Special Linear Combinations Research Articles

Quantum Scars from Zero Modes in an Abelian Lattice Gauge Theory on Ladders.

We consider the spectrum of a U(1) quantum link model where gauge fields are realized as S=1/2 spins and demonstrate a new mechanism for generating quantum many-body scars (high-energy eigenstates that violate the eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body dynamics with local constraints has attracted much attention due to the recent discovery of nonergodic behavior in quantum simulators based on Rydberg atoms. Lattice gauge theories provide natural examples of constrained systems since physical states must be gauge invariant. In our case, the Hamiltonian H=O_{kin}+λO_{pot}, where O_{pot} (O_{kin}) is diagonal (off-diagonal) in the electric flux basis, contains exact midspectrum zero modes at λ=0 whose number grows exponentially with system size. This massive degeneracy is lifted at any nonzero λ but some special linear combinations that simultaneously diagonalize O_{kin} and O_{pot} survive as quantum many-body scars, suggesting an "order-by-disorder" mechanism in the Hilbert space. We give evidence for such scars and show their dynamical consequences on two-leg ladders with up to 56 spins, which may be tested using available proposals of quantum simulators. Results on wider ladders point towards their presence in two dimensions as well.

Kummert's approach to realization on the bidisk

We give a simplified exposition of Kummert's approach to proving that every matrix-valued rational inner function in two variables has a minimal unitary transfer function realization. A slight modification of the approach extends to rational functions which are isometric on the two-torus and we use this to give a largely elementary new proof of the existence of Agler decompositions for every matrix-valued Schur function in two variables. We use a recent result of Dritschel to prove two variable matrix-valued rational Schur functions always have finite-dimensional contractive transfer function realizations. Finally, we prove that two variable matrix-valued polynomial inner functions have transfer function realizations built out of special nilpotent linear combinations.

Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by N gg 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature beta ^{-1}. Given a fixed {1le m ll N}, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order beta , for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

Half-cyclic, dihedral and half-dihedral codes

This note discusses, in elementary terms, linear codes over $$\mathbb {Z}_2$$ which are closed under 2-step cyclic shifts, and classifies them in terms of special linear combinations of polynomials. Codes which are preserved under order-reversing automorphisms are also discussed, and a classification result in terms of special linear combinations of polynomials is obtained.

Nonnegative Hermitian vector bundles and Chern numbers

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of Chern forms are strongly nonnegative. This particularly implies that all the Chern numbers of such a holomorphic vector bundle are nonnegative and can be bounded below and above respectively by two special Chern numbers. As applications, we obtain a family of new results on compact connected complex manifolds which are homogeneous or can be holomorphically immersed into complex tori, some of which improve several classical results.

Genus-zero and genus-one string amplitudes and special multiple zeta values

In this paper we show that in perturbative string theory the genus-one contribution to formal 2-point amplitudes can be related to the genus-zero contribution to 4-point amplitudes. This is achieved by studying special linear combinations of multiple zeta values that appear as coefficients of the amplitudes. We also exploit our results to relate closed strings to open strings at genus one using Brown's single-valued projection, proving a conjecture of Broedel, Schlotterer and the second author.

Completeness and minimality of systems of Bessel functions

|= +∞.At studying of some non-classical boundary-value problems (see [26]–[31]) and generalizedeigenvectors of some linear operators [28], [29] we needed to obtain the analogues of Theorems 1-3 for weighted spaces and establish an approximation properties of the special finite linearcombinations of Bessel functions. We don’t understand to the end the nature of expectedresults for an arbitrary ν ∈ R. For advance in the given direction it is important to investigatein details the simplest model cases ν = −3/2 and ν = 3/2. The case ν = −3/2 was consideredin [27], [30] (see also [26], [28], [31]). Here we consider the case ν = 3/2 more detail. But even inthis case we cannot obtain the all necessary facts. In particular, remains an open one for us theproblem formulated at the end of this paper. In our view, its solution is very important for theconstruction of some spectral theory that is based on the notion of a generalized eigenvector(see [28], [29]).It is well known (see [3], [25, p. 350], [32]) that√zJ

