Finite Geometry

ABSTRACT: Finite geometry explains the surprising symmetry properties of some simple graphic designs--found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the" Miracle Octad Generator" of RT Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

In this work, join and meet algebraic structure which exists in non-near-linear finite geometry are discussed. Lines in non-near-linear finite geometry 2  d were expressed as products of lines in near-linear finite geometry 2 p  (where p is a prime). An existence of lattice between any pair of near-linear finite geometry ( ) 2 , 1, , =  j p j k  of 2  d is confirmed. For | q d , a one-to-one cor- respondence between the set of subgeometry 2 j q  of 2  d and finite geometry 2  d from the subsets of the set ( ) { }  d of divisors of d (where each divisor represents a finite geometry) and set of subsystems ( ) { } Π q (with variables in q  ) of a finite quantum system ( ) Π d with variables in  d and a finite system from the subsets of the set of divisors of d is established.

The spectrum of possible sizes k of complete k-arcs in finite projective planes PG(2, q) is investigated by computer search. Backtracking algorithms that try to construct complete arcs joining the orbits of some subgroup of collineation group P L(3, q) and randomized greedy algorithms are applied. New upper bounds on the smallest size of a complete arc are given for q

We present a deterministic polynomial time algorithm to sample a labeled planar graph uniformly at random. Our approach uses recursive formulae for the exact number of labeled planar graphs with n vertices and m edges, based on a decomposition into 1-, 2-, and 3-connected components. We can then use known sampling algorithms and counting formulae for 3-connected planar graphs.

The aim of this paper is to make sense of Hamada's formula for the dimension of the code generated by the incidence matrix of points and subspaces of a given dimension in a finite projective space. The known results are surveyed and a successful guess is made for the dimension of the dual line code when the field has prime order. Van Lint's proof of the equivalence of the two formulas is given. A further guess on the meaning of the formula is also made and a simple proof due to Glynn is given.

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no -communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum-classical success gap of 0.26, larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of 1, beating the classical bound of 0.73 demonstrating the efficiency of the quantum strategy.

Split Cayley hexagons of order two are distinguished finite geometries living in the three-qubit symplectic polar space in two different forms, called classical and skew. Although neither of the two yields observable-based contextual configurations of their own, classically-embedded copies are found to fully encode contextuality properties of the most prominent three-qubit contextual configurations in the following sense: for each set of unsatisfiable contexts of such a contextual configuration there exists some classically-embedded hexagon sharing with the configuration exactly this set of contexts and nothing else. We demonstrate this fascinating property first on the configuration comprising all 315 contexts of the space and then on doilies, both types of quadrics as well as on complements of skew-embedded hexagons. In connection with the lastmentioned case and elliptic quadrics we also conducted some experimental tests on a Noisy Intermediate Scale Quantum (NISQ) computer to substantiate our theoretical findings.

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022), but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from two to seven. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension two and higher, (ii) non-existence of negative subspaces of dimension three and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank four, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the threequbit polar space we correct and improve the contextuality degree of the full configuration, and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their contextuality and computes their contextuality degree for some small symplectic polar spaces. We show that quadrics in the symplectic polar space W n are contextual for n = 3 , 4 , 5 . The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting i...

For N ≥ 2, an N -qubit doily is a doily living in the N -qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any N > 2. Then we present an effective algorithm for the generation of all N -qubit doilies. Using this algorithm for N = 4 and N = 5, we provide a classification of N -qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about N -qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.

Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we first formulate the contextuality property as the absence of solutions to a linear system. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report first results of a software that generates these subgeometries and decides their contextuality. The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting in quantum game theory.

