Separation Theorem Research Articles

Pair Crossing Number, Cutwidth, and Good Drawings on Arbitrary Point Sets

Abstract Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that $$\textit{cr}(G)=O(\mathop {\textrm{pcr}}(G)^{3/2})$$ cr ( G ) = O ( pcr ( G ) 3 / 2 ) for every graph G, improving the previous best bound by a logarithmic factor. Answering a question of Pach and Tóth, we prove that the bisection width (and, in fact, the cutwidth as well) of a graph G with degree sequence $$d_1,d_2,\dots ,d_n$$ d 1 , d 2 , ⋯ , d n satisfies $$\mathop {\textrm{bw}}(G)=O\big (\sqrt{\mathop {\textrm{pcr}}(G)+\sum _{k=1}^n d_k^2}\big )$$ bw ( G ) = O ( pcr ( G ) + ∑ k = 1 n d k 2 ) . Then we show that there is a constant $$C\ge 1$$ C ≥ 1 such that the following holds: For any graph G of order n and any set S of at least $$n^C$$ n C points in general position on the plane, G admits a straight-line drawing which maps the vertices to points of S and has no more than $$O\left( \log n\cdot \left( \mathop {\textrm{pcr}}(G)+\sum _{k=1}^n d_k^2\right) \right) $$ O log n · pcr ( G ) + ∑ k = 1 n d k 2 crossings. Our proofs rely on a slightly modified version of a separator theorem for string graphs by Lee, which might be of independent interest.

A Proof Using Böhme's Lemma That no Petersen Family Graph has a Flat Embedding

Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.

The separation theorem for mean-field linear quadratic optimal control problem with partial information

Abstract We investigate the separation principle for a class of mean-field stochastic optimal control problems with partial information, where the linear stochastic dynamic system is observed through a linear noisy channel. In our model, the time inconsistency caused by the mean-field terms makes the dynamic programming principle (DPP) inapplicable, which immediately leads to difficulties in deriving the separation principle. Utilizing the optimal filtering equations in several dimensions, we can transform the stochastic optimal control problem with partial information into one with complete details. Subsequently, the Hamilton-Jacobi-Bellman (HJB) equation that can solve the optimal control is obtained with the support of the dynamic programming principle. The separation principle for solving the optimal control of the original problem is derived.

Product structure of graphs with an excluded minor

This paper shows that K t K_t -minor-free (and K s , t K_{s, t} -minor-free) graphs G G are subgraphs of products of a tree-like graph H H (of bounded treewidth) and a complete graph K m K_m . Our results include optimal bounds on the treewidth of H H and optimal bounds (to within a constant factor) on m m in terms of the number of vertices of G G and the treewidth of G G . These results follow from a more general theorem whose corollaries include a strengthening of the celebrated separator theorem of Alon, Seymour, and Thomas [J. Amer. Math. Soc. 3 (1990), 801–808] and the Planar Graph Product Structure Theorem of Dujmović et al. [J. ACM 67 (2020)].

Contributions to Generalized Oscillation Theory of Linear Hamiltonian Systems

In this paper we present several new contributions to the oscillation theory of linear differential equations, in particular of linear Hamiltonian systems, when the traditional Legendre condition is absent. Following our recent work (Discrete Contin. Dyn. Syst. 43(12):4139–4173, 2023), we introduce the multiplicity of a generalized right focal point and derive the corresponding local Sturmian separation theorem. We also examine the relation between the existence of finitely many generalized right focal points, or in the special case the nonexistence of generalized right focal points, with the Legendre condition. As the main tools we use new notions of the minimal multiplicities at a given point and the dual comparative index — an object from matrix analysis or differential geometry (Maslov index theory). Furthermore, we study local limit properties of the dual comparative index and the comparative index and apply them for deriving new oscillation results phrased in terms of the generalized right and left focal points. The investigation of the interplay between generalized right and left focal points leads to conditions characterizing the situation, when in the local Sturmian separation theorem the corresponding multiplicities attain the minimal possible value. This also provides a generalization of the concepts of the right and left proper focal point defined by Kratz (Analysis 23(2):163–183, 2003) and Wahrheit (Int. J. Differ. Equ. 2(2):221–244, 2007) to the setting, which does not impose the Legendre condition. The results are new even for completely controllable linear Hamiltonian systems, including the Sturm–Liouville differential equations of arbitrary even order.

