Invertible symmetric 3 × 3 binary matrices and GQ(2, 4)

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3?×?3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,?4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2,?2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2,?4) as a quadric in PG(5,?2) of projective index one. An interesting physics application of our findings is also mentioned.     

On invariant notions of Segre varieties in binary projective spaces

Invariant notions of a class of Segre varieties S_(m)(2){\mathcal{S}_{(m)}(2)} of PG(2 ^m − 1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S_(m)(2){\mathcal{S}_{(m)}(2)} and is invariant under its projective stabiliser group G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} . By embedding PG(2 ^m − 1, 2) into PG(2 ^m − 1, 4), a basis of the latter space is constructed that is invariant under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant geometric spread of lines of PG(2 ^m − 1, 2). This spread is also related with a G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} -invariant non-singular Hermitian variety. The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under G_S(m)(2){G_{{\mathcal{S}}_{(m)}(2)}} , while the points of PG(7, 2) form five orbits.     

Interaction between model poly(ethylene terephthalate) thin films and weakly ionised oxygen plasma

Model films of poly(ethylene terephthalate) were treated by oxygen plasma in order to quantify the etching rate and estimate the contribution of charged and neutral particles to the reaction probability. Model films with a thickness of 50 nm were deposited on a quartz crystal of a microbalance (QCM) by spin‐coating technique. The samples were exposed to oxygen plasma with the positive ion density of 4 × 10¹⁵ m^?3 and neutral oxygen atom density of 6 × 10²¹ m^?3. The etching rate was determined from the QCM signal and was 4.7 nm s^?1. The etching was found rather inhomogeneous as the atomic force microscopic images showed an increase of the surface roughness as a result of plasma treatment. The model films were completely removed from the surface of the quartz crystals in about 12 s. Knowing the etching rate and the flux of oxygen atoms to the surface allowed for calculation of the reaction probability which was found to be rather low at the value of 1.6 × 10^?4. Copyright © 2011 John Wiley & Sons, Ltd.     

Etching of polyethylene terephthalate thin films by neutral oxygen atoms in the late flowing afterglow of oxygen plasma

Films of polyethylene terephthalate were deposited on quartz crystals and exposed to oxygen atoms to study their etching characteristics and quantify the etching rate. Oxygen (O) atoms were created by passing molecular oxygen through plasma created in a microwave discharge. The discharge power was fixed at 250 W, while the pressure of oxygen was 50 Pa. Before exposure to oxygen atoms, a thin polymer film of polyethylene terephthalate (PET) was deposited uniformly over a crystal with a diameter of 12 mm. The crystal was mounted on a quartz crystal microbalance to accurately determine the thickness of the polymer film. The polymer film was exposed to O atoms in the flowing afterglow. The density of O atoms was measured with a cobalt catalytic probe mounted next to the sample and was determined to be 1.2 × 10²¹ m^–3. Samples were treated with O atoms for different periods of up to 120 min. The thickness of the film decreased linearly with treatment time. After 90 min of treatment, a 65‐nm‐thick polymer film was completely removed. Therefore, the etching rate was 0.5 nm/min, so the interaction probability between an O atom and an atom in the sample was extremely low, just 1.4 × 10^–6. Samples treated for different periods were investigated by atomic force microscopy and X‐ray photoelectron spectroscopy to examine the etching characteristics of O atoms in the flowing afterglow. Copyright © 2012 John Wiley & Sons, Ltd.     

Geometric Hyperplanes of the Near Hexagon L<Subscript>3</Subscript> × GQ(2, 2)

Having in mind their potential quantum physical applications, we classify all geometric hyperplanes of the near hexagon that is a direct product of a line of size three and the generalized quadrangle of order two. There are eight different kinds of them, totalling to 1,023 = 2¹⁰ − 1 = |PG(9, 2)|, and they form two distinct families intricately related with the points and lines of the Veldkamp space of the quadrangle in question.     

Projective line over the finite quotient ring GF(2)[x]/〈x 3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram

In 1993, Mermin gave surprisingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight respectively using what has since been called the Mermin-Peres “magic” square and the Mermin pentagram. The former is a 3×3 array of nine observables commuting pairwise in each row and column and arranged such that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by a similar contradiction. We establish a one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring GF(2) ⊗ GF(2). Under this map, the concept mutually commuting transforms into mutually distant, and the distinguishing character of the third column’s observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are either both zero divisors or both units. The ten operators of the Mermin pentagram correspond to a specific subset of points of the line over GF(2)[x]/〈x³ ™ x〉. But the situation in this case is more intricate because there are two different configurations that seem to serve our purpose equally well. The first one comprises the three distinguished points of the (sub)line over GF(2), their three “Jacobson” counterparts, and the four points whose both coordinates are zero divisors. The other con.guration features the neighborhood of the point (1, 0) (or, equivalently, that of (0, 1)). We also mention some other ring lines that might be relevant to BKS proofs in higher dimensions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 219–227, May, 2007.     

Projective line over the finite quotient ring GF(2)[x]/〈x3 ? x〉 and quantum entanglement: Theoretical background

We consider the projective line over the finite quotient ring R_⋄ ≡ GF(2)[x]/〈x ³ − x〉. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because R_⋄ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence stems from a nontrivial character of the Jacobson radical of the ring. The quotient ring is isomorphic to GF(2) ⊗ GF(2). The projective line over features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we will discuss in detail in Part II of this paper. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 44–53, April, 2007.     

Broadband acoustic scattering measurements of underwater unexploded ordnance (UXO)

In order to evaluate the potential for detection and identification of underwater unexploded ordnance (UXO) by exploiting their structural acoustic response, we carried out broadband monostatic scattering measurements over a full 360 degrees on UXO's (two mortar rounds, an artillery shell, and a rocket warhead) and false targets (a cinder block and a large rock). The measurement band, 1-140 kHz, includes a low frequency structural acoustics region in which the wavelengths are comparable to or larger than the target characteristic dimensions. In general, there are aspects that provide relatively high target strength levels ( approximately -10 to -15 dB), and from our experience the targets should be detectable in this structural acoustics band in most acoustic environments. The rigid body scattering was also calculated for one UXO in order to highlight the measured scattering features involving elastic responses. The broadband scattering data should be able to support feature-based separation of UXO versus false targets and identification of various classes of UXO as well.