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For a general overview of the theory, see General Relativity
|This article/section deals with mathematical concepts appropriate for a student in late university or graduate level.
General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional manifold, which looks locally like Minkowski space, and which acquires curvature due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from Euclidean geometry: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along geodesics in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.
The mathematical expression of the theory of general relativity takes the form of the Einstein field equations, a set of ten nonlinear partial differential equations. While solving these equations is quite difficult, examining them provides valuable insight into the structure and meaning of the theory.
In their general form, the Einstein field equations are written as a single tensor equation in abstract index notation relating the curvature of spacetime to sources of curvature such as energy density and momentum.
In this form, represents the Einstein tensor, is the same gravitational constant that appears in the law of universal gravitation, and is the stress-energy tensor (sometimes referred to as the energy-momentum tensor). The indices and range from zero to three, representing the time coordinate and the three space coordinates in a manner consistent with special relativity. is the cosmological constant.
The left side of the equation — the Einstein tensor — describes the curvature of spacetime in the region under examination. The right side of the equation describes everything in that region that affects the curvature of spacetime.
As we can clearly see even in this simplified form, the Einstein field equations can be solved "in either direction." Given a description of the gravitating matter, energy, momentum and fields in a region of spacetime, we can calculate the curvature of spacetime surrounding that region. On the other hand, given a description of the curvature of a region spacetime, we can calculate the motion of a test particle anywhere within that region.
Even at this level of examination, the fundamental thesis of the general theory of relativity is obvious: motion is determined by the curvature of spacetime, and the curvature of spacetime is determined by the matter, energy, momentum and fields within it.
The right side of the equation: the stress-energy tensor
In the Newtonian approximation, the gravitational vector field is directly proportional to mass. In general relativity, mass is just one of several sources of spacetime curvature. The stress-energy tensor, , includes all of these sources. Put simply, the stress-energy tensor quantifies all the stuff that contributes to spacetime curvature, and thus to the gravitational field.
First we will define the stress-energy tensor technically, then we'll examine what that definition means. In technical terms, the stress energy tensor represents the flux of the component of 4-momentum across a surface of constant coordinate .
Fine. But what does that mean?
In classical mechanics, it's customary to refer to coordinates in space as , and . In general relativity, the convention is to talk instead about coordinates , , , and , where is the time coordinate otherwise called , and the other three are just the , and coordinates. So "a surface of constant coordinate "" simply means a 3-plane perpendicular to the axis.
The flux of a quantity can be visualized as the magnitude of the current in a river: the flux of water is the amount of water that passes through a cross-section of the river in a given interval of time. So more generally, the flux of a quantity across a surface is the amount of that quantity that passes through that surface.
Four-momentum is the special relativity analogue of the familiar momentum from classical mechanics, with the property that the time coordinate of a particle's four-momentum is simply the energy of the particle; the other three components of four-momentum are the same as in classical momentum.
So putting that all together, the stress-energy tensor is the flux of 4-momentum across a surface of constant coordinate. In other words, the stress-energy tensor describes the density of energy and momentum, and the flux of energy and momentum in a region. Since under the mass-energy equivalence principle we can convert mass units to energy units and vice-versa, this means that the stress-energy tensor describes all the mass and energy in a given region of spacetime.
Put even more simply, the stress-energy tensor represents everything that gravitates.
The stress-energy tensor, being a tensor of rank two in four-dimensional spacetime, has sixteen components that can be written as a 4 × 4 matrix.
Here the components have been color-coded to help clarify their physical interpretations.
- energy density, which is equivalent to mass-energy density; this component includes the mass contribution
- , ,
- the components of momentum density
- , ,
- the components of energy flux
The space-space components of the stress-energy tensor are simply the stress tensor from classic mechanics. Those components can be interpreted as:
- , , , , ,
- the components of shear stress, or stress applied tangential to the region
- , ,
- the components of normal stress, or stress applied perpendicular to the region; normal stress is another term for pressure.
Pay particular attention to the first column of the above matrix: the components , , and , are interpreted as densities. A density is what you get when you measure the flux of 4-momentum across a 3-surface of constant time. Put another way, the instantaneous value of 4-momentum flux is density.
Similarly, the diagonal space components of the stress-energy tensor — , and — represent normal stress, or pressure. Not some weird, relativistic pressure, but plain old ordinary pressure, like what keeps a balloon inflated. Pressure also contributes to gravitation, which raises a very interesting observation.
Imagine a box of air, a rigid box that won't flex. Let's say that the pressure of the air inside the box is the same as the pressure of the air outside the box. If we heat the box — assuming of course that the box is airtight — then the temperature of the gas inside will rise. In turn, as predicted by the ideal gas law, the pressure within the box will increase.
The box is now heavier than it was.
More precisely, increasing the pressure inside the box raised the value of the pressure contribution to the stress-energy tensor, which will increase the curvature of spacetime around the box. What's more, merely increasing the temperature alone caused spacetime around the box to curve more, because the kinetic energy of the gas molecules inside the box also contributes to the stress-energy tensor, via the time-time component . All of these things contribute to the curvature of spacetime around the box, and thus to the gravitational field created by the box.
Of course, in practice, the contributions of increased pressure and kinetic energy would be miniscule compared to the mass contribution, so it would be extremely difficult to measure the gravitational effect of heating the box. But on larger scales, such as the sun, pressure and temperature contribute significantly to the gravitational field.
In this way, we can see that the stress-energy tensor neatly quantifies all static and dynamic properties of a region of spacetime, from mass to momentum to electric charge to temperature to pressure to shear stress. Thus, the stress-energy tensor is all we need on the right-hand side of the equation in order to relate matter, energy and, well, stuff to curvature, and thus to the gravitational field.
Example 1: Stress-energy tensor for a vacuum
The simplest possible stress-energy tensor is, of course, one in which all the values are zero.
This tensor represents a region of space in which there is no matter, energy or fields, not just at a given instant, but over the entire period of time in which we're interested in the region. Nothing exists in this region, and nothing happens in this region.
So one might assume that in a region where the stress-energy tensor is zero, the gravitational field must also necessarily be zero. There's nothing there to gravitate, so it follows naturally that there can be no gravitation.
In fact, it's not that simple. We'll discuss this in greater detail in the next section, but even a cursory qualitative examination can tell us there's more going on than that. Consider the gravitational field of an isolated body. A test particle placed somewhere near but outside of the body will move in a geodesic in spacetime, freely falling inward toward the central mass. A test particle with some constant linear velocity component perpendicular to the interval between the particle and the mass will move in a conic section. This is true even though the stress-energy tensor in that region is exactly zero. This much is obvious from our intuitive understanding of gravity: gravity affects things at a distance. But exactly how and why this happens, in the model of the Einstein field equations, is an interesting question which will be explored in the next section.
Example 2: Stress-energy tensor for an ideal dust
Imagine a time-dependent distribution of identical, massive, non-interacting, electrically neutral particles. In general relativity, such a distribution is called a dust. Let's break down what this means.
- time-dependent
- The distribution of particles in our dust is not a constant; that is to say, the particles may be motion. The overall configuration you see when you look at the dust depends on the time at which you look at it, so the dust is said to be time-dependent.
- identical
- The particles that make up our dust are all exactly the same; they don't differ from each other in any way.
- massive
- Each particle in our dust has some rest mass. Because the particles are all identical, their rest masses must also be identical. We'll call the rest mass of an individual particle .
- non-interacting
- The particles don't interact with each other in any way: they don't collide, and they don't attract or repel each other. This is, of course, an idealization; since the particles are said to have mass , they must at least interact with each other gravitationally, if not in other ways. But we're constructing our model in such a way that gravitational effects between the individual particles are so small as to be be negligible. Either the individual particles are very tiny, or the average distance between them is very large. This same assumption neatly cancels out any other possible interactions, as long as we assume that the particles are far enough apart.
- electrically neutral
- In addition to the obvious electrostatic effect of two charged particles either attracting or repelling each other — thus violating our "non-interacting" assumption — allowing the particles to be both charged and in motion would introduce electrodynamic effects that would have to be factored into the stress-energy tensor. We would greatly prefer to ignore these effects for the sake of simplicity, so by definition, the particles in our dust are all electrically neutral.
The easiest way to visualize an ideal dust is to imagine, well, dust. Dust particles sometimes catch the light of the sun and can be seen if you look closely enough. Each particle is moving in apparent ignorance of the rest, its velocity at any given moment dependent only on the motion of the air around it. If we take away the air, each particle of dust will continue moving in a straight line at a constant velocity, whatever its velocity happened to be at the time. This is a good visualization of an ideal dust.
We're now going to zoom out slightly from our model, such that we lose sight of the individual particles that make up our dust and can consider instead the dust as a whole. We can fully describe our dust at any event — where event is defined as a point in space at an instant in time — by measuring the density and the 4-velocity at . If we have those two pieces of information about the dust at every point within it at every moment in time, then there's literally nothing else to say about the dust: it's been fully described.
Let's start by figuring out the density of dust at a the event , as measured from the perspective of an observer moving along with the flow of dust at . The density is calculated very simply:
where is the mass of each particle and is the number of particles in a cubical volume one unit of length on a side centered on . This quantity is called proper density, meaning the density of the dust as measured within the dust's own reference frame. In other words, if we could somehow imagine the dust to measure its own density, the proper density is the number it would get.
Clearly proper density is a function of position, since it varies from point to point within the dust; the dust might be more "crowded" over here, less "crowded" over there. But it's also a function of time, because the configuration of the dust itself is time-dependent. If you measure the proper density at some point in space at one instant of time, then measure it at the same point in space at a different instant of time, you may get a different measurement. By convention, when dealing with a quantity that depends both on position in space and on time, physicists simply say that the quantity is a function of position, with the understanding that they're referring to a "position" in four-dimensional spacetime.
The other quantity we need is 4-velocity. Four-velocity is an extension of three-dimensional velocity (or 3-velocity). In three dimensional space, 3-velocity is a vector with three components. Likewise, in four-dimensional spacetime, 4-velocity is a vector with four components.
Directly measuring 4-velocity is an inherently tricky business, since one of its components describes motion along a "direction" that we cannot see with our eyes: motion through time. The math of special relativity lets us calculate the 4-velocity of a moving particle given only its 3-velocity (with components where ) and the speed of light. The time component of 4-velocity is given by:
and the space components , and by:
where is the boost, or Lorentz factor:
and where , in turn, is the square of the Euclidean magnitude of the 3-velocity vector :
Therefore, if we know the 3-velocity of the dust at event , then we can calculate its 4-velocity. (For more details on the how and why of 4-velocity, refer to the article on special relativity.)
Just as proper density is a function of position in spacetime, 4-velocity also depends on position. The 4-velocity of our dust at a given point in space won't necessarily be the same as the 4-velocity of the dust at another point in space. Likewise, the 4-velocity at a given point at a given time may not be the same as the 4-velocity of the dust at the same point at a different time. It helps to think of 4-velocity as the velocity of the dust through a point in both space and time.
Assembling the stress-energy tensor
Since the density and the 4-velocity fully describe our dust, we have everything we need to calculate the stress-energy tensor.
where the symbol indicates a tensor product. The tensor product of two vectors is a tensor of rank two, so the stress-energy tensor must be a tensor of rank two. In an arbitrary coordinate frame , the contravariant components of the stress-energy tensor for an ideal dust are given by:
From this equation, we can now calculate the contravariant components of the stress-energy tensor for an ideal dust.
Time-time component
We start with the contravariant time-time component :
If we rearrange the terms in this equation slightly, something important becomes apparent:
Recall that is a density quantity, in mass per unit volume. By the mass-energy equivalence principle, we know that . So we can interpret this component of the stress-energy tensor, which is written here in terms of mass-energy, to be equivalent to an energy density.[1]
Off-diagonal components
The off-diagonal components of the tensor — where and are not equal — are calculated this way:
Again, recall that is a quantity of mass per unit volume. Multiplying a mass times a velocity gives momentum, so we can interpret as the density of momentum along the direction, multiplied by constants and . Momentum density is an extremely difficult quantity to visualize, but it's a quantity that comes up over and over in general relativity. If nothing else, one can take comfort in the fact that momentum density is mathematically equivalent to the product of mass density and velocity, both of which are much more intuitive quantities.
