Dimension Formula Research Articles

Dimensions of ‘self-affine sponges’ invariant under the action of multiplicative integers

AbstractLet $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$ . We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $ . We then generalize our results to the same subsets defined in dimension $d \geq 2$ . There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesised version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions F(1)∘…∘F(m) of functions F(l):RN→RN in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Faà di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series.

Orbifold construction and Lorentzian construction of Leech lattice vertex operator algebra

We generalize Conway-Sloane's constructions of the Leech lattice from Niemeier lattices using Lorentzian lattice to holomorphic vertex operator algebras (VOA) of central charge 24. It provides a tool for analyzing the structures and relations among holomorphic VOAs related by orbifold construction. In particular, we are able to get some useful information about certain lattice subVOAs associated with the Cartan subalgebra of the weight one Lie algebra V1. We also obtain a relatively elementary proof that any strongly regular holomorphic VOA of central charge 24 with V1≠0 can be constructed directly by a single orbifold construction from the Leech lattice VOA without using a dimension formula.

An exact and compact formula for the optical intersubband response of finite-barrier quantum wells, wires and dots

The optical response of quantum wells, wires and dots is frequently modelled without including continuum states. This is reasonable in a low-energy regime but fails to capture phenomena such as bound-to-continuum transitions. We show that, in fact, the full optical intersubband response of quantum confined structures can be expressed in an exceedingly simple exact formula for arbitrary barrier height, confinement dimension, and optical frequency. We illustrate the consequences of transitions reaching the continuum for several representative cases, including asymmetric quantum wells. Moreover, we derive a simple modification that may be added to bound-states-only simulation in order to include the continuum. For the asymmetric quantum well, both linear and second-order nonlinear response is obtained. Finally, excellent agreement with measured bound-to-continuum absorption in narrow quantum wells is demonstrated.

EKOR strata for Shimura varieties with parahoric level structure

We study the geometry of reduction modulo p of the Kisin–Pappas integral models for certain Shimura varieties of abelian type with parahoric level structure. We give some direct and geometric constructions for the EKOR (Ekedahl–Kottwitz–Oort–Rapoport) strata on these Shimura varieties, using the theories of G-zips and mixed characteristic local G-Shtukas. We establish several basic properties of these strata, including the smoothness, dimension formula, and closure relation. Moreover, we apply our results to the study of Newton strata and central leaves on these Shimura varieties.

Hausdorff dimension of intersections with planes and general sets

We give conditions on a general family P_{\lambda}\colon \R^n\to\R^m , \lambda \in \Lambda , of orthogonal projections which guarantee that the Hausdorff dimension formula \dim A\cap P_{\lambda}^{-1}\{u\}=s-m holds generically for measurable sets A\subset\R^n with positive and finite s -dimensional Hausdorff measure, s>m , and with positive lower density. As an application we prove for measurable sets A,B\subset\R^n with positive s - and t -dimensional measures, and with positive lower density that if s + (n-1)t/n > n , then \dim A\cap (g(B)+z) = s+t - n for almost all rotations g and for positively many z\in\R^n .

Concrete $L^2$-spectral Analysis of a Bi-weighted $\Gamma$-automorphic Twisted Laplacian

We consider a twisted Laplacian $\Delta_{\nu,\mu}$ on the $n$-complex space associated with the sub-Laplacian of the Heisenberg group $\mathbb{C} \times_{\omega} \mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. The main results to which is aimed this paper concern the spectral theory of $\Delta_{\nu,\mu}^{\Gamma}$ when acting on some $L^2$ space of $\Gamma$-automorphic functions of biweight $(\nu,\mu)$ associated to given cocompat discrete subgroup $\Gamma$ of the additive group $\mathbb{C}^n$. We describe its spectrum proving a stability theorem. Using the Selberg's approach, we give the explicit dimension formula for the corresponding $L^2$-eigenspaces. We also construct a concrete basis of such $L^2$-eigenspaces.

