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Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three‐dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. A review is presented of the current best practice using Delaunay‐reduced unit cells in the six‐dimensional real space of Selling scalar cells S6 and the equivalent three‐dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six‐dimensional real space of Niggli‐reduced unit cells G6. Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths with boundary‐interrupted paths, looking for a path of minimal length. [ABSTRACT FROM AUTHOR]

We consider two sequences of orthogonal polynomials (P n) n ≥ 0 and (Q n) n ≥ 0 such that ∑ j = 1 M a j , n S x D x k P k + n − j (z) = ∑ j = 1 N b j , n D x m Q m + n − j (z) , with k , m , M , N ∈ N , a j , n and b j , n are sequences of complex numbers, 2 S x f (x (s)) = (△ + 2 I) f (z) , D x f (x (s)) = △ △ x (s − 1 / 2) f (z) , z = x (s − 1 / 2) , I is the identity operator, x defines a lattice, and △ f (s) = f (s + 1) − f (s). We show that under some natural conditions, both involved orthogonal polynomials sequences (P n) n ≥ 0 and (Q n) n ≥ 0 are semiclassical whenever k = m. Some particular cases are studied closely where we characterize the continuous dual Hahn and Wilson polynomials for quadratic lattices. [ABSTRACT FROM AUTHOR]

Mathematical approaches to structure model problems have a significant role in expanding our knowledge in our routine life circumstances. To put them into practice, the right formulation, method, systematic representation, and formulation are needed. The purpose of introducing soft graphs is to discretize these fundamental mathematical ideas, which are inherently continuous, and to provide new tools for applying mathematical analysis technology to real-world applications including imperfect and inexact data or uncertainty. Soft rough covering models briefly; SRC-Models), a novel theory that addresses uncertainty. In this present paper, we have introduced two new concepts Li-soft rough covering graphs (Li-SRCGs) and the concept of fixed point of such graphs. Furthermore, we looked into a some algebras that dealt with the fixed points of Li-SRCGs. Applications of the algebraic structures available in covering soft sets to soft graphs may reveal new facets of graph theory. [ABSTRACT FROM AUTHOR]

A dense forest is a set F ⊂ R n with the property that for all ε > 0 there exists a number V (ε) > 0 such that all line segments of length V (ε) are ε -close to a point in F. The function V is called a visibility function of F. In this paper we study dense forests constructed from finite unions of translated lattices (grids). First, we provide a necessary and sufficient condition for a finite union of grids to be a dense forest in terms of the irrationality properties of the matrices defining them. This answers a question raised by Adiceam, Solomon, and Weiss (J Lond Math Soc 105: 1167–1199, 2022). To complement this, we further show that such sets generically admit effective visibility bounds in the following sense: for all η > 0 , there exists a k ∈ N such that almost all unions of k grids are dense forests admitting a visibility function V (ε) ≪ ε - (n - 1) - η . This is arbitrarily close to optimal in the sense that if a finite union of grids admits a visibility function V, then this function necessarily satisfies V (ε) ≫ ε - (n - 1) . One of the main novelties of this work is that the notion of ‘almost all’ is considered with respect to several underlying measures, which are defined according to the Iwasawa decomposition of the matrices used to define the grids. In this respect, the results obtained here vastly extend those of Adiceam, Solomon, and Weiss (2022) who provided similar effective visibility bounds for a particular family of generic unimodular lattices. [ABSTRACT FROM AUTHOR]

Let K and K ′ be arithmetically equivalent number fields, both of degree d. We prove that K and K ′ have the same successive minima, up to a constant depending only on d. We give examples showing that one cannot expect equality. [ABSTRACT FROM AUTHOR]

We present a polynomial time algorithm for breaking NTRU encryption schemes with multiple keys. Our algorithm takes advantage of the specific sampling regime used in NTRU encryption, which samples secret polynomials with a fixed number of coefficients of 1 and - 1 . By constructing an equation system on the secret keys, we are able to recover the unique secret key when n multiple keys sharing a common denominator are given for an extension degree n. This result shows that NTRU encryption schemes with multiple keys can be solved in polynomial time in n. [ABSTRACT FROM AUTHOR]