Note that the off-diagonal components of the tensor are equal to each other:
In other words, in the case of an ideal dust, the stress-energy tensor is said to be symmetric. A rank two symmetric tensor is said to be symmetric if .
Diagonal space components
The diagonal space components of the stress-energy tensor are calculated this way:
In this case, we're multiplying a four-dimensional mass density, , by the square of a component of 4-velocity. By dimensional analysis, we can see:
Recall that the force has units:
If we divide the units of the diagonal space component by the units of force, we get:
So the diagonal space components of the stress-energy tensor come are expressed in terms of force per unit volume. Force per unit area are, of course, the traditional units of pressure in three-dimensional mechanics. So we can interpret the diagonal space components of the stress-energy tensor as the components of "4-pressure"[2] in spacetime.
The big picture
We now know everything we know to assemble the entire stress-energy tensor, all sixteen components, and look at it as a whole.[3]
The large-scale structure of the tensor now becomes apparent. This is the stress-energy tensor of an ideal dust. The tensor is composed entirely out of the proper density and the components of 4-velocity. When velocities are low, the coefficient , even though it's a squared value, remains extremely close to one.
The time-time component includes a mass multiplied by the square of the speed of light, so it has to do with energy. The rest of the top row and left column all include the speed of light as a coefficient, as well as density and velocity; in the case of an ideal dust which is made up of non-interacting particles, the energy flux along any basis direction is the same as the momentum density along that direction. This is not the case in other, less simple models, but it's true here.
The diagonal space components of the tensor represent pressure. For example, the component represents the pressure that would be exerted on a plane perpendicular to the direction.
The off-diagonal space components represent shear stress. The component, for instance, represents the pressure that would be exerted in the direction on a plane perpendicular to the axis.
The overall process for calculating the stress-energy tensor for any system is fairly similar to the example given here. It involves taking into account all the matter and energy in the system, describing how the system evolves over time, and breaking that evolution down into components which represent individual densities and fluxes along different directions relative to a chosen coordinate basis.
As can easily be imagined, the task of constructing a stress-energy tensor for a system of arbitrary complexity can be a very daunting one. Fortunately, gravity is an extremely weak interaction, as interactions go, so on the scales where gravity is interesting, much of the complexity of a system can be approximated. For instance, there is absolutely nothing in the entire universe that behaves exactly like the ideal dust described here; every massive particle interacts, in one way or another, with other massive particles. No matter what, a real system is going to be very much more complex than this approximation. Yet, the ideal dust solution remains a much-used approximation in theoretical physics specifically because gravity is such a weak interaction. On the scales where gravity is worth studying, many distributions of matter, including interstellar nebulae, clusters of galaxies, even the whole universe really do behave very much like an ideal dust.
The left side of the equation: the Einstein curvature tensor
We will recall that the Einstein field equations can be written as a single tensor equation:
The right side of the equation consists of some constants and the stress-energy tensor, described in significant detail in the previous section. The right side of the equation is the "matter" side. All matter and energy in a region of space is described by the right side of the equation.
The left side of the equation, then, is the "space" side. Matter tells space how to curve, and space tells matter how to move. So the left side of the Einstein field equation must necessarily describe the curvature of spacetime in the presence of matter and energy.
Some assumptions about the universe
Before we proceed into a discussion of what curvature is and how the Einstein equation describes it, we must first pause to state some fundamental assumptions about the universe.[4]
The first assumption we're going to make is that spacetime is continuous. In essence, this means that for any event in spacetime — that is, any point in space and moment in time — there exists some local neighborhood of where the intrinsic properties of spacetime differ from those at by only an infinitesimal amount.
The second assumption we're going to make is that spacetime is differentiable everywhere. In other words, the geometry of spacetime doesn't have any sharp creases in it.
If we hold these two assumptions to be true, then a convenient property of spacetime emerges: Given any event , there exists a local neighborhood where spacetime can be treated as flat, that is, having zero curvature. It is not necessarily true that all of spacetime be flat — in fact, it most definitely is not — but given any event in spacetime, there exists some neighborhood around it that is flat. This neighborhood may be arbitrarily small in both time and space, but it is guaranteed to exist as long as our two assumptions remain valid.
With these two assumptions and this convenient property in hand, we will now examine what it means to say that spacetime is curved.
Flatness versus curvature
Let's start by considering the simplest possible geometry:[5] the Euclidean plane.
The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of Cartesian coordinates. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an orthonormal basis.
Note carefully the distinction between the Euclidean plane and Cartesian coordinates. The plane exists as a thing in and of itself, just as a blank piece of paper does. It has certain properties, which we'll get into below. Those properties are intrinsic to the plane. That is, the properties don't have anything to do with the coordinates we project onto the plane. The plane is a geometric object, and the coordinates are the method by which we measure the plane. (The emphasis on the word measure there is not accidental; please keep this idea in the foreground of your mind as we continue.)
Cartesian coordinates are not the only coordinates we can use in the Euclidean plane. For example, instead of having axes that are perpendicular to each other, we could choose axes that are straight lines, but that meet at some non-perpendicular angle. These types of coordinates are called oblique.
For that matter, we're not bound to use straight-line coordinates at all. We could instead choose polar coordinates, wherein every point on the plane is described by a distance from a fixed but arbitrary point and an angle from a fixed but arbitrary direction. Polar coordinates are often more convenient than Cartesian coordinates. For example, when navigating a ship on the ocean, the location of a fixed point is usually described in terms of a bearing and a distance, where the distance is the straight-line distance from the ship to the point, and the bearing is the clockwise angle relative to the direction in which the ship is sailing. Polar coordinates in two and three dimensions are often used in physics for similar reasons.
But there's a fundamental problem with polar coordinates that is not present with Cartesian coordinates. In Cartesian coordinates, every point on the Euclidean plane is identified by exactly one set of real numbers: there is precisely one set of and coordinates for every point, and every point corresponds to precisely one set of coordinates.
This is not true in polar coordinates. What are the unique polar coordinates for the origin? The radial distance is obviously zero, but what is the angle? In actuality, if the radial distance is zero, any angle can be used, and the coordinates will identify the same point. The one-to-one correspondence between points in the plane and pairs of coordinates breaks down at the origin.
In mathematical terms, polar coordinates in the Euclidean plane have a coordinate singularity at the origin. A coordinate singularity is a point in space where ambiguities are introduced, not because of some intrinsic property of space, but because of the coordinate basis you chose.
So clearly there may exist a reason to choose one coordinate system over another when measuring — there's that word again — the Euclidean plane. Polar coordinates have a singularity at the origin — in this case, a point of undefined angle — while Cartesian coordinates have no such singularities anywhere. So there may be good reason to choose Cartesian coordinates over polar coordinates when measuring the Euclidean plane.
Fortunately, this is always possible. The Euclidean plane can always be measured by Cartesian coordinates; that is, coordinates wherein the axes are straight and perpendicular at their intersection, and where lines of constant coordinate — picture the grid on a sheet of graph paper — are always a constant distance apart no matter where you measure them.
Imagine taking a piece of graph paper, which is printed in a pattern that lets us easily visualize the Cartesian coordinate system, and rolling it into a cylinder. Do any creases appear in the paper? No, it remains smooth all over. Do the lines printed on the paper remain a constant distance apart everywhere? Yes, they do. In technical mathematical terms, then, the surface of a cylinder is flat. That is, it can be measured by an orthonormal basis, and there is everywhere a one-to-one correspondence between sets of coordinates and points on the surface. It's possible not to use an orthonormal basis to measure the surface; one might reasonably choose polar coordinates, or some other arbitrary coordinate system, if it's more convenient. But whichever basis is actually used, it's always possible to switch to an orthonormal basis instead.
Now imagine wrapping a sheet of graph paper around a basketball. Does the paper remain smooth? No, if we press it down, creases appear. Do the lines on the paper remain parallel? No, they have to bend in order to conform the paper to the shape of the ball. In the same technical mathematical terms, the surface of a sphere is not flat. It's curved. That is, it is not possible to measure the surface all over using an orthonormal basis.
But what if we focus our attention only on a part of the sphere? What if instead of measuring a basketball, we want to measure the whole Earth? The Earth is a sphere, and therefore its surface is curved and can't be measured all over with Cartesian coordinates. But if we look only at a small section of the surface — a square mile on a side, for instance — then we can project a set of Cartesian coordinates that work just fine. If we choose our region of interest to be sufficiently small, then Cartesian coordinates will fit on the surface to within the limits of our ability to measure the difference.
The surface of a sphere, then, is globally curved, but locally flat. In physicist jargon, the surface of a sphere can be flattened over a sufficiently small region. Not the whole sphere all at once, nor half of it, nor a quarter of it. But a sufficiently small region can be dealt with as if it were a Euclidean plane.
But this brings up an important point. The entire surface of the sphere is curved, and thus can't be approximated with Cartesian coordinates. But a sufficiently small patch of the surface can be approximated with Cartesian coordinates. This implies, then, that "curvedness" isn't an either-or property. Somewhere between the locally flat region of the surface and the entire surface, the amount of curvature goes from none to some value. Curvature, then, must be something we can measure.
The metric tensor
It is a fundamental property of the Euclidean plane that, when Cartesian coordinates are used, the distance between any two points and is given by the following equation:
where and are the distance between and in the and directions, respectively. This is essentially a restatement of the universally known Pythagorean theorem, and in the context of general relativity, it is called the metric equation. Metric, of course, comes from the same linguistic root as the word measure, and since this is the equation we use to measure distances, it makes sense to call it the metric equation.
But this particular metric equation only works on the Euclidean plane with Cartesian coordinates. If we use polar coordinates, this equation won't work.[6] If we're on a curved surface instead of a plane, this equation won't work. This metric equation is only valid on a flat surface with Cartesian coordinates.
Which makes it pretty useless, since so much of physics revolves around curved spacetime and spherical coordinates.
What we need is a generalized metric equation, some way of measuring the interval of any two points regardless of what coordinate system we're using or whether our local geometry is flat or curved.
The metric tensor equation provides this generalization.
If is any vector having components , the length of is given by the following equation:
where is the metric tensor, and and range over the number of dimensions. Recall that Einstein summation notation means that this is actually a sum over indices and . If we assume that we're in the two-dimensional Euclidean plane, the metric tensor equation expands to:
The terms of the metric tensor, then, must be numerical coefficients in the metric equation. We already know what these equations need to be to make the metric equation work in the Euclidean plane with Cartesian coordinates:
Now we can write the metric tensor for the Euclidean plane in Cartesian coordinates in the form of a 2 × 2 matrix:
So in the case of the Euclidean plane with Cartesian coordinates, the metric tensor is the Kronecker delta:
Of course, the same concepts apply if we expand our interest from the plane to three-dimensional Euclidean space with Cartesian coordinates. We just have to let the indices of the Kronecker delta run from 1 to 3.
Which gives us the following metric equation for the length of a vector (omitting terms with zero coefficient):
Which precisely agrees with the Pythagorean theorem in three dimensions. So given a metric tensor for any space and coordinate basis, we can calculate the distance between any two points. The metric tensor, therefore, is what allows us to measure curved space. In a very real sense, the metric tensor describes the shape of both the underlying geometry and the chosen coordinate basis.
But relativity is concerned not with geometrically abstract space; we're interested in very real spacetime, and that requires a slightly different kind of metric.
The local Minkowski metric
Parallel transport and intrinsic curvature
The Riemann and Ricci tensors and the curvature scalar
The Einstein tensor
The cosmological constant
- ↑ Actually rewriting the equation for the time-time component in terms of energy density requires refining our proper density equation into a form that doesn't depend on counting particles in a unit volume. Such a refinement is beyond the scope of this discussion. In less abstract dust solutions, the mass density is usually either assumed to be constant over space (as in the FLRW solution that models a homogenous, isotropic expanding or contracting universe) or is assumed to depend only on the radius of the distribution (as in the LTB solution that models gravitational collapse). At this point, it is sufficient merely to understand that matter density and energy density, and matter flux and energy flux, are equivalent concepts under general relativity.
- ↑ Not a standard term.
- ↑ The stress-energy tensor is practically never written out in matrix form this way, even in textbooks. This is purely for illustration.
- ↑ As we will later see, these assumptions may in fact turn out not to be valid for all of spacetime. It may be more accurate, although significantly less satisfying, to say that we assume these things to be true about spacetime, except where they aren't.
- ↑ Well, the simplest possible interesting geometry, anyway.
- ↑ This is trivial to demonstrate. If is zero and is non-zero, then will be non-zero for a vector that obviously has no length.


