conversely, the Lévy bridge fails to exhibit Markovian behavior when the pinning point has an absolutely continuous part.

Continuous asymmetric Doob inequalities in noncommutative symmetric spaces

Let (M,τ) be a noncommutative probability space equipped with a filtration (Mt)t∈[0,1] whose union is w⁎-dense in M, and let (Et)t∈[0,1] be the associated conditional expectations. We prove in the present paper that if the symmetric space E∈Int[Lp,Lq] with 1<p≤q<2 and E is 2(1−θ)-convex and w-concave with p<w<2, then the following holds:‖(Et(x))t∈[0,1]‖E(M;ℓ∞θ)≤CE,θ‖x‖HEc,x∈HEc(M) provided 1−p/2<θ<1. Similar result holds for x∈HEr(M). Moreover, if E∈Int[Lp,Lq] with 1<p≤q<2 and E is w-concave with 2<w<2p/(2−p), then for each x∈E(M) there exist y, z∈E(M) such that x=y+z and‖(Et(y))t∈[0,1]‖E(M;ℓ∞c)+‖(Et(z))t∈[0,1]‖E(M;ℓ∞r)≤cE‖x‖E(M). These results can be considered as continuous analogues of those due to Randrianantoanina et al. [33]. One of the key ingredients in our proof is a new decomposition theorem of E(M)-modules for general symmetric space E, which extends the known result of Junge and Sherman.

Geometry of K-trivial Moishezon manifolds: decomposition theorem and holomorphic geometric structures

Geometry of K-trivial Moishezon manifolds: decomposition theorem and holomorphic geometric structures

A Study of $\phi$-Pluriharmonicity in Quasi bi-slant Conformal $\xi^\perp$-Submersions From Kenmotsu Manifold

In the present research paper, we look into quasi bi-slant conformal $\xi^{\perp}$-submersions from Kenmotsu manifolds onto Riemannian manifolds as a generalisation of quasi hemi-slant conformal submersions. We investigate the geometry of distributions's leaves in order to explain integrability conditions for distributions. Furthermore, we study of certain decomposition theorems, additionaly provide non-trivial examples of quasi bi-slant conformal $\xi^{\perp}$-submersions from Kenmotsu manifolds. We also look at the $\phi$-pluriharmonicity of quasi bi-slant conformal $\xi^{\perp}$-submersions.

The Brasselet–Schürmann–Yokura Conjecture on L-Classes of Projective Rational Homology Manifolds

Abstract In 2010, Brasselet, Schürmann, and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky–MacPherson $L$-class $L_{*}(X)$ and the Hirzebruch homology class $T_{1,*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions that we find of independent interest.

Reflection and transmission phenomenon of plane waves at the interface of diffusive viscoelastic porous rotating isotropic medium under hall current and nonlocal thermoelastic porous solid

The response of elastic waves upon encountering the boundary between two elastic media is investigated in the present work. Whereas the top medium is thermoelastic isotropic porous, the below medium is thermoelastic rotating porous diffusive. There is an uncoupled transmitted SV-wave propagating through the medium, and the lower medium is rotating with some fixed angular frequency. When the incident wave hits the boundary, it produces four transmitted waves and five coupled quasi-reflected waves. The system is divided into longitudinal and transverse components using the Helmholtz decomposition theorem. Analytical computations of speed and reflection coefficients for transmitted and reflected waves are performed using LS theory. The outcomes are graphically represented for a particular material subject to nonlocal and fractional-order influences. Wave characteristics, such as speed and reflection coefficients for transmitted and reflected waves, are plotted versus angular frequency and angle of incidence using MATLAB programing. The conservation of energy has also been verified. In the absence of rotation, hall current, voids, viscoelasticity, and diffusion in the medium, the previous results in the literature are obtained.

Cascade products and Wheeler automata

The Krohn-Rhodes Decomposition Theorem (KRDT) holds a central position in automata and semigroup theories. It asserts that any finite-state automaton can be broken down into a collection, a cascade, of automata of two simple types (reset and permutation) that, combined, simulate the original automaton.In this paper we show how the cascade product operation and the related decomposition are particularly well-suited for the class of Wheeler automata. In these automata, recently introduced in the context of data compression, states are ordered and transitions map state-intervals to state-intervals. First, we prove that Wheeler DFAs are closed under cascade products in an efficient way: the cascade product of two Wheeler automata is still a Wheeler automaton and has always a number of states which is at most the sum (after removing unreachable states) of the number of states of the two input automata, a result that cannot be achieved for general (even counter-free) automata. Second, we prove that each Wheeler automaton can be decomposed into a cascade of a linear number of reset blocks. Crucially, our line of reasoning avoids the necessity of using full KRDT and proves our results directly by an inductive argument.

Hodge-to-singular correspondence for reduced curves

We study the summands of the decomposition theorem for the Hitchin system for \operatorname{GL}_{n} , in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.

