Fano Plane Research Articles

Decoupling of Background and Resonance Scatterings in Multichannel Quantum Defect Theory and Extraction of Dynamic Parameters from Lu-Fano Plot

Giusti-Suzor and Fano introduced translations of the scales of Lu-Fano plots by phase renormalization in order to decouple the intra- and inter-channel couplings in multichannel quantum defect theory (MQDT). Their theory was further developed by others to deal with systems involving a larger number of channels. In different directions, MQDT was reformulated into forms with a one-to-one correspondence to those in Fano's configuration mixing theory of resonance for photofragmentation processes involving one closed and many open channels. In this study, the theory was further developed to fully reveal the coupling nature, decoupling of the background and resonance scattering in physical scattering matrices as well as to further extract the dynamic parameters undiscovered by Fano and his colleagues. This theory was applied to the photoabsorption spectrum of H 2 observed by Herzberg's group.

Three-qubit operators, the split Cayley hexagon of order two, and black holes

The set of 63 real generalized Pauli matrices of three-qubits can be factored into two subsets of 35 symmetric and 28 antisymmetric elements. This splitting is shown to be completely embodied in the properties of the Fano plane; the elements of the former set being in a bijective correspondence with the 7 points, 7 lines and 21 flags, whereas those of the latter set having their counterparts in 28 anti-flags of the plane. This representation naturally extends to the one in terms of the split Cayley hexagon of order two. 63 points of the hexagon split into 9 orbits of 7 points (operators) each under the action of an automorphism of order 7. 63 lines of the hexagon carry three points each and represent the triples of operators such that the product of any two gives, up to a sign, the third one. Since this hexagon admits a full embedding in a projective 5-space over GF(2), the 35 symmetric operators are also found to answer to the points of a Klein quadric in such space. The 28 antisymmetric matrices can be associated with the 28 vertices of the Coxeter graph, one of two distinguished subgraphs of the hexagon. The P SL2(7) subgroup of the automorphism group of the hexagon is discussed in detail and the Coxeter sub-geometry is found to be intricately related to the E7-symmetric black-hole entropy formula in string theory. It is also conjectured that the full geometry/symmetry of the hexagon should manifest itself in the corresponding black-hole solutions. Finally, an intriguing analogy with the case of Hopf sphere fibrations and a link with coding theory are briefly mentioned.

A hypergraph regularity method for generalized Turán problems

AbstractWe describe a method that we believe may be foundational for a comprehensive theory of generalized Turán problems. The cornerstone of our approach is a quasirandom counting lemma for quasirandom hypergraphs, which extends the standard counting lemma by not only counting copies of a particular configuration but also showing that these copies are evenly distributed. We demonstrate the power of the method by proving a conjecture of Mubayi on the codegree threshold of the Fano plane, that is, any 3‐graph on n vertices for which every pair of vertices is contained in more than n/2 edges must contain a Fano plane, for n sufficiently large. For projective planes over fields of odd size q we show that the codegree threshold is between n/2 − q + 1 and n/2, but for PG2(4) we find the somewhat surprising phenomenon that the threshold is less than (1/2 − ϵ)n for some small ϵ > 0. We conclude by setting out a program for future developments of this method to tackle other problems. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions R} (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R}. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left R}-module R 3 is shown to split into three disjoint sets of cardinalities 9, 9 and 3 according as the number of Jacobson radical entries in the generating vector is 2, 1 or 0, respectively. The corresponding ternion- induced factorization of the lines of the Fano plane sitting in the middle of the Fano- Snowflake is found to dier fundamentally from the natural one, i.e., from that with respect to the Jacobson radical of the Galois field of two elements.

Twin ''Fano-Snowflakes'' over the Smallest Ring of Ternions

Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n+1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 when R is the (unique) non-commutative ring of order eight. The corresponding geometry is dubbed a Fano-Snowflake due to its diagrammatic appearance and the fact that it contains the Fano plane in its center. There exist, in fact, two such configurations - each being tied to either of the two maximal ideals of the ring - which have the Fano plane in common and can, therefore, be viewed as twins. Potential relevance of these noteworthy configurations to quantum information theory and stringy black holes is also outlined.

