text:
### TL;DR In this paper the author discusses the Newtonian views...
Carlo Rovelli is an Italian theoretical physicist and writer. His m...
what does this mean time does not exists ? How it was existing befo...
A typo in the first sentence of the abstract. Absolutely wonderful ...
### Newtonian space and time Isaac Newton considered space and t...
A great lecture by Carlo Rovelli about ***"The Physics and Philosop...
### Einstein's spacetime Einstein's theory of relativity refines...
In Newtonian physics space and time are absolute. They constitute t...
William C. Evans-NEWTON'S BUCKET
This is not what Newton said, rather he said $f = \frac{d}{dt} mv$
> *"After three centuries, Einstein found a new and simpler answer....
What does local mean if you're outside the context of time and space?
The idea of an absolute space acting as a "container" for the physi...
> With or without such an explicit reference to God, for three cent...
A great lecture by Carlo Rovelli about ***"The Physics and Philosop...
Galileo was the first to realize that the physical motion of object...
**What is now?** Time is not absolute, there is not a "now" that co...
> *"Therefore the relational nature of space revealed by GR extends...
Julian Barbour spent the majority of his career studying this quest...
“If this point of view is correct,temporality is an artifact of our...
> *"the idea that there is a ‘‘Present’’ extending all over the uni...
Philosophy and Foundations of Physics
The Ontology of Spacetime
D. Dieks (Editor)
DOI 10.1016/S1871-1774(06)01002-3
25
Chapter 2
The Disappearance of Space and Time
Carlo Rovelli
Centre de Physique Theorique de Luminy, Universite
´
de la Me
´
diterrane
´
e, Marseille,
France
Abstract
I argue that lesson of general relativi ty is that at our present state of knowledge the best
way for making sense of the world is to discard the notions of space and time. New-
tonian space and time can be reinterpreted as aspects of the gravitational ﬁeld, whi ch is
only one among the various dynamical physical ﬁelds making up the world. Physical
ﬁelds do not need to inhabit spacetime in order to exist. The resulting understanding of
the resulting interpretation of temporality, appears to have strong elements of novelty. I
consider the viability of a foundation of our understanding of the world in which space
and time play no role.
1. The ontology of spacetime after relativity
Our understanding of the natural world evolves. We have developed a concep-
tual structure that allows us to apprehend and frame the world that we perceive
and think; but this conceptual structure evolves, driven by experience and ra-
tional investigation. Science is a continuous exploration of novel and more
effective ways for thinking the world. We cannot exit our own way of thinking;
but we can modify it from within, exploring modiﬁcations of our basic as-
sumptions, and testing them for consistence and against experience. This proc-
ess of exploration of the space of the ideas is at the core of theoretical physics.
The notions of space and time of classical physics are a characteristic product
of this process. We own them to a large extent to the work of Newton, in the
17
th
century. Newton defended a novel way of thinking space and time against
the dominant views of his time. This way proved then extraordinarily effective.
The Newtonian notions of space and time have been extensively utilized and
discussed in depth. With time, they have been ‘‘absorbed’’ by our culture at
large, and have become the dominant view.
The relativistic revolution of early 20
th
century, once more due to a remark-
able extent to a single man, has taught us that there is a more effective way of
understanding space and time than the Newtonian one. The novel relativistic
understanding of space and time, however, has not yet been integrated into the
common, and not even into the learned, way of thinking the world. Yes, mental
habits take time to change; but I am often surprised by the excessive attachment
that many thinkers maintain to ideas for understanding the world that have
been clearly proven ineffective. These ideas were useful for a while, roughly in
the three centuries between Newton and Einstein. But we must not mistake a
tool that has proven useful for an eternal truth. There are commonly used
concepts, such as the idea of an ‘‘objective present state of the world’’, that
make no sense, in the light of what we have learned about the universe. Rel-
ativity is not ‘‘contemporary science’’: it is close to a century old. It is certainly
time to take it seriously, discuss it in depth, and get used to it.
A reason for the slow adaptation to the relativistic understanding of space
and time is that the relativistic revolution has happened in more than one step.
The ﬁrst step is special relativity (SR); shortly after came general relativity
(GR)
1
. For a few decades, while SR was blessed by continuous conﬁrmations,
especially from particle physics, GR had spectacular but scarce empirical sup-
port. In this situation, the attention was mostly on the relatively simpler con-
ceptual novelty of SR, leaving GR in a limbo at the borders of our map of
reality. But in the last decades the number and the success of experiments and
observation conﬁrming the physical validity of GR have exploded, and the
theoretical interest in GR has boomed. Today, we cannot leave GR out of the
picture. SR is little more than a minor variation of the Newtonian conceptu-
alization of spacetime. The special relativistic universe is a theoretical model
whose true interest as a fundamental way to understand reality was signiﬁcant
for less than 10 years, between 1905 and 1915. Therefore we must focus on GR
if we want to hold a view of space and time compatible with what we have
understood so far about the natural world.
1
To further complicate matters, the relativistic revolution has not yet ended, because we have
not yet fully unraveled its relation with quantum physics. Knowledge of quantum theory induces
us to certain conceptual choices in understanding relativity, but care should be taken in reading
these hints because we still lack a deﬁnitive synthesis.
C. Rovelli26
The new understanding of spacetime that has emerged from the relativis-
tic revolution differs from the Newtonian picture especially with regard to
the ontological status of spacetime the subject of this book. Newton made
the successful hypothesis that space and time are ﬁxed structured background
entities underlying material reality, which participate in governing the motion
of physical objects. What Einstein has discovered is that Newton had mistaken
a physical ﬁeld for a background entity. The two entities hypostatized by
Newton, space and time, are just a particular local conﬁguration of a physical
entity the gravitational ﬁeld very similar to the electric and the magnetic
ﬁeld.
Einstein’s discovery is that Newtonian space and time and the gravitational
ﬁeld are the same entity. There is a tradition of expressing this discovery saying
that ‘‘there is no gravitational ﬁeld: space and time become dynamical’’. I think
that this is a convoluted and misleading way of thinking, which does not do
justice to Einstein’s discovery, and has the additional ﬂaw of becoming mean-
ingless as soon as we take into account the fact that the gravitational ﬁeld has
quantum properties.
The clean way of expressing Einstein’s discovery is to say that there are no
space and time: there are only dynamical objects. The world is made by dy-
namical ﬁelds. These do not live in, or on, spacetime: they form and exhaust
reality.
One of these ﬁelds is the gravitational ﬁeld. In the regimes in which we can
disregard its dynamics, this ﬁeld interacts with the rest of the physical objects as
if it were a ﬁxed background. This background is what Newton discovered and
called space and time. We can keep using the evocative terminology ‘‘space-
time’’ to indicate the gravitational ﬁeld. But it has practically none of the fea-
tures that characterized space and time. Relativistic spacetime is an entity far
more akin to Maxwell’s electric and magnetic ﬁelds than to Newtonian space.
In classical GR, a given solution of the ﬁeld equations might still have some
vague resemblance to the Newtonian’s notions, since it deﬁnes a ‘‘continuum’’
which things can be imagined ‘‘to inhabit’’. But the only compelling reason for
thinking that ‘‘spacetime’’ is the gravitational ﬁeld, and not say the elec-
tromagnetic ﬁeld, is the contingent fact that we live in a portion of the universe
where the gravitational ﬁeld is sufﬁciently constant for us to use it as a con-
venient reference.
Quantum mechanics reinforces this point of view. A solution of the classical
ﬁeld equations is like a particle trajectory: a notion that only makes physical
sense in the classical limit. The gravitational ﬁeld has quantum properties, and
therefore it cannot deﬁne a spacetime continuum in the small.
Properly speaking, relativity has taught us that the effective way of thinking
about the world in the light of what we have learned so far is to give up the
notions of ‘‘space and time entities’’ entirely. This is not a dramatically radical
The Disappearance of Space and Time 27
view, since it is not far from the way space was commonly conceptualized before
Newton. On the other hand, it has a novel twist of great interest, especially as
far as time, and the relation between time and space, are concerned.
In Newtonian physics, if we take away the dynamical entities, what remains is
space and time. In relativistic physics, if we take away the dynamical entities,
nothing remains. As Whitehead put it, we cannot say that we can have space-
time without dynamical entities, anymore than saying that we can have the cat’s
grin without the cat (Whitehead, 1983).
In the rest of this text, I discuss relativistic spacetime in some more detail. I
start by recalling a few facts about pre-Newtonian western ideas about space
and time. This is important because the Newtonian scheme is often mistaken for
a sort of ‘‘natural’’ understanding of space and time. Nothing is more wrong:
the Newtonian space and time ‘‘entities’’ form a strongly counter-intuitive the-
oretical construction, which met ﬁerce resistance at ﬁrst. Next, I illustrate the
modiﬁcation of the notions of spacetime introduced by SR and GR. I focus on
the Newtonian notions that are to be abandoned. I close by mentioning the
possibility of a proper relativistic foundation of the physical description of the
world where the notions of space and time play no role.
2. Space
There are two traditional ways of understanding space in the western culture: as
an entity or as a relation. ‘‘Space is an entity’’ means that space exists also when
there is nothing else than space. It exists by itself, and objects may move in it.
‘‘Space is a relation’’ means that the world consists entirely of physical objects
(particles, bodies, ﬂuids, ﬁelds y). These objects have the property that they
can be in touch with one another, or not. Space is this ‘‘touch’’, or ‘‘contiguity’’,
or ‘‘adjacency’’ relation between objects. Connected to these two manners of
understanding space, are two manners of understanding motion. If space is an
entity, motion can be deﬁned as going from one part of space to another part of
space. This is denoted by ‘‘absolute motion’’. If space is a relation, motion can
only be deﬁned as going from the contiguity of one object to the contiguity of
another object. This is called ‘‘relative motion’’. For a physicist, the issue is
which of these two ways of thinking about space and motion allows a more
effective description of the world.
The dominant view in the European tradition, from Aristotle to Descartes,
was to understand space and motion as relational. Aristotle, for instance, de-
ﬁnes the spatial location of an object as ‘‘the inner surface of the innermost
object that surrounds the body’’ (Aristotle, Physics, Book IV, Chapter 4[20],
Aristotle, 1952). This is relational space. Descartes deﬁnes motion as ‘‘the
C. Rovelli28
transference of one part of matter or of one body, from the vicinity of those
bodies immediately contiguous to it, and considered at rest, into the vicinity of
some others’’ (Descartes, Principia Philosophiae, Section II-25, p. 51, Descartes,
1983). Aristotle as well insists that motion is relative. He illustrates the point
with the example of a man walking over a boat. The man moves with respect to
the boat, which moves with respect to the water of the river, which moves with
respect to the ground y .
The alternative view of space, as an independent entity, existed since ancient
times (mostly in the Democritean tradition), but became dominant only with
Newton. It is also the way spacetime (rather than space) is understood in SR.
For Newton, space is absolute and motion is absolute: ‘‘So, it is necessary that
the deﬁnition of places, and hence local motion, be referred to some motionless
thing such as extension alone or space, in so far as space is seen truly distinct
from moving bodies’’ (Newton, De gravitatione et Aequipondio Fluidorum, pp.
89–156, Newton, 1962). This is in patent contrast with Descartes deﬁnition,
given above.
It should be noted that the difference between the two points of view is not so
strong as it seems at ﬁrst sight. Starting from a relational point of view, we can
always choose a physical entity, refer all motion to this preferred entity and call
this entity ‘‘space’’. Newton does not miss this point, and in fact he speciﬁes that
what is to be called space has to be ‘‘truly distinct from moving bodies’’. New-
ton thought he had discovered this entity ‘‘truly distinct from moving bodies’’,
the way to detect it and its effects. GR is the realization that the entity dis-
covered by Newton is not at all ‘‘truly distinct from moving bodies’’. In fact, it is
barely distinguishable from the other ﬁelds.
In introducing the idea of absolute space, Newton did not challenge a long
tradition with light heart: he devotes a long initial section of the Principia to
explain the reasons for his choice. Today we can say that the strongest argument
in Newton’s favor is a posteriori: his theoretical construction works extraor-
dinary well. Relational Cartesian and Leibnizian proposals were never as effec-
tive. But this was not Newton’s argument. Newton invokes empirical evidence,
discussing the famous bucket experiment. This experiment proves that there are
physical effects (the bending of the surface of the water in the bucket) that do
not depend on the relative motion of the water with respect to the surrounding
objects (the bucket).
The surface of the water curves when the water rotates: but rotates with
respect to what?
