In this short note we prove that the Winterbottom shape (Winterbottom in Acta Metallurgica 15:303-310, 1967) is a volume-constraint minimizer of the corresponding anisotropic capillary functional.
In this short note we prove that the Winterbottom shape (Winterbottom in Acta Metallurgica 15:303-310, 1967) is a volume-constraint minimizer of the corresponding anisotropic capillary functional.
We establish the equivalence of hyperbolicity and of two other properties for a two-sided linear delay-difference equation and its evolution map. These two properties are the admissibility with respect to various pairs of spaces, and the Ulam–Hyers stability of the equation, again with respect to various spaces. This gives characterizations of important properties of a linear dynamical system in terms of corresponding properties of the autonomous dynamical system determined by the associated evolution map.
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other based on modular forms. We close by proving analogous results for generalized overcubic partitions.
We show that every surjective mapping f between the unit spheres of two real (mathcal {L}^infty (Gamma ))-type spaces satisfies
if and only if f is phase-equivalent to an isometry, i.e., there is a phase-function (varepsilon ) from the unit sphere of the (mathcal {L}^infty (Gamma ))-type space onto ({-1,1}) such that (varepsilon cdot f) is a surjective isometry between the unit spheres of two real (mathcal {L}^infty (Gamma ))-type spaces, and furthermore, this isometry can be extended to a linear isometry on the whole space (mathcal {L}^infty (Gamma )). We also give an example to show that these are not true if “min” is replaced by “max”.
For a given p-variable mean (M :I^p rightarrow I) (I is a subinterval of ({mathbb {R}})), following (Horwitz in J Math Anal Appl 270(2):499–518, 2002) and (Lawson and Lim in Colloq Math 113(2):191–221, 2008), we can define (under certain assumptions) its ((p+1))-variable (beta )-invariant extension as the unique solution (K :I^{p+1} rightarrow I) of the functional equation
$$begin{aligned}&Kbig (M(x_2,dots ,x_{p+1}),M(x_1,x_3,dots ,x_{p+1}),dots ,M(x_1,dots ,x_p)big )&quad =K(x_1,dots ,x_{p+1}), text { for all }x_1,dots ,x_{p+1} in I end{aligned}$$in the family of means. Applying this procedure iteratively we can obtain a mean which is defined for vectors of arbitrary lengths starting from the bivariate one. The aim of this paper is to study the properties of such extensions.
Let (k ge 3). If a multiplicative function f satisfies
for all (a_1, a_2, ldots , a_k in {mathbb {N}}), then f is the identity function. The set of positive cubes is said to be a k-additive uniqueness set for multiplicative functions. But, the condition (k=2) can be satisfied by infinitely many multiplicative functions. In additon, if (k ge 3) and a multiplicative function g satisfies
for all (a_1, a_2, ldots , a_k in {mathbb {N}}), then g is the identity function. However, when (k=2), there exist three different types of multiplicative functions.
Three functional inequalities are shown to uniquely characterize the exponential function. Each of the three inequalities is indispensable in the sense that no two of the three suffice.
In this paper, we consider the 2-adic valuation of integers and provide an alternative representation for the generating function of the number of overpartitions of an integer. As a consequence of this result, we obtain a new formula and a new combinatorial interpretation for the number of overpartitions of an integer. This formula implies a certain type of partitions with restrictions for which we provide two Ramanujan-type congruences and present as open problems two infinite families of linear inequalities. Connections between overpartitions and the game of m-Modular Nim with two heaps are presented in this context.
Let G be a connected graph and let (kge 1) be an integer. Let T be a spanning tree of G. The leaf degree of a vertex (vin V(T)) is defined as the number of leaves adjacent to v in T. The leaf degree of T is the maximum leaf degree among all the vertices of T. Let |E(G)| and (rho (G)) denote the size and the spectral radius of G, respectively. In this paper, we first create a lower bound on the size of G to ensure that G admits a spanning tree with leaf degree at most k. Then we establish a lower bound on the spectral radius of G to guarantee that G contains a spanning tree with leaf degree at most k. Finally, we create some extremal graphs to show all the bounds obtained in this paper are sharp.
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：