Moment Problem Research Articles

Into the EFThedron and UV constraints from IR consistency

Recently it was proposed that the theory space of effective field theories with consistent UV completions can be described as a positive geometry, termed the EFThedron. In this paper we demonstrate that at the core, the geometry is given by the convex hull of the product of two moment curves. This makes contact with the well studied bi-variate moment problem, which in various instances has known solutions, generalizing the Hankel matrices of couplings into moment matrices. We extend these solutions to hold for more general bi-variate problem, and are thus able to obtain analytic expressions for bounds, which closely match (and in some cases exactly match) numerical results from semi-definite programing methods. Furthermore, we demonstrate that crossing symmetry in the IR imposes non-trivial constraints on the UV spectrum. In particular, permutation invariance for identical scalar scattering requires that any UV completion beyond the scalar sector must contain arbitrarily high spins, including at least all even spins ℓ ≤ 28, with the ratio of spinning spectral functions bounded from above, exhibiting large spin suppression. The spinning spectrum must also include at least one state satisfying a bound {m}_J^2<{M}_h^2frac{left({J}^2-12right)left({J}^4-32{J}^2+204right)}{8left(150-43{J}^2+2{J}^4right)} , where J2 = ℓ(ℓ+1), and Mh is the mass of the heaviest spin 2 state in the spectrum.

A NOTE ON OPEN QUESTIONS ASKED TO ANALYSIS AND NUMERICS CONCERNING THE HAUSDORFF MOMENT PROBLEM

Abstract We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function x ∈ L 2 (0, 1) from elements of the infinite dimensional sequence space ` 2 that characterize moments applied to the antiderivative of x. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator A, which is the composition of the compact simple integration operator mapping in L 2 (0, 1) and the non-compact Hausdorff moment operator B(H) mapping from L 2 (0, 1) to ` 2 . There is a seeming contradiction be- tween (a) numerical computations, which show (even for large n) an exponential decay of the singular values for n-dimensional matrices obtained by discretizing the operator A, and (b) a strongly limited smoothness of the well-known kernel k of the Hilbert-Schmidt operator A∗A. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness oc- curs in common just for C∞-kernels k. We recall arguments for the possible occurrence of a polynomial decay of the singular values of A, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.

The quantum moment problem for a classical random variable and a classification of interacting Fock spaces

The fact that any classical random variable with all moments has a quantum decomposition allows to associate to it a family of quantum moments. On the other hand, a classical random variable may have several inequivalent quantum decompositions, which lead to the same classical, but different quantum moments. Even in the simplest Central Limit Theorems (CLT), i.e. those of Bernoulli type, there are examples in which the corresponding quantum moments converge to the canonical quantum moments of the associated classical random variable, and examples in which this is not the case. This poses the problem to find a constructive criterium that characterizes the quantum moments associated to the canonical quantum decomposition (which is unique) with respect to the other ones. Theorem [Formula: see text] of the present paper provides such a criterium. Theorem [Formula: see text] deals with the case when one knows a priori that the quantum moments come from a central limit theorem (the motivation of the present paper arose in this context). It gives only a sufficient condition, but simpler to verify than the necessary and sufficient conditions of Theorem [Formula: see text]. Theorem [Formula: see text] naturally leads to a classification of Interacting Fock Spaces (IFS) into three types. We construct examples showing that all these possibilities can effectively take place. On the way, we prove that all the best known deformations of Heisenberg commutation relations can be obtained as special cases of a general construction within the algebraic approach to the theory of orthogonal polynomials.

Graph Recovery from Incomplete Moment Information

We investigate a class of moment problems, namely recovering a measure supported on the graph of a function from partial knowledge of its moments, as, for instance, in some problems of optimal transport or density estimation. We show that the sole knowledge of first degree moments of the function, namely linear measurements, is sufficient to obtain asymptotically all the other moments by solving a hierarchy of semidefinite relaxations (viewed as moment matrix completion problems) with a specific sparsity-inducing criterion related to a weighted \(\ell _1\)-norm of the moment sequence of the measure. The resulting sequence of optimal solutions converges to the whole moment sequence of the measure which is shown to be the unique optimal solution of a certain infinite-dimensional linear optimization problem (LP). Then one may recover the function by a recent extraction algorithm based on the Christoffel–Darboux kernel associated with the measure. Finally, the support of such a measure supported on a graph is a meager, very thin (hence sparse) set. Therefore, the LP on measures with this sparsity-inducing criterion can be interpreted as an analogue for infinite-dimensional signals of the LP in super-resolution for (sparse) atomic signals.

