Contact numbers for congruent sphere packings in Euclidean 3-space

Continuing the investigations of Harborth (1974) and the author (2002) we study the following two rather basic problems on sphere packings. Recall that the contact graph of an arbitrary finite packing of unit balls (i.e., of an arbitrary finite family of non-overlapping unit balls) in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit balls can have in Euclidean 3-space. Our method for finding lower and upper estimates for the largest contact numbers is a combination of analytic and combinatorial ideas and it is also based on some recent results on sphere packings. Finally, we are interested also in the following more special version of the above problem. Namely, let us imagine that we are given a lattice unit sphere packing with the center points forming the lattice L in Euclidean 3-space (and with certain pairs of unit balls touching each other) and then let us generate packings of n unit balls such that each and every center of the n unit balls is chosen from L. Just as in the general case we are interested in finding good estimates for the largest contact number of the packings of n unit balls obtained in this way.Comment: 18 page

On the most compact regular lattice in large dimensions: A statistical mechanical approach

In this paper I will approach the computation of the maximum density of regular lattices in large dimensions using a statistical mechanics approach. The starting point will be some theorems of Roger, which are virtually unknown in the community of physicists. Using his approach one can see that there are many similarities (and differences) with the problem of computing the entropy of a liquid of perfect spheres. The relation between the two problems is investigated in details. Some conjectures are presented, that need further investigation in order to check their consistency.Comment: 27 page

Lower bounds for measurable chromatic numbers

The Lovasz theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10,..., 24, and we give a new proof that it grows exponentially with the dimension.Comment: 18 pages, (v3) Section 8 revised and some corrections, to appear in Geometric and Functional Analysi

Selected Open Problems in Discrete Geometry and Optimization

A list of questions and problems posed and discussed in September 2011 at the following consecutive events held at the Fields Institute, Toronto: Workshop on Discrete Geometry, Conference on Discrete Geometry and Optimization, and Workshop on Optimization. We hope these questions and problems will contribute to further stimulate the interaction between geometers and optimizers

On the complexity of computing minimum energy consumption broadcast subgraphs

Abstract. We consider the problem of computing an optimal range assignment in a wireless network which allows a specified source station to perform a broadcast operation. In particular, we consider this problem as a special case of the following more general combinatorial optimization problem, called Minimum Energy Consumption Broadcast Subgraph (in short, MECBS): Given a weighted directed graph and a specified source node, find a minimum cost range assignment to the nodes, whose corresponding transmission graph contains a spanning tree rooted at the source node. We first prove that MECBS is not approximable within a constant factor (unless P=NP). We then consider the restriction of MECBS to wireless networks and we prove several positive and negative results, depending on the geometric space dimension and on the distance-power gradient. The main result is a polynomial-time approximation algorithm for the NP-hard case in which both the dimension and the gradient are equal to 2: This algorithm can be generalized to the case in which the gradient is greater than or equal to the dimension.

Formula for unbiased bases

summary:The present paper deals with mutually unbiased bases for systems of qudits in $d$ dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of $1+p$ mutually unbiased bases is given for $d=p$ where $p$ is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group $SU(2)$. A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case when $d = p^e$ ($e \geq 2$), corresponding to the power of a prime integer, is briefly examined. Finally, complete sets of mutually unbiased bases are analysed through a Lie algebraic approach

Addition reactions of some unsaturated systems possessing one or more hetero atoms

