Free multiflows in bidirected and skew-symmetric graphs

A graph (digraph) $G=(V,E)$ with a set $T\subseteq V$ of terminals is called inner Eulerian if each nonterminal node $v$ has even degree (resp. the numbers of edges entering and leaving $v$ are equal). Cherkassky and Lov\'asz showed that the maximum number of pairwise edge-disjoint $T$-paths in an inner Eulerian graph $G$ is equal to $\frac12\sum_{s\in T} \lambda(s)$, where $\lambda(s)$ is the minimum number of edges whose removal disconnects $s$ and $T-\{s\}$. A similar relation for inner Eulerian digraphs was established by Lomonosov. Considering undirected and directed networks with ``inner Eulerian'' edge capacities, Ibaraki, Karzanov, and Nagamochi showed that the problem of finding a maximum integer multiflow (where partial flows connect arbitrary pairs of distinct terminals) is reduced to $O(\log T)$ maximum flow computations and to a number of flow decompositions. In this paper we extend the above max-min relation to inner Eulerian bidirected and skew-symmetric graphs and develop an algorithm of complexity $O(VE\log T\log(2+V^2/E))$ for the corresponding capacitated cases. In particular, this improves the known bound for digraphs. Our algorithm uses a fast procedure for decomposing a flow with O(1) sources and sinks in a digraph into the sum of one-source-one-sink flows.Comment: 21 pages, 4 figures Submitted to a special issue of DA

Optimum Branching Problem Revisited

Given a digraph $G = (V_G, A_G)$, a \emph{branching} in $G$ is a set of arcs $B \subseteq A_G$ such that the underlying undirected graph spanned by $B$ is acyclic and each node in $G$ is entered (\emph{covered}) by at most one arc from $B$. Tarjan developed efficient algorithms (based on the cycle contraction technique) for the following problem: given a digraph $G$ with a \emph{weight} function $w \colon A_G \to \R$, find a branching $B$ of the minimum weight $w(B) := \sum_{a \in B} w(a)$ among all branchings with the maximum ardinality \abs{B}. We generalize this notion as follows: for a digraph $G$ and a matroid \calM_V on $V_G$, a \emph{matroid branching} in $G$ w.r.t. \calM_V is a branching in $G$ such that the covered set of nodes is independent w.r.t. \calM_V. The unweighted (cardinality) problem consists in finding a matroid branching $B$ with \abs{B} maximum. We show that the general cycle contraction approach is applicable to this problem and leads to an efficient algorithm (provided that an oracle is given for testing independence in the matroids arising as the result of the contraction procedure). In the weighted version we are looking for a matroid branching $B$ that minimizes $w(B)$ (for a given weight function $w \colon A_G \to \R$) among all matroid branchings of the maximum cardinality. We show that if \calM_V is a rainbow matroid (that is, nodes of $G$ are marked with colors and it is forbidden to cover more than one node of any color), then there exists an $O(\min(n^2, m \log n))$ method, matching the complexity of Tarjan's algorithm (here n := \abs{V_G}, m := \abs{A_G}).Comment: 12 page

Determination of Low-Energy Parameters of Neutron--Proton Scattering on the Basis of Modern Experimental Data from Partial-Wave Analyses

The triplet and singlet low-energy parameters in the effective-range expansion for neutron--proton scattering are determined by using the latest experimental data on respective phase shifts from the SAID nucleon--nucleon database. The results differ markedly from the analogous parameters obtained on the basis of the phase shifts of the Nijmegen group and contradict the parameter values that are presently used as experimental ones. The values found with the aid of the phase shifts from the SAID nucleon--nucleon database for the total cross section for the scattering of zero-energy neutrons by protons, $\sigma_{0}=20.426$b, and the neutron--proton coherent scattering length, $f=-3.755$fm, agree perfectly with the experimental cross-section values obtained by Houk, $\sigma_{0}=20.436\pm 0.023$b, and experimental scattering-length values obtained by Houk and Wilson, $f=-3.756\pm 0.009$fm, but they contradict cross-section values of $\sigma_{0}=20.491\pm 0.014$b according to Dilg and coherent-scattering-length values of $f=-3.7409\pm 0.0011$fm according to Koester and Nistler.Comment: 17 pages, to be published in Physics of Atomic Nucle

LL-approximation of BB-splines by trigonometric polynomials

This note is a continuation of our papers [1,2], devoted to $L$-approximation of characteristic function of $(-h, h)$ by trigonometric polynomials. In the paper [1] the sharp values of the best approximation for the special values of $h$ were found. In [2] we gave the complete solution of the problem for arbitrary values of $h$. In general case [2] the situation is more deep and results are not so simple as in [1]. For applications to the problem of optimal constants in the Jackson-type inequalities we need, however, results on $L$-approximation of $B$-splines and linear combinations of $B$-splines. Here we present some simple results about $L$-approximation of $B$-splines as well as give the the proof of its sharpness for the special values of $h$.Comment: 6 page

Description of the Low-Energy Doublet Neutron-Deuteron Scattering on the Basis of the Triton Bound and Virtual State Parameters

