N-dimensional Lattice Research Articles

Liouville type theorem for a system of elliptic inequalities on weighted graphs without (p 0)-condition

Abstract In this paper, we study the existence and nonexistence of solutions of a system of inequalities Δ u + h 1 v p ≤ 0 in V , Δ v + h 2 u q ≤ 0 in V , $$\begin{array}{} \displaystyle \begin{cases} \Delta u+h_1v^p\le 0\text{ in } V, \\ \Delta v+h_2 u^q\le 0\text{ in } V, \end{cases} \end{array}$$ where (V, E) is an infinite, connected, locally finite weighted graph, p > 1, q > 1, h 1, h 2 are positive potential functions and Δ is the standard graph Laplacian. We prove that, under some growth assumptions on weighted volume of balls and the existence of a suitable distance on the graph, any nonnegative solution of the above system must be trivial. We also give an application to the N-dimensional integer lattice graph ℤ N and show the sharpness of the obtained result. In particular, our result is a natural extension of the recent result [Monticelli, D. D.—Punzo, F.—Somaglia, J.: Nonexistence results for semilinear elliptic equations on weighted graphs, arXiv:2306.03609, (2023)] from a single inequality to a system of inequalities.

The Second and Fourth Moments of Discrete Gaussian Distributions over Lattices

Let Λ be an n-dimensional lattice. For any n-dimensional vector c and positive real number s, let Ds,c and DΛ,s,c denote the continuous Gaussian distribution and the discrete Gaussian distribution over Λ, respectively. In this paper, we establish the exact relationship between the second and fourth moments centered around c of the discrete Gaussian distribution DΛ,s,c and those of the continuous Gaussian distribution Ds,c, respectively. This provides a quantization form of the result obtained by Micciancio and Regev on the second and fourth moments of discrete Gaussian distribution. Using the relationship, we also derive an uncertainty principle for Gaussian functions, which extend the result of Zheng, Zhao, and Xu. Our proof is based on combination of the idea of Micciancio and Regev and the idea of Zheng, Zhao, and Xu, where the main tool is high-dimensional Fourier transform.

Poset Ramsey Number R(P,Qn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(P,Q_n)$$\\end{document}. II. N-Shaped Poset

Given partially ordered sets (posets) (P,≤P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P, \\le _P)$$\\end{document} and (P′,≤P′)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(P', \\le _{P'})$$\\end{document}, we say that P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P'$$\\end{document} contains a copy of P if for some injective function f:P→P′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:P\\rightarrow P'$$\\end{document} and for any A,B∈P\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A, B\\in P$$\\end{document}, A≤PB\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\le _P B$$\\end{document} if and only if f(A)≤P′f(B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(A)\\le _{P'} f(B)$$\\end{document}. For any posets P and Q, the poset Ramsey number R(P, Q) is the least positive integer N such that no matter how the elements of an N-dimensional Boolean lattice are colored in blue and red, there is either a copy of P with all blue elements or a copy of Q with all red elements. We focus on the poset Ramsey number R(P,Qn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(P, Q_n)$$\\end{document} for a fixed poset P and an n-dimensional Boolean lattice Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document}, as n grows large. It is known that n+c1(P)≤R(P,Qn)≤c2(P)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+c_1(P) \\le R(P,Q_n) \\le c_2(P) n$$\\end{document}, for positive constants c1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_1$$\\end{document} and c2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c_2$$\\end{document}. However, there is no poset P known, for which R(P,Qn)>(1+ϵ)n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(P, Q_n)> (1+\\epsilon )n$$\\end{document}, for ϵ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon >0$$\\end{document}. This paper is devoted to a new method for finding upper bounds on R(P,Qn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(P, Q_n)$$\\end{document} using a duality between copies of Qn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Q_n$$\\end{document} and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that R(N,Qn)=n+Θ(n/logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(\\mathcal {N},Q_n)=n+\\Theta (n/\\log n)$$\\end{document}, for a poset N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {N}$$\\end{document} with four elements A, B, C, and D, such that A<C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A<C$$\\end{document}, B<D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B<D$$\\end{document}, B<C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B<C$$\\end{document}, and the remaining pairs of elements are incomparable.

