Douglas' solution of the Plateau problem

Using ideas suggested by some recent developments in string theory, we give here an elementary demonstration of one of the key steps in Douglas' celebrated proof of the existence of solutions of the Plateau problem in n dimensions.     

Pompeiu's problem on discrete space

Let [unk] be a finite family of finite subsets of the n-dimensional lattice Zⁿ, and let τ denote the group of all translations of Zⁿ. We shall here consider the Pompeiu problem for the family [unk]—namely, to determine when the only function f:Zⁿ → C such that [Formula: see text] is the zero function.     

Solution of the embedding problem and decomposition of symmetric matrices.

A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a general distance matrix by a one-to-one transformation. Embedding of a distance matrix yields a decomposition of the associated symmetric matrix in the form of a sum over outer products of a linear independent system of coordinate vectors. It is shown that such a decomposition exists for every symmetric matrix and that it is numerically stable. From this decomposition, the rank and the numbers of positive, negative, and zero eigenvalues of the symmetric matrix are obtained directly.     

Solution of the pulse width modulation problem using orthogonal polynomials and Korteweg-de Vries equations.

The mathematical underpinning of the pulse width modulation (PWM) technique lies in the attempt to represent "accurately" harmonic waveforms using only square forms of a fixed height. The accuracy can be measured using many norms, but the quality of the approximation of the analog signal (a harmonic form) by a digital one (simple pulses of a fixed high voltage level) requires the elimination of high order harmonics in the error term. The most important practical problem is in "accurate" reproduction of sine-wave using the same number of pulses as the number of high harmonics eliminated. We describe in this paper a complete solution of the PWM problem using Pade approximations, orthogonal polynomials, and solitons. The main result of the paper is the characterization of discrete pulses answering the general PWM problem in terms of the manifold of all rational solutions to Korteweg-de Vries equations.     

The characters of the holomorphic discrete series

This paper summarizes the derivation of an explicit and global formula for the character of any holomorphic discrete series representation of a reductive Lie group G which satisfies certain conditions. The only very restrictive condition is that G/K be a Hermitian symmetric space. (Here K is the maximal compact subgroup of G.)     

Rheology of discrete subaortic stenosis

The discrete form of subaortic stenosis is thought to be an acquired lesion, the aetiology of which may be a combination of factors which include an underlying genetic predisposition, turbulence in the left ventricular outflow tract, and various geometric and anatomical variations of the left ventricular outflow tract. A review of hypotheses relating to its aetiology is provided     

On the problem of stress

After references to the different interpretations of stress in the framework of the stress-demand relations and the activity needs relations a psychophysiologically orientated stress concept is presented which contributed to the explanation of preservation of health and development of health, but also of origination and course of disease. Hereby qualitative peculiarities of the character and the appearance of the stress as well as of the origination of stress are emphasized.     

Identification of discrete functional subregions of the human periaqueductal gray

