A graph $G$ with $p$ vertices and $q$ edges, vertex set $V(G)$ and edge set $E(G)$, is said to be super vertex-graceful (in short SVG), if there exists a function pair $(f, f^+)$ where $f$ is a bijection from $V(G)$ onto $P$, $f^+$ is a bijection from $E(G)$ onto $Q$, $f^+((u, v)) = f(u) + f(v)$ for any $(u, v) \in E(G)$, $$ Q = \begin{cases} \{\pm 1,\dots , \pm \frac 12q\},&\text {if $q$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(q-1)\},&\text {if $q$ is odd,} \end{cases} $$ and $$ P = \begin{cases} \{\pm 1,\dots , \pm \frac 12p\},&\text {if $p$ is even,}\\ \{0, \pm 1, \dots , \pm \frac 12(p-1)\},&\text {if $p$ is odd.} \end{cases} $$ \endgraf We determine here families of unicyclic graphs that are super vertex-graceful.
A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. Some constructions of supermagic labellings of regular graphs are described. Supermagic regular complete multipartite graphs and supermagic cubes are characterized.
Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category $\mathcal {FM}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL(m)$-invariant algebra homomorphism $\nu \colon A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an $\mathcal {FM}_m$-natural transformation $FY\to F^1Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.
Orthomodular implication algebras (with or without compatibility condition) are a natural generalization of Abbott’s implication algebras, an implication reduct of the classical propositional logic. In the paper deductive systems (= congruence kernels) of such algebras are described by means of their restrictions to principal filters having the structure of orthomodular lattices.
The testing of the null hypothesis of no treatment effect against the alternative of increasing treatment effect by means of rank statistics is extended from the classical Friedman random blocks model into an unbalanced design allowing treatments not to be applied simultaneously in each random block. The asymptotic normality of the constructed rank test statistic is proved both in the setting not allowing ties and also for models with presence of ties. As a by-product of the proofs a multiple comparisons rule based on rank statistics is obtained for the case when the null hypothesis of no treatment effect is tested against the general alternative of its negation.
A new method of testing the random closed set model hypothesis (for example: the Boolean model hypothesis) for a stationary random closed set Ξ⊆\rd with values in the extended convex ring is introduced. The method is based on the summary statistics - normalized intrinsic volumes densities of the \ep-parallel sets to Ξ. The estimated summary statistics are compared with theirs envelopes produced from simulations of the model given by the tested hypothesis. The p-level of the test is then computed via approximation of the summary statistics by multinormal distribution which mean and the correlation matrix is computed via given simulations. A new estimator of the intrinsic volumes densities from \cite{MR06} is used, which is especially suitable for estimation of the intrinsic volumes densities of \ep-parallel sets. The power of this test is estimated for planar Boolean model hypothesis and two different alternatives and the resulted powers are compared to the powers of known Boolean model tests. The method is applied on the real data set of a heather incidence.
The main purpose of this paper is to study the hybrid mean value of $\frac {L'}L(1,\chi )$ and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $\sum _{\chi \neq \chi _0} |\tau (\chi )| |\frac {L'}L(1,\chi )|^{2k}$ of $\frac {L'}L$ and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.
Let $a,b,c>0$. We investigate the characterization problem which asks for a classification of all the triples $(a,b,c)$ such that the Weyl-Heisenberg system $\{{\rm e}^{2\pi {\rm i}mbx} \* \chi _{[na,na+c)}\colon m,n\in {\mathbb Z}\}$ is a frame for $L^2({\mathbb R})$. It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b=1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of $a$ being a rational number. We prove some special cases of values for $c$ and $a$ that do not produce a frame, which expands earlier works.
For a simplicial complex $\Delta $ we study the behavior of its $f$- and $h$-triangle under the action of barycentric subdivision. In particular we describe the $f$- and $h$-triangle of its barycentric subdivision $\mathop {\rm sd}(\Delta )$. The same has been done for $f$- and $h$-vector of $\mathop {\rm sd}(\Delta )$ by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the $h$-triangle of $\Delta $ are nonnegative, then the entries of the $h$-triangle of $\mathop {\rm sd}(\Delta )$ are also nonnegative. We conclude with a few properties of the $h$-triangle of $\mathop {\rm sd}(\Delta )$.