Sparse Moments of Univariate Step Functions and Allele Frequency Spectra

We study the univariate moment problem of piecewise-constant density functions on the interval [0,1] and its consequences for an inference problem in population genetics. We show that, up to closure, any collection of n moments is achieved by a step function with at most n − 1 breakpoints and that this bound is tight. We use this to show that any point in the n th coalescence manifold in population genetics can be attained by a piecewise constant population history with at most n − 2 changes. Both the moment cones and the coalescence manifold are projected spectrahedra and we describe the problem of finding a nearest point on them as a semidefinite program.

The Phi _3^4 Measure Has Sub-Gaussian Tails

We provide a very simple argument showing that the Phi ^4_3 measure does have quartic exponential tails, as expected from its formal expression. This shows that the corresponding moment problem is well-posed and provides a simple path to observing non-Gaussianity of the measure.

Tight Bounds for a Class of Data-Driven Distributionally Robust Risk Measures

This paper expands the notion of robust moment problems to incorporate distributional ambiguity using Wasserstein distance as the ambiguity measure. The classical Chebyshev-Cantelli (zeroth partial moment) inequalities, Scarf and Lo (first partial moment) bounds, and semideviation (second partial moment) in one dimension are investigated. The infinite dimensional primal problems are formulated and the simpler finite dimensional dual problems are derived. A principal motivating question is how does data-driven distributional ambiguity affect the moment bounds. Towards answering this question, some theory is developed and computational experiments are conducted for specific problem instances in inventory control and portfolio management. Finally some open questions and suggestions for future research are discussed.

Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere

We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optimization. Using the reformulation-linearization technique (RLT) and Lasserre-type hierarchies, relaxations of the problem are introduced and analyzed. For our analysis we assume throughout the existence of a dual optimal solution as well as strong duality. For the GMP over the simplex we prove a convergence rate of O(1/r) for a linear programming, RLT-type hierarchy, where r is the level of the hierarchy, using a quantitative version of Pólya’s Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020. https://doi.org/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65–92, 2008. https://doi.org/10.1007/s10107-006-0085-1) over the sphere has a convergence rate of O(1/r^2). Moreover, we show the introduced linear RLT-relaxation is a generalization of a hierarchy for minimizing forms of degree d over the simplex, introduced by De Klerk et al. (J Theor Comput Sci 361(2–3):210–225, 2006).

On the Truncated Multidimensional Moment Problems in Cn

We consider the problem of finding a (non-negative) measure μ on B(Cn) such that ∫Cnzkdμ(z)=sk, ∀k∈K. Here, K is an arbitrary finite subset of Z+n, which contains (0,…,0), and sk are prescribed complex numbers (we use the usual notations for multi-indices). There are two possible interpretations of this problem. Firstly, one may consider this problem as an extension of the truncated multidimensional moment problem on Rn, where the support of the measure μ is allowed to lie in Cn. Secondly, the moment problem is a particular case of the truncated moment problem in Cn, with special truncations. We give simple conditions for the solvability of the above moment problem. As a corollary, we have an integral representation with a non-negative measure for linear functionals on some linear subspaces of polynomials.

Inverse Source Identification on Poisson Equation with Cauchy Conditions As a Generalized Problem of Moments

We consider the problem of finding a pair of functions φ( x, t) and w( x, t) that satisfy the equation wxx( x, t) + wtt( x, t) = φ( x, t) under Cauchy boundary conditions. We will see that an approximate solution can be found using the techniques of generalized inverse problem of moments and find dimensions for the error of the estimated solution.

Recovering the Initial Condition of Parabolic Equations from Lateral Cauchy Data As a Generalized Problem of Moments

It will be shown that to recover the initial condition and finding the solution of a parabolic equation with Cauchy conditions can be solved in two steps writing the parabolic equation as an integral equation, which can be solved numerically applying the techniques of inverse generalized moments problem. In a first step, the initial condition is found in approximate form, and in a second step we numerically approximate the solution of the parabolic equation using the initial condition found. The method is illustrated with examples.