Study of irregular dynamics in an economic model: attractor localization and Lyapunov exponents

Cyclicality and instability inherent in the economy can manifest themselves in irregular fluctuations, including chaotic ones, which significantly reduces the accuracy of forecasting the dynamics of the economic system in the long run. We focus on an approach, associated with the identification of a deterministic endogenous mechanism of irregular fluctuations in the economy. Using of a mid-size firm model as an example, we demonstrate the use of effective analytical and numerical procedures for calculating the quantitative characteristics of its irregular limiting dynamics based on Lyapunov exponents, such as dimension and entropy. We use an analytical approach for localization of a global attractor and study limiting dynamics of the model. We estimate the Lyapunov exponents and get the exact formula for the Lyapunov dimension of the global attractor of this model analytically. With the help of delayed feedback control (DFC), the possibility of transition from irregular limiting dynamics to regular periodic dynamics is shown to solve the problem of reliable forecasting. At the same time, we demonstrate the complexity and ambiguity of applying numerical procedures to calculate the Lyapunov dimension along different trajectories of the global attractor, including unstable periodic orbits (UPOs).

Well-posedness and asymptotic behavior of stochastic convective Brinkman–Forchheimer equations perturbed by pure jump noise

This paper is concerned about stochastic convective Brinkman–Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise unique strong solution satisfying the energy equality (Itô’s formula) to SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique. The major difficulty is that an Itô’s formula in infinite dimensions is not available for such systems. This difficulty is overcame by the technique of approximating functions using the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to some technical difficulties, we discuss the global in time regularity results of such strong solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of strong solutions. For large effective viscosity, the exponential stability results (in the mean square and pathwise sense) for stationary solutions is established. Moreover, a stabilization result of SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a unique ergodic and strongly mixing invariant measure for SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of strong solutions.

Two graded rings of Hermitian modular forms

We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in {mathbb {Q}}(sqrt{-7}) and {mathbb {Q}}(sqrt{-11}). In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.

Logarithmic dimension and bases in Whitney spaces

We give a formula for the logarithmic dimension of the generalized Cantor-type set $K$. In the case when the logarithmic dimension of $K$ is smaller than $1$, we construct a Faber basis in the space of Whitney functions $\mathcal{E}(K)$.

Divergent trajectories on products of homogeneous spaces

In this paper, we determine the Hausdorff dimension of the set of points with divergent trajectories on the product of certain homogeneous spaces. The flow is allowed to be weighted with respect to the factors in the product space. The result is derived from its counterpart in Diophantine approximation. In doing this, we introduce a notion of jointly singular matrix tuples, and extend the dimension formula for singular matrices to such matrix tuples.

Some binary products and integer linear programming for [formula omitted]-metric dimension of graphs

A sharp upper bound and a closed formula for the k-metric dimension of the hierarchical product of graphs is proved. Also, sharp lower bounds for the k-metric dimension of the splice and link products of graphs are presented. An integer linear programming model for computing the k-metric dimension as well as a k-metric basis of a given graph is proposed. These results are applied to bound or to compute the k-metric dimension of some classes of graphs that are of interest in mathematical chemistry.

An Estimate of the Growth of Cohomology with Coefficients

For a connected reductive algebraic group over an arbitrary number eld, we consider a nite dimensional algebraic, irreducible representation of the group of its real points. Each adelic locally symmetric space corresponding to a level structure constructed using the group has an associated sheaf induced by this representation. The purpose of this note is to estimate the rate of growth of the total dimension of the pertinent cohomology with coecients as either of the level structure or the associated sheaf varies. We obtain an upper bound on this total dimension. We also obtain a lower bound under certain topological conditions. Both the bounds are consistent with several classical dimension formulae as well as other known results.

