A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence ${2p-1\choose p-1} \equiv 1 \pmod {p^4}$. For such a prime $p$, we establish an expression for ${2p-1\choose p-1}\pmod {p^8}$ given in terms of the sums $R_i:=\sum _{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$ \left ({2p-1\atop p-1}\right ) \equiv 1 -2p \sum _{k=1}^{p-1}\frac {1}{k} -2p^2\sum _{k=1}^{p-1}\frac {1}{k^2}\pmod {p^7}. $$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique of Helou and Terjanian, the above congruence is given as an expression involving the Bernoulli numbers.
Let $p>3$ be a prime, and let $q_p(2)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $$ \sum _{k=1}^{p-1} \frac {1}{k\cdot 2^k} \equiv q_p(2)-\frac {pq_p(2)^2}{2}+ \frac {p^2 q_p(2)^3}{3} -\frac {7}{48} p^2 B_{p-3}\pmod {p^3}, $$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $$ q_p(2)^3 \equiv -3\sum _{k=1}^{p-1} \frac {2^k}{k^3}+ \frac {7}{16} \sum _{k=1}^{(p-1)/2} \frac {1}{k^3} \pmod {p}, $$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum $\sum _{k=1}^{p-1}1/(k^2\cdot 2^k)$ modulo $p^2$ that also generalizes a related Sun's congruence modulo $p$.
A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The connected domatic number of $G$ is the maximum number of pairwise disjoint, connected dominating sets in $V(G)$. We establish a sharp lower bound on the number of edges in a connected graph with a given order and given connected domatic number. We also show that a planar graph has connected domatic number at most 4 and give a characterization of planar graphs having connected domatic number 3.
A dominating set in a graph $G$ is a connected dominating set of $G$ if it induces a connected subgraph of $G$. The minimum number of vertices in a connected dominating set of $G$ is called the connected domination number of $G$, and is denoted by $\gamma _{c}(G)$. Let $G$ be a spanning subgraph of $K_{s,s}$ and let $H$ be the complement of $G$ relative to $K_{s,s}$; that is, $K_{s,s}=G\oplus H$ is a factorization of $K_{s,s}$. The graph $G$ is $k$-$\gamma _{c}$-critical relative to $K_{s,s}$ if $\gamma _{c}(G)=k$ and $\gamma _{c}(G+e)<k$ for each edge $e\in E(H)$. First, we discuss some classes of graphs whether they are $\gamma _{c}$-critical relative to $K_{s,s}$. Then we study $k$-$\gamma _{c}$-critical graphs relative to $K_{s,s}$ for small values of $k$. In particular, we characterize the $3$-$\gamma _{c}$-critical and $4$-$\gamma _{c}$-critical graphs.
For an ordered set $W=\lbrace w_1, w_2, \dots , w_k\rbrace $ of vertices and a vertex $v$ in a connected graph $G$, the representation of $v$ with respect to $W$ is the $k$-vector $r(v|W)$ = ($d(v, w_1)$, $d(v, w_2),\dots ,d(v, w_k))$, where $d(x,y)$ represents the distance between the vertices $x$ and $y$. The set $W$ is a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set for $G$ containing a minimum number of vertices is a basis for $G$. The dimension $\dim (G)$ is the number of vertices in a basis for $G$. A resolving set $W$ of $G$ is connected if the subgraph $<W>$ induced by $W$ is a nontrivial connected subgraph of $G$. The minimum cardinality of a connected resolving set in a graph $G$ is its connected resolving number $\mathop {\mathrm cr}(G)$. Thus $1 \le \dim (G) \le \mathop {\mathrm cr}(G) \le n-1$ for every connected graph $G$ of order $n \ge 3$. The connected resolving numbers of some well-known graphs are determined. It is shown that if $G$ is a connected graph of order $n \ge 3$, then $\mathop {\mathrm cr}(G) = n-1$ if and only if $G = K_n$ or $G = K_{1, n-1}$. It is also shown that for positive integers $a$, $b$ with $a \le b$, there exists a connected graph $G$ with $\dim (G) = a$ and $\mathop {\mathrm cr}(G) = b$ if and only if $(a, b) \notin \lbrace
(1, k)\: k = 1\hspace{5.0pt}\text{or}\hspace{5.0pt}k \ge 3\rbrace $. Several other realization results are present. The connected resolving numbers of the Cartesian products $G \times K_2$ for connected graphs $G$ are studied.
