Partial Spreads Research Articles

New orthotopic mouse models and Ad‐based transcriptionally targeted vectors for squamous cell carcinoma therapy development

Oral squamous cell carcinoma affects about 30,000 Americans each year, and although the overall prognosis has improved in recent years for localized carcinomas, metastases can decrease survival rates by half. For tongue and palate lesions, surgery is usually the initial treatment, but distant metastases can develop at sites where conventional therapies are ineffective (lung, bone, heart). Therefore, our goal has been to develop a fast‐growing orthotopic stromal‐epithelial SCC model with a metastasis profile, in which to test new adenoviral‐based gene therapies. We specifically plan to use systemic delivery of an adenovirus (Ad) that targets expression of therapeutic gene p12/CDK2AP1 to SCC cells using the SCCA2 promoter and a transcriptional amplification system. We developed new orthotopic stromal‐epithelial SCC models for floor of mouth and tongue carcinoma with rapid latency and partial tumor spread to regional lymphnodes. We constructed and are characterizing an Ad‐TSTA‐p12FL bidirectional vector for gene therapy applications. We plan to test in vivo the specificity of the vector, the effectiveness in tumor growth inhibition at both floor of mouth, orthotopic, or distant sites, and following intratumoral or systemic administration. We propose this vector may hold promise as a specific, novel, and robust SCC therapy to treat tumors that have metastasized to distant and thus difficult to treat sites.

Embedding Finite Partial Linear Spaces in Finite Translation Nets

In the 1970’s Paul Erdős and Dominic Welsh independently posed the problem of whether all finite partial linear spaces \({\mathbb{L}}\) are embeddable in finite projective planes. Except for the case when \(\mathbb{L}\) has a unique embedding in a projective plane with few additional points, very little has been done which is directly applicable to this problem. In this paper it is proved that every finite partial linear space \(\mathbb{L}\) is embeddable in a finite translation net generated by a partial spread of a vector space of even dimension. The question of whether every finite partial linear space is embedded in a finite Andre net is also explored. It is shown that for each positive integer n there exist finite partial linear spaces which do not embed in any Andre net of dimension less than or equal to n over its kernel.

The Partial Solution Spread by a Co-evolution Considered Estimation of Distribution Value

The Partial Solution Spread by a Co-evolution Considered Estimation of Distribution Value

Random constructions and density results

In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random techniques used in finite geometry.

The ψ-associate X # of a flat X in PG(n, 2)

For a given hypersurface ψ in PG(n, 2), with equation Q(x) = 0, where Q is a polynomial of (reduced) degree d > 1, a definition is given of the ψ-associate X # of a flat X in PG(n, 2). The definition involves the fully polarized form of the polynomial Q; in the cubic case d = 3 it reads: X # = {zn, 2) | Tx, y, z) = 0 for all x, y}, where T(x, y, z) denotes the alternating trilinear form obtained by completely polarizing Q. Some general results, valid for any degree d and projective dimension n, are first expounded. Thereafter several choices of ψ are visited, but for each choice just a few aspects are highlighted. Despite the partial nature of the survey quite a variety of behaviours of the ψ-associate are uncovered. Many of the choices of ψ which are considered are of cubic hypersurfaces in PG(5, 2). If ψ is the Segre variety it is shown that the 48 planes external to fall into eight pairs of ordered triplets {(P 1, R 1, S 1), (P 2, R 2, S 2)} such that and . Further those lines L of PG(5, 2) which are singular, satisfying that is L # = PG(5.2), are in this case shown to form a complete spread of 21 lines. Another result of note arises in the case where ψ is the underlying 35-set of a non-maximal partial spread Σ5 of five planes in PG(5, 2), where it is shown that one plane is singled out by the property that every line is singular.

Partial ovoids and partial spreads in symplectic and orthogonal polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids and small maximal partial spreads in the classical symplectic and orthogonal polar spaces, and improved upper bounds on the sizes of large maximal partial ovoids and large maximal partial spreads in the classical symplectic and orthogonal polar spaces. An overview of the status regarding these results is given in tables. The similar results for the hermitian classical polar spaces are presented in [J. De Beule, A. Klein, K. Metsch, L. Storme, Partial ovoids and partial spreads in hermitian polar spaces, Des. Codes Cryptogr. (in press)].

