Resolvable Decompositions Research Articles

Resolvent decomposition with applications to semigroups and cosine functions

AbstractFor an operator whose domain is the kernel of a functional that is a finite sum of simpler functionals, we show that the resolvent of the operator can be decomposed as an affine combination of resolvents associated with these simpler functionals. Specifically, given an operator A and a finite number of functionals $$\phi $$ ϕ defined on the domain of A, we prove that the resolvent $$(\lambda - A_{|\ker \sum \phi })^{-1}$$ ( λ - A | ker ∑ ϕ ) - 1 is an affine combination of resolvents $$(\lambda - A_{|\ker \phi })^{-1}$$ ( λ - A | ker ϕ ) - 1 , provided that the sum of the functionals is nontrivial on the kernel of $$\lambda - A$$ λ - A . We use this result to derive generation theorems for semigroups and cosine functions.As applications, we prove that there are cosine functions associated with skew and snapping out Brownian motions on star graphs, and that skew Brownian motion can be approximated by snapping out Brownian motion.

Uniformly resolvable ${P_4, C_k}$-decomposition of $K_n$ - a complete solution

Let $K_n,$ $C_{n}$, and $P_{n}$ respectively denote the complete graph, cycle and path on $n$ vertices. Uniformly resolvable decomposition of $K_n$ is a decomposition of $K_n$ into subgraphs which can be partitioned into factors containing pairwise isomorphic subgraphs. In this paper, we determine necessary and sufficient conditions for the existence of uniformly resolvable decomposition of $K_n$ into $P_4$ and $C_k,~ k\geq3.$

Completing the solution of the directed Oberwolfach problem with cycles of equal length

AbstractIn this paper, we give a solution to the last outstanding case of the directed Oberwolfach problem with tables of uniform length. Namely, we address the two‐table case with tables of equal odd length. We prove that the complete symmetric digraph on vertices, denoted , admits a resolvable decomposition into directed cycles of odd length . This completely settles the directed Oberwolfach problem with tables of uniform length.

On the directed Oberwolfach problem for complete symmetric equipartite digraphs and uniform‐length cycles

AbstractWe examine the necessary and sufficient conditions for a complete symmetric equipartite digraph with parts of size to admit a resolvable decomposition into directed cycles of length . We show that the obvious necessary conditions are sufficient for in each of the following four cases: (i) is even; (ii) ; (iii) and or ; and (iv) , and if , then for a prime .

On uniformly resolvable $(C_4,K_{1,3})$-designs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars.

Complex uniformly resolvable decompositions of K_v

In this paper we consider complex uniformly resolvable decompositions of the complete graph K v into subgraphs such that each resolution class contains only blocks isomorphic to the same graph from a given set ℋ and at least one parallel class is present from each graph of ℋ . We completely determine the spectrum for the cases ℋ = { K 2 , P 3 , K 3 } , ℋ = { P 4 , C 4 } , and ℋ = { K 2 , P 4 , C 4 } .

Resolvable cycle decompositions of complete multigraphs and complete equipartite multigraphs via layering and detachment

AbstractWe construct new resolvable decompositions of complete multigraphs and complete equipartite multigraphs into cycles of variable lengths (and a perfect matching if the vertex degrees are odd). We develop two techniques: layering, which allows us to obtain 2‐factorizations of complete multigraphs from existing 2‐factorizations of complete graphs, and detachment, which allows us to construct resolvable cycle decompositions of complete equipartite multigraphs from existing resolvable cycle decompositions of complete multigraphs. These techniques are applied to obtain new 2‐factorizations of a specified type for both complete multigraphs and complete equipartite multigraphs, with the emphasis on new solutions to the Oberwolfach Problem and the Hamilton–Waterloo Problem. In addition, we show existence of some thickly resolvable cycle decompositions.

Uniformly Resolvable Decompositions of Kv-I into n-Cycles and n-Stars, for Even n

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.

Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States

The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.

Uniformly resolvable $(C_4, K_{1,3})$-designs of order v and

In this paper we consider the uniformly resolvable decompositions of the complete graph $\lambda K_v$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We consider the cases in which all the resolution classes are either $C_4$ or $K_{1,3}$. We prove that this type of system does not exist for $\lambda$ odd and determine completely the spectrum for $\lambda=2$.

Uniformly Resolvable Cycle Decompositions with Four Different Factors

In this paper, we consider uniformly resolvable decompositions of complete graph $$K_v$$ (or $$K_v$$ minus a 1-factor I for even v) into cycles. We will focus on the existence of factorizations of $$K_v$$ or $$K_v-I$$ containing up to four non-isomorphic factors. We obtain all possible solutions for uniform factors involving 4, m, 2m and 4m-cycles with a few possible exceptions when m is odd.