Decomposition and construction of uninorms on the unit interval

By analyzing the behaviour of uninorms on the boundary of the unit square, we present decomposition theorems for uninorms, showing that a conjunctive (resp. disjunctive) uninorm can be decomposed into a disjunctive (resp. conjunctive) uninorm on an upper set (resp. a lower set) and a triangular subnorm (resp. superconorm) on a lower set (resp. an upper set). As an application of the decomposition theorems, we propose several construction methods for uninorms that are not internal on the boundary.

Generating fuzzy sets using rearranged nested family and the mathematical formula in decomposition theorem

Generating fuzzy sets using rearranged nested family and the mathematical formula in decomposition theorem

Distributed State Observer for Systems with Multiple Sensors under Time-Delay Information Exchange.

The issues of state estimations based on distributed observers for linear time-invariant (LTI) systems with multiple sensors are discussed in this paper. We deal with the scenario when the information exchange has known time delays, and aim at designing a distributed observer for each subsystem such that each distributed observer can estimate the system state asymptotically by rejecting the time delay. To begin with, by rewriting the target system in a connecting form, a subsystem which is affected by the time-delay states of other nodes is established. And then, for this subsystem, a distributed observer with time delay is constructed. Moreover, an equivalent state transformation is made for the observer error dynamic system based on the observable canonic decomposition theorem. Further, in order to ensure that the distributed observer error dynamic system is asymptotically stable even if there exists a time delay, a linear matrix inequality (LMI) which is relative to the Laplace matrix is elaborately set up, and a special Lyapunov function candidate based on the LMI is considered. Next, based on the Lyapunov function and Lyapunov stability theory, we prove that the error dynamic system of the distributed observer is asymptotically stable, and the observer gain is determined by a feasible solution of the LMI. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.

True Contraction Decomposition and Almost ETH-Tight Bipartization for Unit-Disk Graphs

We prove a structural theorem for unit-disk graphs, which (roughly) states that given a set \(\mathcal{D}\) of \(n\) unit disks inducing a unit-disk graph \(G_{\mathcal{D}}\) and a number \(p\in[n]\) , one can partition \(\mathcal{D}\) into \(p\) subsets \(\mathcal{D}_{1},\dots,\mathcal{D}_{p}\) such that for every \(i\in[p]\) and every \(\mathcal{D}^{\prime}\subseteq\mathcal{D}_{i}\) , the graph obtained from \(G_{\mathcal{D}}\) by contracting all edges between the vertices in \(\mathcal{D}_{i}\backslash\mathcal{D}^{\prime}\) admits a tree decomposition in which each bag consists of \(O(p+|\mathcal{D}^{\prime}|)\) cliques. Our theorem can be viewed as an analog for unit-disk graphs of the structural theorems for planar graphs and almost-embeddable graphs proved recently by Marx et al. [SODA ’22] and Bandyapadhyay et al. [SODA ’22]. By applying our structural theorem, we give several new combinatorial and algorithmic results for unit-disk graphs. On the combinatorial side, we obtain the first Contraction Decomposition Theorem for unit-disk graphs, resolving an open question in the work by Panolan et al. [SODA ’19]. On the algorithmic side, we obtain a new algorithm for bipartization (also known as odd cycle transversal) on unit-disk graphs, which runs in \(2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}\) time, where \(k\) denotes the solution size. Our algorithm significantly improves the previous slightly subexponential-time algorithm given by Lokshtanov et al. [SODA ’22] which runs in \(2^{O(k^{27/28})}\cdot n^{O(1)}\) time. We also show that the problem cannot be solved in \(2^{o(\sqrt{k})}\cdot n^{O(1)}\) time assuming the Exponential Time Hypothesis, which implies that our algorithm is almost optimal.

Geometric flavors of Quantum Field theory on a Cauchy hypersurface. Part I: Gaussian analysis and other mathematical aspects

In this series of papers we aim to provide a mathematically comprehensive framework to the hamiltonian pictures of quantum field theory in curved spacetimes. Our final goal is to study the kinematics and the dynamics of the theory from the point of differential geometry in infinite dimensions. In this first part we introduce the tools of Gaussian analysis in infinite dimensional spaces of distributions. These spaces will serve the basis to understand the Schrödinger and Holomorphic pictures, over arbitrary Cauchy hypersurfaces, using tools of Hida-Malliavin calculus. Here the Wiener-Ito decomposition theorem provides the QFT particle interpretation. Special emphasis is done in the applications to quantization of these tools in the second part of this paper. We devote a section to introduce Hida test functions as a notion of second quantized test functions. We also analyze of the ingredients of classical field theory modeled as distributions paving the way for quantization procedures that will be analyzed in [3].

