On the coordinates of minimal vectors in a Minkowski-reduced basis

This chapter is intended to give a survey of some recent results which have been discovered after the French version of this book was written, and which are of great importance in the theory of extreme lattices: on the one hand, they allow us to prove that some lattices having a comparatively high dimension are extreme without having any precise description of their minimal vectors (example: 80-dimensional even unimodular lattices of minimum 8, for which two lattices are known from [Bac-Ne2]); on the other hand, they connect the theory of extreme lattices to that of spherical designs, which properly belongs to combinatorics. The basic reference is Venkov’s [Ven3] (English title: “Lattices and Spherical Designs”). We account for some of the results proved in [Ven3] or in some other articles which appeared in the same issue of “L’Enseignement Mathématique” (Bachoc, Martinet, Venkov), and also for results involving group theory, due to Dummigan, Lempken, Schröder and Tiep ([D-T],[L-S-T]).

The theory of concept lattices is an efficient tool for knowledge representation and knowledge discovery, and is applied to many fields successfully. One focus of knowledge discovery is knowledge reduction. Based on the reduction theory of classical formal context, this paper proposes the definition of decision formal context and its reduction theory, which extends the reduction theory of concept lattices. In this paper, strong consistence and weak consistence of decision formal context are defined respectively. For strongly consistent decision formal context, the judgment theorems of consistent sets are examined, and approaches to reduction are given. For weakly consistent decision formal context, implication mapping is defined, and its reduction is studied. Finally, the relation between reducts of weakly consistent decision formal context and reducts of implication mapping is discussed.

Multiband $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ theory is often implemented with the one electron Schr\"odinger equation without spin (single group) as the unperturbed system. The effect of spin is taken into account by considering basis functions formed by a direct product between single-group eigenstates and spinor states (which give rise to the adapted double-group basis after a unitary transformation), with the spin-orbit interaction also treated as a perturbation. The $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ perturbation between these states is calculated using the single-group basis functions. This approach leads to a one-to-one link between occurrence of basis states in the single group with those under the double-group classification, placing constraints on the adapted double-group basis. This paper considers energy eigenstates which form the bases of irreducible representations (IRs) of the double group and derives the direct and remote (L\"owdin term) interaction matrices between the states using perturbation theory and symmetry properties of crystal lattice. The use of general double-group basis functions removes the constraints placed on the adapted double-group basis under the single-group formulation. Together with a change of paradigm in constructing atomic site wave functions using hybridized orbitals (rather than atomic orbitals), it allows direct contributions from $d$ and higher orbitals to the valence band with additional interaction matrices permitted by symmetry. A full description of interactions between states of ${\ensuremath{\Gamma}}_{8}^{\ifmmode\pm\else\textpm\fi{}}$ IRs require two linearly independent matrices and two scaling constants rather than the single matrix and scaling constant under single-group consideration. This formulation is developed from both perturbation theory and the method of invariant approach utilizing the Wigner-Eckart theorem and other group theoretical techniques for calculation of matrix elements. Crystals with diamond lattice are investigated first, with results for zincblende lattice obtained under the compatibility relation between the ${O}_{h}$ and the ${T}_{d}$ groups. We show that a unitary transformation of the ${\ensuremath{\Gamma}}_{8}^{\ensuremath{-}}$ basis of the ${O}_{h}$ group is required before they can be used in ${\ensuremath{\Gamma}}_{8}$ IR of the ${T}_{d}$ group. Consequently, existing data and optical transition selection rules shows that the symmetry assignment of the zone-center conduction band edge state should be ${\ensuremath{\Gamma}}_{6}^{\ensuremath{-}}({\ensuremath{\Gamma}}_{7})$ in Ge (GaAs and other semiconductors with zincblende lattice) with spin-orbit split-off band as origin. In addition to the new interaction matrix between states of ${\ensuremath{\Gamma}}_{8}^{\ifmmode\pm\else\textpm\fi{}}({\ensuremath{\Gamma}}_{8})$ IRs, the form of interband L\"owdin term between ${\ensuremath{\Gamma}}_{8}^{+}({\ensuremath{\Gamma}}_{8})$ and ${\ensuremath{\Gamma}}_{7}^{+}({\ensuremath{\Gamma}}_{7})$ in the Hamiltonian used in the literature is shown to be incorrect. A linear $k$ term between the degenerate valence band, different from those obtained previously, is shown to exist. It modifies the dispersion and density of state in the vicinity of $\ensuremath{\Gamma}$ point but does not lift the Krammer's degeneracy. When quantum well, wires, and dots are considered, operator ordering in the remote interaction emerges naturally by treating wave vector as an operator acting on the envelope functions. This differs from previous schemes based on single-group formulation and a new term, arising from interfacial symmetry breaking, is identified in the valence-band Hamiltonian coupling the degenerate heavy-hole states.

