Morse quasiflats I

This is the first in a series of papers concerned with Morse quasiflats, which are a generalization of Morse quasigeodesics to arbitrary dimension. In this paper we introduce a number of alternative definitions, and under appropriate assumptions on the ambient space we show that they are equivalent and quasi-isometry invariant; we also give a variety of examples. The second paper proves that Morse quasiflats are asymptotically conical and have canonically defined Tits boundaries; it also gives some first applications.

Sufficient Conditions for the Global Rigidity of Periodic Graphs

Tanigawa (2016) showed that vertex-redundant rigidity of a graph implies its global rigidity in arbitrary dimension. We extend this result to periodic frameworks under fixed lattice representations. That is, we show that if a generic periodic framework is vertex-redundantly rigid, in the sense that the deletion of a single vertex orbit under the periodicity results in a periodically rigid framework, then it is also periodically globally rigid. Our proof is similar to the one of Tanigawa, but there are some added difficulties. First, it is not known whether periodic global rigidity is a generic property in dimension $$d>2$$ d > 2 . We work around this issue by using slight modifications of recent results of Kaszanitzky et al. (2021). Secondly, while the rigidity of finite frameworks in $${\mathbb {R}}^d$$ R d on at most d vertices obviously implies their global rigidity, it is non-trivial to prove a similar result for periodic frameworks. This is accomplished by extending a result of Bezdek and Connelly (2002) on the existence of a continuous motion between two equivalent d-dimensional realisations of a single graph in $${\mathbb {R}}^{2d}$$ R 2 d to periodic frameworks. As an application of our result, we give a necessary and sufficient condition for the global rigidity of generic periodic body-bar frameworks in arbitrary dimension. This provides a periodic counterpart to a result of Connelly et al. (2013) regarding the global rigidity of generic finite body-bar frameworks.

The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

Novel method for one-way local distinguishability of generalized Bell states in arbitrary dimension

In this paper the local distinguishability of generalized Bell states in arbitrary dimension is investigated. We firstly study the decomposition of a basis which consists of $d^{2}$ number of generalized Pauli matrices. We discover that this basis is equal to the union of $D$ number of different sets, where $D=\frac{2}{\phi(d)}\sum_{t\in \mathbb{Z}_{d} \atop gcd(t,d)=1}\sum_{i=2}^{\lfloor\frac{d}{t}\rfloor}\phi(i)+1$ and $\phi$ is Euler $\phi$-function. Then we define the generator of the matrices in this decomposition, and exhibit an algorithm to calculate generators of a given set of matrices. This algorithm shows that generators of a given set can be calculated simply and efficiently. Secondly, we show that a set $\mathcal {L}$ of GBSs can be distinguished by one-way LOCC if the cardinality of $\mathcal {G}_{\mathcal {L}}$ is less than $D\phi(d)$, where $\mathcal {G}_{\mathcal {L}}$ is a set of generators of all the elements in difference set of a set $\mathcal {L}$ of GBSs. The previous results in [2004 Phys. Rev. Lett. \textbf{92} 177905; 2019 Phys. Rev. A \textbf{99} 022307; 2021 Quant. Info. Proc. \textbf{20} 52] can be covered by our result. Finally, for the uncovered cases in [2021 Quant. Info. Proc. \textbf{20} 52], we give a new result to partly solve that problem.

Study on energy and information-entropic measures of Hulthén potential in D dimension by an intuitive approximation to centrifugal term

Energy spectrum as well as various information theoretic measures are considered for Hulthén potential in D dimension. For a given ℓ≠0 state, analytic expressions are derived, following a simple intuitive approximation for accurate representation of centrifugal term, within the conventional Nikiforov-Uvarov method. This is derived from a linear combination of two widely used Greene-Aldrich and Pekeris-type approximations. Energy, wave function, normalization constant, expectation value in r and p space, Heisenberg uncertainty relation, entropic moment of order α¯, Shannon entropy, Rényi entropy, disequilibrium, majorization as well as four selected complexity measures like LMC (López-Ruiz, Mancini, Calbert), shape Rényi complexity, Generalized Rényi complexity and Rényi complexity ratio are offered for different screening parameters (δ). The effective potential is described quite satisfactorily throughout the whole domain. Obtained results are compared with theoretical energies available in literature, which shows excellent agreement. Performance of six different approximations to centrifugal term is critically discussed. An approximate analytical expression for critical screening for a specific state in arbitrary dimension is offered. Additionally, some inter-dimensional degeneracy occurring in two states, at different dimension for a particular δ is also uncovered. PACS: 02.60.-x, 03.65.Ca, 03.65.Ge, 03.65.-w Keywords: Hulthén potential, Rényi complexity ratio, Statistical complexity, Majorization, Pekeris approximation, Greene-Aldrich approximation.

Environment-Assisted Shortcuts to Adiabaticity

Envariance is a symmetry exhibited by correlated quantum systems. Inspired by this “quantum fact of life,” we propose a novel method for shortcuts to adiabaticity, which enables the system to evolve through the adiabatic manifold at all times, solely by controlling the environment. As the main results, we construct the unique form of the driving on the environment that enables such dynamics, for a family of composite states of arbitrary dimension. We compare the cost of this environment-assisted technique with that of counterdiabatic driving, and we illustrate our results for a two-qubit model.

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Jacques Tits motivic measure

In this article we construct a new motivic measure called the Jacques Tits motivic measure. As a first main application, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to 2-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $$\{3, 4, 5, 6\}$$ { 3 , 4 , 5 , 6 } , have the same class in the Grothendieck ring of varieties, then they are necessarily birational to each other. As a second main application, we prove that two quadric hypersurfaces (or, more generally, two involution varieties), associated to quadratic forms of dimension 6 or to quadratic forms of arbitrary dimension defined over a base field k with $$I^3(k)=0$$ I 3 ( k ) = 0 , have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that the latter main application also holds for products of quadric hypersurfaces.

Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.