Partial Partitions Research Articles

Discretized configurations and partial partitions

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.

Idempotent Block Splitting on Partial Partitions, I: Isotone Operators

Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.

Adjunctions on the lattices of partitions and of partial partitions

The complete lattice Π(E) of partitions of a space E has been extended into Π*(E), the one of partial partitions of E (where the space covering axiom is removed). We recall the main properties of Π*(E), and exhibit two adjunctions (residuations) between Π(E) and Π*(E). Given two spaces E 1 and E 2 (distinct or equal), we analyse adjunctions between Π*(E 1) and Π*(E 2), in particular those where the lower adjoint applies a set operator to each block of the partial partition; we also show how to build such adjunctions from adjunctions between $${\mathcal{P}(E_2)}$$ and $${\mathcal{P}(E_2)}$$ (the complete lattices of subsets of E 1 and E 2). They are then extended to adjunctions between Π(E 1) and Π(E 2). We obtain as particular case the adjunction on Π(E) that was defined by Serra (for the upper adjoint) and Ronse (for the lower adjoint). We also study dilations from Π*(E 1) to an arbitrary complete lattice L; a particular case is given, for $${L \subseteq [0,+\infty]}$$ , by ultrametrics; then the adjoint erosion provides the corresponding hierarchy. We briefly discuss possible applications in image processing and in data clustering.

Partial Partitions, Partial Connections and Connective Segmentation

In connective segmentation (Serra in J. Math. Imaging Vis. 24(1):83---130, [2006]), each image determines subsets of the space on which it is "homogeneous", in such a way that this family of subset...

Influence of lightweight structural design of office buildings on vibration levels due to people walking and operation of mechanical equipment

West coast buildings typically use lightweight structural designs, even for Class A office building. This means the best floors use lightweight concrete on fluted decks with beam and column spacings of 30 to 35 ft. with open web joists at 10 ft. on center. Class B office building may use columns at 28 and 35 ft. with only two‐layers of plywood on wood open web joists at 24 in. on center. The resulting floor/ceiling assemblies are structurally sound but dynamically very live. Some structural damping would be expected from the introduction of partitions, but with the use of large open areas and partial height partitions, damping is limited. These assemblies have relatively low resonance frequencies and are very susceptible to any sinusoidal input force such as people walking or operation of mechanical equipment. Vibration measurements, made on four buildings after complaints were received, show the source of some of these problems. The results are compared with predictions by Murray et al. and Ungar et al. ...

Free convection in an inclined square cavity with partial partitions on a wavy hot wall

Free convection in an inclined square cavity with partial partitions has been investigated, by numerically solving the partial differential balance equations of mass, momentum, and energy. The analysed configuration has three undulations and three partitions. Investigations have been performed for different inclination angles, partition lengths and Rayleigh numbers, while keeping the Prandtl number constant. The results show that the hot wall geometry with partitions affects the heat transfer rate in the cavity. It has been found in the present numerical study that the mean Nusselt number decreases notably compared with the heat transfer in the square undulated cavity without partitions.

A characterization for a set of partial partitions to define an X-tree

Trees whose vertices are partially labelled by elements of a finite set X provide a natural way to represent partitions of subsets of X. The condition under which a given collection of such partial partitions of X can be represented by a tree has previously been characterized in terms of a chordal graph structure on an underlying intersection graph. In this paper, we obtain a related graph-theoretic characterization for the uniqueness of a tree representation of a set of partial partitions of X.

Natural convection in enclosures with partial partitions

Laminar and turbulent natural convection in enclosures with partial partitions has been studied by a numerical method. Vertical boundaries were isothermal and horizontal boundaries were adiabatic. Two dimensional equations of conservation of mass, momentum and energy, with the Boussinesq approximation are solved using the Simpler method. Various geometrical parameters were: aspect ratio A=0.3 to 0.4, partition position D=0.5−0.6, height of the partitions C= D=0 to 0.15. The Rayleigh number was varied from 10 4 to 10 11. The results are reduced in terms of the normalized Nusselt number as a function of the Rayleigh number, and other non dimensional geometrical parameters. The isotherms and streamlines are produced for various Rayleigh numbers and geometrical conditions. Heat transfer correlations useful for practical design problems have been derived.

A survey of partial difference sets

LetG be a finite group of order ν. Ak-element subsetD ofG is called a (ν,k, λ, μ)-partial difference set if the expressionsgh−1, forg andh inD withg≠h, represent each nonidentity element inD exactly λ times and each nonidentity element not inD exactly μ times. Ife∉D andg∈D iffg−1∈D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.

On a combinatorial problem in group theory

In this paper we continue the study of p-groups G of square order p 2 n and investigate the existence of partial congruence partitions (sets of mutually disjoint subgroups of order p n ) in G. Partial congruence partitions are used to construct translation nets and partial difference sets, two objects studied extensively in finite geometries and combinatorics. We prove that the maximal number of mutually disjoint subgroups of order p n in a group G of order p 2 n cannot be more than ( p n−1 −1)( p−1) −1 provided that n ⩾ 4 and that G is not elementary abelian. This improves an earlier result (D. Hachenberger, J. Algebra 152, 1992, 207–229) and as we do not distinguish the cases p = 2 and p odd in the present paper, we also have a generalization of D. Frohardt's theorem on 2-groups ( J. Algebra 107, 1987, 153–159). Furthermore we study groups of order p 6. We can show that for each odd prime number, there exist exactly four nonisomorphic groups which contain at least p + 2 mutually disjoint subgroups or order p 3. Again, as we do not distinguish between the even and the odd case in advance, we in particular obtain the classification of groups of order 64 which contain at least 4 mutually disjoint subgroups of order 8 found by D. Gluck ( J. Combin. Theory Ser. A 51, 1989, 138–141) and A. P. Sprague ( Mitt. Math. Sem. Giessen 157, 1982, 46–68).