A reformulation of a semiclassical theory that presently seems uniquely capable of interpreting generic complex multiresonant vibrational spectra is presented. Once given the spectroscopic Hamiltonian which reveals the set of possible resonant couplings and its eigenstates, the new and old formulations both yield without any further computation level by level dynamical assignments for the spectra. Computing a simple trajectory in phase space reveals the motions that when quantized yield the assigned levels. The reformulation introduces two new projected representations of the wave functions. The first is in action space and the second in angle space. The projected representations often allow the reduced angle space, where nodal searches are made, to be of lower dimension than formally occurred. In addition the action representation is a similarly lower dimension lattice representation whose discreteness and regularity allow higher reduced dimensions to be studied. The lattice representation is used to produce a significantly more complete and detailed assignment of the thiophosgene spectrum than previously published.

The theory of the concept lattice is an efficient tool for knowledge representation and knowledge discovery, and is applied to many fields successfully. One focus of knowledge discovery is knowledge reduction. This paper proposes the theory of attribute reduction in the concept lattice, which extends the theory of the concept lattice. In this paper, the judgment theorems of consistent sets are examined, and the discernibility matrix of a formal context is introduced, by which we present an approach to attribute reduction in the concept lattice. The characteristics of three types of attributes are analyzed.

Consider \(X\) as a complete separable metric space. This paper demonstrates that, for a typical probability measure, the general lower local dimension is consistently zero, while the general upper local dimension remains infinite. Additionally, the general lower local dimension is almost always zero, and given certain additional conditions on \(X\), a corresponding outcome is established for the general upper local dimension. Furthermore, we establish that the overall Hausdorff dimension of a typical measure on the space \(X\) is consistently zero within any compact space. Simultaneously, under specific additional conditions, we show that the general packing dimension surpasses a predefined value \(s > 0\). More precisely, we show that the local dimension of a typical measure fails to exist. Specifically, the behavior of a typical measure \(\nu\) is so extremely irregular that, for a fixed point \(x \in X\), the generalized local dimension function \[ r \mapsto \frac{\varphi ( \nu(B_r(x)) )}{\psi(r)}, \] of \(\nu\) at \(x\), with respect to the functions \(\varphi\) and \(\psi\), remains divergent as \(r \to 0\), even after being ''averaged'' or ''smoothed out'' by general and powerful averaging methods. These include, for example, higher-order Riesz-Hardy logarithmic averages and Cesàro averages.

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.

General fractal dimensions of typical sets and measures

We study planar two-dimensional quantum systems on a lattice whose Hamiltonian is a sum of local commuting projectors of bounded range. We consider whether or not such a system has a zero energy ground state. To do this, we consider the problem as a one-dimensional problem, grouping all sites along a column into “supersites”; using C ‐algebraic methods (Bravyi and Vyalyi [9]), we can solve this problem if we can characterize the central elements of the interaction algebra on these supersite. Unfortunately, these central elements may be very complex, making brute force impractical. Instead, we show a characterization of these elements in terms of matrix product operators with bounded bond dimension. This bound can be interpreted as a bound on the number of particle types in lattice theories with bounded Hilbert space dimension on each site. Topological order in this approach is related to the existence of certain central elements which cannot be “broken” into smaller pieces without creating an end excitation. Using this bound on bond dimension, we prove that several special cases of this problem are in NP, and we give part of a proof that the general case is in NP. Further, we characterize central elements that appear in certain specific models, including toric code and Levin‐Wen models, as either product operators in the Abelian case or matrix product operators with low bond dimension in the non-Abelian case; this matrix product operator representation may have practical application in engineering the complicated multi-spin interactions in the Levin‐Wen models. 81P16 The subject of Hamiltonian complexity theory is devoted to the study of the computational complexity of various problems in quantum many-body physics. A general framework for this problem is as follows. We consider a quantum system whose Hilbert space is the tensor product of N different Hilbert spaces. Each of these N Hilbert spaces is referred to as the Hilbert space of a “site”. We are interested in the case that each site has Hilbert space dimension that is poly.N/ (indeed, in many practical settings it is O.1/). The Hamiltonian will be a sum of at most poly.N/ terms, where each term in the Hamiltonian acting on at most O.1/ sites. Often, a locality condition is imposed on these terms in the Hamiltonian: there is some given graph and each term in the Hamiltonian only acts on a set of sites which has small diameter with respect to

Lattices are regular arrangements of points in space, whose study appeared in the 19th century in both number theory and crystallography. The goal of lattice reduction is to find useful representations of lattices. A major breakthrough in that field occurred twenty years ago, with the appearance of Lovasz’s reduction algorithm, also known as LLL or L3. Lattice reduction algorithms have since proved invaluable in many areas of mathematics and computer science, especially in algorithmic number theory and cryptology. In this paper, we survey some applications of lattices to cryptology. We focus on recent developments of lattice reduction both in cryptography and cryptanalysis, which followed seminal works of Ajtai and Coppersmith.

