Lattice Operators Research Articles

Holomorphic symplectic fermions

Let V be the even part of the vertex operator super-algebra of r pairs of symplectic fermions. Up to two conjectures, we show that V admits a unique holomorphic extension if r is a multiple of 8, and no holomorphic extension otherwise. This is implied by two results obtained in this paper: (1) If r is a multiple of 8, one possible holomorphic extension is given by the lattice vertex operator algebra for the even self dual lattice $$D_{r}^+$$ with shifted stress tensor. (2) We classify Lagrangian algebras in $$\mathcal {S}\mathcal {F}(\mathfrak {h})$$ , a ribbon category associated to symplectic fermions. The classification of holomorphic extensions of V follows from (1) and (2) if one assumes that $$\mathcal {S}\mathcal {F}(\mathfrak {h})$$ is ribbon equivalent to $${\mathrm {Rep}}(V)$$ , and that simple modules of extensions of V are in one-to-one relation with simple local modules of the corresponding commutative algebra in $$\mathcal {S}\mathcal {F}(\mathfrak {h})$$ .

United lattice fractional integro-differentiation

Abstract New fractional operators of non-integer and integer orders for N-dimensional lattice ℤ N are suggested. The proposed lattice fractional integro-differentiation of positive order can be considered as a lattice analog of partial derivatives of non-integer orders. For negative values of order, it can be considered as a lattice fractional integration. For integer orders, the continuum limit of the united lattice derivatives gives the usual partial derivatives of integer orders. In contrast to the usual lattice derivatives, which are proposed in our recent paper (“Lattice fractional calculus”, Appl. Math. Comp. 257 (2015), 12–33), the suggested lattice operators give the usual partial derivatives for all integer orders. The united lattice integro-differentiation is represented by finite differences with infinite series that describe long-range lattice interactions. The suggested lattice operators of positive orders can be considered as an exact discretization of derivatives of corresponding orders. The basic mathematical feature of these united lattice operators is an implementation of the same algebraic properties that have the differentiation operator.

On Algebraic Properties and Linearity of Owa Operators for Fuzzy Sets

Abstract We deal with an ordered weighted averaging operator (OWA operator) on the set of all fuzzy sets. Our starting point is OWA operator on any lattice introduced in Lizasoain, I.-Moreno,C.: OWA operators defined on complete lattices, Fuzzy Sets and Systems 224 (2013), 36-52; Ochoa, G.-Lizasoain, I.- -Paternain, D.-Bustince, H.-Pal, N. R.: Some properties of lattice OWA operators and their importance in image processing, in: Proc. of the 16th World Congress of the Internat. Systems Assoc.-IFSA ’15 and the 9th Conf. of the European Soc. for Fuzzy Logic and Technology-EUSFLAT ’15 (J. M. Alonso et al., eds.), Atlantis Press, Gijón, Spain, 2015, pp. 1261-1265. We focus on a particular case of lattice, namely that of all normal convex fuzzy sets in [0,1], and study algebraic properties and linearity of the proposed OWA operator. It is shown that the operator is an extension of standard OWA operator for real numbers and it possesses similar algebraic properties as standard one, however, it is neither homogeneous nor shift-invariant, i.e., it is not linear in contrast to the standard OWA operator.

Classification of the vertex operator algebras [formula omitted] of class [formula omitted

In this paper, we prove that an even rootless lattice L is isomorphic to 2A1, 2D4, 2E8, or BW16 if the lattice type vertex operator algebra VL+ is of class S4. In addition, we prove that the vertex operator algebra V2D4+ is of class S5, and the vertex operator algebras V2A1+, V2E8+, and VBW16+ are of class S7.

