We observe that a lobster with diameter at least five has a unique path H = x0, x1, . . . , xm with the property that besides the adjacencies in H both x0 and xm are adjacent to the centers of at least one K1,s, where s > 0, and each xi , 1 ≤ i ≤ m − 1, is adjacent at most to the centers of some K1,s, where s > 0. This path H is called the central path of the lobster. We call K1,s an even branch if s is nonzero even, an odd branch if s is odd and a pendant branch if s = 0. In the existing literature only some specific classes of lobsters have been found to have graceful labelings. Lobsters to which we give graceful labelings in this paper share one common property with the graceful lobsters (in our earlier works) that each vertex xi , 0 ≤ i ≤ m − 1, is even, the degree of xm may be odd or even. However, we are able to attach any combination of all three types of branches to a vertex xi , 1 ≤ i ≤ m, with total number of branches even. Furthermore, in the lobsters here the vertices xi , 1 ≤ i ≤ m, on the central path are attached up to six different combinations of branches, which is at least one more than what we find in graceful lobsters in the earlier works.
We consider a family of conforming finite element schemes with piecewise polynomial space of degree k in space for solving the wave equation, as a model for second order hyperbolic equations. The discretization in time is performed using the Newmark method. A new a priori estimate is proved. Thanks to this new a priori estimate, it is proved that the convergence order of the error is h k + τ 2 in the discrete norms of L∞(0, T ; H1 (Ω)) and W1,∞(0, T ; L 2 (Ω)), where h and τ are the mesh size of the spatial and temporal discretization, respectively. These error estimates are useful since they allow us to get second order time accurate approximations for not only the exact solution of the wave equation but also for its first derivatives (both spatial and temporal). Even though the proof presented in this note is in some sense standard, the stated error estimates seem not to be present in the existing literature on the finite element methods which use the Newmark method for the wave equation (or general second order hyperbolic equations).
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle $\mathbb T$. A set of integers is called $r$-Bohr if it is recurrent for all products of $r$ rotations on $\mathbb T$, and Bohr if it is recurrent for all products of rotations on $\mathbb T$. It is a result due to Katznelson that for each $r\ge 1$ there exist sets of integers which are $r$-Bohr but not $(r+1)$-Bohr. We present new examples of $r$-Bohr sets which are not Bohr, thanks to a construction which is both flexible and completely explicit. Our results are related to an old combinatorial problem of Veech concerning syndetic sets and the Bohr topology on $\mathbb Z$, and its reformulation in terms of recurrence sets which is due to Glasner and Weiss.
About Lehmer’s number, many people have studied its various properties, and obtained a series of interesting results. In this paper, we consider a generalized Lehmer problem: Let p be a prime, and let N(k; p) denote the number of all 1\leqslant a_{i}\leq p-1 such that a_{1}a_{2}...a_{k}\equiv 1 mod p and 2 | ai + āi + 1, i = 1, 2, ..., k. The main purpose of this paper is using the analytic method, the estimate for character sums and trigonometric sums to study the asymptotic properties of the counting function N(k; p), and give an interesting asymptotic formula for it., Han Zhang, Wenpeng Zhang., and Obsahuje seznam literatury
In this paper, we prove new embedding theorems for generalized anisotropic Sobolev spaces, $W_{\Lambda ^{p,q}(w)}^{r_1,\dots ,r_n}$ and $W_{X}^{r_1,\dots ,r_n}$, where $\Lambda ^{p,q}(w)$ is the weighted Lorentz space and $X$ is a rearrangement invariant space in $\mathbb R^n$. The main methods used in the paper are based on some estimates of nonincreasing rearrangements and the applications of $B_p$ weights.
Oscillatory properties of solutions to the system of first-order linear difference equations ∆uk = qkvk ∆vk = −pkuk+1, are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).
In this paper, an equivalence on the class of uninorms on a bounded lattice is discussed. Some relationships between the equivalence classes of uninorms and the equivalence classes of their underlying t-norms and t-conorms are presented. Also, a characterization for the sets admitting some incomparability w.r.t. the U-partial order is given.
Some behavioural aspects of the reproductive biology of Megaselia andrenae Disney, a kleptoparasite of the communal bee Andrena agilissima (Scopoli), were investigated at the nesting site of its host at Isola d'Elba (Italy). The scuttle fly mates more often in the early afternoon, which coincides with the period when the provisioning flights of its host are more frequent. The presence of the flies at the host nesting site, either in copula or single, is lower in the morning. In general only the females enter the host nests immediately after a mate, in a few cases closely followed by the males. When leaving the nest, females refuse to mate again with the males waiting outside. Observations on the ovaries of the females revealed no differences, either in the number of eggs or in the length of the most mature egg, between the individuals collected in copula or when flying alone. The species is sexually dimorphic, the females being larger than males. Male size does not seem to influence the females choice for mating.