We study certain physically-relevant subgeometries of binary symplectic polar spaces W(2N−1,2) of small rank N, when the points of these spaces canonically encode N-qubit observables. Key characteristics of a subspace of such a space W(2N−1,2) are: the number of its negative lines, the distribution of types of observables, the character of the geometric hyperplane the subspace shares with the distinguished (non-singular) quadric of W(2N−1,2) and the structure of its Veldkamp space. In particular, we classify and count polar subspaces of W(2N−1,2) whose rank is N−1. W(3,2) features three negative lines of the same type and its W(1,2)’s are of five different types. W(5,2) is endowed with 90 negative lines of two types and its W(3,2)’s split into 13 types. A total of 279 out of 480 W(3,2)’s with three negative lines are composite, i.e., they all originate from the two-qubit W(3,2). Given a three-qubit W(3,2) and any of its geometric hyperplanes, there are three other W(3,2)’s possessin...

Given the fact that the three-qubit symplectic polar space features three different kinds of observables and each of its labeled Fano planes acquires a definite sign, we found that there are 45 distinct types of Mermin pentagrams in this space. A key element of our classification is the fact that any context of such pentagram is associated with a unique (positive or negative) Fano plane. Several intriguing relations between the character of pentagrams’ three-qubit observables and ‘valuedness’ of associated Fano planes are pointed out. In particular, we find two distinct kinds of negative contexts and as many as four positive ones.

We investigate small geometric configurations that furnish observable-based proofs of the Kochen–Specker theorem. Assuming that each context consists of the same number of observables and each observable is shared by two contexts, it is proved that the most economical proofs are the famous Mermin–Peres square and the Mermin pentagram featuring, respectively, 9 and 10 observables, there being no proofs using less than 9 observables. We also propose a new proof with 14 observables forming a “magic” heptagram. On the other hand, some other prominent small-size finite geometries, like the Pasch configuration and the prism, are shown not to be contextual.

2) be a Segre variety that is N -fold direct product of projective lines of size three. Given two geometric hyperplanes H ′ and H ′′ of S (N ) , let us call the triple {H ′ , H ′′ , H ′ ∆H ′′ } the Veldkamp line of S (N ) . We shall demonstrate, for the sequence 2 ≤ N ≤ 4, that the properties of geometric hyperplanes of S (N ) are fully encoded in the properties of Veldkamp lines of S (N -1) . Using this property, a complete classification of all types of geometric hyperplanes of S (4) is provided. Employing the fact that, for 2 ≤ N ≤ 4, the (ordinary part of) Veldkamp space of S (N ) is P G(2 N -1, 2), we shall further describe which types of geometric hyperplanes of S (N ) lie on a certain hyperbolic quadric Q + 0 (2 N -1, 2) ⊂ P G(2 N -1, 2) that contains the S (N ) and is invariant under its stabilizer group; in the N = 4 case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type LG(4, 8), to the set of 2295 maximal subspaces of the symplectic polar space W(7, 2).

The purpose of this essay related to the string theory is to prove mathematically that we can bring this latter to a framework of space-time with 4 dimensions,and overcome the experimental issue regarding the extra dimensions(we know that the additional spatial dimensions still unproven experimentally at this moment.In addition,to overcome the topological complications of these latter due to using of Calabi-Yau manifolds).
Moreover,to prove that the theory can works(and reach the goal of unification of the four fundamental forces of nature in the same mathematics formalism),without the supersymmetry concept(in fact,we all know that the supersymmetric particles still undiscovered until today,despite the work of the Large Hadron Collider).
For that,i suggest a new axiomatic system called theory of intermediate numbers ables to solve these problems above mentioned,and offers a possible explanation to:
1)-Young's double slit experiment:
The interpretation or the measurement problem linked directly to the deep issue of the true physical nature of the quantum object,because we know according to quantum mechanics that the quantum object is neither a wave(the wave function associated to the quantum system and governed by Erwin Schrödinger's equation for a non relativistic particle,is a probability wave as the physicist Max Born notice before),nor a particle corpuscular.=>In this paper,we will show that the quantum object(i mean each elementary particle of the standard model) is an intermediate vector carrying an amount of energy describe by Max Planck's equation:E=h.f (with h is the Planck's constant and f the frequency of the particle).Therefore,the strings are intermediate vectors(open or closed) vibrate in a 3 dimensional vector space(intermediate vector space with 3 dimensions).
2)-The physical nature of Dark Matter:
Define in this framework,as the force acting by the geometric structure of the fabric of space-time itself on the galaxies and clusters of galaxies(we know according to general relativity,that these cosmic objects bends the fabric of space-time,so in reaction of that the geometric structure of the Universe impose the same force of action on these latter according to Newton's third law of motion,making these cosmic structures stable and showing us Vera Rubin's Rotation Curves).Otherwise,the dark matter is not a non-baryonic matter surrounded the galaxies and clusters of galaxies like a halo of exotic matter or WIMPs(the weakly interacting massive particles not detected as we know by any experiment at now,including by the LZ experiment).
3)-The physical nature of Dark Energy:
Define in this formalism,as the entropy embedded in the geometric structure of the Universe in its quantum scale(we using the second law of thermodynamics to explain this phenomenon).
=> we conclude,that Space-time is a material object,emerges from a quantum network discrete and random represented mathematically by an intermediate vector space with 3 spatial dimensions.