Counting and Detecting Small Subgraphs via Equations

We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let $H$ be a fixed graph with $k$ vertices and an independent set of size $s.$ 1. Detecting if an $n$-vertex graph contains a (not necessarily induced) subgraph isomorphic to $H$ can be done in time $O(n^{\omega(\lceil (k-s)/2 \rceil, 1, \lfloor (k-s)/2 \rfloor )})$, where $\omega (p,q,r)$ is the exponent of fast arithmetic matrix multiplication of an $n^p\times n^q$ matrix by an $n^q\times n^r$ matrix. 2. When $s=2,$ counting the number of (not necessarily induced) subgraphs isomorphic to $H$ can be done in the same time, i.e., in time $O(n^{\omega(\lceil (k-2)/2 \rceil, 1, \lfloor (k-2)/2 \rfloor )}).$ It follows in particular that we can count the nu...

Efficient minimum distance estimation with multiple rates of convergence

This paper extends the asymptotic theory of GMM inference to allow sample counterparts of the estimating equations to converge at (multiple) rates, different from the usual square-root of the sample size. In this setting, we provide consistent estimation of the structural parameters. In addition, we define a convenient rotation in the parameter space (or reparametrization) to disentangle the different rates of convergence. More precisely, we identify special linear combinations of the structural parameters associated with a specific rate of convergence. Finally, we demonstrate the validity of usual inference procedures, like the overidentification test and Wald test, with standard formulas. It is important to stress that both estimation and testing work without requiring the knowledge of the various rates. However, the assessment of these rates is crucial for (asymptotic) power considerations.Possible applications include econometric problems with two dimensions of asymptotics, due to trimming, tail estimation, infill asymptotic, social interactions, kernel smoothing or any kind of regularization.

Approximation of functions with zero integrals over balls by linear combinations of the Laplace operator eigenfunctions

It is proved that the special linear combinations of Bessel functions are dense in the C∞-topology in the space of functions with zero integrals over balls of fixed radii on an arbitrary open domain \( U \subset {\mathbb{R}^n} \). Some generalizations of this result for solutions of some convolution equations of the form f * T = 0, where T is radial, are obtained. Analogous results for rank-one symmetric spaces of the noncompact type are considered.

Identity method for particle number fluctuations and correlations

An incomplete particle identification distorts the observed event-by-event fluctuations of the hadron chemical composition in nucleus-nucleus collisions. A new experimental technique called the identity method was recently proposed. It eliminated the misidentification problem for one specific combination of the second moments in a system of two hadron species. In the present paper, this method is extended to calculate all the second moments in a system with an arbitrary number of hadron species. Special linear combinations of the second moments are introduced. These combinations are presented in terms of single-particle variables and can be found experimentally from the event-by-event averaging. The mathematical problem is then reduced to solving a system of linear equations. The effect of incomplete particle identification is fully eliminated from the final results.

The symmetry properties of stable polynomial Gaussian beam profiles

We consider the symmetry properties of polynomial Gaussian beam profiles (intensity distributions) that remain stable during propagation, apart from being scaled and possibly rotated. These beams are expressed as special linear combinations of the Laguerre–Gaussian modes. Two kinds of symmetries are present: discreet rotational symmetries and mirror symmetries. The symmetry properties are shown to depend on the particular subset of Laguerre–Gaussian modes that is used to construct the stable beam. We demonstrate the symmetry properties of a few examples of stable beams through numerical simulations.

General issues connecting flavor symmetry and supersymmetry

We motivate and construct supersymmetric theories with continuous flavor symmetry, under which the electroweak Higgs doublets transform non-trivially. Flavor symmetry is spontaneously broken at a large mass scale in a sector of gauge-singlet fields; the light Higgs multiplets naturally emerge as special linear combinations that avoid acquiring the generic large mass. Couplings of the light Higgs doublets to light moduli fields from the singlet sector could lead to important effects in the phenomenology of the Higgs sector.