N-Dimensional Lattice Integrable Systems and Their bi-Hamiltonian Structure on the Time Scale Using the R-Matrix Approach

A time scale is a special measure chain that can unify continuous and discrete spaces, enabling the construction of integrable equations. In this paper, with the Lax operator generated by the displacement operator, N-dimensional lattice integrable systems on the time scale are given by the R-matrix approach. The recursion operators of the lattice systems are derived on the time scale. Finally, two integrable hierarchies of the discrete chain with a bi-Hamiltonian structure are obtained. In particular, we give the structure of two-field and four-field systems.

Bounds on the defect of an octahedron in a rational lattice

Consider an arbitrary n-dimensional lattice Λ such that Zn⊂Λ⊂Qn. Such lattices are called rational and can always be obtained by adding m≤n rational vectors to Zn. The defectd(E,Λ) of the standard basis E of Zn (n unit vectors going in the directions of the coordinate axes) is defined as the smallest integer d such that certain (n−d) vectors from E together with some d vectors from the lattice Λ form a basis of Λ.Let ‖⋅‖ be L1-norm on Qn. Suppose that for each non-integer x∈Λ inequality ‖x‖>1 holds. Then the unit octahedron OEn=x∈Rn:‖x‖⩽1 is called admissible with respect to Λ and d(E,Λ) is also called the defect of the octahedron OEn with respect to E and is denoted by d(OEn,Λ).Let dn=maxΛ∈And(OEn,Λ), where An is the set of all rational lattices Λ such that OEn is admissible w.r.t. Λ. In this article we show that n−dn=Θ(logn).Let dnm=maxΛ∈Amd(OEn,Λ), where Am is the set of all rational lattices Λ such that 1) OEn is admissible w.r.t. Λ and 2) Λ can be obtained by adding m rational vectors to Zn: Λ=Zn,a1,…,amZ for some a1,…,am∈Qn. In this article we show that for any 0<ϵ<1 and large enough n we have m<nϵ8(1ϵ+1)⟹dnm<n−n1−ϵ.Finally we show that for any B>0 there exists a positive constant C>0 such that m<Cnloglognlog2n⟹dnm<n−Blogn.

Lattice-Based Analog Mappings for Low-Latency Wireless Sensor Networks

We consider the transmission of spatially correlated analog information in a WSN through fading SIMO MAC with low latency requirements. A lattice-based analog JSCC approach is considered where vectors of consecutive source symbols are encoded at each sensor using an n-dimensional lattice and then transmitted to a multi-antenna central node. We derive a MMSE decoder that accounts for both the multi-dimensional structure of the encoding lattices and the spatial correlation. In addition, a sphere decoder is considered to simplify the required searches over the multi-dimensional lattices. Different lattice-based mappings are approached and the impact of their size and density on performance and latency is analyzed. Results show that, while meeting low-latency constraints, lattice-based analog JSCC provides performance gains and higher reliability with respect to the state-of-the-art JSCC schemes.

Finite-group gauge theories on lattices as Hamiltonian systems with constraints

In this work, we present a brief but insightful overview of the gauge theories, which are defined on n-dimensional lattices by using finite gauge groups, in order to show how they can be interpreted as a Hamiltonian system with constraints, analogous to what happens with the classical (continuous) gauge (field) theories. As this interpretation is not usually explored in the literature that discusses/introduces the concept of lattice gauge theory, but some recent works have been exploring Hamiltonian models in order to support some kind of quantum computation, we use this interpretation to, for example, present a brief geometric view of one class of these models: the Kitaev Quantum Double Models.

Absolute Retracts for Finite Distributive Lattices and Slim Semimodular Lattices

Let n denote a positive integer. We describe the absolute retracts for the following five categories of finite lattices: (1) slim semimodular lattices, which were introduced by G. Grätzer and E. Knapp in (Acta. Sci. Math. (Szeged), 73 445–462 2007), and they have been intensively studied since then, (2) finite distributive lattices (3) at most n-dimensional finite distributive lattices, (4) at most n-dimensional finite distributive lattices with cover-preserving {0,1}-homomorphisms, and (5) finite distributive lattices with cover-preserving {0,1}-homomorphisms. Although the singleton lattice is the only absolute retract for the first category, this result has paved the way to some other classes. For the second category, we prove that the absolute retracts are exactly the finite boolean lattices; this generalizes a 1979 result of J. Schmid. For the third category and also for the fourth, the absolute retracts are the finite boolean lattices of dimension at most n and the direct products of n nontrivial finite chains. For the fifth category, the absolute retracts are the same as those for the second category. Also, we point out that in each of these classes, the algebraically closed lattices and the strongly algebraically closed lattices (investigated by J. Schmid and, in several papers, by A. Molkhasi) are the same as the absolute retracts.