The problem of factoring a composite integer into the product of two distinct primes (the factoring problem) is one of the famous hard problems on which the security of many cryptographic primitives relies. In this paper, we introduce a new cryptographic hard problem (RSA-polynomial problem) and prove that solving the RSA-polynomial problem is at least as hard as solving the factoring problem. As applications of the RSA-polynomial problem, we propose a commitment scheme. The proposed scheme is free of any group-exponentiation and outperforms the previous commitment schemes. Also, using the lattice basis reduction techniques and the RSA-polynomial problem, we propose a method to factor composite integers that are the product of two distinct primes. [ABSTRACT FROM AUTHOR]

In this work, we define Rad− ⊕ −supplemented an d strongly Rad− ⊕ −supplemented lattices an d give some properties of these lattices. We generalize some properties of Rad− ⊕ −supplemented modules to lattices. Let L be a lattice an d 1 = a1 ⊕a2 ⊕. . .⊕an with a1, a2, . . ., an ∈ L. If ai /0 is Rad− ⊕ − supplemented for every i = 1, 2, . . ., n, then L is also Rad− ⊕ − supple- mented. Let L be a distributive Rad−⊕−supplemented lattice. Then 1/u is Rad−⊕−supplemented for every u ∈ L. We also define completely Rad− ⊕ −supplemented lattices an d prove that every Rad− ⊕ −supplemented lattice with SSP property is completely Rad− ⊕ − supplemented. [ABSTRACT FROM AUTHOR]

Let A be a d × d matrix with rational entries which has no eigenvalue λ ∈ C of absolute value | λ | < 1 and let Z d [ A ] be the smallest nontrivial A-invariant Z -module. We lay down a theoretical framework for the construction of digit systems (A , D) , where D ⊂ Z d [ A ] finite, that admit finite expansions of the form x = d 0 + A d 1 + ⋯ + A ℓ − 1 d ℓ − 1 (ℓ ∈ N , d 0 , ... , d ℓ − 1 ∈ D) for every element x ∈ Z d [ A ]. We put special emphasis on the explicit computation of small digit sets D that admit this property for a given matrix A, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way. [ABSTRACT FROM AUTHOR]

It is well known that a line can intersect at most 2n - 1 cells of the n x n chessboard. Here we consider the high-dimensional version: how many cells of the d-dimensional n x . . . x n box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact: if K; L are convex bodies in Rd and K ⊂ L, then the surface area of K is smaller than that of L. [ABSTRACT FROM AUTHOR]

Aspects are opinion targets to extract in aspect-based sentiment analysis. While existing methods can already produce satisfactory extraction results, they suffer when faced with compound aspect terms, typically phrase-level aspect terms that have inner structure and occur infrequently in the training set. This issue can be mainly attributed to the scarcity of training examples targeting compound aspect terms and by the neglect of the syntactic structure of a sentence in the modeling process. In this article, we aim to cope with compound aspect extraction by a two-stage hybrid approach. First, we introduce a conditional generation method for data augmentation in a masked sequence-to-sequence framework, which is controllable to preserve original aspects while generating a new sentence. Second, we propose a constituency lattice structure that is induced from the constituency-based parse tree of a sentence. Experimental results on two review datasets show that this approach can greatly improve the effect of compound aspect extraction. [ABSTRACT FROM AUTHOR]

This work presents the impact of barrier spacer on the structural and optical properties of strain-coupled Stranski-Krastanov (SK) on Sub Monolayer (SML) quantum dot (QD) heterostructures. Various ternary and quaternary materials have been employed as the barrier layer of SK-SML QD heterostructures. In the coupled SK-SML QDs, the residual strain propagates from the SML seed layer towards the top SK dots, introducing defects and dislocations. After employing the ternary (GaAs1-ySby) and quaternary (InxAl0.21Ga1-.0.21-xAs and In0.18Ga0.82As1-ySby) materials, the residual strain reduces, reducing the defects which thereby helps in increasing the crystalline quality of the heterostructure. Nextnano software has been used to compute the structures' strain, energy band profile, probability density functions, and emission wavelength. Two strain components, viz. hydrostatic as well as biaxial strain, have been computed and compared for all the heterostructures to understand the distribution of the strain profile. The emission wavelength is red-shifted for the SK-SML QD heterostructures with ternary and quaternary materials as a barrier layer as compared to that of the GaAs barrier. Moreover, type-I and II energy band profiles are observed for Sb-based barrier material, appropriate for various optoelectronic applications. This is a comparative study of SK-SML QDs with various barrier materials helps minimize the strain and defects and improve the device performance. [ABSTRACT FROM AUTHOR]

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a t-(n,k,λ) combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a t-(n,k,λ) design exist but, in fact, with positive probability there exists a large set of such designs — that is, a partition of the set of k-subsets of [n] into t-(n,k,λ) designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large n, large sets of t-(n,k,λ) designs exist whenever k>12t and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all k>12t.