Newton argues that the only reasonable answer is absolute space. The con-
cavity of the water’s surface is an effect of the absolute circular motion of the
water: the motion with respect to absolute space. This, claims Newton, proves
the existence of absolute space. Newton’s argument is subtle; it has been often
The Disappearance of Space and Time 29
challenged, but it has withstood all attacks for three centuries
2
. It has then
collapsed under Einstein’s alternative and more effective answer.
Newton needed (accelerated) motion, exempliﬁed by the rotation of the water
in the bucket, to be absolute for the foundation of his dynamics. Without this,
Newton’s main law
~
F ¼ m
~
a would not even make sense: what would be the
meaning of the acceleration
~
a?
Opposition to Newton’s absolute space was very strong. Leibniz and his
school argued ﬁercely against absolute motion and absolute acceleration.
Doubts never really disappeared all along the centuries and a feeling kept lin-
gering that something was wrong with Newton’s argument. Ernst Mach re-
turned to the issue suggesting that Newton’s bucket argument could be wrong
because the water does not rotate with respect to absolute space: it rotates with
respect to the full matter content of the universe. But the immense empirical
triumph of Newtonianism could not be overcome. For three centuries.
After three centuries, Einstein found a new and simpler answer. The bending
of the surface of the water is due to the relative motion of the water with respect
to a physical entity: the local gravitational ﬁeld. It is the gravitational ﬁeld, not
Newton’s inert absolute space that tells objects if they are accelerating or not, if
they are rotating or not. There is no inert background entity such as Newtonian
space: there are only dynamical physical entities. Among these are the ﬁelds,
introduced in our physical picture of the world by Faraday and Maxwell.
Among the ﬁelds is the gravitational ﬁeld.
Whether the water surface in Newton’s bucket is concave or ﬂat is not deter-
mined by the motion of the water with respect to absolute space. It is determined
by the physical interaction between the water and the gravitational ﬁeld. Einstein’s
discovery is that Newton had mistaken the gravitational ﬁeld for absolute space.
In Newtonian physics, the spacetime coordinates
~
x and t refer to absolute
space. They can be identiﬁed with the reading of measuring devices carefully
selected as the ones that capture the structure of space and time. This selection is
obtained by monitoring and correcting the ‘‘inertial’’ effects such as the bending
of the water of the water, which signal motion with respect to absolute space.
2
Or course relationalism, i.e., the idea that motion can be deﬁned only in relation to other objects,
should not be confused with Galilean relativity. Galilean relativity is the statement that ‘‘rectilinear
uniform motion’ is a priori indistinguishable from stasis. This is equivalent to saying that velocity
(just velocity!), is only relative to other bodies. Relationalism, on the other hand, holds that any
motion (however zigzagging) is a priori indistinguishable from stasis. The very formulation of Gali-
lean relativity assumes a nonrelational deﬁnition of motion: ‘‘rectilinear and uniform’’ with respect to
what? When Newton claimed that motion with respect to absolute space is real and physical, he, in a
sense, overdid it, by insisting that even rectilinear uniform motion is absolute. This caused a painful
debate, because there are no physical effects of inertial motion (therefore the bucket argument fails
for this particular class of motions). Newton is well aware of this point, which is clearly stated in the
Corollary V5 of the Principia, but he chooses to ignore it in the introduction of the Principia. I think
he did this just to simplify his argument, which was already hard enough for his contemporaries.
C. Rovelli30
Einstein realized that ﬁnding out the pre-GR ‘‘inertial’’
~
x and t is nothing else
than detecting local features of the gravitational ﬁeld. In the theoretical appa-
ratus of GR, on the other hand, the spacetime coordinates
~
x and t have a
completely different status, and it is only an unfortunate historical accident that
they are denoted in the same manner as the pre-general-relativistic inertial co-
ordinates. The relativistic
~
x and t coordinates have no direct physical meaning
(unless we gauge ﬁx them to represent something else). The reading of measuring
devices is identiﬁed with quantities in the theory that are independent of
~
x and t.
More formally, in the mathematics of classical GR we employ a background
‘‘spacetime’’ manifold and describe the ﬁelds as living on this manifold. How-
ever, the diffeomorphism invariance of the theory demands that the localization
on this manifold is pure gauge. That is, it is physically irrelevant. The manifold
is just an artiﬁce for describing a set of ﬁelds and other physical objects whose
only ‘‘localization’’ is with respect to one another.
A state of the universe does not correspond to one ﬁeld conﬁguration over the
spacetime manifold M. It corresponds to an equivalence class under active
diffeomorphisms of ﬁeld conﬁgurations. Therefore localization over M is phys-
ically irrelevant. In fact, M has no physical interpretation. It is a mathematical
device without physical counterpart. It is a gauge artifact. M cannot be inter-
preted as a set of physical ‘‘events’’, or physical spacetime points ‘‘where’’ the
ﬁelds take value. The only possibility of locating points is with respect to the
dynamical ﬁelds and particles of the theory itself. It is meaningless to ask
whether or not the gravitational ﬁeld is ﬂat around the spacetime point A,
because there is no physical entity ‘‘spacetime point A’’. Contrary to Newton,
spacetime points are not entities where particle and ﬁelds live
3
.
The gravitational ﬁeld g
mn
ðxÞ determines a four-dimensional continuum with a
metric structure. Excessive signiﬁcance is often attributed to this structure, as if
distance was an essential property of space, or even an essential property of
reality. This is like an Eskimo thinking that snow is an essential property of the
ground.
We could have developed physics without ever thinking about distances, while
nevertheless retaining the complete predictive and descriptive power of our theories.
We live in physical conditions where atoms form and interact with the gravitational
ﬁeld in such a way that they maintain structures characterized by the fact that the
integral of the gravitational ﬁeld d ¼
R
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
g
mn
dx
m
dx
n
p
along these structures is very
stable. We call this integral d ‘‘distance’’, and we have developed useful mathematics
geometry to describe the structure of these distances.
3
Physical ﬁeld are not attributes of space, anymore than a mosaic is an attribute of the wall. The
wall can be taken away from the mosaic, without necessarilly destroying the mosaic. The clearest
intuition of the nature of a ﬁeld (in particular, a gauge ﬁeld) is the original Faraday’s intuition of a
ﬁeld as a collection of lines.
The Disappearance of Space and Time 31
Geometry has repeatedly been mistaken for an a priori feature of reality.
Euclidean geometry was erroneously thought of as necessary. Later, Riemannian
geometry as well has been erroneously considered necessary. However, there is
no a priori reason for which reality has to be understood as a continuum with
metric properties. Nor, for that matter, as a continuum at all. Indeed, contem-
porary research in quantum gravity points in a very different direction.
Conceptually, what disappears with GR is the idea of space as the ‘‘con-
tainer’’ of the physical world. As mentioned, this disappearance is not so rev-
olutionary after all: to some extent it amounts to return to the pre-Newtonian
view of space as a relation between equal-status physical entities.
Allow me to close this section with a playful observation on the relation
between the shift in our views about space and our overall world conception. In
the pre-Copernican world the cosmic organization of the ‘‘things’’ was hierar-
chically structured. The Heavens above, the Earth below, spheres in order of
decreasing perfection. Objects were located with respect to one another this
served to grant each object a precise ‘‘status’’ in the grand scheme of things,
analogous to the social position of humans. With the Copernican revolution,
this hierarchical structure was lost. Position lost any ranking value. Newton
offered reality a global frame. He offered every object the equal dignity of a
position in a uniform space. For Newton this frame was grounded in God. He
called space the ‘‘sensorium’’ of God: the world as perceived by God. Thus, the
position in space for Newton is, literally, the position of the objects in the eyes
of God. Against the multiplicity of the individual points of view determined by
the observation of the relative motions, absolute space grants a single-organ-
izing principle. According to Newton, our rationality allows us to unveil the
absolute point of view of God (by detecting inertial effects such as the bending
of the surface of the water). With or without such an explicit reference to God,
for three centuries space has been regarded as the preferred Entity with respect
to which all other entities are located. In the 20
th
and 21
st
centuries and with GR
we have been learning that we do not need this frame to keep reality in place.
Reality keeps itself in place. Objects interact with other objects, and this is
reality. Reality is the net of these interactions. We do not need an external entity
to hold this net. We do not need Space, to hold the universe. Maybe the Co-
pernican revolution is ﬁnally being completed.
3. Time
The disappearance of physical time is the second characteristic feature of
the relativistic revolution. The notion of time is harder to deal with than the
notion of space, and represents a more radical step than the disappearance
of space.
C. Rovelli32
Once again, much of the common understanding of time derives from the 17
th
century. Galileo was the ﬁrst to use a mathematical time variable t to formulate
equations describing the motion of terrestrial objects. These are equations for
functions X ðtÞ of time. Newton was well aware that we never measure the
variable t appearing in these equations directly. We use clocks whose reading T
should be taken as a good approximation of the hypothetical ‘‘true’’ time t.As
in the case of spatial measurements, we select better and better clocks by elim-
inating effects that the theory denounces as produced by the difference between
T and t. The relation TðtÞ, between clock reading and true time, can itself be
calculated from the theory, using a mechanical model of the clock. From XðtÞ
and TðtÞ we can compute X ðTÞ, which is the only relation we effectively observe.
Newtonian theory is formulated in terms of the not directly observable quantity
t. The scheme is delicate and involved, but it has worked wonderfully well for
three centuries.
SR takes the ﬁrst step out of the Newtonian understanding of time. SR does
not change the Newtonian hypostatization of absolute space and time, but
destroys the clean distinction between the two.
SR is the discovery that it makes no physical sense to say that two distant
physical events happen ‘‘at the same time’’. It is true that Einstein provides a
deﬁnition of simultaneity, two events A and B, relative to an object O in a given
state of motion
4
. But although this is a useful working deﬁnition, it is a mistake
to give it ontological signiﬁcance. There is nothing in SR that would lead us to
think that A and B have an ontic property of being ‘‘existant at the same time
with respect to O’’, besides satisfying a useful conventional deﬁnition.
To illustrate this point, consider a standard expanding cosmological model.
Its space like surfaces of homogeneity are formed by the events at equal proper
time after the big bang, or equal Friedmann time t
Fr
; these are the surfaces
naturally considered ‘‘simultaneous’’ in cosmology. These surfaces are not equal
time surfaces according to Einstein’s simultaneity deﬁnition
5
. Therefore, in a
cosmological context we have the alternative to call either ‘‘simultaneous’’
events at the same Friedmann time, or events that satisfy Einstein’s deﬁnition of
simultaneity. Both deﬁnitions are useful. The choice between them is a matter of
taste or computational convenience, not a matter of ontology.
The simple physical fact, revealed by SR, is that there are physical events on,
say, Andromeda that have no temporal relation with events on Earth. A small
gravitational wave passing in between could change Einstein’s simultaneity be-
tween us and them by years, without affecting the physics here or there anymore
4
The event A along the trajectory of an observer O is said to be simultaneous to a distant event B
if a light signal emitted by O a time interval T before A and reﬂected by B, returns to O at a time T
after A.
5
I thank Marc Lachieze-Rey for pointing this out to me.
The Disappearance of Space and Time 33
than works on the highway change relations between two cities. The lesson is
that the idea that there exists a ‘‘now’’ all over the universe does not square with
what we know about the universe. At best, we can talk of time lapsed along
individual world lines, or time experienced by individual observers.
The picture of a Universe changing from one global instant to the next is
incompatible with what we know about the world.
What is then ‘‘time’’ in the light of GR? This is a deep and important question
that in my opinion has not yet been sufﬁciently investigated. I offer here what I
think is the most useful answer to this question.
GR inherits from SR the melting of space and time into spacetime. Therefore
the relational nature of space revealed by GR extends to time as well. It follows
that in GR there is no background spacetime and therefore in particular no time
along which things happen. GR teaches us that we must abandon the idea that
the ﬂow of time is an ultimate aspect of reality. The best description we can give
of the world is not in terms of time evolution. The dynamics of GR itself cannot
be cleanly described in terms of evolution in time.
There are many distinct notions of time employed in GR: coordinate time t,
proper time S, clock times T, cosmological time t
Fr
, asymptotic Poincare
´
time y . The last two refer to the description of special solutions of the Einstein
ﬁeld equations only. They are irrelevant in a discussion of the ontology of time,
because a different ontology for different solutions of the same theory is certainly
