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The spread of a graph G is the difference between the largest and smallest eigenvalues of the adjacency matrix of G. In this paper, we consider the family of graphs which contain no K 2 , t -minor. We show that for any t ≥ 2 , there is an integer ξ t such that the maximum spread of an n -vertex K 2 , t -minor-free graph is achieved by the graph obtained by joining a vertex to the disjoint union of ⌊ 2 n + ξ t 3 t ⌋ copies of K t and n − 1 − t ⌊ 2 n + ξ t 3 t ⌋ isolated vertices. The extremal graph is unique, except when t ≡ 4 (mod 12) and 2 n + ξ t 3 t is an integer, in which case the other extremal graph is the graph obtained by joining a vertex to the disjoint union of ⌊ 2 n + ξ t 3 t ⌋ − 1 copies of K t and n − 1 − t (⌊ 2 n + ξ t 3 t ⌋ − 1) isolated vertices. Furthermore, we give an explicit formula for ξ t. [ABSTRACT FROM AUTHOR]

Molybdenum (Mo) based ceramic-metallic (cermet) fuel element is a promising fuel form used in nuclear thermal propulsion (NTP) because of its excellent advantages compared with the tungsten (W) based fuel. Mo, as a matrix material, plays a crucial role in containing the radioactive fission products released from the kernel and hindering the inward diffusion of hydrogen (H 2), thus reducing fuel loss. Therefore, in this study, the phase composition, microstructural evolution, grain growth kinetic, and mechanical properties of nanosized Mo powders fabricated by spark plasma sintering (SPS) are investigated in detail. The results demonstrate that Mo matrix consistently retains the body-centered cubic structure from 1000 to 1700 °C, with no phase transformation occurring. A relative density of 90% can be obtained at the sintering temperature of 1200 °C and >98% of theoretical density is achieved at 1700 °C under the external pressure of 50 MPa. Meanwhile, the average size of Mo grain grows to only about 1 μm from about 60 nm under 1700 °C due to rapid heating and cooling rate of SPS. Kinetic analysis indicates that the grain growth mechanism of nano-Mo powder is controlled by the surface diffusion with an activation energy of 414.16 KJ/mol. Mechanical properties illustrate that the hardness values of the Mo bulk is governed by the porosity fraction under low preparation temperatures (1000 to 1300 °C) and controlled by the Mo grain size at high preparation temperatures (1300 to 1700 °C). At 1300 °C, the corresponding Vickers hardness value is 316.6 Hv and the Rockwell hardness value is HRB 97. Moreover, Mo powders of different size sintered with various techniques are summarized and discussed. This work can enhance our understanding of the crucial role of advanced sintering techniques in combination with fine-grained powders for use in NTP systems. • Effect of the nanosized powder on the properties of the Mo matrix has been investigated using the SPS technique for NTP system. • The performance of the Mo matrix is optimized at lower preparation temperatures using the SPS technique. • This work provides an effective approach to improve the performance of Mo-UO 2 fuel for use in NTP system. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximum spectral radius of outerplanar 3‐uniform hypergraphs. Given a hypergraph ℋ ${\rm{ {\mathcal H} }}$, the shadow of ℋ ${\rm{ {\mathcal H} }}$ is a graph G $G$ with V(G)=V(ℋ) $V(G)=V({\rm{ {\mathcal H} }})$ and E(G)={uv:uv∈hfor someh∈E(H)} $E(G)=\{uv:uv\in \,h\,\text{for\unicode{x02007}some}\,h\in E({\mathscr{H}})\}$. A graph is outerplanar if it can be embedded in the plane such that all its vertices lie on the outer face. A 3‐uniform hypergraph ℋ ${\rm{ {\mathcal H} }}$ is called outerplanar if its shadow has an outerplanar embedding such that every hyperedge of ℋ ${\rm{ {\mathcal H} }}$ is the vertex set of an interior triangular face of the shadow. Cvetković and Rowlinson conjectured in 1990 that among all outerplanar graphs on n $n$ vertices, the graph K1+Pn−1 ${K}_{1}+{P}_{n-1}$ attains the maximum spectral radius. We show a hypergraph analogue of the Cvetković–Rowlinson conjecture. In particular, we show that for sufficiently large n $n$, the n $n$‐vertex outerplanar 3‐uniform hypergraph of maximum spectral radius is the unique 3‐uniform hypergraph whose shadow is K1+Pn−1 ${K}_{1}+{P}_{n-1}$. [ABSTRACT FROM AUTHOR]