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(Michael ho Psellos), Byzantine statesman, scholar, and author, born apparently at Constantinople, 1018; died probably 1078. He attended the schools, afterwards learning jurisprudence from John Xiphilinos, later patriarch (John VIII, 1064-75). Psellus practised law, was appointed judge at Philadelphia, and under the Emperor Michael V (1041-2) became imperial secretary. Under Constantine IX (Monomachos, 1042-54) he became influential in the state. At this time he taught philosophy at the new Academy at Constantinople arousing opposition among ecclesiastical professors by preferring Plato to Aristotle. Psellus gained a great reputation as a philosopher. His pedagogical career was cut short by his appointment as Secretary of State (protosekretis) to Constantine IX. In 1054 he followed Xiphilinos to the monastery of Olympos, in Bithynia, where he took the name Michael. He soon quarrelled with the monks, however, and returned to the capital. He was one of the ambassadors sent to treat with the rebel Isaac Komnenos after the defeat of the imperial army near Nicaea in 1057. When Isaac I (1057-9) entered Constantinople in triumph Psellus had no scruple against transferring allegiance to him. Psellus drew up the indictment against the Patriarch Michael Caerularius in 1059, and preached the enthusiastic panegyric that the government thought advisable after Caerularius's death. Psellus maintained his influence under Constantine X (Dukas, 1059-67); under Michael VII (1071-8) he became chief Minister of State. Famous for oratory as well as for philosophy and statecraft, he preached the panegyric of the Patriarch John Xiphilinos in 1075. A work written in 1096-7 after Psellus's death has a commendatory preface by him. Krumbacher (Byzant. Litteratur., 434) states that the preface may have been written before the work was begun. That Psellus was able maintain his influence under succeeding governments, through revolutions and usurpations, shows his unscrupulous servility to those in power. Krumbacher characterizes him as "grovelling servility, unscrupulousness, insatiable ambition, and unmeasured vanity" (op cit., 435). Nevertheless his many-sided literary work and the elegance of his style give him a chief place among contemporary scholars. Compared with Abertus Magnus and Roger Bacon, he is to Krumbacher "the first man of his time". His important works are: commentary on Aristotle peri hermeneias; treatises on psychology; works on anatomy and medicine, including a poem on medicine and a list of sicknesses; a fragmentary encyclopedia, called "Manifold Teaching" (Didaskalia pantodape); a paraphrase of the Iliad; a poem on Greek dialects; a treatise on the topography of Athens; a poetic compendium of law and an explanation of legal terms. His speeches are famous as examples of style and contain much historical information. His best known panegyrics are on Caerularius, Xiphilinos, and his own mother. About five hundred letters, and a number of rhetorical exercises, poems, epitaphs and occasional writings are extant. His most valuable work is his history (chronographia) from 976 to 1077, forming a continuation to Leo Diaconus.
APA citation. (1911). Michael Psellus. In The Catholic Encyclopedia. New York: Robert Appleton Company. http://www.newadvent.org/cathen/12545a.htm
MLA citation. "Michael Psellus." The Catholic Encyclopedia. Vol. 12. New York: Robert Appleton Company, 1911. <http://www.newadvent.org/cathen/12545a.htm>.
Transcription. This article was transcribed for New Advent by Bryan T. Maguire.
Ecclesiastical approbation. Nihil Obstat. June 1, 1911. Remy Lafort, S.T.D., Censor. Imprimatur. +John Cardinal Farley, Archbishop of New York.
Contact information. The editor of New Advent is Kevin Knight. My email address is webmaster at newadvent.org. Regrettably, I can't reply to every letter, but I greatly appreciate your feedback — especially notifications about typographical errors and inappropriate ads.