unsatisfactory. Clock times are simply the readings of certain physical variables,
which can be locally employed as the independent variable for convenience.
Once again, they have nothing to tell us about the ontology of time. Coordinate
time is unobservable (unless the gauge is ﬁxed, in which case it designates
something else) because of general coordinate invariance. The only residual time
notion that keeps a resemblance of temporality is proper time. Proper time does
not ﬂow uniformly in the universe. It is deﬁned along a world line and, gener-
ically, if two world lines meet twice, the two proper times lapsed between the two
encounters differ. Proper time S depends on the gravitational ﬁeld, which is
inﬂuenced by the interaction with many systems. Typically, harmonic oscilla-
tions are isochronous in S.Therefore,S like the distance d described in the
previous section, is just an observable feature of the gravitational ﬁeld, which is
particularly convenient to use as a stable reference in our environment, when
describing the motion of objects assuming the gravitational ﬁeld ﬁxed. The dy-
namics of the gravitational ﬁeld itself, on the other hand, cannot be naturally
described in terms of evolution in any well-deﬁned preferred time variable.
Instead, we must describe reality in terms of correlations between observables.
We can measure physical quantities around us. The physical theory restricts the
combinations of quantities that we can measure. It predicts relations between
these quantities. Only in particular situations we can choose one quantity as the
C. Rovelli34
independent variable, call it time, and express the others as functions of it. In
general, this may not be possible, and the physical theory gives us constraints on
the values of measurable quantities that we can obtain from physical measure-
ments, not evolution laws in a preferred time variable. Quantum theory assigns
probabilities to such outcomes.
Basic physics without time is viable, it is forced upon us by relativity, and it is
conceptually coherent and consistent with our experience of the world. A com-
plete discussion of the foundations of mechanics in the absence of a notion of
time is given for instance in Rovelli (2002). Remarkably if we give up the idea
that there is a special ‘‘time’’ observable, mechanics takes a far more compact
and elegant form. This shift of point of view is forced upon us by classical GR.
If, in addition, we take quantum theory into account, the spacetime continuum,
with its last vague resemblance to temporality disappears completely, and we
confront the absence of time at the fundamental level in full.
So, where does temporality, with all its peculiar features (‘‘ﬂow’’ of time,
whatever this means, irreversibility, memory, awareness y) come from? I think
that all this has nothing to do with mechanics. It has to do with statistical
mechanics, thermodynamics, perhaps psychology or biology. In Rovelli (1993) I
have developed, in collaboration with Alain Connes, the idea that it may be
possible to recover temporality from statistical mechanics, within an atemporal
mechanical universe (statistical time hypothesis). If this point of view is correct,
temporality is an artifact of our largely incomplete knowledge of the state of the
world, not an ultimate property of reality.
Some people ﬁnd the absence of time difﬁcult to accept. I think that this is just
a sort of nostalgia for the old Newtonian notion of an absolute ‘‘Time’’ along
which everything ﬂows, a notion already shown by SR to be inappropriate for
understanding the real world. I think that the motivation for holding on to
Poincare
´
invariance, to unitary time evolution, to the idea that there is a
‘‘Present’’ extending all over the universe, is only to provide an anchorage for
our familiar notions, which are appropriate to describe the garden of our daily
life. But a bit more at large, these are notions that are inappropriate to describe
this beautiful and surprising world we inhabit.
References
Aristotle (1952). Physics. In P. H. Goetz (Ed.), The works of Aristotle. Chicago: Encyclopedia
Britannica Inc.
Descartes, R. (1983). In: V.R. Miller, & R.P. Miller (Trans.), Principia philosophiae. Dordrecht:
Reidel.
Newton, I. (1962). De Gravitatione et Aequipondio Fluidorum. In A. R. Hall, & M. B. Hall
(Eds), Unpublished papers of Isaac Newton. Cambridge: Cambridge University Press.
The Disappearance of Space and Time 35
Rovelli, C. (1993). Statistical mechanics of gravity and thermodynamical origin of time. Classical
and Quantum Gravity, 10, 1549; (1993). The statistical state of the universe. Classical and
Quantum Gravity, 10, 1567. Connes, A., & Rovelli, C. (1994). Von Neumann algebra auto-
morphisms and time versus thermodynamics relation in general covariant quantum theories.
Classical and Quantum Gravity, 11, 2899.
Rovelli, C. (2002). Partial observables, Physical Review, D65, 124013; A note on the foundation
of relativistic mechanics. I: Relativistic observables and relativistic states, Proceedings of the
15th SIGRAV conference on general relativity and gravitational physics, Rome, September, gr-
qc/0111037; Quantum gravity, Cambridge, MA: Cambridge University Press, 2004.
Whitehead, A. N. (1983). Concept of Nature: The Tarner Lectures Delivered in Trinity College,
November, 1919, 171. Cambridge, UK: Cambridge University Press.
C. Rovelli36
A typo in the first sentence of the abstract. Absolutely wonderful copy-editing from Elsevier as usual! “If this point of view is correct,temporality is an artifact of our largely incomplete knowledge of the state of theworld, not an ultimate property of reality.” “If this point of view is correct,temporality is an artifact of our largely incomplete knowledge of the state of theworld, not an ultimate property of reality.” ### Einstein's spacetime Einstein's theory of relativity refines Newton's law of universal gravitation by providing a unified description of gravity as a geometric property of space and time - **spacetime**. Spacetime is a gravitational field (similar to the electromagnetic field) and its curvature is directly related to the energy and momentum of whatever matter and radiation are present. Space and time are no longer two different absolutes but they are a continuum, unified into a four-dimensional reality that is no longer separated into three spatial dimensions and one time dimension. In Newtonian physics space and time are absolute. They constitute the stage where everything takes place so if you remove all objects space and time remain. In relativistic physics, if we take away the dynamical entities, nothing remains. Space-time as an absolute "container" over which physics happens has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the Universe. The idea of an absolute space acting as a "container" for the physical universe disappears with Einstein's theory of relativity. This disappearance is not that revolutionary since it amounts to return to the pre-Newtonian view of space as a relation between equal-status physical entities - see Aristotle's and Descartes' views above. Galileo was the first to realize that the physical motion of objects could be described by mathematical laws expressing their evolution in time. He was the first to use a mathematical time variable ***t*** to formulate equations describing the motion of terrestrial objects. He noticed that the period of swing of a pendulum is independent of its amplitude. This discovery had important implications for the measurement of time intervals. **What is now?** Time is not absolute, there is not a "now" that common across the universe. Now is a local concept. > *"The lesson is that the idea that there exists a ‘‘now’’ all over the universe does not square with what we know about the universe. At best, we can talk of time lapsed along individual world lines, or time experienced by individual observers."* > With or without such an explicit reference to God, for three centuries space has been regarded as the preferred Entity with respect to which all other entities are located. In the 20th and 21st centuries and with GR we have been learning that we do not need this frame to keep reality in place. Reality keeps itself in place. Objects interact with other objects, and this is reality. Reality is the net of these interactions. We do not need an external entity to hold this net. We do not need Space, to hold the universe. Maybe the Co- pernican revolution is finally being completed. > *"Therefore the relational nature of space revealed by GR extends to time as well. It follows that in GR there is no background spacetime and therefore in particular no time along which things happen. GR teaches us that we must abandon the idea that the flow of time is an ultimate aspect of reality. The best description we can give of the world is not in terms of time evolution. The dynamics of GR itself cannot be cleanly described in terms of evolution in time."* > *"the idea that there is a ‘‘Present’’ extending all over the universe, is only to provide an anchorage for our familiar notions, which are appropriate to describe the garden of our daily life. But a bit more at large, these are notions that are inappropriate to describe this beautiful and surprising world we inhabit."* > ***"Your head is older than your feet"*** - Carlo Rovelli > *"After three centuries, Einstein found a new and simpler answer. The bending of the surface of the water is due to the relative motion of the water with respect to a physical entity: the local gravitational field. It is the gravitational field, not Newton’s inert absolute space that tells objects if they are accelerating or not, if they are rotating or not. There is no inert background entity such as Newtonian space: there are only dynamical physical entities. Among these are the fields, introduced in our physical picture of the world by Faraday and Maxwell. Among the fields is the gravitational field."* Carlo Rovelli is an Italian theoretical physicist and writer. His main field of research is quantum gravity and he is one of the co-creators of loop quantum gravity. He has also worked in the history and philosophy of science. He authored several books on science including "Seven Brief Lessons on Physics" which has been translated into 41 languages and has sold over a million copies worldwide. Learn more about the author here: [Carlo Rovelli](http://www.cpt.univ-mrs.fr/~rovelli/) !["Carlo Rovelli"](https://physicsworld.com/wp-content/uploads/2018/06/PWJul18REV-Carlo_HERO.jpg) ### Newtonian space and time Isaac Newton considered space and time as absolute. According to the Newtonian view space is flat, uniform and distinct from body, a stage where everything happens, and time passes uniformly without regard to whether anything happens in the Universe. This view is often mistaken for a sort of ‘‘natural’’ understanding of space and time but it was proven wrong by Einstein's theory of relativity at the beginning of the 20th century. Flat, uniform fixed stage where everything happens. Read on here: [Newton’s Views on Space, Time, and Motion ](https://plato.stanford.edu/entries/newton-stm/) A great lecture by Carlo Rovelli about ***"The Physics and Philosophy of Time".*** [![physics of time](https://i.imgur.com/JPldvfg.png)](https://www.youtube.com/watch?v=-6rWqJhDv7M) A great lecture by Carlo Rovelli about ***"The Physics and Philosophy of Time".*** Watch it here: [![physics of time](https://i.imgur.com/JPldvfg.png)](https://www.youtube.com/watch?v=-6rWqJhDv7M) This is not what Newton said, rather he said $f = \frac{d}{dt} mv$ what does this mean time does not exists ? How it was existing before ? It Never did !! You can learn more about Aristotle and Descartes' views on space here: - [Aristotle's Doctrines](https://plato.stanford.edu/entries/newton-stm/#2.2) > "According to Aristotle, the universe is a material plenum, finite in extent, bounded by the outermost sphere of the fixed stars. Beyond that there is no void, i.e., empty places, since, as Aristotle defines ‘place’, the place of something is the outermost of “the innermost motionless boundary of what contains it.” Hence, since there are no boundaries outside the outermost celestial sphere, there are no places or space outside of it." - [Descartes' Innovation](https://plato.stanford.edu/entries/newton-stm/#3) > "Motion, according to “the truth of the matter”, is defined to be “the translation of one part of matter, or one body, from the vicinity of those bodies, which are immediately contiguous to it and are viewed as if at rest, to the vicinity of others.” In consequence, Descartes points out, each body has a single motion proper to it (in contrast to the numerous relative motions that can be ascribed to it depending on which other bodies are selected in order to determine its place)." What does local mean if you're outside the context of time and space? William C. Evans-NEWTON'S BUCKET Julian Barbour spent the majority of his career studying this question of what physics would look like without a notion of time, which culminated in several papers together with a popular science book: https://www.wikiwand.com/en/The_End_of_Time_(book) I'm surprised it's not in the references, but it should definitely be noted. ### TL;DR In this paper the author discusses the Newtonian views of space and time and the concept of spacetime as introduced by Einstein's theory of relativity. Newton's ideas were valid and useful for 3 centuries but a useful tool should not be mistaken for an eternal truth. General Relativity refutes the idea of space as an absolute "container" over which physics takes place has no objective physical meaning and instead the gravitational interaction is represented as just one of the fields forming the universe. The idea that there is a "Now" extending all over the universe is not compatible with General Relativity - there is not one time but there are many relative times. The idea of an extending "Now" serves only to provide an anchorage for our familiar notions, which are appropriate to describe our daily lives. Space and time are no longer two different absolutes but they are a continuum, unified into a 4-dimensional reality that is no longer separated into 3 spatial dimensions and one time dimension. This part answers your question: > *"How-ever, the diffeomorphism invariance of the theory demands that the localization on this manifold is pure gauge. That is, it is physically irrelevant. The manifold is just an artiﬁce for describing a set of ﬁelds and other physical objects whose only ‘‘localization’’ is with respect to one another."* There is no absolute reference frame, but objects still have a position **"with respect to one another"**.


