Expansions in Terms of q-Laguerre Heat Polynomials

In a previous paper M. S. Ben Hammouda and Akram Nemri derived criteria for the expansion of solutions u(x, t) from the generalized q-heat equation, in series of polynomial solutions \({p_{n}^{\alpha}}\) , thus extending an analogous theory of the ordinary heat equation developed by P. C. Rosenbloom and D. V. Widder. It is the goal to carry out a parallel study for the q-Laguerre differential heat equation and establish the region of convergence of the series of q-Laguerre heat polynomials and their temperature transforms.     

The partition function modulo 3 in arithmetic progressions

Let \(p(n)\) be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers \(N\) for which \(p(N)\) is not congruent to \(0\;(\mathrm{mod}\;3)\). Radu proved this conjecture in 2010 using the work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono’s conjecture in the special case where the modulus of the arithmetic progression is a power of \(3\) by applying a method of Boylan and Ono and using the work of Bellaïche and Khare generalizing Nicolas and Serre’s results on the local nilpotency of the Hecke algebra.     

An exact method for minimizing the total treatment time in intensity-modulated radiotherapy

In this paper, we solve a combinatorial optimization problem that arises from the treatment planning of a type of radiotherapy where intensity is modulated by multileaf collimators (MLC) in a step-and-shoot manner. In Ernst et al [INFORMS Journal on Computing 21 (4) (2009): 562–574], we proposed an exact method for minimizing the number of MLC apertures needed for a treatment. Our method outperformed the fastest algorithms available at the time. We refer to our method as the CPI method. We now attempt to minimize the total treatment time by modifying our CPI method. This modification involves non-trivial work, as some of the search space elimination schemes used in the CPI method cannot be applied in here. In our numerical experiments, we again compare our new method with the fastest algorithms currently available. There has been significant recent research in this area; hence we compare our method with those published in Wake et al [Computers and Operations Research 36 (2009): 795–810], Ta?kin et al [Operations Research 58 (3) (2010): 674–690] and Cambazard et al [CPAIRO (2010): 1–16]. The numerical comparisons indicate that our method generally outperformed the first two, while being competitive with the third.     

An unconditionally stable spatial sixth-order CCD-ADI method for the two-dimensional linear telegraph equation

The telegraph equation is one of the important models in many physics and engineering. In this work, we discuss the high-order compact finite difference method for solving the two-dimensional second-order linear hyperbolic equation. By using a combined compact finite difference method for the spatial discretization, a high-order alternating direction implicit method (ADI) is proposed. The method is O(τ² + h⁶) accurate, where τ, h are the temporal step size and spatial size, respectively. Von Neumann linear stability analysis shows that the method is unconditionally stable. Finally, numerical examples are used to illustrate the high accuracy of the new difference scheme.     

Exploiting algebraic structure in global optimization and the Belgian chocolate problem

The Belgian chocolate problem involves maximizing a parameter \(\delta \) over a non-convex region of polynomials. In this paper we detail a global optimization method for this problem that outperforms previous such methods by exploiting underlying algebraic structure. Previous work has focused on iterative methods that, due to the complicated non-convex feasible region, may require many iterations or result in non-optimal \(\delta \). By contrast, our method locates the largest known value of \(\delta \) in a non-iterative manner. We do this by using the algebraic structure to go directly to large limiting values, reducing the problem to a simpler combinatorial optimization problem. While these limiting values are not necessarily feasible, we give an explicit algorithm for arbitrarily approximating them by feasible \(\delta \). Using this approach, we find the largest known value of \(\delta \) to date, \(\delta = 0.9808348\). We also demonstrate that in low degree settings, our method recovers previously known upper bounds on \(\delta \) and that prior methods converge towards the \(\delta \) we find.     

Theoretical investigations of the new Cokriging method for variable-fidelity surrogate modeling

Cokriging is a variable-fidelity surrogate modeling technique which emulates a target process based on the spatial correlation of sampled data of different levels of fidelity. In this work, we address two theoretical questions associated with the so-called new Cokriging method for variable-fidelity modeling:

1. (1)
   A mandatory requirement for the well-posedness of the Cokriging emulator is the positive definiteness of the associated Cokriging correlation matrix. Spatial correlations are usually modeled by positive definite correlation kernels, which are guaranteed to yield positive definite correlation matrices for mutually distinct sample points. However, in applications, low-fidelity information is often available at high-fidelity sample points and the Cokriging predictor may benefit from the additional information provided by such an inclusive sampling. We investigate the positive definiteness of the Cokriging covariance matrix in both of the aforementioned cases and derive sufficient conditions for the well-posedness of the Cokriging predictor.
    