Constraint Qualifications and Optimality Conditions for Nonsmooth Semidefinite Multiobjective Programming Problems with Mixed Constraints Using Convexificators

In this article, we investigate a class of non-smooth semidefinite multiobjective programming problems with inequality and equality constraints (in short, NSMPP). We establish the convex separation theorem for the space of symmetric matrices. Employing the properties of the convexificators, we establish Fritz John (in short, FJ)-type necessary optimality conditions for NSMPP. Subsequently, we introduce a generalized version of Abadie constraint qualification (in short, NSMPP-ACQ) for the considered problem, NSMPP. Employing NSMPP-ACQ, we establish strong Karush-Kuhn-Tucker (in short, KKT)-type necessary optimality conditions for NSMPP. Moreover, we establish sufficient optimality conditions for NSMPP under generalized convexity assumptions. In addition to this, we introduce the generalized versions of various other constraint qualifications, namely Kuhn-Tucker constraint qualification (in short, NSMPP-KTCQ), Zangwill constraint qualification (in short, NSMPP-ZCQ), basic constraint qualification (in short, NSMPP-BCQ), and Mangasarian-Fromovitz constraint qualification (in short, NSMPP-MFCQ), for the considered problem NSMPP and derive the interrelationships among them. Several illustrative examples are furnished to demonstrate the significance of the established results.

Приближенный метод синтеза непрерывных систем совместного оценивания и управления на основе SDRE технологи

<p>The problem of approximate synthesis of a closed loop nonlinear continuous system of joint estimation and control is considered. An approach is used based on the application of the idea of the separation theorem for linear dynamical systems. Using the factorization operation, a nonlinear system is transformed into a structure similar to a linear system, and algorithms for synthesizing an optimal linear controller and state observer are applied to the transformed system, a feature of which is the dependence of the matrices included in the corresponding Riccati equations on the state vector. An example of the synthesis of a state observer and a controller is given, demonstrating the application of the proposed algorithm.</p>

Method for adapting radioacoustic sounding systems of the atmosphere

Stations of radioacoustic sounding (RAS) of the atmosphere are a promising means of obtaining information about the altitudinal distribution of meteorological parameters in the Earth's atmosphere used in the process of solving current scientific and applied tasks to ensure aircraft flights, weather forecasting, etc. However, the effectiveness of the existing radio-acoustic means is insufficient, and there are practical needs for the development of appropriate prospective structures and algorithms, which will be implemented during the construction of specific stations designed to solve actual applied tasks. The article presents the synthesis of the RAS systems algorithm adaptation performed by changing the frequency of the sounding radio signal to fulfill the Bragg condition when the emitted acoustic pulse signal moves along the sounding path from the standpoint of optimal control theory. Since in the problem of synthesizing the algorithm for controlling the frequency of a radio signal, the disturbing and leading to violations of the Bragg condition during the propagation of an acoustic packet along the sounding path, as well as the process causing measurement errors, are considered as random, this problem is considered as a problem of stochastic optimal control. It is shown that in accordance with the separation theorem, known from the theory of optimal stochastic control, the method of controlling the frequency of a sounding radio signal should include sequentially peformed operations of forming estimates of the information parameter of the scattered radio signal, optimal linear filtering of the obtained estimates, and deterministic control of the frequency of the sounding radio signal.

Holography of broken U(1) symmetry

We examine the Abelian Higgs model in (d + 1)-dimensional anti-de Sitter space with an ultraviolet brane. The gauge symmetry is broken by a bulk Higgs vacuum expectation value triggered on the brane. We propose two separate Goldstone boson equivalence theorems for the boundary and bulk degrees of freedom. We compute the holographic self-energy of the gauge field and show that its spectrum is either a continuum, gapped continuum, or a discretuum as a function of the Higgs bulk mass. When the Higgs has no bulk mass, the AdS isometries are unbroken. We find in that case that the dual CFT has a non-conserved U(1) current whose anomalous dimension is proportional to the square of the Higgs vacuum expectation value. When the Higgs background weakly breaks the AdS isometries, we present an adapted WKB method to solve the gauge field equations. We show that the U(1) current dimension runs logarithmically with the energy scale in accordance with a nearly-marginal U(1)-breaking deformation of the CFT.

Nonlinear strict cone separation theorems in real normed spaces

Nonlinear strict cone separation theorems in real normed spaces

The Complexity of Finding Fair Many-to-One Matchings

We analyze the (parameterized) computational complexity of “fair” variants of bipartite many-to-one matching, where each vertex from the “left” side is matched to exactly one vertex and each vertex from the “right” side may be matched to multiple vertices. We want to find a “fair” matching, in which each vertex from the right side is matched to a “fair” set of vertices. Assuming that each vertex from the left side has one color modeling its “attribute”, we study two fairness criteria. For instance, in one of them, we deem a vertex set fair if for any two colors, the difference between the numbers of their occurrences does not exceed a given threshold. Fairness is, for instance, relevant when finding many-to-one matchings between students and colleges, voters and constituencies, and applicants and firms. Here colors may model sociodemographic attributes, party memberships, and qualifications, respectively. We show that finding a fair many-to-one matching is NP-hard even for three colors and maximum degree five. Our main contribution is the design of fixed-parameter tractable algorithms with respect to the number of vertices on the right side. Our algorithms make use of a variety of techniques including color coding. At the core lie integer linear programs encoding Hall like conditions. We establish the correctness of our integer programs, based on Frank’s separation theorem [Frank, Discrete Math. 1982]. We further obtain complete complexity dichotomies regarding the number of colors and the maximum degree of each side.