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# Prove that: $\lim\limits_{n\rightarrow \infty}({\frac{10^{\log_2{\log_2{n}}}}{\log_2{n}}})=\infty$

Prove that:

$$\lim\limits_{n\rightarrow \infty}\left({\frac{10^{\log_2{\log_2{n}}}}{\log_2{n}}}\right)=\infty$$

I've tried applying L'hopital to no avail.

Let $\log_2n=m$ to get $$\lim_{n\rightarrow \infty}({\frac{10^{\log_2{\log_2{n}}}}{\log_2{n}}})=\lim_{m\to\infty}\dfrac{10^{\log_2m}}m$$

Again, let $\log_2m=r\iff m = 2^r$

$$\implies\lim_{m\to\infty}\dfrac{10^{\log_2m}}m=\lim_{r\to\infty}\dfrac{10^r}{2^r}=\lim_{r\to\infty}5^r=?$$

One could also use $$10^{\log_2(x)}=2^{\log_2(10)·\log_2(x)}=x^{\log_2(10)}$$ to find that the fraction inside the limit reduces to $$\log_2(n)^{\log_2(10)-1}=\log_2(n)^{\log_2(5)}$$

Another possibility, which doesn't involve substitutions, is to take $\log_2 L$, where $L$ is the limit. The argument then becomes (given the linearity of the operator $\lim$): $$\log_2 L=\log_2\left(\frac{10^{\log_2\log_2 n}}{\log_2 n}\right)=\log_2\log_2 n\cdot\log_210-\log_2\log_2 n=\left(\log_210-1\right)\log_2\log_2 n$$ since $1=\log_22$, it follows that $\log_210-1=\log_25$, which is greater than one. Next, taking $2^{\log_2 L}$ gives the equivalent simplified argument: $$L=2^{\log_2 L}=2^{\log_25\cdot\log_2(\log_2 n)}=2^{\log_2(\log_2 n)^{\log_25}}=(\log_2 n)^{\log_25}$$ Applying the limit as $n\to\infty$ gives the result $L=\infty$, since it's a positive monotone increasing function taken to a positive constant power. $\square$

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# Posting from vim using vimpress

This is my first post from vim using vimpress.Because of having a low bandwidth connection, I was forced to think for an alternative and vim being as versatile it can getto throw answers to everything we can think of. Okay, now getting on with the topic at hand what is vimpress and what does it do ?

Here’s how we start, download the script from vim’s site.Put the files blog.vim and blogsyntax.vim in ~/.vim/plugin folder ,if the folder does not exist create one. Open the blog.vim file and fill in your your username and password and site url in their respective places. Open vim and type in CmdLine

:so ~/.vim/plugin/blog.vim
:so ~/.vim/plugin/blogsyntax.vim

now that you have successfully setup vimpress, you can happily blog away.