subset: commoncrawl


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The manufacturers of automatic luxury watches advise the use of a watch winder, when the watch is not on the wrist. If such a watch is updated manually each and every time, it will gradually affect and damage the mechanism.
In association with SwissKubik, the Swiss Leader of watch winders, Lucrin is proposing this exclusive watch winder for 2 watches, which can be designed as per your wish and tastes. We are the only ones who propose more than a hundred of possible combinations in terms of colours and leathers with various textures (smooth and granulated cow leather, goat leather, real ostrich leather, alligator-style calfskin). You may pick the colour of the thread used and even choose to go for a beautiful engraving, for instance, your initials or your watch name or model.
Of course, each of the two watch winders is programmable according to the brand of the watch that it will welcome – the 2 motors function independently. You can set up your SwissKubik via the SwissKubik app from Google Applications.
SwissKubik by Lucrin Watch Winder Technical Specifications:
- Bidirectional rotating case to wind automatic watches
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- Powered by 4 standard C-size 1.5V alkaline batteries.
-Standard adjustable watch support with security clip. (Small support option also available)
-Pause/standby mode configured with watch in vertical position (12:00)
-3 year worldwide warranty


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The ruling came after Superior Court Judge Robert Perry chose to throw out the 2010 jury verdict that manager Howard K. Stern and psychiatrist Khristine Eroshevich conspired to obtain prescription drugs for the troubled model by using false names.
The judge found it was not unusual in the celebrity world Smith inhabited for fake names to be used to protect privacy.
The Second District Court of Appeals reversed those findings, saying there was enough evidence to support the conspiracy convictions. It said Stern cannot be tried but Eroshevich might get another trial.
Lawyers called the decision confusing.
Stern and Eroshevich were convicted Oct. 28, 2010, in the case of the reality TV star and former Playboy model, but the bulk of the charges against them were thrown out at their sentencing hearing.
In September of last year, prosecutors filed an unusual appeal saying Judge Perry was biased and abused his discretion when he dismissed the few counts on which Stern and Eroshevich were convicted.
A jury previously acquitted them on most of the 11 charges involving drug prescriptions given to Smith in the months before she died in Florida on Feb. 8, 2007, of an accidental drug overdose. The defendants were not charged with causing her death.
The two guilty verdicts against Eroshevich and Stern on conspiracy counts involved the use of false names to obtain prescription pain medications for Smith. Eroshevich also was convicted of one count of obtaining Vicodin under a false name.
Perry dismissed the conspiracy counts and reduced the one conviction against Eroshevich to a misdemeanor with a sentence of one year probation.
Prosecutors argued in the appeal that Perry was biased against them and had created a celebrity exception to the law. They noted that Perry accused a prosecutor of ethical violations during the case and said that the case had been overcharged.
Perry suggested authorities had chosen the wrong case to prove their point and indicated they did not understand the legislative intent of the law involving prescription drugs.
A co-defendant, Dr. Sandeep Kapoor, who was Smith's physician, was acquitted of all charges.
The defendants' nine-week trial was the final act of the long-running drama centering on the blonde beauty's troubled life, which was documented on reality TV, in tabloids and in trial testimony.


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Part three of a three part series.
Don't let that tree outside your kitchen window fool you. Though it may not look like it's doing much more than adding a little shade to your yard, it is, in fact, working to curb global warming. Through the process of photosynthesis, green plants use the energy from the sun to draw down carbon dioxide and combine it with water to create the carbohydrates plants need to grow and animals need to live. While a single tree does not have much effect on the atmospheric levels of carbon dioxide, en masse plants draw down millions of tons of this potent greenhouse gas and effectively help keep the Earth cool.
Given their positive effect on the climate, one would think that we'd
want to keep the world's plants, especially those in our backyards,
intact. But each year humans destroy untold amounts of plants and fertile soil
through the process of urbanization. Every time a subdivision is built, a strip
mall is erected, or a road is laid, the local vegetation is uprooted and the
soil is turned. Though new grass and trees may sprout afterwards, this newly
grown canopy of vegetation is usually much less dense.
|Urbanization tends to the reduce vegetation density. These two images show city lights and Normalized
Difference Vegetation Index (NDVI) for the eastern United States. Bright areas of the lights image are more urbanized. Dark green regions of the
vegetation image represent dense vegetation, while brown indicates sparse vegetation. Between the two images, urban areas correspond to regions with sparse vegetation.
(Lights image data courtesy Marc Imhoff of NASA GSFC and Christopher Elvidge of NOAA NGDC. NDVI data courtesy Compton Tucker and Bob Mahoney of
Tracking the precise impact of urbanization on vegetation, however, is no simple task. Urbanization moves relatively fast and its outlines are often hard to discern. It's also difficult to gauge what the urbanized areas were like before the cities were erected. Recently, a group of researchers at NASA's Goddard Space Flight Center led by biologist and remote sensing specialist Marc Imhoff found a way to overcome these obstacles. Using satellite images of city lights at night, they constructed a map of the urbanized areas of the United States. They then retrieved vegetation density readings of present day American cities as well as simulated readings of the landscapes that pre-dated these cities. By combining the vegetation data with the urbanization maps, Imhoff was able to calculate the effects of urbanization on many types of ecosystems across the country.


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Jarmila Groth and Juan Ignacio Chela dropped by the Miami Metro Zoo before they begin competing at the Sony Ericsson Open in Key Biscayne, Fla. Groth, the 28th seed, will open her 2011 Miami run with a match against Yaroslava Shvedova, who took out Sara Errani 6-3, 5-7, 7-6 (5). Groth shares the bottom half of the draw with Clijsters, Vika, and Schiavone, among others. And on the men's side, Chela will play the winner of the Feli Lopez/Berankis match. He has a potential fourth-round match against Rafael Nadal.
Draws: See all the Miami draws here.
(image via Getty Images)


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I've been holding off on any blogposts about the G20 summit garbage.
It's too bad, because with a few differences I would have looked almost exactly like this. So, uh, here it goes...
Le tout rightosphere up north has been riven by the actions and inactions of the Toronto Police during the G-20. P M Jaworkski sees it as the difference between libertarian conservatives and law-&-order conservatives - or perhaps between "life, liberty and the pursuit of happiness", and the less full-throated Canadian founding cry of "peace, order and good government". Except, of course, that peace, order and good government were noticeable mainly by their absence in Toronto last weekend.
Kathy Shaidle suggests it may be a rural/urban thing: She's a city gal; many of her detractors at Small Dead Animals seem to be rural dwellers. I don't know about that. In my corner of New Hampshire, I have very cordial relations with my one-man police department. His son and mine will be off at camp together next week. I have known a lot of police officers in northern New England over the years, and I have a lot of respect for the difference they can make to the rhythms of life in a small town. Because of that, I don't have any problem agreeing with Jay Currie that Officer #3478 "pretty much illustrates how not to deal with a law abiding member of the general public".
That's all we're saying, but it's nevertheless quite a lot: For one thing, this isn't Dragnet or Naked City. These days, everything's a camera - a telephone's a camera, a hi-fi player is a camera, just about anything larger than a Tootsie Roll is a camera. So the police ought to expect that every time they're out on the streets, especially in the downcore core of a major metropolitan area, someone somewhere will be filming it. The rest of us have had to get used to it. Just up the street from where Mike Brock was set upon by Toronto cops, I came out of the committee room at the Ontario Parliament after testifying on the "human rights" regime and had a bit of chit-chat in the corridor, and some guy with a cellphone had it up on the Internet in nothing flat. That's just the way it is. And, if the alternatives are either living with it or browbeating citizens engaged in non-illegal activity, then the police should learn to live with it.
Secondly, Officer #3478's response is simply not an appropriate way for a minor municipal functionary to talk to a member of the public. It demonstrates a defective understanding of the relationship between law enforcement and the citizenry - admittedly in a footling matter, but, left unchecked, who's to say where it could lead? Why, one day, you never know, it might lead to a chief of police who lies to the public and makes up his own laws.
My own approach to officialdom is summed up here. One reason I oppose the paramilitarization of the police is that it encourages the mindset on display in Toronto last weekend. Its defenders, in many comments at Small Dead Animals, say, well, sorry, but it's necessary in order to keep the Queen's peace. I think not. The disruption to the Queen's peace, in Queen's Park, in the shadow of the building that embodies the idea of a state accountable to its citizens, came from cops rending the air with their cheery cry of "I DON'T GIVE A FUCK WHAT YOU THINK". Actually, the very minimum that law-abiding citizens are entitled to demand from police officers is that they give a fuck what we think. One thing I happen to think is that no citizen going about his lawful business should have to put up with a police officer yelling "fuck" from the get-go.
In further defence of the Toronto Police, commenters have said, well, Brock and Shaidle were asking for it, making provocative statements about the accountability of public servants and wearing black T-shirts and whatnot. "After Saturday, police were jumpy", writes Ella at Small Dead Animals.
Oh, right. They're the fellows with the guns and the billion-dollar security budget, but they're the "jumpy" ones? That's great news, isn't it? I've encountered "jumpy" cops in Third World crapholes many times over the years: You choose your words with care, because it's not yet clear whether you're dealing with merely an insecure twerp or a thug who's itching to club you to a pulp. I don't think the residents of First World cities (which Toronto still is, just about) should be expected to perform similar psychological evaluations if they decide to take a stroll around Queen's Park of a weekend.
The other argument advanced at SDA is that, well, it was G-20, you should trim your sails accordingly. Really? 99.999 per cent of Torontonians were not participating in the summit. Unlike the Olympics or the World Cup, it's not a public event. It was a private (and pointless) affair imposed on the public, and if the price of that imposition is that Bill Blair gets upgraded to some rinky-dink Latin-American caudillo for a week then the only appropriate response of self-respecting citizens should be screw you.
Finally, Kate McMillan, for whom I have enormous respect and with whom I rarely disagree, argues that what happened in Toronto was little more than the "profiling" we conversatives claim to be in favour of, at least when it comes to young men from Pakistani madrassahs. I think not. When you're beating up Guardian reporters and harrassing Mike Brock, that's pretty much a failure of "profiling" by definition, isn't it? Or, at any rate, as inept "profiling" as that of the Metropolitan Police who, after the July 7th Tube bombings, decided to "profile" a Brazilian electrician and put five bullets in his head. If you reflexively defend lazy and incompetent policing, you'll soon find yourself having to defend lazy, incompetent and murderous policing.
And that's my biggest problem with "law-&-order conservatives": They seem to think when the coppers are kicking around Mike Brock, that just shows they're doing their job. Au contraire, the ten minutes they're kicking around Mike Brock is ten minutes they're not doing their job, ten minutes they're not devoting to the guys they should be kicking around - just as pulling over the octogenarian nun for secondary screening doesn't demonstrate the rigor of homeland security but rather the waste of limited resources. A cop who shouts "FUCK!" at Mike Brock isn't communicating the authority of the state so much as its insecurity. Or do you really think a bozo baying uncontrollably into the summer night on University Avenue is going to be the guy who spots the fellow with the darting eyes and the lumpy midriff?