Let G be an r-uniform hypergraph of order n. For each p ≥ 1 , the p-spectral radius λ (p) (G) is defined as λ (p) (G) := max | x 1 | p + ⋯ + | x n | p = 1 r ∑ { i 1 , ... , i r } ∈ E (G) x i 1 ⋯ x i r . The p-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when p = r, and λ (r) (G) is called the spectral radius of G. The α-normal labelling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is 1 while the product of all corner labels at any edge is α. Since then, this method has been used by many researchers in studying λ (r) (G). In this paper, we extend Lu and Man's α-normal labelling method to the p-spectral radii of uniform hypergraphs for p ≠ r ; and find some applications. [ABSTRACT FROM AUTHOR]

Molybdenum (Mo)-based uranium dioxide (UO 2) fuel is considered one of the most promising fuel forms for deep-space exploration. The rolling process has been proposed for the preparation of Mo matrix due to its excellent performance. However, there has been limited research on the ultra-high temperature stability of Mo matrix used in nuclear thermal propulsion (NTP). Therefore, in this study, annealing experiments were carried out on rolled Mo plates within a relatively wide temperature range of 1200 to 2300 °C for 1 h to investigate the effects of temperature on the phase composition, microstructure, and mechanical property. The results showed that, despite the body-centered cubic structure of the Mo plates remaining unchanged without undergoing phase transition after high-temperatures, the intensity of the crystal diffraction peaks experienced a significant increase. The preferred growth orientation shifted from the (110) direction at lower temperatures to the (200) direction at 2300 °C. The grain shape of the rolled Mo plates changed from elongated to equiaxed after annealing, and the grain size increased from the initial 1 μm to tens of microns. No discernible surface pores or fracture bubbles were observed when annealed at 1200 to 2100 °C. However, subsequent annealing at 2300 °C revealed a proliferation of polyhedral bubbles, ranging in size from tens of nanometers to several microns, distributed both in grain boundaries and within grains. Nevertheless, there is no apparent change in the density from 1200 to 2300 °C. Bright-field TEM micrographs show that after annealing at 2300 °C, the dislocation density is significantly reduced, and the dislocation shape evolves from screw dislocation to long and straight line. In addition, the Vickers hardness value of the rolled Mo plates undergoes a significant decrease, from 247.1 to 199.6 Hv, after annealing at 1200 °C. Subsequently, in the temperature range of 1200 to 2300 °C, a gradual decline in the hardness value is observed. This work will deepen our understanding of the high-temperature resistance of rolled Mo plates and provide data to support their applications in ultra-high-temperature conditions. • Effect of the ultra-high temperatures (1200–2300 °C) on the properties of the rolled Mo has been studied for NTP systems. • Phase composition, microstructure evolution, and hardness variation of the rolled Mo are investigated in detail. • This work advances our knowledge on the high-temperature resistance of rolled Mo plates used in NTP systems. [ABSTRACT FROM AUTHOR]

The fissile fuel agglomeration and loss in the tungsten (W) based ceramic-metallic (cermet) fuel element are the major challenges for application in nuclear thermal propulsion (NTP). The previous research indicated that a W coating deposited on the surface of uranium dioxide (UO 2) by the fluidized bed chemical vapor deposition (CVD) technique is an efficient way to achieve the uniform distribution of fuel particles and reduce fuel loss simultaneously. However, the W layers' resulting thicknesses differ, which is not conducive to achieving high fuel loading. Therefore, in this study, the appropriate thickness of the W coating is first investigated by COMSOL Multiphysics modeling. Through contrastive analyses, the W layer with a thickness of about 5–10 μm is considered to be the optimal choice, where the maximum loading capacity of the fissile fuel can be achieved with a volume fraction of about 60%. Based on the above calculations, the CVD technique is employed to produce the optimized thickness of the W layer by precisely adjusting the deposition time at 900 °C. This approach is taken after determining the critical influencing factor that governs the performance of the W layer. Experimental results show that the highly dense W layer presents a uniform thickness and fine-grained size. As the only structural and protective layer, the application of this W coating provides a barrier for blocking the inward diffusion of hydrogen (H 2) to stop the detrimental chemical reactions between UO 2 and H 2 , thus lowering fuel loss. The spark plasma sintering (SPS) demonstrates that zirconium dioxide (ZrO 2) particles as the surrogate for UO 2 exhibit a homogeneous distribution without any direct contact in the W matrix because of the presence of the W coating. Despite the fact that the interface between the W coating and the W matrix exhibits remarkable bonding, making it difficult to clearly discern the boundaries due to surface diffusion and atomic binding, the existence and integrity of the W coating after high-temperature sintering is unequivocally confirmed through interface TEM observations and the uniform dispersion of coated particles within the W-Y 2 O 3 matrix. Furthermore, the fine-grained W coating has been demonstrated to effectively hinder the diffusion of H 2 isotopes through permeation and diffusion tests. This comprehensive study not only aids in establishing the optimal geometrical thickness for the W coating but also underscores its pivotal role in W-based cermet fuels for NTP system. • Effect of different thicknesses of the W layer on the performance of the cermet fuel is studied. • A highly dense and fine-grained W coating with a thickness of about 10 μm is produced. • The fine-grained W layer is favorable for achieving an even distribution of fuel particles and blocking H 2 diffusion. [ABSTRACT FROM AUTHOR]