Shot noise and higher current moments in dimensions 2, 3 and 4

AbstractShot noise and higher current moments TM = 〈Tr (t†t)M〉 are studied within the Anderson model of disordered conductors in dimensions d = 2, 3 and 4. Here t denotes a transmission matrix. We calculate the conductance G = T1, shot noise S = T1–T2, Fano factor F = S/G and ratios CM = TM /G, and we show indications that all of them are as good order parameters as the conductance itself. In the limit of infinite system size, two limiting values of and CM are found; the stable one in the metallic regime, and the unstable one, characterizing the critical point for d > 2. We present analytical expressions for both limiting values, together with a compact formula for current cumulants at 3D criticality. Our data confirm also Nazarov's microscopic theory [Phys. Rev. B 52, 4720 (1995)], as we show numerically for special linear combinations of TM. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

B and Bsdecay constants from QCD Duality at three loops

Using special linear combinations of finite energy sum rules which minimize the contribution of the unknown continuum spectral function, we compute the decay constants of the pseudoscalar mesons B and B_s. In the computation, we employ the recent three loop calculation of the pseudoscalar two-point function expanded in powers of the running bottom quark mass. The sum rules show remarkable stability over a wide range of the upper limit of the finite energy integration. We obtain the following results for the pseudoscalar decay constants: f_B=178 \pm 14 MeV and f_{B_s}=200 \pm 14 MeV. The results are somewhat lower than recent predictions based on Borel transform, lattice computations or HQET. Our sum rule approach of exploiting QCD quark hadron duality differs significantly from the usual ones, and we believe that the errors due to theoretical uncertainties are smaller.

Special linear combinations of orthogonal polynomials

The paper lists a number of problems that motivate consideration of special linear combinations of polynomials, orthogonal with the weight p ( x ) on the interval ( a , b ) . We study properties of the polynomials, as well as the necessary and sufficient conditions for their orthogonality. The special linear combinations of Chebyshev orthogonal polynomials of four kinds with absolutely constant coefficients hold a distinguished place in the class of such linear combinations.

FERMION STATES ON THE SPHERE S2

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The igenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with half-integer angular momenta. They are special linear combinations of conventional spherical spinors.

Linear Combinations of Gammaoperators in L p — Spaces

In 1951 P.L. Butzer defined in his thesis linear combinations of Bernstein polynomials. This type of combinations was extended to other operators by several authors. In this paper we define special linear combinations of Gammaoperators and we will prove a global direct result and an equivalence theorem. The proofs are based on a remarkable connection between the derivatives of the kernel of Gammaoperators and Laguerre polynomials and moreover between the moments of the Gammaoperators and Laguerre polynomials.

Polynomial evaluation and associated polynomials

We show a connection between the Clenshaw algorithm for evaluating a polynomial \(q_n\), expanded in terms of a system of orthogonal polynomials, and special linear combinations of associated polynomials. These results enable us to get the derivatives of \(q_n(z)\) analogously to the Horner algorithm for evaluating polynomials in monomial representations. Furthermore we show how a polynomial \(\hat q_n(z)\) given in monomial (!) representation can be evaluated for \(z\in{\Bbb C}\) using the Clenshaw algorithm without complex arithmetic. From this we get a connection between zeros of polynomials expanded in terms of Chebyshev polynomials and the corresponding polynomials in monomial representation with the same coefficients.

A note on fast Fourier transforms for nonequispaced grids

In this paper, we are concerned with fast Fourier transforms for nonequispaced grids. We propose a general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the polynomials by special linear combinations of translates of suitable functions ϕ. We derive estimates for the approximation error. In particular, we improve the estimates given by Dutt and Rokhlin [7]. As a practical consequence, we obtain a criterion for the choice of the parameters involved in the fast transforms.