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been introduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a t-(n,k,λ) combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a t-(n,k,λ) design exist but, in fact, with positive probability there exists a large set of such designs — that is, a partition of the set of k-subsets of [n] into t-(n,k,λ) designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large n, large sets of t-(n,k,λ) designs exist whenever k>12t and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all k>12t.

One of the significant post-quantum cryptographic candidates is the NTRU public key cryptosystem. It operates on polynomial rings, where the parameter largely determines the security of the system. Although NTRU is being studied currently, it has a long and well-established history. There are several lattice-based attacks on NTRU-like systems that exploit the special structures of the rings used in these systems. The aim of this paper is to analyze the original NTRU, NTRU Encrypt, and NTRU Primes encryption schemes by structuring their common elements and showing the strongest hybrid attack using both lattice reduction and meet-in-the-middle (MITM) search on them. Furthermore, it is noted that, ignoring a polynomial factor of the not-well-studied cost of Block Korkin-Zolotarev (BKZ) algorithm, we estimate the security of the construction of encryption keys and show that by balancing lattice reduction costs and a MITM search cost, one can achieve better performance than using any of these methods on their own. Unlike previous studies, we found the way to ignore polynomial impact 22-24 from BKZ loops with multiple shortest vector problem (SVP) and the factor of 27 was omitted from the cost of one step in guessing the SVP.

The osseointegration in/around additively manufactured (AM) lattice structures of a new titanium alloy, Ti–19Nb–14Zr, was evaluated. Different lattices with increasingly high sidewalls gradually closing them were manufactured and implanted in sheep. After removal, the bone–interface implant (BII) and bone–implant contact (BIC) were studied from 3D X-ray computed tomography images. Measured BII of less than 10 µm and BIC of 95% are evidence of excellent osseointegration. Since AM naturally leads to a high-roughness surface finish, the wettability of the implant is increased. The new alloy possesses an increased affinity to the bone. The lattice provides crevices in which the biological tissue can jump in and cling. The combination of these factors is pushing ossification beyond its natural limits. Therefore, the quality and speed of the ossification and osseointegration in/around these Ti–19Nb–14Zr laterally closed lattice implants open the possibility of bone spline key of prostheses. This enables the stabilization of the implant into the bone while keeping the possibility of punctual hooks allowing the implant to be removed more easily if required. Thus, this new titanium alloy and such laterally closed lattice structures are appropriate candidates to be implemented in a new generation of implants. [ABSTRACT FROM AUTHOR]

In a recent work (Wang et al. 2020), a partial order ⪯, a join operation ⊔ and a meet operation ⊓ of probabilistic linguistic term sets (PLTSs) were introduced and it was proved that L1 ⊓ L2 ⪯ L1 ⪯ L1 ⊔ L2 and L1 ⊓ L2 ⪯ L2 ⪯ L1 ⊔ L2. In this paper, we demonstrate that its join and meet operations are not satisfy the above requirement. To satisfy this requirement, we modify its join and meet operations. Moreover, we define a negation operation of PLTSs based on the partial order ⪯. The combinations of the proposed negation, the modified join and meet operations yield a bounded, distributive lattice over PLTSs. Meanwhile, we also define a new join operation and a new meet operation which, together with the negation operation, yield a bounded De Morgan over PLTSs. [ABSTRACT FROM AUTHOR]

Cellular materials, in particular metal or polymer foams and beam lattices, are characterized by a complex architecture where the material is concentrated in slender struts. This study is concerned with the construction of numerically efficient sample‐specific digital twin for the analysis and characterization of the mechanical behavior of such meta‐materials. A method to build an explicit, light, and analysis‐suitable isogeometric beam model directly from digital images is proposed. More precisely, as a main contribution, the virtual image correlation method is extended to the case of multiple interconnected beams with varying cross‐sections. Special care is also given to the initialization for better efficiency and robustness. Synthetic and real examples of increasing complexity, that is, curved beams, lattices, and foams, are presented to account for the performance of the developed workflow from digital images to an isogeometric twin for mechanical modeling and simulation. [ABSTRACT FROM AUTHOR]