• The spherical cermet element is proposed for nuclear thermal propulsion (NTP) to alleviate metallurgical bond issues. • The reactor using spherical cermet elements is optimized to satisfy design requirements for NTP. • The reactor is analyzed to obtain neutronics and thermal hydraulics performances and evaluated its feasibility for NTP. Due to the advantages of high thrust and high specific impulse, nuclear thermal propulsion (NTP) is a preferred near-term rocket engine technology for future Mars missions. This article introduces the reactor design based on spherical cermet fuel element, with the aim of studying the feasibility of applying this new fuel in NTP system. First, design requirements are determined based on the unique features of NTP. In terms of neutronics, the effect of rocket launch accidents and fuel loss caused by high-temperature hydrogen are considered. Thermal hydraulics refer to the requirements of NASA's Mars Design Reference Architecture 5.0 (DRA 5.0) for impulse, thrust, and thrust to weight ratio. Second, based on the requirements above, the optimal reactor configuration parameters are obtained with the goal of maximizing the thrust to weight ratio: reactor radius 51.1 cm, height 106 cm, aspect ratio 1.04, reflector 13 cm, and reactor mass 3940 kg. Finally, the performances of the optimal reactor design are analyzed. The neutron spectrum shows a characteristic of fast neutron spectrum, and fuel loss analysis shows that the core can withstand a fuel mass loss of about 4 wt%. In addition, the core axial and radial power distribution are calculated, with an overall power peak factor of 1.26. The thermal hydraulic calculation provides a range of core operating conditions that satisfy the design requirements of DRA 5.0 and temperature, with power of 620 ∼ 950 MW and flow rate of 16 ∼ 24 kg/s. Compared with the ANL200/2000 reactor using prismatic cermet, it has the advantage of higher specific impulse. The main disadvantage is the large pressure drop, which prevents it from being applied to the scenarios of high power and high thrust. Therefore, for NTP reactor using spherical cermet elements, it should be applied to low to medium power scenarios. [ABSTRACT FROM AUTHOR]

An edge-colored graph G is called rainbow if every edge of G receives a different color. The anti-Ramsey number of t edge-disjoint rainbow spanning trees, denoted by r(n,t), is defined as the maximum number of colors in an edge-coloring of Kn containing no t edge-disjoint rainbow spanning trees. Jahanbekam and West [J. Graph Theory, 2014] conjectured that for any fixed t, r(n,t)=(n-22)+t whenever n≥2t+2≥6. In this paper, we prove this conjecture. We also determine r(n,t) when n=2t+1. Together with previous results, this gives the anti-Ramsey number of t edge-disjoint rainbow spanning trees for all values of n and t. [ABSTRACT FROM AUTHOR]

Abstract For any r -order { 0 , 1 } -tensor A with e ones, we prove that the spectral radius of A is at most e r − 1 r with the equality holds if and only if e = k r for some integer k and all ones forms a principal sub-tensor 1 k × ⋯ × k. We also prove a stability result for general tensor A with e ones where e = k r + l with relatively small l. Using the stability result, we completely characterized the tensors achieving the maximum spectral radius among all r -order { 0 , 1 } -tensor A with k r + l ones, for − r − 1 ≤ l ≤ r , and k sufficiently large. [ABSTRACT FROM AUTHOR]

For r ≥ 3 , let f r : [ 0 , ∞ ) → [ 1 , ∞ ) be the unique analytic function such that f r ( ( k r ) ) = ( k − 1 r − 1 ) for any k ≥ r − 1 . We prove that the spectral radius of an r -uniform hypergraph H with e edges is at most f r ( e ) . The equality holds if and only if e = ( k r ) for some positive integer k and H is the union of a complete r -uniform hypergraph K k r and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs. [ABSTRACT FROM AUTHOR]