Have you written a response to this? Let me know the URL


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# Tag Info

4

Below is a cross section of the skin between the passenger windows of a Boeing 747-100 (N602US). It's 2 layers, each layer is ~.075" thick. This is the first iteration of the 747 (50 years ago) so might be different on later series as they improved the design and reduced weight. Source: own work

0

Using a delta wing for a glider would not be ideal because delta wings have a lower Coefficient of lift as well as a lower aspect ratio. However you can find wing span by using the formula in the link attached: https://www.grc.nasa.gov/www/k-12/airplane/geom.html AR=s^2/A when you use this formula you can get a idea on how well it would glide. For example ...

3

It's not just for the looks. The A350 is the first Airbus ever to feature curved cocpit windows to reduce drag, and according to Airbus, the black "mask" contributes to ease of maintenance and harmonization of the thermal condition of the windows: "The perfectly curved shape of the nose helps the air flow hug the surface, in the least turbulent manner, ...

1

There are three approaches to this need. First, you could build what amounts to a tandem wing. If you check the CoG location for a Rutan Quickie or its many derivatives, I'm pretty sure you'll find it's between the front (lower) and rear (upper) wings. Likewise, a Flying Flea (aka Pou de Ciel) has the CoG much further aft, relative to the forward/upper ...

0

They certainly do. My airplane is built 100% out of 3d- printed plastic. It's really handy. Everything is easily replaceable. Crashes are no longer any concern. I've saved a template of my cranium too, just in case it might someday be need to be replaced.

3

If they have to pilot it manually, without training or guidance, a passenger has about as much chance to land an airliner as the airliner has to just happen to land on its own as it runs out of fuel. At least there will be less of it to burn. It's not a ground vehicle, you can't wing it, and the runway is tiny compared to the sky. Talkdown landings are a ...

0

Aluminum and titanium are used in aircraft because they are light weight materials that are strong and durable like steel, however they are non-ferrous unlike steel meaning no rust. The two metals are usually composited with another material such as carbon, composites are known for being stronger than using one base material because it will alter the ...

3

I've boldfaced the parts of this abstract that answer the question. Abstract: Ever since the advent of fiber reinforced polymer materials in the field of Aerospace, Metal-FRP stacks started to gain importance due to their superior fatigue performance, phenomenal low weight and good specific strength. However the machining, specially drilling these multi ...

1

The F-16 is unstable at low speeds, and stable at higher Mach numbers. It has a flight control computer though, which stabilizes the aircraft using the feedback from pitch rate (so from the view of the pilot an F-16 acts similarly as a stable aircraft). This applies to both positive and negative attitude disturbances, so for both + and - G loads perceived by ...

3

Robin makes a lot of great points, but since we are talking on a purely hypothetical case lets have fun and take it to supersonic speeds. Let's assume that somehow batteries have the required energy density to be used to run some sort of low-bypass electric turbofan to propel our aircraft to high speeds where the neutral point of the aircraft starts ...

9

There are a few electric aircraft now, and because batteries hold far less energy than fuel, they're low-performance and short-range compared to normal aircraft. If we imagine that at some point in the future we'll have batteries that can match fuel, the answer would be: Only slightly. Back in the 60's and 70's, airliners regularly carried a flight ...

1

Pre WW2 French Aircraft had the throttle reversed - i.e. Moving the throttle rearwards INCREASED engine power, and moving the throttle forwards DECREASED engine power. American Curtiss H-75 Hawks exported to France had the modifications made on re-assembly in France - something for any American company test-pilot in France to remember....

2

John K.s answer applies to lithium batteries. They're quite convenient in many ways, but they clearly don't work for this application. Fuel cells on the other hand are more reasonable. A fuel cell produces electricity from fuel, such as hydrogen or ethanol. As such, it doesn't have the weight penalties of batteries. Additionally, this mostly solves the ...

7

The top wing is always the most efficient, as the lower ones suffer reduced lift due to interference from the air flowing beneath the one above. Ailerons will therefore be most effective on the top wing. If they do the job OK then there is no need for any more. Reasons for adding more below include: The upper set are likely to be (or found to be) inadequate....

1

A random guess without any supporting information. (please edit or delete if this turns out to be inaccurate) I would guess the logistics of manufacturing. It could the space needed for handling the aircraft in and/or out of manufacturing buildings. The B-2 program was wildly expensive due to the large number of new technologies involved. So I guess it was ...

6

Well, to use the same airframe, you'd basically have to find a way to replace the fuel volume in the wings with battery volume, and batteries are somewhere around 15-20% of the power density of fuel, as the answers in this ASE post lay out pretty well. So if you had an electric fan that made the required thrust, and somehow figured out how to replace all ...

8

Aspect ratio. Look at your two images: both wings offer the same wingspan, but the former requires a lot more wing area for it. Wing area is weight. More wingspan for less wing area means better fuel efficiency and thus better range. Poor aspect ratio, leading to excess induced drag, is a major disadvantage of delta wings. Much of modern stealth design is ...

3

Yes you simply fly with the pressurization system set to a mode that holds the outflow valve open so the air conditioning air being pumped in can't "inflate" the cabin; it all just goes overboard. Most transport aircraft have a specific QRH procedure for "unpressurized flight", which among other things, limits the maximum altitude to 10000 ft, the altitude ...

1

If you have zero sideslip, then the moment coefficients will be similar between the two configurations. The only difference would come from difference in drag and its effect on moments, which would be small. If you have non-zero sideslip, then the roll moment coefficients will be different due to dihedral effect.

1

For a first approximation, the coefficients will be similar enough, but not exactly identical. As you correctly point out the moment arms will be the same, but the interference effects of the wing-fuselage joint may not. That is to say, your assumption that the lift distribution will remain unchanged will likely not hold out in reality. The stability ...

0

Remember that when you are analysing stability, you are interested in the derivatives by the parameter in question, not in the absolute magnitude. In longitudinal motion, you are mostly interested in how the balance changes when AoA changes. It is understood that in static motion, the absolute moment will be trimmed out (normally by tail) - otherwise it ...

3

So let’s assume the aircraft has a $C_{m_a}$ curve as you show. When you start to pull up you will feel nose down force on the yoke because of negative pitching moment. As you further increase the pull, the force required should increase also. However in your picture at some point the negative pitching moment $C_m$ decreases (in magnitude), still negative ...

4

There are several factors to consider, and depending on your preferences there is not one ideal size and place for a canard. For a first rough concept look at similar designs and use the same canard volume. That is the product of canard area and the distance between its and the wing's neutral points, relative to the aircraft's reference parameters. V_{... 0 This diagram is from the book Design for Flying by David Thurston and might be of use. 9 Some fatigue related issues that are not directly tied to departures and arrivals, like wing bending from gust loads, but may be randomly continuous throughout the flight, need an hour limit. Taking it to a comical extreme, if the airplane flew for 3 years straight without ever landing because it had a new pixie dust powered engine, it would have close to ... 2 Aerodynamic Centre (AC) is particularly handy when we analyse the dynamics, that is, changes. For instance, changes due to disturbances: stability. First thing, for analysis, we split all forces into forces acting where we want plus moments caused by the fact that these forces actually act somewhere else. And then when we write our balance equation, we need ... 1 An airplane achieves steady flight at a given airspeed only if the total pitching moment about its CG is zero. This is achieved with a particular fixed tail angle/elevator angle/canard angle. If the AOA corresponding to the total lift at this airspeed is 0 AOA, then so be it. It's just that for typical airplanes at operational speeds (away from V_{FE} for ... 2 The first certified stable aeroplane flew in 1910 at (and without) the hands of J W Dunne. It was also the first tailless swept plane to fly, sort of a biplane flying wing except that everything was piled in between the wings so not a true flying wing. Contemporaries Handley Page and Igo Etrich were obliged to add tails to their more bird-like attempts. ... 0 It's because anhedral wings remove one of the biggest disadvantages of dihedral wings - maneuverability. Although dihedral wings give roll stability by pushing up on the wing which is lowered while banking, it also means that whenever the pilot is rolling to one particular direction, the wings have a tendency to oppose that by rolling the aircraft to the ... 0 To fly, a wing must maintain its angle of attack within a certain range. If the AOA is too low, the wing produces no lift. If the AOA is too high, air moving over the top cannot follow the wing's contour, separates, and causes a reduction in lift (aerodynamic stall). Designs that employ passive stability to maintain an appropriate AOA will make the aircraft ... 4 The problem which this gap is solving has to do with boundary layer air as @ymb1 suggested but performance is not the only complication. Boundary layer is not only slower, in some regimes of flight it can become highly turbulent compared to the air that is offset from the aircraft's surface and is freely ingested by the aircraft's intakes. Mixing the two ... 1 If you're talking about flex thrust, then NO - it is not at all a problem. It is basically a FAKE temperature which the pilots provide the FADEC with so that the FADEC purposely reduces the amount of thrust generated by the engine. This can improve engine lift by reducing wear. But remember, PILOT'S ARE NOT SUPPOSED TO USE FLEX THRUST DURING ROUGH WEATHER. ... 1 No it's not dangerous. Flex thrust simply exploits all of the runway available, taking advantage of any length that is surplus to the minimum required at max thrust, to allow takeoff with less than maximum thrust to save wear and tear and fuel (wear and tear is a big deal taking off in a sandy/dusty environment and use of flex has a significant effect on ... 6 For one reason or another, the main gear was made longer that originally planned and it was decided to avoid the cost and weight also extending the nose gear to match. The nose down angle while rolling along the ground is no big deal one way or the other, except for looking bad and disconcerting passengers who wonder why the plane is tilted forward. Lots ... 4 Boeing B-52G Stratofortress In the design of the B-52G, considerable attention was paid to reducing the structural weight. Different materials were used in the construction of the airframe, and the wing structure was extensively redesigned. The most visible difference was a vertical tail which was reduced in size. The height was reduced from 48 feet 3 ... 3 Earlier B-52s did not carry all their fuel in the fuselage. They also carried their fuel in the wings using rubber bladder-type wing tanks. The new B52G “wet wing” was a weight savings measure which also increased the fuel capacity over the bladder-type in-wing tanks. The metal fatigue of the new wing was due to a poor structural design which was supposed ... 1 Your general formulation seems fine. Two things to point out: If you choose to sample only half the wing, then you cannot utilize the relation: C_L=A_1\pi A. You would have to do the integration: C_L=\frac{2}{V_\infty S}\int^{b/2}_{-b/2}{\Gamma(y)dy}. That being said, maybe you can simplify the expression and get a closed-form result? Haven't tried ... 3 The tail only looks backward when people can usually see it: in the pattern, parked, or flying formation with a photographer in a slower aircraft. Rather than using trim tabs to push control surfaces out of alignment (which adds drag), the entire empennage pivots for pitch trim adjustments. To trim the nose up at low speed, the leading edge of the horizontal ... 1 Why wouldn't an airline choose the MD-90, which was more efficient? Airliners are usually retired because the airframe reaches the end of its useful life, not because the engines are "obsolete." Towards the end of an airliner's production run, the manufacturing process has been optimized and the cost to the manufacturer is lowest. They are willing to give ... 0 It sure can; in fact, they are equivalent (with an offset of 1G). In the body frame, the vertical equation of motion is:Z+mg\cos\theta\cos\phi=m(\dot{w}+pv-qu) Accelerometers can't measure inertial or gravitational forces, so vertical acceleration as measured is $N_z=Z/m$.