Algebraic Structure:
The intermediate numbers are the solution of the following system of two equations:
X+X=0.
X+X=2X; with X is real number,but different to zero(this system of 2 equations is the mathematical representation of the problem of the interference phenomenon of Young's double slit experiment and the problem of the physical nature of light,because we have:
X+X=0 <=> photon + photon = dark spot on the screen with probability of occurrence equal to 50%.also,we have:
X+X=2X <=> photon + photon = bright spot on the screen with probability of occurrence equal to 50%: more explanation in the paper below).Indeed,this system of equations does not admit a solution in the group of real numbers(R*) or the group of complex numbers(C*).But,it does in the group of intermediate numbers(H).
Thus,the group of intermediate numbers is a set where each element is the inverse of itself by the addition law(+):
The law of internal composition of the abelian group(H,+).

We prove a generalization of the Droz-Farny line theorem with or-thologic triangles. In 1899, Arnold Droz-Farny [3] discovered and published without proof the following beautiful result. Theorem 1 (Droz-Farny line theorem). Two perpendicular lines 1 and 2 pass through the orthocenter of triangle ABC. If 1 intersects BC, CA, AB at D 1 , E 1 , F 1 , and 2 intersects BC, CA, AB at D 2 , E 2 , F 2 , then the midpoints of D 1 D 2 , E 1 E 2 , F 1 F 2 are collinear. The line containing these midpoints is called the Droz-Farny line associated with 1 and 2. There are a number of generalizations of the Droz-Farny line theorem, notably by van Lamoen [5] (see also [4, 8]), Thas, [10], and Bradley [2]. The most recent one is by Letrouit [6]. If directly similar triangles DD 1 D 2 , EE 1 E 2 , F F 1 F 2 are constructed on D 1 D 2 , E 1 E 2 , F 1 F 2 respectively, then D, E, F are collinear (see also Nicollier [7]). In this note we prove a generalization associated with a pair of orthologic triang...

By Markowitz geometry we mean the intersection theory of ellipsoids and affine subspaces in a real finite-dimensional linear space. In the paper we give a meticulous and self-contained treatment of this arch-classical subject, which lays a solid mathematical groundwork of Markowitz mean-variance theory of efficient portfolios in economics.

We present numerical simulations of closed wavy Taylor vortices and of helicoidal wavy spirals in the Taylor-Couette system. These wavy structures appearing via a secondary bifurcation out of Taylor vortex flow and out of spiral vortex flow, respectively, mediate transitions between Taylor and spiral vortices and vice versa. Structure, dynamics, stability and bifurcation behaviour are investigated in quantitative detail as a function of Reynolds numbers and wave numbers for counter-rotating as well as corotating cylinders. These results are obtained by solving the Navier-Stokes equations subject to axial periodicity for a radius ratio η = 0.5 with a combination of a finite differences method and a Galerkin method.

