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The Turán number of a graph H, denoted by ex(n, H), is the maximum number of edges of an n-vertex simple graph having no H as a subgraph. Let Sℓ denote the star on ℓ + 1 vertices, and let k · Sℓ denote k disjoint copies of Sℓ. Erdős and Gallai determined the value ex(n, k · S1) for all positive integers k and n. Yuan and Zhang determined the value ex(n, k · S2) and characterized all extremal graphs for all positive integers k and n. Recently, Lan et al. determined the value ex(n, 2 · S3) for all positive integers n, and Li and Yin determined the values ex(n, k · Sℓ) for k = 2, 3 and all positive integers ℓ and n. In this paper, we further determine the value ex(n, 4 · Sℓ) for all positive integers ℓ and almost all n, improving one of the results of Lidický et al.

Let S=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}), where a1,…,am{a}_{1},\ldots ,{a}_{m} and b1,…,bn{b}_{1},\ldots ,{b}_{n} are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E)G=\left(X\cup Y,E) such that a1,…,am{a}_{1},\ldots ,{a}_{m} and b1,…,bn{b}_{1},\ldots ,{b}_{n} are the degrees of the vertices in XX and YY, respectively. In this case, GG is referred to as a realization of SS. Given a bigraphic pair SS, and a complete bipartite graph Ks,t{K}_{s,t}, we say that SS is a potentially Ks,t{K}_{s,t}-bigraphic pair if some realization of SS contains Ks,t{K}_{s,t} as a subgraph (with ss vertices in the part of size mm and tt in the part of size nn). Ferrara et al. (Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n) to be the minimum integer kk such that every bigraphic pair S=(a1,…,am;b1,…,bn)S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n}) with σ(S)=a1+⋯+am≥k\sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge k is a potentially Ks,t{K}_{s,t}-bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determined σ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n) for n≥m≥9s4t4n\ge m\ge 9{s}^{4}{t}^{4}. In this paper, we further determine σ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n) for n≥m≥sn\ge m\ge s and n+m≥2t2+t+sn+m\ge 2{t}^{2}+t+s. As two corollaries, if n≥m≥t2+t+s2n\ge m\ge {t}^{2}+\frac{t+s}{2} or if n≥m≥sn\ge m\ge s and n≥2t2+tn\ge 2{t}^{2}+t, the values σ(Ks,t,m,n)\sigma \left({K}_{s,t},m,n) are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

A new simplified practical method for one‐dimensional nonlinear large‐strain consolidation of saturated homogenous soils is proposed. The derivation processes of the proposed method are introduced first, with a modification of Terzaghi's theory from a novel perspective to solve large‐strain consolidation problems. Verification checks of the proposed method with other solutions are then conducted. The proposed method is different from Lekha's solution because Lekha's analytical solution is based on the small strain theory. For linear consolidation, the proposed method shows excellent agreement with the Consolidation Settlement 2 (CS2) model. For nonlinear large‐strain consolidation, the new method is in good agreement with the CS2 model when Cc/Ck ≤ 1. After that, optimization of the proposed nonlinear solution is carried out for Cc/Ck > 1 with a more precise average constant coefficient of consolidation used in the simplified practical method, and good agreement is obtained between the solutions from the proposed method and the CS2 model. Overall, the proposed simplified method provides practical, reliable, and efficient solutions for analyzing linear and nonlinear large‐strain consolidation. [ABSTRACT FROM AUTHOR]

Vacuum preloading via prefabricated horizontal drains (PHDs) is an efficient technique to accelerate the consolidation of dredged soft soils. However, this technique lacks a clear and concise theoretical solution to the PHD vacuum consolidation analysis for slurry. To address this issue, this study proposes a novel analytical solution based on a mathematical logical procedure. The governing equations are first derived based on square PHD consolidation unit cells given the assumption of equal strain, taking into account various drainage conditions at the boundaries and the horizontal and vertical spacings of PHD. A seepage direction coefficient, directly determined through PHD spacing without additional identification, is further introduced into the derived formulation to enhance the solution to be more consistent with the results of free strain consolidation analysis. Then, the proposed analytical solution is validated based on three laboratory and field tests. The proposed solution is successfully applied to estimate the consolidation behavior at a site in Aomori Prefecture, Japan, involving PHD vacuum preloading. All results demonstrate that the proposed analytical solution is applicable for the design and management of PHD vacuum preloading construction projects and practically useful for engineers. Furthermore, the influences of the large strain effect, variable hydraulic conductivity, and variable compressibility on the consolidation analysis are discussed. [ABSTRACT FROM AUTHOR]