In order to improve the security and increase the discharge burnup of the reactor, nuclear materials UN and U3Si2 have been considered for their higher densities and breeding ratios compared with uranium dioxide. A specific modeling of Tristructural isotropic particles with UN and U3Si2 as nuclear fuels was developed, and its failure probability was evaluated under irradiation conditions typical of a Molten Salt Reactor. The dimensional changes of the fuel particle kernel and layers were given. The stresses in the pyrolytic carbon and silicon carbide layers caused by the kernel gas pressure under normal operating conditions are calculated alone with the failure probability of the coated particle. The finding of this study will facilitate the development of new coated particulate fuels as well as support fuel parameters for design and construction of nuclear reactors. [ABSTRACT FROM AUTHOR]

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot and Wakefield. The properties of these polynomials need to be further explored. In this paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide with Motzkin polynomials, which was conjectured by Gedeon. As a byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig polynomials of wheel matroids and whirl matroids. We prove the real-rootedness of the Kazhdan-Lusztig polynomials of these matroids, thus providing positive evidence for a conjecture due to Gedeon, Proudfoot and Young. Based on the results on the Kazhdan-Lusztig polynomials, we also determine the Z -polynomials of fan matroids, wheel matroids and whirl matroids, and prove their real-rootedness, thus providing further evidence in support of a conjecture of Proudfoot, Xu, and Young. [ABSTRACT FROM AUTHOR]

Let 2 [ n ] denote the power set of [ n ] , where [ n ] = { 1 , 2 , … , n } . A collection B ⊂ 2 [ n ] forms a d -dimensional Boolean algebra if there exist pairwise disjoint sets X 0 , X 1 , … , X d ⊆ [ n ] , all non-empty with perhaps the exception of X 0 , so that B = { X 0 ∪ ⋃ i ∈ I X i : I ⊆ [ d ] } . Let b ( n , d ) be the maximum cardinality of a family F ⊂ 2 X that does not contain a d -dimensional Boolean algebra. Gunderson, Rödl, and Sidorenko proved that b ( n , d ) ≤ c d n − 1 / 2 d ⋅ 2 n where c d = 10 d 2 − 2 1 − d d d − 2 − d . In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets F with F ⊆ 2 [ n ] is defined by h n ( F ) = ∑ F ∈ F 1 / ( n | F | ) . We prove the following Turán type theorem. If F ⊆ 2 [ n ] contains no d -dimensional Boolean algebra, then h n ( F ) ≤ 2 ( n + 1 ) 1 − 2 1 − d for sufficiently large n . This result implies b ( n , d ) ≤ C n − 1 / 2 d ⋅ 2 n , where C is an absolute constant independent of n and d . With some modification, the ideas in Gunderson, Rödl, and Sidorenko's proof can be used to obtain this result. We apply the new bound on b ( n , d ) to improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras. [ABSTRACT FROM AUTHOR]

Let F be a family of subsets of { 1 , … , n } . We say that F is P -free if the inclusion order on F does not contain P as an induced subposet. The Turán function of P , denoted La ⁎ ( n , P ) , is the maximum size of a P -free family of subsets of { 1 , … , n } . We show that La ⁎ ( n , P ) ≤ ( 4 r + O ( r ) ) ( n ⌊ n / 2 ⌋ ) if P is an r -element poset of height at most 2. We also show that La ⁎ ( n , S r ) = ( r + O ( r ) ) ( n ⌊ n / 2 ⌋ ) where S r is the standard example on 2 r elements, and that La ⁎ ( n , 2 [ 2 ] ) ≤ ( 2.583 + o ( 1 ) ) ( n ⌊ n / 2 ⌋ ) , where 2 [ 2 ] is the 2-dimensional Boolean lattice. [ABSTRACT FROM AUTHOR]

Abstract: The crown is a height-2 poset whose Hasse diagram is a cycle of length 2t. A family of subsets of is -free if is not a weak subposet of . Let be the largest size of -free families of subsets of . De Bonis–Katona–Swanepoel proved . Griggs and Lu proved that for all even . In this paper, we prove for all odd . [Copyright &y& Elsevier]