The introduction reviews the Michael addition, and shows how the addition of dimethyl acetylenedicarboxylate to heterocycles can be explained if initial attack gives a zwitterion, = +N-Co = −Co, which can then react further in several ways. A summary of how the acetylenic ester combines with various heterocycles is given, and brief mention is made of the related Diels-Alder and 1, 3-dipolar additions. Chapter One discusses the reaction of 2-phenylpyridine with dimethyl acetylenedicarboxylate. The initial product has structure [1], which,on heating above its melting point, isomerises to [2], these structures being assigned by consideration of the spectral properties. Heating [1] with excess acetylenic ester gives the 9a-vinyl-9aH-quinolizine, [3], whose spectra resemble those of [1] rather than those of [2]. The 9a-vinyl- 9aH-quinolizines, [4] and [5], are obtained from the acetylenic ester and 2-vinyl- and 2-methyl-6-vinylpyridine, respectively. A minor product from 2-vinylpyridine is [6], and a possible mode of formation is proposed. In Chapter Two, the addition of dimethyl acetylenedicarboxylate to azoles is discussed. Thiazole and 4-methylthiazole give the adducts [7] and [8], while a minor product of the thiazole reaction is [9]. Objections to alternative structures1 for [7] and some analogues are put forward. Benzoxazole forms the adduct [10] which can be reduced to the dihydroderivative [ll]. 2-Methylbenzoxazole and the acetylenic ester give the compound [12], a possible reaction scheme being postulated, and the similar compounds [13] and [14] are obtained from l-alkyl-2-methyl-benziminazoles. The nuclear magnetic resonance spectra of these adducts are discussed, and compared with those of [15] and [16], previously prepared.2 1, 2-Dimethylbenziminazole gives a second adduct which has been shown to be [l7]. The sole adduct from 2-ethyl-1-methylbenziminazole and the acetylenic ester is the analogous compound [18]. Comparison of the nuclear magnetic resonance and ultraviolet absorption spectra of these compounds with those of a minor product from the reaction of 1-methylbenziminazole and dimethyl acetylenedicarboxylate, indicate it has the structure [19], but the major product of the reaction is [20]. A similar compound [21] is obtained from benziminazole. Attempts to convert [19] into [20], or both into the same perchlorate, were unsuccessful. Two adducts [22] and [23] are obtained from 1-methylpyrazole, while 1-methylindazole and the acetylenic ester give [24]. These structures are based mainly on spectral evidence. 1-Methyl- and 1-benzylbenzotriazole react with dimethyl acetylene-dicarboxylate to give 1:2 molar adducts, which are formulated as [25] and [26], on the basis of their reduction and oxidation products, and their respective spectra. Structure [25] is preferred to [27], which might be expected by analogy with earlier results, because of its ready loss of CH2, characteristic of vinyl ethers. A method of formation of [25] and related compounds is suggested. l-Methyl-l,2,4-triazole and the acetylenic ester form the compound [28], but the N-methyl derivative of 3-methyl-l,2,4-triazole gives three 1:2 molar adducts, for which no definite structures could be proposed. The reactions between some diazines and dimethyl acetylenedicarboxylate are considered in Chapter Three. In acetonitrile, the sole products from 2-methyl- and 2,6-dimethylpyrazine and the ester are [29] and [3O], while the similar compounds [31], [32], and [33] are minor products when pyridazine, 3-methylpyridazine, and 1-methylphthalazine are used. The structures [31] and [32], previously described , are confirmed by nuclear magnetic resonance spectroscopy, which was also used to assign structures to the major products of the reactions of the acetylenic ester with pyridazine and its 3-methyl derivative, [34] and [35] respectively. [36], the analogue of [35] is obtained from 3,6-dimethylpyridazine and the ester, but the major product of this reaction is [37], identified by its proton resonance spectrum, which is similar to those of the compounds [12]-[16]. Unlike 3-methylpyridazine, 1-methylphthalazine gives a 1:2 molar adduct with a hydrogen at the bridgehead, [38], and this is isomerised to [39] in acid solution. In aprotic solvents, phthalazine, quinazoline, and quinoxaline give no crystalline products with dimethyl acetylenedicarboxylate, but, in methanol, phthalazine gives [40]. Phenazine and the acetylenic ester in methanol form the adduct [4l], the structure being supported by the proton resonance spectrum, consistent with a symmetrical molecule, and derivatives obtained on reduction and bromination. Chapter four is concerned with alkylquinoxalines and methyl- substituted diazines, whose methyl groups take part in the reactions with acetylenic esters. Thus, 2-methylquinoxaline and dimethyl acetylene dicarboxylate give two isomeric 1:2 molar adducts, [42] and [43]. These structures are based partly on analogy with the "red adduct" from quinaldine4 , and partly on the chemistry and spectra of the compounds. The nuclear magnetic resonance spectrum of [42] is discussed in detail, and a theoretical spectrum of the CH2-CH system has been calculated5, excellent agreement being found between calculated and observed spectra for both methyl and ethyl esters. Reduction of [42] gives [45], and again agreement between observed and calculated proton resonance spectra is found. 2,3~Dimethylquinoxaline forms the methyl analogue of [43] i.e. [44], and the differences in the CH2-CHCOOMe systems of [42] and [44] are used to justify the differences in the respective proton resonance spectra. Attempts to degrade these adducts were unsuccessful. Reduction of [44] gives the 5,6-dihydro derivative, while bromination of [42] gives the 2,3-dibromo derivative, and bromination of [44] gives bromo-compounds, substituted in the benzene ring and/or the methyl side chain. An interesting reaction is the conversion of [44] to [43] by selenium dioxide. Possible ways in which these adducts could be formed are discussed. 4,6-Dimethyl- and 2,4,6-trimethylpyrimidine and 4-methylquinazoline form 1:2 molar adducts with dimethyl acetylenedicarboxylate, whose nuclear magnetic resonance spectra show the presence of CH2-CH groupings. Structures [46], [47], and [48] are proposed for these compounds, although, the CH2-CH systems may be reversed. A 1:3 molar adduct from 2,4-dimethyl-quinazoline and the acetylenic ester is formulated as [49], on the basis of its nuclear magnetic resonance spectrum. 2,3,5,6-Tetramethylpyrazine and the acetylenic ester give a compound, whose structure is considered to be [50], although alternatives are discussed. The similar compound [51] is obtained in small yield from 2,3-dimethylquinoxaline, while 2,5-dimethylpyrazine and dimethyl acetylene-dioarboxylate give an adduct, whose properties support the structure [52]. A scheme is suggested for the formation of this compound. In Chapter Five, adducts from the acetylenic ester and some heterocyoles containing a free amino group are shown to be one of two types. 4-Methyliminazole and indazole form adducts like [53], while indazole, 3-methylindazole, 1,2,3-triazole, benzotriazole, 1,2,4-triazole, and 3-methyl-1,2,4-triazole give adducts of the type [54]. The position of attachment of the side chain cannot be decided, except in the adducts of indazole and 3-methylindazole, [55] and [56] respectively. Unlike the other succinates, [55] and [56] show two sets of peaks in their nuclear magnetic resonance spectra, indicating the asymmetry of the molecules. Diazoaminobenzene and dimethyl acetylenedicarboxylate give three adducts, two of which correspond to [53] and [54], while the structure [57] is suggested for the third adduct.</p

The origin and migration of cortical neurones: new vistas

The principal neuronal types of the cerebral cortex are the excitatory pyramidal cells, which project to distant targets, and the inhibitory nonpyramidal cells, which are the cortical interneurones. This article reviews evidence suggesting that these two neuronal types are generated in distinct proliferative zones. Pyramidal cells are derived from the neuroepithelium in the cortical ventricular zone, and use the processes of radial glia in order to migrate and take their positions in the cortex in an 'inside-out' sequence. Relatively few nonpyramidal cells are generated in the cortical neuroepithelium: the majority is derived from the ganglionic eminence of the ventral telencephalon. These nonpyramidal neurones use tangential migratory paths to reach the cortex, probably travelling along axonal bundles of the developing corticofugal fibre system