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When Haiti struck we were all shaken. We all pitched in even in these hard times and we we made small donations add up to hundreds of millions of dollars in hours. In the face of devastation of this magnitude we said that we would not only build back, we would build back better.
Hundreds of blue and white tarp-covered shacks crowd a low-lying, flood-prone ravine at Marassa 14, a camp where 3,000 people live outside the capital of Haiti. But since January, we have seen little happen. Now, almost 5 months after the quake, only around 7000 people have been moved to safer housing while hundreds of thousands of families still live in 12000 tent cities across the country.
With so many still living in harms way, still dying from diseases caused by dirty water alone, Alonzo Mourning hopes that by remaining active in his local community and by telling these stories of success, that he might inspire others to play a part.
To Mourning, a better Haiti isn't good enough
By TIM REYNOLDS (AP)
PORT-AU-PRINCE, Haiti — As the charter jet began backing away from the terminal, Ginel Thermosey slowly turned around to shake hands with the medical student seated one row behind him.
Had he not made the trip from Port-au-Prince to Miami, Thermosey would have died within two weeks of leukemia. In a few days, with some donated treatment and medication, the 20-year-old will have a new lease on life.
These are the stories Alonzo Mourning insists the world needs to hear, the success stories from Haiti, where the earthquakes that struck 4 1/2 months ago took everything from people who had nothing to begin with. No one knows for sure how many people died, how many were never found, how many could have been saved.
Deep down, Mourning fears the world has already forgotten.
The work isn't complete in Haiti, he says. It's only beginning.
"These are human beings," Mourning said, overlooking the tent hospital at the edge of Port-au-Prince's airport. "These are children that are suffering, that need help. That's what moved me to come, to do this, to continue to provide help as much as I can and continue to reach out to others, so others can be made aware that they, too, can play a part."
For that reason, every couple weeks, Mourning makes the short flight from Miami to Port-au-Prince. The retired Miami Heat star is among the many spreading the tale of Project Medishare, the not-for-profit group from the University of Miami that already had spent nearly two decades trying to improve the quality of health care in the impoverished nation.
Millions have been raised.
Many millions more are needed. Things are better in Haiti, for certain — yet still terrible. Haitians are dying daily from drinking filthy water, or from hunger, or disease. Some parts of the city are so gripped by crime and desperation, visitors are told not to even think about venturing that way because no one's safety is guaranteed.
"There still is a need for major help," Heat coach Erik Spoelstra said. "It is our responsibility."
On Mourning's latest trip, a down-and-back jaunt Saturday, Spoelstra, former Heat guard Tim Hardaway and Memphis Grizzlies star Rudy Gay made the trip with him, along with several other members of the Heat organization.
None had seen Haiti before.
Nothing could prepare them, either.
"Even flying in on the plane, seeing the devastation shocked me," Gay said. "It's easy to just hear about it, but to see it firsthand, it's really humbling."
Spoelstra held a premature baby in his left hand, the child so tiny that it rested its head on the tips of two of the coach's fingers. Gay was approached by a woman at the tent hospital, who invited him to hold her newborn daughter — then begged him to take her back to the U.S., saying she lacked the means to give her child a life in Haiti. Hardaway was overwhelmed by the sight of ailing children.
"We are the luckiest people on earth," Hardaway would say softly later, sunglasses hiding the emotion welling in his eyes. "We should be counting our lucky stars every day. This is tough."
Somehow, among the Project Medishare staff, morale remains shockingly high.
Volunteers get shuttled in every Saturday for a seven-day stay. Anything longer, Medishare staff has learned, becomes just too tough for many to bear. Upon arrival, they are greeted by Tom Koulouris, who has run the tent hospital since its very first days after the earthquake.
"We have some of the best food in Port-au-Prince catered twice a day," Koulouris is saying to the new volunteers. "At lunchtime, it consists of beans and rice. In the evening, it's rice and beans."
The hospital is not just for those affected by the earthquake. When people need medical attention in Haiti, they are usually taken to Project Medishare's facility. Twenty-three American visitors were treated recently for trauma injuries after their truck toppled. Machete wounds, premature babies, mothers in desperate need, disease, all part of the daily lineup.
"We are the safety net of Port-au-Prince," Koulouris said.
Koulouris goes home this week. His mission will be complete. The tent hospital is closing, thankfully. Patients began being moved about 15 minutes away Sunday to a more secure structure — an absolute necessity now that hurricane season has arrived in the Atlantic. Any storm with tropical or hurricane-force winds hitting the tent hospital would have almost surely destroyed everything.
He rattles off the stats, proudly: The tent hospital has treated over 20,000 patients, performed more than 1,500 major surgical procedures. It's so high-tech, they can even fit people for prosthetic limbs; 37 of those have been issued so far, another 500 are on the way, as many as 4,500 are needed.
Through it all, Koulouris has not stopped to reflect on what's transpired.
"I'm afraid if I do that here, I'll probably fall apart," he said. "I'll wait until I get home."
Soon, Thermosey will be back home as well. He can thank Anika Mirick for that.
Mirick is a first-year medical student from the University of Central Florida, who just finished her one-week stay at the tent hospital. Thermosey was one of her patients.
"I was crying every day," Mirick said.
Desperate to find a way to save him, Mirick made calls and pleaded with doctors in Orlando, Fla. to save his life. They agreed without hesitation. And within one day, Thermosey had his 30-day visa allowing him to enter the U.S., a minor miracle in itself.
When that plane was leaving, Mirick was the person Thermosey thanked first, extending his hand, intravenous tubes protruding from it.
"He will survive," Mirick said. "We just saved a life."
It's the moments like those that Mourning comes to see.
Perhaps the signature moment of his career was a blocked shot against Dallas in Miami's title-clinching Game 6 of the 2006 NBA finals, when he flew over two people to slap Jason Terry's layup away, fell to the floor, got up and shouted at teammates — with an expletive tossed in — "What are we doing?"
He's shouting the same now.
He wants Warren Buffett to listen. Bill Gates, too. Anyone, anywhere, Mourning wants to tell them all, Haiti needs help more than they probably realize.
So he comes back, vowing not to stop until he has nothing left to offer.
"Things have progressed tremendously since Jan. 12," Mourning said. "But there's so much more work to be done."
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WASHINGTON — Over six frightening months, a deadly germ untreatable by most antibiotics spread in the nation's leading research hospital. Pretty soon, a patient a week was catching the bug. Scientists at the National Institutes of Health locked down patients, cleaned with bleach, even ripped out plumbing — and still the germ persisted.
By the end, 18 people harbored the dangerous germ, and six died of bloodstream infections from it. Another five made it through the outbreak only to die from the diseases that brought them to NIH's world-famous campus in the first place.
It took gene detectives teasing apart the bacteria's DNA to solve the germ's wily spread, a CSI-like saga with lessons for hospitals everywhere as they struggle to contain the growing threat of superbugs.
It all stemmed from a single patient carrying a fairly new superbug known as KPC — Klebsiella pneumoniae that resists treatment by one of the last lines of defense, antibiotics called carbapenems.
"We never want this to happen again," said Dr. Tara Palmore, deputy hospital epidemiologist at the NIH Clinical Center.
Infections at health care facilities are one of the nation's leading causes of preventable death, claiming an estimated 99,000 lives a year. They're something of a silent killer, as hospitals fearful of lawsuits don't like to publicly reveal when they outfox infection control — yet no hospital is immune.
Wednesday, government researchers published an unusually candid account of last year's outbreak, with some advice: Fast sequencing of a germ's genome, its full DNA, may be essential. It can reveal how drug-resistant bacteria are spreading so that doctors can protect other patients.
"This is not an easy story to tell," said Dr. Julie Segre, a senior investigator at NIH's National Human Genome Research Institute. She led the genetic sleuthing that found the bug hiding in sink drains and, most chilling, even in a ventilator that had been cleaned with bleach.
Infection-control specialists at other hospitals called this detailed anatomy of an outbreak, published in the journal Science Translational Medicine, important to share.
"They were able to demonstrate that this sneaky little bug was able to stay alive and get transmitted in ways they hadn't quite predicted before they had the detailed genetic information," said Dr. Sara Cosgrove, associate hospital epidemiologist at Johns Hopkins University. "It's very revealing."
"Absolutely this could happen in any hospital," said Dr. Deverick Anderson, co-director of a Duke University infection control network that advises smaller community hospitals.
"This is really exciting stuff, cutting-edge technology, to try and better understand how these infections get spread," he added. That in turn may lead to new protections, important because "there's something that's very, very wrong about going to a hospital and becoming more ill."
Normally, the Klebsiella bacteria live in human intestines and don't harm people with healthy immune systems. But the multidrug-resistant strain named KPC has emerged over the past decade to become a fast-growing threat in intensive care units, spreading easily between very ill people and killing half of those it sickens. Worse, people can carry KPC without symptoms unless the germs slip into the urinary tract or bloodstream — theirs or the person's in the next bed — through a catheter or surgical wound.
The 243-bed NIH Clinical Center, in Washington's suburbs, is a unique hospital, only treating people enrolled in government research studies.
So on June 13, 2011, a research nurse carefully checked the medical records as a New York City hospital transferred a study participant who had become critically ill with a rare lung disease. The nurse found that the patient had KPC as well.
The woman went into strict isolation: Everyone entering her room donned a protective gown and gloves and rigorously washed their hands. Her medical equipment got special decontamination. All other patients in the ICU had their throats and groins tested regularly to see if the bug was spreading.
All seemed OK. The woman recovered, and went home on July 15.
Fast forward three weeks. Now a man with cancer has KPC despite never crossing paths with Patient No. 1. Ten days later, a woman with an immune disease fell ill, too. Both died of the infection.
Did they arrive carrying their own KPC bacteria, or did that first patient's germ somehow escape into the hospital? Standard tests couldn't tell. Segre, the geneticist, turned to DNA.
As bacteria multiply, mistakes appear and are repaired in their genetic code. Sequencing that genome allowed Segre to follow differences in single genetic letters like a trail of the germ's transmission and evolution.
Sure enough, the KPC originated from the New York patient despite NIH's precautions. Testing bacteria from the 17 additional patients who ultimately caught it shows the KPC was transmitted three separate times from Patient No. 1, and then spread more widely.
Even this sophisticated technology couldn't prove exactly how transmission occurred. But it turns out that Patient 3 had been in the ICU at the same time as the New York woman and really was the next infected, silently carrying the bug longer before becoming sick. That was enough time for Patient 3's infection to spread to Patient 2, who just got sick faster.
Meanwhile, NIH was making big changes. All the ICU patients underwent more invasive testing, using rectal swabs, to check for silent germ carriers. A new wall created a separate ICU to house them. Doctors, nurses, even janitors assigned there could work nowhere else, and monitors were paid to make sure everyone followed infection-control rules.
Yet a patient a week was either becoming infected or found to be a silent carrier of the same KPC strain.
"Honestly, we were very scared at that point," Segre recalled.
Test after test never found the bug on hospital workers' hands. Tainted objects like the ventilator couldn't be ruled out — but NIH adopted more complex and expensive decontamination, using robot-like machines to spray germ-killing hydrogen peroxide into the tiniest of crevices in all affected rooms and equipment.
Still, November brought more bad news: The outbreak strain had escaped the ICU, as two patients who'd never been there now were carrying it. A new isolation room was built, and all 200-plus patients in the hospital started undergoing rectal testing.
The outbreak now is over, the last carrier found in December. But NIH isn't dropping its guard. The isolation room remains, used every time one of the seven outbreak survivors returns to the hospital for their ongoing research studies — because they still carry the strain. Those rectal tests continue, hospital-wide once a month, to be sure no new KPC strain sneaks in.
Bacterial sequencing is becoming fast and cheap enough for most large hospitals to use during tough outbreaks, said Dr. Lance Peterson, microbiology and infectious disease director at NorthShore University HealthSystem in Evanston, Ill.
But another lesson is how much it takes to guard against these bugs sneaking in in the first place. Peterson said his hospital does weekly rectal testing of every ICU patient as a precaution.
"There's better technology becoming available for your hospital to prevent these bacteria from spreading, and this is what you should expect from your hospital," he said.