On maximal partial spreads of [formula omitted

A lower bound for the size of a maximal partial spread of H ( 2 n + 1 , q 2 ) is given. For H ( 2 n + 1 , q 2 ) in general, and for H ( 5 , q 2 ) in particular, new upper bounds for this size are also obtained. In [A. Aguglia, A. Cossidente, G.L. Ebert, Complete spans on Hermitian varieties, in: Proceedings of the Conference on Finite Geometries (Oberwolfach, 2001), vol. 29, 2003, pp. 7–15.], maximal partial spreads of H ( 3 , q 2 ) and H ( 5 , q 2 ) have been constructed from spreads of W 3 ( q ) and W 5 ( q ) , respectively; the construction for H ( 5 , q 2 ) will be generalized to H ( 4 n + 1 , q 2 ) , n ⩾ 1 , thus yielding examples of maximal partial spreads of H ( 4 n + 1 , q 2 ) for all n ⩾ 1 .

On the ranks of bent functions

The rank of a bent function is the 2-rank of the associated symmetric 2-design. In this paper, it is shown that it is an invariant under the equivalence relation among bent functions. Some upper and lower bounds of ranks of general bent functions, Maiorana–McFarland bent functions and Desarguesian partial spread bent functions are given. As a consequence, it is proved that almost every Desarguesian partial spread bent function is not equivalent to any Maiorana–McFarland bent function.

Partial ovoids and partial spreads in hermitian polar spaces

We present improved lower bounds on the sizes of small maximal partial ovoids in the classical hermitian polar spaces, and improved upper bounds on the sizes of large maximal partial spreads in the classical hermitian polar spaces. Of particular importance is the presented upper bound on the size of a maximal partial spread of H(3,q 2). For q = 2,3, the presented upper bound is sharp. For q = 3, our results confirm via theoretical arguments properties, deduced by computer searches performed by Ebert and Hirschfeld, for the largest partial spreads of H(3,9). An overview of the status regarding these results is given in two summarizing tables. The similar results for the classical symplectic and orthogonal polar spaces are presented in De Beule et al. [8].

Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

AbstractMaximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H(3,q2) and H(4,q2) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H(4,q2). We also construct in H(3,q2), q=22h+1, h≥ 1, maximal partial spreads of size smaller than the size q2+1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H(4,q2) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008

On the smallest maximal partial ovoids and spreads of the generalized quadrangles W(q) and Q(4,q)

We present results on the size of the smallest maximal partial ovoids and on the size of the smallest maximal partial spreads of the generalized quadrangles W ( q ) and Q ( 4 , q ) .

The maximum size of a partial spread in [formula omitted] is [formula omitted

In this paper, we show that the largest maximal partial spreads of the hermitian variety H ( 5 , q 2 ) consist of q 3 + 1 generators. Previously, it was only known that q 4 is an upper bound for the size of these partial spreads. We also show for q ⩾ 7 that every maximal partial spread of H ( 5 , q 2 ) contains at least 2 q + 3 planes. Previously, only the lower bound q + 1 was known.

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q4,q of PG(4, q) mapping suitably PG(3, q) over Q4,q. More precisely the points of PG(3, q) are the lines of Q4,q; the lines of PG(3, q) are the tangent cones of Q4,q and the reguli of the hyperbolic quadrics hyperplane section of Q4,q. A plane of PG(3, q) is the set of lines of Q4,q meeting a fixed line of Q4,q. We remark that this representation is valid also for a projective space \({\mathbb{P}}_{{3,{\mathbf{k}}}} \) over any field K and we apply the above representation to construct maximal partial spreads \({\mathfrak{F}}\) in PG(3, q). For q even we get new cardinalities for \({\mathfrak{F}}.\) For q odd the cardinalities are partially known.

Maximal arcs in PG(2, q) and partial flocks of the quadratic cone

Abstract In this paper we show that there are several other structures that arise from the functions associated with the maximal arcs of Mathon type. So it is shown that maximal arcs of Mathon type are equivalent to additive partial flocks of the quadratic cone in PG(3,q) and to additive partial q-clans. Further they yield partial ovoids of Q +(5,q) partial spreads of lines of PG(3,q), translation k-arcs of PG(2,q) and m-ovoids of a certain classical generalized quadrangle.