Abstract: The spectral radius of a graph G is the largest eigenvalue of its adjacency matrix. Woo and Neumaier discovered that a connected graph G with is either a dagger, an open quipu, or a closed quipu. The reverse statement is not true. Many open quipus and closed quipus have spectral radii greater than . In this paper we proved the following results. For any open quipu G on n vertices () with spectral radius less than , its diameter satisfies . This bound is tight. For any closed quipu G on n vertices () with spectral radius less than , its diameter satisfies . The upper bound is tight while the lower bound is asymptotically tight. Let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter D. For and , we proved that is the unique graph obtained by attaching two paths of lengths and to a pair of antipodal vertices of the even cycle . This result is tight. Thus we settled a conjecture of Cioabaˇ–van Dam–Koolen–Lee, who previously proved a special case for and n large enough. [Copyright &y& Elsevier]

Abstract: Let be a triangle-free graph with maximum degree at most 3. Staton proved that the independence number of is at least . Heckman and Thomas conjectured that Staton’s result can be strengthened into a bound on the fractional chromatic number of , namely . Recently, Hatami and Zhu proved that . In this paper, we prove . [Copyright &y& Elsevier]

Abstract: The spectral radius of a graph G is the largest eigenvalue of its adjacency matrix . For a fixed integer , let be a graph with minimal spectral radius among all connected graphs on n vertices with diameter . Let be a tree obtained from a path of p vertices () by linking one pendant path at for each . For , were determined in the literature. Cioabaˇ et al. conjectured for fixed , is in the family . For , they conjectured and . In this paper, we settle their conjectures positively. Note that any tree in is uniquely determined by its internal path lengths. For any non-negative integers , let with , for . (Here we assume and .) Let . For any integer and sufficiently large n, we proved that must be one of the trees with the parameters satisfying for and . Moreover, for and for . These results are best possible as shown by cases , where are completely determined here. Moreover, if is divisible by and n is sufficiently large, then where . [Copyright &y& Elsevier]

Meyniel conjectured that the cop number c( G) of any connected graph G on n vertices is at most for some constant C. In this article, we prove Meyniel's conjecture in special cases that G has diameter 2 or G is a bipartite graph of diameter 3. For general connected graphs, we prove , improving the best previously known upper-bound O( n/ ln n) due to Chiniforooshan. [ABSTRACT FROM AUTHOR]

Abstract: This paper is motivated by a recent result of Wolf on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in , where p is a prime number. Wolf proved that there is a 2-coloring of with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic. In this paper, we present an explicit and simple construction of a 2-coloring with fewer monochromatic 4-APs than random 2-colorings. This problem leads us to consider the minimum number of monochromatic 4-APs in for general n. We obtain both lower bound and upper bound on the minimum number of monochromatic 4-APs in . Wolf proved that any 2-coloring of has at least monochromatic 4-APs. We improve this lower bound to . Our method for naturally apply to the similar problem on . In 2008, Parillo, Robertson, and Saracino constructed a 2-coloring of with 14.6% fewer monochromatic 3-APs than random 2-colorings. In 2010, Butler, Costello, and Graham used a new method to construct a 2-coloring of with 17.35% fewer monochromatic 4-APs (and 26.8% fewer monochromatic 5-APs) than random 2-colorings. Our construction gives a 2-coloring of with 33.33% fewer monochromatic 4-APs (and 57.89% fewer monochromatic 5-APs) than random 2-colorings. [Copyright &y& Elsevier]

Abstract: The Randić index of a graph is defined as the sum of over all edges of , where and are the degrees of vertices and , respectively. Let be the diameter of when is connected. Aouchiche et al. (2007)  conjectured that among all connected graphs on vertices the path achieves the minimum values for both and . We prove this conjecture completely. In fact, we prove a stronger theorem: If is a connected graph, then , with equality if and only if is a path with at least three vertices. [Copyright &y& Elsevier]

For a fixed set of positive integers R , we say H is an R -uniform hypergraph , or R -graph , if the cardinality of each edge belongs to R. For a graph G = (V , E) , a hypergraph H is called a Berge - G , if there is an injection i : V (G) → V (H) and a bijection f : E (G) → E (H) such that for all e = u v ∈ E (G) , we have { i (u) , i (v) } ⊆ f (e). We define the cover Turán number of a graph G , denoted as ex ˆ R (n , B G) , as the maximum number of edges in the 2-shadow of an n -vertex Berge- G -free R -graph, where the 2-shadow H of a hypergraph H is a graph such that an edge e ∈ E (H) if and only if e ⊆ h for some h ∈ E (H). In this paper, we show an Erdős–Stone–Simonovits-type upper bound on the cover Turán number of graphs and determine the cover Turán density of all graphs when the host hypergraph is 3-uniform. [ABSTRACT FROM AUTHOR]