2. (2)
   The approximation quality of the Cokriging predictor is highly dependent on a number of model- and hyper-parameters. These parameters are determined by the method of maximum likelihood estimation. For standard Kriging, closed-form optima of the model parameters along hyper-parameter profile lines are known. Yet, these do not readily transfer to the setting of Cokriging, since additional parameters arise, which exhibit a mutual dependence. In previous work, this obstacle was tackled via a numerical optimization. Here, we derive closed-form optima for all Cokriging model parameters along hyper-parameter profile lines. The findings are illustrated by numerical experiments.
    

     

Polynomial convergence of two higher order interior-point methods for $$P_*(\kappa )$$-LCP in a wide neighborhood of the central path

In this paper, we propose two interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problems (\(P_*(\kappa )\)-LCPs): a high order large update path following method and a high order corrector–predictor method. Both algorithms generate sequences of iterates in the wide neighborhood \((\mathcal {N}_{2,\tau }^-(\alpha ))\) of the central path introduced by Ai and Zhang. The methods do not depend on the handicap \(\kappa \) of the problem so that they work for any \(P_*(\kappa )\)-LCP . They have \(O((1 +\kappa )\sqrt{n}L)\) iteration complexity, the best-known iteration complexity obtained so far by any interior-point method for solving \(P_*(\kappa )\)-LCP. The high order corrector–predictor algorithm is superlinearly convergent with Q-order \((m_p+1)\) for problems that admit a strict complementarity solution and \((m_p+1)/2\) for general problems, where \(m_p\) is the order of the predictor step.     

A recursive construction for simple <Emphasis Type="Italic">t</Emphasis>-designs using resolutions

This work presents a recursive construction for simple t-designs using resolutions of the ingredient designs. The result extends a construction of t-designs in our recent paper van Trung (Des Codes Cryptogr 83:493–502, 2017). Essentially, the method in van Trung (Des Codes Cryptogr 83:493–502, 2017) describes the blocks of a constructed design as a collection of block unions from a number of appropriate pairs of disjoint ingredient designs. Now, if some pairs of these ingredient t-designs have both suitable s-resolutions, then we can define a distance mapping on their resolution classes. Using this mapping enables us to have more possibilities for forming blocks from those pairs. The method makes it possible for constructing many new simple t-designs. We give some application results of the new construction.     

Optimal decay rates of classical solutions for the full compressible MHD equations

In this paper, we are concerned with optimal decay rates for higher-order spatial derivatives of classical solutions to the full compressible MHD equations in three-dimensional whole space. If the initial perturbation is small in \({H^3}\)-norm and bounded in \({L^q(q\in \left[1, \frac{6}{5} \right))}\)-norm, we apply the Fourier splitting method by Schonbek (Arch Ration Mech Anal 88:209–222, 1985) to establish optimal decay rates for the second-order spatial derivatives of solutions and the third-order spatial derivatives of magnetic field in \({L^2}\)-norm. These results improve the work of Pu and Guo (Z Angew Math Phys 64:519–538, 2013).     

An unexpected Ramanujan-type congruence modulo 7 for 4-colored generalized Frobenius partitions

In this paper, we present an unexpected Ramanujan-type congruence modulo 7 for \(c\phi _4(n)\), which denotes the number of generalized Frobenius partitions of n with 4 colors. This work extends the recent work of Lin on \(c\phi _4\) modulo 7.     

Global solutions for a nonlinear wave equation

In this work the existence of a global solution for the mixed problem associated to the nonlinear equation

is proved in a Hilbert space framework by using Galerkin method.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Iteration complexity of randomized block-coordinate descent methods for minimizing a composite function

In this paper we develop a randomized block-coordinate descent method for minimizing the sum of a smooth and a simple nonsmooth block-separable convex function and prove that it obtains an $\varepsilon $ -accurate solution with probability at least $1-\rho $ in at most $O((n/\varepsilon ) \log (1/\rho ))$ iterations, where $n$ is the number of blocks. This extends recent results of Nesterov (SIAM J Optim 22(2): 341–362, 2012), which cover the smooth case, to composite minimization, while at the same time improving the complexity by the factor of 4 and removing $\varepsilon $ from the logarithmic term. More importantly, in contrast with the aforementioned work in which the author achieves the results by applying the method to a regularized version of the objective function with an unknown scaling factor, we show that this is not necessary, thus achieving first true iteration complexity bounds. For strongly convex functions the method converges linearly. In the smooth case we also allow for arbitrary probability vectors and non-Euclidean norms. Finally, we demonstrate numerically that the algorithm is able to solve huge-scale $\ell _1$ -regularized least squares problems with a billion variables.     