ITERATED PRIORITY ARGUMENTS IN DESCRIPTIVE SET THEORY

AbstractWe present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff–Kuratowski and Wadge theorems on the structure of $\mathbf {\Delta }^0_\xi $ , Louveau and Saint Raymond’s separation theorem, and Louveau’s separation theorem.

Optimal finite-dimensional controller of the stochastic differential object’s state by its output II. Stochastic measurements and separation theorem

Consideration is continued of the problem of the inertial control law by the output synthesis of a continuous nonlinear stochastic plant, which is optimal on average and on a finite time interval, and works with the desired speed. An algorithm for synthesizing the optimal structure of a dynamic controller of a selected finite order, obtained in the first part of the article for the case of accurate measurements of a of the control object’s state variables part, is presented. Its application is demonstrated in detail for the case when the state variables of an object are measured with random errors. Using the example of a linear-quadratic-Gaussian problem, it is shown that the proposed controller of the corresponding order also satisfies the well-known separation theorem.

Optimal Finite-Dimensional Controller of a Stochastic Differential Object’s State by Its Output. II. Stochastic Measurements and Separation Theorem

Optimal Finite-Dimensional Controller of a Stochastic Differential Object’s State by Its Output. II. Stochastic Measurements and Separation Theorem

Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

Nonlinear Cone Separation Theorems in Real Topological Linear Spaces

Diagonal solutions for a class of linear matrix inequality

<p>In this paper, we present a characterization of diagonal solutions for a class of linear matrix inequalities. We consider linear hybrid time-delay systems and explore the conditions under which these systems are positive and asymptotically stable. Specifically, we investigate the existence of positive diagonal solutions for a linear inequality when the system matrices are Metzler and nonnegative. Using various mathematical tools, including the Schur complement and separation theorems, we derive necessary and sufficient conditions for the stability of these systems. Our results extend existing stability criteria and provide new insights into the stability analysis of positive time-delay systems.</p>

The nearest point problems in fuzzy quasi-normed spaces

<abstract> <p>Motivated by the fact that the fuzzy quasi-normed space provides a suitable framework for complexity analysis and has important roles in discussing some questions in theoretical computer science, this paper aims to study the nearest point problems in fuzzy quasi-normed spaces. First, by using the theory of dual space and the separation theorem of convex sets, the properties of the fuzzy distance from a point to a set in a fuzzy quasi-normed space are studied comprehensively. Second, more properties of the nearest point are given, and the existence, uniqueness, characterizations, and qualitative properties of the nearest points are obtained. The results obtained in this paper are of great significance for expanding the application fields of optimization theory.</p> </abstract>

On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory

<abstract><p>In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.</p></abstract>

On Proper Separation Theorems by Means of the Quasi-Relative Interior with Applications

In this paper, we establish several proper separation theorems for an element and a convex set and for two convex sets in terms of their quasi-relative interiors. Then, we prove that the separation theorem given by [Cammaroto, F and B Di Bella (2007). On a separation theorem involving the quasi-relative. Proceedings of the Edinburgh Mathematical Society, 50(3), 605–610] in Theorem 2.5, is in fact a proper separation theorem for two convex sets in which the classical interior is replaced by the quasi-relative interior. Besides, we extend some known results in the literature, such as [Adán, M and V Novo (2004). Proper efficiency in vector optimization on real linear spaces. Journal of Optimization Theory and Applications, 121, 515–540] in Theorem 2.1 and [Edwards, R (1965). Functional Analysis: Theory and Applications. New York: Reinhart and Winston] in Corollary 2.2.2, through the quasi-relative interior and the quasi-interior, respectively. As an application, we provide Karush–Kuhn–Tucker multipliers for quasi-relative solutions of vector optimization problems. Several examples are given to illustrate the obtained results.

Fisher's Separation Theorem and its Limitations

When businesses make investment decisions, they often evaluate the Net Present Value (NPV) of investment projects. For investors, a project is deemed worthwhile if its NPV is greater than zero. This phenomenon can be attributed to the Fisher Separation Theorem, which asserts that in perfectly efficient capital markets without transaction costs, a single interest rate prevails. In this context, investors' production choices adhere strictly to objective market regulations aimed at maximizing attainable wealth, entirely detached from individual subjective preferences. This, in turn, enables both borrowers and lenders to base their consumption and investment decisions on the prevailing market interest rate, thereby encouraging a clear distinction between investment and financing choices. Therefore, a company's investment decisions can be discussed separately from its financing decisions. So, the enterprise's investment decision and financing decision can be discussed separately. When enterprises invest, enterprises do not have to consider the subjective consumption preferences of different investors, because investors' consumption preferences can be satisfied accordingly through perfect capital market operation. Because of Fisher's Separation Theorem, which requires the market to be transaction cost-free, it also has some limitations.