OAR@UM Collection:

Insertion of continuous set-valued mappings

Sun, 01 Jan 2023 00:00:00 GMT

Title: Insertion of continuous set-valued mappings Authors: Gutev, Valentin Abstract: An interesting result about the existence of “intermediate” set-valued mappings between pairs of such mappings was obtained by Nepomnyashchii. His construction was for a paracompact domain, and he remarked that his result is similar to Dowker’s insertion theorem and may represent a generalisation of this theorem. In the present paper, we characterise the τ-paracompact normal spaces by this set-valued “insertion” property and for τ=ω, i.e. for countably paracompact normal spaces, we show that it is indeed equivalent to the mentioned Dowker’s theorem. Moreover, we obtain a similar result for τ-collectionwise normal spaces and show that for normal spaces, i.e. for ω-collectionwise normal spaces, our result is equivalent to the Katětov-Tong insertion theorem. Several related results are obtained as well.

Solution to a problem of Katona on counting cliques of weighted graphs

Mon, 01 Jan 2024 00:00:00 GMT

Title: Solution to a problem of Katona on counting cliques of weighted graphs Authors: Borg, Peter; Feghali, Carl; Pellerin, Rémi Abstract: A subset I of the vertex set V ( G ) of a graph G is called a k -clique independent set of G if no k vertices in I form a k -clique of G . An independent set is a 2-clique independent set. Let π k ( G ) denote the number of k -cliques of G . For a function w : V ( G ) → { 0 , 1 , 2 , … } , let G ( w ) be the graph obtained from G by replacing each vertex v by a w ( v ) -clique K v and making each vertex of K u adjacent to each vertex of K v for each edge { u , v } of G . For an integer m ≥ 1 , consider any w with ∑ v ∈ V ( G ) w ( v ) = m . For U ⊆ V ( G ) , we say that w is uniform on U if w ( v ) = 0 for each v ∈ V ( G ) ∖ U and, for each u ∈ U , w ( u ) = m / | U | or w ( u ) = m / | U | . Katona asked if π k ( G ( w ) ) is smallest when w is uniform on a largest k -clique independent set of G . He placed particular emphasis on the Sperner graph B n , given by V ( B n ) = { X : X ⊆ { 1 , … , n } } and E ( B n ) = { { X , Y } : X ⊊ Y ∈ V ( B n ) } . He provided an affirmative answer for k = 2 (and any G ). We determine graphs for which the answer is negative for every k ≥ 3 . These include B n for n ≥ 2 . Generalizing Sperner’s Theorem and a recent result of Qian, Engel and Xu, we show that π k ( B n ( w ) ) is smallest when w is uniform on a largest independent set of B n . We also show that the same holds for complete multipartite graphs and chordal graphs. We show that this is not true of every graph, using a deep result of Bohman on triangle-free graphs.

On middle box products and paracompact cardinals

Mon, 01 Jan 2024 00:00:00 GMT

Title: On middle box products and paracompact cardinals Authors: Buhagiar, David; Džamonja, Mirna Abstract: The paper gives several sufficient conditions on the paracompactness of box products with an arbitrary number of factors and boxes of arbitrary size. The former include results on generalised metrisability and Sikorski spaces. Of particular interest are products of the type <𝜅 □ 2𝜆, where we prove that for a regular uncountable cardinal κ, if <𝜅 □ 2𝜆 is paracompact for every 𝜆 ≥𝜅, then κ is at least inaccessible. The case of the products of the type <𝜅 □ 𝑋𝜆 for κ singular has not been studied much in the literature and we offer various results. The question if <𝜅 □ 2𝜆 can be paracompact for all λ when κ is singular has been partially answered but remains open in general.

Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

Mon, 01 Jan 2024 00:00:00 GMT

Title: Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers Authors: Baysal, Onur; Hasanov, Alemdar; Kawano, Alexandre Abstract: In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation ρ ( x ) u t t + μ ( x ) u t + ( r ( x ) u x x ) x x = 0 , subject to the clamped boundary conditions u ( 0 , t ) = u x ( 0 , t ) = 0 at x = 0 , and the boundary conditions ( − r ( x ) u x x ) x = ℓ = k r u x ( ℓ , t ) + k a u x t ( ℓ , t ) , ( − ( r ( x ) u x x ) x ) x = ℓ = − k d u ( ℓ , t ) − k v u t ( ℓ , t ) at x = ℓ , is analyzed. The boundary conditions at x = ℓ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient μ ( x ) ≥ 0 , and the boundary springs k r , k d ≥ 0 and dampers k a , k v ≥ 0 . Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.