1

I'd guess that they wanted to avoid an acute angle between the lower part of the vertical stabiliser and the horizontal - not for structural reasons but to limit interference drag. They could have made the top piece vertical and just kicked out the bottom part, but then the rudder would have to have been made in two parts (or the hinge wouldn't have worked!)...

4

The decrease in fatigue is not a property that's automatically conferred to the plane by the introduction of wet wing - it's a product of smart design. Usually the decision to use a wet wing is made during the initial design of a plane, and the support structure of the wing is built accordingly, so that it can take the dynamic strains that occur upon ...

5

I'm fairly sure (but have no sources to confirm) that the reasons are purely structural. The vertical stabilisers are simply set at a right angle to the horizontal one. This is just the easiest thing to do, and it also provides better clearance with deflected lower rudders. The aerodynamic consequences of this (for such small angles) are pretty minor. Now ...

2

Yes it's because of dispersion. With wing mounted guns you normally "harmonize" them to converge at a point, say 300 yards or 600 yards to have a sweet spot where most of the projectiles are passing through a small area to maximize the fire concentration. But at ranges closer and farther, the streams of projectiles are more dispersed and their effects are ...

28

No. It doesn't have an APU — it has two of them. They are located close to the main landing gears. From An-225 Mriya is the world’s largest aircraft (English version): Auxiliary power plant consisting of two TA-12 turbofans installed in the left and right chassis fairings provides independent power to all systems and starts the engines. I think "...

1

Also define obsolete. The 737, one of the (if not The) most successful airliner in the world was designed about the same time (1960s) and it shows no sign of seeing its production being stopped. About narrow body = obsolete, I think you will find members of the CRJ (four-across seating [1]) and ERJ families that are not only are narrowbodies but look very ...

3

Not necessarily if the aircraft flies slowly so forces stay small. But it is better to have balanced control surfaces. Now imagine that you fly through a vertical gust and the aircraft pitches up, then down. Since it rotates around its center of gravity, the tail will make the largest excursions from a straight line. Now imagine the elevator has no mass ...

4

Flutter is an energy storage and release phenomenon. Surface moves, applies a load to the fixed surface it's attached to, fixed surface bends under the load, stores some of the energy like a spring, then releases it in moving back to its original position, and some of this released energy is transferred to the moving surface attached to it, which is also ...

5

The MD80 was not obsolete at the time of its first flight and the decade thereafter. The market situation for 150 pax aeroplanes around 1980: Boeing B727-200: first flight 1967. Three JT8D-7/9/11 engines. MD-80: first flight 1979. Two JT8D-200 engines. The first generation of JT8D had a bypass ratio of 0.7, the JT8D-200 engines had an increased bypass ...

2

Regarding the aspect of obsolence: in contemporary personal land transport, the Otto engine is, by pretty much all means, obsolete. There are very few cases in which the Otto engine outperforms an all-electric drivetrain, but these cases hardly make up more than 10% of the field. Yet... The Otto engine prevails. Why? The infrastructure is there. The ...

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# Non-recursivity of language of TMs which have equivalent TMs of smaller and larger description length

Prove that the language

$$L=\{\langle M \rangle \mid \exists M_1, M_2 : L(M_1)=L(M_2)=L(M) \text{ and } |\langle M_1 \rangle| < |\langle M \rangle| < |\langle M_2\rangle| \}$$

is not recursive.

• What do you think? What have you tried, and where did you get stuck? – Yuval Filmus Jan 19 '17 at 14:38
• I have tried the recrusion theory but I can not get the answer – Sikelef Jan 19 '17 at 15:20
• Hint: If you could decide $L$, you could compute (more or less) the Kolmogorov complexity of a string, which you shouldn't be able to do. – Yuval Filmus Jan 19 '17 at 16:58
• We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might find this page helpful in improving your question. – D.W. Jan 19 '17 at 17:21
• @YuvalFilmus I think I can define Kolmogorov complexity of $w$ as the shortest encoding of a TM that accepts $\{ w \}$. So if I could decide $L$, then I would be able to tell whether a specific TM that accepts some $\{ w \}$ is the shortest TM that accepts $\{ w \}$. But if it isn't the shortest, then even if I test whether all shorter TMs are in $L$, I wouldn't know which of them accept $\{ w \}$, and so it wouldn't help me in finding the Kolmogorov complexity of $w$... Am I misinterpreting your hint? – Oren Milman May 22 '19 at 12:16

First, let us notice that in the definition of $$L$$, a machine $$M_2$$ accepting the same language as $$M$$ but having a longer description always exists, so only the condition on $$M_1$$ is pertinent.
Suppose that $$L$$ were computable. Consider the following Turing machine $$T_\ell$$. The machine enumerates all Turing machines of size at least $$2^\ell$$, until it finds a machine $$M$$ such that $$\langle M \rangle \notin L$$ (such a machine must exist, since there are infinitely many computable languages but only finitely many Turing machines of size less than $$2^\ell$$). Then it transfers control to the machine $$M$$, i.e., it runs $$M$$ on the input.
By construction, $$L(T_\ell) = L(M)$$. Since $$\langle M \rangle \notin L$$, necessarily $$|\langle T_\ell \rangle| \geq |\langle M \rangle| \geq 2^\ell$$. However, by hardcoding $$\ell$$ into a Turing machine in which $$\ell$$ is an additional input, we see that $$|\langle T_\ell \rangle| = O(\ell)$$. We obtain a contradiction for large enough $$\ell$$.


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# Residually nilpotent vs residually p

A well-known theorem of Gruenberg implies that a finitely generated residually torsion-free nilpotent group is residually $p$-finite for all primes $p$. What about the converse?

Question: Are there any examples of a finitely generated group $G$ with the properties that $G$ is residually $p$-finite for all primes $p$, but $G$ is not residually torsion-free nilpotent?

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Right now I can't even find an example residually-$p$ for infinitely many $p$ and not RTFN. – YCor Feb 27 '13 at 17:16
@Yves: Neither can I. For any finite collection of primes $\Pi$, ${\rm SL}(3,\mathbb{Z})$ has a finite index subgroup $K$ which is residually-$p$ for each $p \in\Pi$. But of course $K$ is not RTFN, as it has property (T). – Ashot Minasyan Feb 27 '13 at 17:37

Browsing through the archive of solved problems of Kourovka Notebook, I accidentally saw that the same question was asked by Yu.V. Kuz'min in 1999 (see question 14.52). Apparently the required examples were produced a long time ago by B. Hartley (Hartley, B., `On residually finite p-groups.' Symposia Mathematica, Vol. XVII (Convegno sui Gruppi Infiniti, INDAM, Roma, 1973), pp. 225–234. Academic Press, London, 1976. MR0419611 (54 #7629)).

Hartley showed that for any set of primes $\Pi$ there exists a $3$-generated center-by-metabelian group $G$ which is not residually torsion-free nilpotent and which is residually $p$-finite iff $p \in \Pi$.

-


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## Elementary Algebra

Published by Cengage Learning

# Chapter 2 - Real Numbers - Chapter 2 Review Problem Set: 59

#### Answer

$x-3$

#### Work Step by Step

The word "minus" means subtraction. Since the problem says $x$ $\text{minus 3}$, we know that we have to subtract $3$ from $x$. In order to express this algebraically, we write $x-3$.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## C Specification

To record a dispatch using non-zero base values for the components of WorkgroupId, call:

// Provided by VK_VERSION_1_1
void vkCmdDispatchBase(
VkCommandBuffer                             commandBuffer,
uint32_t                                    baseGroupX,
uint32_t                                    baseGroupY,
uint32_t                                    baseGroupZ,
uint32_t                                    groupCountX,
uint32_t                                    groupCountY,
uint32_t                                    groupCountZ);

or the equivalent command

// Provided by VK_KHR_device_group
void vkCmdDispatchBaseKHR(
VkCommandBuffer                             commandBuffer,
uint32_t                                    baseGroupX,
uint32_t                                    baseGroupY,
uint32_t                                    baseGroupZ,
uint32_t                                    groupCountX,
uint32_t                                    groupCountY,
uint32_t                                    groupCountZ);

## Parameters

• commandBuffer is the command buffer into which the command will be recorded.

• baseGroupX is the start value for the X component of WorkgroupId.