The midbrain periaqueductal gray (PAG) region is organized into distinct subregions that coordinate survival-related responses during threat and stress [Bandler R, Keay KA, Floyd N, Price J (2000) Brain Res 53 (1):95–104]. To examine PAG function in humans, researchers have relied primarily on functional MRI (fMRI), but technological and methodological limitations have prevented researchers from localizing responses to different PAG subregions. We used high-field strength (7-T) fMRI techniques to image the PAG at high resolution (0.75 mm isotropic), which was critical for dissociating the PAG from the greater signal variability in the aqueduct. Activation while participants were exposed to emotionally aversive images segregated into subregions of the PAG along both dorsal/ventral and rostral/caudal axes. In the rostral PAG, activity was localized to lateral and dorsomedial subregions. In caudal PAG, activity was localized to the ventrolateral region. This shifting pattern of activity from dorsal to ventral PAG along the rostrocaudal axis mirrors structural and functional neurobiological observations in nonhuman animals. Activity in lateral and ventrolateral subregions also grouped with distinct emotional experiences (e.g., anger and sadness) in a factor analysis, suggesting that each subregion participates in distinct functional circuitry. This study establishes the use of high-field strength fMRI as a promising technique for revealing the functional architecture of the PAG. The techniques developed here also may be extended to investigate the functional roles of other brainstem nuclei.The periaqueductal gray (PAG) is a small tube-shaped region of the midbrain involved in survival-related responses and homeostatic regulation important for affective responses and stress (1–3). Subregions of the PAG underlie distinct, coordinated behavioral responses to threat. For example, stimulation in the lateral/dorsolateral portion produces active-coping responses (e.g., “fight” or “flight”) that involve increasing heart rate and arterial pressure, redistribution of the blood to the limbs, and a fast-acting, nonopioid-mediated analgesia. Stimulation in the ventrolateral portion produces passive-coping responses (i.e., disengagement, freezing) that involve reduced heart rate, decreased reactivity to the environment, and a longer-term, opioid-mediated analgesic response. These responses occur even when inputs to PAG from the cortex are severed (1, 4).The considerable animal literature on the critical role of the PAG in coordinating emotional responses has led to a surge of interest in studying the PAG in humans. The PAG plays a central role in neurobiologically inspired theories of human emotion (5), the neural circuitry underlying depression and anxiety (3, 6), autonomic regulation (7), and pain (8–11). To examine PAG function in humans, researchers have relied primarily on functional MRI (fMRI). To date, dozens of human neuroimaging studies have observed increased activation in the vicinity of the PAG during administration of painful and aversive stimuli (8, 12–16) and across a variety of emotional states (17).Unfortunately however, standard fMRI is fundamentally limited in its resolution, making it uncertain which fMRI results lie in the PAG and which lie in other nearby nuclei. The overarching issue is size and shape. The PAG is small and is shaped like a hollow cylinder with an external diameter of ∼6 mm, a height of ∼10 mm, and an internal diameter of ∼2–3 mm. The cerebral aqueduct, which runs through the middle, can prevent detecting activations within the PAG [type II errors (18)] and also can create artificial activations that appear to be in the PAG but are not [type I errors (19)], making the PAG particularly challenging to image among the subcortical nuclei. Standard smoothing and normalization procedures, even with high-resolution scanning, incorporate signal from the aqueduct (Fig. 1). This signal can be overpowering. The variability of signal in the aqueduct can be an order of magnitude greater than that of the surrounding PAG. (Figs. S1 and S2).Open in a separate windowFig. 1.The PAG imaged at high resolution. The transaxial slice on the left shows the PAG from a functional scan at ultra-high field strength (7-T) and high resolution (0.75 mm isotropic). Scanning the PAG at lower resolutions prevents clear separation of the PAG from the aqueduct and surrounds. (A) The mean functional image for a single run at the 0.75-mm isotropic resolution used in this study shows the PAG crisply as indicated by the red arrow. (B) Downsampling the image to a resolution of 1.5 mm isotropic begins to blur the boundary between the PAG and its surrounds because of partial-volume effects. (C) Further downsampling the image to a resolution of 3 mm isotropic eliminates the ability to distinguish PAG from the aqueduct with any degree of confidence. (D–F) Smoothing with a standard 4-mm kernel further increases the partial-volume effects that blend signal from PAG with the aqueduct and surrounds, as shown for 0.75 mm isotropic (D), 1.5 mm isotropic (E), and 3 mm isotropic (F) resolutions. Most neuroimaging studies use a 3-mm isotropic resolution with a 4-mm or higher smoothing kernel. We addressed these issues by separating PAG voxels from the aqueduct before additional image processing (i.e., using the image shown in A) so that only voxels within the PAG are incorporated into later stages of analysis. The top of the transaxial image corresponds to the anterior portion of the head; the bottom corresponds to the posterior portion of the head.Standard neuroimaging techniques also are fundamentally limited in capturing the remarkable functional organization that is internal to the PAG. In addition to being differentiated into columns (1, 4), the PAG also is organized rostrocaudally. In caudal PAG, neurons that contain endogenous opioids and neuropeptides involved in nonopioid analgesia are concentrated in the ventrolateral columns, whereas in rostral PAG this concentration is greater in the lateral and dorsomedial columns (20, 21). Mirroring this distribution, administration of anxiogenic drugs produces greater neural activity in caudal, ventrolateral PAG and rostral, dorsolateral PAG [as measured from c-Fos expression (22)]. Connections from the central nucleus of the amygdala terminate more extensively in lateral and dorsal rostral PAG and ventrolateral caudal PAG (23). The ability to resolve which of these circuits is involved in a given behavior is crucial for understanding the implications of PAG activity in a given situation and for mapping homologies across species.Resolving activity to subregions of the PAG requires greater precision than provided by any study to date. Indeed, the overwhelming majority of previous studies have used 1.5- or 3-T MRI systems that typically cannot exceed an isotropic resolution of 1.5–2 mm without incurring significant losses in the signal-to-noise ratio (SNR) and significant increases in image distortion. This resolution already introduces substantial partial-volume effects (Fig. 1) that merge signal from the PAG with the exceedingly variable signal in the aqueduct (Fig. S2). Standard intersubject normalization and smoothing procedures that generally increase SNR but diminish localization accuracy further aggravate these issues. Ultimately, standard fMRI techniques—even those using high-resolution scanning—are incapable of capturing the functional organization of the PAG in humans.To overcome these obstacles, we used ultra-high field (7-T) fMRI combined with 32-channel parallel imaging (24), which can boost sensitivity by as much as an order of magnitude compared with low-field strength magnets (1.5–3 T) and volume coils (25). Higher-field strength magnets also provide greater sensitivity to the susceptibility effects that underlie the blood-oxygenation level–dependent (BOLD) signal measured in fMRI (26) and greater sensitivity to microvasculature (27, 28). At a nominal voxel resolution of 0.75 mm isotropic (Fig. 1A), we isolated the PAG directly from the functional scans and separated activity in the PAG from activity in the aqueduct (Fig. S1).We examined activity in the PAG while participants viewed aversive images (29), which included images of burn victims, gory injuries, and other content related to threat, harm, and loss, or while they viewed neutral images. Although prior studies have demonstrated activity in the vicinity of the PAG during the viewing of aversive images (15), we first tested whether the PAG definitively showed greater activity during to these images when signal was separated from the aqueduct and surrounding brainstem nuclei. Next, we used two approaches to test whether activity was localized to subregions of the PAG. In one approach, we segmented the PAG along dorsal/ventral and rostral/caudal divisions and examined whether activity was localized within specific segments. In a second but related approach, we performed analyses on a voxel-by-voxel basis to test whether high-field strength imaging can map the functional architecture of PAG at the voxel level. Given that discrete voxels identified in this approach necessarily would reside within just a few millimeters of each other, we performed a factor analysis to discover whether they carried similar or unique information during emotional responses.     