Projective, affine, and abelian colorings of cubic graphs

We develop an idea of a local 3- edge-coloring of a cubic graph, a generalization of the usual 3-edge-coloring. We allow for an unlimited number of colors but require that the colors of two edges meeting at a vertex always determine the same third color. Local 3-edge-colorings are described in terms of colorings by points of a partial Steiner triple system such that the colors meeting at each vertex form a triple of the system. An important place in our investigation is held by the two smallest non-trivial Steiner triple systems, the Fano plane PG ( 2 , 2 ) and the affine plane AG ( 2 , 3 ) . For i = 4 , 5 , and 6 we identify certain configurations F i and A i of i lines of the Fano plane and the affine plane, respectively, and prove a theorem saying that a cubic graph admits an F i -coloring if and only if it admits an A i -coloring. Among consequences of this is the result of Holroyd and Škoviera [F. Holroyd, M. Škoviera, Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91 (2004) 57–66] that the edges of every bridgeless cubic graph can be colored by using points and blocks of any non-trivial Steiner triple system S . Another consequence is that every bridgeless cubic graph has a proper edge-coloring by elements of any abelian group of order at least 12 such that around each vertex the group elements sum to 0. We also propose several conjectures concerning edge-coloring of cubic graphs and relate them to several well-known conjectures. In particular, we show that both the Cycle Double Cover Conjecture and the Fulkerson Conjecture can be formulated as a coloring problem in terms of known geometric configurations — the Desargues configuration and the Cremona–Richmond configuration, respectively.

Using Algebraic Properties of Minimal Idempotents for Exhaustive Computer Generation of Association Schemes

During the past few years we have obtained several new computer classification results on association schemes and in particular distance regular and strongly regular graphs. Central to our success is the use of two algebraic constraints based on properties of the minimal idempotents $E_{i}$ of these association schemes : the fact that they are positive semidefinite and that they have known rank. Incorporating these constraints into an actual isomorph-free exhaustive generation algorithm turns out to be somewhat complicated in practice. The main problem to be solved is that of numerical inaccuracy: we do not want to discard a potential solution because a value which is close to zero is misinterpreted as being negative (in the first case) or nonzero (in the second). In this paper we give details on how this can be accomplished and also list some new classification results that have been recently obtained using this technique: the uniqueness of the strongly regular $(126,50,13,24)$ graph and some new examples of antipodal distance regular graphs. We give an explicit description of a new antipodal distance regular $3$-cover of $K_{14}$, with vertices that can be represented as ordered triples of collinear points of the Fano plane.

E7and the tripartite entanglement of seven qubits

In quantum information theory, it is well known that the tripartite entanglement of three qubits is described by the group [SL(2, C)] 3 and that the entanglement measure is given by Cayley's hyperdeterminant. This has provided an analogy with certain N = 2 supersymmetric black holes in string theory, whose entropy is also given by the hyperdeterminant. In this paper, we extend the analogy to N = 8. We propose that a particular tripartite entanglement of seven qubits, encoded in the Fano plane, is described by the exceptional group E 7 (C) and that the entanglement measure is given by Cartan's quartic E 7 invariant.

Total domination of graphs and small transversals of hypergraphs

The main result of this paper is that every 4-uniform hypergraph on n vertices and m edges has a transversal with no more than (5n + 4m)/21 vertices. In the particular case n = m, the transversal has at most 3n/7 vertices, and this bound is sharp in the complement of the Fano plane. Chvátal and McDiarmid [5] proved that every 3-uniform hypergraph with n vertices and edges has a transversal of size n/2. Two direct corollaries of these results are that every graph with minimal degree at least 3 has total domination number at most n/2 and every graph with minimal degree at least 4 has total domination number at most 3n/7. These two bounds are sharp.