In this study, the non-linear dynamics of Taylor-Couette flow in a very small-aspect-ratio wide-gap annulus in a counter-rotating regime under the influence of radial through-flow are investigated by solving its full three-dimensional Navier-Stokes equations. Depending on the intensity of the radial flow, either an axisymmetric (pure m = 0 mode) pulsating flow structure or an axisymmetric axially propagating vortex will appear below the centrifugal instability threshold of the circular Couette flow. In contrast to earlier detected propagating vortices, these break Z2 symmetry and exist as a pair of degenerated solutions. Either pulsating or propagating solutions originate owing to the here considered small aspect ratio of the system, resulting in a constant competition between two-and one-cell states. With the variation of radial flow, a pulsating flow appears as a supercritical Hopf bifurcation from a stationary twin-pair state, persisting as the main feature. Thereafter, it underg...

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs mea...

New types of upper bounds for the smallest size t2(2, q) of a complete arc in the projective plane P G(2, q) are proposed. The value t2(2, q) = d(q) √ q ln q, where d(q) < 1 is a decreasing function of q, is used. The case d(q) < α/ ln βq + γ, where α, β, γ are positive constants independent of q, is considered. It is shown that t2(2, q) < (2/ ln 1 10 q + 0.32) √ q ln q if q ≤ 67993, q prime, and q ∈ R, where R is a set of 27 values in the region 69997...110017. Also, for q ∈ [9311, 67993], q prime, and q ∈ R, it is shown that √ q(ln q) a 1 −bq < t2(2, q) < √ q(ln q) a 2 −bq , a1 = 0.771, a2 = 0.752, b = 2.2 • 10 −7. In addition, our results allow us to conjecture that these estimates hold for all q. An algorithm FOP using any fixed order of points in P G(2, q) is proposed for constructing complete arcs. The algorithm is based on an intuitive postulate that P G(2, q) contains a sufficient number of relatively small complete arcs. It is shown that the type of order on the points of P G(2, q) is not relevant.

In a user-private information retrieval (UPIR) scheme, a set of users collaborate to retrieve files from a database without revealing to observers which participant in the scheme requested the file. To achieve privacy, users retrieve files from the database in response to anonymous requests posted to message spaces; assuming that each message space can be accessed by a subset of the participants in the scheme. Privacy with respect to the database is easily achieved, but privacy with respect to coalitions of other users within the scheme is sensitive to the choice of incidence structure determining which users can access each message space. Earlier schemes were based on pairwise balanced designs and symmetric designs, and involved at most one step of message passing to retrieve a file. We propose a new class of UPIR schemes based on generalised quadrangles (GQs), which need up to two steps of message passing in each file retrieval. We introduce a new message passing protocol in which messages are encrypted. Even using this protocol, previously proposed schemes are compromised by finite coalitions of users. We construct a family of GQ-UPIR schemes which maintain privacy with high probability even when O(n 1/2−) users collude, where n is the total number of users in the scheme. We also show that a UPIR scheme based on any family of generalised quadrangles is secure against coalitions of O(n 1/4−) users.

ABSTRACT: Finite geometry explains the surprising symmetry properties of some simple graphic designs--found, for instance, in quilts. Links are provided for applications to sporadic simple groups (via the&quot; Miracle Octad Generator&quot; of RT Curtis), to the connection between orthogonal Latin squares and projective spreads, and to symmetry of Walsh functions.

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs mea...

We look at some techniques for constructing permutation arrays using projections in finite projective spaces and the geometry of arcs in the finite projective plane. We say a permutation array P A(n, d) has length n and minimum distance d when it consists of a collection of permutations on n symbols that pairwise agree in at most n − d coordinate positions. Such arrays can also be viewed as non-linear codes and are used in powerline communication. While our techniques likely do not produce optimal arrays, we are able to construct examples of codes for certain parameter sets where no known constructions were previously known.

We present new constructions for (n, w,) optical orthogonal codes (OOC) using techniques from finite projective geometry. In one case codewords correspond to (q − 1)-arcs contained in Baer subspaces (and, in general, kth-root subspaces) of a projective space. In the other construction, we use sublines isomorphic to PG(2, q) lying in a projective plane isomorphic to PG(2, q k), k > 1. Our construction yields for each > 1 an infinite family of OOCs which, in many cases, are asymptotically optimal with respect to the Johnson bound.