A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator

A novel reduced-order model for time-varying nonlinear flows arising from a resolvent decomposition based on the time-mean flow is proposed. The inputs required for the model are the mean-flow field and a small set of velocity time-series data obtained at isolated measurement points, which are used to fix relevant frequencies, amplitudes and phases of a limited number of resolvent modes that, together with the mean flow, constitute the reduced-order model. The technique is applied to derive a model for the unsteady three-dimensional flow in a lid-driven cavity at a Reynolds number of 1200 that is based on the two-dimensional mean flow, three resolvent modes selected at the most active spanwise wavenumber, and either one or two velocity probe signals. The least-squares full-field error of the reconstructed velocity obtained using the model and two point velocity probes is of the order of 5 % of the lid velocity, and the dynamical behaviour of the reconstructed flow is qualitatively similar to that of the complete flow.

Uniformly resolvable decompositions of [formula omitted] into paths on two, three and four vertices

In this paper we consider uniformly resolvable decompositions of the complete graph Kv, i.e., decompositions of Kv whose blocks can be partitioned into factors and each factor contains pairwise isomorphic blocks. We determine necessary and sufficient conditions for the existence of a uniformly resolvable decomposition of Kv into paths on two, three and four vertices.

Resolvent decomposition theorems and Kolmogorov <italic>Q</italic>-process

The main aim of this paper is to use the resolvent decomposition theorems to analysing the Kol- mogorov Q -process which has nitely many instantaneous states. After obtaining an elegant condition for existence and uniqueness, we focus on discussing all kinds of properties of such subtle process. In particular, we show that the Kolmogorov Q -process is always recurrent and then an easy-checking condition for positive recurrence is ob- tained. The closed form for the limiting distribution is explicitly expressed. The symmetry and reversibility of such process are also discussed and well-answered.

Resolvable 3-star designs

Let Kv be the complete graph of order v and F be a set of 1-factors of Kv. In this article we study the existence of a resolvable decomposition of Kv−F into 3-stars when F has the minimum number of 1-factors. We completely solve the case in which F has the minimum number of 1-factors, with the possible exception of v∈{40,44,52,76,92,100,280,284,328,332,428,472,476,572}.

On the origin of frequency sparsity in direct numerical simulations of turbulent pipe flow

The possibility of creating reduced-order models for canonical wall-bounded turbulent flows based on exploiting energy sparsity in frequency domain, as proposed by Bourguignon et al. [Phys. Fluids 26, 015109 (2014)], is examined. The present letter explains the origins of energetically sparse dominant frequencies and provides fundamental information for the design of such reduced-order models. The resolvent decomposition of a pipe flow is employed to consider the influence of finite domain length on the flow dynamics, which acts as a restriction on the possible wavespeeds in the flow. A forcing-to-fluctuation gain analysis in the frequency domain reveals that large sparse peaks in amplification occur when one of the possible wavespeeds matches the local wavespeed via the critical layer mechanism. A link between amplification and energy is provided through the similar characteristics exhibited by the most energetically relevant flow structures, arising from a dynamic mode decomposition of direct numerical simulation data, and the resolvent modes associated with the most amplified sparse frequencies. These results support the feasibility of reduced-order models based on the selection of the most amplified modes emerging from the resolvent model, leading to a novel computationally efficient method of representing turbulent flows.

Maximum uniformly resolvable decompositions of [formula omitted] and [formula omitted] into 3-stars and 3-cycles

Let Kv denote the complete graph of order v and Kv−I denote Kv minus a 1-factor. In this article we investigate uniformly resolvable decompositions of Kv and Kv−I into r classes containing only copies of 3-stars and s classes containing only copies of 3-cycles. We completely determine the spectrum in the case where the number of resolution classes of 3-stars is maximum.

Uniformly resolvable decompositions of [formula omitted] into [formula omitted] and [formula omitted] graphs

In this paper we consider the uniformly resolvable decompositions of the complete graph Kv, or the complete graph minus a 1-factor as appropriate, into subgraphs such that each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either P3s or K3s.

On the Directed Oberwolfach Problem with Equal Cycle Lengths

We examine the necessary and sufficient conditions for a complete symmetric digraph $K_n^\ast$ to admit a resolvable decomposition into directed cycles of length $m$. We give a complete solution for even $m$, and a partial solution for odd $m$.

The properties of collision branching processes with immigration and instantaneous resurrection

We consider a modified collision branching process incorporating with both independent immigration and instantaneous resurrection. The existence criterion of the process is firstly considered, and also we use the resolvent decomposition theorem to prove that there is a unique CBP-IIR.