We prove that for a homogeneous linear partial differential operator $\mathcal {A}$ of order $k \le 2$ and an integrable map $f$ taking values in the essential range of that operator, there exists a function $u$ of special bounded variation satisfying

\[ \mathcal{A} u(x)= f(x) \qquad \text{almost everywhere}. \]

$\mathbf {R}^N$

$0 \le m < N$

$m$

$(m+1)$

An explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

Natural scenes contain a variety of visual cues that facilitate boundary perception (e.g., luminance, contrast, and texture). Here we explore whether single neurons in early visual cortex can process both contrast and texture cues. We recorded neural responses in cat A18 to both illusory contours formed by abutting gratings (ICs, texture-defined) and contrast-modulated gratings (CMs, contrast-defined). We found that if a neuron responded to one of the two stimuli, it also responded to the other. These neurons signaled similar contour orientation, spatial frequency, and movement direction of the two stimuli. A given neuron also exhibited similar selectivity for spatial frequency of the fine, stationary grating components (carriers) of the stimuli. These results suggest that the cue-invariance of early cortical neurons extends to different kinds of texture or contrast cues, and might arise from a common nonlinear mechanism.

In this paper, by use of affine biquadratic elements, we constructand analyze a finite volume element scheme for elliptic equations onquadrilateral meshes. The scheme is shown to be of second-order inH 1-norm, provided that each quadrilateral in partition is almosta parallelogram. Numerical experiments are presented to confirm theusefulness and efficiency of the method.

Neurons in the mammalian visual cortex have been found to respond to second-order features which are not defined by changes in luminance over the retina (Albright, 1992; Zhou & Baker, 1993, 1994, 1996; Mareschal & Baker, 1998a,b). The detection of these stimuli is most often accounted for by a separate nonlinear processing stream, acting in parallel to the linear stream in the visual system. Here we examine the two-dimensional spatial properties of these nonlinear neurons in area 18 using envelope stimuli, which consist of a high spatial-frequency carrier whose contrast is modulated by a low spatial-frequency envelope. These stimuli would fail to elicit a response in a conventional linear neuron because they are designed to contain no spatial-frequency components overlapping the neuron's luminance defined passband. We measured neurons' responses to these stimuli as a function of both the relative spatial frequencies and relative orientations of the carrier and envelope. Neurons' responses to envelope stimuli were narrowband to the carrier spatial frequency, with optimal values ranging from 8- to 30-fold higher than the envelope spatial frequencies. Neurons' responses to the envelope stimuli were strongly dependent on the orientation of the envelope and less so on the orientation of the carrier. Although the selectivity to the carrier orientation was broader, neurons' responses were clearly tuned, suggesting that the source of nonlinear input is cortical. There was no fixed relationship between the optimal carrier and envelope spatial frequencies or orientations, such that nonlinear neurons responding to these stimuli could perhaps respond to a variety of stimuli defined by changes in scale or orientation.

Two classes of ergodic stationary multitype spatial point processes are constructed. These processes have the property that interactions between the types exist, but cannot be detected using standard second-order methods of analysis. Simulations indicate that the interactions can, however, be detected by using ‘empty space' techniques.