Kaon BSMB-parameters using improved staggered fermions fromNf=2+1unquenched QCD

We present results for the matrix elements of the additional $\Delta S=2$ operators that appear in models of physics beyond the Standard Model (BSM), expressed in terms of four BSM $B$-parameters. Combined with experimental results for $\Delta M_K$ and $\epsilon_K$, these constrain the parameters of BSM models. We use improved staggered fermions, with valence HYP-smeared quarks and $N_f=2+1$ flavors of "asqtad" sea quarks. The configurations have been generated by the MILC collaboration. The matching between lattice and continuum four-fermion operators and bilinears is done perturbatively at one-loop order. We use three lattice spacings for the continuum extrapolation: $a\approx 0.09$, $0.06$ and $0.045\;$fm. Valence light-quark masses range down to $\approx m_s^{\rm phys}/13$ while the light sea-quark masses range down to $\approx m_s^{\rm phys}/20$. Compared to our previous published work, we have added four additional lattice ensembles, leading to better controlled extrapolations in the lattice spacing and sea-quark masses. We report final results for two renormalization scales, $\mu=2\;\text{GeV}$ and $3\;\text{GeV}$, and compare them to those obtained by other collaborations. Agreement is found for two of the four BSM $B$-parameters ($B_2$ and $B_3^\text{SUSY}$). The other two ($B_4$ and $B_5$) differ significantly from those obtained using RI-MOM renormalization as an intermediate scheme, but are in agreement with recent preliminary results obtained by the RBC-UKQCD collaboration using RI-SMOM intermediate schemes.

Perturbative renormalization of neutron-antineutron operators

Two-loop anomalous dimensions and one-loop renormalization scheme matching factors are calculated for six-quark operators responsible for neutron-antineutron transitions. When combined with lattice QCD determinations of the matrix elements of these operators, our results can be used to reliably predict the neutron-antineutron vacuum transition time, $\tau_{n\bar{n}}$, in terms of basic parameters of baryon-number violating beyond-the-Standard-Model theories. The operators are classified by their chiral transformation properties, and a basis in which there is no operator mixing due to strong interactions is identified. Operator projectors that are required for non-perturbative renormalization of the corresponding lattice QCD six-quark operator matrix elements are constructed. A complete calculation of $\delta m = 1/\tau_{n\bar{n}}$ in a particular beyond-the-Standard-Model theory is presented as an example to demonstrate how operator renormalization and results from lattice QCD are combined with experimental bounds on $\delta m$ to constrain the scale of new baryon-number violating physics. At the present computationally accessible lattice QCD matching scale of $\sim$ 2 GeV, the next-to-next-to-leading-order effects calculated in this work correct the leading-order plus next-to-leading-order $\delta m$ predictions of beyond-the-Standard-Model theories by $< 26\%$. Next-to-next-to-next-to-leading-order effects provide additional unknown corrections to predictions of $\delta m$ that are estimated to be $< 7\%$.

2-cyclic permutations of lattice vertex operator algebras

The irreducible modules of the 2-cyclic permutation orbifold models of lattice vertex operator algebras of rank one are classified, and the quantum dimensions of irreducible modules and the fusion rules are determined.

Quantitative orness for lattice OWA operators

This paper deals with OWA (ordered weighted average) operators defined on any complete lattice endowed with a t-norm and a t-conorm and satisfying a certain finiteness local condition. A parametrization of these operators is suggested by introducing a quantitative orness measure for each OWA operator, based on its proximity to the OR operator. The meaning of this measure is analyzed for some concrete OWA operators used in color image reduction, as well as for some OWA operators used in a medical decision making process.

Boundary algebras and Kac modules for logarithmic minimal models

Virasoro Kac modules were originally introduced indirectly as representations whose characters arise in the continuum scaling limits of certain transfer matrices in logarithmic minimal models, described using Temperley–Lieb algebras. The lattice transfer operators include seams on the boundary that use Wenzl–Jones projectors. If the projectors are singular, the original prescription is to select a subspace of the Temperley–Lieb modules on which the action of the transfer operators is non-singular. However, this prescription does not, in general, yield representations of the Temperley–Lieb algebras and the Virasoro Kac modules have remained largely unidentified. Here, we introduce the appropriate algebraic framework for the lattice analysis as a quotient of the one-boundary Temperley–Lieb algebra. The corresponding standard modules are introduced and examined using invariant bilinear forms and their Gram determinants. The structures of the Virasoro Kac modules are inferred from these results and are found to be given by finitely generated submodules of Feigin–Fuchs modules. Additional evidence for this identification is obtained by comparing the formalism of lattice fusion with the fusion rules of the Virasoro Kac modules. These are obtained, at the character level, in complete generality by applying a Verlinde-like formula and, at the module level, in many explicit examples by applying the Nahm–Gaberdiel–Kausch fusion algorithm.