Classification of partitions of all triples on ten points into copies of Fano and affine planes

We continue our study of partitions of the full set of v 3 triples chosen from a v -set into copies of the Fano plane PG ( 2 , 2 ) (Fano partitions) or copies of the affine plane AG ( 2 , 3 ) (affine partitions) or into copies of both of these planes (mixed partitions). The smallest cases for which such partitions can occur are v = 8 where Fano partitions exist, v = 9 where affine partitions exist, and v = 10 where both affine and mixed partitions exist. The Fano partitions for v = 8 and the affine partitions for v = 9 and 10 have been fully classified, into 11, two and 77 isomorphism classes, respectively. Here we classify (1) the sets of i pairwise disjoint affine planes for i = 1 , … , 7 , and (2) the mixed partitions for v = 10 into their 22 isomorphism classes. We consider the ways in which these partitions relate to the large sets of AG ( 2 , 3 ) .

Codegree problems for projective geometries

The codegree density γ ( F ) of an r-graph F is the largest number γ such that there are F-free r-graphs G on n vertices such that every set of r − 1 vertices is contained in at least ( γ − o ( 1 ) ) n edges. When F = PG 2 ( 2 ) is the Fano plane Mubayi showed that γ ( F ) = 1 / 2 . This paper studies γ ( PG m ( q ) ) for further values of m and q. In particular we have an upper bound γ ( PG m ( q ) ) ⩽ 1 − 1 / m for any projective geometry. We show that equality holds whenever m = 2 and q is odd, and whenever m = 3 and q is 2 or 3. We also give examples of 3-graphs with codegree densities equal to 1 − 1 / k for all k ⩾ 1 .

Strings, black holes, the tripartite entanglement of seven qubits, and the Fano plane

Recently it has been observed that the group ${E}_{7}$ can be used to describe a special type of quantum entanglement of seven qubits partitioned into seven tripartite systems. Here we show that this curious type of entanglement is entirely encoded into the discrete geometry of the Fano plane. We explicitly work out the details concerning a qubit interpretation of the ${E}_{7}$ generators as representatives of tripartite protocols acting on the 56-dimensional representation space. Using these results we extend further the recently studied analogy between quantum information theory and supersymmetric black holes in four-dimensional string theory. We point out that there is a dual relationship between entangled subsystems containing three and four tripartite systems. This relationship is reflected in the structure of the expressions for the black hole entropy in the $N=4$ and $N=2$ truncations of the ${E}_{7(7)}$ symmetric area form of $N=8$ supergravity. We conjecture that a similar picture based on other qubit systems might hold for black hole solutions in magic supergravities.

Time-dependent formation of the profile of the He2s2pPo1state excited by a short laser pulse

We produce quantitatively accurate data for the energy- and time-dependent formation of the profile of the differential ionization probability of the He $1{s}^{2}\phantom{\rule{0.2em}{0ex}}^{1}S$ ground state from the coherent excitation and decay of the doubly excited He $2s2p\phantom{\rule{0.2em}{0ex}}^{1}P^{o}$ resonance state induced by a pulse of duration of $450\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ and field strength in the range $F=0.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}F=0.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ Two general methods were applied. One is analytic, using Fano's configuration interaction in the continuum in the framework of first order time-dependent perturbation theory. The other is numerical, using the state-specific expansion approach for the nonperturbative solution of the time-dependent Schr\"odinger equation. Electronic structures and electron correlation are incorporated via the use of state-specific wave functions for the initial state, the resonance state, and the continuum of scattering states. The results from the two methods are in perfect agreement, with a small discrepancy starting at $F=0.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}1}\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ The weak field analytic formulas show explicitly the dependence of the profile formation on the pulse characteristics. In the limit of large times, the system becomes stationary and the computed resonance state profile yields the Fano asymmetry parameter of $q=\ensuremath{-}2.8$, with energy ${E}_{r}=60.20\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and width $\ensuremath{\Gamma}=0.038\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. These values agree with previously published ones obtained from time-independent calculations and from photoabsorption measurements of the type initiated by Madden and Codling in 1963.

Fuzzy projective spreads of fuzzy projective spaces

In this paper, we define the notion of a fuzzy spread of a fuzzy projective space. We classify the fuzzy line spreads of the smallest finite projective space, the Fano plane, and prove a general existence theorem for line spreads of arbitrary finite projective planes. We then extend the classical relation between designs and statistics to an application of the fuzzy spreads in statistics, by considering projective spaces as 2-designs. A fuzzy spread then gives a blueprint of how to make groups for test procedures where certain kinds of objects are mixed together and in each group one kind is over-represented.