Searching for maximal partial ovoids and spreads in generalized quadrangles

Searching for maximal partial ovoids and spreads in generalized quadrangles

Partial spreads in PG(4,2) and flats in PG(9,2) external to the Grassmannian [formula omitted

We consider the following ‘ even hyperplane construction’ of flats in the projective space PG ( 9 , 2 ) = P ( ∧ 2 V ( 5 , 2 ) ) which are external to the Grassmannian G 1 , 4 , 2 of lines of PG(4,2). Let the Grassmann image in G 1 , 4 , 2 of a partial spread S r = { μ 1 , … , μ r } in PG(4,2) be C r = { m 1 , … , m r } . Then C r is an r-cap on G 1 , 4 , 2 . Using the recent classification [N.A. Gordon, R. Shaw, L.H. Soicher, Classification of Partial Spreads in PG(4,2), pp. 63, available from: http://www.hull.ac.uk/maths/people/rs/staffdetails.html] of partial spreads in PG(4,2), we determine those partial spreads S r such that the projective space E ( C r ) = 〈 C r 〉 even generated by the r 2 points m ij ≔ m i + m j is an external flat. We show that, in this simple manner, we may construct seven out of the ten GL(5,2)-orbits of external flats.

Local Schur’s Lemma and Commutative Semifields

A set of linear maps R subset GL(V,K), V a finite vector space over a field K, is regular if to each x,y in V* there corresponds a unique element R in R such that R(x)=y. In this context, Schur's lemma implies that R =R \cup {l} is a field if (and only if) it consists of pairwise commuting elements. We consider when R is locally commutative: at some µ e V*, AB(µ)=BA(µ) for all A,B \in \ R, and R has been normalized to contain the identity. We show that such locally commutative R are equivalent to commutative semifields, generalizing a result of Ganley, and hence characterizing commutative semifield spreads within the class of translation planes. This enables the determination of the orders |V| for which all locally commutative R on V are (globally) commutative. Similarly, we determine a sharp upperbound for the maximum size of the Schur kernel associated with strictly locally commutative R. We apply our main result to demonstrate the existence of a partial spread of degree 5, with nominated shears axis, that cannot be extend to a commutative semifield spread. Finally, we note that although local commutativity for a regular linear set R implies that the set of Lie products [R, R] consists entirely of singular maps, the converse is false.

On the Spectrum of the Sizes of Maximal Partial Line Spreads in PG(2n,q), n ≥ 3

A lot of research has been done on the spectrum of the sizes of maximal partial spreads in PG(3,q) [P. Govaerts and L. Storme, Designs Codes and Cryptography, Vol. 28 (2003) pp. 51--63; O. Heden, Discrete Mathematics, Vol. 120 (1993) pp. 75--91; O. Heden, Discrete Mathematics, Vol. 142 (1995) pp. 97--106; O. Heden, Discrete Mathematics, Vol. 243 (2002) pp. 135--150]. In [A. Gacs and T. Sz?nyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123--129], results on the spectrum of the sizes of maximal partial line spreads in PG(N,q), N? 5, are given. In PG(2n,q), n ? 3, the largest possible size for a partial line spread is q2n-1+q2n-3+...+q3+1. The largest size for the maximal partial line spreads constructed in [A. Gacs and T. Sz?nyi, Designs Codes and Cryptography, Vol. 29 (2003) pp. 123--129] is (q2n+1?q)/(q2?1)?q3+q2?2q+2. This shows that there is a non-empty interval of values of k for which it is still not known whether there exists a maximal partial line spread of size k in PG(2n,q). We now show that there indeed exists a maximal partial line spread of size k for every value of k in that interval when q ? 9.

Maximality Properties of Stable Partial Spreads and of Shear Planes

Shear planes are special stable planes which can be derived from stable partial spreads by a construction analogous to the construction of (punctured) dual translation planes from spreads. We give a sufficient condition for the maximality of shear planes in the category of stable planes and open embeddings. The condition is expressed in terms of the defining partial spread. Moreover, we give several examples of connected and disconnected shear planes in dimension 4, 8 and 16 which are maximal stable planes and have large automorphism groups. Most of these examples are direct limits of open subplanes of dual projective translation planes.

Psoas compartment block in children: Part I - description of the technique*

Until recently only small series of psoas compartment blocks (PCB) in children have been reported. A high incidence of epidural spread as an important side effect was noted. A series of 100 consecutive blocks using new standardized landmarks is reported. In 100 consecutive children (5.9-106 kg) PCB was performed using a nerve stimulator. In the lateral position, the needle was inserted between the medial 2/3 and the lateral 1/3 on a line from the spinous process of L4 to the posterior superior iliac spine. The blocks were performed by residents in training under supervision of one specific designated pediatric anesthetist. All blocks were clinically successful. In 64% only one attempt was necessary, in 26 patients two attempts were necessary and in 10 patients more than two attempts were necessary. In 16% a vascular puncture occurred. In one patient with a dense unilateral block a partial epidural spread occurred. No serious complications were observed during the perioperative period. The described new technique has a very high success rate with no relevant side effects. Although only one case of epidural spread occurred, PCB remains an invasive technique with the potential for serious complications.