On stabilizability of evolution systems of partial differential equations on ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>×[0,+∞) by time-delayed feedback controlsby time-delayed feedback controls

In this work we obtain sufficient conditions for stabilizability by time-delayed feedback controls for the system

$\frac{{\partial w\left( {x,t} \right)}}{{\partial t}} = A(D_x )w(x,t) - A(D_x )u(x,t), x \in \mathbb{R}^n , t > h, $

where D _x =(-i?/?x ₁,...-i?/?x _n ), A(σ) and B(σ) are polynomial matrices (m×m), det B(σ)≡0 on ? ⁿ , w is an unknown function, u(·,t)=P(D _x )w(·,t?h) is a control, h>0. Here P is an infinite differentiable matrix (m×m), and the norm of each of its derivatives does not exceed Γ(1+|σ|²)^γ for some Γ, γ∈? depending on the order of this derivative. Necessary conditions for stabilizability of this system are also obtained. In particular, we study the stabilizability problem for the systems corresponding to the telegraph equation, the wave equation, the heat equation, the Schrödinger equation and another model equation. To obtain these results we use the Fourier transform method, the Lojasiewicz inequality and the Tarski—Seidenberg theorem and its corollaries. To choose an appropriate P and stabilize this system, we also prove some estimates of the real parts of the zeros of the quasipolynomial det {Iλ-A(σ)+B(σ)P(σ)e ^-hλ.

     

Gradient shrinking Ricci solitons of half harmonic Weyl curvature

Gradient Ricci solitons and metrics with half harmonic Weyl curvature are two natural generalizations of Einstein metrics on four-manifolds. In this paper we prove that if a metric has structures of both gradient shrinking Ricci soliton and half harmonic Weyl curvature, then except for three examples, it has to be an Einstein metric with positive scalar curvature. Precisely, we prove that a four-dimensional gradient shrinking Ricci soliton with \(\delta W^{\pm }=0\) is either Einstein, or a finite quotient of \(S^3\times \mathbb {R}\), \(S^2\times \mathbb {R}^2\) or \(\mathbb {R}^4\). We also prove that a four-dimensional gradient Ricci soliton with constant scalar curvature is either Kähler–Einstein, or a finite quotient of \(M\times \mathbb {C}\), where M is a Riemann surface. The method of our proof is to construct a weighted subharmonic function using curvature decompositions and the Weitzenböck formula for half Weyl curvature, and the method was motivated by previous work (Gursky and LeBrun in Ann Glob Anal Geom 17:315–328, 1999; Wu in Einstein four-manifolds of three-nonnegative curvature operator 2013; Trans Am Math Soc 369:1079–1096, 2017; Yang in Invent Math 142:435–450, 2000) on the rigidity of Einstein four-manifolds with positive sectional curvature, and previous work (Cao and Chen in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013; Catino in Math Ann 35:629–635, 2013) on the rigidity of gradient Ricci solitons.     

Special issue: recent progress on iterative methods for large systems of equations

Let A x = b be a large and sparse system of linear equations where A is a nonsingular matrix. An approximate solution is frequently obtained by applying preconditioned iterations. Consider the matrix B = A + P Q ^T where \(P, Q \in \mathbb {R}^{n \times k}\) are full rank matrices. In this work, we study the problem of updating a previously computed preconditioner for A in order to solve the updated linear system B x = b by preconditioned iterations. In particular, we propose a method for updating a Balanced Incomplete Factorization preconditioner. The strategy is based on the computation of an approximate Inverse Sherman-Morrison decomposition for an equivalent augmented linear system. Approximation properties of the preconditioned matrix and an analysis of the computational cost of the algorithm are studied. Moreover, the results of the numerical experiments with different types of problems show that the proposed method contributes to accelerate the convergence.     

A note on the independent domination number versus the domination number in bipartite graphs

Let γ(G) and i(G) be the domination number and the independent domination number of G, respectively. Rad and Volkmann posted a conjecture that i(G)/γ(G) ≤ Δ(G)/2 for any graph G, where Δ(G) is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than Δ(G)/2 are provided as well.     

Implicit QR for rank-structured matrix pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of Hermitian matrices is adjusted to work with matrix pencils arising from zerofinding problems for polynomials expressed in Chebyshev-like bases. The modified QZ algorithm computes the generalized eigenvalues of certain $N\times N$ rank structured matrix pencils using $O(N^2)$ flops and $O(N)$ memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.     

A modified weak Galerkin finite element methods for convection–diffusion problems in 2D

In this paper, we develop a modified weak Galerkin finite element method on arbitrary grids for convection–diffusion problems in two dimensions based on our previous work (Wang et al., J Comput Appl Math 271, 319–327, 2014), in which we only considered second order Poisson equations and thus only introduced a modified weak gradient operator. This method, called MWG-FEM, is based on a modified weak gradient operator and weak divergence operator which is put forward in this paper. Optimal order error estimates are established for the corresponding MWG-FEM approximations in both a discrete \(H^1\) norm and the standard \(L^2\) norm. Numerical results are presented to demonstrate the robustness, reliability, and accuracy of the MWG-FEM.