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 physical application of our findings is also mentioned.

It is demonstrated that the magic three-qubit Veldkamp line occurs naturally within the Veldkamp space of a combinatorial Grassmannian of type G 2 (7), V (G 2 (7)). The lines of the ambient symplectic polar space are those lines of V (G 2 (7)) whose cores feature an odd number of points of G 2 (7). After introducing the basic properties of three different types of points and seven distinct types of lines of V (G 2 (7)), we explicitly show the combinatorial Grassmannian composition of the magic Veldkamp line; we first give representatives of points and lines of its core generalized quadrangle GQ(2, 2), and then additional points and lines of a specific elliptic quadric Q − (5, 2), a hyperbolic quadric Q + (5, 2), and a quadratic cone Q(4, 2) that are centered on the GQ(2, 2). In particular, each point of Q + (5, 2) is represented by a Pasch configuration and its complementary line, the (Schläfli) double-six of points in Q − (5, 2) comprise six Cayley-Salmon configurations and six Desargues configurations with their complementary points, and the remaining Cayley-Salmon configuration stands for the vertex of Q(4, 2).

This study investigated by an analytical method the flow that develops in the gap between concentric rotating cylinders when the Taylor number Ta exceeds the first critical value. Concentric cylinders rotating at the speed ratio µ = 0 are investigated over the radius ratio range 0.20 ≤ η ≤ 0.95. This range includes configurations characterised by a larger annular gap width d than classical journal bearing test cases and by a Taylor number beyond the first critical Taylor number at which Taylor vortices develop. The analysis focuses on determining the parameters for the direct transition from axisymmetric Couette flow to wavy Taylor vortex flow. The results show a marked change in trend as the radius ratio η reduces below 0.49 and 0.63 for the azimuthal wave-numbers m = 2 and 3 respectively. The axial wavenumber increases so that the resulting wavy Taylor vortex flow is characterised by vortex structures elongated in the radial direction, with a meridional cross-section that is significantly elliptical. The linear stability analysis of the perturbation equations suggests this instability pattern is neutrally stable. Whereas a direct transition from axisymmetric Couette flow is not necessarily the only route for the onset of wavy Taylor vortex flow, the significant difference between the predicted pattern at large gap widths and classical wavy Taylor vortex flow merits further investigation.

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022), but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from two to seven. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension two and higher, (ii) non-existence of negative subspaces of dimension three and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank four, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the threequbit polar space we correct and improve the contextuality degree of the full configuration, and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their contextuality and computes their contextuality degree for some small symplectic polar spaces. We show that quadrics in the symplectic polar space W n are contextual for n = 3, 4, 5. The proofs we consider involve more contexts and observables than the smallest known proofs. This intermediate size property of those proofs is interesting for experimental tests, but could also be interesting in quantum game theory.

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

For N ≥ 2, an N-qubit doily is a doily living in the N-qubit symplectic polar space. These doilies are related to operator-based proofs of quantum contextuality. Following and extending the strategy of Saniga et al. (Mathematics 9 (2021) 2272) that focused exclusively on three-qubit doilies, we first bring forth several formulas giving the number of both linear and quadratic doilies for any N > 2. Then we present an effective algorithm for the generation of all N-qubit doilies. Using this algorithm for N = 4 and N = 5, we provide a classification of N-qubit doilies in terms of types of observables they feature and number of negative lines they are endowed with. We also list several distinguished findings about N-qubit doilies that are absent in the three-qubit case, point out a couple of specific features exhibited by linear doilies and outline some prospective extensions of our approach.

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus g, we write down a collection of polynomials in genus g theta constants, such that their common zero locus contains the locus of Jacobians of genus g curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for genus (g − 1) theta constants.