To address the issues of difficult sharing, data tampering, and privacy leakage in traditional electronic medical record storage schemes, a blockchain-based electronic medical record searchable encrypted storage, efficient sharing, and verifiable scheme has been constructed. A user identity authentication system was built using zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) to enhance user identity privacy and anonymity. A searchable encryption and verification scheme for electronic medical records was achieved by proposing a lattice-based searchable encryption scheme based on an improved Merkle tree. A dual verification search scheme was constructed through smart contracts based on this scheme. The corresponding encrypted data was outsourced to the interplanetary file system (IPFS) to reduce storage costs. Simulation tests were conducted on the performance of encryption algorithms, smart contract time and cost, and off-chain storage performance. The system has been shown to meet the application requirements of electronic medical record systems and improve data integrity, search efficiency, and identity confidentiality. [ABSTRACT FROM AUTHOR]

Let K = Q (ζ 8) be the complex multiplication field over Q of extension degree 4. We give an integral lattice construction on Q (ζ 8) induced from binary codes. We define a theta series using these lattices and discuss its relation with the complete weight enumerator of a binary code. If C is a binary Type II code of length l, we find that the complete weight enumerator of C gives a Jacobi form of weight l and the index 2l over the maximal totally real subfield k = Q (ζ 8 + ζ 8 - 1) of K. Also, we see that Hilbert-Siegel modular form of weight l and genus g can be seen in terms of the complete joint weight enumerator for codes C j , 1 ≤ j ≤ g over F 2 . [ABSTRACT FROM AUTHOR]

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical (12 , 6) Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated. [ABSTRACT FROM AUTHOR]

Steinberg's fixed point theorem states that given a finite complex reflection group the stabilizer subgroup of a point is generated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus classify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem. [ABSTRACT FROM AUTHOR]

We establish conditions under which lattices in certain simple Lie groups are profinitely solitary in the absolute sense, so that the commensurability class of the profinite completion determines the commensurability class of the group among finitely generated residually finite groups. While cocompact lattices are typically not absolutely solitary, we show that noncocompact lattices in \operatorname {Sp}(n,\mathbb {R}), G_{2(2)}, E_8(\mathbb {C}), F_4(\mathbb {C}), and G_2(\mathbb {C}) are absolutely solitary if a well-known conjecture on Grothendieck rigidity is true. [ABSTRACT FROM AUTHOR]

Information-theoretic quantities reveal dependencies among variables in the structure of joint, marginal, and conditional entropies while leaving certain fundamentally different systems indistinguishable. Furthermore, there is no consensus on the correct higher-order generalisation of mutual information (MI). In this manuscript, we show that a recently proposed model-free definition of higher-order interactions among binary variables (MFIs), such as mutual information, is a Möbius inversion on a Boolean algebra, except of surprisal instead of entropy. This provides an information-theoretic interpretation to the MFIs, and by extension to Ising interactions. We study the objects dual to mutual information and the MFIs on the order-reversed lattices. We find that dual MI is related to the previously studied differential mutual information, while dual interactions are interactions with respect to a different background state. Unlike (dual) mutual information, interactions and their duals uniquely identify all six 2-input logic gates, the dy- and triadic distributions, and different causal dynamics that are identical in terms of their Shannon information content. [ABSTRACT FROM AUTHOR]

A CM-order is a reduced order equipped with an involution that mimics complex conjugation. The Witt-Picard group of such an order is a certain group of ideal classes that is closely related to the "minus part" of the class group. We present a deterministic polynomial-time algorithm for the following problem, which may be viewed as a special case of the principal ideal testing problem: given a CM-order, decide whether two given elements of its Witt - Picard group are equal. In order to prevent coefficient blow-up, the algorithm operates with lattices rather than with ideals. An important ingredient is a technique introduced by Gentry and Szydlo in a cryptographic context. Our application of it to lattices over CM-orders hinges upon a novel existence theorem for auxiliary ideals, which we deduce from a result of Konyagin and Pomerance in elementary number theory.

As quantum computing advances rapidly, guaranteeing the security of cryptographic protocols resistant to quantum attacks is paramount. Some leading candidate cryptosystems use the learning with errors (LWE) problem, attractive for its simplicity and hardness guaranteed by reductions from hard computational lattice problems. Its algebraic variants, ring-learning with errors (RLWE) and polynomial learning with errors (PLWE), gain efficiency over standard LWE, but their security remains to be thoroughly investigated. In this work, we consider the “smearing” condition, a condition for attacks on PLWE and RLWE introduced in Elias et al. We expand upon some questions about smearing posed by Elias et al. and show how smearing is related to the coupon collector’s problem. Furthermore, we develop an algorithm for computing probabilities related to smearing. Finally, we present a smearing-based algorithm for solving the PLWE problem.