• baseGroupY is the start value for the Y component of WorkgroupId.

• baseGroupZ is the start value for the Z component of WorkgroupId.

• groupCountX is the number of local workgroups to dispatch in the X dimension.

• groupCountY is the number of local workgroups to dispatch in the Y dimension.

• groupCountZ is the number of local workgroups to dispatch in the Z dimension.

## Description

When the command is executed, a global workgroup consisting of groupCountX × groupCountY × groupCountZ local workgroups is assembled, with WorkgroupId values ranging from [baseGroup*, baseGroup* + groupCount*) in each component. vkCmdDispatch is equivalent to vkCmdDispatchBase(0,0,0,groupCountX,groupCountY,groupCountZ).

Valid Usage
• VUID-vkCmdDispatchBase-magFilter-04553
If a VkSampler created with magFilter or minFilter equal to VK_FILTER_LINEAR and compareEnable equal to VK_FALSE is used to sample a VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_SAMPLED_IMAGE_FILTER_LINEAR_BIT

• VUID-vkCmdDispatchBase-mipmapMode-04770
If a VkSampler created with mipmapMode equal to VK_SAMPLER_MIPMAP_MODE_LINEAR and compareEnable equal to VK_FALSE is used to sample a VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_SAMPLED_IMAGE_FILTER_LINEAR_BIT

• VUID-vkCmdDispatchBase-None-06479
If a VkImageView is sampled with depth comparison, the image view’s format features must contain VK_FORMAT_FEATURE_2_SAMPLED_IMAGE_DEPTH_COMPARISON_BIT

• VUID-vkCmdDispatchBase-None-02691
If a VkImageView is accessed using atomic operations as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_STORAGE_IMAGE_ATOMIC_BIT

• VUID-vkCmdDispatchBase-None-02692
If a VkImageView is sampled with VK_FILTER_CUBIC_EXT as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_SAMPLED_IMAGE_FILTER_CUBIC_BIT_EXT

• VUID-vkCmdDispatchBase-filterCubic-02694
Any VkImageView being sampled with VK_FILTER_CUBIC_EXT as a result of this command must have a VkImageViewType and format that supports cubic filtering, as specified by VkFilterCubicImageViewImageFormatPropertiesEXT::filterCubic returned by vkGetPhysicalDeviceImageFormatProperties2

• VUID-vkCmdDispatchBase-filterCubicMinmax-02695
Any VkImageView being sampled with VK_FILTER_CUBIC_EXT with a reduction mode of either VK_SAMPLER_REDUCTION_MODE_MIN or VK_SAMPLER_REDUCTION_MODE_MAX as a result of this command must have a VkImageViewType and format that supports cubic filtering together with minmax filtering, as specified by VkFilterCubicImageViewImageFormatPropertiesEXT::filterCubicMinmax returned by vkGetPhysicalDeviceImageFormatProperties2

• VUID-vkCmdDispatchBase-flags-02696
Any VkImage created with a VkImageCreateInfo::flags containing VK_IMAGE_CREATE_CORNER_SAMPLED_BIT_NV sampled as a result of this command must only be sampled using a VkSamplerAddressMode of VK_SAMPLER_ADDRESS_MODE_CLAMP_TO_EDGE

• VUID-vkCmdDispatchBase-OpTypeImage-07027
For any VkImageView being written as a storage image where the image format field of the OpTypeImage is Unknown, the view’s format features must contain VK_FORMAT_FEATURE_2_STORAGE_WRITE_WITHOUT_FORMAT_BIT

• VUID-vkCmdDispatchBase-OpTypeImage-07028
For any VkImageView being read as a storage image where the image format field of the OpTypeImage is Unknown, the view’s format features must contain VK_FORMAT_FEATURE_2_STORAGE_READ_WITHOUT_FORMAT_BIT

• VUID-vkCmdDispatchBase-OpTypeImage-07029
For any VkBufferView being written as a storage texel buffer where the image format field of the OpTypeImage is Unknown, the view’s buffer features must contain VK_FORMAT_FEATURE_2_STORAGE_WRITE_WITHOUT_FORMAT_BIT

• VUID-vkCmdDispatchBase-OpTypeImage-07030
Any VkBufferView being read as a storage texel buffer where the image format field of the OpTypeImage is Unknown then the view’s buffer features must contain VK_FORMAT_FEATURE_2_STORAGE_READ_WITHOUT_FORMAT_BIT

• VUID-vkCmdDispatchBase-None-02697
For each set n that is statically used by the VkPipeline bound to the pipeline bind point used by this command, a descriptor set must have been bound to n at the same pipeline bind point, with a VkPipelineLayout that is compatible for set n, with the VkPipelineLayout used to create the current VkPipeline, as described in [descriptorsets-compatibility]

• VUID-vkCmdDispatchBase-maintenance4-06425
If the maintenance4 feature is not enabled, then for each push constant that is statically used by the VkPipeline bound to the pipeline bind point used by this command, a push constant value must have been set for the same pipeline bind point, with a VkPipelineLayout that is compatible for push constants, with the VkPipelineLayout used to create the current VkPipeline, as described in [descriptorsets-compatibility]

• VUID-vkCmdDispatchBase-None-02699
Descriptors in each bound descriptor set, specified via vkCmdBindDescriptorSets, must be valid if they are statically used by the VkPipeline bound to the pipeline bind point used by this command

• VUID-vkCmdDispatchBase-None-02700
A valid pipeline must be bound to the pipeline bind point used by this command

• VUID-vkCmdDispatchBase-commandBuffer-02701
If the VkPipeline object bound to the pipeline bind point used by this command requires any dynamic state, that state must have been set or inherited (if the VK_NV_inherited_viewport_scissor extension is enabled) for commandBuffer, and done so after any previously bound pipeline with the corresponding state not specified as dynamic

• VUID-vkCmdDispatchBase-None-02859
There must not have been any calls to dynamic state setting commands for any state not specified as dynamic in the VkPipeline object bound to the pipeline bind point used by this command, since that pipeline was bound

• VUID-vkCmdDispatchBase-None-02702
If the VkPipeline object bound to the pipeline bind point used by this command accesses a VkSampler object that uses unnormalized coordinates, that sampler must not be used to sample from any VkImage with a VkImageView of the type VK_IMAGE_VIEW_TYPE_3D, VK_IMAGE_VIEW_TYPE_CUBE, VK_IMAGE_VIEW_TYPE_1D_ARRAY, VK_IMAGE_VIEW_TYPE_2D_ARRAY or VK_IMAGE_VIEW_TYPE_CUBE_ARRAY, in any shader stage

• VUID-vkCmdDispatchBase-None-02703
If the VkPipeline object bound to the pipeline bind point used by this command accesses a VkSampler object that uses unnormalized coordinates, that sampler must not be used with any of the SPIR-V OpImageSample* or OpImageSparseSample* instructions with ImplicitLod, Dref or Proj in their name, in any shader stage

• VUID-vkCmdDispatchBase-None-02704
If the VkPipeline object bound to the pipeline bind point used by this command accesses a VkSampler object that uses unnormalized coordinates, that sampler must not be used with any of the SPIR-V OpImageSample* or OpImageSparseSample* instructions that includes a LOD bias or any offset values, in any shader stage

• VUID-vkCmdDispatchBase-uniformBuffers-06935
If any stage of the VkPipeline object bound to the pipeline bind point used by this command accesses a uniform buffer, and that stage was created without enabling either VK_PIPELINE_ROBUSTNESS_BUFFER_BEHAVIOR_ROBUST_BUFFER_ACCESS_EXT or VK_PIPELINE_ROBUSTNESS_BUFFER_BEHAVIOR_ROBUST_BUFFER_ACCESS_2_EXT for uniformBuffers, and the robustBufferAccess feature is not enabled, that stage must not access values outside of the range of the buffer as specified in the descriptor set bound to the same pipeline bind point

• VUID-vkCmdDispatchBase-storageBuffers-06936
If any stage of the VkPipeline object bound to the pipeline bind point used by this command accesses a storage buffer, and that stage was created without enabling either VK_PIPELINE_ROBUSTNESS_BUFFER_BEHAVIOR_ROBUST_BUFFER_ACCESS_EXT or VK_PIPELINE_ROBUSTNESS_BUFFER_BEHAVIOR_ROBUST_BUFFER_ACCESS_2_EXT for storageBuffers, and the robustBufferAccess feature is not enabled, that stage must not access values outside of the range of the buffer as specified in the descriptor set bound to the same pipeline bind point

• VUID-vkCmdDispatchBase-commandBuffer-02707
If commandBuffer is an unprotected command buffer and protectedNoFault is not supported, any resource accessed by the VkPipeline object bound to the pipeline bind point used by this command must not be a protected resource

• VUID-vkCmdDispatchBase-None-06550
If the VkPipeline object bound to the pipeline bind point used by this command accesses a VkSampler or VkImageView object that enables sampler Y′CBCR conversion, that object must only be used with OpImageSample* or OpImageSparseSample* instructions

• VUID-vkCmdDispatchBase-ConstOffset-06551
If the VkPipeline object bound to the pipeline bind point used by this command accesses a VkSampler or VkImageView object that enables sampler Y′CBCR conversion, that object must not use the ConstOffset and Offset operands

• VUID-vkCmdDispatchBase-None-04115
If a VkImageView is accessed using OpImageWrite as a result of this command, then the Type of the Texel operand of that instruction must have at least as many components as the image view’s format

• VUID-vkCmdDispatchBase-OpImageWrite-04469
If a VkBufferView is accessed using OpImageWrite as a result of this command, then the Type of the Texel operand of that instruction must have at least as many components as the buffer view’s format

• VUID-vkCmdDispatchBase-SampledType-04470
If a VkImageView with a VkFormat that has a 64-bit component width is accessed as a result of this command, the SampledType of the OpTypeImage operand of that instruction must have a Width of 64

• VUID-vkCmdDispatchBase-SampledType-04471
If a VkImageView with a VkFormat that has a component width less than 64-bit is accessed as a result of this command, the SampledType of the OpTypeImage operand of that instruction must have a Width of 32