Many graphs arising in various information networks exhibit the "power law" behavior –the number of vertices of degree k is proportional to k[sup -β] for some positive β.We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely$ (1+ o(1))\sqrt{m} $$*FORMULAINSPRINGER$ where m is the maximum degree. Moreover, the klargest eigenvalues of a random power law graph with exponent β have power law distribution with exponent2β if the maximum degree is sufficiently large, where k is a function depending on β, mand d, the average degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm[sup 3-β] for some constant c depending on β and the average degree. This resultfollows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and$ \tilde{d} $$*FORMULAINSPRINGER$,the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely [ABSTRACT FROM AUTHOR]

We consider a family of random graphs with a given expected degree sequence. Each edge is chosen independently with probability proportional to the product of the expected degrees of its endpoints. We examine the distribution of the sizes/volumes of the connected components which turns out depending primarily on the average degree d and the second-order average degree d[sup ~] . Here d[sup ~] denotes the weighted average of squares of the expected degrees. For example, we prove that the giant component exists if the expected average degree d is at least 1, and there is no giant component if the expected second-order average degree d[sup ~] is at most 1. Examples are given to illustrate that both bounds are best possible. [ABSTRACT FROM AUTHOR]

Given a set R , a hypergraph is R -uniform if the size of every hyperedge belongs to R. A hypergraph H is called covering if every vertex pair is contained in some hyperedge in H. In this note, we show that every covering [3]-uniform hypergraph on n ≥ 6 vertices contains a Berge cycle C s for any 3 ≤ s ≤ n. As an application, we determine the maximum Lagrangian of k -uniform Berge- C t -free hypergraphs and Berge- P t -free hypergraphs. [ABSTRACT FROM AUTHOR]

Discusses the sheets and rigid orbit covers of exceptional Lie groups. Induced orbit covers and sheets; Levi subgroups of exceptional groups; Proving the conjecture that the sheets of an exceptional Lie group are disjoint on the basis of the calculating result.

The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias et al. (2016) [4]. Let U m , d denote the uniform matroid of rank d on a set of m + d elements. Gedeon et al. (2017) [7] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of U m , d using equivariant Kazhdan-Lusztig polynomials. In this paper we give an alternative explicit formula, which allows us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of U m , d for 2 ≤ m ≤ 15 and all d 's. The case m = 1 was previously proved by Gedeon et al. (2017) [8]. We further determine the Z -polynomials of all U m , d 's and prove the real-rootedness of the Z -polynomials of U m , d for 2 ≤ m ≤ 15 and all d 's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of U m , d 's without using the equivariant Kazhdan-Lusztig polynomials. [ABSTRACT FROM AUTHOR]

For a fixed set of positive integers R , we say H is an R -uniform hypergraph, or R -graph, if the cardinality of each edge belongs to R. An R -graph H is covering if every vertex pair of H is contained in some hyperedge. For a graph G = (V , E) , a hypergraph H is called a Berge - G , denoted by B G , if there is an injection i : V (G) → V (H) and a bijection f : E (G) → E (H) such that for all e = u v ∈ E (G) , we have { i (u) , i (v) } ⊆ f (e). In this note, we define a new type of Ramsey number, namely the cover Ramsey number , denoted as R ˆ R (B G 1 , B G 2) , as the smallest integer n 0 such that for every covering R -uniform hypergraph H on n ≥ n 0 vertices and every 2-edge-coloring (blue and red) of H , there is either a blue Berge- G 1 or a red Berge- G 2 subhypergraph. We show that for every k ≥ 2 , there exists some c k such that for any finite graphs G 1 and G 2 , R (G 1 , G 2) ≤ R ˆ k (B G 1 , B G 2) ≤ c k ⋅ R (G 1 , G 2) 3 . Moreover, we show that for each positive integer d and k , there exists a constant C = C (d , k) such that if G is a graph on n vertices with maximum degree at most d , then R ˆ k (B G , B G) ≤ C n. [ABSTRACT FROM AUTHOR]