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The Sandhurst Collect (Almighty God whose son, the Lord of all life)
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Set in marble above the west door of the Royal Memorial Chapel, RMA Sandhurst, and in 'The Collects of the British Army'
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June 8, 2009
The Potter and the Clay
Mary Southerland
Today's truth
Isaiah 64:8 "But LORD, you are our father. We are like clay, and you are the potter; your hands made us all" (NCV).
Friend to Friend
When our daughter, Danna, was in preschool, one of her favorite activities was playing with clay. She often pulled her little step stool over to the closet where the clay was stored and pulled out the boxes of brightly colored clay. With great concentration and effort, her little fingers would then pry each box open and then she went to work.
For hours at a time, Danna would pound, mold, stretch, pinch and ultimately create her own clay world. Animals, people, flowers and aliens would emerge from the small lumps of clay. When she finished, Danna called me and asked me to "put them up." That meant it was time to exhibit her clay creations for her brother and father to see when they came home.
Danna's creations would remain on display until they grew hard and brittle. We would then break them apart, sprinkle them with water and put them back into their containers, allowing the clay to grow soft again.
When it comes to spiritual matters – God is the Potter, the Creator, and we are the clay. The world says we should never give up our rights because if we do, we will lose everything. God says, "Give up your rights and you will gain everything." When we surrender, we gain everything that is important and eternal – including God's perfect plan and purpose for our life.
In order for God to make us into what He wants us to be, we have to become pliable and willing - like clay. We must come to Him, placing ourselves in His hands, the Potter's hands. Total surrender demands that we come with no agenda, no pre-set conditions – giving Him permission to bend us, break us and change us.
We sometimes come to God, with a hardened heart and brittle spirit. Life has broken and damaged us and the wrong hands have molded us. Remember, how we come to Him is not nearly as important as the fact that we come. As we give up control to Him, like a Master Craftsman, He begins to work patiently and lovingly, with a specific plan and one-of-a-kind design in mind. He works through circumstances as well as people, through joy and pain, victory and defeat. He filters every part of life through His plan for us and uses it to create a life of worth and meaning.
Oh, we have tried to live life with our own agenda and plan and have experienced discontentment and frustration. Our soul longs for more and our heart cries out, "Is this all there is?" The truth is that nothing and no one but God can bring us the peace and joy for which we searching. Are you ready to become the clay, surrendering your broken life to the Potter's loving and purpose-filled hands? Today is the day!
Let's pray
Lord, I am tired of trying to live life in my own power. Right now, I give up and surrender my life completely to You. I choose Your plan over mine. I give You permission to mold me and make me what You want me to be. I will wait before You and seek Your direction.
In Jesus' name,
Now it's your turn
What has been the plan – according to you – for your life up to this point?
What kind of results has that plan produced?
Write a prayer of surrender to God and record it in your journal. Be sure to include the following steps:
Recognize that you have been trying to live life according to your plan – your agenda.
Admit that your refusal to surrender your life to God has brought only frustration and emptiness.
Right now – surrender to God.
What action do you need to take in order to obey God:
_____Join a local church
_____Get involved in a small group bible study
_____Read the Bible daily
_____Pray constantly
_____Seek God in everything I do
More from the Girls
I have to admit I am not very good at giving up. I was taught never to give up; to be strong and to depend on no one but myself. In fact, the very thought of surrender seems like a weakness to me and is completely opposite to my human strength and will. Exactly! Did you know that my name, Mary, literally means bitter, but when broken, sweet? Unless I am in a constant state of surrender to God, I will surely become bitter, brittle and broken. Today, I choose to surrender my heart and life to Him.
Need help in your walk with Christ? Beginning June 22, Mary's online bible study, Light for the Journey, begins a new topic, "How to Get Past Your Past." Check it out at www.marysoutherland.com and be sure to visit Mary's online store for MP3s, CDs, books and bible studies that will help you grow in God.
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Girlfriends in God
Matthews, NC 28106
Click here to learn more about hosting a Girlfriends in God conference in your area or having one of the GiGs speak at your next women's event.
Click here to learn more about hosting a Girlfriends in God conference in your area or having one of the GiGs speak at your next women's event.
Originally published Monday, 08 June 2009.


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Singapore, 4 June 2010 – StarHub today announced the launch of Singapore's second solar-powered mobile base station (BTS) on the IKEA Building at Alexandra Road, following the successful deployment of StarHub's first on StarHub Green at Ubi last year.
The latest BTS system will support all mobile voice and data traffic in and around the Queensway/Jalan Bukit Merah/Alexandra road junction, which includes Anchor Point and Queensway Shopping Centre.
Mr Peter Cook, vice president of StarHub's Integrated Network Engineering said, "Since our first deployment in August 2009, we have seen a good reduction in energy consumption by at least 58%. Given this success, we are presently looking at alternate ways to reduce our overall reliance on power from the utility grid by off-loading the energy requirements to alternate renewable sources, or upgrading our mobile network with the latest energy-efficient technologies, even as our current network evolves to more advanced ones such as HSPA+ with Dual Carrier, and LTE. Eventually, we believe we can cost-effectively deploy such green or energy-efficient technologies across our entire mobile network."
Under this solution, the solar panels supply the BTS power needs during the day. On cloudy or rainy days, or at night, the BTS will revert to using energy from the utility grid. A single BTS using this solar solution will enable StarHub to reduce its CO2 emissions by about 2.5 tonnes yearly.
Enhancements to 3G Radio and Core Mobile Networks
In addition to the solar-powered BTS, StarHub today announced further mobile network improvements by upgrading its 3G radio network to be more energy-efficient. This upgrade, which utilises the latest fourth-generation base station technologies from Huawei, allows the mobile base station power consumption to be cut by more than 10%. StarHub has also enhanced its mobile core network with the latest technologies from Nokia Siemens Networks, resulting in a reduction in power consumption of at least 40% in this area. Collectively, these initiatives will save on average more than 1000 tonnes of CO2 emissions per year.
These initiatives are part of StarHub's programme to introduce green or energy-efficient projects in order to lower its overall power consumption in its operations.
For more information on StarHub, visit www.starhub.com.