A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors

There are various reasons why a naive analog of the Maeda conjecture has to fail for Drinfeld cusp forms. Focussing on double cusp forms and using the link found by Teitelbaum between Drinfeld cusp forms and certain harmonic cochains, we observed a while ago that all obvious counterexamples disappear for certain Hecke-invariant subquotients of spaces of Drinfeld cusp forms of fixed weight, which can be defined naturally via representation theory. The present work extends Teitelbaum’s isomorphism to an adelic setting and to arbitrary levels, it makes precise the impact of representation theory, it relates certain intertwining maps to hyperderivatives of Bosser-Pellarin, and it begins an investigation into dimension formulas for the subquotients mentioned above. We end with some numerical data for A={mathbb {F}}_3[t] that displays a new obstruction to an analog of a Maeda conjecture by discovering a conjecturally infinite supply of {mathbb {F}}_3(t)-rational eigenforms with combinatorially given (conjectural) Hecke eigenvalues at the prime t.

Thermodynamic formalism for invariant measures in iterated function systems with overlaps

We study images of equilibrium (Gibbs) states for a class of non-invertible transformations associated to conformal iterated function systems (IFSs) with overlaps [Formula: see text]. We prove exact dimensionality for these image measures, and find a dimension formula using their overlap numbers. In particular, we obtain a geometric formula for the dimension of self-conformal measures for IFSs with overlaps, in terms of the overlap numbers. This implies a necessary and sufficient condition for dimension drop. If [Formula: see text] is a self-conformal measure, then [Formula: see text] if and only if the overlap number [Formula: see text]. Examples are also discussed.

The points with dense orbit under the β-expansions of different bases

It was conjectured by Furstenberg that for any x∈[0,1]∖Q, dimH{2nx(mod1):n≥1}¯+dimH{3nx(mod1):n≥1}¯≥1. Where dimH denotes the Hausdorff dimension and A¯ denotes the closure of a set A. Can we obtain analogous dimension formula when considering the point under the β-expansions of different bases? In this paper, we are aiming at giving explicit non-normal numbers for which the above dimensional formula holds. We illustrate our result as follows: For any β1>1 and β2>1,there exists a Cantor set E composed of real numbers in the unit interval such that 1. Each x∈E is not β1-normal; 2. For each x∈E, {Tβ1nx:n≥1}¯=[0,1] and {Tβ2nx:n≥1}¯=[0,1]; 3. dimHE=1. Where Tβ denotes the β-transformation.

R-GROUP AND WHITTAKER SPACE OF SOME GENUINE REPRESENTATIONS

AbstractFor a unitary unramified genuine principal series representation of a covering group, we study the associated R-group. We prove a formula relating the R-group to the dimension of the Whittaker space for the irreducible constituents of such a principal series representation. Moreover, for certain saturated covers of a semisimple simply connected group, we also propose a simpler conjectural formula for such dimensions. This latter conjectural formula is verified in several cases, including covers of the symplectic groups.

Anomalous dimensions of effective theories from partial waves

On-shell amplitude methods have proven to be extremely efficient for calculating anomalous dimensions. We further elaborate on these methods to show that, by the use of an angular momentum decomposition, the one-loop anomalous dimensions can be reduced to essentially a sum of products of partial waves. We apply this to the SM EFT, and show how certain classes of anomalous dimensions have their origin in the same partial-wave coefficients. We also use our result to obtain a generic formula for the one-loop anomalous dimensions of nonlinear sigma models at any order in the energy expansion, and apply our method to gravity, where it proves to be very advantageous even in the presence of IR divergencies.

On the Strong Metric Dimension of Annihilator Graphs of Commutative Rings

For a connected graph G(V, E), a vertex $$w\in V(G)$$ strongly resolves two vertices $$u, v \in V(G)$$ if there exists a shortest $$u-w$$ path containing v or a shortest $$v-w$$ path containing u. A set S of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is a simple graph with the vertex set $$Z(R)^*=Z(R){\setminus }\{0\}$$ , and two distinct vertices x and y are adjacent if and only if $$ann_R(xy)\ne ann_R(x)\cup ann_R(y)$$ . In this paper, we study the strong metric dimension of annihilator graphs associated with commutative rings and some strong metric dimension formulae for annihilator graphs are given.