In this paper we give a new, simple and explicit proof of a theorem of Baker. This states that every odd diml'nsional projective space over the field of two elements admits a I-packing of I-spreads, i.e. a partition of its lines into families of mutual disjoint lines whose union covers the space. This I-packing may be generated from anyone of its spreads by repeated application of a fixed collineation.

In this paper we give a new, simple and explicit proof of a theorem of Baker. This states that every odd diml'nsional projective space over the field of two elements admits a I-packing of I-spreads, i.e. a partition of its lines into families of mutual disjoint lines whose union covers the space. This I-packing may be generated from anyone of its spreads by repeated application of a fixed collineation.

The authors previously found a model of universal quantum computation by making use of the coset structure of subgroups of a free group G with relations. A valid subgroup H of index d in G leads to a ‘magic’ state ψ in d-dimensional Hilbert space that encodes a minimal informationally complete quantum measurement (or MIC), possibly carrying a finite ‘contextual’ geometry. In the present work, we choose G as the fundamental group π 1 ( V ) of an exotic 4-manifold V, more precisely a ‘small exotic’ (space-time) R 4 (that is homeomorphic and isometric, but not diffeomorphic to the Euclidean R 4 ). Our selected example, due to S. Akbulut and R. E. Gompf, has two remarkable properties: (a) it shows the occurrence of standard contextual geometries such as the Fano plane (at index 7), Mermin’s pentagram (at index 10), the two-qubit commutation picture G Q ( 2 , 2 ) (at index 15), and the combinatorial Grassmannian Gr ( 2 , 8 ) (at index 28); and (b) it allows the interpretation of MICs mea...

Flow characterization in a Taylor Couette system was made by investigating the radial velocity component with Ultrasonic Doppler Velocimetry based flow mapping. With the technique presented in this work, it is possible to measure the radial velocity components for variable axial position in a Couette cell within Taylor vortex flow (TVF), wavy vortex flow (WVF), modulated vortex flow (MVF) as well as spiral vortex domains in a conical shaped gap. The resulting maps for the different flow states show the location of vortices in the annular gap between the inner and outer cylinder. Cylindrical and conical concentrically rotating inner bodies were applied and respective flow patterns were analyzed. The method uses a stroboscopic triggering to synchronize flow measurements and rotational motion. The oscillation frequency f of unsteady motion in WVF, MVF, and spirals can be obtained from the power spectrum of velocity. The UVP transducer was preferably positioned in radial direction, perpendicular to the surface of the inner rotating body for measuring the radial velocity component. At the same time, the transducer was moved with constant velocity vertically along the outer

This purpose is about a computational fluid dynamics investigation of a step varying in annular space effect on Taylor vortices flow and velocity in cylindrical Taylor-Couette system. Three cases are considered, the first geometric configuration GCi where the change in radius of the outer cylinder, and the half endplate is replaced by fluid close to the inner cylinder. The second geometrical configuration GCo with a step change in radius of inner cylinder and the half end-plate is replaced by a fluid close to outer cylinder. The last case CGb, the system has a change in both cylinders. The bottom endplate is attached to the outer cylinder, which are held fixed, the inner one is rotating, and a cylinder radius ratio of η = 0.66. Hence, the Taylor vortices flow development in geometrical configuration GCo for different Reynolds number values and Axial profile of dimensionless tangential velocity component midway across the gap for Τa = = 59.42, a numerical method based on the finite volume method is employed in simulations.

Subspace codes are codes in which the codewords consist of subspaces of a vector space V (n, q) of dimension n over the finite field of order q. These codes now receive a lot of attention because of their relevance for transmission of information in wireless networks. Since codewords are subspaces of a vector space V (n, q), they also can be interpreted as being subspaces of the projective space PG(n − 1, q) of dimension n − 1 over the finite field of order q. This implies that geometrical techniques can be used to construct subspace codes and to investigate properties of subspace codes. In this talk, a number of geometrical results on subspace codes will be presented to show that the theory of subspace codes is a new interesting research domain for finite geometers.