Forney has proposed an iterated construction called the squaring construction for simplified derivation and representation of the Barnes-Wall lattices. He used as a starting partition chain the two-dimensional infinite two-way partition...Z/sup 2// Z/sup 2// RZ/sup 2// 2Z/sup 2// 2RZ/sup 2/...with minimum Euclidean distances 1/1/1/2/4/8/..., where R is a two-dimensional rotation operator. We apply this construction to the one-dimensional infinite two-way partition...Z/Z/2Z/4Z/8Z...with minimum distance...1/1/2/4/8/...which has clearly the same properties as the previous partition. The resulting lattices of dimension N=2/sup n/ for the l/sub 1/-distance can therefore be regarded as the duals of the Barnes-Wall lattices of dimension 2N for the Euclidean distance. Since the 2-depth of each of these lattices is equal to n they necessarily contain the 2/sup n/Z/sup N/ lattice. The coset representatives of these lattices in /spl nu/2/sup n/Z/sup N/, where /spl nu/ is an arbitrary nonnegative integer, are good codes for the Lee distance since they outperform the negacyclic codes in low dimensions. Maximum likelihood (ML) soft detection can be performed easily on these lattices and codes since they have a simple trellis structure.

Register synthesis for multi-sequences has significance for the security of word-oriented stream ciphers. Feedback with carry shift registers (FCSRs) are promising alternatives to linear feedback shift registers for the design of stream ciphers. In this paper, we solve the FCSR synthesis problem for multi-sequences by two rational approximation algorithms using lattice theory. One is based on the lattice reduction greedy algorithm proposed by Nguyen and Stehle (ACM Trans Algorithms (TALG) 5(4):46, 2009). The other is based on the LLL algorithm which is a polynomial time lattice reduction algorithm. Both of these rational approximation algorithms can find the smallest common FCSR for a given multi-sequence but with different numbers of known terms. When the number of sequences within the multi-sequence is less than or equal to 3, the former is suggested because it has better time complexity and fewer terms are needed. Otherwise, the latter will have better time complexity.

Recently, lattice theory has been widely used for integer ambiguity resolution in the Global Navigation Satellite System (GNSS). When using lattice theory to deal with integer ambiguity, we need to reduce the correlation between lattice bases to ensure the efficiency of the solution. Lattice reduction is divided into scale reduction and basis vector exchange. The scale reduction has no direct impact on the subsequent search efficiency, while the basis vector exchange directly impacts the search efficiency. Hence, Lenstra-Lenstra-Lovász (LLL) is applied in the ambiguity resolution to improve the efficiency. And based on Householder transformation, the HLLL improved method is also used. Moreover, to improve the calculation speed further, a Pivoting Householder LLL (PHLLL) method based on Householder orthogonal transformation and rotation sorting is proposed here. The idea of PHLLL method is as follows: First, a sort matrix is introduced into the lattice basis reduction process to sort the original matrix. Then, the sorted matrix is used for Householder transformation. After transformation, it needs to be sorted again, until the diagonal elements in the matrix meet the ascending order. In addition, when using the Householder image operator for orthogonalization, the old column norm is modified to obtain a new norm, reducing the number of column norm calculations. Compared with the LLL reduction algorithm and HLLL reduction algorithm, the experimental results show that the PHLLL algorithm has higher reduction efficiency and effectiveness. The theoretical superiority of the algorithm is proved.

We discuss recent work on the development and analysis of low-concentration series. For many models, the recent breakthrough in the extremely efficient no- free-end method of series generation facilitates the derivation of 15th-order series for multiple moments in general dimension. The 15th-order series have been obtained for lattice animals, percolation, and the Edwards-Anderson Ising spin glass. In the latter cases multiple moments have been found. From complete graph tables through to 13th order, general dimension 13th-order series have been derived for the resistive susceptibility, the moments of the logarithms of the distribution of currents in resistor networks, and the average transmission coefficient in the quantum percolation problem, 11th-order series have been found for several other systems, including the crossover from animals to percolation, the full resistance distribution, nonlinear resistive susceptibility and current distribution in dilute resistor networks, diffusion on percolation clusters, the dilute Ising model, dilute antiferromagnet in a field, and random field Ising model and self-avoiding walks on percolation clusters. Series for the dilute spin-1/2 quantum Heisenberg ferromagnet are in the process of development. Analysis of these series gives estimates for critical thresholds, amplitude ratios, and critical exponents for all dimensions. Where comparisons are possible, our series results are in good agreement with bothe-expansion results near the upper critical dimension and with exact results (when available) in low dimensions, and are competitive with other numerical approaches in intermediate realistic dimensions.

This note is intended as an introduction to the functorial formulation of quantum field theories with defects.After some remarks about models in general dimension, we restrict ourselves to two dimensions -the lowest dimension in which interesting field theories with defects exist.We study in some detail the simplest example of such a model, namely a topological field theory with defects which we describe via lattice TFT.Finally, we give an application in algebra, where the defect TFT provides us with a functorial definition of the centre of an algebra.This involves changing the target category of commutative algebras into a bicategory.Throughout this paper, we emphasise the role of higher categories -in our case bicategories -in the description of field theories with defects.