Optimal template banks

When searching for new gravitational-wave or electromagnetic sources, the $n$ signal parameters (masses, sky location, frequencies,...) are unknown. In practice, one hunts for signals at a discrete set of points in parameter space, with a computational cost that is proportional to the number of these points. If that is fixed, the question arises, where should the points be placed in parameter space? The current literature advocates selecting the set of points (called a "template bank") whose Wigner-Seitz (also called Vorono\"i) cells have the smallest covering radius ($\equiv$ smallest maximal mismatch). Mathematically, such a template bank is said to have "minimum thickness". Here, for realistic populations of signal sources, we compute the fraction of potential detections which are "lost" because the template bank is discrete. We show that at fixed computational cost, the minimum thickness template bank does not maximize the expected number of detections. Instead, the most detections are obtained for a bank which minimizes a particular functional of the mismatch. For closely spaced templates, the fraction of lost detections is proportional to a scale-invariant "quantizer constant" G, which measures the average squared distance from the nearest template, i.e., the average expected mismatch. This provides a straightforward way to characterize and compare the effectiveness of different template banks. The template bank which minimizes G is mathematically called the "optimal quantizer", and maximizes the expected number of detections. We review optimal quantizer and minimum thickness template banks that are built as n-dimensional lattices, showing that even the best of these offer only a marginal advantage over template banks based on the humble cubic lattice.

Topological Quantum Codes from Lattices Partition on the n-Dimensional Flat Tori.

In this work, we show that an n-dimensional sublattice of an n-dimensional lattice induces a tessellation in the flat torus , where the group G is isomorphic to the lattice partition . As a consequence, we obtain, via this technique, toric codes of parameters , and from the lattices , and , respectively. In particular, for , if is either the lattice or a hexagonal lattice, through lattice partition, we obtain two equivalent ways to cover the fundamental cell of each hexagonal sublattice of hexagonal lattices , using either the fundamental cell or the Voronoi cell . These partitions allow us to present new classes of toric codes with parameters and color codes with parameters in the flat torus from families of hexagonal lattices in .

Arbitrary synthetic dimensions via multiboson dynamics on a one-dimensional lattice

The synthetic dimension, a research topic of both fundamental significance and practical applications, has been attracting increasing attention in recent years. In this paper, we propose a theoretical framework to construct arbitrary synthetic dimensions, or N-boson synthetic lattices, using multiple bosons on one-dimensional lattices. We show that a one-dimensional lattice hosting N indistinguishable bosons can be mapped to a single boson on an N-dimensional lattice with high symmetry. Band structure analyses on this N-dimensional lattice can then be mathematically performed to predict the existence of exotic eigenstates and the motion of N-boson wave packets. As illustrative examples, we demonstrate the edge states in two-boson Su-Schrieffer-Heeger synthetic lattices without interactions, interface states in two-boson Su-Schrieffer-Heeger synthetic lattices with interactions, and weakly bound triplon states in three-boson tight-binding synthetic lattices with interactions. The interface states and weakly bound triplon states have not been thoroughly understood in previous research. Our proposed theoretical framework hence provides an interesting perspective to explore the multiboson dynamics on lattices with boson-boson interactions, and opens up a future avenue in the field of multiboson manipulation in quantum engineering.

Materialized View Selection Using Swap Operator Based Particle Swarm Optimization

The data warehouse is a key data repository of any business enterprise that stores enormous historical data meant for answering analytical queries. These queries need to be processed efficiently in order to make efficient and timely decisions. One way to achieve this is by materializing views over a data warehouse. An n-dimensional star schema can be mapped into an n-dimensional lattice from which Top-K views can be selected for materialization. Selection of such Top-K views is an NP-Hard problem. Several metaheuristic algorithms have been used to address this view selection problem. In this paper, a swap operator-based particle swarm optimization technique has been adapted to address such a view selection problem.

Note on a theorem of Davenport

Let Λ be a 𝑛-dimensional lattice, and 𝑐1, . . . , 𝑐𝑛−1 be any 𝑛 − 1 vectors in 𝑛-dimensional real Euclidean space. We show that there exists a basis 𝛼1, . . . ,𝛼𝑛 of Λ such that $$|𝛼𝑖 − 𝑁𝑐𝑖| = 𝑂(log^2 𝑁), (1 2, where the constant implied by the 𝑂 symbol depends only on Λ and 𝑐1, . . . , 𝑐𝑛−1.

Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields

Abstract In 2006, Hubert, Mauduit and Sárközy extended the notion of binary sequences to n-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, Gyarmati, Mauduit and Sárközy extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order k, family complexity, collision and avalanche effect. Results indicate that such binary lattices are “good,” and their families possess a nice structure in terms of family complexity, collision and avalanche effect.

Concentration wave method for description of all possible energy states of A2BC ordered alloy

The procedure for describing all possible energy states of an alloy based on the concentration wave method proposed by Khachaturian is employed to describe all possible energy states of ordered A 2 BC alloy based on N-dimensional lattice. Within the framework of this method, a complete enumeration of the structures obtained by the superposition of N plane concentration waves with all possible wave vectors is carried out, provided that the given stoichiometry is preserved. For each such superposition, the order parameters on the first I coordination spheres are calculated, thereby determining the point in the I-dimensional order parameter space corresponding to the given structure. For the case of I = 2 it is shown that a complete enumeration of all structures generated by one plane concentration wave fills a non-convex figure in the space of two order parameters.

N-Dimensional congruent lattices using necklaces

This work introduces the n-dimensional congruent lattices using necklaces, a general methodology to generate uniform distributions in multidimensional modular spaces. The formulation presented in this manuscript constitutes the mathematical foundation of the most used satellite constellation designs, including Walker Constellations, and Lattice and Necklace Flower Constellations. These constellation design models are based on Number Theory and allow to obtain distributions that have some interesting properties of uniformity and large number of symmetries. This work includes the complete formulation of the methodology, proofs for existence and uniqueness of the distribution definitions, and several theorems that focus on the counting possibilities of design for the most common cases of study.

Threshold of discrete Schrödinger operators with delta potentials on n-dimensional lattice

ABSTRACT Eigenvalue behaviours of Schrödinger operator defined on n-dimensional lattice with n + 1 delta potentials are studied. It can be shown that lower threshold eigenvalue and lower threshold resonance appear for , and lower super-threshold resonance appears for n = 1.

Families of lattice polytopes of mixed degree one

It has been shown by Soprunov that the normalized mixed volume (minus one) of an n-tuple of n-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined n-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension n≥4, showing that all but finitely many n-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex Δn−1. Furthermore, we give a complete solution in dimension n=3. In the course of this we show that our finiteness result does not extend to dimension n=3, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle Δ2.

On difference graphs and the local dimension of posets

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is n while its local dimension is only 3. Hiraguchi (1955) proved that the maximum dimension of a poset of order n is n∕2. However, we find a very different result for local dimension, proving a bound of Θ(n∕logn). This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the n-dimensional Boolean lattice is Ω(n∕logn) and make progress toward resolving an analogue of the removable pair conjecture for local dimension.

The faster lattice sieving for SVP

The security of lattice-based cryptography is based on the hardness of the difficult problems on lattice, especially the famous shortest vector problem. There are many famous heuristic lattice sieving algorithms to solve SVP, such as the Gauss Sieve, NV Sieve, which deal the full-rank n-dimensional lattice from start. Inspired by the idea of rank reduction, in this paper we present new technique on lattice sieving to make the algorithm solve the SVP faster. We split the basis of “bigger” lattice into several blocks according to some rule until the sublattice is small enough. Then we recursively sieved on these sub-lattices to get short vector lists. With the short vector lists, we can find the shortest vector when the algorithm recoverd the original lattice. This lead to obviously speedup with the recursive procedure without extra space overhead. Compared with the Gauss Sieve, the new method converge faster, and achieved at least 70% acceleration.