Abstract. A main issue in superstring theory are the superstring measures. D’Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic measure. They also suggested a generalisation to higher genus. We showed that their approach works, with a minor modification, in genus three and we announced a positive result also in genus four. Here we give the modular form in genus four explicitly. Recently S. Grushevsky published this result as part of a more general approach. 1.

In the simplest and original case of study of the Taylor-Couette TC problems, the fluid is contained between a fixed outer cylinder and a concentric inner cylinder which rotates at constant angular velocity. Much of the works done has been concerned on steady rotating cylinder(s) i.e. rotating cylinders with constant velocity and the various transitions that take place as the cylinder(s) velocity (ies) is (are) steadily increased. On this work, we concentrated our attention in the case in which the inner cylinder velocity is not constant, but oscillates harmonically (in time) clockwise and counterclockwise while the outer cylinder is maintained fixed. Our aim is to attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case and if there are any reversing or non reversing flows apparition. If the modulation amplitude is large enough to destabilize the circular Couette flow, two classes of axisymmetric Taylor vortex flow are possible: reversing Taylor Vortex Flow (RTVF) and Non-Reversing Taylor Vortex Flow (NRTVF). Our work presents an experimental investigation of the effect of oscillatory Couette-Taylor flow on the instantaneous and local mass transfer and wall shear rates evolutions, i.e. the impact of vortices at wall; and the detection of any RTVF and/or NRTVF apparition. The vortices may manifest themselves by the presence of timeoscillations of mass transfer and wall shear rates; this generally corresponds to an instability apparition even for steady rotating cylinder. On laminar CT flow, the time-evolution of wall shear rate is linear. It can be presented as a linear function of the angular velocity. For a mean Taylor number corresponding to a laminar Couette flow, a modulation frequency F = 0.1 Hz and an amplitude respectively β = 0.53 andβ = 1.08 are sufficient to destabilize the laminar CT flow, Taylor vortices appear. Comparing to a steady rotational velocity case, oscillatory flow accelerates the instability apparition, i.e. the mean critical Taylor number corresponds to the transition is smaller than that of the steady rotational case. The vortices direction can be deduced from the sign of the instantaneous wall shear rate time evolution.

In the simplest and original case of study of the Taylor-Couette TC problems, the fluid is contained between a fixed outer cylinder and a concentric inner cylinder which rotates at constant angular velocity. Much of the works done has been concerned on steady rotating cylinder(s) i.e. rotating cylinders with constant velocity and the various transitions that take place as the cylinder(s) velocity (ies) is (are) steadily increased. On this work, we concentrated our attention in the case in which the inner cylinder velocity is not constant, but oscillates harmonically (in time) clockwise and counterclockwise while the outer cylinder is maintained fixed. Our aim is to attempt to answer the question if the modulation makes the flow more or less stable with respect to the vortices apparition than in the steady case and if there are any reversing or non reversing flows apparition. If the modulation amplitude is large enough to destabilize the circular Couette flow, two classes of axisymmetric Taylor vortex flow are possible: reversing Taylor Vortex Flow (RTVF) and Non-Reversing Taylor Vortex Flow (NRTVF). Our work presents an experimental investigation of the effect of oscillatory Couette-Taylor flow on the instantaneous and local mass transfer and wall shear rates evolutions, i.e. the impact of vortices at wall; and the detection of any RTVF and/or NRTVF apparition. The vortices may manifest themselves by the presence of timeoscillations of mass transfer and wall shear rates; this generally corresponds to an instability apparition even for steady rotating cylinder. On laminar CT flow, the time-evolution of wall shear rate is linear. It can be presented as a linear function of the angular velocity. For a mean Taylor number corresponding to a laminar Couette flow, a modulation frequency F = 0.1 Hz and an amplitude respectively β = 0.53 andβ = 1.08 are sufficient to destabilize the laminar CT flow, Taylor vortices appear. Comparing to a steady rotational velocity case, oscillatory flow accelerates the instability apparition, i.e. the mean critical Taylor number corresponds to the transition is smaller than that of the steady rotational case. The vortices direction can be deduced from the sign of the instantaneous wall shear rate time evolution.