Rapid loss of lakes on the Mongolian Plateau

Lakes are widely distributed on the Mongolian Plateau and, as critical water sources, have sustained Mongolian pastures for hundreds of years. However, the plateau has experienced significant lake shrinkage and grassland degradation during the past several decades. To quantify the changes in all of the lakes on the plateau and the associated driving factors, we performed a satellite-based survey using multitemporal Landsat images from the 1970s to 2000s, combined with ground-based censuses. Our results document a rapid loss of lakes on the plateau in the past decades: the number of lakes with a water surface area >1 km² decreased from 785 in the late 1980s to 577 in 2010, with a greater rate of decrease (34.0%) in Inner Mongolia of China than in Mongolia (17.6%). This decrease has been particularly pronounced since the late 1990s in Inner Mongolia and the number of lakes >10 km² has declined by 30.0%. The statistical analyses suggested that in Mongolia precipitation was the dominant driver for the lake changes, and in Inner Mongolia coal mining was most important in its grassland area and irrigation was the leading factor in its cultivated area. The deterioration of lakes is expected to continue in the following decades not only because of changing climate but also increasing exploitation of underground mineral and groundwater resources on the plateau. To protect grasslands and the indigenous nomads, effective action is urgently required to save these valuable lakes from further deterioration.The Mongolian Plateau, located in the hinterland of temperate Asia, sustains the eastern part of the Eurasian Steppe (1). The Inner Mongolia Autonomous Region of China (Inner Mongolia hereafter) and the entire territory of Mongolia (formerly the Mongolian People’s Republic) constitute its core region, with an area of about 2.75 million km² and a population of about 28 million (2–4). The plateau is dotted with numerous lakes surrounded by vast grasslands (Fig. 1), which have nourished the Mongolian people and created a unique Mongolian nomadic civilization (5). Many of these lakes on the plateau are internationally important wetlands for threatened species and migratory waterfowls, 13 of which are designated to be protected by the Ramsar Convention (6) (SI Appendix, Text S1).Open in a separate windowFig. 1.Distribution of lakes with water surface area >1 km² on the Mongolian Plateau. Inset shows the study area.However, a number of lakes have shrunk remarkably in recent decades as a result of intensive human activities and climate change. The shrinkage and drying up of lakes have exacerbated the deterioration of regional environment, which has directly threatened the livelihood of local people. Because of the degradation of lakes and grasslands, the plateau has become one of the major sources of sand–dust storms in northern China (7, 8), and dust from this region was even detected in North America in 1998 (9). Although several previous works have examined the changes in some lakes on the plateau (10, 11), a collective study of changes in the lakes across the plateau has not been performed. Using 1,240 available scenes of multitemporal images of Landsat Multispectral Scanner (MSS), Thematic Mapper (TM), and Enhanced Thematic Mapper (ETM+) from the late 1970s to 2010, combined with information on climate, topography, land use, human activity, and high-resolution Google Earth images, we established a database of Mongolian lakes (MONLAKE) for the entire plateau (for details see Materials and Methods and SI Appendix, Text S2 and Table S1). Using this database, we explored the changes in the lakes over the past three decades and investigated their possible driving forces.