Additive manufacturing is catalyzing a new class of volumetrically varying lattice structures in which the dynamic mechanical response can be tailored for a specific application. Simultaneously, a diversity of materials is now available as feedstock including elastomers, which provide high viscoelasticity and increased durability. The combined benefits of complex lattices coupled with elastomers is particularly appealing for anatomy-specific wearable applications such as in athletic or safety equipment. In this study, Siemens' DARPA TRADES-funded design and geometry-generation software, Mithril, was leveraged to design vertically-graded and uniform lattices, the configurations of which offer varying degrees of stiffness. The designed lattices were fabricated in two elastomers using different additive manufacturing processes: (a) vat photopolymerization (with compliant SIL30 elastomer from Carbon) and (b) thermoplastic material extrusion (with Ultimaker™ TPU filament providing increased stiffness). Both materials provided unique benefits with the SIL30 material offering compliance suitable for lower energy impacts and the Ultimaker™ TPU offering improved protection against higher impact energies. Moreover, a hybrid lattice combination of both materials was evaluated and demonstrated the simultaneous benefits of each, with good performance across a wider range of impact energies. This study explores the design, material, and process space for manufacturing a new class of comfortable, energy-absorbing protective equipment to protect athletes, consumers, soldiers, first responders, and packaged goods. [ABSTRACT FROM AUTHOR]

In this paper, we propose a partial ordering ⪯ on the set of ordered weighted averaging (OWA) operators. Based on this relation ⪯, we introduce the negation, conjunction and disjunction operations, and establish a bounded De Morgan lattice equipped with an involutive negation for OWA operators. Finally, we develop a similarity measure between OWA operators based on the ordering ⪯. [ABSTRACT FROM AUTHOR]

The main idea behind lattice sieving algorithms is to reduce a sufficiently large number of lattice vectors with each other so that a set of short enough vectors is obtained. It is therefore natural to study vectors which cannot be reduced. In this work we give a concrete definition of an irreducible vector and study the properties of the set of all such vectors. We show that the set of irreducible vectors is a subset of the set of Voronoi relevant vectors and study its properties. For extremal lattices this set may contain as many as 2 n vectors, which leads us to define the notion of a complete system of irreducible vectors, whose size can be upper-bounded by the kissing number. One of our main results shows that modified heuristic sieving algorithms heuristically approximate such a set (modulo sign). We provide experiments in low dimensions which support this theory. Finally we give some applications of this set in the study of lattice problems such as SVP, SIVP and CVPP. The introduced notions, as well as various results derived along the way, may provide further insights into lattice algorithms and motivate new research into understanding these algorithms better. [ABSTRACT FROM AUTHOR]

In this paper, we develop and formalize a set of tools to efficiently compute inner-products homomorphically over the ring Z p for small modulus p. This allows us to use smaller modulus size in various cryptographic primitives based on lattices, which is desirable both from a security and efficiency points of view. Our approach is to directly express the computation of inner-products as a special form of branching program and then homomorphically evaluate them. This is in contrast to previous works that first invoked the Barrington's theorem to convert the circuit computing the inner-product indirectly into a branching program. The direct computation of branching programs substantially improves the efficiency compared to previous methods since the invocation of the Barrington's theorem incurred a significant efficiency loss. We obtain the following concrete applications as a result of our technique. (1) We propose new attribute-based encryption (ABE ) schemes with improved efficiency for several useful predicates in NC 1 . While previous approaches typically require either a super-polynomial modulus or invocation of the Barrington's theorem, we manage to make the modulus size polynomially small without relying on any of these. Consequently, we obtain more efficient and secure schemes. (2) We propose a new tightly secure identity-based encryption (IBE ) scheme from lattices with small polynomial modulus. Compared with the construction by Boyen and Li (Asiacrypt 404–434, Springer, Heidelberg, 2016, 10.1007/978-3-662-53890-6_14), our scheme achieves much better efficiency, albeit assuming the security of a special type of pseudorandom function (PRF ) by Boneh et al. (TCC 535–554, Springer, Heidelberg, 2018, 10.1007/978-3-540-70936-7_29). [ABSTRACT FROM AUTHOR]

In this paper, we classify Eulerian posets of small rank which are pleasant and further show that there is no pleasant Eulerian poset of rank less than are equal to 7. [ABSTRACT FROM AUTHOR]