Configurations of lines and models of Lie algebras

The automorphism groups of the 27 lines on the smooth cubic surface or the 28 bitangents to the general quartic plane curve are well-known to be closely related to the Weyl groups of E 6 and E 7 . We show how classical subconfigurations of lines, such as double-sixes, triple systems or Steiner sets, are easily constructed from certain models of the exceptional Lie algebras. For e 7 and e 8 we are lead to beautiful models graded over the octonions, which display these algebras as plane projective geometries of subalgebras. We also interpret the group of the bitangents as a group of transformations of the triangles in the Fano plane, and show how this allows to realize the isomorphism PSL ( 3 , F 2 ) ≃ PSL ( 2 , F 7 ) in terms of harmonic cubes.

Fano colourings of cubic graphs and the Fulkerson Conjecture

A Fano colouring is a colouring of the edges of a cubic graph by points of the Fano plane such that the colours of any three mutually adjacent edges form a line of the Fano plane. It has recently been shown by Holroyd and Škoviera [Colouring of cubic graphs by Steiner triple systems, J. Combin. Theory Ser. B 91 (2004) 57–66] that a cubic graph has a Fano colouring if and only if it is bridgeless. In this paper we prove that six, and conjecture that four, lines of the Fano plane are sufficient to colour any bridgeless cubic graph. We establish connections of our conjecture to other conjectures concerning bridgeless cubic graphs, in particular to the well-known conjecture of Fulkerson about the existence of a double covering by 1-factors in every bridgeless cubic graph.

The Turán Number Of The Fano Plane

Let PG2(2) be the Fano plane, i. e., the unique hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. In this paper we prove that for sufficiently large n, the maximum number of edges in a 3-uniform hypergraph on n vertices not containing a Fano plane is $$ex{\left( {n,PG_{2} {\left( 2 \right)}} \right)} = {\left( {\begin{array}{*{20}c} {n} \\ {3} \\ \end{array} } \right)} - {\left( {\begin{array}{*{20}c} {{{\left\lfloor {n/2} \right\rfloor }}} \\ {3} \\ \end{array} } \right)} - {\left( {\begin{array}{*{20}c} {{{\left\lceil {n/2} \right\rceil }}} \\ {3} \\ \end{array} } \right)}.$$ Moreover, the only extremal configuration can be obtained by partitioning an n-element set into two almost equal parts, and taking all the triples that intersect both of them. This extends an earlier result of de Caen and Furedi, and proves an old conjecture of V. Sos. In addition, we also prove a stability result for the Fano plane, which says that a 3-uniform hypergraph with density close to 3/4 and no Fano plane is approximately 2-colorable.

The co-degree density of the Fano plane

It is shown that every n vertex triple system with every pair of vertices lying in at least ( 1 / 2 + o ( 1 ) ) n triples contains a copy of the Fano plane. The constant 1 / 2 is sharp. This is the first triple system for which such a (nonzero) constant is determined.

Triple Systems Not Containing a Fano Configuration

A Fano configuration is the hypergraph of 7 vertices and 7 triplets defined by the points and lines of the finite projective plane of order 2. Proving a conjecture of T. Sos, the largest triple system on $n$ vertices containing no Fano configuration is determined (for $n> n_1$). It is 2-chromatic with $\binom{n}{3}-\binom{\lfloor n/2 \rfloor}{3} -\binom{\lceil n/2 \rceil}{3}$ triples. This is one of the very few nontrivial exact results for hypergraph extremal problems.

The Turán problem for projective geometries

We consider the following Turán problem. How many edges can there be in a ( q + 1 ) -uniform hypergraph on n vertices that does not contain a copy of the projective geometry PG m ( q ) ? The case q = m = 2 (the Fano plane) was recently solved independently and simultaneously by Keevash and Sudakov (The Turán number of the Fano plane, Combinatorica, to appear) and Füredi and Simonovits (Triple systems not containing a Fano configuration, Combin. Probab. Comput., to appear). Here we obtain estimates for general q and m via the de Caen–Füredi method of links combined with the orbit-stabiliser theorem from elementary group theory. In particular, we improve the known upper and lower bounds in the case q = 2 , m = 3 .