Low-energy doublet neutron-deuteron scattering is described on the basis of the triton bound and virtual state parameters - the energies and the nuclear vertex constants of these states. The van Oers-Seagrave formula is derived from the Bargmann representation of the S matrix for a system having two states. The presence of a pole in this formula is shown to be a direct corollary of the existence of a low-energy triton virtual state. Simple explicit expressions for the nd scattering length and for the pole of the function $k\cot \delta$ are obtained in terms of the triton bound and virtual state parameters. Numerical calculations of the nd low-energy scattering parameters show their high sensitivity to variations in the asymptotic normalization constant of the virtual state $C_{v}^{2}$. The $C_{v}^{2}$ value fitted in our model to the experimental result for the nd scattering length is $C_{v}^{2}=0.0592$.Comment: 15 pages, 1 figur

P-matrix Description of Interaction of Two Charged Hadrons And Low-energy Nuclear-Coulomb Scattering Parameters

The scattering of two charged strongly interacting particles is described on the basis of the P-matrix approach. In the P matrix, it is proposed to isolate explicitly the background term corresponding to purely Coulomb interaction, whereby it becomes possible to improve convergence of the expansions used and to obtain a correct asymptotic behavior of observables at high energies. The expressions for the purely Coulomb background P matrix, its poles and residues, and purely Coulomb eigenfunctions of the P-matrix approach are obtained. The nuclear-Coulomb low-energy scattering parameters of two charged hadrons are investigated on the basis of this approach combined with the method of isolating the background P matrix. Simple explicit expressions for the nuclear-Coulomb scattering length and effective range in terms of the residual P matrix are derived. These expressions give a general form of the nuclear-Coulomb low-energy scattering parameters for models of finite-range strong interaction. Specific applications of the general expressions derived in this study are exemplified by considering some exactly solvable models of strong interaction containing hard core repulsion, and, for these models, the nuclear-Coulomb low-energy scattering parameters for arbitrary values of the orbital angular momentum are found explicitly. In particular, the nuclear-Coulomb scattering length and effective range are obtained explicitly for the boundary-condition model, the model of a hard-core delta-shell potential, the Margenau model, and the model of hard-core square-well potential.Comment: 23 page

Study of the Low-Energy Characteristics of Neutron-Neutron Scattering in the Effective-Range Approximation

The influence of the mass difference between the charged and neutral pions on the low-energy characteristics of nucleon-nucleon interaction in the $^{1}S_{0}$ spin-singlet state is studied within the framework of the effective-range approximation. By making use of the experimental singlet neutron-proton scattering parameters and the experimental value of neutron-neutron virtual-state energy, the following values were obtained for the neutron-neutron scattering length and effective range: $a_{nn}=-16.59(117)$fm, $r_{nn}=2.83(11)$fm. The calculated neutron-neutron scattering length $a_{nn}$ is in good agreement with one of the two well known and differing experimental values of this quantity, and the calculated effective range $r_{nn}$ is also in good agreement with present-day experimental results.Comment: 12 pages, 1 table. Slightly modified version of the article published in Physics of Atomic Nucle

On the Connection Between the Charged and Neutral Pion-Nucleon Coupling Constants in the Yukawa Model

In the Yukawa model for nuclear forces, a simple relation between the charged and neutral pion-nucleon coupling constants is derived. The relation implies that the charged pion-nucleon coupling constant is larger than the neutral one since the np interaction is stronger than the pp interaction. The derived value of the charged pion-nucleon constant shows a very good agreement with one of the recent experimental values. The relative splitting between the charged and neutral pion-nucleon coupling constants is predicted to be practically the same as that between the charged and neutral pion masses. The charge dependence of the NN scattering length arising from the mass difference between the charged and neutral pions is also analyzed.Comment: 19 pages. Slightly modified and typos corrected version of the article published in PEPAN Letter

A combinatorial algorithm for the planar multiflow problem with demands located on three holes

We consider an undirected multi(commodity)flow demand problem in which a supply graph is planar, each source-sink pair is located on one of three specified faces of the graph, and the capacities and demands are integer-valued and Eulerian. It is known that such a problem has a solution if the cut and (2,3)-metric conditions hold, and that the solvability implies the existence of an integer solution. We develop a purely combinatorial strongly polynomial solution algorithm.Comment: 16 pages, 4 figure

On Weighted Multicommodity Flows in Directed Networks

Let $G = (VG, AG)$ be a directed graph with a set $S \subseteq VG$ of terminals and nonnegative integer arc capacities $c$. A feasible multiflow is a nonnegative real function $F(P)$ of "flows" on paths $P$ connecting distinct terminals such that the sum of flows through each arc $a$ does not exceed $c(a)$. Given $\mu \colon S \times S \to \R_+$, the \emph{$\mu$-value} of $F$ is $\sum_P F(P) \mu(s_P, t_P)$, where $s_P$ and $t_P$ are the start and end vertices of a path $P$, respectively. Using a sophisticated topological approach, Hirai and Koichi showed that the maximum $\mu$-value multiflow problem has an integer optimal solution when $\mu$ is the distance generated by subtrees of a weighted directed tree and $(G,S,c)$ satisfies certain Eulerian conditions. We give a combinatorial proof of that result and devise a strongly polynomial combinatorial algorithm.Comment: 12 page