For an ordered k-decomposition D = {G1, G2, . . . , Gk} of a connected graph G and an edge e of G, the D-code of e is the k-tuple cD(e) = (d(e, G1), d(e, G2), . . . , d(e, Gk)), where d(e, Gi) is the distance from e to Gi . A decomposition D is resolving if every two distinct edges of G have distinct D-codes. The minimum k for which G has a resolving k-decomposition is its decomposition dimension dimd(G). A resolving decomposition D of G is connected if each Gi is connected for 1 ≤ i ≤ k. The minimum k for which G has a connected resolving k-decomposition is its connected decomposition number cd(G). Thus 2 ≤ dimd(G) ≤ cd(G) ≤ m for every connected graph G of size m ≥ 2. All nontrivial connected graphs with connected decomposition number 2 or m are characterized. We provide bounds for the connected decomposition number of a connected graph in terms of its size, diameter, girth, and other parameters. A formula for the connected decomposition number of a nonpath tree is established. It is shown that, for every pair a, b of integers with 3 ≤ a ≤ b, there exists a connected graph G with dimd(G) = a and cd(G) = b.
The paper characterizes the evolution, structure and areal distribution of the large-scale background magnetic fields in the solar photosphere. The direction of the horizontal streaming of the solar photospheric plasma was found. Active regions are formed mainly in places where the global circulation displays maximum
velocity. Filaments occur in the areas with a high value of velocity gradient perpendicular to the filament axis. The relationship between the weak background and the strong local magnetic fields is documented.
In this paper we consider a product preserving functor $\mathcal F$ of order $r$ and a connection $\Gamma $ of order $r$ on a manifold $M$. We introduce horizontal lifts of tensor fields and linear connections from $M$ to $\mathcal F(M)$ with respect to $\Gamma $. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.
The inferior cerebellar peduncle (ICP) is an important role in motor control, such as coordination of movement control of balance, posture, and gait. In the current study, using diffusion tensor tractography (DTT), we attempted to investigate the connectivity of the ICP in normal subjects. Forty healthy subjects were recruited for this study. DTTs were acquired using a sensitivity-encoding head coil at 1.5 Tesla. A seed region of interest was drawn at the ICP using the FMRIB Software Library. Connectivity was defined as the incidence of connection between the ICP and target brain regions at the threshold of 5, 25, and 50 streamlines. The ICP showed 100% connectivity to the vestibular nucleus, reticular formation, pontine tegmentum, and posterior lobe of the cerebellum, irrespective of thresholds. In contrast, the ICP showed more than 70% connectivity with the target brain regions at the threshold of 5 streamlines that is to the thalamus (100 %), anterior lobe of the cerebellum (100 %), pedunculopontine nucleus (95.0 %), red nucleus (92.5 %), primary somatosensory cortex (86.3 %), and primary motor cortex (75.0 %). According to our findings, the ICP had high connectivity, mainly with the sensory-motor related areas. We believe that the methodology and results of this study would be useful in investigation of the neural network associated with the sensory-motor system and brain plasticity following brain injury and other diseases.
The eigenvalues of graphs are related to many of its combinatorial properties. In his fundamental work, Fiedler showed the close connections between the Laplacian eigenvalues and eigenvectors of a graph and its vertex-connectivity and edge-connectivity. We present some new results describing the connections between the spectrum of a regular graph and other combinatorial parameters such as its generalized connectivity, toughness, and the existence of spanning trees with bounded degree., Sebastian M. Cioabă, Xiaofeng Gu., and Obsahuje seznam literatury