• VUID-vkCmdDispatchBase-SampledType-04472
If a VkBufferView with a VkFormat that has a 64-bit component width is accessed as a result of this command, the SampledType of the OpTypeImage operand of that instruction must have a Width of 64

• VUID-vkCmdDispatchBase-SampledType-04473
If a VkBufferView with a VkFormat that has a component width less than 64-bit is accessed as a result of this command, the SampledType of the OpTypeImage operand of that instruction must have a Width of 32

• VUID-vkCmdDispatchBase-sparseImageInt64Atomics-04474
If the sparseImageInt64Atomics feature is not enabled, VkImage objects created with the VK_IMAGE_CREATE_SPARSE_RESIDENCY_BIT flag must not be accessed by atomic instructions through an OpTypeImage with a SampledType with a Width of 64 by this command

• VUID-vkCmdDispatchBase-sparseImageInt64Atomics-04475
If the sparseImageInt64Atomics feature is not enabled, VkBuffer objects created with the VK_BUFFER_CREATE_SPARSE_RESIDENCY_BIT flag must not be accessed by atomic instructions through an OpTypeImage with a SampledType with a Width of 64 by this command

• VUID-vkCmdDispatchBase-OpImageWeightedSampleQCOM-06971
If OpImageWeightedSampleQCOM is used to sample a VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_2_WEIGHT_SAMPLED_IMAGE_BIT_QCOM

• VUID-vkCmdDispatchBase-OpImageWeightedSampleQCOM-06972
If OpImageWeightedSampleQCOM uses a VkImageView as a sample weight image as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_2_WEIGHT_IMAGE_BIT_QCOM

• VUID-vkCmdDispatchBase-OpImageBoxFilterQCOM-06973
If OpImageBoxFilterQCOM is used to sample a VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_2_BOX_FILTER_SAMPLED_BIT_QCOM

• VUID-vkCmdDispatchBase-OpImageBlockMatchSSDQCOM-06974
If OpImageBlockMatchSSDQCOM is used to read from an VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_2_BLOCK_MATCHING_BIT_QCOM

If OpImageBlockMatchSADQCOM is used to read from an VkImageView as a result of this command, then the image view’s format features must contain VK_FORMAT_FEATURE_2_BLOCK_MATCHING_BIT_QCOM

If OpImageBlockMatchSADQCOM or OpImageBlockMatchSSDQCOM is used to read from a reference image as result of this command, then the specified reference coordinates must not fail integer texel coordinate validation.

• VUID-vkCmdDispatchBase-OpImageWeightedSampleQCOM-06977
If OpImageWeightedSampleQCOM, OpImageBoxFilterQCOM, OpImageBlockMatchSSDQCOM, or OpImageBlockMatchSADQCOM uses a VkSampler as a result of this command, then the sampler must have been created with VK_SAMPLER_CREATE_IMAGE_PROCESSING_BIT_QCOM.

• VUID-vkCmdDispatchBase-OpImageWeightedSampleQCOM-06978
If any command other than OpImageWeightedSampleQCOM, OpImageBoxFilterQCOM, OpImageBlockMatchSSDQCOM, or OpImageBlockMatchSADQCOM uses a VkSampler as a result of this command, then the sampler must not have been created with VK_SAMPLER_CREATE_IMAGE_PROCESSING_BIT_QCOM.

• VUID-vkCmdDispatchBase-None-07288
Any shader invocation executed by this command must terminate

• VUID-vkCmdDispatchBase-commandBuffer-02712
If commandBuffer is a protected command buffer and protectedNoFault is not supported, any resource written to by the VkPipeline object bound to the pipeline bind point used by this command must not be an unprotected resource

• VUID-vkCmdDispatchBase-commandBuffer-02713
If commandBuffer is a protected command buffer and protectedNoFault is not supported, pipeline stages other than the framebuffer-space and compute stages in the VkPipeline object bound to the pipeline bind point used by this command must not write to any resource

• VUID-vkCmdDispatchBase-commandBuffer-04617
If any of the shader stages of the VkPipeline bound to the pipeline bind point used by this command uses the RayQueryKHR capability, then commandBuffer must not be a protected command buffer

• VUID-vkCmdDispatchBase-baseGroupX-00421
baseGroupX must be less than VkPhysicalDeviceLimits::maxComputeWorkGroupCount[0]

• VUID-vkCmdDispatchBase-baseGroupX-00422
baseGroupY must be less than VkPhysicalDeviceLimits::maxComputeWorkGroupCount[1]

• VUID-vkCmdDispatchBase-baseGroupZ-00423
baseGroupZ must be less than VkPhysicalDeviceLimits::maxComputeWorkGroupCount[2]

• VUID-vkCmdDispatchBase-groupCountX-00424
groupCountX must be less than or equal to VkPhysicalDeviceLimits::maxComputeWorkGroupCount[0] minus baseGroupX

• VUID-vkCmdDispatchBase-groupCountY-00425
groupCountY must be less than or equal to VkPhysicalDeviceLimits::maxComputeWorkGroupCount[1] minus baseGroupY

• VUID-vkCmdDispatchBase-groupCountZ-00426
groupCountZ must be less than or equal to VkPhysicalDeviceLimits::maxComputeWorkGroupCount[2] minus baseGroupZ

• VUID-vkCmdDispatchBase-baseGroupX-00427
If any of baseGroupX, baseGroupY, or baseGroupZ are not zero, then the bound compute pipeline must have been created with the VK_PIPELINE_CREATE_DISPATCH_BASE flag

Valid Usage (Implicit)
• VUID-vkCmdDispatchBase-commandBuffer-parameter
commandBuffer must be a valid VkCommandBuffer handle

• VUID-vkCmdDispatchBase-commandBuffer-recording
commandBuffer must be in the recording state

• VUID-vkCmdDispatchBase-commandBuffer-cmdpool
The VkCommandPool that commandBuffer was allocated from must support compute operations

• VUID-vkCmdDispatchBase-renderpass
This command must only be called outside of a render pass instance

• VUID-vkCmdDispatchBase-videocoding
This command must only be called outside of a video coding scope

Host Synchronization
• Host access to commandBuffer must be externally synchronized

• Host access to the VkCommandPool that commandBuffer was allocated from must be externally synchronized

Command Properties
Command Buffer Levels Render Pass Scope Video Coding Scope Supported Queue Types Command Type

Primary
Secondary

Outside

Outside

Compute

Action

## Document Notes

This page is extracted from the Vulkan Specification. Fixes and changes should be made to the Specification, not directly.

Copyright 2014-2022 The Khronos Group Inc.


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# Math Help - Basic Related rates question

1. ## Basic Related rates question

Hi guys,

I have a basic Cal I related rates question and am unsure how to correctly solve. Can you check my solution? The part I'm a little confused about is where I differentiated both sides. Can you do as I did?
THanks for any help,
james

If two resistors of $R_1$ and $R_2$ are connected in parallel in an electric circuit to make an $R$-ohm resister, the value of $R$ can be found from the equation:
$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}$
If $R_1$ is decreasing at the rate of $3$ ohms/sec and $R_2$ is increasing at a rate of $5$ ohms/sec at what rate is $R$ changing when $R_1 = 60$ ohms and $R_2 = 50$ ohms

Substituing $R_1 = 60$ and $R_2 = 50$ gives that $R = \frac{11}{300}$

Differentiating both sides with respect to time gives:

$-\frac{1}{R^2}\left(\frac{dR}{dt}\right) = -\frac{1}{R_1^2}\left(\frac{dR_1}{dt}\right) -\frac{1}{R_2^2}\left(\frac{dR_2}{dt}\right)$

Substituting all known values into the above equation give:
$-\frac{300}{11}\left(\frac{dR}{dt}\right) = -\frac{1}{60^2}(-3) - \frac{1}{50^2}(5)$

Then we solve for $\frac{dR}{dt}$ to obtain the answer.

2. finish. solve for $\frac{dR}{dt}$ ...

$\frac{dR}{dt} = \frac{11}{300}\left(-\frac{3}{60^2} + \frac{5}{50^2}\right)$


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# Quarterly Journal of Austrian Economics

Home | Mises Library | Apriorism, Introspection, and the Axiom of Action: A Realist Solution

## Apriorism, Introspection, and the Axiom of Action: A Realist Solution

07/30/2014

Volume 10, No. 3 (2007)

This article deals with the epistemological bases for the axiom of action and more particularly with man’s capacity to have an a priori knowledge. The simplest argument in favour of the true a priori character of the axiom of action is the introspective one. It is sufficient to look at oneself to see that it is obvious that man acts intentionally. A part of the Austrian school, after Lachman, estimated that this introspective argument was insufficient. They preferred to go the way proposed by Max Weber and to base social sciences on the construction of ideal types. Nevertheless, this article shows that by defining the introspection as an ideal type we solve neither the induction problem nor the historicity of intellectual categories problem (1). The ideality of types creates assumptions (Utopia) whereas their typicity creates historical categories. The theory of the ideal type of Schütz seeks well to leave historicity while insisting on the ideality. On the contrary, the economic hermeneutics of Lavoie focuses its attention on historicity. To leave these difficulties, it is preferable to return to the philosophical sources of Menger, i.e. to the realistic aristotelian  tradition and more particularly to the thomist one. In the second part of the paper we confirm coming back to the philosophical origin of Menger’s theory, the aristotelian and the thomist school. Thomist philosophy of knowledge resulted in defining introspection as a form of qualitative induction, and to support that the axiom of the action is a judgement, i.e. the  combination of two  universalities known thanks to the capacity of abstraction of man’s intelligence. In that way, it is possible to think of the axiom of action as of a historical category and universality. To affirm that  man has reasons to act (intention) is at the same time a discovery and a truth proposal a priori.

Author:

Contact François Facchini

François Facchini is associate researcher at the University of Paris 1, CES-MATISSE, France.