Absence of $l^1$ eigenfunctions for lattice operators with fast local periodic approximation

We show that a lattice Schrödinger operator \Delta+v whose potential v\colon \bf Z^d\to\bf C admits fast local approximation by periodic functions does not have l^1 eigenfunctions. In particular, it does not exhibit Anderson localization. A special case of this result pertaining to quasi-periodic potentials states: Let V\colon \bf R^d\to\bf C be a (1,\ldots,1) -periodic function satisfying the Hölder condition. There is such \theta&gt;0 that if real numbers \alpha_1,\ldots,\alpha_d satisfy the inequality \|n_1\alpha_1\|+\cdots+\|n_d\alpha_d\| &lt; \theta^{n_1\ldots n_d} for infinitely many d -tuples (n_1,\ldots,n_d)\in\bf N^d ( \|\cdot\| is the distance from a real number to the nearest integer), then the operator \Delta+v with v(x)=V(\alpha_1 x_1,\ldots,\alpha_d x_d) has no nontrivial eigenfunctions in l^1(\bf Z^d) . This statement contrasts the result of J. Bourgain: Anderson localization for quasi-periodic lattice Schrödinger operators on \bf Z^d , d arbitrary, Geom. Funct. Anal. \bf {17} (2007), 682–706.

Application of a ℤ₃-orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices

By applying Miyamoto’s Z 3 \mathbb {Z}_{3} -orbifold construction to the lattice vertex operator algebras associated to Niemeier lattices and their automorphisms of order 3 3 , we construct holomorphic vertex operator algebras of central charge 24 24 whose Lie algebras of the weight one spaces are of types A 2 , 3 6 A_{2,3}^6 , E 6 , 3 G 2 , 1 3 E_{6,3}G_{2,1}^{3} , and A 5 , 3 D 4 , 3 A 1 , 1 3 A_{5,3}D_{4,3}A_{1,1}^{3} , which correspond to No. 6, No. 17, and No. 32 on Schellekens’ list, respectively.

Grounded fixpoints and their applications in knowledge representation

In various domains of logic, researchers have made use of a similar intuition: that facts (or models) can be derived from the ground up. They typically phrase this intuition by saying, e.g., that the facts should be grounded, or that they should not be unfounded, or that they should be supported by cycle-free arguments, et cetera. In this paper, we formalise this intuition in the context of algebraical fixpoint theory. We define when a lattice element x∈L is grounded for lattice operator O:L→L. On the algebraical level, we investigate the relationship between grounded fixpoints and the various classes of fixpoints of approximation fixpoint theory, including supported, minimal, Kripke–Kleene, stable and well-founded fixpoints. On the logical level, we investigate groundedness in the context of logic programming, autoepistemic logic, default logic and argumentation frameworks. We explain what grounded points and fixpoints mean in these logics and show that this concept indeed formalises intuitions that existed in these fields. We investigate which existing semantics are grounded. We study the novel semantics for these logics that is induced by grounded fixpoints, which has some very appealing properties, not in the least its mathematical simplicity and generality. Our results unveil a remarkable uniformity in intuitions and mathematics in these fields.

Modeling the electrical conduction in DNA nanowires: charge transfer and lattice fluctuation theories.

An analytical approach is proposed for the investigation of the conductivity properties of DNA. The charge mobility of DNA is studied based on an extended Peyrard-Bishop-Holstein model when the charge carrier is also subjected to an external electrical field. We have obtained the values of some of the system parameters, such as the electron-lattice coupling constant, by using the mean Lyapunov exponent method. On the other hand, the electrical current operator is calculated directly from the lattice operators. Also, we have studied Landauer resistance behavior with respect to the external field, which could serve as the interface between chaos theory tools and electronic concepts. We have examined the effect of two types of electrical fields (dc and ac) and variation of the field frequency on the current flowing through DNA. A study of the current-voltage (I-V) characteristic diagram reveals regions with a (quasi-)Ohmic property and other regions with negative differential resistance (NDR). NDR is a phenomenon that has been observed experimentally in DNA at room temperature. We have tried to study the affected agents in charge transfer phenomena in DNA to better design nanostructures.