NaC-MgCl2 eutectic salt and α-Al2O3 are selected as phase change material (PCM) and encapsulation material, respectively. Six kinds of NaC-MgCl2 eutectic salt/α-Al2O3 ceramic medium-high temperature composite phase change material (CPCM) with different mass ratios are prepared by spark plasma sintering (SPS) process. The microstructure and thermodynamic properties of CPCM are analyzed by scanning electron microscope (SEM), electron probe micro-analysis (EPMA), X-ray diffraction (XRD), high temperature laser thermal conductivity meter and differential scanning calorimeter-thermogravimetric analyzer (TG-DSC). The changing and discharging rates of CPCM are analyzed by finite element simulation. The results show that after sintered at 550 °C, α-Al2O3 ceramic plays a good role in encapsulation and effectively reduces the weight loss rate of NaC-MgCl2 eutectic salt at high temperature. Compared with NaCl-MgCl2 eutectic salt, the overall thermal conductivity of CPCM is significantly improved, and the phase change time is greatly shortened, indicating that the heat charging and discharging rates of CPCM are significantly improved. The medium-high temperature CPCM has dense microstructure, and the distribution of all elements in CPCM is natively uniform. The distribution of Na and Mg elements is superposition, indicating that there is no chemical reaction between NaCl-MgCl2 eutectic salt and α-Al2O3, and CPCM has good chemical stability. The melting point of NaC-MgCl2 eutectic salt/α-Al2O3 ceramic medium-high temperature CPCM is 408.7 °C, the latent heat of phase transition is 142.6 J/g, and the thermal stability is good at 500-650 °C. This paper provides a novel experimental basis for the preparation and characterization of eutectic salt/ceramic CPCM. [ABSTRACT FROM AUTHOR]

Magnolia kwangsiensis, a dioecious tree native to China, is recognized not only for its status as an at-risk species but also for its potential in therapeutic applications courtesy of its bioactive compounds. However, the genetic underpinnings of its leaf development and compound biosynthesis are not well documented. Our study aims to bridge this knowledge gap through comparative transcriptomics, analyzing gene expression through different leaf maturation stages. We studied the transcriptome of M. kwangsiensis leaves by applying RNA sequencing at juvenile, tender, and mature phases. We identified differentially expressed genes (DEGs) to explore transcriptional changes accompanying the developmental trajectory. Our analysis delineates the transcriptional landscape of over 20,000 genes with over 6000 DEGs highlighting significant transcriptional shifts throughout leaf maturation. Mature leaves demonstrated upregulation in pathways related to photosynthesis, cell wall formation, and polysaccharide production, affirming their structural integrity and specialized metabolic functions. Our GO and KEGG enrichment analyses underpin these findings. Furthermore, we unveiled coordinated gene activity correlating development with synthesizing therapeutically relevant polysaccharides. We identified four novel glycosyltransferases potentially pivotal in this synergistic mechanism. Our study uncovers the complementary evolutionary forces that concurrently sculpt structural and chemical defenses. These genetic mechanisms calibrate leaf tissue resilience and biochemical efficacy. [ABSTRACT FROM AUTHOR]

Let G be a graph. We say that a hypergraph H is a Berge- G if there is a bijection ϕ : E (G) → E (H) such that e ⊆ ϕ (e) for all e ∈ E (G). For any r -uniform hypergraph H and a real number p ≥ 1 , the p -spectral radius λ (p) (H) of H is defined as λ (p) (H) : = max x ∈ R n , ‖ x ‖ p = 1 ⁡ r ∑ { i 1 , i 2 , ... , i r } ∈ E (H) x i 1 x i 2 ⋯ x i r . In this paper, we study the p -spectral radius of Berge- G hypergraphs. We determine the 3-uniform hypergraphs with maximum p -spectral radius for p ≥ 1 among Berge- G hypergraphs when G is a path, a cycle or a star. [ABSTRACT FROM AUTHOR]

We propose a directed acyclic hypergraph framework for a probabilistic graphical model that we call Bayesian hypergraphs. The space of directed acyclic hypergraphs is much larger than the space of chain graphs. Hence Bayesian hypergraphs can model much finer factorizations than Bayesian networks or LWF chain graphs and provide simpler and more computationally efficient procedures for factorizations and interventions. Bayesian hypergraphs also allow a modeler to represent causal patterns of interaction such as Noisy-OR graphically (without additional annotations). We introduce global, local and pairwise Markov properties of Bayesian hypergraphs and prove under which conditions they are equivalent. We also extend the causal interpretation of LWF chain graphs to Bayesian hypergraphs and provide corresponding formulas and a graphical criterion for intervention. [ABSTRACT FROM AUTHOR]