The Riesz potential fs(r)=r−s$f_s(r)=r^{-s}$ is known to be an important building block of many interactions, including Lennard‐Jones–type potentials fn,mLJ(r):=ar−n−br−m$f_{n,m}^{\rm {LJ}}(r):=a r^{-n}-b r^{-m}$, n>m$n>m$ that are widely used in molecular simulations. In this paper, we investigate analytically and numerically the minimizers among three‐dimensional lattices of Riesz and Lennard‐Jones energies. We discuss the minimality of the body‐centered‐cubic (BCC) lattice, face‐centered‐cubic (FCC) lattice, simple hexagonal (SH) lattices, and hexagonal close‐packing (HCP) structure, globally and at fixed density. In the Riesz case, new evidence of the global minimality at fixed density of the BCC lattice is shown for s<0$s<0$ and the HCP lattice is computed to have higher energy than the FCC (for s>3/2$s>3/2$) and BCC (for s<3/2$s<3/2$) lattices. In the Lennard‐Jones case with exponents 3

Given a lattice L ⊆ Z m and a subset A ⊆ R m , we say that a point in A is lonely if it is not equivalent modulo L to another point of A. We are interested in identifying lonely points for specific choices of L when A is a dilated standard simplex, and in conditions on L which ensure that the number of lonely points is unbounded as the simplex dilation goes to infinity. [ABSTRACT FROM AUTHOR]

Several cryptosystems based on the Ring Learning with Errors (RLWE) problem have been proposed within the NIST post-quantum cryptography standardization process, e.g., NewHope. Furthermore, there are systems like Kyber which are based on the closely related MLWE assumption. Both previously mentioned schemes result in a non-zero decryption failure rate (DFR). The combination of encryption and decryption for these kinds of algorithms can be interpreted as data transmission over a noisy channel. To the best of our knowledge this paper is the first work that analyzes the capacity of this channel. We show how to modify the encryption schemes such that the input alphabets of the corresponding channels are increased. In particular, we present lower bounds on their capacities which show that the transmission rate can be significantly increased compared to standard proposals in the literature. Furthermore, under the common assumption of stochastically independent coefficient failures, we give lower bounds on achievable rates based on both the Gilbert-Varshamov bound and concrete code constructions using BCH codes. By means of our constructions, we can either increase the total bitrate (by a factor of 1.84 for Kyber and by factor of 7 for NewHope) while guaranteeing the same DFR or for the same bitrate, we can significantly reduce the DFR for all schemes considered in this work (e.g., for NewHope from 2−216 to 2−12769). [ABSTRACT FROM AUTHOR]

The past 50 years witnessed the inventions of more than one thousand power electronic topologies. However, we still lack a common standard to compare and classify various topologies. Moreover, topologies are often created through intuition rather than systematic approaches. The intuition of voltage and current sharing, along with modularity and scalability, gives rise to cascaded-bridge and multi-parallleled converters, respectively. Nevertheless, power converters with both high-voltage and large-current capabilities are still obscure yet important. This article first investigates the symmetries of circuit topologies through group theory. Notably, symmetries lie in twofold—individual modules and connections. Symmetries not only help to classify topologies but also play an important role in the creation of new topologies. Based on wallpaper groups, this article proposes the lattice converters, which readily scale to manage high voltages and large currents. The proposed lattice converters benefit from 1) modularity and scalability, 2) boundless current and voltage ratings, 3) multiple input/output ports, 4) sensorless voltage balancing, 5) high power quality, and 6) flexible control. Simulation and experimental results demonstrate the effectiveness and superiority of the proposed theory and lattice converters. [ABSTRACT FROM AUTHOR]

In real life, data are often of poor quality as a result, for instance, of uncertainty, mismeasurements, missing values or bad inputs. This issue hampers an implicit yet crucial operation of every database management system: equality testing. Indeed, equality is, in the end, a context-dependent operation with a plethora of interpretations. In practice, the treatment of different types of equality is left to programmers, who have to struggle with those interpretations in their code. We introduce a lattice-based declarative framework to address this problem. It allows specification of numerous semantics for equality at a high level of abstraction. To go beyond tuple equality, we study functional dependencies (FDs) in the light of our framework. First, we define abstract FDs, generalizing classical FDs. These lead to the consideration of particular interpretations of equality: realities. Building upon realities and possible/certain answers, we introduce possible/certain FDs and give some complexity results. [ABSTRACT FROM AUTHOR]