Bounds on a class of partial partitions of a vector space over gf(2):a graph theoretical approach

A collection Q of linearly independent w-suhicfs of the n-dimensional vector space V(n) over GF(2) is a w-quilt if whenever X and Y are distinct elements of Q, then X is disjoint from the linear span of Y. The main problem is to determine the maximum possibility cardinality of a w-quilt in V(n) for fixed w and n. Here a graph T(Q) is associated with each quilt Q. The connected components of T(Q) are shown to be complete graphs and the structure of the subquilts corresponding to these components is completely determined. By use of Ramsey type arguments these results are shown to lead to new upper bounds on the cardinality of a w-quilt in V(n) where n = w + 2, a case of particular interest.

Constructions of large translation nets with nonabelian translation groups

In this paper the first infinite series of translation nets with nonabelian translation groups and a large number of parallel classes are constructed. For that purpose we investigate partial congruence partitions (PCPs) with at least one normal component.Two series correspond to partial congruence partitions containing one normal elementary abelian component. The construction results by using some basic facts about the first cohomology group of the translation group G regarded as an extension of the normal component which itself is a group of central translations.The other series correspond to partial congruence partitions containing two normal nonabelian components. The constructions are based on the well known automorphism method which leads to so-called splitting translation nets. By investigating the Suzuki groups Sz(q), the protective unitary groups PSU(3, q 2) and the Ree groups R(q) as doubly transitive permutation groups, we obtain examples of nonabelian groups admitting a large number of pairwise orthogonal fixed-point-free group automorphisms.

Numerical calculation of equivalent moment of inertia for a fluid in a cylindrical container with partitions

AbstractA 3‐D numerical code was developed to solve the irrotational flow of an ideal fluid inside a moving cylindrical container with partitions. The problem is formulated in terms of two Poisson equations, the velocity equation and the pressure equation. These are expressed in terms of finite differences and solved by a procedure of line over relaxation. From the fluid pressure we computed the resultant moment on the container which was then expressed in terms of the equivalent moment of inertia. The code was checked against analytical results of Moiseyev and we found an agreement of 2–5 per cent even for a sparse mesh. We then demonstrated the capability of the code by computing the equivalent moment of inertia for various configurations with full and partial partitions.

Acoustical characteristics of partitioned spaces under a modular grid of vertical sound absorptive surfaces

A facility requiting great flexibility in the size and arrangement of partitioned spaces for evaluating various computer systems was designed with fixed lighting and air conditioning diffusers. Sound absorption is provided by a 5‐ft (1.5 m) by 5‐ft grid of vertical acoustical panels consisting of fibrous glass and perforated metal facing approximately 15 in. (0.38 m) high and 2 in. thick. Measured room to room noise attenuation and the effective ceiling sound absorption coefficient compare favorably with a conventional continuous horizontal acoustical ceiling and partial partitions which extend to 2 to 3 ft below the ceiling. The open grid system eliminates the need to change the duct and diffuser layout and permits unrestricted access to the open plenum space for modifications of services and maintenance. Fire protection systems can also be hidden from view in the darkly painted plenum. The room to room noise reduction values of such an open grid system can be predicted by a simple model.

Concepts and Acoustics of Open-Plan Offices and Schools

The present popularity of open planning for offices and schools is surprising in terms of supposedly establishing criteria for privacy and sound isolation. It is very difficult to approach these criteria in an “open” plan. Precise model measurements define attainable steps in sound isolation as one uses extreme absorption on available surfaces, and then introduces functional obstacles as sound barriers, partial partitions, and even specific acoustical reflectors to improve the ratio of desired to objectionable communication. Much can be accomplished with transparent partitions to preserve visual openness and improve isolation. The magnitude of background noise to mask residual unwanted communication and noise exceeds generally tolerable levels and suggests use of programmed familiar sounds selected for acceptability and without distraction. Economy will not be a feature of the acoustically acceptable open plan. Knowledgeable school planners stress flexibility more than openness; hence, folding and readily movable partitioning may flourish. Since architects generally will not respect the acoustical sophistication, nor will owners like the cost, the alternative will probably be either passing of the fad or evolution of consideration and discretion in voice communication.

Discussions

Dr F. E. E ames said that Dr Adams had been kind enough to show him much of the material. The palaeontological material had consisted entirely of thin sections of hard limestones, and the speaker considered that, in the study of so many random sections of fossils, Dr Adams had shown commendable caution in the frequent use of 'sp.' and 'cf.' The speaker felt, however, that there might be some doubt about the identification of the genus Dictyoconus. None of the specimens Dr Adams had shown him had the characteristic features of the genus Dictyoconus : the high conic form, the distinctly coiled early portion with noticeable proloculus, and the well-developed two orders of partial partitions in the chambers of the outer cortical layer of the core. He hoped that the range of the genus would not be unequivocally extended up into the Oligocene on the basis of the occurrence of this species. The speaker understood that there was a tentative suggestion that a small thickness of beds in the Melinau section that immediately overlay richly num-mulitic Middle Oligocene (d stage) and underlay beds satisfactorily equated to the Indonesian e stage (lower part of Lower Miocene) were of Upper Oligocene age and should be included with the e stage. Some authors in the last few decades (e.g., van der Vlerk) had suggested that the Indonesian e stage was possibly of Chattian (Upper Oligocene) age in its lower part. Although it might have been thought, as a result of the speaker's own work,

Malaria Studies in Greece

Malaria Studies in Greece