Solving a Class of Two-dimensional Nonlinear Volterra Integral Equations As a Generalized Problem of Moments

It will be shown that solve an equation two-dimensional Volterra nonlinear can be solved numerically applying the techniques of inverse generalized moments problem in two steps writing the Volterra's equation as a Klein-Gordon equation of the form wtt – wxx = H(x, t), which H(x, t) it is unknown. In a first step, H(x, t) is numerically approximate, and in a second step we numerically approximate the solution of Klein-Gordon equation using the H(x, t) previously approximated. The method is illustrated with examples.

On the statement and solvability of the $l$-problem of moments for fractional systems

On the statement and solvability of the $l$-problem of moments for fractional systems

On the similarity of complex symmetric operators to perturbations of restrictions of normal operators

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove the similarity to rank-one perturbations of restrictions of normal operators. The main tools are a truncated moment problem in $\mathbb{C}$, and some objects similar to objects from the theory of spectral problems for Jacobi matrices.

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

Using moment approximations to study the density of jump driven SDEs

In order to study the regularity of the density of a solution of a infinite activity jump driven stochastic differential equation we consider the following two-step approximation method. First, we use the solution of the moment problem in order to approximate the small jumps by another whose Lévy measure has finite support. In a second step we replace the approximation of the first two moments by a small noise Brownian motion based on the Assmussen-Rosiński approach. This approximation needs to satisfy certain properties in order to apply the “balance” method which allows the study of densities for the solution process based on Malliavin Calculus for the Brownian motion. Our results apply to situations where the Lévy measure is absolutely continuous with respect to the Lebesgue measure or purely atomic measures or combinations of them.

Hausdorff moment problem and nonlinear time optimality

A complete analytic solution for the time-optimal control problem for nonlinear control systems of the form ẋ1 = u, ẋj = ẋ1j−1, j = 2, …, n, is obtained for arbitrary n. In the paper we present the following surprising observation: this nonlinear optimality problem leads to a truncated Hausdorff moment problem, which gives analytic tools for finding the optimal time and optimal controls. Being homogeneous, the mentioned system approximates affine systems from a certain class in the sense of time optimality. Therefore, the obtained results can be used for solving the time-optimal control problem for systems from this class.

Method of moments in the problem of inversion of the Laplace transform and its regularization

Integral equations of the first kind are considered, which belong to the class of ill-posed problems. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to illconditioned systems of linear algebraic equations, in which the unknowns are the coefficients of the expansion in a series in the shifted Legendre polynomials of some function that simply expresses in terms of the sought original. This function is found as a solution to a certain finite moment problem in a Hilbert space. To obtain a reliable solution to the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated, focused on a priori low degree of smoothness of the desired original. The results of numerical experiments are presented, confirming the effectiveness of the proposed inversion algorithm.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

Society-based Grey Wolf Optimizer for large scale Combined Heat and Power Economic Dispatch problem considering power losses

Combined Heat and Power Economic Dispatch (CHPED) reflects a momentous optimization and operation problem in the power systems for optimally allocating the produced heat and power to the committed Power-only (PO), Heat-only (HO), and CHP units. In this work, a new society-based optimization algorithm using social hierarchy of grey wolves, namely Society-based Grey Wolf Optimizer (SGWO) is proposed for the CHPED problem. This scheme divides the group of wolves to some societies so that each society has its own leader. These leaders follow the dominant wolf and also guide their societies in which the classification of wolves is based on the social hierarchy. Moreover a novel attacking and hunting the prey is suggested to change the effects of dominant, ordinate, and subordinate wolves in their new roles in societies. In order to verify the capabilities of the SGWO, simulations are conducted through two large-scale CHPED problems with the related challenges like Valve-Point Loading Effect (VPLE) and Prohibited Zones (PZ) of PO units, mutual dependency of produced heat and power of CHP units, and especially transmission power losses. Moreover, it is evaluated on twenty-three standard functions to verify its stability on different low- and high-dimensional functions. Comparisons based on the obtained solutions by different methods demonstrate the robustness and superior performance of the presented technique to fast provide better optimum point (more economical benefits) and solution quality meeting all equality and inequality constraints. In addition, the results of CHPED indicate that ≈0.5% reduction in the production cost results in up to $2.6−$34×106 increase in Annual Cost Saving (ACS). Also, the transmission losses and PZs of POs can increase the production cost by about 1.4% which leads to over $1.3×107 reduction in ACS in comparison with ignoring them. Moreover, this condition increases the computational time by about 35% while the proposed method can still be up to 13−26 times faster than the other analysed algorithms.