A Posteriori Error Estimates for Nonconforming Approximations of Evolutionary Convection-Diffusion Problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.Comment: 10 page

A Neural Model of How the Brain Represents and Compares Multi-Digit Numbers: Spatial and Categorical Processes

Both animals and humans are capable of representing and comparing numerical quantities, but only humans seem to have evolved multi-digit place-value number systems. This article develops a neural model, called the Spatial Number Network, or SpaN model, which predicts how these shared numerical capabilities are computed using a spatial representation of number quantities in the Where cortical processing stream, notably the Inferior Parietal Cortex. Multi-digit numerical representations that obey a place-value principle are proposed to arise through learned interactions between categorical language representations in the What cortical processing stream and the Where spatial representation. It is proposed that learned semantic categories that symbolize separate digits, as well as place markers like "tens," "hundreds," "thousands," etc., are associated through learning with the corresponding spatial locations of the Where representation, leading to a place-value number system as an emergent property of What-Where information fusion. The model quantitatively simulates error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. In the Where cortical process, it is proposed that transient responses to inputs are integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence. Neural mechanisms are defined which give rise to an ordered spatial numerical map ordering and Weber law characteristics as emergent properties. The dynamics of numerical comparison are encoded in activity pattern changes within this spatial map. Such changes cause a "directional comparison wave" whose properties mimic data about numerical comparison. These model mechanisms are variants of neural mechanisms that have elsewhere been used to explain data about motion perception, attention shifts, and target tracking. Thus, the present model suggests how numerical representations may have emerged as specializations of more primitive mechanisms in the cortical Where processing stream. The model's What-Where interactions can explain human psychophysical data, such as error rates and reaction times, about multi-digit (base 10) numerical stimuli, and describe how such a competence can develop through learning. The SpaN model and its explanatory range arc compared with other models of numerical representation.Defense Advanced Research Projects Agency and the Office of Naval Research (N00014-95-1-0409); National Science Foundation (IRI-97-20333

A Neural Model of How The Brain Represents and Compares Numbers

Many psychophysical experiments have shown that the representation of numbers and numerical quantities in humans and animals is related to number magnitude. A neural network model is proposed to quantitatively simulate error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. Transient responses to inputs arc integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence. The dynamics of numerical comparison are encoded in activity pattern changes within this spatial map. Such changes cause a "directional comparison wave" whose properties mimic data about numerical comparison. These model mechanisms are variants of neural mechanisms that have elsewhere been used to explain data about motion perception, attention shifts, and target tracking. Thus, the present model suggests how numerical representations may have emerged as specializations of more primitive mechanisms in the cortical Where processing stream.National Science Foundation (IRI-97-20333); Defense Advanced research Projects Agency and the Office of Naval Research (N00014-95-1-0409); National Institute of Health (1-R29-DC02952-01

A Neural Model of Multidigit Numerical Representation and Comparison

The Extended Spatial Number Network (ESpaN) is a neural model that simulates processing of high-level numerical stimuli such as multi-digit numbers. The ESpaN model targets the explanation of human psychophysical data, such as error rates and reaction times, about multi-digit (base 10) numerical stimuli, and describes how such a competence can develop through learning. The model suggests how the brain represents and processes an open-ended set of numbers and their regularities, such as the place-value structure, with finite resources in the brain. The model does that by showing how a multi-digit spatial number map forms through interactions with learned semantic categories that symbolize separate digits, as well as place markers like "tens," "hundreds," "thousands," etc. When number-stimuli are presented to the network, they trigger learning of associations between specific semantic categories and corresponding spatial locations of the spatial number map that together build a multi-digit spatial representation. Training of the network is aimed at portraying the process of development of human numerical competence during the first years of a child's life. The earlier SpaN model proposed a spatial number map, which both human and animal possess in their Where cortical processing stream, that can explain many data about analog numerical representation and comparison. The ESpaN model shows how learned cognitive categories in the What cortical processing stream can extend numerical competence to multi-digit numbers with a place-value structure. The ESpaN model hereby suggests how cortical cognitive and spatial processes can utilize a learned What-and-Where interstream interaction to control the development of multidigit numerical abilities.National Science Foundation (IRI-97-20333); Defense Advanced Research Projects Agency and the Office of Naval Research (NOOOI4-95-I-0409

Verifications of primal energy identities for variational problems with obstacles

We discuss error identities for two classes of free boundary problems generated by obstacles. The identities suggest true forms of the respective error measures which consist of two parts: standard energy norm and a certain nonlinear measure. The latter measure controls (in a weak sense) approximation of free boundaries. Numerical tests confirm sharpness of error identities and show that in different examples one or another part of the error measure may be dominant.Comment: 8 pages, 2 figures, conference paper: LSSC (Large-Scale scientific computing), Sozopol, Bulgaria, 2017. The final version will be published at Springe