A multivariate process quality correlation diagnosis method based on grouping technique

Correlation diagnosis in multivariate process quality management is an important and challenging issue. In this paper, a new diagnostic method based on quality component grouping is proposed. Firstly, three theorems describing the properties of the covariance matrix of multivariate process quality are established based on the statistical viewpoint of product quality, to prove the correlation decomposition theorem, which decomposes the correlation of all the quality components into a series of correlations of components pairs, and then by using the factor analysis method, all quality components are grouped in order to maximize the correlations in the same groups and minimize the ones between different groups. Finally, on the basis of correlations between different groups are ignored, T2 control charts of component pairs in the same groups are established to form the diagnostic model. Theoretical analysis and practice prove that for the multivariate process quality whose the correlations between different components vary considerably, the grouping technique enables the size of the correlation diagnostic model to be drastically reduced, thus allowing the proposed method can be used as a generalized theoretical model for the correlation diagnosis.

Revealing the Correlation Between Lyapunov Exponent and Modulus of an n-Dimensional Nondegenerate Hyperchaotic Map

For their good randomness and long iteration periods, chaotic maps have been widely used in cryptography. Recently, we have revealed the correlation between Lyapunov exponent and sequence randomness of multidimensional chaotic maps based on modular operation. Since the modular operation can realize the boundedness of chaotic state points, it is important to further reveal the deterministic correlation between Lyapunov exponent and modulus. First, we constructed an [Formula: see text]-dimensional nondegenerate hyperchaotic map model with the desired Lyapunov exponents. Then, we gave the existence and uniqueness proof of quadrature rectangle decomposition theorem and revealed the correlation between Lyapunov exponent and modulus. The novelty lies in that (1) in order to realize the irreversibility of the iterative processes of chaotic maps, we constructed a chaotic map based on modular exponentiation, and its inverse function is the discrete logarithm problem; and (2) we reveal for the first time the correlation between Lyapunov exponent and modulus, and give the lower bound of the modulus of the nondegenerate chaotic map. In addition, to verify the effectiveness of the scheme, we constructed four-dimensional and five-dimensional chaotic maps, respectively, and analyzed their dynamical behaviors, and the results revealed that there exist linear or nonlinear correlation between Lyapunov exponent and modulus.

On the Computation of the Cohomological Invariants of Bott–Samelson Resolutions of Schubert Varieties

Let X⊆G/B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\subseteq G/\\mathcal {B}$$\\end{document} be a Schubert variety in a flag manifold and let π:X~→X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi : \ ilde{X} \\rightarrow X$$\\end{document} be a Bott–Samelson resolution of X. In this paper, we prove an effective version of the decomposition theorem for the derived pushforward Rπ∗QX~\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R \\pi _{*} \\mathbb {Q}_{\ ilde{X}}$$\\end{document}. As a by-product, we obtain recursive procedure to extract Kazhdan–Lusztig polynomials from the polynomials introduced by Deodhar [7], which does not require prior knowledge of a minimal set. We also observe that any family of equivariant resolutions of Schubert varieties allows to define a new basis in the Hecke algebra and we show a way to compute the transition matrix, from the Kazhdan–Lusztig basis to the new one.

Deterministic fuzzy two-dimensional on-line tessellation automata and their languages

The main work of this paper is to extend deterministic two-dimensional on-line tessellation automata (D2OTAs) to the fuzzy setting. We call these new automata models deterministic fuzzy two-dimensional on-line tessellation automata (DF2OTAs) and focus on some of their properties. Concretely, we firstly give the definitions of DF2OTAs and the fuzzy picture languages recognized by them. Next, the closure properties of the collection of fuzzy picture languages recognized by DF2OTAs are concentrated on under some familiar operations. The decomposition theorem, the representation theorem and the Pumping lemma, which are studied in the theory of fuzzy string automata and the related languages, are also considered scrupulously in the framework of DF2OTAs and their languages. Then, we put forward the concepts of transition-accessible DF2OTAs and state-accessible DF2OTAs, and conclude that transition-accessible DF2OTAs are special state-accessible DF2OTAs and DF2OTAs are equivalent to state-accessible DF2OTAs. Finally, state reduction relations on state-accessible DF2OTAs are defined to study the problem of the state reduction of DF2OTAs. Given a state-accessible DF2OTA A, in order to construct the factor DF2OTA of A with respect to a state reduction relation on A such that this factor one is equivalent to A and possesses a fewer states, we design a polynomial-time algorithm to compute the largest state reduction relation and then construct the corresponding factor automaton.

Conformal bi-slant ξ⊥-Riemannian submersion: a note in contact geometry

This study explores conformal bi-slant ξ⊥-Riemannian submer- sion where the total manifold is a contact metric manifold, more specifically a Sasakian manifold. To illustrate this study, few non-trivial examples are discussed. Meanwhile, we addressed the conditions for integrability of anti- invariant and slant distributions and determine the conditions for the map that must be met in order to be totally geodesic. Furthermore, some decomposition theorems for the fibres as well as for the total space are discussed.

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem