Dimension Formula Research Articles

Schellekens' list and the very strange formula

In [45] (see also [20]) Schellekens proved that the weight-one space V1 of a strongly rational, holomorphic vertex operator algebra V of central charge 24 must be one of 71 Lie algebras. During the following three decades, in a combined effort by many authors, it was proved that each of these Lie algebras is realised by such a vertex operator algebra and that, except for V1={0}, this vertex operator algebra is uniquely determined by V1. Uniform proofs of these statements were given in [42,26].In this paper we give a fundamentally different, simpler proof of Schellekens' list of 71 Lie algebras. Using the dimension formula in [42] and Kac's “very strange formula” [28] we show that every strongly rational, holomorphic vertex operator algebra V of central charge 24 with V1≠{0} can be obtained by an orbifold construction from the Leech lattice vertex operator algebra VΛ. This suffices to restrict the possible Lie algebras that can occur as weight-one space of V to the 71 of Schellekens.Moreover, the fact that each strongly rational, holomorphic vertex operator algebra V of central charge 24 comes from the Leech lattice Λ can be used to classify these vertex operator algebras by studying properties of the Leech lattice. We demonstrate this for 43 of the 70 non-zero Lie algebras on Schellekens' list, omitting those cases that are too computationally expensive.

Compactification of Drinfeld moduli spaces as moduli spaces of 𝐴-reciprocal maps and consequences for Drinfeld modular forms

We construct a compactification of the moduli space of Drinfeld modules of rank r r and level N N as a moduli space of A A -reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on N N that are satisfied for a cofinal set of ideals N N . In the special case where A = F q [ t ] A=\mathbb {F}_q[t] and N = ( t n ) N=(t^n) , we obtain a presentation for the graded ideal of Drinfeld cusp forms of level N N and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect similar results in general, but the proof will require more ideas.

Numerical technique for fractional variable-order differential equation of fourth-order with delay

In this paper, we construct a new efficient numerical scheme for variable fractional order differential equation of fourth order with delay. Here, we propose to use parametric quintic spline in the spatial dimension and L2−1σ formula for time dimension. The stability, convergence, and solvability are rigorously proved using discrete energy method. Our proposed scheme improves convergence in both aspects (spatial-dimension and time-dimension) in comparison to those earlier work. Numerical simulation is carried out using the MATLAB software to demonstrate the effectiveness of our scheme.

On the metric dimension of a total graph of non-zero annihilating ideals

Abstract Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0} such that Ir = (0). Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by Ω(R) as the graph with the vertex-set A(R)*, the set of all non-zero annihilating ideals of R and two distinct vertices I, J are joined if and only if I +J is also an annihilating ideal of R. In this paper, we study the metric dimension of Ω(R) and provide metric dimension formulas for Ω(R).

Obtaining the Practical Formula for the Trim-tab Dimensions to Reach the Minimum Drag for Planing Boat

Trim-tab is a device for controlling and stabilizing planing boats in different sea conditions. This paper presents to find dimensions of the trim-tab at various speeds of the boat to reach the minimum drag. The speed of the boat is from15 m/s to 30m/s. Drag of the boat is calculated by Stavisky’s method. The minimum drag is found based on the optimum the trim angle. A Matlab program code is prepared for these calculations and all the all results such as drag, dimensions of the trim-tab, longitudinal center of pressure and time angle at different speeds of the boat are presented and discussed.

ASSOUAD DIMENSION AND SPECTRUM OF HOMOGENEOUS PERFECT SETS

The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar. 107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established.

On the metric dimension of strongly annihilating-ideal graphs of commutative rings

Abstract Let 𝒭 be a commutative ring with identity and 𝒜(𝒭) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of 𝒭 is defined as the graph SAG(𝒭) with the vertex set 𝒜 (𝒭)* = 𝒜 (𝒭) \{0} and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) ≠ (0) and J ∩ Ann(I) ≠ (0). In this paper, we study the metric dimension of SAG(𝒭) and some metric dimension formulae for strongly annihilating-ideal graphs are given.

Dimension formula for the affine Deligne–Lusztig variety $$X(\mu , b)$$

The study of certain union $$X(\mu , b)$$ of affine Deligne–Lusztig varieties in the affine flag varieties arose from the study of Shimura varieties with Iwahori level structure. In this paper, we give an explicit dimension formula for $$X(\mu , b)$$ associated to sufficiently large dominant coweight $$\mu $$ .

The characteristic masses of Niemeier lattices

Let L be an integral lattice in the Euclidean space R n and W an irreducible representation of the orthogonal group of R n. We give an implemented algorithm computing the dimension of the subspace of invariants in W under the isometry group O(L) of L. A key step is the determination of the number of elements in O(L) having any given characteristic polynomial, a datum that we call the characteristic masses of L. As an application, we determine the characteristic masses of all the Niemeier lattices, and more generally of any even lattice of determinant ≤ 2 in dimension n ≤ 25. For Niemeier lattices, as a verification, we provide an alternative (human) computation of the characteristic masses. The main ingredient is the determination, for each Niemeier lattice L with non-empty root system R, of the G(R)-conjugacy classes of the elements of the umbral subgroup O(L)/W(R) of G(R), where G(R) is the automor-phism group of the Dynkin diagram of R, and W(R) its Weyl group. These results have consequences for the study of the spaces of au-tomorphic forms of the definite orthogonal groups in n variables over Q. As an example, we provide concrete dimension formulas in the level 1 case, as a function of the weight W , up to dimension n = 25.

Noncommutative inclusion–exclusion, representations of left regular bands and the Tsetlin library

We find a semigroup [Formula: see text], whose category of partial representations contains the representation category [Formula: see text] of the free left regular band [Formula: see text]. We use this to construct a resolution for the absolute kernel of a representation of [Formula: see text], for which the kernel [Formula: see text] of the Markov operation in the Tsetlin library model is a prominent example. We obtain a formula for the dimension of the absolute kernel, generalizing the equality of the dimension of [Formula: see text] to the number of derangements of order [Formula: see text].

Assouad Dimensions and Lower Dimensions of Some Moran Sets

We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c ∗ > 0 .

Enumerating higher-dimensional operators with on-shell amplitudes

We establish a simple formula for the minimal dimension of operators leading to any helicity amplitude. It eases the systematic enumeration of independent operators from the construction of massless non-factorizable on-shell amplitudes. Little-group constraints can then be solved algorithmically for each helicity configuration to extract a complete set of spinor structures with lowest dimension. Occasionally, further reduction using momentum conservation, on-shell conditions and Schouten identities is required. A systematic procedure to account for the latter is presented. Dressing spinor structures with dot products of momenta finally yields the independent Lorentz structures for each helicity amplitude. We apply these procedures to amplitudes involving particles of spins 0,1/2,1,2. Spin statistics and elementary selection rules due to gauge symmetry lead to an enumeration of operators involving gravitons and standard-model particles, in the effective field theory denoted GRSMEFT. We also list the independent spinor structures generated by operators involving standard-model particles only. In both cases, we cover operators of dimension up to eight.

The full four-loop cusp anomalous dimension in mathcal{N} = 4 super Yang-Mills and QCD

We present the complete formula for the cusp anomalous dimension at four loops in QCD and in maximally supersymmetric Yang-Mills. In the latter theory it is given by {left.{Gamma}_{mathrm{cusp},mathrm{A}}right|}_{alpha_s^4}=-{left(frac{alpha_sN}{pi}right)}^4left[frac{73{pi}^6}{20160}+frac{zeta_3^2}{8}+frac{1}{N^2}left(frac{31{pi}^6}{5040}+frac{9{zeta}_3^2}{4}right)right]. Our approach is based on computing the correlation function of a rectangular light-like Wilson loop with a Lagrangian insertion, normalized by the expectation value of the Wilson loop. In maximally supersymmetric Yang-Mills theory, this ratio is a finite function of a cross-ratio and the coupling constant. We compute it to three loops, including the full colour dependence. Integrating over the position of the Lagrangian insertion gives the four-loop Wilson loop. We extract its leading divergence, which determines the four-loop cusp anomalous dimension. Finally, we employ a supersymmetric decomposition to derive the last missing ingredient in the corresponding QCD result.

On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras

Using techniques from the theory of mock modular forms and harmonic Maaß forms, especially Weierstrass mock modular forms, we establish several dimension formulas for certain holomorphic, strongly rational vertex operator algebras, complementing previous work by van Ekeren, Möller, and Scheithauer.

On the Hadamard product of degenerate subvarieties

We consider generic degenerate subvarieties X_i \subset \mathbb P^n . We determine an integer N , depending on the varieties, and for n geq N we compute dimension and degree formulas for the Hadamard product of the varieties X_i . Moreover, if the varieties X_i are smooth, their Hadamard product is smooth too. For n < N , if the X_i are generically d_i -parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus.

Homological dimension formulas for trivial extension algebras

This paper is the first one of a series of papers in which we study a finitely graded Iwanaga-Gorenstein (IG) algebras A=⨁i=0ℓAi and their representation theory. Every finitely graded algebra is graded Morita equivalent to a trivial extension algebra A=Λ⊕C with the grading deg⁡Λ=0,deg⁡C=1. Hence, in principle, representation theoretic study of finitely graded algebras is reduced to that of trivial extension algebras. The main result of this paper is a necessary and sufficient condition that a trivial extension algebra A=Λ⊕C to be an IG-algebra. We also give a description of the kernel Ker ϖ of a canonical functor ϖ:Db(modΛ)→SingZA. These results play important roles in the subsequent works.To this end, we study complexes of graded projective (injective) A-modules and establish formulas for the projective (injective) dimension of a graded A-modules in terms of Λ and C. As a byproducts, we obtain a modern expression of the classical formula for the global dimension of a trivial extension algebra by Palmer-Roos and Löfwall that is written in complicated classical derived functors.

Dual Complex of Log Fano Pairs and Its Application to Witt Vector Cohomology

AbstractWe prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro–Fujino type and a rational point formula in Dimension 3.

X2 series of universal quantum dimensions

The antisymmetric square of the adjoint representation of any simple Lie algebra is equal to the sum of adjoint and X2 representations. We present universal formulae for quantum dimensions of an arbitrary Cartan power of X2, and analyze them for singular cases and permuted universal Vogel’s parameters. X2 has been the only representation in the decomposition of the square of the adjoint with unknown universal series. Application to universal knot polynomials is discussed.

Fractal catastrophes

We analyse the spatial inhomogeneities (‘spatial clustering’) in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes (‘caustics’) in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor (‘fractal catastrophes’) make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.

Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps

For the class of graph-directed self-similar measures on $\mathbf{R}$, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.