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Thursday, October 14, 2010
ありがとう Todd Selby!
ありがとう The Selby for this mention in Elle UK!
Thank you also lovely Victoria for this scan!
17 comments:
oooh congrats! that is uber amazing :DReplyDelete
Yay! But it's no secret that you are awesome!ReplyDelete
Congrats Sandwich! That is a nice little plug there! He must be creative! Is he going to help you get your fantastic guide Tokyo out there.ReplyDelete
great great great!!!!!ReplyDelete
omedetou ! such lovely news !ReplyDelete
WOAH! congratulations!ReplyDelete
So over the moon for you. xxReplyDelete
how exciting!!ReplyDelete
Congrats :D Your blog deserves it.ReplyDelete
P.s. On a side note, I love that Todd Shelby shares my love for bad movies. And that he mentions yet another one to add to my list to watch - looks epic.
AWWWWWW I would be speechless too!!! Cheers to you!ReplyDelete
so cooool!ReplyDelete
I've found you blog via urbanoutfitters blog, I definetely love you!
tonight I'm leaving from Rome (Italy) and flight to Tokyo!!!
I'so happy!!!!
WOW! Very cool indeed.ReplyDelete
Hello Cavaan, thank you so much :)ReplyDelete
Hello Hello Tiger! nawww shucks you are a sweetie! xoxo
Rhys! Hellooooo! It's 12:08am!!! YOUR TANJOUBI!!! Omedetou!!!
Mette thank you pretty one!
Thanks lovely Gini! How long till you come to Tokyo now?
Thank you WL! It was such a nice surprise!
Thank you Shinya. I was very lucky to be included!
Thank you Vic! xoxo
Hi Camila, thank you for your nice comment!
nawww thanks Jennier! That's sweet of you to say! Oh goodie you can enjoy watching a nice movie next rainy weekend! I am still to buy a TV here in Tokyo! But I am missing watching movies in English! Sometimes I watch TV at the gym though! Mostly cooking shows!
Thank you Shellie!
Thanks so much Random Elephant!
Oh thank you very much Fania, I am happy you can enjoy Hello Sandwich! How exciting you are in Tokyo at the moment too! I hope you're having a fantastic time!!!
Hello Alan, thank you very much!
Love Love
Congrats, Ebony! That's such a big compliment, but very much well-deserved! It's nice to know I'm in such good company with my blog-following ;)ReplyDelete
Chic chic!ReplyDelete
Just discovered your blog, I travel just looking at your pictures.
i really enjoy for this picture in articleReplyDelete
Model rambut sebahu


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EN: Preskoči na glavno vsebino
Number of submitted applications (2022)
EN: Skoči na filtre seznama
EN: Filtriranje se bo izvedlo ob spremembi izbire filtra in ne zahteva dodatne potrditve.
Where can I conclude a marriage? How can I ensure parental authority and children's rights? How can I arrange childcare? How can I adopt a child?
When must I renew my identity card or passport? Where can I obtain a driving licence? What is a digital certificate?
How to enrol a child in primary school? Where do I apply for a secondary school? How do I obtain a scholarship? What are the conditions for obtaining ...
How can I obtain an old-age pension; state retirement pension? What are my rights from the employment relationship etc.?
How can I obtain financial social assistance? How do I report death? How can I obtain a health insurance card?
How to obtain a tax identification number? How to submit a tax return form for tax on property?
How can I inquire as to whether I am eligible to vote? How do I report a public event?
How do I obtain a visa; information for travellers?
How to change the actual use of land? How to obtain a suitable building permit etc.?
Agricultural and rural development; organic farming; integrated production; obtaining status of a farmer etc.
Where can a roadworthiness test of a vehicle be conducted? How to register a vehicle?
In a few seconds, you will be automatically redirected to the old website of eUprava, but you can also click on the "Continue" button.
In a few seconds, you will be automatically redirected to the old website of eUprava, but you can also click on the "Continue" button.


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The website of ChaosFish. It's the home of "Little Bit Adventure" that contain various LBA modding information and some translations of the game. And also a Hebrew LBA page.
The predecessor of the Magicball Network. The World of LBA, launched in the summer of 1997, because a very large site with a lot of content. It later, at the end of 2000, grew out to the new place of the LBA community called the Magicball Network.
This is a mirror of the original site, not everything works but most of it does. The World of LBA is no longer updated, the MBN has replaced it.


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Home Page Chess Life Online 2009 May 2010 U.S. Championships Announced at Closing
|2010 U.S. Championships Announced at Closing
|May 20, 2009
The closing ceremony of the 2009 U.S. Chess Championship featured opera singing by GM Emil Sutovsky and speeches by the top three finishers, GM Hikaru Nakamura, GM-elect Robert Hess and GM Alexander Onischuk. The highpoint of the night was the announcement made by founder+president Rex Sinquefield, who started the Chess Club and Scholastic Center of Saint Louis with his wife Jeanne: There is an agreement to host the 2010 U.S. Chess Championships in Saint Louis. Details to come.
For now enjoy a final photo gallery by Betsy Dynako, official event photographer.
Fan Meets Fan
There were many happy players at the closing ceremony, due to the big announcement about 2010, the amazing view of Saint Louis and flowing streams of red and white. But IM Irina Krush was beyond thrilled. A few months prior, she was introduced to the Saint Louis rapper Oreo.con's Lil Wayne diss on youtube.com. "I just got hooked on this guy." In fact, Irina was so into Oreo.con that she had memorized some of his raps, one of which she performed at an Italian restaurant during the Championships. But she never expected to meet him until after round eight, when IM David Pruess contacted Irina with some numbers. She tried one, "I didnt recognize the voice, and asked 'Who is this?' and he said '.con." I'm not sure what I said after that. I guess I was in some shock. We talked for a while, turned out he'd even heard of my name cause of the Kasparov vs. the world game ten years ago!" Oreo.con also attended the closing ceremony but unfortunately, he did not rap this time. Perhaps for the U.S. Women's Championship in October?
Look for upcoming post-event blogs by Jennifer Shahade and GM Joel Benjamin and video interviews with Hikaru Nakamura and Robert Hess. Also read part I of Hikaru Nakamura's wrap-up on his personal blog.


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A trademark may be defined as a word, letter, device, or symbol, as well as any combination of these, that is used in connection with merchandise and that points distinctly to the origin of the goods.
Certificates of registration of trademarks are issued under the seal of the Patent and Trademark Office and may be registered by the owner if he or she is engaged in interstate or foreign commerce. Federal jurisdiction over trademarks arises under the commerce clause of the Constitution. Effective Nov. 16, 1989, applications to register may also be based on a "bona fide intention to use the mark in commerce." Trademarks may be registered by foreign owners who comply with U.S. law, as well as by citizens of foreign countries with which the United States has treaties relating to trademarks. U.S. citizens may register trademarks in foreign countries by complying with the laws of those countries. The right to registration and protection of trademarks in many foreign countries is guaranteed by treaties.
General jurisdiction in trademark cases involving federal registrations is given to federal courts. Adverse decisions of examiners on applications for registration are appealable to the Trademark Trial and Appeal Board, whose affirmances and decisions in inter partes proceedings are subject to court review. Before adopting a trademark, a person should make a search of prior marks to avoid unwittingly infringing upon them.
The duration of a trademark registration is ten years, but it may be renewed indefinitely for 10-year periods, provided the trademark is still in use at the time of expiration.
The application fee for registering is $325 per class.
Here are the facts and trivia that people are buzzing about.