Land reclamations often demand large quantities of fill materials. However, many countries commonly engaged in land reclamation are facing an escalating scarcity of natural fill sources. To address this issue, this study proposes a sustainable land reclamation approach that involves dredged marine sediments as fill materials. The dredged marine sediments are improved by a combined method that incorporates the use of horizontal drains with vacuum preloading for preliminary soil treatment and vertical drains with vacuum preloading for further improvement. A field trial has been designed and conducted to verify the performance of the proposed preliminary treatment of marine sediment slurry using prefabricated horizontal drains (PHDs) and vacuum preloading under a "membrane-free" condition. Field measurements, including settlement, pore pressure, effective earth pressure, vacuum pressure, water content, and undrained shear strength, were systematically recorded during the field trial period. After a four-month treatment, the average water content of the filled soil was slightly below 50%, which is one-third of the initial water content. Owing to the combined effect of the vacuum preloading and drying process, a crust layer with the undrained shear strength of 40 kPa was formed on the top of the treated soil. Below the crust layer, the average undrained shear strength of the treated marine deposits was approximately 32 kPa. Overall, the notable effect of using PHDs and the "membrane-free" vacuum preloading method to rapidly consolidate marine deposits slurry has been successfully demonstrated with field monitoring under the site trial condition. [ABSTRACT FROM AUTHOR]

In triaxial tests, it is of great importance to accurately measure the radial strain of a specimen. However, devices for radial strain measurement are relatively fewer than the ones for vertical strain measurement. In this study, a new measurement device based on the fiber Bragg grating (FBG) was developed to determine the small and large radial strains of a soil specimen. The new device includes two yokes and two compass-type mechanisms, which are mounted on diametrically opposite sides of the specimen. Metal strips bonded with FBG sensors were chosen as sensing elements and installed at two ends of two legs of a compass-type mechanism so that the small deformation of the specimen could be amplified by a constant ratio to a more detectable value with a higher signal-noise ratio. Considering the temperature sensitivity of FBG and the real working environment, the new device was carefully calibrated underwater. Validation tests on an intact triaxial specimen demonstrated that the new device can work reliably and accurately under both static and cyclic loadings. Key words: triaxial test, radial strain, small strain measurement, cyclic, fiber Bragg grating (FBG). Dans les essais triaxiaux, il est tres important de mesurer avec precision la deformation radiale d'un echantillon. Cependant, les dispositifs de mesure de la deformation radiale sont relativement moins nombreux que ceux de mesure de la deformation verticale. Dans cette etude, un nouveau dispositif de mesure base sur le reseau de Bragg a fibres (FBG) a ete developpe pour determiner les petites et grandes deformations radiales d'un echantillon de sol. Le nouveau dispositif comprend deux jougs et deux mecanismes de type compas, qui sont montes sur des cotes diametralement opposes de l'echantillon. Des bandes de metal collees avec des capteurs FBG ont ete choisies comme elements de detection et installees aux deux extremites des deux jambes d'un mecanisme de type boussole afin que la petite deformation du specimen puisse etre amplifiee par un rapport constant a une valeur plus detectable avec un rapport signal-bruit plus eleve. Compte tenu de la sensibilite a la temperature du FBG et de l'environnement de travail reel, le nouveau dispositif a ete soigneusement calibre sous l'eau. Des tests de validation sur un echantillon triaxial intact ont demontre que le nouveau dispositif peut fonctionner de maniere fiable et precise sous des charges statiques et cycliques. [Traduit par la Redaction] Mots-cles: test triaxial, deformation radiale, mesure de la petite deformation, cyclique, reseau de Bragg a fibres (FBG)., Introduction In conventional triaxial tests, it is generally acknowledged that determining the small-strain stiffness of a soil specimen accurately has been hindered by its seating and bedding errors as well [...]