An r -matrix is a matrix with symbols in { 0 , 1 , ... , r − 1 }. A matrix is simple if it has no repeated columns. Let F be a finite set of r -matrices. Let forb (m , r , F) denote the maximum number of columns possible in a simple r -matrix A that has no submatrix which is a row and column permutation of any F ∈ F. Many investigations have involved r = 2. For general r , Füredi and Sali proved that forb (m , r , F) is polynomial in m if and only if for every pair i , j ∈ { 0 , 1 , ... , r − 1 } there is a matrix in F whose entries are only i or j. Let T ℓ (r) denote the set of 4 ( r 2 ) r -matrices of the following form. For a pair i , j ∈ { 0 , 1 , ... , r − 1 } we form four ℓ × ℓ matrices, namely the matrix with i 's on the diagonal and j 's off the diagonal and the matrix with i 's on and above the diagonal and j 's below the diagonal and the two matrices with the roles of i , j reversed. Anstee and Lu determined that forb (m , r , T ℓ (r)) is a constant. Let F be a finite set of 2-matrices. In the present paper we investigate the question whether forb (m , r , T ℓ (r) ﹨ T ℓ (2) ∪ F) is Θ (forb (m , 2 , F)) and settle this in the affirmative for some cases with F = { F } , including most 2-columned F. [ABSTRACT FROM AUTHOR]

Abstract: Given a finite poset P, we consider the largest size of a family of subsets of that contains no (weak) subposet P. This problem has been studied intensively in recent years, and it is conjectured that exists for general posets P, and, moreover, it is an integer. For let denote the k-diamond poset . We study the average number of times a random full chain meets a P-free family, called the Lubell function, and use it for to determine for infinitely many values k. A stubborn open problem is to show that ; here we make progress by proving (if it exists). [Copyright &y& Elsevier]

The fissile fuel loss in tungsten (W) based ceramic-metal (Cermet) element used in nuclear thermal propulsion is of great importance. In this study, the W-Y 2 O 3 composite matrix was manufactured tentatively by Spark Plasma Sintering (SPS) to relieve the fuel loss. Performance evaluation of hydrogen (H 2) permeation and diffusion behaviors in the matrix was carried out by a gas permeation device. The results indicate that the diffusion coefficient of H 2 in the pure W matrix is nearly 10 times larger than that in the W-Y 2 O 3 matrix at a test temperature of 823 K. Based on the results, a careful investigation of the fuel loss mechanism of the Y 2 O 3 dispersed W matrix was performed. SEM and TEM showed that the fine Y 2 O 3 particles are distributed along grain boundaries of the W ternary-phase and present a distinct transition region of about 10 nm between the Y 2 O 3 and W phases. In particular, the crystallographic orientation relationship suggests that a semi-coherent structure is formed at the W/Y 2 O 3 phase interface: (0 1 −1) W || (1 0 −1) Y 2 O 3 and [0 1 1] W || [1 1 1] Y 2 O 3. This semi-coherent structure is favorable for binding the two phases together tightly, thus hindering the inward diffusion of H 2 greatly. Moreover, the improved density and refined size of the W grains are achieved. Therefore, there is no doubt that the dispersion of Y 2 O 3 is beneficial for lowering the H 2 diffusion rate in the W matrix and thus reducing the fuel loss effectively. This work furthers our understanding of the key role of Y 2 O 3 in the W matrix used in nuclear thermal propulsion. • The effect and influence mechanism of Y 2 O 3 on the W matrix has been studied for use in nuclear thermal propulsion. • The semi-coherent structure facilitates tight binding of the two phases, thus blocking the inward diffusion of H 2 greatly. • The H 2 diffusivity in pure W is 10 times higher than in W-Y 2 O 3 matrix at 823 K, which is favorable for reducing fuel loss. [ABSTRACT FROM AUTHOR]

In this study, a ternary salt/porous Si 3 N 4 composite was prepared as a high-temperature shape-stabilized phase change material by pressure impregnation. In this composite, the ternary salt composed of 50 wt% sodium chloride (NaCl), 30 wt% potassium chloride (KCl), and 20 wt% magnesium chloride (MgCl 2) was acted as the high-temperature phase change material. A porous Si 3 N 4 ceramic was acted as the skeleton material and heat transfer enhancer. The porosity infiltration ratio of the pores reached 89.19%, demonstrating the good chemical compatibility of Si 3 N 4 with chloride salts. The thermal conductivity of the composite was over six times greater than that of pure ternary salts. The thermal physical properties of the composite were stable even after 300 thermal cycles. The coefficients of thermal conductivity obtained by theoretical analysis and calculation were compared to the experimental values. All the results showed that this composite is a promising candidate as a shape-stabilized high-temperature phase change heat storage material. • Ternary chloride salt/Si 3 N 4 composite phase change material (PCM) was prepared. • The composite PCM showed good chemical stability and thermal cycling stability. • The thermal conductivity of composite was more than six times than that of salt. • Heat transfer rate of the composite PCM was significantly improved. • Phase change process of composite PCM was simulated by finite element method. [ABSTRACT FROM AUTHOR]