text: SIEMENS FOUNDATION AND DISCOVERY EDUCATION ANNOUNCE WINNERS OF THE 2014 SIEMENS WE CAN CHANGE THE WORLD CHALLENGE Chanika Colbert, Siemens Foundation770-751-2073, firstname.lastname@example.com Christina Scripps, Discovery Education240-662-6502, firstname.lastname@example.com — Premier Competition Empowers Students Nationwide To Tackle the Country's Most Pressing Environmental Issues and Earn More Than $300,000 in Scholarships, Savings Bonds and More – ISELIN, N.J., April 22, 2014—Today, the Siemens Foundation and Discovery Education announced the national winners of the sixth annual Siemens We Can Change the World Challenge, the nation's premier national environmental sustainability challenge that provides K-12 students with the tools and inspiration to improve their schools, communities and world. As part of the 2014 Challenge, elementary, middle and high school students across the country identified environmental issues in their schools and communities and created replicable solutions using digital curriculum designed by Discovery Education. An esteemed panel of environmental advocates, STEM professionals, science teachers and professors selected this year's winners based on each project's viable solution to a real world environmental challenge. Projects were also evaluated on their ability to engage the support of the local community and ability to be replicated globally. "Now, more than ever, environmental issues are at the forefront of our national discussion. These students bring fresh perspectives with new and exciting solutions that affect change," said David Etzwiler, CEO of the Siemens Foundation. "The Siemens Foundation and Siemens' 53,000 employees are inspired by the innovation of these students." "Participating in the Siemens We Can Change the World Challenge gave my students an opportunity to put the science they learn in the classroom to work in the real world," said Scott Bolen, teacher and mentor to the first place high school winners of the 2013 Siemens We Can Change the World Challenge. "By the end of the project my students were not only advancing their science skills, they were developing a passion for science and the environment that will serve them well as both students and citizens for years to come." In addition to other prizes, the high school first place winning team receives $50,000 in scholarship money. Each student on the middle school first place winning team will receive a $10,000 savings bond along with a Discovery Student Adventure trip to Alaska. The first place elementary school winners in K-2 and 3-5 are each invited to a Discovery Education assembly focusing on sustainability, at which time a $5,000 grant will be given to each school. Past winners have presented their projects at the United Nations and the White House Science Fair, to the Chief Administrator of the EPA, and to other national, regional and local officials. Many projects have gone on to change schools' policies and inspire new ideas for politicians to craft green legislation. High School Winners First Place Winners: Water Bottle Warriors – A Novel Biosensor and Filter from Recycled Bottles for Water PollutionNorth County Senior High SchoolGlen Burnie, MD Second Place Winners: Vehicle Vindicators – Cars Producing Oxygen: The Power of PhotosynthesisMarvin Ridge High SchoolWaxhaw, NC Third Place Winners: Team Excelsior – An Inquiry Into Sustainable Solutions for Colorado Coal Mine Waste RevegetationAnimas Charter High SchoolDurango, CO Middle School Winners First Place Winners: Zero Waste Team Delaware – Toward Zero WasteCab Calloway School of the ArtsWilmington, DE Second Place Winners: Detergent Detectives – Let's Switch to Environmentally Friendly DetergentsNovi Middle SchoolNovi, MI Third Place Winners: Hudson Lake Doctors – Hudson Lake Doctors Doctoring our Lakes and RiversHudson Middle SchoolHudson, WI Elementary School Winners Grades 3-5 First Place Winners: Lynn Woods Grade 3 – Stop Dumping and Start HelpingLynn Woods Elementary SchoolLynn, MA Second Place Winners: King's 4th Grade – Green Team Challenge – Plastic Waste ReductionKing's Elementary SchoolSeattle, WA Third Place Winners: Zero Waste – Peabody TeamPine Ridge Elementary SchoolAuberry, CA Elementary School Winners Grades K-2 First Place Winners: Earth Agents – Precious WaterSnyder Elementary SchoolSnyder, TX Second Place Winners: The Zero-Waste Zealots – Reducing TrashKindi Mont AcademyNaperville, IL Third Place Winners: Riverside Elementary Gifted – Grade 2 – The Textile Troops: Operation Main StreetRiverside School DistrictTaylor, PA For additional details about prizes, visit: • High School: http://www.wecanchange.com/high-school/about-challenge/prizes/ • Middle School: http://www.wecanchange.com/middle-school/about-challenge/prizes/ • Elementary School: http://www.wecanchange.com/elementary-school/about-challenge/prizes/ For more information on the Challenge, the national winners and their projects, visit www.wecanchange.com. About Siemens We Can Change the World Challenge The Siemens We Can Change the World Challenge (www.wecanchange.com) is the premier national sustainability challenge. It empowers students in grades K-12 to develop and share environmental solutions that change the world. The Challenge is sponsored by the Siemens Foundation and is administered by Discovery Education to educate, empower and engage elementary, middle and high school students and teachers to become "Agents of Change" in improving their schools, communities and world. To date, nearly 100,000 students have participated, creatively engaging their communities to find solutions to a local environmental issue or global energy challenge. All participants research their issue using scientific investigation and create a replicable green solution using web-based tools powered by Discovery Education. For more information, visit www.wecanchange.com. About the Siemens Foundation The Siemens Foundation supports educational initiatives in the areas of science, technology, engineering and mathematics (STEM) in the United States. Its signature programs include the Siemens Competition in Math, Science & Technology, Siemens We Can Change the World Challenge, a sustainability challenge which encourages K-12 students to develop innovative green solutions for environmental issues and the Siemens STEM Academy, a national educator professional development program designed to support educators in their efforts to foster student achievement in STEM fields. The Foundation's mission is based on the culture of innovation, research and educational support that is the hallmark of Siemens' U.S. companies. For further information, visit www.siemens-foundation.org or follow @sfoundation. About Discovery Education Discovery Education is the global leader in standards-based digital content and professional development for K-12, transforming teaching and learning with award-winning digital textbooks, multimedia content that supports the implementation of Common Core, professional development, assessment tools, and the largest professional learning community of its kind. Available in over half of all U.S. schools and primary schools in England, community colleges and in 50 countries around the world, Discovery Education partners with districts, states and like-minded organizations to captivate students, empower teachers, and transform classrooms with customized solutions that accelerate academic achievement. Discovery Education is powered by Discovery Communications (NASDAQ: DISCA, DISCB, DISCK), the number one nonfiction media company in the world. Explore the future of education at www.discoveryeducation.com.


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Attention Conservation Notice: HSR supporters have undermined their own vision through their cynical lack of faith in voters and democratic process. Yeah, democracy is tedious. And you don't get what you want when you want it. That's the price.
For those of you who have been watching the California HSR saga, the Superior Court Judge Michael Kenny issued his second ruling in the Kings County Lawsuit filed earlier this year. To recap, the lawsuit alleged multiple things (and since I'm not a lawyer, I will probably mess this up, but here goes)
1. The state does not have a finance plan that satisfies Prop 1a passed in 2008, which the first link of statewide HSR system (520 km if I remember right), and that link would operate without taxpayer subsidy. (Whose idea was it to put THAT in the initiative? Yes, HSR systems make money around the world; but they usually do so after government eats most of the early capital cost risks and babies the systems along in their early years. Is that really so wrong, when you think about it? Isn't one of the roles of government to absorb risk?)
2. The "blended system" plan that came about after the CHSRA was forced to admit the real costs of the proposed high speed rail system isn't what Prop 1a promised voters. (The cost estimates went from $32 billion at the time of the vote, which was laughable, to $43 billion, and then, after much dusting up with People Who Can Do Math the far-more-accurate $83 to $117 billion, which caused an outcry, which caused them to rescope the project to the blended system, resulting about $76 billion. This rescoping is going to bite them, too. Yeah, I know, but keep up, would you?)
3. And therefore, the state is not allowed to spend out of the $9 billion of bonds that Prop 1a authorized, which also means they can't spend from the federal pots of money they obtained either.
Sigh. That's only on Part 1. Part 2 of the lawsuit will be argued in 2014, when they take on whether the blended system on the table captures what voters were promised. As it stands now, plenty of people are annoyed; what they voted for was a high speed rail system. They didn't vote for a proposition that builds HSR in the central valley and then improves Caltrain and Amtrack in the big urban markets. The revised cost estimates made some folks furious; the revised construction plan annoys plenty of others who note that the resulting system is not likely to be particularly high speed. (It would be higher speed than now. Does that help?)
So the Judge heard arguments in June and issued his first ruling in favor of King's County and Fukuda in November, but he gave the Attorney General time to make arguments for why the state should be allowed to spend out those resources. Again, I'm not a lawyer, but my sense would have been that this precise moment, having lost, would have been a good time to grovel and scramble to find a potential source of funding to fix the gap. Instead, the Attorney General tried to argue that the court does not have the authority to make a decision, aka as "you're not the boss of me," and that the right to determine whether funds may be spent resides with the legislature.
Last week, the judge issued his final decision, and it's not good for HSR: the state can't spend 1a money, and they can't enjoin Federal money. Since they've already spent about $1 billion, Californians are on the hook for that anyway.
Reporting on Kenny's ruling has been, generally, petulant, like Kenny is a mean meany mean pants who is capriciously throwing up obstacles to a great vision.
Only he's not. The judge is enforcing the terms of the ballot box initiatives supporters wrote and voted for.
From the beginning, the supporters of the project have overemphasized what damage criticism might do–note their continued flouncing around about the Hyperloop, which is the least of their problems at this point–and underemphasized the possibility that, in managing and using criticisms that arise, voters might still be brought around to understanding–and perhaps event wanting–public infrastructure of this scale and scope. Instead, they pursued assumed the worst of voters–that voters are spoiled brats incapable of signing on to a vision unless it was tarted up with unrealistic promises about how the projects work. I have said it a million times on this blog: I think voters would have been open to persuasion, even if it took longer than supporters wanted. Instead of trusting the democratic process, project supporters did everything they could to subvert it. In politics, William Galston once noted, one does occasionally have the obligation to play hardball. But here? Now, the procedural tide has turned on them, and they will have no choice but to go back to the very voters they have trusted so little–and whose faith in the project they have done much to undermine.


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- Home
- English Language Learning
- Catalogue
- Adult & Young Adult
- Cambridge English Empower
Cambridge English Empower Elementary
Cambridge English Empower Elementary
Combo A with Online Assessment
English Type
British English
Replaced by 9781108962001
£31.14 (+VAT) Print/online bundleReplaced by 9781108962001Unavailable
1 Static online product, 1 Paperback
1 Static online product, 1 Paperback
Print/online bundle
Part of
Date Published
December 2015
CEF Level
Product description
Included in Combo A is the first half of the Student's Book and the print Workbook (with key) plus Online Assessment and Personalised Practice.
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face2face Second edition is the flexible, easy-to-teach, 6-level course (A1 to C1) for busy teachers who want to get their adult and young adult learners to communicate with confidence.Authors: Chris Redston, Gillie CunninghamEnglish Type: British EnglishCEF Level: B2Publication date: November 2017
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