In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.

The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

Research in warehouse optimization has gotten increased attention in the last few years due to e-commerce. The warehouse contains a waste range of different products. Due to the nature of the individual order, it is challenging to plan the picking list to optimize the material flow in the process. There are also challenges in minimizing costs and increasing production capacity, and this complexity can be defined as a multidisciplinary optimization problem with an IDF nature. In recent years the use of parallel computing using GPGPUs has become increasingly popular due to the introduction of CUDA C and accompanying applications in, e.g., Python. 

In the case study at the company in the field of retail, a case study including a system design optimization (SDO) resulted in an increase in throughput with well over 20% just by clustering different categories and suggesting in which sequence the orders should be picked during a given time frame. 

The options provided by implementing a distributed high-performance computing network based on GPUs for subsystem optimization have shown to be fruitful in developing a functioning SDO for warehouse optimization. The toolchain can be used for designing new warehouses or evaluating and tuning existing ones. 

With insight from linguistics that degrees of text cohesion are similar to forces in physics, and the frequent use of the energy concept in text categorization by machine learning, we consider the applicability of particle-wave duality to semantic content inherent in index terms. Wave-like interpretations go back to the regional nature of such content, utilizing functions for its representation, whereas content as a particle can be conveniently modelled by position vectors. Interestingly, wave packets behave like particles, lending credibility to the duality hypothesis. We show in a classical mechanics framework how metaphorical term mass can be computed.

In this paper, we consider a system modeling an axially moving viscoelastic string under a spatiotemporally varying tension. A mechanism consisted of a hydraulic touch-roll actuator pointed at the right boundary to suppress the transverse vibrations. We adopt the multiplier method to design a boundary control law and to prove an exponential stability result. However, this result is obtained provided that the lower bound of the tension in the string is larger than its time derivative. The effectiveness of the proposed control law is demonstrated via simulations. 

Finite-difference (FD) modelling of seismic waves in the vicinity of dipping interfaces gives rise to artefacts. Examples are phase and amplitude errors, as well as staircase diffractions. Such errors can be reduced in two general ways. In the first approach, the interface can be anti-aliased (i.e. with an anti-aliased step-function, or a lowpass filter). Alternatively, the interface may be replaced with an equivalent medium (i.e. using Schoenberg & Muir (SM) calculus or orthorhombic averaging). We test these strategies in acoustic, elastic isotropic, and elastic anisotropic settings. Computed FD solutions are compared to analytical solutions. We find that in acoustic media, anti-aliasing methods lead to the smallest errors. Conversely, in elastic media, the SM calculus provides the best accuracy. The downside of the SM calculus is that it requires an anisotropic FD solver even to model an interface between two isotropic materials. As a result, the computational cost increases compared to when using isotropic FD solvers. However, since coarser grid spacings can be used to represent the dipping interfaces, the two effects (an expensive FD solver on a coarser FD grid) equal out. Hence, the SM calculus can provide an efficient means to reduce errors, also in elastic isotropic media.

Semidiscrete finite element approximation of the linear stochastic wave equation (LSWE) with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multidimensional domains and spatially correlated noise. Numerical examples illustrate the theory. 

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag--Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.  

This paper suggests a modification of the Conformal Prediction framework for regression that will strengthen the associated guarantee of validity. We motivate the need for this modification and argue that our conformal regressors are more closely tied to the actual error distribution of the underlying model, thus allowing for more natural interpretations of the prediction intervals. In the experimentation, we provide an empirical comparison of our conformal regressors to traditional conformal regressors and show that the proposed modification results in more robust two-tailed predictions, and more efficient one-tailed predictions.

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order p and prove convergence of order p, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.

Standard discontinuous Galerkin methods, based on piecewise polynomials of degree q=0,1, are considered for temporal semi-discretization for second-order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results. 

An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is presented such that it can be used for adaptive strategies based on dual weighted residual methods. A posteriori error estimates based on weighted global projections and local projections are also proved.   

A hyperbolic integro-differential equation is considered, as a model problem, where the convolution kernel is assumed to be either smooth or no worse than weakly singular. Well-posedness of the problem is studied in the context of semigroup of linear operators, and regularity of any order is proved for smooth kernels. A continuous space–time finite element method of order 1 is formulated for the problem. Stability of the discrete dual problem is proved, which is used to obtain optimal order a priori estimates via duality arguments. The theory is illustrated by an example. 

We study a second order hyperbolic initial-boundary value partial differential equation (PDE) with memory that results in an integro-differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016 

A hyperbolic type integro-differential equation with two weakly singular kernels is considered together with mixed homogeneous Dirichlet and non-homogeneous Neumann boundary conditions. Existence and uniqueness of the solution is proved by means of Galerkin's method. Regularity estimates are proved and the limitations of the regularity are discussed. The approach presented here is also used to prove regularity of any order for models with smooth kernels, that arise in the theory of linear viscoelasticity, under the appropriate assumptions on data.

In this work, we investigate the asymptotic behavior of solutions of a viscoelastic flexible marine riser with vessel dynamics. Under a suitable control applied at the top end of the riser, we establish explicit decay rates for a large class of relaxation functions. In particular, exponentially and polynomially (or power type) decaying functions are included in this class. Our method is based on the multiplier technique. Numerical simulations justifying the effectiveness of the proposed boundary control to suppress the vibrations of the flexible marine riser are provided.

A hierarchy of semidefinite programming (SDP) relaxations approximates the global optimum of polynomial optimization problems of noncommuting variables. Generating the relaxation, however, is a computationally demanding task, and only problems of commuting variables have efficient generators. We develop an implementation for problems of noncommuting problems that creates the relaxation to be solved by SDPA -- a high-performance solver that runs in a distributed environment. We further exploit the inherent sparsity of optimization problems in quantum physics to reduce the complexity of the resulting relaxations. Constrained problems with a relaxation of order two may contain up to a hundred variables. The implementation is available in Python. The tool helps solve problems such as finding the ground state energy or testing quantum correlations.

The Trotter-Suzuki approximation leads to an efficient algorithm for solving the time-dependent Schrödinger equation. Using existing highly optimized CPU and GPU kernels, we developed a distributed version of the algorithm that runs efficiently on a cluster. Our implementation also improves single node performance, and is able to use multiple GPUs within a node. The scaling is close to linear using the CPU kernels, whereas the efficiency of GPU kernels improve with larger matrices. We also introduce a hybrid kernel that simultaneously uses multicore CPUs and GPUs in a distributed system. This kernel is shown to be efficient when the matrix size would not fit in the GPU memory. Larger quantum systems scale especially well with a high number nodes. The code is available under an open source license.

In this paper we introduce a MapReduce-based implementation of self-organizing maps that performs compute-bound operations on distributed GPUs. The kernels are optimized to ensure coalesced memory access and effective use of shared memory. We have performed extensive tests of our algorithms on a cluster of eight nodes with two NVidia Tesla M2050 attached to each, and we achieve a 10x speedup for self-organizing maps over a distributed CPU algorithm.

With the emergence of high-performance computing instances in the cloud, massive scale computations have become available to technically every organization. Digital libraries typically employ a data-intensive infrastructure, but given the resources, advanced services based on data and text mining could be developed. A fundamental issue is the ease of development and integration of such services. We demonstrate the feasibility by providing a case study on a visual machine learning algorithm with MapReduce running in the cloud in a small cluster.