Grounded Fixpoints

Algebraical fixpoint theory is an invaluable instrument for studying semantics of logics. For example, all major semantics of logic programming, autoepistemic logic, default logic and more recently, abstract argumentation have been shown to be induced by the different types of fixpoints defined in approximation fixpoint theory (AFT). In this paper, we add a new type of fixpoint to AFT: a grounded fixpoint of lattice operator O : L → L is defined as a lattice element x ∈ L such that O(x) = x and for all v ∈ L such that O(v ∧ x) ≤ v, it holds that x ≤ v. On the algebraical level, we show that all grounded fixpoints are minimal fixpoints approximated by the well-founded fixpoint and that all stable fixpoints are grounded. On the logical level, grounded fixpoints provide a new mathematically simple and compact type of semantics for any logic with a (possibly non-monotone) semantic operator. We explain the intuition underlying this semantics in the context of logic programming by pointing out that grounded fixpoints of the immediate consequence operator are interpretations that have no non-trivial unfounded sets. We also analyse the complexity of the induced semantics. Summarised, grounded fixpoint semantics is a new, probably the simplest and most compact, element in the family of semantics that capture basic intuitions and principles of various non-monotonic logics.

A novel approach to nonperturbative renormalization of singlet and nonsinglet lattice operators

A novel method for nonperturbative renormalization of lattice operators is introduced, which lends itself to the calculation of renormalization factors for nonsinglet as well as singlet operators. The method is based on the Feynman–Hellmann relation, and involves computing two-point correlators in the presence of generalized background fields arising from introducing additional operators into the action. As a first application, and test of the method, we compute the renormalization factors of the axial vector current Aμ and the scalar density S for both nonsinglet and singlet operators for Nf=3 flavors of SLiNC fermions. For nonsinglet operators, where a meaningful comparison is possible, perfect agreement with recent calculations using standard three-point function techniques is found.

Parafermionic conformal field theory on the lattice

Finding the precise correspondence between lattice operators and the continuum fields that describe their long-distance properties is a largely open problem for strongly interacting critical points. Here, we solve this problem essentially completely in the case of the three-state Potts model, which exhibits a phase transition described by a strongly interacting ‘parafermion’ conformal field theory. Using symmetry arguments, insights from integrability, and extensive simulations, we construct lattice analogues of nearly all the relevant and marginal physical fields governing this transition. This construction includes chiral fields such as the parafermion. Along the way we also clarify the structure of operator product expansions between order and disorder fields, which we confirm numerically. Our results both suggest a systematic methodology for attacking non-free field theories on the lattice and find broader applications in the pursuit of exotic topologically ordered phases of matter.

Matching lattice and continuum four-fermion operators with nonrelativistic QCD and highly improved staggered quarks

We match continuum and lattice heavy-light four-fermion operators at one loop in perturbation theory. For the heavy quarks we use nonrelativistic QCD and for the massless light quarks the highly improved staggered quark action. We include the full set of $\Delta B=2$ operators relevant to neutral $B$ mixing both within and beyond the Standard Model and match through order $\alpha_s$, $\Lambda_{\mathrm{QCD}}/M_b$, and $\alpha_s/(aM_b)$.

On Some Semi-Intuitionistic Logics

Semi-intuitionistic logic is the logic counterpart to semi-Heyting algebras, which were defined by H. P. Sankappanavar as a generalization of Heyting algebras. We present a new, more streamlined set of axioms for semi-intuitionistic logic, which we prove translationally equivalent to the original one. We then study some formulas that define a semi-Heyting implication, and specialize this study to the case in which the formulas use only the lattice operators and the intuitionistic implication. We prove then that all the logics thus obtained are equivalent to intuitionistic logic, and give their Kripke semantics.

Lattice vertex algebras over fields of prime characteristic

Based on a work of Dong and Griess, we study a class of vertex algebras over fields of prime characteristic obtained from integral forms in lattice vertex operator algebras over complex field. In particular, the generating set is given, a bilinear form is obtained, and the simplicity is studied. Finally, the vertex operator algebra structure is characterized.

A generalized Kac–Moody algebra of rank 14

We construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12. The BRST-cohomology group of this vertex algebra is a new generalized Kac–Moody algebra of rank 14. We determine its root space multiplicities and a set of simple roots.